COMC 2016年真题

A1.Pat has ten tests to write in the school year. He can obtain

a maximum score of 100 on each test. The average score

of Pat’s first eight tests is 80 and the average score of all of Pat’s tests is N. What is the maximum possible value of N?

A2.A square is inscribed in a circle, as shown in the figure.

If the area of the circle is 16π cm2 and the area of the square is S cm2, what is the value of S?

A3.Determine the pair of real numbers x, y which satisfy the

system of equations:

A4.Three males and two females write their names on sheets of paper, and randomly arrange them in order, from left to right.

What is the probability that all of the female names appear to the right of all the male names?

B1.If the cubic equation x3 − 10x2 + Px − 30 = 0 has three

positive integer roots, determine the value of P.

B2.The squares of a 6 × 6 square grid are each labelled with a

point value. As shown in the diagram below, the point value of

the square in row i and column j is i × j.

A path in the grid is a sequence of squares, such that consecutive squares share an edge and no square occurs twice in the sequence. The score of a path is the sum of the point values of all squares in the path.

Determine the highest possible score of a path that begins with the bottom left corner of the grid and ends with the top right corner.

B3.A hexagon ABCDEF has AB = 18cm, BC = 8cm, CD = 10cm, DE = 15cm, EF = 20cm, FA = 1cm, ∠FAB = 90°,

∠CDE = 90° and BC is parallel to EF . Determine the area of

this hexagon, in cm2.

B4.Let n be a positive integer. Given a real number x, let

be the greatest integer less than or equal to x. For example,

 and . Define a sequence a1, a2, a3, . . .

where a1 = n and  

,

for all integers m ≥ 2. The sequence stops when it reaches zero.

The number n is said to be lucky if 0 is the only number in the

sequence that is divisible by 3.

For example, 7 is lucky, since a1 = 7, a2 = 2, a3 = 0, and none of 7, 2 are divisible by 3. But 10 is not lucky, since a1 = 10, a2 = 3,

a3 = 1, a4 = 0, and a2 = 3 is divisible by 3. Determine the number of lucky positive integers less than or equal to 1000.

C1..

A sequence of three numbers a, b, c form an arithmetic sequence if the difference between successive terms in the sequence is the same. That is, when b − a = c − b.

(a) The sequence 2, b, 8 forms an arithmetic sequence. Determine b.

(b) Given a sequence a, b, c, let d1 be the non-negative number to increase or decrease b by so that the result is an arithmetic sequence and let d2 be the positive number to increase or

decrease c by so that the result is an arithmetic sequence.

For example, if the three-term sequence is 3, 10, 13, then we need to decrease 10 to 8 to make the arithmetic sequence 3, 8,

13. We decreased b by 2, so d1 = 2. If we change the third term, we need to increase 13 to 17 to make the arithmetic sequence 3, 10, 17. We increased 13 by 4, so d2 = 4.

Suppose the original three term sequence is 1, 13, 17. Determine d1 and d2.

(c) Define d1, d2 as in part (b). For all three-term sequences, prove that 2d1 = d2.

C2.Alice and Bob play a game, taking turns, playing on a row of n seats. On a player’s turn, he or she places a coin on any seat provided there is no coin on that seat or on an adjacent seat. Alice moves first. The player who does not have a valid move loses the game.

(a) Show that Alice has a winning strategy when n = 5.

(b) Show that Alice has a winning strategy when n = 6.

(c) Show that Bob has a winning strategy when n = 8.

C3.Let A = (0, a), O = (0, 0), C = (c, 0), B = (c, b), where a, b, c are positive integers. Let P = (p, 0) be the point on line segment OC that minimizes the distance AP + PB, over all

choices of P. Let X = AP + PB.

(a) Show that this minimum distance is  

(b) If c = 12, find all pairs (a, b) for which a, b, p, and X are positive integers.

(c) If a, b, p, X are all positive integers, prove that there exists an n ≥ 3 that divides both a and b.

C4.Two lines intersect at a point Q at an angle θ◦, where 0 < θ < 180. A frog is originally at a point other than Q on the angle bisector of this angle. The frog alternately jumps over these two lines, where a jump over a line results in the frog landing at a point which is the reflection across the line of the frog’s jumping point.

The frog stops when it lands on one of the two lines.

(a) Suppose θ = 90°. Show that the frog never stops.

(b) Suppose θ = 72°. Show that the frog eventually stops.

(c) Determine the number of integer values of θ, with 0 < θ° < 180°, for which the frog never stops.