9512.net

甜梦文库

甜梦文库

当前位置：首页 >> 学科竞赛 >> # 2016年欧几里得数学竞赛真题

The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca

Euclid Contest

Tuesday, April 12, 2016

(in North America and South America)

Wednesday,

April 13, 2016

(outside of North America and South America)

Time:

21 2 hours

c 2016 University of Waterloo Do not open this booklet until instructed to do so.

Number of questions: 10

Each question is worth 10 marks

Calculators are allowed, with the following restriction: you may not use a device that has internet access, that can communicate with other devices, or that contains previously stored information. For example, you may not use a smartphone or a tablet. Parts of each question can be of two types: 1. SHORT ANSWER parts indicated by ? worth 3 marks each ? full marks given for a correct answer which is placed in the box ? part marks awarded only if relevant work is shown in the space provided 2. FULL SOLUTION parts indicated by ? ? ? ? worth the remainder of the 10 marks for the question must be written in the appropriate location in the answer booklet marks awarded for completeness, clarity, and style of presentation a correct solution poorly presented will not earn full marks

WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED. ? Extra paper for your ?nished solutions supplied by your supervising teacher must be inserted into your answer booklet. Write your name, school name, and question number on any inserted pages. √ ? Express calculations and answers as exact numbers such as π + 1 and 2, etc., rather than as 4.14 . . . or 1.41 . . ., except where otherwise indicated. Do not discuss the problems or solutions from this contest online for the next 48 hours. The name, grade, school and location, and score range of some top-scoring students will be published on our website, cemc.uwaterloo.ca. In addition, the name, grade, school and location, and score of some top-scoring students may be shared with other mathematical organizations for other recognition opportunities.

NOTE: 1. Please read the instructions on the front cover of this booklet. 2. Write all answers in the answer booklet provided. 3. For questions marked , place your answer in the appropriate box in the answer booklet and show your work. For questions marked , provide a well-organized solution in the answer booklet. Use mathematical statements and words to explain all of the steps of your solution. Work out some details in rough on a separate piece of paper before writing your ?nished solution. Diagrams are not drawn to scale. They are intended as aids only. While calculators may be used for numerical calculations, other mathematical steps must be shown and justi?ed in your written solutions and speci?c marks may be allocated for these steps. For example, while your calculator might be able to ?nd the x-intercepts of the graph of an equation like y = x3 ? x, you should show the algebraic steps that you used to ?nd these numbers, rather than simply writing these numbers down.

4.

5. 6.

A Note about Bubbling Please make sure that you have correctly coded your name, date of birth and grade on the Student Information Form, and that you have answered the question about eligibility.

1.

(a) What is the average of the integers 5, 15, 25, 35, 45, 55? (b) If x2 = 2016, what is the value of (x + 2)(x ? 2)? (c) In the diagram, points P (7, 5), Q(a, 2a), and R(12, 30) lie on a straight line. Determine the value of a.

y

R (12, 30) Q (a, 2a) P (7, 5)

x

2. (a) What are all values of n for which 25 n = ? 9 n

(b) What are all values of x for which (x ? 3)(x ? 2) = 6 ? (c) At Willard’s Grocery Store, the cost of 2 apples is the same as the cost of 3 bananas. Ross buys 6 apples and 12 bananas for a total cost of $6.30. Determine the cost of 1 apple.

3.

(a) In the diagram, point B is on AC , point F is on DB , and point G is on EB .

D q? A

p?

F

r?

s?

G

E

t? u?

B

C

What is the value of p + q + r + s + t + u? (b) Let n be the integer equal to 1020 ? 20. What is the sum of the digits of n? (c) A parabola intersects the x-axis at P (2, 0) and Q(8, 0). The vertex of the parabola is at V , which is below the x-axis. If the area of V P Q is 12, determine the coordinates of V .

4.

7 (a) Determine all angles θ with 0? ≤ θ ≤ 180? and sin2 θ + 2 cos2 θ = 4 .

(b) The sum of the radii of two circles is 10 cm. The circumference of the larger circle is 3 cm greater than the circumference of the smaller circle. Determine the di?erence between the area of the larger circle and the area of the smaller circle.

5.

(a) Charlotte’s Convenience Centre buys a calculator for $p (where p > 0), raises its price by n%, then reduces this new price by 20%. If the ?nal price is 20% higher than $p, what is the value of n? (b) A function f is de?ned so that if n is an odd integer, then f (n) = n ? 1 and if n is an even integer, then f (n) = n2 ? 1. For example, if n = 15, then f (n) = 14 and if n = ?6, then f (n) = 35, since 15 is an odd integer and ?6 is an even integer. Determine all integers n for which f (f (n)) = 3.

6.

1 x (a) What is the smallest positive integer x for which = for some positive 32 10y integer y ? (b) Determine all possible values for the area of a right-angled triangle with one side length equal to 60 and with the property that its side lengths form an arithmetic sequence. (An arithmetic sequence is a sequence in which each term after the ?rst is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9 are the ?rst four terms of an arithmetic sequence.)

7.

(a) Amrita and Zhang cross a lake in a straight line with the help of a one-seat kayak. Each can paddle the kayak at 7 km/h and swim at 2 km/h. They start from the same point at the same time with Amrita paddling and Zhang swimming. After a while, Amrita stops the kayak and immediately starts swimming. Upon reaching the kayak (which has not moved since Amrita started swimming), Zhang gets in and immediately starts paddling. They arrive on the far side of the lake at the same time, 90 minutes after they began. Determine the amount of time during these 90 minutes that the kayak was not being paddled. (b) Determine all pairs (x, y ) of real numbers that satisfy the system of equations x

1 2

+ y ? 2x2

5 2

= 0 = 0

y

+x?y

8.

(a) In the diagram, ABCD is a parallelogram. Point E is on DC with AE perpendicular to DC , and point F is on CB with AF perpendicular to CB . If AE = 20, AF = 32, and cos(∠EAF ) = 1 3 , determine the exact value of the area of quadrilateral AECF . (b) Determine all real numbers x > 0 for which log4 x ? logx 16 =

7 6

A

32 20

B

F C

D

E

? logx 8

9.

(a) The string AAABBBAABB is a string of ten letters, each of which is A or B , that does not include the consecutive letters ABBA. The string AAABBAAABB is a string of ten letters, each of which is A or B , that does include the consecutive letters ABBA. Determine, with justi?cation, the total number of strings of ten letters, each of which is A or B , that do not include the consecutive letters ABBA. (b) In the diagram, ABCD is a square. Points E and F are chosen on AC so that ∠EDF = 45? . If AE = x, EF = y , and F C = z , prove that y 2 = x2 + z 2 .

A x E y

45?

B

F z C

D

10.

Let k be a positive integer with k ≥ 2. Two bags each contain k balls, labelled with the positive integers from 1 to k . Andr? e removes one ball from each bag. (In each bag, each ball is equally likely to be chosen.) De?ne P (k ) to be the probability that the product of the numbers on the two balls that he chooses is divisible by k . (a) Calculate P (10). (b) Determine, with justi?cation, a polynomial f (n) for which f (n) ? P (n) ≥ 2 for all positive integers n with n ≥ 2, and n f (n) ? P (n) = 2 for in?nitely many positive integers n with n ≥ 2. n (A polynomial f (x) is an algebraic expression of the form m f (x) = am x + am?1 xm?1 + · · · + a1 x + a0 for some integer m ≥ 0 and for some real numbers am , am?1 , . . . , a1 , a0 .) 2016 (c) Prove there exists a positive integer m for which P (m) > . m

2016 Euclid Contest (English)

The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca

For students... Thank you for writing the 2016 Euclid Contest! Each year, more than 220 000 students from more than 60 countries register to write the CEMC’s Contests. If you are graduating from secondary school, good luck in your future endeavours! If you will be returning to secondary school next year, encourage your teacher to register you for the 2016 Canadian Senior Mathematics Contest, which will be written in November 2016. Visit our website cemc.uwaterloo.ca to ?nd ? Free copies of past contests ? Math Circles videos and handouts that will help you learn more mathematics and prepare for future contests ? Information about careers in and applications of mathematics and computer science For teachers... Visit our website cemc.uwaterloo.ca to ? Obtain information about our 2016/2017 contests ? Look at our free online courseware for high school students ? Learn about our face-to-face workshops and our web resources ? Subscribe to our free Problem of the Week ? Investigate our online Master of Mathematics for Teachers ? Find your school’s contest results

更多相关文章：
*2016数学*思维方式与创新期末考试答案

*2016数学*思维方式与创新期末考试答案_教育学_高等教育_教育专区。惊天刚考的*试题*...C、*欧几里得* ? D、罗巴切夫斯基 正确答案: D 我的答案:D 44 设 R 是一个...*2016*届高三试*数学*(理)*试题*及答案

*2016*届高三试*数学*(理)*试题*及答案_高三*数学*_*数学*_高中教育_教育专区。*2016* 届...1 D. 2 *2016* ?1 5. 公元前 3 世纪,古希腊*欧几里得*在《几何原本》里提出:...**网络课程***数学*大观*2016年*期末考试答案

网络课程*数学*大观*2016年*期末考试答案_理学_高等教育_...D、不一定 我的答案:D 二、判断题 1 密码必须...我的答案: √ 35 整个宇宙空间符合*欧几里得*几何。 ...*2016数学*建模考试答案_图文

*2016数学*建模考试答案_学科*竞赛*_初中教育_教育专区。西南大学网络与继续教育学院...更严 格的说,从*欧几里得*空间 Rn 到 R 的函数的梯度是在 Rn 某一点最佳的...*2016年*《*数学*史与*数学*教育》尔雅期末考试答案

*2016年*《*数学*史与*数学*教育》尔雅期末考试答案_教育学...*欧几里得* 我的答案:C 39 李善兰凭借()获得了麦都思...判断题(题数:50,共 50.0 分) 1 中国第一本微...*数学*思维与创新*2016*下半年考试答案

*数学*思维与创新*2016*下半年考试答案 一、 单选题(题数:50,共 50.0 分) 1 ...*欧几里得* 我的答案:D 11 特征为 2 的域是 1.0 分 A、 Z B、 Z2 C、...*2016年数学*文化顾沛尔雅通识考试

*2016年数学*文化顾沛尔雅通识考试_教育学_高等教育_...A、 *欧几里得* ? B、 希尔伯特 ? C、 D、 柯西 ...D、 以上全部是 我的答案:D 二、判断题(题数:...*2016年*3月2016届高三第二次全国大联考(新课标Ⅰ卷)理数...

*2016年*3月2016届高三第二次全国大联考(新课标Ⅰ卷)理数卷(正式考试版)_英语...8 ? 4 4.我国*数学*史上有一部堪与*欧几里得*《几何原本》媲美的书,这就是历来...*2016年*丘维声《*数学*的思维方式与创新》期末考试*试题*及...

*2016年*丘维声《*数学*的思维方式与创新》期末考试*试题*及答案_教育学_高等教育_教育...(1.0 分)1.0 分 A、 牛顿 B、 笛卡尔 C、 阿基米德 D、 *欧几里得* 我的...*2016年*3月2016届高三第二次全国大联考(新课标Ⅰ卷)理数...

2 是 4.我国*数学*史上有一部堪与*欧几里得*《几何原本》媲美的书,这就是历来被尊为算经之首的《九 章算术》 ,其中卷第五《商功》有一道关于圆柱体的体积*试题*... 更多相关标签：
2016aime数学竞赛真题 欧几里得数学竞赛 欧几里得数学竞赛官网 欧几里得数学竞赛2017 欧几里得数学竞赛前25 欧几里得数学竞赛难度 欧几里得数学竞赛奖项 欧几里得数学竞赛成绩

网络课程

2 是 4.我国