Competition · AMC preparation · step 4 of 4

AMC 10 2013B: 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 2013B #1 grade 5+ arithmetic
What is (2+4+6)/(1+3+5) - (1+3+5)/(2+4+6)?
AMC 10 2013B #2 grade 4+ geometry-2d
Mr. Green measures his rectangular garden by walking two of the sides and finds that it is 15 steps by 20 steps. Each of…
AMC 10 2013B #3 grade 7+ algebra
On a particular January day, the high temperature in Lincoln, Nebraska, was 16 degrees higher than the low temperature,…
AMC 10 2013B #4 grade 4+ counting
When counting from 3 to 201, 53 is the 51^st number counted. When counting backwards from 201 to 3, 53 is the n^th numbe…
AMC 10 2013B #5 grade 7+ algebra
Positive integers a and b are each less than 6. What is the smallest possible value for 2 · a - a · b?
AMC 10 2013B #6 grade 6+ rate-ratio
The average age of 33 fifth-graders is 11. The average age of 55 of their parents is 33. What is the average age of all…
AMC 10 2013B #7 grade 8+ geometry-2d
Six points are equally spaced around a circle of radius 1. Three of these points are the vertices of a triangle that is…
AMC 10 2013B #8 grade 6+ rate-ratio
Ray's car averages 40 miles per gallon of gasoline, and Tom's car averages 10 miles per gallon of gasoline. Ray and Tom…
AMC 10 2013B #9 grade 6+ number-theory
Three positive integers are each greater than 1, have a product of 27000, and are pairwise relatively prime. What is the…
AMC 10 2013B #10 grade 7+ algebra
A basketball team's players were successful on 50% of their two-point shots and 40% of their three-point shots, which re…
AMC 10 2013B #11 grade 8+ algebra
Real numbers x and y satisfy the equation x²+y²=10x-6y-34. What is x+y?
AMC 10 2013B #12 grade 7+ probability
Let S be the set of sides and diagonals of a regular pentagon. A pair of elements of S are selected at random without re…
AMC 10 2013B #13 grade 4+ counting
Jo and Blair take turns counting from 1 to one more than the last number said by the other person. Jo starts by saying `…
AMC 10 2013B #14 grade 8+ algebra
Define a♣ b=a²b-ab². Which of the following describes the set of points (x, y) for which x♣ y=y♣ x?
AMC 10 2013B #15 grade 8+ geometry-2d
A wire is cut into two pieces, one of length a and the other of length b. The piece of length a is bent to form an equil…
AMC 10 2013B #16 grade 8+ geometry-2d
In triangle ABC, medians AD and CE intersect at P, PE=1.5, PD=2, and DE=2.5. What is the area of AEDC?
AMC 10 2013B #17 grade 6+ logic, number-theory
Alex has 75 red tokens and 75 blue tokens. There is a booth where Alex can give two red tokens and receive in return a s…
AMC 10 2013B #18 grade 6+ counting
The number 2013 has the property that its units digit is the sum of its other digits, that is 2+0+1=3. How many integers…
AMC 10 2013B #19 grade 8+ algebra
The real numbers c,b,a form an arithmetic sequence with a ≥ b ≥ c ≥ 0. The quadratic ax²+bx+c has exactly one root. What…
AMC 10 2013B #20 grade 6+ number-theory
The number 2013 is expressed in the form 2013 = (a₁!a₂!...a_m!)/(b₁!b₂!...b_n!), where a₁ ≥ a₂ ≥ … ≥ a_m and b₁ ≥ b₂ ≥ ……
AMC 10 2013B #21 grade 7+ counting
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that eac…
AMC 10 2013B #22 grade 7+ counting
The regular octagon ABCDEFGH has its center at J. Each of the vertices and the center are to be associated with one of t…
AMC 10 2013B #23 grade 8+ geometry-2d
In triangle ABC, AB=13, BC=14, and CA=15. Distinct points D, E, and F lie on segments BC, CA, and DE, respectively, such…
AMC 10 2013B #24 grade 6+ number-theory
A positive integer n is nice if there is a positive integer m with exactly four positive divisors (including 1 and m) su…
AMC 10 2013B #25 grade 6+ number-theory
Bernardo chooses a three-digit positive integer N and writes both its base-5 and base-6 representations on a blackboard.…

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