Competition · AMC preparation · step 4 of 4

AMC 10 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 10 2013A #1 grade 5+ arithmetic
A taxi ride costs $1.50 plus $0.25 per mile traveled. How much does a 5-mile taxi ride cost?
AMC 10 2013A #2 grade 5+ arithmetic
Alice is making a batch of cookies and needs 21/2 cups of sugar. Unfortunately, her measuring cup holds only 1/4 cup of…
AMC 10 2013A #3 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 10 2013A #4 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 g…
AMC 10 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 10 2013A #6 grade 1+ arithmetic
Joey and his five brothers are ages 3, 5, 7, 9, 11, and 13. One afternoon two of his brothers whose ages sum to 16 went…
AMC 10 2013A #7 grade 7+ counting
A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History…
AMC 10 2013A #8 grade 8+ algebra
What is the value of (2²⁰¹⁴+2²⁰¹²)/(2²⁰¹⁴-2²⁰¹²) ?
AMC 10 2013A #9 grade 6+ arithmetic
In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on 20% of…
AMC 10 2013A #10 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 10 2013A #11 grade 7+ counting
A student council must select a two-person welcoming committee and a three-person planning committee from among its memb…
AMC 10 2013A #12 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 10 2013A #13 grade 6+ number-theory
How many three-digit numbers are not divisible by 5, have digits that sum to less than 20, and have the first digit equa…
AMC 10 2013A #14 grade 3+ geometry-3d
A solid cube of side length 1 is removed from each corner of a solid cube of side length 3. How many edges does the rema…
AMC 10 2013A #15 grade 7+ geometry-2d
Two sides of a triangle have lengths 10 and 15. The length of the altitude to the third side is the average of the lengt…
AMC 10 2013A #16 grade 8+ geometry-2d
A triangle with vertices (6, 5), (8, -3), and (9, 1) is reflected about the line x=8 to create a second triangle. What i…
AMC 10 2013A #17 grade 6+ number-theory
Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beat…
AMC 10 2013A #18 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 10 2013A #19 grade 6+ number-theory
In base 10, the number 2013 ends in the digit 3. In base 9, on the other hand, the same number is written as (2676)₉ and…
AMC 10 2013A #20 grade 8+ geometry-2d
A unit square is rotated 45° about its center. What is the area of the region swept out by the interior of the square?
AMC 10 2013A #21 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 10 2013A #22 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 10 2013A #23 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 10 2013A #24 grade 8+ counting
Central High School is competing against Northern High School in a backgammon match. Each school has three players, and…
AMC 10 2013A #25 grade 8+ counting
All 20 diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the…

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