当前位置:首页 >> 学科竞赛 >>

1998滑铁卢竞赛试题


www.linstitute.net

Canadian Mathematics Competition
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

IBM Canada Ltd.

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

Calculators are permitted, provided they are non-programmable and without graphic displays. Do not open this booklet until instructed to do so. The paper consists of 10 questions, each worth 10 marks. Parts of each question can be of two types. SHORT ANSWER parts are worth 2 marks each (questions 1-2) or 3 marks each (questions 3-7). FULL SOLUTION parts are worth the remainder of the 10 marks for the question. Instructions for SHORT ANSWER parts 1. SHORT ANSWER parts are indicated like this: 2. Enter the answer in the appropriate box in the answer booklet. For these questions, full marks will be given for a correct answer which is placed in the box. Part marks will be awarded only if relevant work is shown in the space provided in the answer booklet. Instructions for FULL SOLUTION parts 1. FULL SOLUTION parts are indicated like this: 2. Finished solutions must be written in the appropriate location in the answer booklet. Rough work should be done separately. If you require extra pages for your finished solutions, foolscap will be supplied by your supervising teacher. Insert these pages into your answer booklet. 3. Marks are awarded for completeness, clarity, and style of presentation. A correct solution poorly presented will not earn full marks. NOTE: At the completion of the contest, insert the information sheet inside the answer booklet.

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 .

www.linstitute.net

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.


相关文章:
1998年全国高中学生化学竞赛(初赛)试题
1998 年全国高中学生化学竞赛(初赛)试题评析 1.某芳香烃 A,分子式 C9H12。在光照下用 Br2 溴化 A 得到两种一溴衍生物(B1 和 B2) ,产率约为 1 ︰1。在...
1998年全国高中学生化学竞赛(决赛)理论试题答案
1998 年全国高中学生化学竞赛(决赛)标准答案与评分标准 共 6 页 5 第五题 (评判分: 10 分) 1. n (Cu2+) : n (BCPF) = 1 : 1 原因: 曲线 1 ...
1998年全国初中数学竞赛试卷(含答案)
1998年全国初中数学竞赛试卷(含答案)_初三数学_数学_初中教育_教育专区。编辑了1998年全国初中数学竞赛试题及解析 1998 年全国初中数学竞赛试卷一、选择题: (每小...
1998年第十二届全国高中学生化学竞赛(初赛)试题及答案
1998 年全国高中学生化学竞赛初赛试题及答案 共 6 页 1 1998 年全国高中学生化学竞赛(初赛)试题题号 满分 得分 1.某芳香烃 A,分子式 C9H12。在光照下用 ...
1998-2012年全国初中数学竞赛试题及答案
1998-2012年全国初中数学竞赛试题及答案_学科竞赛_初中教育_教育专区。2011 年全国初中数学竞赛试题及答案一、选择题(共 5 小题,每小题 7 分,共 35 分. 每...
1998年江苏省中学生化学竞赛试题
www.jsccn.com 1998年江苏省中学生化学竞赛试题原子量:Fe:56 O:16 N:14 Na:23 H: 1 K: 39 Al:27 P:31 Cl:35.5 S:32 Ag:108 Mn:55 Cu:64 Ba...
1998年全国初中数学竞赛试题及解答
1998 年全国初中数学竞赛试题及解答 一、选择题(本题共 5 小题,每小题 6 分,满分 30 分) 选择题 1.已知 a,b,c 都是实数,并且 a>b>c,那么下列式子中...
1998-1999学年度天津市初二数学竞赛决赛试题
1998-1999 学年度天津市初二数学竞赛决赛试题一.如图,立方体的每一个面上都有一个自然数,已知相对的两个面上二数之和相等。如果 2 2 2 13,9,3 的对面的数...
1998-1999学年度天津市初二数学竞赛预赛试题
珍藏已久的历年数学竞赛试题,已经很那找到的。现在拿出来和大家一起分享一下。隐藏>> 1998-1999 学年度天津市初二数学竞赛预赛试题一、计算: ( ? 1 ) ? 7 ...
全国初中数学竞赛_1998~2012_试题集锦(附解答)
1998 年全国初中数学竞赛试卷 一、选择题: (每小题 6 分,共 30 分) 1、已知 a、b、c 都是实数,并且 a ? b ? c ,那么下列式子中正确的是( (A) ab...
更多相关标签: