# 1998滑铁卢竞赛试题

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An activity of The Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario

Euclid Contest (Grade 12)
for the Awards

Tuesday, April 21, 1998
C.M.C. Sponsors: C.M.C. Supporters: C.M.C. Contributors:
The Great-West Life Assurance Company

Northern Telecom (Nortel) Manulife Financial

Canadian Institute of Actuaries

Equitable Life of Canada

Sybase Inc. (Waterloo)

Chartered Accountants

Time: 2 1 hours
2

? 1998 Waterloo Mathematics Foundation

www.linstitute.net NOTE:

1. 2. 3.

Please read the instructions on the front cover of this booklet. Place all answers in the answer booklet provided. For questions marked “ ”, full marks will be given for a correct answer which is placed

4.

in the box in the answer booklet. Part marks will be awarded only if relevant work is shown in the space provided in the answer booklet. It is expected that all calculations and answers will be expressed as exact numbers such as 4π, 2 + 7 , etc., except where otherwise indicated.

1.

(a) (b) (c)

If one root of x 2 + 2 x – c = 0 is x = 1, what is the value of c? If 2 2 x – 4 = 8 , what is the value of x? Two perpendicular lines with x-intercepts – 2 and 8 intersect at (0, b) . Determine all values of b. The vertex of y = ( x – 1)2 + b has coordinates (1, 3) . What is the y-intercept of this parabola? What is the area of ? ABC with vertices A( – 3, 1) , B(5, 1) and C(8, 7) ? In the diagram, the line y = x + 1 intersects the parabola y = x – 3 x – 4 at the points P and Q. Determine the coordinates of P and Q.
2

2.

(a)

(b) (c)

y

Q

P

O

x

3.

(a) (b)

The graph of y = m x passes through the points (2, 5) and (5, n) . What is the value of mn ? Jane bought 100 shares of stock at \$10.00 per share. When the shares increased to a value of \$N each, she made a charitable donation of all the shares to the Euclid Foundation. She received a tax refund of 60% on the total value of her donation. However, she had to pay a tax of 20% on the increase in the value of the stock. Determine the value of N if the difference between her tax refund and the tax paid was \$1000. n – 3? Consider the sequence t1 = 1, t2 = –1 and tn = ? t where n ≥ 3. What is the value ? n – 1? n – 2 of t1998 ?

4.

(a)

(b)

The nth term of an arithmetic sequence is given by tn = 555 – 7n . If Sn = t1 + t2 + ... + tn , determine the smallest value of n for which Sn < 0 .

www.linstitute.net 5.

(a)

A square OABC is drawn with vertices as shown. Find the equation of the circle with largest area that can be drawn inside the square. C(– 2, 2)

y B(0, 4) A(2, 2) x

O(0, 0)

(b)

In the diagram, DC is a diameter of the larger circle centred at A, and AC is a diameter of the smaller circle centred at B. If DE is tangent to the smaller circle at F, and DC = 12, determine the length of DE .

F D

E

A

B

C

6.

(a)

In the grid, each small equilateral triangle has side length 1. If the vertices of ?WAT are themselves vertices of small equilateral triangles, what is the area of ?WAT ?

W

A T

(b)

In ? ABC , M is a point on BC such that BM = 5 and MC = 6 . If AM = 3 and AB = 7, determine the exact value of AC . B 5

A 7 M y 2
....

3 6 C

7.

(a)

The function f ( x ) has period 4. The graph of one period of y = f ( x ) is shown in the diagram. Sketch the graph of y=
1 2

[ f ( x – 1) + f ( x + 3)] , for – 2 ≤ x ≤ 2 .
–3

–2

0

–1 –2

1

2

3

x

(b)

If x and y are real numbers, determine all solutions ( x, y) of the system of equations

.

.... .

x 2 – xy + 8 = 0 x 2 – 8x + y = 0 .

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8.

(a)

In the graph, the parabola y = x 2 has been translated to the position shown. Prove that de = f .

y

( – d , 0) (0, – f )
(b) In quadrilateral KWAD , the midpoints of KW and AD are M and N respectively. If MN = ( AW + DK ) , prove that WA is
1 2

(e, 0)

x

y W M A N D x

parallel to KD .

K

9.

Consider the first 2 n natural numbers. Pair off the numbers, as shown, and multiply the two members of each pair. Prove that there is no value of n for which two of the n products are equal. 1 2 3 . . . (n – 2) (n – 1) n (n + 1) (n + 2) (n + 3) . . . (2n – 1) 2n
...

10.

The equations x 2 + 5 x + 6 = 0 and x 2 + 5 x – 6 = 0 each have integer solutions whereas only one of the equations in the pair x 2 + 4 x + 5 = 0 and x 2 + 4 x – 5 = 0 has integer solutions. (a) Show that if x 2 + px + q = 0 and x 2 + px – q = 0 both have integer solutions, then it is possible to find integers a and b such that p2 = a 2 + b 2 . (i.e. ( a, b, p) is a Pythagorean triple). (b) Determine q in terms of a and b.