2017 AIME I 真题

2017 AIME I Problems

Problem 1

Fifteen distinct points are designated on ∆ABC: the 3 vertices A, B, and C; 3 other points on side $\overline{AB}$ ; 4 other points on side $\overline{BC}$ ; and 5 other points on side $\overline{CA}$ . Find the number of triangles with positive area whose vertices are among these 15 points.

Problem 2

When each of 702, 787, and 855 is divided by the positive integer m, the remainder is always the positive integer r. When each of 412, 722, and 815 is divided by the positive integer n, the remainder is always the positive integer s ≠ r. Find m + n + r + s.

Problem 3

For a positive integer n, let dn be the units digit of 1 + 2 + ... + n. Find the remainder when is divided by 1000.

Problem 4

A pyramid has a triangular base with side lengths 20, 20, and 24. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length 25. The volume of the pyramid is $m\sqrt{n}$ , where m and n are positive integers, and n is not divisible by the square of any prime. Find m + n.

Problem 5

A rational number written in base eight is $\underline{a} \underline{b} . \underline{c} \underline{d}$ , where all digits are nonzero. The same number in base twelve is $\underline{b} \underline{b} . \underline{b} \underline{a}$ . Find the base-ten number $\underline{a} \underline{b} \underline{c}$ .

Problem 6

A circle circumscribes an isosceles triangle whose two congruent angles have degree measure x. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is . Find the difference between the largest and smallest possible values of x.

Problem 7

For nonnegative integers a and b with a + b ≤ 6, let . Let S denote the sum of all T(a, b), where a and b are nonnegative integers with a + b ≤ 6. Find the remainder when S is divided by 1000.

Problem 8

Two real numbers a and b are chosen independently and uniformly at random from the interval (0, 75). Let and be two points on the plane with OP = 200. Let Q and R be on the same side of line OP such that the degree measures of ∠POQ and ∠POR are a and b respectively, and ∠OQP and ∠ORP are both right angles. The probability that QR ≤ 100 is equal to , where m and n are relatively prime positive integers. Find m + n.

Problem 9

Let a10 = 10, and for each integer n > 10 let an = 100an-1 + n. Find the least n > 10 such that an is a multiple of 99.

Problem 10

Let z1 = 18 + 83i, z2 = 18 + 39i, and z3 = 78 + 99i, where $i = \sqrt{-1}$ . Let z be the unique complex number with the properties that is a real number and the imaginary part of z is the greatest possible. Find the real part of z.

Problem 11

Consider arrangements of the numbers 1, 2, 3, ..., 9 in a 3 x 3 array. For each such arrangement, let a1, a2, and a3 be the medians of the numbers in rows 1, 2, and 3 respectively, and let m be the median of {a1, a2, a3}. Let Q be the number of arrangements for which m = 5. Find the remainder when Q is divided by 1000.

Problem 12

Call a set S product-free if there do not exist a, b, c∈S(not necessarily distinct) such that ab = c. For example, the empty set and the set {16, 20) are product-free, whereas the sets (4, 16} and (2, 8, 16} are not product-free. Find the number of product-free subsets of the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

Problem 13

$\sum_{m = 2}^{2017} Q(m)$ For every m ≥ 2 , let Q(m) be the least positive integer with the following property: For every n ≥ Q(m), there is always a perfect cube k3 in the range n < k3 ≤ m ∙ n. Find the remainder when is divided by 1000.

Problem 14

Let a > 1 and x > 1 satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$ . Find the remainder when x is divided by 1000.

Problem 15

The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths $2\sqrt3$ , 5, and $\sqrt{37}$ , as shown, is , where m, n, and p are positive integers, m and n are relatively prime, and p is not divisible by the square of any prime. Find m + n + p.