Competition · AMC preparation · step 4 of 4

AMC 8 2010: all 25 problems

Each problem has a solution worked from first principles and the first grade (by CCSS standards) that can solve it.

AMC 8 2010 #1 grade 2+ arithmetic
At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are 11 students in…
AMC 8 2010 #2 grade 6+ arithmetic
If a @ b = (a× b)/(a+b) for a,b positive integers, then what is 5 @10?
AMC 8 2010 #3 grade 6+ rate-ratio
The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the hi…
AMC 8 2010 #4 grade 6+ arithmetic
What is the sum of the mean, median, and mode of the numbers 2,3,0,3,1,4,0,3?
AMC 8 2010 #5 grade 6+ arithmetic
Alice needs to replace a light bulb located 10 centimeters below the ceiling in her kitchen. The ceiling is 2.4 meters a…
AMC 8 2010 #6 grade 4+ geometry-2d
Which of the following figures has the greatest number of lines of symmetry?
AMC 8 2010 #7 grade 3+ arithmetic
Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pa…
AMC 8 2010 #8 grade 6+ rate-ratio
As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction 1/2 mile in fron…
AMC 8 2010 #9 grade 6+ rate-ratio
Ryan got 80% of the problems correct on a 25-problem test, 90% on a 40-problem test, and 70% on a 10-problem test. What…
AMC 8 2010 #10 grade 7+ geometry-2d
Six pepperoni circles will exactly fit across the diameter of a 12-inch pizza when placed. If a total of 24 circles of p…
AMC 8 2010 #11 grade 6+ rate-ratio
The top of one tree is 16 feet higher than the top of another tree. The heights of the two trees are in the ratio 3:4. I…
AMC 8 2010 #12 grade 6+ arithmetic
Of the 500 balls in a large bag, 80% are red and the rest are blue. How many of the red balls must be removed so that 75…
AMC 8 2010 #13 grade 6+ geometry-2d
The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shortest side is 30%…
AMC 8 2010 #14 grade 4+ number-theory
What is the sum of the prime factors of 2010?
AMC 8 2010 #15 grade 6+ arithmetic
A jar contains 5 different colors of gumdrops. 30% are blue, 20% are brown, 15% are red, 10% are yellow, and other 30 gu…
AMC 8 2010 #16 grade 8+ geometry-2d
A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle…
AMC 8 2010 #17 grade 6+ geometry-2d
The diagram shows an octagon consisting of 10 unit squares. The portion below PQ is a unit square and a triangle with ba…
AMC 8 2010 #18 grade 7+ geometry-2d
A decorative window is made up of a rectangle with semicircles at either end. The ratio of AD to AB is 3:2. And AB is 30…
AMC 8 2010 #19 grade 8+ geometry-2d
The two circles pictured have the same center C. Chord AD is tangent to the inner circle at B, AC is 10, and chord AD ha…
AMC 8 2010 #20 grade 7+ counting, logic
In a room, 2/5 of the people are wearing gloves, and 3/4 of the people are wearing hats. What is the minimum number of p…
AMC 8 2010 #21 grade 6+ arithmetic
Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read 1/5 of the pag…
AMC 8 2010 #22 grade 6+ arithmetic, number-theory
The hundreds digit of a three-digit number is 2 more than the units digit. The digits of the three-digit number are reve…
AMC 8 2010 #23 grade 8+ geometry-2d
Semicircles POQ and ROS pass through the center O. What is the ratio of the combined areas of the two semicircles to the…
AMC 8 2010 #24 grade 8+ arithmetic
What is the correct ordering of the three numbers, 10⁸, 5¹², and 2²⁴?
AMC 8 2010 #25 grade 4+ counting
Everyday at school, Jo climbs a flight of 6 stairs. Jo can take the stairs 1, 2, or 3 at a time. For example, Jo could c…

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