Euclid 2012年真题

(a) John buys 10 bags of apples, each of which contains 20 apples. If he eats 8 apples a day, how many days will it take him to eat the 10 bags of apples?

(b) Determine the value of

sin(0°) + sin(60°) + sin(120°) + sin(180°) + sin(240°) + sin(300°) + sin(360°)

(c) A set of integers has a sum of 420, and an average of 60. If one of the integers in the set is 120, what is average of the remaining integers in the set?

(a) If ax + ay = 4 and x + y = 12, what is the value of a?

(b) If the lines with equations 4x + 6y = 5 and 6x + ky = 3 are parallel, what is the value of k?

x + y = 0

x2 - y = 2

(a) A 200 g solution consists of water and salt. 25% of the total mass of the solution is salt. How many grams of water need to be added in order to change the solution so that it is 10% salt by mass?

(b) The correct formula for converting a Celsius temperature (C) to a Fahrenheit temperature (F) is given by .

To approximate the Fahrenheit temperature, Gordie doubles C and then adds 30 to get f.

If f < F, the error in the approximation is F - f; otherwise, the error in the approximation is f - F. (For example, if F = 68 and

f = 70, the error in the approximation is f - F = 2.)

Determine the largest possible error in the approximation that Gordie would make when converting Celsius temperatures C with -20 ≤ C ≤ 35.

(a) The horizontal line y = k intersects the parabola with equation y = 2(x-3)(x-5) at points A and B. If the length of line segment AB is 6, what is the value of k?

(b) Determine three pairs (a, b) of positive integers for which

(3a + 6a + 9a + 12a + 15a) + (6b + 12b + 18b + 24b + 30b)

is a perfect square.

(a) Triangle ABC has vertices A(0, 5), B(3, 0) and C(8, 3). Determine the measure of ∠ACB.

(b) In the diagram, P QRS is an isosceles

trapezoid with P Q = 7, P S = QR = 8,

and SR = 15. Determine the length of the

diagonal P R.

(a) Blaise and Pierre will play 6 games of squash. Since they are equally skilled, each is equally likely to win any given game. (In squash, there are no ties.) The probability that each of them will win 3 of the 6 games is . What is the probability that Blaise will win more games than Pierre?

(b) Determine all real values of x for which

3x+2 + 2x+2 + 2x = 2x+5 + 3x

(a) In the diagram, ∆ABC has AB = AC and

∠BAC < 60° . Point D is on AC with BC = BD.

Point E is on AB with BE = ED. If ∠BAC = θ,

determine ∠BED in terms of θ.

(b) In the diagram, the ferris wheel has a

diameter of 18 m and rotates at a constant

rate. When Kolapo rides the ferris wheel and

is at its lowest point, he is 1 m above the

ground. When Kolapo is at point P that is

16 m above the ground and is rising, it takes

him 4 seconds to reach the highest point, T.

He continues to travel for another 8 seconds

reaching point Q. Determine Kolapo’s height

above the ground when he reaches point Q.

(a) On Saturday, Jimmy started painting his toy

helicopter between 9:00 a.m. and 10:00 a.m.

When he finished between 10:00 a.m. and

11:00 a.m. on the same morning, the hour

hand was exactly where the minute hand

had been when he started, and the minute

hand was exactly where the hour hand had

been when he started. Jimmy spent t hours

painting. Determine the value of t.

(b) Determine all real values of x such that

log5x+9(x2 + 6x + 9) + log x+3 (5x2 + 24x + 27) = 4

(a) An auditorium has a rectangular array of chairs. There are exactly 14 boys seated in each row and exactly 10 girls seated in each column. If exactly 3 chairs are empty, prove that there are at least 567 chairs in the auditorium.

(b) In the diagram, quadrilateral ABCD

has points M and N on AB and DC,

respectively, with . Line

segments AN and DM intersect at P,

while BN and CM intersect at Q.

Prove that the area of quadrilateral PMQN equals the sum of the areas of ∆APD and ∆BQC.

10.

For each positive integer N, an Eden sequence from {1, 2, 3, . . . , N} is defined to be a sequence that satisfies the following conditions:

(i) each of its terms is an element of the set of consecutive integers {1, 2, 3, . . . , N},

(ii) the sequence is increasing, and

(iii) the terms in odd numbered positions are odd and the terms in even numbered positions are even.

For example, the four Eden sequences from {1, 2, 3} are

1 3 1, 2 1, 2, 3

(a) Determine the number of Eden sequences from {1, 2, 3, 4, 5}.

(b) For each positive integer N, define e(N) to be the number of Eden sequences from {1, 2, 3, . . . , N}. If e(17) = 4180 and e(20) = 17710, determine e(18) and e(19).