COMC 2018年真题

A1.Suppose x is a real number such that x(x + 3) = 154. Determine the value of (x + 1)(x + 2).

A2.Let v, w, x, y, and z be five distinct integers such that 45 = v × w × x × y × z. What is the sum of the integers?

A3.Points (0, 0) and (, ) are the endpoints of a diameter of circle Γ. Determine the other x intercept of Γ.

A4.In the sequence of positive integers, starting with 2018, 121, 16, ... each term is the square of the sum of digits of the previous term. What is the 2018th term of the sequence?

B1.Let , where a and b are positive integers. Determine the value of a + b.

B2.Let ABCD be a square with side length 1. Points X and Y are on sides BC and CD respectively such that the areas of triangles ABX, XCY , and Y DA are equal. Find the ratio of the area of ∆AXY to the area of ∆XCY .

B3.The doubling sum function is defined by

For example, we have

D(5, 3) = 5 + 10 + 20 = 35

and

D(11, 5) = 11 + 22 + 44 + 88 + 176 = 341.

Determine the smallest positive integer n such that for every integer i between 1 and 6, inclusive, there exists a positive integer ai such that D(ai , i) = n.

B4.Determine the number of 5-tuples of integers (x1, x2, x3, x4, x5) such that

(a) xi ≥ i for 1 ≤ i ≤ 5;

(b)

C1.At Math-ee -Mart, cans of cat food are

arranged in an pentagonal pyramid of 15

layers high, with 1 can in the top layer,

5 cans in the second layer, 12 cans in the

third layer, 22 cans in the fourth layer etc,

so that the kth layer is a pentagon with k

cans on each side.

(a) How many cans are on the bottom, 15th, layer of this pyramid?

(b) The pentagonal pyramid is rearranged into a prism consisting of 15 identical layers. How many cans are on the bottom layer of the prism?

layers, each of which has a shape of a

triangle. (The number of cans in a triangular

layer is one of the triangular numbers:

1,3,6,10,...) For example, a prism could be

composed of the following layers:

Prove that a pentagonal pyramid of cans with any number of layers l ≥ 2 can be rearranged (without a deficit or leftover) into a triangular prism of cans with the same number of layers l.

C2.Alice has two boxes A and B. Initially box A contains n coins and box B is empty. On each turn, she may either move a coin from box A to box B, or remove k coins from box A, where k is the current number of coins in box B. She wins when box A is empty.

(a) If initially box A contains 6 coins, show that Alice can win in 4 turns.

(b) If initially box A contains 31 coins, show that Alice cannot win in 10 turns.

C3.Consider a convex quadrilateral ABCD. Let rays BA and CD intersect at E, rays DA and CB intersect at F, and the diagonals AC and BD intersect at G. It is given that the triangles DBF and DBE have the same area.

(a) Prove that EF and BD are parallel.

(b) Prove that G is the midpoint of BD.

C4.Given a positive integer N, Matt writes N in decimal on a blackboard, without writing any of the leading 0s. Every minute he takes two consecutive digits, erases them, and replaces

them with the last digit of their product. Any leading zeroes created this way are also erased. He repeats this process for as long as he likes. We call the positive integer M obtainable from N if starting from N, there is a finite sequence of moves that Matt can make to produce the number M. For example, 10 is obtainable from 251023 via

251023 → 25106 → 106 → 10

(a) Show that 2018 is obtainable from 2567777899.

(b) Find two positive integers A and B for which there is no positive integer C such that both A and B are obtainable from C.

(c) Let S be any finite set of positive integers, none of which contains the digit 5 in its decimal representation. Prove that there exists a positive integer N for which all elements of S are obtainable from N.