COMC 2015年真题

A1.A palindrome is a number where the digits read the same forwards or backwards, such as 4774 or 505. What is the smallest palindrome that is larger than 2015?

A2.In the picture below, there are four triangles labelled S, T, U, and V. Two of the triangles will be coloured red and the other two triangles will be coloured blue. How many ways can the triangles be coloured such that the two blue triangles have a common side?

A3.In the given figure, ABCD is a square with sides of length 4, and Q is the midpoint of CD. AB’C’D’ is reflected along the line AQ to give the square AB’C’D’. The two squares overlap in the

quadrilateral ADQD’. Determine the area of quadrilateral ADQD’.

A4.The area of a rectangle is 180 units2 and the perimeter is 54 units. If the length of each side of the rectangle is increased by six units, what is the area of the resulting rectangle?

B1.Given an integer n ≥ 2, let f(n) be the second largest positive divisor of n. For example, f(12) = 6 and f(13) = 1. Determine the largest positive integer n such that f(n) = 35.

B2.Let ABC be a right triangle with ∠BCA = 90°. A circle with diameter AC intersects the hypotenuse AB at K. If BK:AK = 1:3, find the measure of the angle ∠BAC.

B3.An arithmetic sequence is a sequence where each term after the first is the sum of the previous term plus a constant value. For example, 3, 7, 11, 15,... is an arithmetic sequence.

S is a sequence which has the following properties:

• The first term of S is positive.

• The first three terms of S form an arithmetic sequence.

• If a square is constructed with area equal to a term in S, then the perimeter of that square is the next term in S.

Determine all possible values for the third term of S.

B4.A farmer has a flock of n sheep, where 2000 ≤ n ≤ 2100. The farmer puts some number of the sheep into one barn and the rest of the sheep into a second barn. The farmer realizes that if she were to select two different sheep at random from her flock, the probability that they are in different barns is exactly . Determine the value of n.

C1.A quadratic polynomial f(x) = x2 + px + q, with p and q real numbers, is said to be a double-up polynomial if it has two real roots, one of which is twice the other.

(a) If a double-up polynomial f(x) has p = −15, determine the value of q.

(b) If f(x) is a double-up polynomial with one of the roots equal to 4, determine all possible values of p + q.

C2.Let O = (0, 0), Q = (13, 4), A = (a, a), B = (b, 0), where a and b are positive real numbers with b ≥ a. The point Q is on the line segment AB.

(a) Determine the values of a and b for which Q is the midpoint of AB.

(b) Determine all values of a and b for which Q is on the line segment AB and the triangle OAB is isosceles and right-angled.

(c) There are infinitely many line segments AB that contain the point Q. For how many of these line segments are a and b both integers?

C3.(a) If n = 3, determine all integer values of m such that m2 + n2 + 1 is divisible by m − n + 1 and m + n + 1.

(b) Show that for any integer n there is always at least one integer value of m for which m2 + n2 + 1 is divisible by both m − n + 1 and m + n + 1.

(c) Show that for any integer n there are only a finite number of integer values m for which m2 + n2 + 1 is divisible by both m − n + 1 and m + n + 1.

C4.Mr. Whitlock is playing a game with his math class to teach them about money. Mr. Whitlock’s math class consists of n ≥ 2 students, whom he has numbered from 1 to n. Mr. Whitlock gives mi ≥ 0 dollars to student i, for each 1 ≤ i ≤ n, where each mi is an integer and m1 + m2 + ··· + mn ≥ 1.

We say a student is a giver if no other student has more money than they do and we say a student is a receiver if no other student has less money than they do. To play the game, each student who is a giver, gives one dollar to each student who is a receiver (it is possible for a student to have a

negative amount of money after doing so). This process is repeated until either all students have the same amount of money, or the students reach a distribution of money that they had previously reached.

(a) Give values of n, m1, m2,...,mn for which the game ends with at least one student having a negative amount of money, and show that the game does indeed end this way.

(b) Suppose there are n students. Determine the smallest possible value kn such that if m1 +m2 + ··· + mn ≥ kn then no player will ever have a negative amount of money.

(c) Suppose n = 5. Determine all quintuples (m1, m2, m3, m4, m5), with m1 ≤ m2 ≤ m3 ≤ m4 ≤ m5, for which the game ends with all students having the same amount of money.