COMC 2013年真题
A1.Determine the positive integer n that satisfies the following equation:
A2.Determine the positive integer k for which the parabola y = x2 − 6 passes through the point (k, k).
A3.In the figure below, the circles have radii 1, 2, 3, 4, and 5. The total area that is contained inside an odd number of these circles is mπ for a positive number m. What is the value of m?
A4.A positive integer is said to be bi-digital if it uses two different digits, with each digit used exactly twice. For example, 1331 is bi-digital, whereas 1113, 1111, 1333, and 303 are not.
Determine the exact value of the integer b, the number of bi-digital positive integers.
B1.Given a triangle ABC, X, Y are points on side AB, with X closer to A than Y , and Z is a point on side AC such that XZ is parallel to Y C and Y Z is parallel to BC. Suppose AX = 16 and XY = 12. Determine the length of Y B.
B2.There is a unique triplet of positive integers (a, b, c) such that a ≤ b ≤ c and
Determine a + b + c.
B3.Teams A and B are playing soccer until someone scores 29 goals. Throughout the game the score is shown on a board displaying two numbers – the number of goals scored by A
and the number of goals scored by B. A mathematical soccer fan noticed that several times throughout the game, the sum of all the digits displayed on the board was 10. (For example,
a score of 12 : 7 is one such possible occasion). What is the maximum number of times throughout the game that this could happen?
B4.Let a be the largest real value of x for which x3 − 8x2 − 2x + 3 = 0. Determine the integer closest to a2.
C1.In the diagram, ∆AOB is a triangle with coordinates
O = (0, 0), A = (0, 30), and B = (40, 0). Let C be the point on
AB for which OC is perpendicular to AB.
(a) Determine the length of OC.
(b) Determine the coordinates of point C.
(c) Let M be the centre of the circle passing through O, A, and B. Determine the length of CM.
C2.(a) Determine all real solutions to a2 + 10 = a + 102.
(b) Determine two positive real numbers a, b > 0 such that a ≠ b and a2 + b = b2 + a.
(c) Find all triples of real numbers (a, b, c) such that a2 + b2 + c = b2 + c2 + a = c2 + a2 + b.
C3.Alphonse and Beryl play the following game. Two positive integers m and n are written on the board. On each turn, a player selects one of the numbers on the board, erases it, and
writes in its place any positive divisor of this number as long as it is different from any of the numbers previously written on the board. For example, if 10 and 17 are written on the board, a player can erase 10 and write 2 in its place. The player who cannot make a move loses. Alphonse goes first.
(a) Suppose m = 240 and n = 351. Determine which player is always able to win the game
and explain the winning strategy.
(b) Suppose m = 240 and n = 251. Determine which player is always able to win the game and explain the winning strategy.
C4.For each real number x, let [x] be the largest integer less than or equal to x. For example, [5] = 5, [7.9] = 7 and [−2.4] = −3. An arithmetic progression of length k is a sequence
a1, a2,...,ak with the property that there exists a real number
b such that ai+1 − ai = b for each 1 ≤ i ≤ k − 1.
Let α > 2 be a given irrational number. Then S = {[n · α] : n ∈ℤ}, is the set of all integers that are equal to [n · α] for some integer n.
(a) Prove that for any integer m ≥ 3, there exist m distinct numbers contained in S which form an arithmetic progression of length m.
(b) Prove that there exist no infinite arithmetic progressions contained in S.
