Competition · AMC preparation · step 4 of 4

AMC 12 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 12 2005A #1 grade 6+ arithmetic
Two is 10 % of x and 20 % of y. What is x - y? (A) 1 (B) 2 (C) 5 (D) 10 (E) 20
AMC 12 2005A #2 grade 7+ arithmetic
The equations 2x + 7 = 3 and bx - 10 = -2 have the same solution x. What is the value of b?
AMC 12 2005A #3 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 12 2005A #4 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 12 2005A #5 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 12 2005A #6 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 12 2005A #7 grade 8+ geometry-2d
Square EFGH is inside square ABCD so that each side of EFGH can be extended to pass through a vertex of ABCD. Square ABC…
AMC 12 2005A #8 grade 6+ number-theory
Let A,M, and C be digits with (100A+10M+C)(A+M+C) = 2005. What is A? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
AMC 12 2005A #9 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 12 2005A #10 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 12 2005A #11 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 12 2005A #12 grade 8+ number-theory
A line passes through A(1,1) and B(100,1000). How many other points with integer coordinates are on the line and strictl…
AMC 12 2005A #13 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 ne…
AMC 12 2005A #14 grade 7+ probability
On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled…
AMC 12 2005A #15 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 12 2005A #16 grade 9+ geometry-2d
Three circles of radius s are drawn in the first quadrant of the xy-plane. The first circle is tangent to both axes, the…
AMC 12 2005A #17 grade 10+ geometry-3d
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is…
AMC 12 2005A #18 grade 10+ number-theory
Call a number "prime-looking" if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbe…
AMC 12 2005A #19 grade 7+ counting
A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. For example…
AMC 12 2005A #20 grade 9+ algebra, counting
For each x in [0,1], define f(x) = 2x, if 0 ≤ x ≤ 1/2 2-2x, if 1/2 < x ≤ 1. Let f^([2])(x) = f(f(x)), and f^([n + 1])(x)…
AMC 12 2005A #21 grade 11+ algebra, number-theory
How many ordered triples of integers (a,b,c), with a ≥ 2, b ≥ 1, and c ≥ 0, satisfy both log_ab = c²⁰⁰⁵ and a + b + c =…
AMC 12 2005A #22 grade 9+ geometry-3d
A rectangular box P is inscribed in a sphere of radius r. The surface area of P is 384, and the sum of the lengths of it…
AMC 12 2005A #23 grade 11+ probability
Two distinct numbers a and b are chosen randomly from the set {2, 2², 2³, …, 2²⁵}. What is the probability that log_ab i…
AMC 12 2005A #24 grade 11+ algebra
Let P(x)=(x-1)(x-2)(x-3). For how many polynomials Q(x) does there exist a polynomial R(x) of degree 3 such that P(Q(x))…
AMC 12 2005A #25 grade 12+ geometry-3d, counting
Let S be the set of all points with coordinates (x,y,z), where x, y, and z are each chosen from the set {0,1,2}. How man…

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