Euclid 2015年真题

1. (a) What is value of ?

(b) If , what is the value of 3x + 8?

2. (a) If , what is the value of a?

(b) Two circles have the same centre. The

radius of the smaller circle is 1. The area

of the region between the circles is equal

to the area of the smaller circle. What is

the radius of the larger circle?

(c) There were 30 students in Dr. Brown’s class. The average mark of the students in the class was 80. After two students dropped the class, the average mark of the remaining students was 82. Determine the average mark of the two students who dropped the class.

3. (a) In the diagram, BD = 4 and point

C is the midpoint of BD. If point A is

placed so that △ABC is equilateral,

what is the length of AD?

(b) △MNP has vertices M(1, 4), N(5, 3), and P (5, c). Determine the sum of the two values of c for which the area of

△MNP is 14.

4. (a) What are the x-intercepts and the y-intercept of the graph with equation y = (x - 1)(x - 2)(x - 3) - (x - 2)(x - 3)(x - 4)?

(b) The graphs of the equations y = x3 - x2 + 3x - 4 and y = ax2 - x - 4 intersect at exactly two points. Determine all possible values of a.

5. (a) In the diagram, ∠CAB = 90◦ .

Point D is on AB and point E is on

AC so that AB = AC = DE, DB = 9,

and EC = 8. Determine the length of DE.

(b) Ellie has two lists, each consisting of 6 consecutive positive integers. The smallest integer in the first list is a, the smallest integer in the second list is b, and a < b. She makes a third list which consists of the 36 integers formed by multiplying

each number from the first list with each number from the second list. (This third list may include some repeated numbers.) If

• the integer 49 appears in the third list,

• there is no number in the third list that is a multiple of 64, and

• there is at least one number in the third list that is larger than 75, determine all possible pairs (a, b).

6. (a) A circular disc is divided into

36 sectors. A number is written in

each sector. When three consecutive

sectors contain a, b and c in that

order, then b = ac. If the number 2

is placed in one of the sectors and

the number 3 is placed in one of the adjacent sectors, as shown, what is the sum of the 36 numbers on the disc?

(b) Determine all values of x for which 0 < < 7 .

7. (a) In the diagram, ACDF is a rectangle

with AC = 200 and CD = 50. Also,

△FBD and △AEC are congruent

triangles which are right-angled at

B and E, respectively. What is the area

of the shaded region?

(b) The numbers a1, a2, a3, . . . form an arithmetic sequence with a1 ≠ a2. The three numbers a1, a2, a6 form a geometric sequence in that order. Determine all possible positive integers k for which the three numbers a1, a4, ak also form a geometric

sequence in that order.

(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 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. For example, 3, 6, 12 is a geometric sequence with three terms.)

8. (a) For some positive integers k, the parabola with equation intersects the circle with equation x2 +y2 = 25 at exactly three distinct points A, B and C.

Determine all such positive integers k for which the area of △ABC is an integer.

(b) In the diagram, △XYZ is isosceles

with XY = XZ = a and Y Z = b where

b < 2a. A larger circle of radius R is

inscribed in the triangle (that is, the circle

is drawn so that it touches all three sides

of the triangle). A smaller circle of radius r

is drawn so that it touches XY , XZ and the

larger circle. Determine an expression for

in terms of a and b.

9. Consider the following system of equations in which all logarithms have base 10:

(log x)(log y) ) - 3 log 5y - log 8x = a

(log y)(log z) ) - 4 log 5y - log 16z = b

(log z)(log x) ) - 4 log 8x - 3 log 625z = c

(a) If a = -4, b = 4, and c = -18, solve the system of equations.

(b) Determine all triples (a, b, c) of real numbers for which the system of equations has an infinite number of solutions (x, y, z).

10. For each positive integer n ≥ 1, let Cn be the set containing the n smallest positive integers; that is, Cn = {1, 2, . . . , n - 1, n}. For example, C4 = {1, 2, 3, 4}. We call a set, F, of subsets of Cn a Furoni family of Cn if no element of F is a subset of another element of F.

(a) Consider A = {{1, 2}, {1, 3}, {1, 4}}. Note that A is a Furoni family of C4.

Determine the two Furoni families of C4 that contain all of the elements of A and to which no other subsets of C4 can be added to form a new (larger) Furoni family.

(b) Suppose that n is a positive integer and that F is a Furoni family of Cn. For each non-negative integer k, define ak to be the number of elements of F that contain exactly k integers. Prove that

(The sum on the left side includes n + 1 terms.)

(Note: If n is a positive integer and k is an integer with 0 ≤ k ≤ n, then is the number of subsets of Cn that contain exactly k integers, where 0! = 1 and, if m is a positive integer, m! represents the product of the integers from 1 to m, inclusive.)

(c) For each positive integer n, determine, with proof, the number of elements in the largest Furoni family of Cn (that is, the number of elements in the Furoni family that contains the maximum possible number of subsets of Cn).