2018 AIME II 真题

2018 AIME II Problems

Problem 1

Points A, B, and C lie in that order along a straight path where the distance from A to C is 1800 meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at A and running toward C, Paul starting at B and running toward C, and Eve starting at C and running toward A. When Paul meets Eve, he turns around and runs toward A. Paul and Ina both arrive at B at the same time. Find the number of meters from A to B.


Problem 2

Let a0 = 2, a1 = 5, and a2 = 8, and for n > 2 define an recursively to be the remainder when 4(an-1 + an-2 + an-3) is divided by 11. Find a2018 ∙ a2020 ∙ a2022.


Problem 3

Find the sum of all positive integers b < 1000 such that the base-b integer 36b is a perfect square and the base-b integer 27b is a perfect cube.


Problem 4

In equiangular octagon CAROLINE, $CA = RO = LI = NE =$$\sqrt{2}$ and AR = OL = IN = EC = 1. The self-intersecting octagon CORNELIA enclosed six non-overlapping triangular regions. Let K be the area enclosed by CORNELIA, that is, the total area of the six triangular regions. Then img3, where a and b are relatively prime positive integers. Find a + b.


Problem 5

Suppose that x, y, and z are complex numbers such that xy = -80 -320i, yz = 60, and zx = -96 + 24i, where $i$$=$$\sqrt{-1}$. Then there are real numbers a and b such that x + y + z = a + bi. Find a2 + b2.


Problem 6

A real number $a$is chosen randomly and uniformly from the interval

[-20, 18]. The probability that the roots of the polynomial

x4 + 2ax3 + (2a - 2)x2 + (-4a + 3)x - 2 are all real can be written in the form img8 , where m and n are relatively prime positive integers. Find m + n.


Problem 7

Triangle ABC has side lengths AB = 9, $BC =$$5\sqrt{3}$, and AC = 12. Points A = P0, P1, P2,... P2450 = B are on segment $\overline{AB}$ with Pk between Pk-1 and Pk+1 for k = 1, 2, ..., 2449, and points A = Q0, Q1, Q2,... Q2450 = C are on segment $\overline{AC}$ with Qk between Qk-1 and Qk+1 for k = 1, 2, ..., 2449. Furthermore, each segment $\overline{P_{k}Q_{k}}$, k = 1, 2, ..., 2449, is parallel to $\overline{BC}$. The segments cut the triangle into 2450 regions, consisting of 2449 trapezoids and 1 triangle. Each of the 2450 regions has the same area. Find the number of segments $\overline{P_{k}Q_{k}}$, k = 1, 2, ..., 2450, that have rational length.


Problem 8

A frog is positioned at the origin of the coordinate plane. From the point (x, y), the frog can jump to any of the points (x + 1, y), (x + 2, y), (x, y + 1), or (x, y + 2). Find the number of distinct sequences of jumps in which the frog begins at (0, 0) and ends at (4, 4).


Problem 9

Octagon ABCDEFGH with side lengths AB = CD = EF = GH = 10 and BC = DE = FG = HA = 11 is formed by removing 6-8-10 triangles from the corners of a 23 x 27 rectangle with side $\overline{AH}$ on a short side of the rectangle, as shown. Let J be the midpoint of $\overline{AH}$, and partition the octagon into 7 triangles by drawing segments $\overline{JB}$, $\overline{JC}$, $\overline{JD}$, $\overline{JE}$, $\overline{JF}$, and $\overline{JG}$. Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.

[asy] unitsize(6); pair P = (0, 0), Q = (0, 23), R = (27, 23), SS = (27, 0); pair A = (0, 6), B = (8, 0), C = (19, 0), D = (27, 6), EE = (27, 17), F = (19, 23), G = (8, 23), J = (0, 23/2), H = (0, 17); draw(P--Q--R--SS--cycle); draw(J--B); draw(J--C); draw(J--D); draw(J--EE); draw(J--F); draw(J--G); draw(A--B); draw(H--G); real dark = 0.6; filldraw(A--B--P--cycle, gray(dark)); filldraw(H--G--Q--cycle, gray(dark)); filldraw(F--EE--R--cycle, gray(dark)); filldraw(D--C--SS--cycle, gray(dark)); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(J); dot(H); defaultpen(fontsize(10pt)); real r = 1.3; label("$A$", A, W*r); label("$B$", B, S*r); label("$C$", C, S*r); label("$D$", D, E*r); label("$E$", EE, E*r); label("$F$", F, N*r); label("$G$", G, N*r); label("$H$", H, W*r); label("$J$", J, W*r); [/asy]



Problem 10

Find the number of functions f(x) from {1, 2, 3, 4, 5,} to {1, 2, 3, 4, 5,} that satisfy f(f(x)) = f(f(f(x))) for all x in {1, 2, 3, 4, 5,}.


Problem 11

Find the number of permutations of 1, 2, 3, 4, 5, 6 such that for each k with 1 ≤ k ≤ 5, at least one of the first k terms of the permutation is greater than k.


Problem 12

Let ABCD be a convex quadrilateral with AB = CD = 10, BC = 14, and $AD = 2\sqrt{65}$. Assume that the diagonals of ABCD intersect at point P, and that the sum of the areas of triangles APB and CPD equals the sum of the areas of triangles BPC and APD. Find the area of quadrilateral ABCD.


Problem 13

Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is img26 where m and n are relatively prime positive integers. Find m + n.


Problem 14

The incircle ω of triangle ABC is tangent to $\overline{BC}$ at X. Let Y ≠ X be the other intersection of $\overline{AX}$ with ω. Points P and Q lie on $\overline{AB}$ and $\overline{AC}$, respectively, so that $\overline{PQ}$ is tangent to ω at Y. Assume that AP = 3, PB = 4, AC = 8, and $AQ = \dfrac{m}{n}$, where m and n are relatively prime positive integers. Find m + n.


Problem 15

Find the number of functions f from (0, 1, 2, 3, 4, 5, 6} to the integers such that f(0) = 0, f(6) = 12, and |x - y| ≤ |f(x) - f(y)| ≤ 3|x - y| for all x and y in (0, 1, 2, 3, 4, 5, 6}.




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