Competition · AMC preparation · step 4 of 4

AMC 10 2010A: all 25 problems

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

AMC 10 2010A #1 grade 6+ arithmetic
Mary's top book shelf holds five books with the following widths, in centimeters: 6, 1/2, 1, 2.5, and 10. What is the av…
AMC 10 2010A #2 grade 6+ geometry-2d
Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectan…
AMC 10 2010A #3 grade 6+ arithmetic
Tyrone had 97 marbles and Eric had 11 marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with tw…
AMC 10 2010A #4 grade 5+ arithmetic
A book that is to be recorded onto compact discs takes 412 minutes to read aloud. Each disc can hold up to 56 minutes of…
AMC 10 2010A #5 grade 7+ geometry-2d
The area of a circle whose circumference is 24π is kπ. What is the value of k?
AMC 10 2010A #6 grade 6+ arithmetic
For positive numbers x and y the operation ♠ (x,y) is defined as ♠ (x,y) = x-1/y What is ♠ (2,♠ (2,2))?
AMC 10 2010A #7 grade 8+ geometry-2d
Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She th…
AMC 10 2010A #8 grade 6+ arithmetic
Tony works 2 hours a day and is paid $0.50 per hour for each full year of his age. During a six month period Tony worked…
AMC 10 2010A #9 grade 4+ number-theory
A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers x and x+32 are…
AMC 10 2010A #10 grade 4+ arithmetic
Marvin had a birthday on Tuesday, May 27 in the leap year 2008. In what year will his birthday next fall on a Saturday?
AMC 10 2010A #11 grade 7+ algebra
The length of the interval of solutions of the inequality a ≤ 2x + 3 ≤ b is 10. What is b - a?
AMC 10 2010A #12 grade 8+ geometry-3d
Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a…
AMC 10 2010A #13 grade 7+ rate-ratio
Angelina drove at an average rate of 80 kmh and then stopped 20 minutes for gas. After the stop, she drove at an average…
AMC 10 2010A #14 grade 8+ geometry-2d
Triangle ABC has AB=2 · AC. Let D and E be on AB and BC, respectively, such that ∠ BAE = ∠ ACD. Let F be the intersectio…
AMC 10 2010A #15 grade 2+ logic
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whos…
AMC 10 2010A #16 grade 7+ geometry-2d, number-theory
Nondegenerate △ ABC has integer side lengths, BD is an angle bisector, AD = 3, and DC=8. What is the smallest possible v…
AMC 10 2010A #17 grade 7+ geometry-3d
A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of…
AMC 10 2010A #18 grade 7+ probability
Bernardo randomly picks 3 distinct numbers from the set {1,2,3,...,7,8,9} and arranges them in descending order to form…
AMC 10 2010A #19 grade 8+ geometry-2d
Equiangular hexagon ABCDEF has side lengths AB=CD=EF=1 and BC=DE=FA=r. The area of △ ACE is 70% of the area of the hexag…
AMC 10 2010A #20 grade 8+ geometry-3d
A fly trapped inside a cubical box with side length 1 meter decides to relieve its boredom by visiting each corner of th…
AMC 10 2010A #21 grade 7+ algebra
The polynomial x³-ax²+bx-2010 has three positive integer roots. What is the smallest possible value of a?
AMC 10 2010A #22 grade 7+ counting
Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in…
AMC 10 2010A #23 grade 7+ probability
Each of 2010 boxes in a line contains a single red marble, and for 1 ≤ k ≤ 2010, the box in the kth position also contai…
AMC 10 2010A #24 grade 8+ number-theory
The number obtained from the last two nonzero digits of 90! is equal to n. What is n?
AMC 10 2010A #25 grade 7+ number-theory
Jim starts with a positive integer n and creates a sequence of numbers. Each successive number is obtained by subtractin…

A parent dashboard for the family lives at sensimlab.com.