Competition · AMC preparation · step 4 of 4

AMC 12 2014A: 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 2014A #1 grade 8+ arithmetic
What is 10·(1/2+1/5+1/10)⁻¹?
AMC 12 2014A #2 grade 6+ algebra
At the theater children get in for half price. The price for 5 adult tickets and 4 child tickets is $24.50. How much wou…
AMC 12 2014A #3 grade 2+ logic, counting
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house…
AMC 12 2014A #4 grade 7+ rate-ratio
Suppose that a cows give b gallons of milk in c days. At this rate, how many gallons of milk will d cows give in e days?
AMC 12 2014A #5 grade 6+ arithmetic
On an algebra quiz, 10% of the students scored 70 points, 35% scored 80 points, 30% scored 90 points, and the rest score…
AMC 12 2014A #6 grade 7+ number-theory, algebra
The difference between a two-digit number and the number obtained by reversing its digits is 5 times the sum of the digi…
AMC 12 2014A #7 grade 11+ algebra, pattern
The first three terms of a geometric progression are √3, ³√3, and ⁶√3. What is the fourth term?
AMC 12 2014A #8 grade 7+ algebra
A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1: 10% off the…
AMC 12 2014A #9 grade 6+ algebra
Five positive consecutive integers starting with a have average b. What is the average of 5 consecutive integers that st…
AMC 12 2014A #10 grade 8+ geometry-2d
Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side len…
AMC 12 2014A #11 grade 7+ rate-ratio
David drives from his home to the airport to catch a flight. He drives 35 miles in the first hour, but realizes that he…
AMC 12 2014A #12 grade 11+ geometry-2d
Two circles intersect at points A and B. The minor arcs AB measure 30° on one circle and 60° on the other circle. What i…
AMC 12 2014A #13 grade 7+ counting
A fancy bed and breakfast inn has 5 rooms, each with a distinctive color-coded decor. One day 5 friends arrive to spend…
AMC 12 2014A #14 grade 9+ algebra, number-theory
Let a<b<c be three integers such that a,b,c is an arithmetic progression and a,c,b is a geometric progression. What is t…
AMC 12 2014A #15 grade 7+ number-theory, counting
A five-digit palindrome is a positive integer with respective digits abcba, where a is non-zero. Let S be the sum of all…
AMC 12 2014A #16 grade 6+ number-theory
The product (8)(888…8), where the second factor has k digits, is an integer whose digits have a sum of 1000. What is k?
AMC 12 2014A #17 grade 8+ geometry-3d
A 4× 4× h rectangular box contains a sphere of radius 2 and eight smaller spheres of radius 1. The smaller spheres are e…
AMC 12 2014A #18 grade 11+ number-theory
The domain of the function f(x)=log_(1/2)(log₄(log_(1/4)(log₁₆(log_(1/16)x)))) is an interval of length m/n, where m and…
AMC 12 2014A #19 grade 8+ algebra, number-theory
There are exactly N distinct rational numbers k such that |k|<200 and 5x²+kx+12=0 has at least one integer solution for…
AMC 12 2014A #20 grade 11+ geometry-2d
In △ BAC, ∠ BAC=40°, AB=10, and AC=6. Points D and E lie on AB and AC respectively. What is the minimum possible value o…
AMC 12 2014A #21 grade 11+ algebra
For every real number x, let ⌊ x⌋ denote the greatest integer not exceeding x, and let f(x)=⌊ x⌋(2014^(x-⌊ x⌋)-1). The s…
AMC 12 2014A #22 grade 8+ number-theory, counting
The number 5⁸⁶⁷ is between 2²⁰¹³ and 2²⁰¹⁴. How many pairs of integers (m,n) are there such that 1≤ m≤ 2012 and 5ⁿ<2^m<2…
AMC 12 2014A #23 grade 8+ number-theory, pattern
The fraction 1/99²=0.b_(n-1)b_(n-2)… b₂b₁b₀, where n is the length of the period of the repeating decimal expansion. Wha…
AMC 12 2014A #24 grade 9+ algebra, counting
Let f₀(x)=x+|x-100|-|x+100|, and for n≥ 1, let f_n(x)=|f_(n-1)(x)|-1. For how many values of x is f₁₀₀(x)=0?
AMC 12 2014A #25 grade 10+ geometry-2d, number-theory
The parabola P has focus (0,0) and goes through the points (4,3) and (-4,-3). For how many points (x,y)∈ P with integer…

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