Euclid 2017年真题

1. (a) There is one pair (a, b) of positive integers for which 5a + 3b = 19. What are the values of a and b?

(b) How many positive integers n satisfy 5 < 2n < 2017?

(c) Jimmy bought 600 Euros at the rate of 1 Euro equals $1.50. He then converted his 600 Euros back into dollars at the rate of $1.00 equals 0.75 Euros. How many fewer dollars did Jimmy have after these two transactions than he had before these two transactions?

2. (a) What are all values of x for which x ≠ 0 and x ≠ 1 and

(b) In a magic square, the numbers in each

row, the numbers in each column, and the

numbers on each diagonal have the same

sum. In the magic square shown, what are

the values of a, b and c?

(ii) Determine one pair (a, b) of positive integers for which a > 1 and b > 1 and ab = 9559.

3. (a) In the diagram, △ABC is right-angled

at B and △ACD is right-angled at A. Also,

AB = 3, BC = 4, and CD = 13. What is the

area of quadrilateral ABCD?

(b) Three identical rectangles PQRS, WTUV

and XW V Y are arranged, as shown, so that

RS lies along T X. The perimeter of each of

the three rectangles is 21 cm. What is the

perimeter of the whole shape?

(c) One of the faces of a rectangular prism has area 27 cm2 . Another face has area 32 cm2 . If the volume of the prism is 144 cm3 , determine the surface area of the prism in cm2.

4. (a) The equations y = a(x - 2)(x + 4) and y = 2(x - h)2 + k represent the same parabola. What are the values of

a, h and k ?

(b) In an arithmetic sequence with 5 terms, the sum of the squares of the first 3 terms equals the sum of the squares of the last 2 terms. If the first term is 5, determine all possible values of the fifth term.

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9, 11 is an arithmetic sequence with five terms.)

5. (a) Dan was born in a year between 1300 and 1400. Steve was born in a year between 1400 and 1500. Each was born on April 6 in a year that is a perfect square. Each lived for 110 years. In what year while they were both alive were their ages both perfect squares on April 7?

(b) Determine all values of k for which the points A(1, 2), B(11, 2) and C(k, 6) form a right-angled triangle.

6. (a) The diagram shows two hills that

meet at O. One hill makes a 30° angle

with the horizontal and the other hill

makes a 45° angle with the horizontal.

Points A and B are on the hills so that OA = OB = 20 m. Vertical poles BD and AC are connected by a straight cable CD. If AC = 6 m, what is the length of BD for which CD is as short as possible?

(b) If cos θ = tan θ, determine all possible values of sin θ, giving your answer(s) as simplified exact numbers.

7. (a) Linh is driving at 60 km/h on a long straight highway parallel to a train track. Every 10 minutes, she is passed by a train travelling in the same direction as she is. These trains depart from the station behind her every 3 minutes and all travel at the same constant speed. What is the constant speed of the trains, in km/h?

(b) Determine all pairs (a, b) of real numbers that satisfy the following system of equations:

Give your answer(s) as pairs of simplified exact numbers.

8. (a) In the diagram, line segments AC

and DF are tangent to the circle at B and E,

respectively. Also, AF intersects the circle

at P and R, and intersects BE at Q, as

shown. If ∠CAF = 35°, ∠DF A = 30° , and

∠F P E = 25° , determine the measure of

∠P EQ.

(b) In the diagram, ABCD and P NCD are

squares of side length 2, and P NCD is

perpendicular to ABCD. Point M is chosen

on the same side of P NCD as AB so

that △PMN is parallel to ABCD, so that

∠PMN = 90° , and so that PM = MN.

Determine the volume of the convex solid ABCDPMN.

9. A permutation of a list of numbers is an ordered arrangement of the numbers in that list. For example, 3, 2, 4, 1, 6, 5 is a permutation of 1, 2, 3, 4, 5, 6. We can write this permutation as a1, a2, a3, a4, a5, a6, where a1 = 3, a2 = 2, a3 = 4, a4 = 1, a5 = 6,

and a6 = 5.

(a) Determine the average value of

|a1 - a2| + |a3 - a4|

over all permutations a1, a2, a3, a4 of 1, 2, 3, 4.

(b) Determine the average value of

a1 - a2 + a3 - a4 + a5 - a6 + a7

over all permutations a1, a2, a3, a4, a5, a6, a7 of 1, 2, 3, 4, 5, 6, 7.

|a1 - a2| + |a3 - a4| + · · · + |a197 - a198| + |a199 - a200| (∗)

over all permutations a1, a2, a3, ..., a199, a200 of 1, 2, 3, 4, . . . , 199, 200. (The sum labelled (∗) contains 100 terms of the form |a2k-1 - a2k|.)

10. Consider a set S that contains m ≥ 4 elements, each of which is a positive integer and no two of which are equal. We call S boring if it contains four distinct integers a, b, c, d such that a + b = c + d. We call S exciting if it is not boring. For example, {2, 4, 6, 8, 10} is boring since 4 + 8 = 2 + 10. Also, {1, 5, 10, 25, 50} is exciting.

(a) Find an exciting subset of {1, 2, 3, 4, 5, 6, 7, 8} that contains exactly 5 elements.

(b) Prove that, if S is an exciting set of m ≥ 4 positive integers, then S contains an integer greater than or equal to

(c) Define rem(a, b) to be the remainder when the positive integer a is divided by the positive integer b. For example, rem(10, 7) = 3, rem(20, 5) = 0, and rem(3, 4) = 3. Let n be a positive integer with n ≥ 10. For each positive integer k with 1 ≤ k ≤ n, define xk = 2n · rem(k2 , n) + k. Determine, with proof, all positive integers n ≥ 10 for which the set {x1, x2, . . . , xn - 1, xn} of n integers is exciting.