{"version":19,"lastUpdated":"2026-05-28","questions":[{"id":"amc8_001","contestType":"AMC 8","difficulty":3,"text":"How many positive integers less than 100 are divisible by 3 or 5 but not by both?","options":["32","33","40","48","53"],"correctIndex":2,"explanation":"Numbers divisible by 3 or 5 but not both = (divisible by 3) + (divisible by 5) - 2×(divisible by 15) = 33 + 19 - 2×6 = 40.","topic":"Number Theory"},{"id":"amc8_002","contestType":"AMC 8","difficulty":2,"text":"What is 7/8 expressed as a decimal rounded to the nearest hundredth?","options":["0.87","0.875","0.88","0.90","0.8750"],"correctIndex":2,"explanation":"7 ÷ 8 = 0.875. The thousandths digit is 5, so we round the hundredths up from 7 to 8 → 0.88.","topic":"Algebra"},{"id":"amc8_003","contestType":"AMC 8","difficulty":4,"text":"A square has side length 10. An equilateral triangle is drawn outwardly on one side. What is the straight-line distance from either endpoint of the opposite side of the square to the farthest vertex of the triangle?","options":["10 + 5√3","10√2 + 5√3","10√(2 + √3)","10 + 10√3","15√2"],"correctIndex":2,"explanation":"The straight-line distance is 10√(2 + √3).","solutionSteps":["Draw the square with side length 10, and draw the equilateral triangle outward on the top side of the square.","The equilateral triangle has side length 10, so its altitude is 5√3 by the 30-60-90 triangle ratio.","From an endpoint of the opposite side of the square to the vertical line through the triangle's top vertex, the horizontal distance is half the side length: 5.","The vertical distance is the square's height plus the triangle's altitude: 10 + 5√3.","Use the Pythagorean theorem: d² = 5² + (10 + 5√3)².","Expand: d² = 25 + (100 + 100√3 + 75) = 200 + 100√3 = 100(2 + √3).","Therefore d = √[100(2 + √3)] = 10√(2 + √3)."],"diagram":{"type":"square_triangle"},"topic":"Geometry"},{"id":"amc8_004","contestType":"AMC 8","difficulty":3,"text":"Five distinct students must sit in a row with two particular students next to each other. How many arrangements are possible?","options":["48","96","120","240","480"],"correctIndex":0,"explanation":"Treat the two particular students as one block. Then there are 4 objects to arrange, so 4! arrangements, and the two students inside the block can switch places in 2 ways. The total is 4! × 2 = 48.","topic":"Counting & Probability"},{"id":"amc8_005","contestType":"AMC 8","difficulty":4,"text":"The sum of three consecutive even integers is 48. What is the largest of the three?","options":["14","16","18","20","22"],"correctIndex":2,"explanation":"Let the three numbers be x-2, x, x+2 (where x is the middle even integer). Then (x-2) + x + (x+2) = 3x = 48 → x = 16. The numbers are 14, 16, and 18. The largest is 18.","topic":"Algebra"},{"id":"amc8_006","contestType":"AMC 8","difficulty":5,"text":"In a regular octagon with side length 1, what is the length of a longer diagonal (skipping two vertices)?","options":["1 + √2","2 + √2","1 + 2√2","3","2√2"],"correctIndex":0,"explanation":"Longer diagonal in unit regular octagon equals 1 + √2.","diagram":{"type":"regular_polygon","sides":8,"sideLabel":"1","diagonalLabel":"longer diagonal"},"topic":"Geometry"},{"id":"mc_001","contestType":"MATHCOUNTS","difficulty":5,"text":"How many 4-digit numbers have digits that sum to 10 and are strictly increasing?","options":["1","120","210","286","715"],"correctIndex":0,"explanation":"A strictly increasing 4-digit number is determined by its four distinct digits in order. The only 4 distinct positive digits that sum to 10 are 1, 2, 3, and 4, giving the number 1234.","topic":"Number Theory"},{"id":"mc_002","contestType":"MATHCOUNTS","difficulty":4,"text":"A bag contains 5 red, 4 blue, and 3 green marbles. Two marbles are drawn without replacement. What is the probability both are the same color?","options":["13/66","19/66","23/66","29/66","31/66"],"correctIndex":1,"explanation":"P both red + both blue + both green = (5/12)(4/11) + (4/12)(3/11) + (3/12)(2/11) = 19/66.","topic":"Counting & Probability"},{"id":"mc_003","contestType":"MATHCOUNTS","difficulty":6,"text":"Find the number of positive integer solutions to x + y + z = 12 where x ≤ y ≤ z.","options":["12","15","18","21","24"],"correctIndex":0,"explanation":"Because x, y, and z are positive and x ≤ y ≤ z, this asks for the partitions of 12 into exactly 3 positive parts. Listing them gives 12 possibilities: (1,1,10), (1,2,9), (1,3,8), (1,4,7), (1,5,6), (2,2,8), (2,3,7), (2,4,6), (2,5,5), (3,3,6), (3,4,5), and (4,4,4).","topic":"Number Theory"},{"id":"mc_004","contestType":"MATHCOUNTS","difficulty":5,"text":"A circle of radius 5 is inscribed in an equilateral triangle. What is the side length of the triangle?","options":["10√3","15","10 + 5√3","15√3","30"],"correctIndex":0,"explanation":"Height h = 3r = 15 for incircle radius r = h/3. Side = (2/√3) × 15 = 10√3.","diagram":{"type":"triangle","variant":"equilateral_incircle"},"topic":"Geometry"},{"id":"sb_001","contestType":"Science Bowl","difficulty":2,"text":"In a forest ecosystem, if the rabbit population drops sharply due to disease, what most likely happens to the fox population in the short term?","options":["Increases","Decreases","Stays the same","Foxes eat plants instead","No effect"],"correctIndex":1,"explanation":"Foxes are predators that rely heavily on rabbits. Sharp drop in prey → short-term decrease in predator numbers.","topic":"Science"},{"id":"sb_002","contestType":"Science Bowl","difficulty":3,"text":"Which gas is the most abundant in Earth's atmosphere by volume?","options":["Oxygen","Carbon dioxide","Nitrogen","Argon","Water vapor"],"correctIndex":2,"explanation":"Nitrogen makes up approximately 78% of the atmosphere by volume.","topic":"Science"},{"id":"sb_003","contestType":"Science Bowl","difficulty":4,"text":"What is the pH of a neutral aqueous solution at 25°C?","options":["0","7","14","1","Depends on the solute"],"correctIndex":1,"explanation":"Pure water at 25°C has [H+] = 10^-7 M, so pH = 7.","topic":"Science"},{"id":"sb_004","contestType":"Science Bowl","difficulty":5,"text":"In the human circulatory system, which chamber of the heart pumps oxygenated blood to the body?","options":["Right atrium","Right ventricle","Left atrium","Left ventricle","Aorta"],"correctIndex":3,"explanation":"The left ventricle has the thickest wall and pumps blood into the aorta for systemic circulation.","topic":"Science"},{"id":"sb_005","contestType":"Science Bowl","difficulty":3,"text":"Which of the following is a chemical change rather than a physical change?","options":["Melting ice","Dissolving sugar in water","Burning wood","Boiling water","Crushing a can"],"correctIndex":2,"explanation":"Burning wood involves combustion — new substances (CO2, ash, water) are formed. The others are physical changes.","topic":"Science"},{"id":"amc8_007","contestType":"AMC 8","difficulty":5,"text":"A rectangle has perimeter 36 and area 80. What is the length of its diagonal?","options":["√82","√85","2√41","√97","10√2"],"correctIndex":2,"explanation":"The rectangle has side lengths 10 and 8, so its diagonal is 2√41.","solutionSteps":["Let the rectangle have length l and width w. From the perimeter, 2l + 2w = 36, so l + w = 18.","The area gives lw = 80. The two side lengths must multiply to 80 and add to 18.","The pair 10 and 8 works because 10 × 8 = 80 and 10 + 8 = 18.","Use the Pythagorean theorem on the rectangle's diagonal: d² = 10² + 8² = 100 + 64 = 164.","Therefore d = √164 = √(4 × 41) = 2√41."],"topic":"Geometry"},{"id":"mc_005","contestType":"MATHCOUNTS","difficulty":6,"text":"How many positive integers n ≤ 1000 have the property that the sum of the digits of n is divisible by 9?","options":["111","112","113","222","333"],"correctIndex":0,"explanation":"Digital root cycles every 9. Roughly 1/9 of numbers satisfy. Exact count up to 1000 is 111 (including adjustments for 999 and 1000).","topic":"Number Theory"},{"id":"sb_006","contestType":"Science Bowl","difficulty":4,"text":"Which particle is responsible for mediating the electromagnetic force?","options":["Gluon","Photon","W boson","Z boson","Graviton"],"correctIndex":1,"explanation":"The photon is the gauge boson that carries the electromagnetic interaction.","topic":"Science"},{"id":"amc8_008","contestType":"AMC 8","difficulty":4,"text":"If 3^x = 81 and 2^y = 64, what is x + y?","options":["7","8","9","10","11"],"correctIndex":3,"explanation":"3^4 = 81 → x=4. 2^6=64 → y=6. x+y=10.","topic":"Algebra"},{"id":"mc_006","contestType":"MATHCOUNTS","difficulty":5,"text":"A fair six-sided die is rolled twice. What is the probability the product of the two rolls is greater than 20?","options":["1/6","5/18","7/18","4/9","1/3"],"correctIndex":0,"explanation":"Total 36 possible outcomes. Pairs with product > 20: (4,6), (6,4), (5,5), (5,6), (6,5), (6,6) — 6 outcomes. Probability = 6/36 = 1/6.","topic":"Counting & Probability"},{"id":"amc8_009","contestType":"AMC 8","difficulty":3,"text":"What is the value of 3^4 + 4^3?","options":["81","97","145","193","256"],"correctIndex":2,"explanation":"3^4 = 81 and 4^3 = 64. 81 + 64 = 145.","topic":"Algebra"},{"id":"amc8_010","contestType":"AMC 8","difficulty":4,"text":"A bag contains 8 red marbles and 5 blue marbles. If one marble is drawn at random, what is the probability it is not red?","options":["5/13","8/13","3/8","5/8","13/8"],"correctIndex":0,"explanation":"Probability not red = blue / total = 5/13.","topic":"Counting & Probability"},{"id":"amc8_011","contestType":"AMC 8","difficulty":5,"text":"How many positive three-digit numbers have the property that the middle digit is the average of the other two digits?","options":["36","45","100","121","243"],"correctIndex":1,"explanation":"Let the number be 100a + 10b + c, where a is 1-9 and c is 0-9. The middle digit b=(a+c)/2 is an integer exactly when a and c have the same parity. There are 5 odd choices for a and 5 odd choices for c, plus 4 even choices for a and 5 even choices for c, giving 5×5 + 4×5 = 45 numbers.","topic":"Number Theory"},{"id":"mc_007","contestType":"MATHCOUNTS","difficulty":5,"text":"What is the sum of the first 20 positive even integers?","options":["210","380","400","420","440"],"correctIndex":3,"explanation":"The 20th even integer is 40. Sum = 20 × 21 = 420 (or n(n+1) for first n evens).","topic":"Algebra"},{"id":"mc_008","contestType":"MATHCOUNTS","difficulty":6,"text":"In how many ways can 5 people be seated around a circular table?","options":["24","60","120","720","3125"],"correctIndex":0,"explanation":"For circular permutations of n distinct objects: (n-1)! = 4! = 24.","topic":"Counting & Probability"},{"id":"sb_007","contestType":"Science Bowl","difficulty":3,"text":"What is the chemical symbol for the element with atomic number 79?","options":["Ag","Au","Fe","Hg","Pb"],"correctIndex":1,"explanation":"Gold (Au) has atomic number 79.","topic":"Science"},{"id":"sb_008","contestType":"Science Bowl","difficulty":4,"text":"Which planet in our solar system has the most confirmed moons?","options":["Jupiter","Saturn","Uranus","Neptune","Mars"],"correctIndex":1,"explanation":"Saturn has the most confirmed moons of any planet in the solar system, ahead of Jupiter.","topic":"Science"},{"id":"amc8_012","contestType":"AMC 8","difficulty":3,"text":"If a rectangle has length 12 and width 5, what is its area?","options":["17","34","60","120","144"],"correctIndex":2,"explanation":"Area = length × width = 12 × 5 = 60.","topic":"Geometry"},{"id":"mc_009","contestType":"MATHCOUNTS","difficulty":5,"text":"What is the greatest common divisor of 48, 72, and 120?","options":["6","8","12","24","48"],"correctIndex":3,"explanation":"GCD of 48 and 72 is 24. GCD(24, 120) = 24.","topic":"Number Theory"},{"id":"sb_009","contestType":"Science Bowl","difficulty":2,"text":"What is the powerhouse of the cell?","options":["Nucleus","Ribosome","Mitochondria","Golgi apparatus","Lysosome"],"correctIndex":2,"explanation":"Mitochondria are known as the powerhouse of the cell because they produce ATP.","topic":"Science"},{"id":"amc8_013","contestType":"AMC 8","difficulty":4,"text":"A shirt is on sale for 20% off. If the original price was $45, what is the sale price?","options":["$9","$25","$36","$41","$63"],"correctIndex":2,"explanation":"20% of 45 is 9. Sale price = 45 - 9 = 36.","topic":"Algebra"},{"id":"mc_010","contestType":"MATHCOUNTS","difficulty":6,"text":"What is the number of distinct arrangements of the letters in the word 'MISSISSIPPI'?","options":["34650","39916800","1247400","5040","11"],"correctIndex":0,"explanation":"11! / (4! × 4! × 2!) = 39916800 / (24 × 24 × 2) = 34650.","topic":"Counting & Probability"},{"id":"sb_010","contestType":"Science Bowl","difficulty":5,"text":"What is the SI unit of electric current?","options":["Volt","Ohm","Watt","Ampere","Coulomb"],"correctIndex":3,"explanation":"The ampere (A) is the SI base unit of electric current.","topic":"Science"},{"id":"amc8_014","contestType":"AMC 8","difficulty":3,"text":"The average of five numbers is 12. If four of the numbers are 8, 10, 14, and 16, what is the fifth number?","options":["8","10","12","14","16"],"correctIndex":2,"explanation":"Sum of the five numbers = 12 × 5 = 60. 8+10+14+16=48. Fifth number = 60 - 48 = 12.","topic":"Algebra"},{"id":"amc8_015","contestType":"AMC 8","difficulty":4,"text":"A recipe calls for 3 cups of flour for every 2 cups of sugar. How many cups of flour are needed for 10 cups of sugar?","options":["12","13","14","15","16"],"correctIndex":3,"explanation":"Flour : Sugar = 3 : 2. For 10 cups sugar: (3/2) × 10 = 15 cups of flour.","topic":"Algebra"},{"id":"amc8_016","contestType":"AMC 8","difficulty":3,"text":"What is the least common multiple of 6 and 8?","options":["12","18","24","48","2"],"correctIndex":2,"explanation":"Multiples of 6: 6,12,18,24... Multiples of 8: 8,16,24... LCM is 24.","topic":"Number Theory"},{"id":"amc8_017","contestType":"AMC 8","difficulty":5,"text":"A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?","options":["13","14","15","17","60"],"correctIndex":0,"explanation":"By Pythagorean theorem: √(5² + 12²) = √(25 + 144) = √169 = 13.","diagram":{"type":"triangle","variant":"right"},"topic":"Geometry"},{"id":"amc8_018","contestType":"AMC 8","difficulty":4,"text":"John is 4 years older than Mary. In 5 years, John will be twice as old as Mary is now. How old is Mary now?","options":["3","5","7","9","11"],"correctIndex":3,"explanation":"Let Mary’s age now = x. John = x+4. In 5 years: (x+4)+5 = 2x → x+9 = 2x → x=9.","topic":"Algebra"},{"id":"amc8_019","contestType":"AMC 8","difficulty":3,"text":"If 40% of a number is 24, what is 75% of the same number?","options":["30","36","45","48","60"],"correctIndex":2,"explanation":"40% of number = 24 → number = 24 / 0.4 = 60. 75% of 60 = 45.","topic":"Algebra"},{"id":"mc_011","contestType":"MATHCOUNTS","difficulty":5,"text":"What is the sum of the interior angles of a regular octagon?","options":["720°","900°","1080°","1260°","1440°"],"correctIndex":2,"explanation":"Sum of interior angles = (n-2) × 180° = 6 × 180 = 1080° for an octagon.","topic":"Geometry"},{"id":"mc_012","contestType":"MATHCOUNTS","difficulty":6,"text":"How many positive integers less than or equal to 1000 are divisible by 3 but not by 5?","options":["200","267","300","333","400"],"correctIndex":1,"explanation":"Div by 3: floor(1000/3)=333. Div by 15: floor(1000/15)=66. 333-66=267.","topic":"Number Theory"},{"id":"mc_013","contestType":"MATHCOUNTS","difficulty":5,"text":"If 2^x = 32 and 3^y = 81, what is x + y?","options":["5","6","7","8","9"],"correctIndex":4,"explanation":"2^5=32, 3^4=81. x=5, y=4. x+y=9.","topic":"Algebra"},{"id":"sb_011","contestType":"Science Bowl","difficulty":3,"text":"Which vitamin is produced when human skin is exposed to sunlight?","options":["Vitamin A","Vitamin B12","Vitamin C","Vitamin D","Vitamin K"],"correctIndex":3,"explanation":"UVB rays from sunlight trigger the production of Vitamin D in the skin.","topic":"Science"},{"id":"sb_012","contestType":"Science Bowl","difficulty":4,"text":"What is the most abundant element in the Earth's crust by mass?","options":["Iron","Silicon","Oxygen","Aluminum","Calcium"],"correctIndex":2,"explanation":"Oxygen makes up about 46% of the Earth's crust by mass.","topic":"Science"},{"id":"sb_013","contestType":"Science Bowl","difficulty":5,"text":"In physics, what is the SI unit of force?","options":["Joule","Watt","Newton","Pascal","Hertz"],"correctIndex":2,"explanation":"The newton (N) is the SI unit of force (kg·m/s²).","topic":"Science"},{"id":"sb_014","contestType":"Science Bowl","difficulty":3,"text":"Which blood type is known as the universal donor?","options":["A","B","AB","O negative","O positive"],"correctIndex":3,"explanation":"Type O negative blood can be given to anyone because it has no A, B, or Rh antigens.","topic":"Science"},{"id":"sb_015","contestType":"Science Bowl","difficulty":4,"text":"What is the name of the process by which plants convert light energy into chemical energy?","options":["Respiration","Photosynthesis","Transpiration","Fermentation","Osmosis"],"correctIndex":1,"explanation":"Photosynthesis is the process plants use to convert sunlight, CO2, and water into glucose and oxygen.","topic":"Science"},{"id":"amc8_020","contestType":"AMC 8","difficulty":2,"text":"What is the units digit of 7 raised to the 25th power?","options":["1","3","5","7","9"],"correctIndex":3,"explanation":"The units digits of successive powers of 7 cycle every 4: 7, 9, 3, 1. Since 25 mod 4 = 1, the units digit is the same as 7^1, which is 7.","topic":"Number Theory"},{"id":"amc8_021","contestType":"AMC 8","difficulty":3,"text":"How many positive divisors does 180 have?","options":["12","15","18","20","24"],"correctIndex":2,"explanation":"Prime factorization: 180 = 2^2 × 3^2 × 5^1. Number of divisors = (2+1) × (2+1) × (1+1) = 3 × 3 × 2 = 18.","topic":"Number Theory"},{"id":"amc8_022","contestType":"AMC 8","difficulty":3,"text":"A 4 by 4 grid is formed by unit squares. How many squares of all possible sizes are contained in the figure?","options":["16","20","25","30","36"],"correctIndex":3,"explanation":"Count by size: 1×1 squares: 16; 2×2: 9; 3×3: 4; 4×4: 1. Total = 16 + 9 + 4 + 1 = 30.","topic":"Geometry"},{"id":"amc8_023","contestType":"AMC 8","difficulty":3,"text":"A store increases the price of an item by 25% and then decreases the new price by 20%. What is the overall percent change from the original price?","options":["0%","5% increase","5% decrease","10% increase","10% decrease"],"correctIndex":0,"explanation":"A 25% increase multiplies the price by 1.25. A 20% decrease multiplies by 0.8. 1.25 × 0.8 = 1.0, so the final price is exactly the original price (0% net change).","topic":"Algebra"},{"id":"amc8_024","contestType":"AMC 8","difficulty":4,"text":"Red and blue marbles are in the ratio 5:3. After 4 red marbles are removed, the ratio becomes 3:2. How many blue marbles are in the bag?","options":["12","18","24","30","36"],"correctIndex":2,"explanation":"Let red = 5k and blue = 3k. After 4 red marbles are removed, (5k - 4) / 3k = 3/2. Cross-multiplying gives 10k - 8 = 9k, so k = 8. Blue marbles = 3 × 8 = 24.","topic":"Algebra"},{"id":"amc8_025","contestType":"AMC 8","difficulty":4,"text":"Painter A can paint a room in 6 hours working alone. Painter B can paint the same room in 8 hours working alone. How many hours will it take them to paint the room if they work together?","options":["3 hours","3.5 hours","24/7 hours","4 hours","14 hours"],"correctIndex":2,"explanation":"A paints 1/6 of the room per hour; B paints 1/8 per hour. Combined rate = 1/6 + 1/8 = 7/24 room per hour. Time required = 1 / (7/24) = 24/7 hours.","topic":"Algebra"},{"id":"amc8_026","contestType":"AMC 8","difficulty":3,"text":"A cyclist travels 30 miles to a destination at 15 mph and returns by the same route at 10 mph. What is the average speed for the round trip?","options":["12 mph","12.5 mph","13 mph","14 mph","15 mph"],"correctIndex":0,"explanation":"Total distance = 60 miles. Time going = 30/15 = 2 hours. Time returning = 30/10 = 3 hours. Total time = 5 hours. Average speed = 60 miles / 5 hours = 12 mph.","topic":"Algebra"},{"id":"amc8_027","contestType":"AMC 8","difficulty":5,"text":"What is the measure, in degrees, of the smaller angle formed by the hour and minute hands of a clock at 2:30?","options":["60","75","90","105","150"],"correctIndex":3,"explanation":"At 2:00 the hour hand is at 60°. By 2:30 the hour hand has moved an additional 15° (30 min × 0.5° per min). The minute hand is at 180°. The difference is |60 + 15 - 180| = 105°. The smaller angle is 105°.","topic":"Geometry"},{"id":"amc8_028","contestType":"AMC 8","difficulty":4,"text":"Eight friends attend a reunion. Each friend shakes hands with every other friend exactly once. How many handshakes take place?","options":["21","28","36","56","64"],"correctIndex":1,"explanation":"The number of ways to choose 2 people out of 8 to shake hands is the combination C(8,2) = (8 × 7) / 2 = 28.","topic":"Counting & Probability"},{"id":"amc8_029","contestType":"AMC 8","difficulty":3,"text":"Three-fifths of the students in a class play a musical instrument. Of the students who play an instrument, two-thirds play the piano. What fraction of the entire class plays the piano?","options":["1/5","2/5","3/10","4/15","2/3"],"correctIndex":1,"explanation":"The fraction that play piano = (3/5) × (2/3) = 2/5 of the entire class.","topic":"Algebra"},{"id":"amc8_030","contestType":"AMC 8","difficulty":4,"text":"How many prime numbers are there between 1 and 50, inclusive?","options":["13","14","15","16","17"],"correctIndex":2,"explanation":"The primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. There are 15 primes.","topic":"Number Theory"},{"id":"amc8_031","contestType":"AMC 8","difficulty":2,"text":"Today is Wednesday. What day of the week will it be exactly 100 days from today?","options":["Friday","Saturday","Sunday","Monday","Tuesday"],"correctIndex":0,"explanation":"100 divided by 7 is 14 weeks with a remainder of 2 days (98 + 2). Wednesday plus 2 days is Friday.","topic":"Number Theory"},{"id":"amc8_032","contestType":"AMC 8","difficulty":5,"text":"A grid is composed of 3 rows and 4 columns of unit rectangles. A straight diagonal line is drawn from the southwest corner to the northeast corner. How many unit rectangles does the diagonal cross?","options":["4","5","6","7","8"],"correctIndex":2,"explanation":"In an m-by-n grid the number of unit squares crossed by the main diagonal is given by m + n − gcd(m, n). Here m=3, n=4, gcd=1, so 3 + 4 − 1 = 6.","diagram":{"type":"rect_grid_diagonal","rows":3,"cols":4,"aspectRatio":1.55},"topic":"Geometry"},{"id":"amc8_033","contestType":"AMC 8","difficulty":3,"text":"The sum of five consecutive odd integers is 85. What is the middle integer?","options":["13","15","17","19","21"],"correctIndex":2,"explanation":"Let the middle integer be m. The five numbers are m−4, m−2, m, m+2, m+4. Their sum is 5m = 85, so m = 17.","topic":"Algebra"},{"id":"amc8_034","contestType":"AMC 8","difficulty":4,"text":"How many positive integers less than or equal to 100 are divisible by 3 or by 4?","options":["42","45","50","58","67"],"correctIndex":2,"explanation":"Multiples of 3: floor(100/3) = 33. Multiples of 4: floor(100/4) = 25. Multiples of 12: floor(100/12) = 8. By inclusion-exclusion: 33 + 25 − 8 = 50.","topic":"Number Theory"},{"id":"amc8_035","contestType":"AMC 8","difficulty":5,"text":"Two positive numbers have a sum of 36 and a product of 288. What is the larger of the two numbers?","options":["12","18","20","24","28"],"correctIndex":3,"explanation":"Let the numbers be x and 36−x. Then x(36−x) = 288 → x² − 36x + 288 = 0. The discriminant is 144, so roots are (36 ± 12)/2, which are 24 and 12. The larger is 24.","topic":"Algebra"},{"id":"amc8_036","contestType":"AMC 8","difficulty":4,"text":"A store increases the price of a jacket by 30% and then offers a 30% discount on the new price. The final price after the discount is $91. What was the original price before any changes?","options":["$70","$80","$90","$100","$130"],"correctIndex":3,"explanation":"The net effect is multiplication by 1.30 × 0.70 = 0.91. Therefore the original price = 91 / 0.91 = $100.","topic":"Algebra"},{"id":"amc8_037","contestType":"AMC 8","difficulty":3,"text":"Red and blue cards in a box are in the ratio 2:3. After 4 red cards are added, there are equal numbers of red and blue cards. How many cards were in the box originally?","options":["12","15","20","25","30"],"correctIndex":2,"explanation":"Let red = 2k and blue = 3k. After adding: 2k + 4 = 3k. Solving gives k = 4. Original total = 5k = 20 cards.","topic":"Algebra"},{"id":"amc8_038","contestType":"AMC 8","difficulty":2,"text":"What is the units digit of 3 raised to the 17th power?","options":["1","3","7","9","5"],"correctIndex":1,"explanation":"The units digits of powers of 3 cycle every 4: 3, 9, 7, 1. 17 divided by 4 gives remainder 1, so the units digit is the same as 3^1, which is 3.","topic":"Number Theory"},{"id":"amc8_039","contestType":"AMC 8","difficulty":3,"text":"On a grid you must move only right or up. How many different paths are there from the bottom-left corner to the top-right corner of a figure that requires exactly 3 right and 2 up moves?","options":["6","8","10","12","15"],"correctIndex":2,"explanation":"The total number of moves is 5, and you choose 3 positions out of 5 for the right moves (the rest are up). This is the binomial coefficient C(5,3) = 10.","diagram":{"type":"grid_paths","right":3,"up":2,"aspectRatio":1.45},"topic":"Counting & Probability"},{"id":"amc8_040","contestType":"AMC 8","difficulty":3,"text":"Group A has 5 more members than Group B. Group B has 3 more members than Group C. If the three groups together have 47 people, how many people are in the largest group?","options":["12","15","17","20","22"],"correctIndex":3,"explanation":"Let C = x. Then B = x + 3 and A = x + 8. Total: x + (x+3) + (x+8) = 3x + 11 = 47 → 3x = 36 → x = 12. A has 20 people.","topic":"Algebra"},{"id":"amc8_041","contestType":"AMC 8","difficulty":4,"text":"The ratio of red to blue tokens is 5:3. After 6 more red tokens are added the ratio becomes 2:1. How many blue tokens are there?","options":["12","15","18","21","24"],"correctIndex":2,"explanation":"Let red = 5k, blue = 3k. (5k + 6) / 3k = 2/1 → 5k + 6 = 6k → k = 6. Blue = 3 × 6 = 18.","topic":"Algebra"},{"id":"amc8_042","contestType":"AMC 8","difficulty":3,"text":"Three points are plotted on the coordinate plane: (0,0), (8,0), and (2, y). If the area of the triangle they form is 20, what is the value of y?","options":["2","4","5","8","10"],"correctIndex":2,"explanation":"Base along x-axis is 8 units. Area = (1/2) × base × height = (1/2) × 8 × y = 20 → 4y = 20 → y = 5.","diagram":{"type":"triangle","variant":"coordinate"},"topic":"Geometry"},{"id":"amc8_043","contestType":"AMC 8","difficulty":5,"text":"Three fair six-sided dice are rolled. How many of the 216 possible outcomes have a product that is divisible by 4?","options":["108","120","135","162","180"],"correctIndex":2,"explanation":"For the product to be divisible by 4, the three rolls must contribute at least two factors of 2. Count the complement: all rolls odd gives 3^3 = 27 outcomes, and exactly one roll contributes a single factor of 2 (rolling 2 or 6) while the other two are odd gives 3 × 2 × 3 × 3 = 54 outcomes. So 81 outcomes fail, and 216 - 81 = 135 outcomes work.","topic":"Counting & Probability"},{"id":"amc8_044","contestType":"AMC 8","difficulty":3,"text":"First, 1/3 of a pie is eaten. Then 1/5 of what remains is eaten. What fraction of the original pie is left?","options":["1/3","2/5","5/12","8/15","1/2"],"correctIndex":3,"explanation":"After 1/3 of the pie is eaten, 2/3 remains. Then 1/5 of the remaining 2/3 is eaten, leaving 4/5 of that portion: (2/3) × (4/5) = 8/15 of the original pie.","topic":"Algebra"},{"id":"amc8_045","contestType":"AMC 8","difficulty":4,"text":"How many 3-cent and 5-cent stamps can be used to make exactly 47 cents (order does not matter)?","options":["3","4","5","6","7"],"correctIndex":0,"explanation":"Solve 3x + 5y = 47 with x,y ≥ 0 integers. Checking y values, y must be 1, 4, or 7, giving (x,y) = (14,1), (9,4), and (4,7). There are 3 combinations.","topic":"Number Theory"},{"id":"amc8_046","contestType":"AMC 8","difficulty":5,"text":"Two concentric circles have radii r and R. The area of the ring between them equals the area of the inner circle. If r=5, what is R?","options":["5√2","6","7","5√3","10"],"correctIndex":0,"explanation":"πR² − πr² = πr² → R² = 2r² → R = r√2 = 5√2.","diagram":{"type":"circle","variant":"concentric"},"topic":"Geometry"},{"id":"amc8_047","contestType":"AMC 8","difficulty":3,"text":"Mia is 4 years older than Leo. In 3 years Mia will be twice as old as Leo is now. How old is Leo now?","options":["5","6","7","8","9"],"correctIndex":2,"explanation":"Leo = x. Mia = x+4. In 3 years: x+7 = 2x → x=7.","topic":"Algebra"},{"id":"amc8_048","contestType":"AMC 8","difficulty":4,"text":"A 2-by-3 grid of unit squares is drawn (3 horizontal lines and 4 vertical lines). How many rectangles of any size are in the figure?","options":["18","24","30","36","48"],"correctIndex":0,"explanation":"Horizontal choices: C(3,2)=3. Vertical choices: C(4,2)=6. Total rectangles = 3×6=18.","topic":"Geometry"},{"id":"amc8_049","contestType":"AMC 8","difficulty":5,"text":"A standard deck has 52 cards. Two cards are drawn at random without replacement. What is the probability both are face cards (J, Q, or K)?","options":["11/221","33/221","11/663","1/17","3/26"],"correctIndex":0,"explanation":"There are 12 face cards. P = (12/52)*(11/51) = 132/2652 = 11/221.","topic":"Counting & Probability"},{"id":"amc8_050","contestType":"AMC 8","difficulty":4,"topic":"Number Theory","text":"How many positive divisors of 360 are divisible by 6?","options":["8","10","12","15","18"],"correctIndex":2,"explanation":"Since 360 = 2^3 × 3^2 × 5, a divisor divisible by 6 must include at least one factor of 2 and one factor of 3. The exponent of 2 has 3 choices (1,2,3), the exponent of 3 has 2 choices (1,2), and the exponent of 5 has 2 choices (0,1). The total is 3 × 2 × 2 = 12.","solutionSteps":["Factor 360 as 2^3 × 3^2 × 5.","A divisor is divisible by 6 only if it contains 2^1 and 3^1 at minimum.","Choose the exponent of 2: 1, 2, or 3, giving 3 choices.","Choose the exponent of 3: 1 or 2, giving 2 choices.","Choose the exponent of 5: 0 or 1, giving 2 choices.","Multiply the independent choices: 3 × 2 × 2 = 12."]},{"id":"amc8_051","contestType":"AMC 8","difficulty":5,"topic":"Geometry","text":"A right triangle has legs of length 9 and 12. What is the length of the altitude from the right angle to the hypotenuse?","options":["24/5","36/5","45/4","9","12"],"correctIndex":1,"explanation":"The hypotenuse is 15. The area is (1/2)(9)(12)=54. Using the hypotenuse as the base, 54 = (1/2)(15)h, so h = 108/15 = 36/5.","solutionSteps":["Find the hypotenuse using the Pythagorean theorem: 9^2 + 12^2 = 225, so the hypotenuse is 15.","Compute the area using the legs: area = (1/2)(9)(12) = 54.","Use the hypotenuse as the base and call the altitude h.","Then 54 = (1/2)(15)h.","Solving gives h = 108/15 = 36/5."]},{"id":"amc8_052","contestType":"AMC 8","difficulty":4,"topic":"Counting & Probability","text":"A bag contains 3 red, 2 blue, and 1 green marble. Two marbles are drawn without replacement. What is the probability that exactly one of the two marbles is red?","options":["1/2","3/5","2/3","3/4","4/5"],"correctIndex":1,"explanation":"There are C(6,2)=15 ways to choose 2 marbles. To choose exactly one red, pick 1 of the 3 red marbles and 1 of the 3 non-red marbles, giving 3 × 3 = 9 favorable pairs. The probability is 9/15 = 3/5."},{"id":"amc8_053","contestType":"AMC 8","difficulty":3,"topic":"Algebra","text":"The sequence 7, 11, 15, 19, ... continues by adding 4 each time. How many terms of the sequence are less than or equal to 100?","options":["22","23","24","25","26"],"correctIndex":2,"explanation":"The nth term is 7 + 4(n-1). We need 7 + 4(n-1) ≤ 100, so 4(n-1) ≤ 93 and n-1 ≤ 23.25. Thus the largest whole number n is 24."},{"id":"amc8_054","contestType":"AMC 8","difficulty":4,"topic":"Geometry","text":"A 3-by-4 grid of unit squares is drawn. How many rectangles of all possible sizes are contained in the grid?","options":["24","36","48","60","72"],"correctIndex":3,"explanation":"A 3-by-4 grid has 4 horizontal grid lines and 5 vertical grid lines. A rectangle is determined by choosing 2 horizontal lines and 2 vertical lines. This gives C(4,2) × C(5,2) = 6 × 10 = 60 rectangles."},{"id":"amc8_055","contestType":"AMC 8","difficulty":3,"topic":"Number Theory","text":"What is the units digit of 2^2026?","options":["2","4","6","8","0"],"correctIndex":1,"explanation":"The units digits of powers of 2 cycle as 2, 4, 8, 6. Since 2026 leaves a remainder of 2 when divided by 4, the units digit is the second number in the cycle, which is 4."},{"id":"amc8_056","contestType":"AMC 8","difficulty":5,"topic":"Algebra","text":"Pipe A can fill a tank in 12 minutes, Pipe B can fill it in 20 minutes, and a drain can empty it in 30 minutes. If both pipes and the drain are open, how many minutes will it take to fill the tank?","options":["8","10","12","15","20"],"correctIndex":1,"explanation":"The combined rate is 1/12 + 1/20 - 1/30 tank per minute. With common denominator 60, this is 5/60 + 3/60 - 2/60 = 6/60 = 1/10 tank per minute. The tank fills in 10 minutes."},{"id":"amc8_057","contestType":"AMC 8","difficulty":3,"topic":"Geometry","text":"What is the measure of each interior angle of a regular pentagon?","options":["72°","90°","108°","120°","135°"],"correctIndex":2,"explanation":"The sum of the interior angles of a pentagon is (5-2)×180° = 540°. A regular pentagon has 5 equal interior angles, so each one is 540°/5 = 108°.","diagram":{"type":"regular_polygon","sides":5}},{"id":"amc8_058","contestType":"AMC 8","difficulty":4,"topic":"Counting & Probability","text":"A code consists of two different letters chosen from A, B, and C followed by two different digits chosen from 0, 1, 2, 3, and 4. How many such codes are possible?","options":["60","90","100","120","150"],"correctIndex":3,"explanation":"There are 3 choices for the first letter and 2 choices for the second letter, since the letters must be different. There are 5 choices for the first digit and 4 choices for the second digit. The total number of codes is 3 × 2 × 5 × 4 = 120."},{"id":"amc8_059","contestType":"AMC 8","difficulty":2,"topic":"Algebra","text":"The average of six numbers is 12. Five of the numbers are 8, 10, 10, 12, and 15. What is the sixth number?","options":["15","16","17","18","19"],"correctIndex":2,"explanation":"If the average of six numbers is 12, their total is 6 × 12 = 72. The five known numbers sum to 8 + 10 + 10 + 12 + 15 = 55. The missing number is 72 - 55 = 17."},{"id":"amc8_060","contestType":"AMC 8","difficulty":3,"topic":"Geometry","text":"A rectangle has side lengths 6 and 8. If a diagonal is drawn, what is the perimeter of one of the two right triangles formed?","options":["20","22","24","26","28"],"correctIndex":2,"explanation":"The diagonal is the hypotenuse of a right triangle with legs 6 and 8. Its length is √(6^2 + 8^2) = √100 = 10. The perimeter of one triangle is 6 + 8 + 10 = 24."},{"id":"amc8_061","contestType":"AMC 8","difficulty":4,"topic":"Algebra","text":"If a:b = 2:5 and b:c = 10:3, what is a:c?","options":["2:3","3:4","4:3","5:6","20:15"],"correctIndex":2,"explanation":"Make the b-values match. The ratio a:b = 2:5 can be scaled by 2 to get a:b = 4:10. Since b:c = 10:3, we can combine them to get a:b:c = 4:10:3. Therefore a:c = 4:3."},{"id":"amc8_062","contestType":"AMC 8","difficulty":4,"topic":"Number Theory","text":"How many two-digit positive integers are divisible by 4 but not by 6?","options":["14","15","16","17","18"],"correctIndex":0,"explanation":"The two-digit multiples of 4 run from 12 to 96, so there are (96-12)/4 + 1 = 22 of them. A number divisible by both 4 and 6 is divisible by lcm(4,6)=12. There are 8 two-digit multiples of 12, so the count is 22 - 8 = 14.","solutionSteps":["Two-digit multiples of 4 are 12, 16, ..., 96.","There are (96 - 12) / 4 + 1 = 22 such numbers.","Numbers divisible by both 4 and 6 are multiples of 12.","Two-digit multiples of 12 are 12, 24, ..., 96, giving 8 numbers.","So the count is 22 - 8 = 14."]},{"id":"amc8_063","contestType":"AMC 8","difficulty":4,"topic":"Geometry","text":"A square has area 50. What is the length of its diagonal?","options":["5√2","10","10√2","25","50"],"correctIndex":1,"explanation":"If the square has side length s, then s^2 = 50. The diagonal of a square is s√2, so the diagonal is √50 × √2 = √100 = 10."},{"id":"amc8_064","contestType":"AMC 8","difficulty":3,"topic":"Counting & Probability","text":"A spinner has 8 equal sections numbered 1 through 8. What is the probability that a spin lands on a number that is prime or odd?","options":["1/2","5/8","3/4","7/8","1"],"correctIndex":1,"explanation":"The odd numbers are 1, 3, 5, and 7. The prime numbers are 2, 3, 5, and 7. Their union is {1,2,3,5,7}, which has 5 outcomes out of 8. The probability is 5/8."},{"id":"amc8_065","contestType":"AMC 8","difficulty":5,"topic":"Algebra","text":"A number is increased by 20%, and the result is then decreased by 25%. The final result is 90. What was the original number?","options":["90","96","100","108","120"],"correctIndex":2,"explanation":"Increasing by 20% multiplies by 1.2, and then decreasing by 25% multiplies by 0.75. The net multiplier is 1.2 × 0.75 = 0.9. If 0.9 times the original number is 90, the original number is 100."},{"id":"amc8_066","contestType":"AMC 8","difficulty":4,"topic":"Geometry","text":"A circle has circumference 18π. What is its area?","options":["36π","54π","72π","81π","162π"],"correctIndex":3,"explanation":"The circumference is 2πr = 18π, so r = 9. The area is πr^2 = π(9^2) = 81π."},{"id":"amc8_067","contestType":"AMC 8","difficulty":5,"topic":"Counting & Probability","text":"How many integers from 1 to 200 inclusive are divisible by exactly one of 3 and 5?","options":["80","93","94","107","120"],"correctIndex":0,"explanation":"Multiples of 3: floor(200/3)=66. Multiples of 5: floor(200/5)=40. Multiples of both are multiples of 15: floor(200/15)=13. Divisible by exactly one means 66 + 40 - 2×13 = 80."},{"id":"amc8_068","contestType":"AMC 8","difficulty":3,"topic":"Algebra","text":"The sum of four consecutive integers is 94. What is the largest of the four integers?","options":["22","23","24","25","26"],"correctIndex":3,"explanation":"Let the four integers be n, n+1, n+2, and n+3. Their sum is 4n+6=94, so 4n=88 and n=22. The largest is n+3=25."},{"id":"amc8_069","contestType":"AMC 8","difficulty":4,"topic":"Geometry","text":"A rectangular prism has dimensions 2, 3, and 4. What is its surface area?","options":["24","36","48","52","72"],"correctIndex":3,"explanation":"The surface area of a rectangular prism is 2(lw + lh + wh). With dimensions 2, 3, and 4, this is 2(2×3 + 2×4 + 3×4) = 2(6 + 8 + 12) = 52."},{"id":"mc_014","contestType":"MATHCOUNTS","difficulty":5,"text":"How many positive integers n ≤ 200 have the property that the sum of the digits of n is a multiple of 3?","options":["66","67","68","69","70"],"correctIndex":0,"explanation":"A number has digit sum divisible by 3 exactly when the number itself is divisible by 3. The positive multiples of 3 up to 200 are 3, 6, ..., 198, so there are 198 ÷ 3 = 66.","topic":"Number Theory"},{"id":"mc_015","contestType":"MATHCOUNTS","difficulty":6,"text":"In how many ways can 6 distinct people be seated at a round table if two particular people must sit next to each other?","options":["48","60","120","240","720"],"correctIndex":0,"explanation":"Treat the two particular people as one block. Then there are 5 units around the table, which can be arranged in (5-1)! = 24 circular arrangements. The two people inside the block can switch places, so the total is 24 × 2 = 48.","topic":"Counting & Probability"},{"id":"mc_016","contestType":"MATHCOUNTS","difficulty":5,"text":"What is the greatest common divisor of 84, 126, and 210?","options":["14","21","28","42","63"],"correctIndex":3,"explanation":"GCD(84,126)=42. GCD(42,210)=42.","topic":"Number Theory"},{"id":"mc_017","contestType":"MATHCOUNTS","difficulty":6,"text":"A bag contains 6 red, 5 blue, and 4 green marbles. Three marbles are drawn without replacement. What is the probability that all three are different colors?","options":["1/5","4/15","24/91","2/7","3/10"],"correctIndex":2,"explanation":"There are C(15,3) = 455 ways to draw 3 marbles. To get all three colors, choose 1 red, 1 blue, and 1 green: 6 × 5 × 4 = 120 ways. The probability is 120/455 = 24/91.","topic":"Counting & Probability"},{"id":"mc_018","contestType":"MATHCOUNTS","difficulty":5,"text":"The sum of the interior angles of a regular polygon is 1260 degrees. How many sides does the polygon have?","options":["7","8","9","10","11"],"correctIndex":2,"explanation":"(n-2)*180 = 1260 → n-2 = 7 → n=9.","diagram":{"type":"regular_polygon","sides":9},"topic":"Geometry"},{"id":"sb_016","contestType":"Science Bowl","difficulty":3,"text":"Which of the following is the smallest unit of matter that retains the properties of an element?","options":["Proton","Neutron","Electron","Atom","Molecule"],"correctIndex":3,"explanation":"An atom is the smallest particle of an element that still has all the chemical properties of that element.","topic":"Science"},{"id":"sb_017","contestType":"Science Bowl","difficulty":4,"text":"In the human body, which structure is primarily responsible for filtering blood and producing urine?","options":["Liver","Kidney","Spleen","Pancreas","Heart"],"correctIndex":1,"explanation":"The kidneys filter waste products from the blood and produce urine.","topic":"Science"},{"id":"sb_018","contestType":"Science Bowl","difficulty":2,"text":"Which planet is known as the 'Red Planet'?","options":["Venus","Jupiter","Mars","Saturn","Mercury"],"correctIndex":2,"explanation":"Mars appears reddish due to iron oxide (rust) on its surface.","topic":"Science"},{"id":"sb_019","contestType":"Science Bowl","difficulty":4,"text":"What is the chemical formula for table salt?","options":["NaCl","KCl","CaCl2","Na2CO3","MgSO4"],"correctIndex":0,"explanation":"Table salt is sodium chloride, NaCl.","topic":"Science"},{"id":"sb_020","contestType":"Science Bowl","difficulty":3,"text":"Which type of energy is stored in a stretched rubber band?","options":["Kinetic energy","Thermal energy","Potential energy","Chemical energy","Nuclear energy"],"correctIndex":2,"explanation":"A stretched rubber band has elastic potential energy that can be converted to kinetic energy when released.","topic":"Science"},{"id":"amc8_070","contestType":"AMC 8","difficulty":3,"topic":"Geometry","text":"A triangle has base 14 and height 9. What is its area?","options":["46","54","63","72","126"],"correctIndex":2,"explanation":"The area of a triangle is (1/2)bh. Here the area is (1/2)(14)(9) = 7 × 9 = 63."},{"id":"amc8_071","contestType":"AMC 8","difficulty":4,"topic":"Geometry","text":"A rectangle has side lengths 5 and 12. What is its perimeter plus the length of one diagonal?","options":["30","34","39","47","60"],"correctIndex":3,"explanation":"The perimeter is 2(5+12)=34. The diagonal has length √(5^2+12^2)=13. The requested sum is 34+13=47."},{"id":"amc8_072","contestType":"AMC 8","difficulty":2,"topic":"Algebra","text":"If 3x + 7 = 31, what is x?","options":["6","7","8","9","10"],"correctIndex":2,"explanation":"Subtract 7 from both sides to get 3x = 24. Dividing by 3 gives x = 8."},{"id":"amc8_073","contestType":"AMC 8","difficulty":4,"topic":"Algebra","text":"Adult tickets cost $8 and child tickets cost $5. A group buys 10 tickets for $71. How many adult tickets did they buy?","options":["5","6","7","8","9"],"correctIndex":2,"explanation":"Let a be the number of adult tickets. Then 10-a child tickets were bought. The cost equation is 8a + 5(10-a) = 71, so 3a + 50 = 71 and a = 7."},{"id":"amc8_074","contestType":"AMC 8","difficulty":3,"topic":"Number Theory","text":"What is the units digit of 7^2027?","options":["1","3","5","7","9"],"correctIndex":1,"explanation":"The units digits of powers of 7 cycle as 7, 9, 3, 1. Since 2027 leaves a remainder of 3 when divided by 4, the units digit is the third number in the cycle, which is 3."},{"id":"amc8_075","contestType":"AMC 8","difficulty":4,"topic":"Number Theory","text":"How many positive integers less than or equal to 150 are divisible by 4 or by 9?","options":["45","48","49","53","57"],"correctIndex":2,"explanation":"There are floor(150/4)=37 multiples of 4 and floor(150/9)=16 multiples of 9. Multiples of both are multiples of 36, and floor(150/36)=4. By inclusion-exclusion, the count is 37+16-4=49."},{"id":"amc8_076","contestType":"AMC 8","difficulty":4,"topic":"Counting & Probability","text":"How many distinct arrangements are there of the letters in the word LEVEL?","options":["20","24","30","60","120"],"correctIndex":2,"explanation":"The word LEVEL has 5 letters, with two L's and two E's repeated. The number of distinct arrangements is 5!/(2!2!) = 120/4 = 30."},{"id":"amc8_077","contestType":"AMC 8","difficulty":3,"topic":"Counting & Probability","text":"Two fair coins are flipped and one fair six-sided die is rolled. What is the probability of getting exactly one head and an even number on the die?","options":["1/8","1/4","1/3","1/2","3/4"],"correctIndex":1,"explanation":"The probability of exactly one head in two coin flips is 2/4 = 1/2. The probability of an even die roll is 3/6 = 1/2. The events are independent, so the probability is (1/2)(1/2)=1/4."},{"id":"amc8_078","contestType":"AMC 8","difficulty":3,"topic":"Trigonometry","text":"In a right triangle, one acute angle has sine 3/5. What is the cosine of the other acute angle?","options":["2/5","3/5","4/5","5/3","5/4"],"correctIndex":1,"explanation":"The two acute angles in a right triangle are complementary. The cosine of one acute angle equals the sine of the other acute angle. Therefore the cosine of the other acute angle is 3/5."},{"id":"amc8_079","contestType":"AMC 8","difficulty":4,"topic":"Trigonometry","text":"A right triangle has legs 6 and 8 and hypotenuse 10. For the acute angle opposite the side of length 6, what is the tangent of the angle?","options":["3/5","3/4","4/3","5/4","5/3"],"correctIndex":1,"explanation":"Tangent is opposite divided by adjacent. The angle is opposite the side of length 6, and the other leg has length 8, so tan = 6/8 = 3/4."},{"id":"sb_021","contestType":"Science Bowl","difficulty":3,"topic":"Science","text":"Which part of a plant cell is primarily responsible for photosynthesis?","options":["Nucleus","Mitochondrion","Chloroplast","Cell wall","Ribosome"],"correctIndex":2,"explanation":"Chloroplasts contain chlorophyll and are the organelles where photosynthesis takes place in plant cells."},{"id":"sb_022","contestType":"Science Bowl","difficulty":4,"topic":"Science","text":"In an electric circuit, what quantity is measured in ohms?","options":["Voltage","Current","Resistance","Power","Charge"],"correctIndex":2,"explanation":"The ohm is the SI unit of electrical resistance."},{"id":"amc8_080","contestType":"AMC 8","difficulty":3,"topic":"Geometry","text":"A trapezoid has bases of length 8 and 14 and height 5. What is its area?","options":["40","45","50","55","70"],"correctIndex":3,"explanation":"The area of a trapezoid is (1/2)(b1+b2)h. Here the area is (1/2)(8+14)(5) = 11×5 = 55."},{"id":"amc8_081","contestType":"AMC 8","difficulty":3,"topic":"Geometry","text":"A circle has area 49π. What is its circumference?","options":["7π","14π","21π","28π","49π"],"correctIndex":1,"explanation":"Area is πr^2 = 49π, so r=7. The circumference is 2πr = 14π."},{"id":"amc8_082","contestType":"AMC 8","difficulty":4,"topic":"Geometry","text":"A cube has edge length 3. What is its surface area?","options":["27","36","45","54","81"],"correctIndex":3,"explanation":"A cube has 6 congruent square faces. Each face has area 3^2 = 9, so the surface area is 6×9 = 54."},{"id":"amc8_083","contestType":"AMC 8","difficulty":2,"topic":"Algebra","text":"If 2x - 5 = 17, what is x?","options":["6","8","10","11","12"],"correctIndex":3,"explanation":"Add 5 to both sides to get 2x = 22. Dividing by 2 gives x = 11."},{"id":"amc8_084","contestType":"AMC 8","difficulty":3,"topic":"Algebra","text":"Apples and oranges are in the ratio 3:5. If there are 64 pieces of fruit altogether, how many are oranges?","options":["24","32","36","40","48"],"correctIndex":3,"explanation":"The total ratio is 3+5=8 parts. Since 64 fruit represent 8 parts, each part is 8 fruit. Oranges are 5 parts, so there are 5×8 = 40 oranges."},{"id":"amc8_085","contestType":"AMC 8","difficulty":3,"topic":"Algebra","text":"The nth term of a sequence is 4n + 1. What is the 20th term?","options":["77","80","81","84","85"],"correctIndex":2,"explanation":"Substitute n=20 into 4n+1 to get 4(20)+1 = 81."},{"id":"amc8_086","contestType":"AMC 8","difficulty":4,"topic":"Number Theory","text":"How many positive divisors does 420 have?","options":["16","20","24","28","32"],"correctIndex":2,"explanation":"Factor 420 = 2^2 × 3 × 5 × 7. The number of positive divisors is (2+1)(1+1)(1+1)(1+1)=24."},{"id":"amc8_087","contestType":"AMC 8","difficulty":3,"topic":"Number Theory","text":"What is the remainder when 5^3 + 7^2 is divided by 6?","options":["0","1","2","3","4"],"correctIndex":0,"explanation":"Compute 5^3 + 7^2 = 125 + 49 = 174. Since 174 is divisible by 6, the remainder is 0."},{"id":"amc8_088","contestType":"AMC 8","difficulty":4,"topic":"Number Theory","text":"How many positive integers less than 100 are divisible by 7 but not by 2?","options":["6","7","8","9","14"],"correctIndex":1,"explanation":"The multiples of 7 less than 100 are 7, 14, ..., 98, for 14 multiples total. Half of them are even multiples and half are odd multiples, so 7 are not divisible by 2."},{"id":"amc8_089","contestType":"AMC 8","difficulty":2,"topic":"Counting & Probability","text":"How many ways are there to choose a 2-person team from 6 students?","options":["12","15","18","24","30"],"correctIndex":1,"explanation":"Choosing a 2-person team from 6 students gives C(6,2) = (6×5)/2 = 15."},{"id":"amc8_090","contestType":"AMC 8","difficulty":4,"topic":"Counting & Probability","text":"On a grid, a path from one corner to the opposite corner requires exactly 4 moves right and 3 moves up. How many such paths are possible?","options":["21","28","35","42","70"],"correctIndex":2,"explanation":"There are 7 total moves. Choose which 3 of the 7 moves are up, giving C(7,3)=35 paths."},{"id":"amc8_091","contestType":"AMC 8","difficulty":3,"topic":"Counting & Probability","text":"Two fair six-sided dice are rolled. What is the probability that their sum is 7?","options":["1/12","1/9","1/6","1/4","1/3"],"correctIndex":2,"explanation":"There are 36 equally likely outcomes. The outcomes with sum 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), so the probability is 6/36 = 1/6."},{"id":"amc8_092","contestType":"AMC 8","difficulty":3,"topic":"Trigonometry","text":"In a right triangle, an acute angle has sine 5/13. What is the cosine of that same angle?","options":["5/13","12/13","13/12","5/12","12/5"],"correctIndex":1,"explanation":"If sin = 5/13, then the opposite side can be 5 and the hypotenuse 13. The adjacent side is 12 by the 5-12-13 triangle, so cos = 12/13."},{"id":"amc8_093","contestType":"AMC 8","difficulty":2,"topic":"Trigonometry","text":"In a 45-45-90 triangle, what is the tangent of either acute angle?","options":["0","1/2","1","√2","2"],"correctIndex":2,"explanation":"In a 45-45-90 triangle, the two legs are equal. Tangent is opposite divided by adjacent, so tan 45° = 1."},{"id":"amc8_094","contestType":"AMC 8","difficulty":4,"topic":"Trigonometry","text":"A 10-foot ladder leans against a wall and reaches a height of 8 feet. How far is the foot of the ladder from the wall?","options":["4","5","6","8","10"],"correctIndex":2,"explanation":"The ladder is the hypotenuse of a right triangle. If the height is 8 and the hypotenuse is 10, the horizontal distance is √(10^2-8^2)=√36=6."},{"id":"sb_023","contestType":"Science Bowl","difficulty":3,"topic":"Science","text":"Which type of blood cell primarily carries oxygen through the body?","options":["Platelet","Red blood cell","White blood cell","Neuron","Stem cell"],"correctIndex":1,"explanation":"Red blood cells contain hemoglobin, which binds and transports oxygen."},{"id":"sb_024","contestType":"Science Bowl","difficulty":4,"topic":"Science","text":"Which law states that an object in motion remains in motion unless acted on by a net external force?","options":["Newton's first law","Newton's second law","Newton's third law","Ohm's law","Boyle's law"],"correctIndex":0,"explanation":"Newton's first law is the law of inertia: motion stays unchanged unless a net external force acts."},{"id":"sb_025","contestType":"Science Bowl","difficulty":3,"topic":"Science","text":"Which layer of Earth is liquid and lies beneath the mantle?","options":["Crust","Inner core","Outer core","Lithosphere","Asthenosphere"],"correctIndex":2,"explanation":"The outer core is a liquid layer mostly made of iron and nickel beneath the mantle."},{"id":"amc8_095","contestType":"AMC 8","difficulty":3,"topic":"Geometry","text":"What is the measure of each interior angle of a regular hexagon?","options":["90°","108°","120°","135°","144°"],"correctIndex":2,"explanation":"The sum of the interior angles of a hexagon is (6-2)×180° = 720°. A regular hexagon has 6 equal angles, so each angle is 720°/6 = 120°."},{"id":"amc8_096","contestType":"AMC 8","difficulty":4,"topic":"Algebra","text":"A number is decreased by 15%, and the result is 68. What was the original number?","options":["75","80","82","85","90"],"correctIndex":1,"explanation":"After a 15% decrease, 85% of the original number remains. If 0.85x = 68, then x = 68/0.85 = 80."},{"id":"amc8_097","contestType":"AMC 8","difficulty":3,"topic":"Number Theory","text":"How many positive integers less than 100 have digit sum divisible by 9?","options":["9","10","11","12","13"],"correctIndex":2,"explanation":"A number has digit sum divisible by 9 exactly when the number itself is divisible by 9. The positive multiples of 9 less than 100 are 9, 18, ..., 99, giving 11 numbers."},{"id":"amc8_098","contestType":"AMC 8","difficulty":3,"topic":"Counting & Probability","text":"A code consists of one letter chosen from A, B, C, and D followed by two digits from 0 through 9. Digits may repeat. How many codes are possible?","options":["40","80","100","200","400"],"correctIndex":4,"explanation":"There are 4 choices for the letter, 10 choices for the first digit, and 10 choices for the second digit. The total is 4×10×10 = 400."},{"id":"amc8_099","contestType":"AMC 8","difficulty":2,"topic":"Geometry","text":"A parallelogram has base 13 and height 6. What is its area?","options":["19","38","72","78","156"],"correctIndex":3,"explanation":"The area of a parallelogram is base times height. Here the area is 13×6 = 78."},{"id":"amc8_100","contestType":"AMC 8","difficulty":4,"topic":"Algebra","text":"The ratio of girls to boys in a club is 5:4. If there are 45 students in the club, how many boys are there?","options":["16","18","20","25","30"],"correctIndex":2,"explanation":"The total ratio is 5+4=9 parts. Since 45 students represent 9 parts, each part is 5 students. Boys are 4 parts, so there are 4×5 = 20 boys."},{"id":"amc8_101","contestType":"AMC 8","difficulty":4,"topic":"Counting & Probability","text":"A bag contains 4 red marbles and 3 blue marbles. A marble is drawn, replaced, and then another marble is drawn. What is the probability that the two marbles are the same color?","options":["12/49","16/49","24/49","25/49","31/49"],"correctIndex":3,"explanation":"With replacement, the draws are independent. The probability of two red is (4/7)(4/7)=16/49, and the probability of two blue is (3/7)(3/7)=9/49. The total is 25/49."},{"id":"amc8_102","contestType":"AMC 8","difficulty":3,"topic":"Number Theory","text":"What is the greatest common divisor of 96 and 144?","options":["12","24","36","48","72"],"correctIndex":3,"explanation":"The common divisors include 48, and 96 = 2×48 while 144 = 3×48. No larger number can divide both, so the greatest common divisor is 48."},{"id":"amc8_103","contestType":"AMC 8","difficulty":3,"topic":"Trigonometry","text":"In a 30-60-90 triangle, the hypotenuse has length 14. What is the length of the shorter leg?","options":["4","7","7√2","7√3","14√3"],"correctIndex":1,"explanation":"In a 30-60-90 triangle, the hypotenuse is twice the shorter leg. If the hypotenuse is 14, the shorter leg is 7."},{"id":"amc8_104","contestType":"AMC 8","difficulty":3,"topic":"Geometry","text":"A rectangular prism has side lengths 5, 6, and 7. What is its volume?","options":["18","105","180","210","420"],"correctIndex":3,"explanation":"The volume of a rectangular prism is length×width×height. Here the volume is 5×6×7 = 210."},{"id":"amc8_105","contestType":"AMC 8","difficulty":3,"topic":"Algebra","text":"The sum of four consecutive odd integers is 112. What is the largest of the four integers?","options":["25","27","29","31","33"],"correctIndex":3,"explanation":"Let the four odd integers be n, n+2, n+4, and n+6. Their sum is 4n+12=112, so 4n=100 and n=25. The largest is 25+6=31."},{"id":"amc8_106","contestType":"AMC 8","difficulty":4,"topic":"Counting & Probability","text":"How many ways can 3 books be arranged in a row if they are chosen from 5 different books?","options":["10","20","30","60","125"],"correctIndex":3,"explanation":"There are 5 choices for the first book, 4 for the second, and 3 for the third. The total is 5×4×3 = 60 arrangements."},{"id":"amc8_107","contestType":"AMC 8","difficulty":2,"topic":"Number Theory","text":"What is the least common multiple of 12 and 18?","options":["24","30","36","54","72"],"correctIndex":2,"explanation":"The multiples of 12 are 12, 24, 36, ... and the multiples of 18 are 18, 36, ... . The least common multiple is 36."},{"id":"amc8_108","contestType":"AMC 8","difficulty":4,"topic":"Algebra","text":"A taxi charges a $4 starting fee plus $3 per mile. If a ride costs $28, how many miles was the ride?","options":["6","7","8","9","10"],"correctIndex":2,"explanation":"Subtract the starting fee: 28 - 4 = 24 dollars came from mileage. At $3 per mile, the ride was 24/3 = 8 miles."},{"id":"amc8_109","contestType":"AMC 8","difficulty":4,"topic":"Trigonometry","text":"A right triangle has side lengths 8, 15, and 17. For the acute angle opposite the side of length 8, what is the sine of the angle?","options":["8/17","8/15","15/17","15/8","17/8"],"correctIndex":0,"explanation":"Sine is opposite divided by hypotenuse. The angle is opposite the side of length 8, and the hypotenuse is 17, so sin = 8/17."},{"id":"sb_026","contestType":"Science Bowl","difficulty":3,"topic":"Science","text":"What type of biological molecule speeds up chemical reactions in living organisms?","options":["Enzyme","Lipid","DNA","Starch","Water"],"correctIndex":0,"explanation":"Enzymes are biological catalysts that speed up chemical reactions without being consumed."},{"id":"sb_027","contestType":"Science Bowl","difficulty":4,"topic":"Science","text":"In DNA, adenine most commonly pairs with which base?","options":["Adenine","Cytosine","Guanine","Thymine","Uracil"],"correctIndex":3,"explanation":"In DNA base pairing, adenine pairs with thymine, while cytosine pairs with guanine."},{"id":"sb_028","contestType":"Science Bowl","difficulty":3,"topic":"Science","text":"What is the SI unit of energy?","options":["Newton","Joule","Watt","Pascal","Volt"],"correctIndex":1,"explanation":"The joule is the SI unit of energy."},{"id":"amc8_110","contestType":"AMC 8","difficulty":3,"topic":"Trigonometry","text":"In a right triangle, an acute angle has opposite side 5 and hypotenuse 13. What is the cosine of the angle?","options":["5/13","12/13","5/12","12/5","13/12"],"correctIndex":1,"explanation":"The missing adjacent side is 12 by the 5-12-13 right triangle. Cosine equals adjacent over hypotenuse, so cos = 12/13.","solutionSteps":["Use the Pythagorean theorem: adjacent^2 + 5^2 = 13^2.","The adjacent side is sqrt(169 - 25) = sqrt(144) = 12.","Cosine is adjacent divided by hypotenuse, so the value is 12/13."]},{"id":"amc8_111","contestType":"AMC 8","difficulty":3,"topic":"Trigonometry","text":"A ladder 10 feet long leans against a wall. Its foot is 6 feet from the wall. What is the sine of the angle the ladder makes with the ground?","options":["3/5","4/5","5/4","4/3","5/3"],"correctIndex":1,"explanation":"The ladder is the hypotenuse. The height on the wall is sqrt(10^2 - 6^2) = 8, so sine = opposite/hypotenuse = 8/10 = 4/5.","solutionSteps":["Model the wall, ground, and ladder as a right triangle.","The height is sqrt(100 - 36) = 8 feet.","For the ground angle, opposite = 8 and hypotenuse = 10, so sin = 4/5."]},{"id":"amc8_112","contestType":"AMC 8","difficulty":4,"topic":"Trigonometry","text":"In a 30-60-90 triangle, the shortest side is 7. What is the length of the hypotenuse?","options":["7√2","7√3","14","14√3","21"],"correctIndex":2,"explanation":"In a 30-60-90 triangle, the sides are x, x√3, and 2x. If the shortest side is 7, the hypotenuse is 14.","solutionSteps":["Recall the side ratio 1 : √3 : 2.","The shortest side corresponds to x, so x = 7.","The hypotenuse is 2x = 14."]},{"id":"amc8_113","contestType":"AMC 8","difficulty":4,"topic":"Trigonometry","text":"In a 45-45-90 triangle, each leg has length 9. What is the hypotenuse?","options":["9","9√2","18","18√2","81"],"correctIndex":1,"explanation":"A 45-45-90 triangle has side ratio 1 : 1 : √2. With leg 9, the hypotenuse is 9√2.","solutionSteps":["Both legs are equal in a 45-45-90 triangle.","The hypotenuse is leg times √2.","So the hypotenuse is 9√2."]},{"id":"amc8_114","contestType":"AMC 8","difficulty":4,"topic":"Trigonometry","text":"A right triangle has legs 9 and 12. For the acute angle opposite the side of length 9, what is the tangent?","options":["3/4","4/3","3/5","4/5","5/3"],"correctIndex":0,"explanation":"Tangent is opposite divided by adjacent. For the angle opposite the side 9, the adjacent leg is 12, so tan = 9/12 = 3/4.","solutionSteps":["Identify the opposite side as 9.","The other leg, 12, is adjacent to the angle.","tan = opposite/adjacent = 9/12 = 3/4."]},{"id":"amc8_115","contestType":"AMC 8","difficulty":5,"topic":"Trigonometry","text":"An angle θ in a right triangle has sin θ = 3/5. If the hypotenuse is 20, what is the length of the side opposite θ?","options":["8","10","12","15","16"],"correctIndex":2,"explanation":"Since sin θ = opposite/hypotenuse = 3/5 and the hypotenuse is 20, the opposite side is (3/5)×20 = 12.","solutionSteps":["Write sin θ = opposite/hypotenuse.","Substitute 3/5 = opposite/20.","Multiply to get opposite = 12."]},{"id":"amc8_116","contestType":"AMC 8","difficulty":5,"topic":"Trigonometry","text":"A flagpole casts a shadow 12 meters long. The angle from the tip of the shadow to the top of the pole has tangent 5/6. How tall is the pole?","options":["8","9","10","12","15"],"correctIndex":2,"explanation":"Tangent equals height divided by shadow length. If tan = 5/6 and the shadow is 12, then height/12 = 5/6, so height = 10.","solutionSteps":["The opposite side is the pole height and the adjacent side is the shadow.","Set height/12 = 5/6.","Solve: height = 12 × 5/6 = 10 meters."]},{"id":"amc8_117","contestType":"AMC 8","difficulty":5,"topic":"Trigonometry","text":"In a right triangle, cos A = 7/25. What is sin A?","options":["7/24","24/25","24/7","18/25","25/24"],"correctIndex":1,"explanation":"cos A = adjacent/hypotenuse = 7/25. The opposite side is sqrt(25^2 - 7^2) = 24, so sin A = 24/25.","solutionSteps":["Treat adjacent = 7 and hypotenuse = 25.","Find opposite: sqrt(625 - 49) = sqrt(576) = 24.","sin A = opposite/hypotenuse = 24/25."]},{"id":"amc8_118","contestType":"AMC 8","difficulty":4,"topic":"Counting & Probability","text":"A fair coin is flipped 4 times. What is the probability of getting exactly 3 heads?","options":["1/16","1/8","1/4","3/8","1/2"],"correctIndex":2,"explanation":"There are 2^4 = 16 equally likely outcomes. Exactly 3 heads can occur in C(4,3) = 4 ways, so the probability is 4/16 = 1/4.","solutionSteps":["Count all outcomes: 2^4 = 16.","Choose which 3 of the 4 flips are heads: C(4,3) = 4.","Probability = 4/16 = 1/4."]},{"id":"amc8_119","contestType":"AMC 8","difficulty":4,"topic":"Counting & Probability","text":"How many three-digit numbers can be formed using the digits 1, 2, 3, 4, and 5 if no digit is repeated?","options":["10","30","60","75","125"],"correctIndex":2,"explanation":"There are 5 choices for the hundreds digit, then 4 choices, then 3 choices, giving 5×4×3 = 60 numbers.","solutionSteps":["Choose the hundreds digit in 5 ways.","Choose the tens digit in 4 remaining ways.","Choose the ones digit in 3 remaining ways, for 5×4×3 = 60."]},{"id":"amc8_120","contestType":"AMC 8","difficulty":5,"topic":"Counting & Probability","text":"A bag contains 3 red marbles and 5 blue marbles. Two marbles are drawn without replacement. What is the probability both are red?","options":["3/28","1/7","9/64","3/8","2/7"],"correctIndex":0,"explanation":"The chance the first is red is 3/8. Then 2 red marbles remain out of 7 total, so the chance is (3/8)(2/7) = 3/28.","solutionSteps":["First red: 3 out of 8.","Second red after one red is removed: 2 out of 7.","Multiply: (3/8)(2/7) = 6/56 = 3/28."]},{"id":"amc8_121","contestType":"AMC 8","difficulty":5,"topic":"Counting & Probability","text":"How many different arrangements are there of the letters in LEVEL?","options":["20","30","40","60","120"],"correctIndex":1,"explanation":"LEVEL has 5 letters, with L repeated twice and E repeated twice. The number of arrangements is 5!/(2!2!) = 30.","solutionSteps":["Start with 5! arrangements if all letters were distinct.","Divide by 2! for the two Ls and by 2! for the two Es.","5!/(2!2!) = 120/4 = 30."]},{"id":"amc8_122","contestType":"AMC 8","difficulty":4,"topic":"Counting & Probability","text":"A spinner has 6 equal sections numbered 1 through 6. What is the probability of spinning a prime number?","options":["1/6","1/3","1/2","2/3","5/6"],"correctIndex":2,"explanation":"The primes from 1 to 6 are 2, 3, and 5. That is 3 favorable outcomes out of 6, so the probability is 1/2.","solutionSteps":["List the prime numbers: 2, 3, 5.","There are 3 favorable sections out of 6 total.","The probability is 3/6 = 1/2."]},{"id":"amc8_123","contestType":"AMC 8","difficulty":5,"topic":"Counting & Probability","text":"How many ways can 4 students be chosen from a club of 9 students?","options":["36","84","126","3024","6561"],"correctIndex":2,"explanation":"Choosing 4 from 9 is C(9,4) = 9×8×7×6/(4×3×2×1) = 126.","solutionSteps":["Order does not matter, so use combinations.","C(9,4) = 9!/(4!5!).","Compute 9×8×7×6/(4×3×2×1) = 126."]},{"id":"amc8_124","contestType":"AMC 8","difficulty":3,"topic":"Geometry","text":"A rectangle has length 11 and width 7. What is its perimeter?","options":["18","36","42","77","154"],"correctIndex":1,"explanation":"The perimeter of a rectangle is 2(length + width). Here it is 2(11 + 7) = 36.","solutionSteps":["Add length and width: 11 + 7 = 18.","Double that sum for the two pairs of sides.","Perimeter = 2 × 18 = 36."]},{"id":"amc8_125","contestType":"AMC 8","difficulty":4,"topic":"Geometry","text":"A circle has diameter 18. What is its radius?","options":["6","9","18","27","36"],"correctIndex":1,"explanation":"The radius is half the diameter. Half of 18 is 9.","solutionSteps":["Diameter is twice the radius.","So radius = diameter/2.","18/2 = 9."]},{"id":"amc8_126","contestType":"AMC 8","difficulty":4,"topic":"Geometry","text":"The area of a triangle is 48 and its base is 12. What is its height?","options":["4","6","8","10","12"],"correctIndex":2,"explanation":"Using area = (1/2)bh, we have 48 = (1/2)(12)h = 6h, so h = 8.","solutionSteps":["Use A = (1/2)bh.","Substitute 48 = (1/2)(12)h.","Then 48 = 6h, so h = 8."]},{"id":"amc8_127","contestType":"AMC 8","difficulty":5,"topic":"Geometry","text":"A square has area 81. What is the length of its diagonal?","options":["9","9√2","18","18√2","81√2"],"correctIndex":1,"explanation":"The square side length is sqrt(81) = 9. A square diagonal is side×√2, so the diagonal is 9√2.","solutionSteps":["Find the side length: sqrt(81) = 9.","The diagonal forms a 45-45-90 triangle with the sides.","Diagonal = 9√2."]},{"id":"amc8_128","contestType":"AMC 8","difficulty":5,"topic":"Geometry","text":"A trapezoid has bases 8 and 14 and height 5. What is its area?","options":["40","45","50","55","70"],"correctIndex":3,"explanation":"The area of a trapezoid is (1/2)(sum of bases)(height). This is (1/2)(8 + 14)(5) = 55.","solutionSteps":["Add the bases: 8 + 14 = 22.","Multiply by the height: 22 × 5 = 110.","Take half: 110/2 = 55."]},{"id":"amc8_129","contestType":"AMC 8","difficulty":4,"topic":"Number Theory","text":"What is the least common multiple of 18 and 24?","options":["6","36","48","72","432"],"correctIndex":3,"explanation":"18 = 2×3^2 and 24 = 2^3×3. The LCM uses the highest powers: 2^3×3^2 = 72.","solutionSteps":["Prime factor: 18 = 2×3^2 and 24 = 2^3×3.","Use the highest power of each prime.","LCM = 2^3×3^2 = 72."]},{"id":"amc8_130","contestType":"AMC 8","difficulty":4,"topic":"Number Theory","text":"How many positive factors does 72 have?","options":["6","8","10","12","16"],"correctIndex":3,"explanation":"Since 72 = 2^3×3^2, the number of positive factors is (3+1)(2+1) = 12.","solutionSteps":["Prime factor 72 as 2^3×3^2.","For p^a q^b, the number of factors is (a+1)(b+1).","So 72 has (3+1)(2+1) = 12 factors."]},{"id":"amc8_131","contestType":"AMC 8","difficulty":3,"topic":"Number Theory","text":"What is the remainder when 247 is divided by 9?","options":["1","2","3","4","5"],"correctIndex":3,"explanation":"The nearest multiple of 9 below 247 is 243. Since 247 - 243 = 4, the remainder is 4.","solutionSteps":["Find a nearby multiple of 9: 9×27 = 243.","Subtract from 247: 247 - 243 = 4.","So the remainder is 4."]},{"id":"amc8_132","contestType":"AMC 8","difficulty":4,"topic":"Number Theory","text":"What is the greatest common divisor of 84 and 126?","options":["14","21","28","42","84"],"correctIndex":3,"explanation":"84 = 2^2×3×7 and 126 = 2×3^2×7. The common prime factors are 2×3×7 = 42.","solutionSteps":["Prime factor both numbers.","Use the lowest shared powers: 2, 3, and 7.","The GCD is 2×3×7 = 42."]},{"id":"amc8_133","contestType":"AMC 8","difficulty":3,"topic":"Algebra","text":"If 4x - 7 = 21, what is x?","options":["5","6","7","8","9"],"correctIndex":2,"explanation":"Add 7 to both sides to get 4x = 28. Divide by 4 to get x = 7.","solutionSteps":["Start with 4x - 7 = 21.","Add 7: 4x = 28.","Divide by 4: x = 7."]},{"id":"amc8_134","contestType":"AMC 8","difficulty":4,"topic":"Algebra","text":"If 3 more than twice a number is 17, what is the number?","options":["5","6","7","8","10"],"correctIndex":2,"explanation":"Let the number be n. Then 2n + 3 = 17, so 2n = 14 and n = 7.","solutionSteps":["Let n be the number.","Translate the sentence: 2n + 3 = 17.","Solve to get n = 7."]},{"id":"amc8_135","contestType":"AMC 8","difficulty":5,"topic":"Algebra","text":"The average of five numbers is 18. Four of the numbers are 12, 16, 20, and 25. What is the fifth number?","options":["15","17","19","21","23"],"correctIndex":1,"explanation":"If the average of five numbers is 18, their sum is 90. The four known numbers sum to 73, so the fifth is 90 - 73 = 17.","solutionSteps":["Total sum = 5 × 18 = 90.","Known sum = 12 + 16 + 20 + 25 = 73.","Fifth number = 90 - 73 = 17."]},{"id":"amc8_136","contestType":"AMC 8","difficulty":4,"topic":"Algebra","text":"If y = 2x + 5 and x = 6, what is y?","options":["11","12","15","17","22"],"correctIndex":3,"explanation":"Substitute x = 6 into y = 2x + 5 to get y = 2(6) + 5 = 17.","solutionSteps":["Substitute x = 6.","Compute y = 2(6) + 5.","y = 17."]},{"id":"sb_029","contestType":"Science Bowl","difficulty":3,"topic":"Science","text":"Which planet in our solar system is known for its large rings?","options":["Mercury","Venus","Earth","Saturn","Mars"],"correctIndex":3,"explanation":"Saturn is famous for its large, bright ring system made mostly of ice and rock particles.","solutionSteps":["Identify the planet most associated with a visible ring system.","Saturn has the most prominent rings.","Therefore the answer is Saturn."]},{"id":"sb_030","contestType":"Science Bowl","difficulty":4,"topic":"Science","text":"What organelle is the main site of photosynthesis in plant cells?","options":["Nucleus","Mitochondrion","Chloroplast","Ribosome","Vacuole"],"correctIndex":2,"explanation":"Photosynthesis occurs mainly in chloroplasts, which contain chlorophyll.","solutionSteps":["Photosynthesis is the process plants use to capture light energy.","Chlorophyll captures light inside chloroplasts.","So the organelle is the chloroplast."]},{"id":"sb_031","contestType":"Science Bowl","difficulty":3,"topic":"Science","text":"What force pulls objects toward the center of Earth?","options":["Friction","Gravity","Magnetism","Tension","Buoyancy"],"correctIndex":1,"explanation":"Gravity is the attractive force that pulls objects with mass toward Earth.","solutionSteps":["The force that attracts masses to each other is gravity.","Near Earth, gravity pulls objects downward toward Earth’s center.","So the answer is gravity."]},{"id":"sb_032","contestType":"Science Bowl","difficulty":4,"topic":"Science","text":"In an electrical circuit, what quantity is measured in amperes?","options":["Voltage","Current","Resistance","Power","Energy"],"correctIndex":1,"explanation":"Electric current is the rate of flow of electric charge, and it is measured in amperes.","solutionSteps":["Amperes are the SI unit abbreviated A.","They measure electric current.","Current is charge flow per unit time."]},{"id":"mc_019","contestType":"MATHCOUNTS","difficulty":5,"topic":"Counting & Probability","text":"A code consists of two different letters followed by one digit. If letters are chosen from A, B, C, D and digits from 0 through 9, how many codes are possible?","options":["40","80","100","120","160"],"correctIndex":3,"explanation":"There are 4 choices for the first letter and 3 remaining choices for the second letter. There are 10 choices for the digit, so 4×3×10 = 120.","solutionSteps":["Choose the first letter in 4 ways.","Choose a different second letter in 3 ways.","Choose the digit in 10 ways, for 4×3×10 = 120."]},{"id":"mc_020","contestType":"MATHCOUNTS","difficulty":5,"topic":"Algebra","text":"If 2a + 3b = 31 and a = 8, what is b?","options":["3","4","5","6","7"],"correctIndex":2,"explanation":"Substitute a = 8 to get 16 + 3b = 31. Then 3b = 15, so b = 5.","solutionSteps":["Replace a with 8.","2(8) + 3b = 31 becomes 16 + 3b = 31.","Subtract 16 and divide by 3 to get b = 5."]},{"id":"amc8_137","contestType":"AMC 8","difficulty":4,"topic":"Geometry","text":"A rectangle has length 14 and width 6. What is the area of the triangle formed by one diagonal and two adjacent sides of the rectangle?","options":["20","36","42","48","84"],"correctIndex":2,"explanation":"A diagonal divides a rectangle into two congruent triangles. The rectangle area is 14×6 = 84, so each triangle has area 84/2 = 42.","solutionSteps":["Find the rectangle area: 14×6 = 84.","A diagonal splits the rectangle into two equal-area triangles.","Triangle area = 84 ÷ 2 = 42."]},{"id":"amc8_138","contestType":"AMC 8","difficulty":3,"topic":"Algebra","text":"The sum of three consecutive integers is 72. What is the largest of the three integers?","options":["22","23","24","25","26"],"correctIndex":3,"explanation":"Let the integers be x - 1, x, and x + 1. Their sum is 3x = 72, so x = 24 and the largest is 25.","solutionSteps":["Represent the middle integer as x.","The three integers are x - 1, x, and x + 1.","Their sum is 3x = 72, so x = 24.","The largest integer is x + 1 = 25."]},{"id":"amc8_139","contestType":"AMC 8","difficulty":4,"topic":"Counting & Probability","text":"A fair coin is flipped 4 times. What is the probability of getting exactly 3 heads?","options":["1/16","1/4","3/8","1/2","5/8"],"correctIndex":1,"explanation":"There are 2^4 = 16 equally likely outcomes. Exactly 3 heads can happen in 4 ways, depending on which flip is tails. The probability is 4/16 = 1/4.","solutionSteps":["There are 2^4 = 16 possible flip sequences.","To get exactly 3 heads, choose which 1 of the 4 flips is tails.","That gives 4 favorable sequences.","Probability = 4/16 = 1/4."]},{"id":"amc8_140","contestType":"AMC 8","difficulty":4,"topic":"Number Theory","text":"What is the remainder when 7^4 is divided by 10?","options":["1","3","5","7","9"],"correctIndex":0,"explanation":"The last digits of powers of 7 cycle as 7, 9, 3, 1. Since 4 lands on the fourth term of the cycle, 7^4 has remainder 1 when divided by 10.","solutionSteps":["Track the last digit of powers of 7.","7^1 ends in 7, 7^2 ends in 9, 7^3 ends in 3, and 7^4 ends in 1.","Therefore the remainder modulo 10 is 1."]},{"id":"amc8_141","contestType":"AMC 8","difficulty":4,"topic":"Trigonometry","text":"In a right triangle, an acute angle has opposite side 5 and hypotenuse 13. What is the sine of the angle?","options":["5/13","12/13","5/12","12/5","13/5"],"correctIndex":0,"explanation":"Sine of an angle in a right triangle is opposite divided by hypotenuse. Thus sin = 5/13.","solutionSteps":["Recall sin(angle) = opposite/hypotenuse.","The opposite side is 5 and the hypotenuse is 13.","So sin = 5/13."]},{"id":"amc8_142","contestType":"AMC 8","difficulty":5,"topic":"Geometry","text":"A circle has circumference 18π. What is its area?","options":["18π","36π","81π","162π","324π"],"correctIndex":2,"explanation":"Circumference is 2πr. If 2πr = 18π, then r = 9. The area is πr^2 = 81π.","solutionSteps":["Use C = 2πr.","2πr = 18π, so r = 9.","Area = πr² = π(9²) = 81π."]},{"id":"amc8_143","contestType":"AMC 8","difficulty":4,"topic":"Algebra","text":"If 3x - 4 = 2x + 9, what is x?","options":["5","9","11","13","15"],"correctIndex":3,"explanation":"Subtract 2x from both sides to get x - 4 = 9. Add 4 to get x = 13.","solutionSteps":["Start with 3x - 4 = 2x + 9.","Subtract 2x from both sides: x - 4 = 9.","Add 4 to both sides: x = 13."]},{"id":"amc8_144","contestType":"AMC 8","difficulty":5,"topic":"Counting & Probability","text":"How many three-digit positive integers have all digits odd?","options":["25","75","100","125","500"],"correctIndex":3,"explanation":"Each digit can be one of 1, 3, 5, 7, 9, giving 5 choices for each of the hundreds, tens, and ones digits. Total = 5×5×5 = 125.","solutionSteps":["Odd digit choices are 1, 3, 5, 7, and 9.","There are 5 choices for each of the three positions.","Total integers = 5×5×5 = 125."]},{"id":"amc8_145","contestType":"AMC 8","difficulty":5,"topic":"Number Theory","text":"How many positive divisors does 72 have?","options":["8","10","12","14","18"],"correctIndex":2,"explanation":"Prime factorize 72 = 2^3 × 3^2. The number of positive divisors is (3+1)(2+1) = 12.","solutionSteps":["Factor 72 = 2³×3².","A divisor can use 0, 1, 2, or 3 factors of 2: 4 choices.","It can use 0, 1, or 2 factors of 3: 3 choices.","Total divisors = 4×3 = 12."]},{"id":"amc8_146","contestType":"AMC 8","difficulty":5,"topic":"Trigonometry","text":"In a 30-60-90 triangle, the side opposite the 30° angle is 7. What is the hypotenuse?","options":["7√3","14","14√3","21","28"],"correctIndex":1,"explanation":"In a 30-60-90 triangle, the hypotenuse is twice the side opposite 30°. Thus the hypotenuse is 2×7 = 14.","solutionSteps":["Use the 30-60-90 side ratio: 1 : √3 : 2.","The side opposite 30° is the short leg.","Hypotenuse = 2 × 7 = 14."]},{"id":"amc8_147","contestType":"AMC 8","difficulty":5,"topic":"Geometry","text":"A square has area 50. What is the length of its diagonal?","options":["5","5√2","10","10√2","25"],"correctIndex":2,"explanation":"If the square side is s, then s^2 = 50. The diagonal is s√2 = √50·√2 = √100 = 10.","solutionSteps":["Let the side length be s, so s² = 50.","The diagonal of a square is s√2.","Diagonal = √50 × √2 = √100 = 10."]},{"id":"amc8_148","contestType":"AMC 8","difficulty":6,"topic":"Counting & Probability","text":"A committee of 3 students is chosen from 5 girls and 4 boys. How many committees have exactly 2 girls?","options":["20","30","40","60","80"],"correctIndex":2,"explanation":"Choose 2 of the 5 girls and 1 of the 4 boys. This gives C(5,2)×C(4,1) = 10×4 = 40.","solutionSteps":["Choose 2 girls from 5: C(5,2) = 10.","Choose 1 boy from 4: C(4,1) = 4.","Multiply independent choices: 10×4 = 40."]},{"id":"amc8_149","contestType":"AMC 8","difficulty":5,"topic":"Number Theory","text":"What is the least positive integer n such that 18n is a perfect square?","options":["2","3","6","18","72"],"correctIndex":0,"explanation":"Since 18 = 2×3^2, the factor 2 has an odd exponent. Multiplying by n = 2 gives 36, which is a perfect square. No smaller positive integer works, so n = 2.","solutionSteps":["Prime factorize 18 = 2×3².","A perfect square has even exponents in its prime factorization.","The exponent of 2 is odd, so multiply by one more 2.","Thus n = 2, and 18n = 36."]},{"id":"amc8_150","contestType":"AMC 8","difficulty":4,"topic":"Algebra","text":"A number is increased by 20% and then decreased by 20%. What percent of the original number remains?","options":["80%","84%","96%","100%","104%"],"correctIndex":2,"explanation":"Increasing by 20% multiplies by 1.2. Decreasing by 20% then multiplies by 0.8. The product is 1.2×0.8 = 0.96, so 96% remains.","solutionSteps":["Start with an original value x.","After a 20% increase, the value is 1.2x.","After a 20% decrease, the value is 0.8(1.2x) = 0.96x.","So 96% of the original remains."]},{"id":"amc8_151","contestType":"AMC 8","difficulty":5,"topic":"Trigonometry","text":"A right triangle has legs 9 and 12. What is the cosine of the angle opposite the side of length 9?","options":["3/5","4/5","5/4","9/12","12/13"],"correctIndex":1,"explanation":"The hypotenuse is √(9^2 + 12^2) = 15. For the angle opposite the side 9, the adjacent leg is 12, so cosine = 12/15 = 4/5.","solutionSteps":["Find the hypotenuse: √(81 + 144) = √225 = 15.","The angle opposite side 9 has adjacent leg 12.","cosine = adjacent/hypotenuse = 12/15 = 4/5."]},{"id":"amc8_152","contestType":"AMC 8","difficulty":5,"topic":"Geometry","text":"The angles of a triangle are in the ratio 2:3:4. What is the measure of the largest angle?","options":["40°","60°","80°","90°","100°"],"correctIndex":2,"explanation":"The ratio parts sum to 2 + 3 + 4 = 9. Since triangle angles sum to 180°, each part is 20°. The largest angle is 4×20° = 80°.","solutionSteps":["The ratio has 2 + 3 + 4 = 9 total parts.","Each part is 180° ÷ 9 = 20°.","The largest angle is 4 parts: 4×20° = 80°."]},{"id":"mc_021","contestType":"MATHCOUNTS","difficulty":5,"topic":"Counting & Probability","text":"How many different arrangements are there of the letters in the word LEVEL?","options":["20","30","60","120","240"],"correctIndex":1,"explanation":"LEVEL has 5 letters, with L repeated twice and E repeated twice. The number of arrangements is 5!/(2!2!) = 120/4 = 30.","solutionSteps":["Start with 5! arrangements of five positions.","Divide by 2! for the two Ls and by 2! for the two Es.","5!/(2!2!) = 120/4 = 30."]},{"id":"mc_022","contestType":"MATHCOUNTS","difficulty":5,"topic":"Number Theory","text":"What is the greatest common divisor of 84 and 126?","options":["14","21","28","42","84"],"correctIndex":3,"explanation":"84 = 2×2×3×7 and 126 = 2×3×3×7. Their common prime factors are 2×3×7 = 42.","solutionSteps":["Factor 84 = 2²×3×7.","Factor 126 = 2×3²×7.","Use the common factors with the smaller exponents: 2×3×7 = 42."]},{"id":"mc_023","contestType":"MATHCOUNTS","difficulty":6,"topic":"Trigonometry","text":"In a right triangle, tan A = 3/4. What is sin A?","options":["3/5","4/5","3/4","4/3","5/3"],"correctIndex":0,"explanation":"If tan A = opposite/adjacent = 3/4, use a 3-4-5 right triangle. The hypotenuse is 5, so sin A = opposite/hypotenuse = 3/5.","solutionSteps":["Tangent is opposite/adjacent, so the legs can be 3 and 4.","A 3-4 right triangle has hypotenuse 5.","Sine is opposite/hypotenuse, so sin A = 3/5."]},{"id":"mc_024","contestType":"MATHCOUNTS","difficulty":5,"topic":"Algebra","text":"If x/3 + x/4 = 7, what is x?","options":["8","10","12","14","16"],"correctIndex":2,"explanation":"Use common denominator 12: x/3 + x/4 = 4x/12 + 3x/12 = 7x/12. So 7x/12 = 7, and x = 12.","solutionSteps":["Use common denominator 12.","x/3 + x/4 = 4x/12 + 3x/12 = 7x/12.","Solve 7x/12 = 7 to get x = 12."]},{"id":"sb_033","contestType":"Science Bowl","difficulty":4,"topic":"Science","text":"Which part of an atom has a negative electric charge?","options":["Proton","Neutron","Electron","Nucleus","Photon"],"correctIndex":2,"explanation":"Electrons are the negatively charged particles in atoms. Protons are positive, and neutrons have no charge.","solutionSteps":["Atoms contain protons, neutrons, and electrons.","Protons are positive and neutrons are neutral.","Electrons are negative."]},{"id":"sb_034","contestType":"Science Bowl","difficulty":4,"topic":"Science","text":"What type of energy is stored in a stretched rubber band?","options":["Thermal energy","Elastic potential energy","Chemical energy","Nuclear energy","Sound energy"],"correctIndex":1,"explanation":"A stretched rubber band stores elastic potential energy because it can do work when it returns to its original shape.","solutionSteps":["Stretching changes the rubber band shape.","Energy stored due to deformation is elastic potential energy.","So the answer is elastic potential energy."]},{"id":"sb_035","contestType":"Science Bowl","difficulty":5,"topic":"Science","text":"Which blood vessels carry blood away from the heart?","options":["Veins","Arteries","Capillaries","Alveoli","Neurons"],"correctIndex":1,"explanation":"Arteries carry blood away from the heart. Veins carry blood back toward the heart.","solutionSteps":["Remember: arteries away.","Arteries carry blood away from the heart.","Veins return blood to the heart."]},{"id":"sb_036","contestType":"Science Bowl","difficulty":5,"topic":"Science","text":"What is the SI unit of force?","options":["Joule","Watt","Newton","Pascal","Volt"],"correctIndex":2,"explanation":"The SI unit of force is the newton, named after Isaac Newton.","solutionSteps":["Force is measured in SI units.","The SI unit of force is the newton.","Its symbol is N."]}]}