当前位置:首页 >> 学科竞赛 >>



Euclid Contest
Tuesday, April 12, 2016
(in North America and South America)

Wednesday, April 13, 2016
(outside of North America and South America)


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.


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.


(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.

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

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.


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

D q? A





t? u?



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 .


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.


(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.


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.)


(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




(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

32 20





? logx 8


(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


F z C



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)


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


第十三届“小机灵杯”数学竞赛初赛试题解析(三年级...2.西方最早发现勾股定理的数学家之一是欧几里得。 ...文档贡献者 13651641095 贡献于2016-06-16 ...
数学竞赛_学科竞赛_初中教育_教育专区。一、整除的定义...又称欧几里得算法。 二、倍数和约数 1 两个整数 A...说明的情况下,它表示所学过的数,并且能使题设有...
欧几里得除法,非负最小完全剩余类,高斯函数,费马小定理,欧拉函数,孙子定理,格点...数学竞赛题 暂无评价 1页 免费 初一数学竞赛 2页 免费喜欢此文档的还喜欢 ...
全国高中数学联赛试题新规则和考试范围_学科竞赛_高中...3.初等数论 同余,欧几里得除法,裴蜀定理,完全剩余系...2016年全国高中数学联赛... 11页 2下载券 高中...
2016年数学的思维方式与创新期末考试_理学_高等教育_...(题数:50,共 50.0 分) 1 如果 d 是被除数和...C、 欧几里得 ? ? D、 罗巴切夫斯基 ? 正确答案:...
3、 ZFC 公理系统中具备规定集合的九条公理。 4、 欧几里 一、判断题 1、...A、结构、大小 C、动量、位置 B、结构、位置 D、动量、能量 15、数学的基石...
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 年全国第二次大联考高考数学模拟试卷 (新课标Ⅰ) (理科)一、选择题(本...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...
2016-2017年高二下期中考理科数学试题及答案 - 普宁二中 2016--2017 学年度第二学期期中考 高二级理科数学试卷 命题人:陈木茂 审题人:舒有汉 祝考试顺利! 一....
滑铁卢大学举办费马国际数学竞赛及欧几里得数学竞赛。...试题不再使用,于是将迪卡尔(法国著名数学家)数学竞赛...©2016 Baidu 使用百度前必读 | 文库协议 | 广告...

All rights reserved Powered by 甜梦文库 9512.net

copyright ©right 2010-2021。