Competition · AMC preparation · step 4 of 4

AMC 12 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 12 2013B #1 grade 7+ algebra
On a particular January day, the high temperature in Lincoln, Nebraska, was 16 degrees higher than the low temperature,…
AMC 12 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 12 2013B #3 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 12 2013B #4 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 12 2013B #5 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 12 2013B #6 grade 8+ algebra
Real numbers x and y satisfy the equation x²+y²=10x-6y-34. What is x+y?
AMC 12 2013B #7 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 12 2013B #8 grade 8+ geometry-2d
Line l₁ has equation 3x - 2y = 1 and goes through A = (-1, -2). Line l₂ has equation y = 1 and meets line l₁ at point B.…
AMC 12 2013B #9 grade 8+ number-theory
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides 12!?
AMC 12 2013B #10 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 12 2013B #11 grade 8+ geometry-3d, pattern
Two bees start at the same spot and fly at the same rate in the following directions. Bee A travels 1 foot north, then 1…
AMC 12 2013B #12 grade 7+ counting
Cities A, B, C, D, and E are connected by roads AB, AD, AE, BC, BD, CD, and DE. How many different routes are there from…
AMC 12 2013B #13 grade 8+ geometry-2d
The internal angles of quadrilateral ABCD form an arithmetic progression. Triangles ABD and DCB are similar with ∠ DBA =…
AMC 12 2013B #14 grade 7+ counting
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that eac…
AMC 12 2013B #15 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 12 2013B #16 grade 8+ geometry-2d
Let ABCDE be an equiangular convex pentagon of perimeter 1. The pairwise intersections of the lines that extend the side…
AMC 12 2013B #17 grade 9+ algebra
Let a,b, and c be real numbers such that a+b+c=2, and a²+b²+c²=12 What is the difference between the maximum and minimum…
AMC 12 2013B #18 grade 6+ logic, pattern
Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbar…
AMC 12 2013B #19 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 12 2013B #20 grade 11+ geometry-2d
For 135° < x < 180°, points P=(cos x, cos² x), Q=(cot x, cot² x), R=(sin x, sin² x) and S =(tan x, tan² x) are the verti…
AMC 12 2013B #21 grade 11+ geometry-2d, counting
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines…
AMC 12 2013B #22 grade 11+ algebra
Let m>1 and n>1 be integers. Suppose that the product of the solutions for x of the equation 8(log_n x)(log_m x)-7log_n…
AMC 12 2013B #23 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.…
AMC 12 2013B #24 grade 11+ geometry-2d
Let ABC be a triangle where M is the midpoint of AC, and CN is the angle bisector of ∠ACB with N on AB. Let X be the int…
AMC 12 2013B #25 grade 11+ algebra, counting
Let G be the set of polynomials of the form P(z)=zⁿ+c_(n-1)zⁿ⁻¹+…+c₂z²+c₁z+50, where c₁,c₂,…, c_(n-1) are integers and P…

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