Euclid 2013年真题

(a) What is the smallest positive integer x for which is an integer?

(b) The average of 3 and 11 is a. The average of a and b is 11. What is the value of b?

(c) Charlie is 30 years older than his daughter Bella. Charlie is also six times as old as Bella. Determine Charlie’s age.

(a) If with x ≠ 0 and y ≠ 0, what is the value of ?

(b) For which positive integer n are both and true?

so that AH is perpendicular to BC. Also,

AB = 10, AH = 8, and the area of ∆ABC

is 84. Determine the perimeter of ∆ABC.

(a) In the Fibonacci sequence, 1, 1, 2, 3, 5, . . ., each term after the second is the sum of the previous two terms. How many of the first 100 terms of the Fibonacci sequence are odd?

(b) In an arithmetic sequence, the sum of the first and third terms is 6 and the sum of the second and fourth terms is 20. Determine the tenth term in the sequence.

(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 are the first four terms of an arithmetic sequence.)

(a) How many positive integers less than 1000 have only odd digits?

(b) Determine all ordered pairs (a, b) that satisfy the following system of equations.

(a) Tanner has two identical dice. Each die has six faces which are numbered 2, 3, 5, 7, 11, 13. When Tanner rolls the two dice, what is the probability that the sum of the numbers on the top faces is a prime number?

(b) In the diagram, V is the vertex of

the parabola with equation y=-x2+4x+1.

Also, A and B are the points of

intersection of the parabola and

the line with equation y = -x + 1. Determine the value of AV 2 + BV2 - AB2.

(a) In the diagram, ABC is a quarter of a circular

pizza with centre A and radius 20 cm. The piece

of pizza is placed on a circular pan with A, B and

C touching the circumference of the pan, as shown.

What fraction of the pan is covered by the piece of pizza?

(b) The deck AB of a sailboat is 8 m long. Rope

extends at an angle of 60° from A to the top (M)

of the mast of the boat. More rope extends at an

angle of θ from B to a point P that is 2 m below M,

as shown. Determine the height MF of the mast,

in terms of θ.

(a) If , what is the numerical value of sin x?

(b) Determine all linear functions f(x) = ax + b such that if g(x) = f -1(x) for all values of x, then f(x) - g(x) = 44 for all values of x. (Note: f-1 is the inverse function of f.)

(a) Determine all pairs (a, b) of positive integers for which a3 + 2ab = 2013.

(b) Determine all real values of x for which log2 (2x-1 + 3x+1) = 2x - log2 (3x).

(a) Square W XY Z has side length 6 and is drawn,

as shown, completely inside a larger square

EFGH with side length 10, so that the squares

do not touch and so that WX is parallel to EF.

Prove that the sum of the areas of trapezoid

EFXW and trapezoid GHZY does not depend

on the position of WXY Z inside EFGH.

(b) A large square ABCD is drawn, with a second

smaller square PQRS completely inside it so

that the squares do not touch. Line segments

AP, BQ, CR, and DS are drawn, dividing

the region between the squares into four non

overlapping convex quadrilaterals, as shown. If

the sides of P QRS are not parallel to the sides of ABCD, prove that the sum of the areas of quadrilaterals APSD and BCRQ equals the sum of the areas of quadrilaterals ABQP and

CDSR. (Note: A convex quadrilateral is a quadrilateral in which the measure of each of the four interior angles is less than 180◦ .)

10.

A multiplicative partition of a positive integer n ≥ 2 is a way of writing n as a product of one or more integers, each greater than 1. Note that we consider a positive integer to be a multiplicative partition of itself. Also, the order of the factors in a partition does not matter; for example, 2 × 3 × 5 and 2 × 5 × 3 are considered to be the same partition of 30. For each positive integer n ≥ 2, define P(n) to be the number of multiplicative partitions of n. We also define P(1) = 1. Note that P(40) = 7, since the multiplicative partitions of 40 are 40, 2 × 20, 4 × 10,