COMC 2017年真题
A1.The average of the numbers 2, 5, x, 14, 15 is x. Determine the value of x.
A2.An equilateral triangle has sides of length 4cm. At each vertex, a circle with radius 2cm is drawn, as shown in the figure below. The total area of the shaded regions of the three circles
is a × π cm2 . Determine a.
A3.Two 1 × 1 squares are removed from a 5 × 5 grid as shown.
Determine the total number of squares of various sizes on the grid.
A4.Three positive integers a, b, c satisfy
4a × 5b × 6c = 88 × 99 × 1010 .
Determine the sum of a + b + c.
B1.Andrew and Beatrice practice their free throws in basketball. One day, they attempted a total of 105 free throws between them, with each person taking at least one free throw. If Andrew made exactly 1/3 of his free throw attempts and Beatrice made exactly 3/5 of her free throw attempts, what is the highest number of successful free throws they could have
made between them?
B2.There are twenty people in a room, with a men and b women. Each pair of men shakes hands, and each pair of women shakes hands, but there are no handshakes between a man and a woman. The total number of handshakes is 106. Determine the value of a × b.
B3.Regular decagon (10-sided polygon) ABCDEFGHIJ has area 2017 square units. Determine the area (in square units) of the rectangle CDHI.
B4.Numbers a, b and c form an arithmetic sequence if b - a = c - b. Let a, b, c be positive integers forming an arithmetic sequence with a < b < c. Let f(x) = ax2 + bx + c. Two distinct real numbers r and s satisfy f(r) = s and f(s) = r. If rs = 2017, determine the smallest possible value of a.
C1.For a positive integer n, we define function P(n) to be the sum of the digits of n plus the number of digits of n. For example, P(45) = 4 + 5 + 2 = 11. (Note that the first digit of n
reading from left to right, cannot be 0).
(a) Determine P(2017).
(b) Determine all numbers n such that P(n) = 4.
(c) Determine with an explanation whether there exists a number n for which P(n) ) P(n + 1) > 50.
C2.A function f(x) is periodic with period T > 0 if f(x + T) = f(x) for all x. The smallest such number T is called the least period. For example, the functions sin(x) and cos(x) are periodic with least period 2π.
(a) Let a function g(x) be periodic with the least period T = π. Determine the least period of g(x/3).
(b) Determine the least period of H(x) = sin(8x) + cos(4x)
(c) Determine the least periods of each of G(x) = sin(cos(x)) and F(x) = cos(sin(x)).
C3.Let XYZ be an acute-angled triangle. Let s be the side-length of the square which has two adjacent vertices on side Y Z, one vertex on side XY and one vertex on side XZ. Let h be
the distance from X to the side Y Z and let b be the distance from Y to Z.
(a) If the vertices have coordinates X = (2, 4), Y = (0, 0)
and Z = (4, 0), find b, h and s.
(b) Given the height h = 3 and s = 2, find the base b.
(c) If the area of the square is 2017, determine the minimum area of triangle XYZ.
C4.Let n be a positive integer and Sn = {1, 2, . . . , 2n - 1, 2n}. A perfect pairing of Sn is defined to be a partitioning of the 2n numbers into n pairs, such that the sum of the two numbers in each pair is a perfect square. For example, if n = 4, then a perfect pairing of S4 is (1, 8),(2, 7),(3, 6),(4, 5). It is not necessary for each pair to sum to the same perfect square.
(a) Show that S8 has at least one perfect pairing.
(b) Show that S5 does not have any perfect pairings.
(c) Prove or disprove: there exists a positive integer n for which Sn has at least 2017 different perfect pairings. (Two pairings that are comprised of the same pairs written in a different order are considered the same pairing.)
