Competition · AMC preparation · step 4 of 4

AMC 12 2018A: 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 2018A #1 grade 6+ rate-ratio
A large urn contains 100 balls, of which 36 % are red and the rest are blue. How many of the blue balls must be removed…
AMC 12 2018A #2 grade 6+ arithmetic
While exploring a cave, Carl comes across a collection of 5-pound rocks worth $14 each, 4-pound rocks worth $11 each, an…
AMC 12 2018A #3 grade 7+ counting
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day…
AMC 12 2018A #4 grade 6+ rate-ratio
Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at…
AMC 12 2018A #5 grade 9+ algebra
What is the sum of all possible values of k for which the polynomials x² - 3x + 2 and x² - 5x + k have a root in common?
AMC 12 2018A #6 grade 8+ algebra
For positive integers m and n such that m+10<n+1, both the mean and the median of the set {m, m+4, m+10, n+1, n+2, 2n} a…
AMC 12 2018A #7 grade 8+ number-theory
For how many (not necessarily positive) integer values of n is the value of 4000· (2/5)ⁿ an integer?
AMC 12 2018A #8 grade 8+ geometry-2d
All of the triangles in the diagram below are similar to isosceles triangle ABC, in which AB=AC. Each of the 7 smallest…
AMC 12 2018A #9 grade 11+ algebra
Which of the following describes the largest subset of values of y within the closed interval [0,π] for which sin(x+y)≤…
AMC 12 2018A #10 grade 8+ algebra
How many ordered pairs of real numbers (x,y) satisfy the following system of equations? x+3y =3 ||x|-|y|| =1
AMC 12 2018A #11 grade 8+ geometry-2d
A paper triangle with sides of lengths 3,4, and 5 inches, as shown, is folded so that point A falls on point B. What is…
AMC 12 2018A #12 grade 4+ number-theory
Let S be a set of 6 integers taken from {1,2,…,12} with the property that if a and b are elements of S with a<b, then b…
AMC 12 2018A #13 grade 7+ number-theory
How many nonnegative integers can be written in the form a₇·3⁷+a₆·3⁶+a₅·3⁵+a₄·3⁴+a₃·3³+a₂·3²+a₁·3¹+a₀·3⁰, where a_i∈ {-1…
AMC 12 2018A #14 grade 11+ algebra
The solution to the equation log_3x 4 = log_2x 8, where x is a positive real number other than 1/3 or 1/2, can be writte…
AMC 12 2018A #15 grade 8+ counting
A scanning code consists of a 7 × 7 grid of squares, with some of its squares colored black and the rest colored white.…
AMC 12 2018A #16 grade 8+ algebra
Which of the following describes the set of values of a for which the curves x²+y²=a² and y=x²-a in the real xy-plane in…
AMC 12 2018A #17 grade 8+ geometry-2d
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In…
AMC 12 2018A #18 grade 8+ geometry-2d
Triangle ABC with AB=50 and AC=10 has area 120. Let D be the midpoint of AB, and let E be the midpoint of AC. The angle…
AMC 12 2018A #19 grade 11+ number-theory
Let A be the set of positive integers that have no prime factors other than 2, 3, or 5. The infinite sum 1/1 + 1/2 + 1/3…
AMC 12 2018A #20 grade 10+ geometry-2d
Triangle ABC is an isosceles right triangle with AB=AC=3. Let M be the midpoint of hypotenuse BC. Points I and E lie on…
AMC 12 2018A #21 grade 11+ algebra
Which of the following polynomials has the greatest real root?
AMC 12 2018A #22 grade 11+ geometry-2d
The solutions to the equations z²=4+4√15i and z²=2+2√3i, where i=√(-1), form the vertices of a parallelogram in the comp…
AMC 12 2018A #23 grade 10+ geometry-2d
In △ PAT, ∠ P=36°, ∠ A=56°, and PA=10. Points U and G lie on sides TP and TA, respectively, so that PU=AG=1. Let M and N…
AMC 12 2018A #24 grade 9+ probability
Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is…
AMC 12 2018A #25 grade 8+ number-theory
For a positive integer n and nonzero digits a, b, and c, let A_n be the n-digit integer each of whose digits is equal to…

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