Competition · AMC preparation · step 4 of 4

AMC 12 2007A: 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 2007A #1 grade 6+ arithmetic
One ticket to a show costs $20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buy…
AMC 12 2007A #2 grade 5+ geometry-3d
An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a…
AMC 12 2007A #3 grade 6+ algebra
The larger of two consecutive odd integers is three times the smaller. What is their sum?
AMC 12 2007A #4 grade 6+ rate-ratio
Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her…
AMC 12 2007A #5 grade 7+ algebra
Last year Mr. Jon Q. Public received an inheritance. He paid 20% in federal taxes on the inheritance, and paid 10% of wh…
AMC 12 2007A #6 grade 8+ geometry-2d
Triangles ABC and ADC are isosceles with AB=BC and AD=DC. Point D is inside triangle ABC, angle ABC measures 40 degrees,…
AMC 12 2007A #7 grade 7+ algebra
Let a, b, c, d, and e be five consecutive terms in an arithmetic sequence, and suppose that a+b+c+d+e=30. Which of a, b,…
AMC 12 2007A #8 grade 10+ geometry-2d
A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from t…
AMC 12 2007A #9 grade 7+ rate-ratio
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he…
AMC 12 2007A #10 grade 8+ geometry-2d
A triangle with side lengths in the ratio 3 : 4 : 5 is inscribed in a circle with radius 3. What is the area of the tria…
AMC 12 2007A #11 grade 6+ number-theory
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively…
AMC 12 2007A #12 grade 7+ probability
Integers a, b, c, and d, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. Wha…
AMC 12 2007A #13 grade 9+ geometry-2d, algebra
A piece of cheese is located at (12,10) in a coordinate plane. A mouse is at (4,-2) and is running up the line y=-5x+18.…
AMC 12 2007A #14 grade 7+ number-theory, algebra
Let a, b, c, d, and e be distinct integers such that (6-a)(6-b)(6-c)(6-d)(6-e)=45 What is a+b+c+d+e?
AMC 12 2007A #15 grade 7+ arithmetic
The set {3,6,9,10} is augmented by a fifth element n, not equal to any of the other four. The median of the resulting se…
AMC 12 2007A #16 grade 7+ counting, number-theory
How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?
AMC 12 2007A #17 grade 11+ algebra
Suppose that sin a + sin b = √(5/3) and cos a + cos b = 1. What is cos (a - b)?
AMC 12 2007A #18 grade 11+ algebra
The polynomial f(x) = x⁴ + ax³ + bx² + cx + d has real coefficients, and f(2i) = f(2 + i) = 0. What is a + b + c + d?
AMC 12 2007A #19 grade 8+ geometry-2d, algebra
Triangles ABC and ADE have areas 2007 and 7002, respectively, with B = (0,0), C = (223,0), D = (680,380), and E = (689,3…
AMC 12 2007A #20 grade 11+ geometry-3d
Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the r…
AMC 12 2007A #21 grade 11+ algebra
The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function f(x)=ax²+bx+c are equal.…
AMC 12 2007A #22 grade 4+ number-theory
For each positive integer n, let S(n) denote the sum of the digits of n. For how many values of n is n + S(n) + S(S(n))…
AMC 12 2007A #23 grade 11+ algebra, geometry-2d
Square ABCD has area 36, and AB is parallel to the x-axis. Vertices A, B, and C are on the graphs of y = log_ax, y = 2lo…
AMC 12 2007A #24 grade 11+ algebra
For each integer n>1, let F(n) be the number of solutions to the equation sinx=sin(nx) on the interval [0,π]. What is su…
AMC 12 2007A #25 grade 9+ counting, pattern
Call a set of integers spacy if it contains no more than one out of any three consecutive integers. How many subsets of…

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