Euclid 2021年真题

2. (a) What is the sum of the digits of the integer equal to (103 + 1)2?

(b) A bakery sells small and large cookies. Before a price increase, the price of each small cookie is $1.50 and the price of each large cookie is $2.00. The price of each small cookie is increased by 10% and the price of each large cookie is increased by 5%. What is the percentage increase in the total cost of a purchase of 2 small cookies and 1 large cookie?

(c) Qing is twice as old as Rayna. Qing is 4 years younger than Paolo. The average age of Paolo, Qing and Rayna is 13. Determine their ages.

3. (a) In the diagram, PQRS is a quadrilateral. What is its perimeter?

(b) In the diagram, A has coordinates (0, 8). Also, the midpoint of AB is M(3, 9) and the midpoint of BC is N(7, 6). What is the slope of AC?

(c) The parabola with equation y = -2x2 + 4x + c has vertex V(1, 18). The parabola intersects the y-axis at D and the x-axis at E and F. Determine the area of △DEF.

4. (a) If 3(8x) + 5(8x) = 261, what is the value of the real number x?

(b) For some real numbers m and n, the list 3n2, m2, 2(n + 1)2 consists of three consecutive integers written in increasing order. Determine all possible values of m.

5. (a) Chinara starts with the point (3, 5), and applies the following three-step process, which we call P:

Step 1: Reflect the point in the x-axis.

Step 2: Translate the resulting point 2 units upwards.

Step 3: Reflect the resulting point in the y-axis.

As she does this, the point (3, 5) moves to(3, -5) then to (3, -3), and then to (-3, -3).

Chinara then starts with a different point S0. She applies the three-step process P to the point S0 and obtains the point S1. She then applies P to S1 to obtain the point S2. She applies P four more times, each time using the previous output of P to be the new input, and eventually obtains the point S6(-7, -1). What are the coordinates of the point S0?

(b) In the diagram, ABDE is a rectangle, ∆BCD is equilateral, and AD is parallel to BC. Also, AE = 2x for some real number x.

(i) Determine the length of AB in terms of x.

(ii) Determine positive integers r and s for which .

6. (a) Suppose that n > 5 and that the numbers t1, t2, t3, . . . , tn-2, tn-1, tn form an arithmetic sequence with n terms. If t3 = 5, tn-2 = 95, and the sum of all n terms is 1000, what is the value of n?

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant, called the common difference. For example, 3, 5, 7, 9 are the first four terms of an arithmetic sequence.)

(b) Suppose that a and r are real numbers. A geometric sequence with first term a and common ratio r has 4 terms. The sum of this geometric sequence is . A second geometric sequence has the same first term a and the same common ratio r, but has 8 terms. The sum of this second geometric sequence is . Determine all possible values for a.

(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, called the common ratio. For example, 3, -6, 12, -24 are the first four terms of a geometric sequence.)

7. (a) A bag contains 3 green balls, 4 red balls,and no other balls. Victor removes balls randomly from the bag, one at a time, and places them on a table. Each ball in the bag is equally likely to be chosen each time that he removes a ball. He stops removing balls when there are two balls of the same colour on the table. What is the probability that, when he stops, there is at least 1 red ball and at least 1 green ball on the table?

(b) Suppose that f(a) = 2a2 - 3a + 1 for all real numbers a and  for all b > 0. Determine all θ with 0 ≤ θ ≤ 2π for which f(g(sin θ)) = 0.

8. (a) Five distinct integers are to be chosen from the set {1, 2, 3, 4, 5, 6, 7, 8} and placed in some order in the top row of boxes in the diagram. Each box that is not in the top row then contains the product of the integers in the two boxes connected to it in the row directly above. Determine the number of ways in which the integers can be chosen and placed in the top row so that the integer in the bottom box is 9 953 280 000.

(b) Prove that the integer  is a perfect square. (In this fraction, the numerator is the product of the factorials of the integers from 1 to 400, inclusive.)

9. (a) Suppose that a = 5 and b = 4. Determine all pairs of integers (K, L) for which K2 + 3L2 = a2 + b2 - ab.

(b) Prove that, for all integers K and L, there is at least one pair of integers (a, b) for which K2 + 3L2 = a2 + b2 - ab.

(c) Prove that, for all integers a and b, there is at least one pair of integers (K, L) for which K2 + 3L2 = a2 + b2 - ab.

10. (a) In the diagram, eleven circles of four different sizes are drawn. Each circle labelled W has radius 1, each circle labelled X has radius 2, the circle labelled Y has radius 4, and the circle labelled Z has radius r. Each of the circles labelled W or X is tangent to three other circles. The circle labelled Y is tangent to all ten of the other circles. The circle labelled Z is tangent to three other circles. Determine positive integers s and t for which .

(b) Suppose that c is a positive integer. Define f(c) to be the number of pairs (a, b) of positive integers with c < a < b for which two circles of radius a, two circles of radius b, and one circle of radius c can be drawn so that 

l                      each circle of radius a is tangent to both circles of radius b and to the circle of radius c, and

l                      each circle of radius b is tangent to both circles of radius a and to the circle of radius c,

as shown. Determine all positive integers c for which f(c) is even.