Competition · AMC preparation · step 4 of 4

AMC 10 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 10 2018A #1 grade 8+ arithmetic
What is the value of (((2+1)⁻¹+1)⁻¹+1)⁻¹+1?
AMC 10 2018A #2 grade 7+ arithmetic
Liliane has 50% more soda than Jacqueline, and Alice has 25% more soda than Jacqueline. What is the relationship between…
AMC 10 2018A #3 grade 6+ arithmetic
A unit of blood expires after 10!=10· 9 · 8 … 1 seconds. Yasin donates a unit of blood at noon of January 1. On what day…
AMC 10 2018A #4 grade 7+ counting
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day…
AMC 10 2018A #5 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 10 2018A #6 grade 6+ arithmetic
Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a…
AMC 10 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 10 2018A #8 grade 7+ arithmetic
Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent co…
AMC 10 2018A #9 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 10 2018A #10 grade 8+ arithmetic
Suppose that real number x satisfies √(49-x²)-√(25-x²)=3What is the value of √(49-x²)+√(25-x²)?
AMC 10 2018A #11 grade 7+ probability
When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be…
AMC 10 2018A #12 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 10 2018A #13 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 10 2018A #14 grade 8+ arithmetic
What is the greatest integer less than or equal to (3¹⁰⁰+2¹⁰⁰)/(3⁹⁶+2⁹⁶)?
AMC 10 2018A #15 grade 8+ geometry-2d
Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at poin…
AMC 10 2018A #16 grade 8+ geometry-2d
Right triangle ABC has leg lengths AB=20 and BC=21. Including AB and BC, how many line segments with integer length can…
AMC 10 2018A #17 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 10 2018A #18 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 10 2018A #19 grade 7+ probability
A number m is randomly selected from the set {11,13,15,17,19}, and a number n is randomly selected from {1999,2000,2001,…
AMC 10 2018A #20 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 10 2018A #21 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 10 2018A #22 grade 6+ number-theory
Let a, b, c, and d be positive integers such that gcd(a, b)=24, gcd(b, c)=36, gcd(c, d)=54, and 70<gcd(d, a)<100. Which…
AMC 10 2018A #23 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 10 2018A #24 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 10 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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