Euclid 2018年真题

1. (a) If x = 11, what is the value of x + (x + 1) + (x + 2) + (x + 3)?

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

(c) The total cost of one chocolate bar and two identical packs of gum is $4.15. One chocolate bar costs $1.00 more than one pack of gum.

Determine the cost of one chocolate bar.

2.

(a) A five-digit integer is made using each of the digits 1, 3, 5, 7, 9. The integer is greater than 80 000 and less than 92 000. The units (ones) digit is 3. The hundreds and tens digits, in that order, form a two-digit integer that is divisible by 5. What is the five-digit integer?

(b) In the diagram, point D is on AC so that

BD is perpendicular to AC. Also, AB = 13,

BC = 12 and BD = 12. What is the

length of AC?

(c) In the diagram, square OABC has side

length 6. The line with equation y = 2x

intersects CB at D. Determine the area of

the shaded region.

3. (a) What is the value of ?

(b) There is exactly one pair (x, y) of positive integers for which . What is this pair (x, y)?

(c) The line with equation y = mx + 2 intersects the parabola with equation y = ax2 + 5x - 2 at the points P(1, 5) and Q. Determine

(i) the value of m,

(ii) the value of a, and

(iii) the coordinates of Q.

4. a) The positive integers 34 and 80 have exactly two positive common divisors, namely 1 and 2. How many positive integers n with 1 ≤ n ≤ 30 have the property that n and 80 have exactly two positive common divisors?

(b) A function f is defined so that

• f(1) = 1,

• if n is an even positive integer, then , and

• if n is an odd positive integer with n > 1, then f(n) = f(n - 1) + 1.

For example, f(34) = f(17) and f(17) = f(16) + 1.

Determine the value of f(50).

5. (a) The perimeter of equilateral △PQR is 12. The perimeter of regular hexagon ST UV W X is also 12. What is the ratio of the area of △PQR to the area of ST UV W X?

(b) In the diagram, sector AOB is  of an

entire circle with radius AO = BO = 18.

The sector is cut into two regions with a

single straight cut through A and point P on

OB. The areas of the two regions are equal.

Determine the length of OP.

6. (a) For how many integers k with 0 < k < 18 is？

(b) In the diagram, a straight, flat road joins A to B.

Karuna runs from A to B, turns around instantly, and runs back to A. Karuna runs at 6 m/s. Starting at the same time as Karuna, Jorge runs from B to A, turns around instantly, and runs back to B. Jorge runs from B to A at 5 m/s and from A to B at 7.5 m/s. The distance from A to B is 297 m and each runner takes exactly 99 s to run their route. Determine the two values of t for which Karuna and Jorge are at the same place on the road after running for t seconds.

7. (a) Eight people, including triplets Barry, Carrie and Mary, are going for a trip in four canoes. Each canoe seats two people. The eight people are to be randomly assigned to the four canoes in pairs. What is the probability that no two of Barry,

Carrie and Mary will be in the same canoe?

(b) Diagonal WY of square WXYZ has slope 2. Determine the sum of the slopes of WX and XY .

8. (a) Determine all values of x such that .

(b) In the diagram, rectangle P QRS is placed inside rectangle ABCD in two different ways: first, with Q at B and R at C; second, with P on AB, Q on BC, R on CD, and S on DA.

If AB = 718 and P Q = 250, determine the length of BC.

9. An L-shaped triomino is composed of three unit squares, as shown:

Suppose that H and W are positive integers. An H ×W rectangle can be tiled if the rectangle can be completely covered with non-overlapping copies of this triomino

(each of which can be rotated and/or translated) and the sum of the areas of these non-overlapping triominos equals the area of the rectangle (that is, no triomino is partly outside the rectangle). If such a rectangle can be tiled, a tiling is a specific configuration of triominos that tile the rectangle.

(a) Draw a tiling of a 3 × 8 rectangle.

(b) Determine, with justification, all integers W for which a 6 × W rectangle can be tiled.

(c) Determine, with justification, all pairs (H, W) of integers with H ≥ 4 and W ≥ 4 for which an H × W rectangle can be tiled.

10. In an infinite array with two rows, the numbers in the top row are denoted . . . , A-2, A-1, A0, A1, A2, . . . and the numbers in the bottom row are denoted . . . , B-2, B-1, B0, B1, B2, . . .. For each integer k, the entry Ak is directly above the entry Bk in the array, as shown:

For each integer k, Ak is the average of the entry to its left, the entry to its right, and the entry below it; similarly, each entry Bk is the average of the entry to its left, the entry to its right, and the entry above it.

(a) In one such array, A0 = A1 = A2 = 0 and A3 = 1.

Determine the value of A4.

The maximum mark on this part is 2 marks.

(b) In another such array, we define Sk = Ak + Bk for each integer k.

Prove that Sk+1 = 2Sk - Sk-1 for each integer k.

The maximum mark on this part is 2 marks.

(c) Consider the following two statements about a third such array:

(P) If each entry is a positive integer, then all of the entries in the array are equal.

(Q) If each entry is a positive real number, then all of the entries in the array are equal.

Prove statement (Q).

The maximum mark on this part is 6 marks.

A complete proof of statement (Q) will earn the maximum of 6 marks for part (c), regardless of whether any attempt to prove (P) is made.

A complete proof of statement (P) will earn 2 of the 6 possible marks for part (c).

In such a case, any further progress towards proving (Q) would be assessed for partial marks towards the remaining 4 marks.

Students who do not fully prove either (P) or (Q) will have their work assessed for partial marks.