Euclid 2014年真题

(a) What is the value of ?

(b) In the diagram, the angles of ∆ABC are

shown in terms of x. What is the value of x?

(c) Lisa earns two times as much per hour as Bart. Lisa works 6 hours and Bart works 4 hours. They earn $200 in total. How much does Lisa earn per hour?

(a) The semi-circular region shown has

radius 10. What is the perimeter of the region?

(b) The parabola with equation y = 10(x + 2)(x - 5) intersects the x-axis at points P and Q. What is the length of line segment

P Q?

(c) The line with equation y = 2x intersects the line segment joining C(0, 60) and D(30, 0) at the point E. Determine the coordinates of E.

(a) Jimmy is baking two large identical

triangular cookies, ∆ABC and ∆DEF.

Each cookie is in the shape of an isosceles

right-angled triangle. The length of the

shorter sides of each of these triangles is 20 cm. He puts the cookies on a rectangular baking tray so that A, B, D, and E are at the vertices of the rectangle, as shown. If the distance between parallel sides AC and DF is 4 cm, what is the width BD of the tray?

(b) Determine all values of x for which .

(a) Determine the number of positive divisors of 900, including 1 and 900, that are perfect squares. (A positive divisor of 900 is a positive integer that divides exactly into 900.)

(b) Points A(k, 3), B(3, 1) and C(6, k) form an isosceles triangle. If ∠ABC = ∠ACB, determine all possible values of k.

(a) A chemist has three bottles, each containing a mixture of acid and water:

• bottle A contains 40 g of which 10% is acid,

• bottle B contains 50 g of which 20% is acid, and

• bottle C contains 50 g of which 30% is acid.

She uses some of the mixture from each of the bottles to create a mixture with mass 60 g of which 25% is acid. Then she mixes the remaining contents of the bottles to create a new mixture. What percentage of the new mixture is acid?

(b) Suppose that x and y are real numbers with 3x + 4y = 10. Determine the minimum possible value of x2 + 16y2.

(a) A bag contains 40 balls, each of which is black or gold. Feridun reaches into the bag and randomly removes two balls. Each ball in the bag is equally likely to be removed. If the probability that two gold balls are removed is , how many of

the 40 balls are gold?

(b) The geometric sequence with n terms t1, t2, . . . , tn-1, tn has t1tn = 3. Also, the product of all n terms equals 59 049 (that is, t1t2 · · ·tn-1tn = 59 049). Determine the value of n.

(A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a constant. For example, 3, 6, 12 is a geometric sequence with three terms.)

(a) If , what is the value of x + y?

(b) Determine all real numbers x for which

(a) In the diagram, ∠ACB = ∠ADE = 90◦ . If AB = 75, BC = 21, AD = 20, and CE = 47, determine the exact length of BD.

(b) In the diagram, C lies on BD. Also, ∆ABC and ∆ECD are equilateral triangles. If M is the midpoint of BE and N is the midpoint of AD, prove that ∆MNC is equilateral.

(a) Without using a calculator, determine positive integers m and n for which

(The sum on the left side of the equation consists of 89 terms of the form sin6 x°, where x takes each positive integer value from 1 to 89.)

(b) Let f(n) be the number of positive integers that have exactly n digits and whose digits have a sum of 5. Determine, with proof, how many of the 2014 integers f(1), f(2), . . . , f(2014) have a units digit of 1.

10.

Fiona plays a game with jelly beans on the number line. Initially, she has N jelly beans, all at position 0. On each turn, she must choose one of the following moves:

• Type 1: She removes two jelly beans from position 0, eats one, and puts the other at position 1.

• Type i, where i is an integer with i ≥ 2: She removes one jelly bean from position i - 2 and one jelly bean from position i - 1, eats one, and puts the other at position i.

The positions of the jelly beans when no more moves are possible is called the final state. Once a final state is reached, Fiona is said to have won the game if there are at most three jelly beans remaining, each at a distinct position and no two at

consecutive integer positions. For example, if N = 7, Fiona wins the game with the sequence of moves

Type 1, Type 1, Type 2, Type 1, Type 3

which leaves jelly beans at positions 1 and 3. A different sequence of moves starting with N = 7 might not win the game.

(a) Determine an integer N for which it is possible to win the game with one jelly bean left at position 5 and no jelly beans left at any other position.

(b) Suppose that Fiona starts the game with a fixed unknown positive integer N.

Prove that if Fiona can win the game, then there is only one possible final state.