Euclid 2019年真题

1. (a) Joyce has two identical jars. The first jar is full of water and contains 300 mL of water. The second jar is full of water. How much water, in mL, does the second jar contain?

(b) What integer a satisfies ?

2. (a) In the diagram, two small circles of

radius 1 are tangent to each other and to

a larger circle of radius 2. What is the area

of the shaded region?

(b) Kari jogs at a constant speed of 8 km/h. Mo jogs at a constant speed of 6 km/h. Kari and Mo jog from the same starting point to the same finishing point along a straight road. Mo starts at 10:00 a.m. Kari and Mo both finish at 11:00 a.m.

At what time did Kari start to jog?

The line with equation y = mx + b passes through the point (9,2). Determine the value of b.

3. (a) Michelle calculates the average of the following numbers:

5, 10, 15, 16, 24, 28, 33, 37

Daphne removes one number and calculates the average of the remaining numbers. The average that Daphne calculates is one less than the average that Michelle calculates. Which number does Daphne remove?

(b) If , what is the value of x?

4. (a) In the diagram, △ABC has a right angle at

B and point D lies on AB. If DB = 10, ∠ACD = 30°

and ∠CDB = 60° , what is the length of AD?

(b) The points A(d, -d) and B(-d + 12, 2d - 6) both

lie on a circle centered at the origin. Determine the

possible values of d.

5. (a) Determine the two pairs of positive integers (a, b) with a < b that satisfy the equation .

(b) Consider the system of equations:

Determine the number of integers k with k ≥ 0 for which there is at least one pair of integers (c, d) that is a solution to the system.

6. (a) A regular pentagon covers part of

another regular polygon, as shown. This

regular polygon has n sides, five of which

are completely or partially visible. In the

diagram, the sum of the measures of the

angles marked a° and b° is 88° . Determine

the value of n.

(The side lengths of a regular polygon are all equal, as are the measures of its interior angles.)

(b) In trapezoid ABCD, BC is parallel to AD

and BC is perpendicular to AB. Also, the

lengths of AD, AB and BC, in that order,

form a geometric sequence. Prove that AC is perpendicular to BD.

(A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant.)

7. (a) Determine all real numbers x for which 2 log2(x - 1) = 1 - log2(x + 2).

(b) Consider the function f(x) = x2 - 2x. Determine all real numbers x that satisfy the equation f(f(f(x))) = 3.

8. (a) A circle has centre O and radius 1.

Quadrilateral ABCD has all 4 sides tangent

to the circle at points P, Q, S, and T, as shown.

Also, ∠AOB = ∠BOC = ∠COD = ∠DOA. If

AO = 3, determine the length of DS.

(b) Suppose that x satisfies and .

Determine all possible values of sin 2x, expressing your answers in the form where a, b, c, d are integers.

9. For positive integers a and b, define .

For example, the value of f(1, 2) is 3.

(a) Determine the value of f(2, 5).

(b) Determine all positive integers a for which f(a, a) is an integer.

(d) Determine four pairs of positive integers (a, b), with 2 < a < b, for which f(a, b) is an integer.

10. (a) Amir and Brigitte play a card game. Amir starts with a hand of 6 cards: 2 red, 2 yellow and 2 green. Brigitte starts with a hand of 4 cards: 2 purple and 2 white.

Amir plays first. Amir and Brigitte alternate turns. On each turn, the current player chooses one of their own cards at random and places it on the table. The cards remain on the table for the rest of the game. A player wins and the game ends when they have placed two cards of the same colour on the table. Determine the probability that Amir wins the game.

(b) Carlos has 14 coins, numbered 1 to 14. Each coin has exactly one face called “heads”. When flipped, coins 1, 2, 3, . . . , 13, 14 land heads with probabilities h1, h2, h3, . . . , h13, h14, respectively. When Carlos flips each of the 14 coins exactly

once, the probability that an even number of coins land heads is exactly . Must there be a k between 1 and 14, inclusive, for which ? Prove your answer.

prints the digits 1, 2, 3, 4, 5, 6 with probability p1, p2, p3, p4, p5, p6, respectively.

Lis’s machine prints the digits 1, 2, 3, 4, 5, 6 with probability q1, q2, q3, q4, q5, q6, respectively. Each of the machines prints one digit. Let S(i) be the probability that the sum of the two digits printed is i. If S(2) = S(12) = S(7) and S(7) > 0, prove that p1 = p6 and q1 = q6.