2018 AMC 10A

All 25 problems, 75 minutes, scored 6 / 1.5 / 0. The answer key and the idea behind each problem are on this page too, folded away until you ask for them.

Take it timed (75 min)Print the paper PDF 194 KB Blank answer sheet Enter answers from paper

The problems

Show the answer key, topics and key insights

This gives away all 25 answers and the idea behind each one. Sit the paper first if you mean to.

The same key as a PDF 127 KB — a separate file from the paper, so one can be handed over without the other.

#	Answer	Topic	Key insight
1	B	Exponents, Logarithms & Radicals	A power of -1 just means flip the fraction upside down. Start at the innermost brackets and repeat: flip, then add 1.
2	A	Ratios, Percents & Averages	Pick an easy amount for Jacqueline, like 100. Then Liliane has 150 and Alice has 125, and you just compare 150 with 125.
3	E	Clocks, Calendars & Time	One day is 60 times 60 times 24 seconds. Split 10! into those same factors and the leftover is exactly 42 days, which lands on noon of February 12.
4	E	Arrangements with Restrictions	First choose which three periods hold math, with gaps between them; only four choices work. Then put the three different courses into those periods in 6 orders.
5	D	Absolute Value & Inequalities	Every statement is false, so flip each one: less than 6, more than 5, more than 4. The numbers that pass all three lie strictly between 5 and 6.
6	B	Ratios, Percents & Averages	Out of every 100 votes, 65 are likes and 35 are dislikes, so each 100 votes adds 30 to the score. A score of 90 needs 300 votes.
7	E	Exponents, Logarithms & Radicals	4000 has five 2s and three 5s. Positive n uses up a 5 each step, negative n uses up a 2. Count both directions and n = 0.
8	C	Linear Equations & Word Problems	Call the number of 5-cent coins x. Write the other two counts using x, and the total value becomes one equation in x.
9	E	Similar & Congruent Triangles	Four small triangles fit side by side along DE, so triangle ADE is a small one scaled up 4 times, with area 4 times 4 = 16.
10	A	Algebraic Manipulation	The two numbers under the roots differ by 24. Big minus small, times big plus small, equals that difference, so 3 times the sum is 24.
11	E	Distributions & Stars and Bars	Each die shows at least 1, so seven dice already give 7. Only 3 extra points are left to hand out, and there are just three ways to split 3.
12	C	Absolute Value & Inequalities	Write x using y from the line. The line then splits into three stretches where x and y keep their signs, and each stretch gives simple equations.
13	D	Transformations & Symmetry	The crease is the line through the middle of AB at a right angle to it. It forms a small right triangle shaped like the paper.
14	A	Exponents, Logarithms & Radicals	The 3-powers dominate, so the value is near 3 to the fourth, 81. The small 2-power terms pull it just below 81, not above.
15	D	Circles	A touching point lies on the line through both centers. Stretching the center segments from 8 to 13 turns triangle OPQ into triangle OAB.
16	D	Triangles: Area & Pythagorean	Sliding along the hypotenuse, the segment shrinks from 20 to the height (about 14.5), then grows to 21, hitting each whole number once per side.
17	C	Divisibility & Factors	A small smallest number bans all its multiples, and leftover pairs like 4 and 8 still clash. Smallest 2 or 3 allows only 5 numbers.
18	D	Bases & Digits	Digits -1, 0, 1 with powers of 3 make every whole number from -3280 to 3280 exactly once; the nonnegative ones are 0 through 3280.
19	E	Modular Arithmetic	Only the last digit of m matters: 1 always works, 5 never, 9 needs an even exponent, 3 and 7 need a multiple of 4.
20	B	Paths & Grids	All the rotations and flips together fold the grid onto one triangular slice, one-eighth of the square, holding 10 cells. Color those freely: 2^10 ways, minus all-black and all-white.
21	E	Coordinate Geometry	Put y + a in place of x squared. The equation in y factors: one root is the parabola's bottom point, the other adds two points when a > 1/2.
22	D	GCD & LCM	Follow the 2s and 3s along the chain. The gcd of d and a has exactly one 2 and one 3, which forces it to be 78.
23	D	Triangles: Area & Pythagorean	Join the square's outer corner to the far corners. The pieces are the square, a height-2 triangle on the hypotenuse, and two thin triangles.
24	D	Triangle Centers & Cevians	Since AB is 5 times AC, the bisector splits both triangle ABC and half-size triangle ADE into areas 5 to 1.
25	D	Bases & Digits	Using R = 11...1, the equation becomes (9c minus a squared) times R = b minus 2c. Two values of n force both sides to be 0.

2018 AMC 10A. Problems © Mathematical Association of America, American Mathematics Competitions (maa.org/amc), licensed CC BY-NC-SA 4.0. Typeset and annotated by AMC 10 Step by Step (amcstep.com) and shared under the same licence. Not endorsed by the MAA.