Competition · AMC preparation · step 4 of 4

AMC 10 2005A: 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 2005A #1 grade 6+ arithmetic
While eating out, Mike and Joe each tipped their server $2. Mike tipped 10% of his bill and Joe tipped 20% of his bill.…
AMC 10 2005A #2 grade 7+ arithmetic
For each pair of real numbers a ≠ b, define the operation ⋆ as (a ⋆ b) = (a+b)/(a-b). What is the value of ((1 ⋆ 2) ⋆ 3)…
AMC 10 2005A #3 grade 7+ arithmetic
The equations 2x + 7 = 3 and bx - 10 = -2 have the same solution x. What is the value of b?
AMC 10 2005A #4 grade 8+ geometry-2d
A rectangle with a diagonal of length x is twice as long as it is wide. What is the area of the rectangle?
AMC 10 2005A #5 grade 4+ arithmetic
A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four.…
AMC 10 2005A #6 grade 6+ arithmetic
The average (mean) of 20 numbers is 30, and the average of 30 other numbers is 20. What is the average of all 50 numbers…
AMC 10 2005A #7 grade 7+ rate-ratio
Josh and Mike live 13 miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike s…
AMC 10 2005A #8 grade 8+ geometry-2d
In the figure, the length of side AB of square ABCD is √50, E is between B and H, and BE = 1. What is the area of the in…
AMC 10 2005A #9 grade 7+ probability
Three tiles are marked X and two other tiles are marked O. The five tiles are randomly arranged in a row. What is the pr…
AMC 10 2005A #10 grade 8+ algebra
There are two values of a for which the equation 4x² + ax + 8x + 9 = 0 has only one solution for x. What is the sum of t…
AMC 10 2005A #11 grade 6+ geometry-3d
A wooden cube n units on a side is painted red on all six faces and then cut into n³ unit cubes. Exactly one-fourth of t…
AMC 10 2005A #12 grade 7+ geometry-2d
The figure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilat…
AMC 10 2005A #13 grade 8+ arithmetic
How many positive integers n satisfy the following condition: (130n)⁵⁰ > n¹⁰⁰ > 2²⁰⁰ ?
AMC 10 2005A #14 grade 6+ arithmetic
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
AMC 10 2005A #15 grade 8+ arithmetic
How many positive cubes divide 3! · 5! · 7!?
AMC 10 2005A #16 grade 6+ arithmetic
The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is 6. How many…
AMC 10 2005A #17 grade 6+ geometry-2d
In the five-sided star shown, the letters A, B, C, D, and E are replaced by the numbers 3, 5, 6, 7, and 9, although not…
AMC 10 2005A #18 grade 7+ probability
Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win e…
AMC 10 2005A #19 grade 8+ geometry-2d
Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45°, as shown.…
AMC 10 2005A #20 grade 8+ geometry-2d
An equiangular octagon has four sides of length 1 and four sides of length √2/2, arranged so that no two consecutive sid…
AMC 10 2005A #21 grade 7+ arithmetic
For how many positive integers n does 1+2+…+n evenly divide 6n?
AMC 10 2005A #22 grade 6+ arithmetic
Let S be the set of the 2005 smallest positive multiples of 4, and let T be the set of the 2005 smallest positive multip…
AMC 10 2005A #23 grade 8+ geometry-2d
Let AB be a diameter of a circle and C be a point on AB with 2 · AC = BC. Let D and E be points on the circle such that…
AMC 10 2005A #24 grade 8+ number-theory
For each positive integer m > 1, let P(m) denote the greatest prime factor of m. For how many positive integers n is it…
AMC 10 2005A #25 grade 6+ geometry-2d
In △ ABC we have AB = 25, BC = 39, and AC = 42. Points D and E are on AB and AC respectively, with AD = 19 and AE = 14.…

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