Competition · AMC preparation · step 4 of 4

AMC 12 2013A: 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 2013A #1 grade 6+ geometry-2d
Square ABCD has side length 10. Point E is on BC, and the area of bigtriangleup ABE is 40. What is BE?
AMC 12 2013A #2 grade 4+ logic, counting
A softball team played ten games, scoring 1,2,3,4,5,6,7,8,9, and 10 runs. They lost by one run in exactly five games. In…
AMC 12 2013A #3 grade 6+ rate-ratio
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are…
AMC 12 2013A #4 grade 8+ algebra
What is the value of (2²⁰¹⁴+2²⁰¹²)/(2²⁰¹⁴-2²⁰¹²)?
AMC 12 2013A #5 grade 4+ arithmetic
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Doroth…
AMC 12 2013A #6 grade 6+ rate-ratio
In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on 20% of…
AMC 12 2013A #7 grade 4+ pattern, algebra
The sequence S₁, S₂, S₃, …, S₁₀ has the property that every term beginning with the third is the sum of the previous two…
AMC 12 2013A #8 grade 8+ algebra
Given that x and y are distinct nonzero real numbers such that x+2/x = y + 2/y, what is xy?
AMC 12 2013A #9 grade 8+ geometry-2d
In △ ABC, AB=AC=28 and BC=20. Points D,E, and F are on sides AB, BC, and AC, respectively, such that DE and EF are paral…
AMC 12 2013A #10 grade 8+ number-theory
Let S be the set of positive integers n for which 1/n has the repeating decimal representation 0.ab = 0.ababab…, with a…
AMC 12 2013A #11 grade 8+ geometry-2d
Triangle ABC is equilateral with AB=1. Points E and G are on AC and points D and F are on AB such that both DE and FG ar…
AMC 12 2013A #12 grade 11+ geometry-2d
The angles in a particular triangle are in arithmetic progression, and the side lengths are 4,5,x. The sum of the possib…
AMC 12 2013A #13 grade 8+ geometry-2d
Let points A = (0, 0), B = (1, 2), C=(3, 3), and D = (4, 0). Quadrilateral ABCD is cut into equal area pieces by a line…
AMC 12 2013A #14 grade 11+ algebra, pattern
The sequence log₁₂162, log₁₂x, log₁₂y, log₁₂z, log₁₂1250 is an arithmetic progression. What is x?
AMC 12 2013A #15 grade 7+ counting
Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distribute…
AMC 12 2013A #16 grade 7+ algebra
A, B, C are three piles of rocks. The mean weight of the rocks in A is 40 pounds, the mean weight of the rocks in B is 5…
AMC 12 2013A #17 grade 6+ number-theory
A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows. The k^th pirate to tak…
AMC 12 2013A #18 grade 8+ geometry-3d
Six spheres of radius 1 are positioned so that their centers are at the vertices of a regular hexagon of side length 2.…
AMC 12 2013A #19 grade 7+ geometry-2d
In △ ABC, AB = 86, and AC=97. A circle with center A and radius AB intersects BC at points B and X. Moreover BX and CX h…
AMC 12 2013A #20 grade 7+ counting, number-theory
Let S be the set {1,2,3,...,19}. For a,b ∈ S, define a succ b to mean that either 0 9. How many ordered triples (x,y,z)…
AMC 12 2013A #21 grade 11+ algebra
Consider A = log (2013 + log (2012 + log (2011 + log (… + log (3 + log 2) … )))). Which of the following intervals conta…
AMC 12 2013A #22 grade 7+ probability
A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no…
AMC 12 2013A #23 grade 8+ geometry-2d
ABCD is a square of side length √3 + 1. Point P is on AC such that AP = √2. The square region bounded by ABCD is rotated…
AMC 12 2013A #24 grade 11+ geometry-2d
Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. W…
AMC 12 2013A #25 grade 11+ algebra, counting
Let f : C → C be defined by f(z) = z² + iz + 1. How many complex numbers z are there such that Im(z) > 0 and both the re…

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