2015加拿大滑铁卢大学高斯数学竞赛(CEMC)

2015 AMC 12B竞赛真题Problem 1What is the value of ?Problem 2Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?Problem 3Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?Problem 4David, Hikmet, Jack, Marta, Rand, and Todd were in a 12-person race with 6 other people. Rand finished 6 places ahead of Hikmet. Marta finished 1 place behind Jack. David finished 2 places behind Hikmet. Jack finished 2 places behind Todd. Todd finished 1 place behind Rand. Marta finished in 6th place. Who finished in 8th place?Problem 5The Tigers beat the Sharks 2 out of the 3 times they played. They then played more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for ?Problem 6Back in 1930, Tillie had to memorize her multiplication facts fromto . The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd?Problem 7A regular 15-gon has lines of symmetry, and the smallest positive angle for which it has rotational symmetry is degrees. What is ?Problem 8What is the value of ?Problem 9Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a playerknocks the bottle off the ledge is , independently of what has happened before. What is the probability that Larry wins the game?Problem 10How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles?Problem 11The line forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?Problem 12Let , , and be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation?Problem 13Quadrilateral is inscribed in a circle withand . What is ?Problem 14A circle of radius 2 is centered at . An equilateral triangle with side4 has a vertex at . What is the difference between the area of the regionthat lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle?Problem 15At Rachelle's school an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA on the four classes she is taking is computed as the total sum of points divided by 4. She is certain that she will get As in both Mathematics and Science, and at least a C in each of English and History.She thinks she has a chance of getting an A in English, and a chance of getting a B. In History, she has a chance of getting an A, and a chanceof getting a B, independently of what she gets in English. What is the probability that Rachelle will get a GPA of at least 3.5?Problem 16A regular hexagon with sides of length 6 has an isosceles triangle attached to each side. Each of these triangles has two sides of length 8. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid?Problem 17An unfair coin lands on heads with a probability of . When tossed times,the probability of exactly two heads is the same as the probability of exactly three heads. What is the value of ?Problem 18For every composite positive integer , define to be the sum of the factors in the prime factorization of . For example, because the prime factorization of is , and . What is the range of the function , ?Problem 19In , and . Squares and are constructed outside of the triangle. The points , , , and lie on a circle. What is the perimeter of the triangle?Problem 20For every positive integer , let be the remainder obtained when is divided by 5. Define a functionrecursively as follows:What is ?Problem 21Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose that Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of ?Problem 22Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same chair and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done?Problem 23A rectangular box measures , where , , and are integers and. The volume and the surface area of the box are numerically equal. How many ordered triples are possible?Problem 24Four circles, no two of which are congruent, have centers at , , , and , and points and lie on all four circles. The radius of circleis times the radius of circle , and the radius of circle is timesthe radius of circle . Furthermore, and . Let be the midpoint of . What is ?Problem 25A bee starts flying from point . She flies inch due east to point . For , once the bee reaches point , she turns counterclockwise and then flies inches straight to point . When the bee reaches she is exactly inches away from , where , , andare positive integers and and are not divisible by the square of any prime. What is ?2015 AMC 12B竞赛真题答案1.C2.b3.a4.b5.b6.a7.d8.d9.c 10.c 11.e 12.d 13.b 14.d 15.d 16.c 17.d 18.d 19.c 20.b 21.d 22.d 23.b 24.d 25.b。

在撰写这篇文章之前，我首先要对“canadamo数学竞赛知识点”进行全面评估，以确保文章的深度和广度兼具。

在这篇文章中，我将从简到繁地分析并探讨canadamo数学竞赛的知识点，帮助你更深入地理解这个主题。

canadamo是加拿大数学奥林匹克(Canadian Mathematical Olympiad)的缩写，是加拿大国内最具权威性和影响力的数学竞赛之一。

参加canadamo数学竞赛，不仅能锻炼学生的数学能力，更能培养学生的逻辑思维和解决问题的能力。

我将从基础知识点开始，逐步深入，全面探讨canadamo数学竞赛的重要知识点。

1. 数论- 数论是canadamo数学竞赛中的重要知识点之一。

它涉及整数的性质、因数分解、同余方程等内容。

在canadamo数学竞赛中，数论题目常常涉及数字性质的推导和证明，考查选手的数学逻辑推理能力。

2. 几何- 几何是canadamo数学竞赛的另一个重要知识点。

它包括平面几何和立体几何两部分，涉及角度、边长、面积、体积等概念。

在canadamo数学竞赛中，几何题目常常涉及图形的性质和相似性的判断，考查选手的几何分析能力和空间想象能力。

3. 代数- 代数是canadamo数学竞赛的核心知识点之一。

它涉及方程、不等式、多项式、数列等内容。

在canadamo数学竞赛中，代数题目常常涉及函数的性质和变量的关系，考查选手的代数运算能力和推理能力。

4. 组合数学- 组合数学是canadamo数学竞赛的另一个重要知识点。

它包括排列、组合、概率等内容。

在canadamo数学竞赛中，组合数学题目常常涉及排列组合的计算和概率问题的推导，考查选手的组合分析能力和概率计算能力。

总结回顾：通过对canadamo数学竞赛知识点的全面评估，我们可以看到，数论、几何、代数和组合数学是其重要的知识点。

参加canadamo数学竞赛不仅需要掌握这些知识点，还需要灵活运用，并具备深入思考和解决问题的能力。

大学生学科竞赛种类调研明细目录：全国大学生数学建模竞赛 .................................................美国大学生数学建模竞赛（MCM/ICM） ......................................全国大学生数学竞赛 .....................................................丘成桐大学生数学竞赛 .....................................................国际大学生物理竞赛 .....................................................8中国大学生物理学术竞赛（CUPT） .........................................9南京大学青年物理学家锦标赛（NYPT） ......................................国际全局轨道优化竞赛 .....................................................全国深空轨道设计竞赛 .....................................................全国大学生英语竞赛 .....................................................国家大学生创新性实验计划 .................................................挑战杯系列赛事 .........................................................大学生学术科技作品展 .....................................................基础学科论坛 .............................................................学生学科竞赛项目一览表： 序号 类别学科竞赛名称学科竞赛主办单位备注（请注明该项赛事举办年度情况）分类说明1数学类 全国大学生数学建模大赛（CUMCM)教育部高教司、中国工业与应用数学学会每年9月 国家级2 美国大学生数学建模竞赛（MCM/ICM）美国数学及其应用联合会每年2月 国际级3 全国大学生数学竞赛The ChineseMathematicsCompetitions（CMC）中国数学会、国防科学技术大学每年10月 国家级4丘成桐大学生数学竞赛丘成桐教授 每年一次 国际级 5 物理类 国际大学物美国物理学每年11月 国际级理竞赛（UPC）会、天文学会6 中国大学生物理学术竞赛（CUPT)教育部中国物理学会/全国高等院校每年8月 国家级7 南京大学青年物理学家锦标赛（NYPT）物理学院 每年 校级8航天类 国际全局轨道优化竞赛（GTOC）1到2年 国际级9全国空间轨道设计竞赛中国力学学会 国家级10 英语类全国大学生英语竞赛 高校外语教学指导委员会每年4月 国家级11科研学术类 国家大学生创新性实验计划教育部 每年 国家级12 挑战杯系列赛事（TheChallenge共青团中央、中国科协、教育部和全国学每个项目每两年一届国家级Cup） 联、地方省级人民政府13 南京大学大学生学术科技作品展校团委、教务处、科技处、社科处、创新创业学院校级14基础学科论坛 南京大学基础学科论坛组委会每年3月 校级全国大学生数学建模竞赛主办：中国工业与应用数学学会(CSIAM)教育部高等教育司校管理部门：教务处校承办单位：数学系竞赛时间：每年9月竞赛简介：数模竞赛是由美国工业与应用数学学会在1985年发起的一项大学生竞赛活动。

A Note about BubblingPlease 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 value of 102−92 10+9?(b)If x+1x+4=4,what is the value of3x+8?(c)If f(x)=2x−1,determine the value of(f(3))2+2(f(3))+1.2.(a)If √a+√a=20,what is the value of a?(b)Two circles have the same centre.The radiusof the smaller circle is1.The area of theregion between the circles is equal to the areaof the smaller circle.What is the radius ofthe larger circle?(c)There were30students in Dr.Brown’s class.The average mark of the studentsin the class was80.After two students dropped the class,the average mark of the remaining students was82.Determine the average mark of the two students who dropped the class.3.(a)In the diagram,BD=4and point C is themidpoint of BD.If point A is placed sothat ABC is equilateral,what is the lengthof AD?AB C D(b) MNP has vertices M(1,4),N(5,3),and P(5,c).Determine the sum of thetwo values of c for which the area of MNP is14.4.(a)What are the x-intercepts and the y-intercept of the graph with equationy=(x−1)(x−2)(x−3)−(x−2)(x−3)(x−4)?(b)The graphs of the equations y=x3−x2+3x−4and y=ax2−x−4intersectat exactly two points.Determine all possible values of a.5.(a)In the diagram,∠CAB=90◦.Point Dis on AB and point E is on AC so thatAB=AC=DE,DB=9,and EC=8.Determine the length of DE.ABC D E(b)Ellie has two lists,each consisting of6consecutive positive integers.The smallestinteger in thefirst list is a,the smallest integer in the second list is b,and a<b.She makes a third list which consists of the36integers formed by multiplying each number from thefirst list with each number from the second list.(This third list may include some repeated numbers.)If•the integer49appears in the third list,•there is no number in the third list that is a multiple of64,and•there is at least one number in the third list that is larger than75, determine all possible pairs(a,b).6.(a)A circular disc is divided into36sectors.A number is writtenin each sector.When three consecutivesectors contain a ,b and c in thatorder,then b =ac .If the number 2is placed in one of the sectors andthe number 3is placed in one of theadjacent sectors,as shown,what is thesum of the 36numbers on the disc?23. . .. . . a b c . . . (b)Determine all values of x for which 0<x 2−11x +1<7.7.(a)In the diagram,ACDF is a rectanglewith AC =200and CD =50.Also,F BD and AEC are congruenttriangles which are right-angledat Band E ,respectively.What is the areaof the shaded region?F (b)The numbers a 1,a 2,a 3,...form an arithmetic sequence with a 1=a 2.The threenumbers a 1,a 2,a 6form a geometric sequence in that order.Determine all possiblepositive integers k for which the three numbers a 1,a 4,a k also form a geometricsequence in that order.(An arithmetic sequence is a sequence in which each term after the first is obtainedfrom the previous term by adding a constant.For example,3,5,7,9are the firstfour terms of an arithmetic sequence.A geometric sequence is a sequence in which each term after the first is obtainedfrom the previous term by multiplying it by a non-zero constant.For example,3,6,12is a geometric sequence with three terms.)8.(a)For some positive integers k ,the parabola with equation y =x 2k−5intersects the circle with equation x 2+y 2=25at 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, XY Z is isosceles withXY =XZ =a and Y Z =b where b <2a .A larger circle of radius R is inscribed inthe triangle (that is,the circle is drawn sothat it touches all three sides of the triangle).A smaller circle of radius r is drawn so thatit touches XY ,XZ and the larger circle.Determine an expression for R r in terms of a and b .b a a X Y Z9.Consider the following system of equations in which all logarithms have base 10:(log x )(log y )−3log 5y −log 8x =a(log y )(log z )−4log 5y −log 16z =b(log z )(log x )−4log 8x −3log 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 equationshas an infinite number of solutions (x,y,z ).10.For each positive integer n ≥1,let C n be the set containing the n smallest positiveintegers;that is,C n ={1,2,...,n −1,n }.For example,C 4={1,2,3,4}.We call a set,F ,of subsets of C n a Furoni family of C n 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 C 4.Determine the two Furoni families of C 4that contain all of the elements of A and to which no other subsets of C 4can 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 C n .For eachnon-negative integer k ,define a k to be the number of elements of F that contain exactly k integers.Prove thata 0 n 0 +a 1 n 1 +a 2 n 2 +···+a n −1 n n −1 +a n nn≤1(The sum on the left side includes n +1terms.)(Note:If n is a positive integer and k is an integer with 0≤k ≤n ,then n k =n !k !(n −k )!is the number of subsets of C n that contain exactly k integers,where 0!=1and,if m is a positive integer,m !represents the product of the integers from 1to m ,inclusive.)(c)For each positive integer n ,determine,with proof,the number of elements in thelargest Furoni family of C n (that is,the number of elements in the Furoni family that contains the maximum possible number of subsets of C n ).Euclid Contest(English) 2015。

加拿大高斯数学竞赛是加拿大国际数学竞赛之一，由加拿大滑铁卢大学数学系的加拿大数学与计算机中心（CEMC）举办，通常也被称为Waterloo数学竞赛。

该竞赛以德国著名的数学家、物理学家和天文学家JohannCarl Friedrich Gauss的名字命名。

高斯数学竞赛作为加拿大初中阶段最高级别的数学竞赛之一，每年有大量的海内外注册学校组织7-8年级对数学极有兴趣的学生参加，其竞赛内容的范围均在加拿大各省的教学大纲范围内。

竞赛分值：满分150分，10道题，每道题5分（基本上是送分题）；Section B: 10道题，每道题6分（考察数学基础）；Section C: 5道题，每道题8分（难度偏高的赛事题）。

该竞赛是加拿大官方的，也是最负盛名的数学竞赛。

Canadian Intermediate Mathematics Contest NOTE:1.Please read the instructions on the front cover of this booklet.2.Write solutions in the answer booklet provided.3.It is expected that all calculations and answers will be expressed as exact numbers such as 4π,2+√7,etc.,rather than as 12.566...or4.646....4.While calculators may be used for numerical calculations,other mathematical steps must be shown and justified in your written solutions and specific marks may be allocated for these steps.For example,while your calculator might be able to find the x -intercepts of the graph of an equation like y =x 3−x ,you should show the algebraic steps that you used to find these numbers,rather than simply writing these numbers down.5.Diagrams are not drawn to scale.They are intended as aids only.6.No student may write both the Canadian Senior Mathematics Contest and the Canadian Intermediate Mathematics Contest in the same year.PART AFor each question in Part A,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.1.There are 200people at the beach,and 65%of these people are children.If 40%of the children are swimming,how many children are swimming?2.If x +2y =14and y =3,what is the value of 2x +3y ?3.In the diagram,ABCD is a rectangle with points Pand Q on AD so that AB =AP =P Q =QD .Also,point R is on DC with DR =RC .If BC =24,whatis the area of P QR ?A B CD P Q R 4.At a given time,the depth of snow in Kingston is 12.1cm and the depth of snow in Hamilton is 18.6cm.Over the next thirteen hours,it snows at a constant rate of2.6cm per hour in Kingston and at a constant rate of x cm per hour in Hamilton.At the end of these thirteen hours,the depth of snow in Kingston is the same as the depth of snow in Hamilton.What is the value of x ?5.Scott stacks golfballs to make a pyramid.The first layer,or base,of the pyramid is a square of golfballs and rests on a flat table.Each golfball,above the first layer,rests in a pocket formed by four golfballs in the layer below (as shown in Figure 1).Each layer,including the first layer,is completely filled.For example,golfballs can be stacked into a pyramid with 3levels,as shown in Figure 2.The four triangular faces of the pyramid in Figure 2include a total of exactly 13different golfballs.Scott makes a pyramid in which the four triangular faces include a total of exactly 145different golfballs.How many layers does this pyramid have?Figure 1Figure 26.A positive integer is a prime number if it is greater than1and has no positive divisorsother than1and itself.For example,the number5is a prime number because its only two positive divisors are1and5.The integer43797satisfies the following conditions:•each pair of neighbouring digits(read from left to right)forms a two-digit primenumber,and•all of the prime numbers formed by these pairs are different,because43,37,79,and97are all different prime numbers.There are many integers with more thanfive digits that satisfy both of these conditions.What is the largest positive integer that satisfies both of these conditions?PART BFor each question in Part B,your solution must be well organized and contain words of explanation or justification.Marks are awarded for completeness,clarity,and style of presentation.A correct solution,poorly presented,will not earn full marks.1.(a)Determine the average of the six integers22,23,23,25,26,31.(b)The average of the three numbers y+7,2y−9,8y+6is27.What is the valueof y?(c)Four positive integers,not necessarily different and each less than100,have anaverage of94.Determine,with explanation,the minimum possible value forone of these integers.2.(a)In the diagram, P QR is right-angled at R.If P Q=25and RQ=24,determine the perimeterand area of P QR.PQR(b)In the diagram, ABC is right-angled at C withAB=c,AC=b,and BC=a.Also, ABC has perimeter144and area504.Determine all possible values of c.(You may use the facts that,for any numbers x and y, (x+y)2=x2+2xy+y2and(x−y)2=x2−2xy+y2.)AB C ab cCanadian Intermediate Mathematics Contest(English)20143.Vicky starts with a list(a,b,c,d)of four digits.Each digit is0,1,2,or3.Vickyenters the list into a machine to produce a new list(w,x,y,z).In the new list,w is the number of0s in the original list,while x,y and z are the numbers of1s,2s and3s,respectively,in the original list.For example,if Vicky enters(1,3,0,1),the machine produces(1,2,0,1).(a)What does the machine produce when Vicky enters(2,3,3,0)?(b)Vicky enters(a,b,c,d)and the machine produces the identical list(a,b,c,d).Determine all possible values of b+2c+3d.(c)Determine all possible lists(a,b,c,d)with the property that when Vicky enters(a,b,c,d),the machine produces the identical list(a,b,c,d).(d)Vicky buys a new machine into which she can enter a list of ten digits.Eachdigit is0,1,2,3,4,5,6,7,8,or9.The machine produces a new list whoseentries are,in order,the numbers of0s,1s,2s,3s,4s,5s,6s,7s,8s,and9s inthe original list.Determine all possible lists,L,of ten digits with the propertythat when Vicky enters L,the machine produces the identical list L.。

欧几里得数学竞赛奖项设置
欧几里得数学竞赛（Euclid Mathematics Contest）是由加拿大滑铁卢大学的数学院（Centre for Education in Mathematics and Computing, CEMC）主办的一项国际性高中数学竞赛。

该竞赛为全球高中生提供了一个展示数学才能的平台，并设置了以下奖项：
个人奖项：
Certificate of Distinction：颁发给在全球参赛者中排名前25%的学生。

Contest Medal：由CEMC决定，通常授予每个学校表现最优秀的学生。

Honour Rolls：根据成绩分设不同的荣誉榜，如全国荣誉榜、省级荣誉榜等。

团队奖项：
虽然主要以个人形式参加，但竞赛可能也会基于学校或地区团队整体成绩进行评价，并设立相应的团队奖项。

区域奖项：
根据成绩，可能会评出不同等级的奖项，比如针对加拿大区域的Zone、Provincial和National级别奖项。

其他表彰：
高分选手可能还会获得额外的证书或其他形式的表彰。

需要注意的是，具体的奖项设置以及获奖标准可能会随着年份的不同有所调整，请参考当年竞赛官方发布的最新公告和规则。

2015高考后加拿大留学：精算师的摇篮----滑铁卢大学在加拿大最受学生欢迎的大学“Best of All”排名中排名第一，“最具创新精神”大学第一，第一家推出Co-op课程的大学，有5300多家公司于滑铁卢大学合作，是目前拥有最多Co-op课程的大学。

前任滑铁卢校长戴维·约翰斯顿，为现任的加拿大总督。

滑铁卢大学重要优势：1.高科技地区2.共150多所研究院(其中理论物理研究所和量子纳米计算中心全球闻名)3.Intellectual property(学校提供资金赞助学生的创新和想法，学校不占有学生的盈利成果)4.黑莓手机是1984年滑铁卢大学在校学生想法中诞生滑铁卢全日制在校学生人数35100，1700名教授，无线网全校园覆盖，拥有6大学院：健康学院、文学院、理学院、工程学院、环境学院、数学学院。

健康学院是全世界“公共场合禁烟”的最早提出者。

文学院是六学院中最大的学院，包括了社会科学，语言类等各种专业。

工程学院在世界排名27位，并且全部专业均配有带薪实习，其中的建筑专业是知名之一并有专门的独立校园。

环境学院，标志为黄色安全帽，提供可持续发展与金融相结合的特色专业，城市规划也非常知名。

理学院的心理学在北美排名第三。

数学学院：标志为粉红色领带(其创始人喜欢佩戴粉红色领带，2010年去世)，是世界最大的数学专业学院，有丰富的带薪实习课程，相关知名人士：比尔盖茨多次访问，微软员工大多来自于滑铁卢，黑莓创始人也是滑铁卢毕业生，黑莓手机加密很好，谷歌最早的投资人是滑铁卢校友。

数学学院目前有6000名学生在读，1600国际生，其中有600-700名中国学生，与其他大学不同，滑铁卢的教授必须教授本科课程。

数学学院有3栋教学楼，其中今年刚刚投入使用的一栋专门作为精算、统计专业的学生使用，一层有专门的股票、证券交易大厅供学生实践。

迪拜校区：只有数学和工程，前两年在迪拜就读，后两年在滑铁卢市就读。

学费比加拿大便宜很多。

滑铁卢数学竞赛高中试题一、选择题1. 已知函数\( f(x) = ax^2 + bx + c \)，其中\( a, b, c \)为实数，且\( f(1) = 2 \)，\( f(-1) = 0 \)，\( f(2) = 6 \)。

求\( a \)的值。

2. 一个圆的半径为5，圆心位于原点，求圆上点\( P(3,4) \)到圆心的距离。

3. 若\( \sin(\alpha + \beta) = \frac{1}{2} \)，\( \cos(\alpha + \beta) = \frac{\sqrt{3}}{2} \)，且\( \alpha \)在第二象限，\( \beta \)在第一象限，求\( \sin(\alpha) \)的值。

二、填空题1. 计算\( \int_{0}^{1} x^2 dx \)。

2. 若\( \log_{2}8 = n \)，则\( n \)的值为______。

3. 一个等差数列的前三项分别为2，5，8，求该数列的第10项。

三、解答题1. 证明：对于任意正整数\( n \)，\( 1^3 + 2^3 + ... + n^3 =\frac{n^2(n+1)^2}{4} \)。

2. 一个矩形的长是宽的两倍，若矩形的周长为24，求矩形的面积。

3. 已知一个等比数列的前三项分别为3，9，27，求该数列的第5项。

四、应用题1. 一个工厂每天生产相同数量的零件，如果每天生产100个零件，工厂可以在30天内完成订单。

如果每天生产150个零件，工厂可以在20天内完成订单。

求工厂每天实际生产的零件数量。

2. 一个圆环的外圆半径是内圆半径的两倍，且圆环的面积为π。

求外圆的半径。

五、证明题1. 证明：对于任意实数\( x \)，\( \cos(x) + \cos(2x) + \cos(3x) \)可以表示为一个单一的余弦函数。

六、开放性问题1. 考虑一个无限大的棋盘，每个格子可以放置一个硬币。

Canadian Intermediate Mathematics Contest NOTE:1.Please read the instructions on the front cover of this booklet.2.Write solutions in the answer booklet provided.3.Express calculations and answers as exact numbers such as π+1and √2,etc.,rather than as4.14...or 1.41...,except where otherwise indicated.4.While calculators may be used for numerical calculations,other mathematical steps must be shown and justified in your written solutions and specific marks may be allocated for these steps.For example,while your calculator might be able to find the x -intercepts of the graph of an equation like y =x 3−x ,you should show the algebraic steps that you used to find these numbers,rather than simply writing these numbers down.5.Diagrams are not drawn to scale.They are intended as aids only.6.No studentmay write both the Canadian Senior Mathematics Contest and the Canadian Intermediate Mathematics Contest in the same year.PART AFor each question in Part A,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.1.Stephanie has 1000eggs to pack into cartons of 12eggs.While packingthe eggs,she breaks n eggs.The unbroken eggs completely fill a collection of cartons with no eggs left over.If n <12,what is the value of n ?2.In the diagram,ABCD is a square with side length 4.Points P ,Q ,R ,and S are the midpoints of the sides ofthe square,as shown.What is the area of the shadedregion?3.In the diagram,line segments AB ,CD and EF parallel,and points A and E lie on CG and respectively.If ∠GAB =100◦,∠CEF =30◦,∠ACE =x ◦,what is the value of x ?H4.If 12x =4y +2,determine the value of the expression 6y −18x +7.5.Determine the largest positive integer n with n <500for which 6048(28n )is a perfect cube (that is,it is equal to m 3for some positive integer m ).6.A total of2015tickets,numbered1,2,3,4,...,2014,2015,are placed in an emptybag.Alfie removes ticket a from the bag.Bernice then removes ticket b from the bag.Finally,Charlie removes ticket c from the bag.They notice that a<b<c and a+b+c=2018.In how many ways could this happen?PART BFor each question in Part B,your solution must be well organized and contain words of explanation or justification.Marks are awarded for completeness,clarity,and style of presentation.A correct solution,poorly presented,will not earn full marks.1.At Cornthwaite H.S.,many students enroll in an after-school arts program.Theprogram offers a drama class and a music class.Each student enrolled in the program is in one class or both classes.(a)This year,41students are in the drama class and28students are in the musicclass.If15students are in both classes,how many students are enrolled in theprogram?(b)In2014,a total of80students enrolled in the program.If3x−5students werein the drama class,6x+13students were in the music class,and x studentswere in both classes,determine the value of x.(c)In2013,half of the students in the drama class were in both classes and one-quarter of the students in the music class were in both classes.A total ofN students enrolled in the program in2013.If N is between91and99,inclusive,determine the value of N.2.Alistair,Conrad,Emma,and Salma compete in a three-sport race.They each swim2km,then bike40km,andfinally run10km.Also,they each switch instantly from swimming to biking and from biking to running.(a)Emma has completed113of the total distance of the race.How many kilometershas she travelled?(b)Conrad began the race at8:00a.m.and completed the swimming portion in30minutes.Conrad biked12times as fast as he swam,and ran3times as fast as he swam.At what time did hefinish the race?(c)Alistair and Salma also began the race at8:00a.m.Alistairfinished theswimming portion in36minutes,and then biked at28km/h.Salmafinished the swimming portion in30minutes,and then biked at24km/h.Alistair passed Salma during the bike portion.At what time did Alistair pass Salma?2015Canadian Intermediate Mathematics Contest (English)3.Heron’s Formula says that if a triangle has side lengths a ,b and c ,then its area equals s (s −a )(s −b )(s −c ),where s =12(a +b +c )is called the semi-perimeter ofthe triangle.A B Ca b c(a)In the diagram, ABC has side lengthsAB =20,BC =99,and AC =101.If his the perpendicular distance from A toBC ,determine the value of h .A BC 9910120h (b)In the diagram,trapezoid P QRS hasP S parallel to QR .Also,P Q =7,QR =40,RS =15,and P S =20.Ifx is the distance between parallel sidesP S and QR ,determine the value of x .(c)The triangle with side lengths 3,4and 5has the following five properties:•its side lengths are integers,•the lengths of its two shortest sides differ by one,•the length of its longest side and the semi-perimeter differ by one,•its area is an integer,and •its perimeter is less than 200.Determine all triangles that have thesefive properties.。

滑铁卢数学竞赛滑铁卢数学竞赛是加拿大一项著名的数学竞赛活动，每年都吸引了许多有志于挑战自己数学能力的学生参加。

它的历史可以追溯到1967年，从那以后，滑铁卢数学竞赛已经成为了全球最重要的数学竞赛之一。

滑铁卢数学竞赛分为不同的级别，包括高中水平的离散数学竞赛和全国高中生数学竞赛等。

它不仅仅考察了学生的计算能力，更注重培养学生的数学思维能力和解决问题的能力。

滑铁卢数学竞赛的题目涵盖了数学的各个领域，如代数、几何、数论和组合数学等。

这些题目往往以形式化和抽象的方式出现，需要参赛学生进行深入的分析和推理。

竞赛要求学生在有限的时间内回答一系列问题，并用严谨的数学推导来解决问题。

滑铁卢数学竞赛的题目难度非常高，需要具备扎实的数学基础和高超的解题能力。

参赛学生需要理解问题的本质，并能够找到解题的关键步骤。

在竞赛中，学生往往需要面对复杂的数学问题，需要运用各种数学知识和技巧来解答。

参加滑铁卢数学竞赛对于学生的数学能力和素质有着很高的要求。

它不仅考察了学生的记忆和计算能力，更注重培养学生的数学思维和解决问题的能力。

通过参加竞赛，学生可以提高自己的数学水平，拓宽数学思维的广度和深度。

滑铁卢数学竞赛为学生提供了一个展示自己数学才能的舞台。

通过竞赛，学生可以与其他优秀的数学爱好者交流和切磋，共同进步。

竞赛的结果不仅是学生们的荣誉和成绩，更是他们自信心的提升和未来学习、发展的动力。

作为一项重要的学科竞赛，滑铁卢数学竞赛不仅在加拿大，而且在国际上都享有较高的声誉。

许多优秀的数学家和科学家都曾经参加过滑铁卢数学竞赛，这为他们日后的学术研究和职业发展奠定了坚实的基础。

总之，滑铁卢数学竞赛是一个激励学生充分发挥数学潜能、提高数学能力和培养创新思维的重要平台。

通过参加竞赛，学生可以不断挑战自我，锻炼解决问题的能力，为未来的学习和职业发展奠定良好的基础。

滑铁卢数学竞赛的成功举办，为数学教育和科学研究做出了重要贡献。

北美驯鹿国际数学竞赛题随着科技的发展和经济的全球化，数学作为一门基础学科在各行各业中扮演着越来越重要的角色。

为了提高青少年的数学能力和水平，各国纷纷组织了国际数学竞赛。

其中，北美驯鹿国际数学竞赛题备受关注。

本文将详细介绍北美驯鹿国际数学竞赛题的历史、特点、难度和意义。

一、历史北美驯鹿国际数学竞赛为一项面向中小学生的国际数学竞赛。

它始于2003年，由加拿大的一群数学教育者和爱好者共同发起。

竞赛的名称源自加拿大驯鹿的文化，也反映了竞赛的国际性。

二、特点北美驯鹿国际数学竞赛题有以下特点：1.难度适中。

与国际数学奥林匹克竞赛相比，北美驯鹿国际数学竞赛题的难度要相对低一些。

这也为更多的中小学生提供了参加国际数学竞赛的机会。

2.类型多样。

竞赛包括单项、团队和口语等多个方面，使得学生不仅能够锻炼数学能力，还能够提高其他技能。

3.注重实用性。

竞赛的题目都是以实际问题为背景而设计的，旨在培养学生将数学知识应用于实际生活的能力。

三、难度北美驯鹿国际数学竞赛题的难度和其它国际数学竞赛相比难度要低一些，但是也并不简单。

难度的选择是根据竞赛的群体性考虑的，让更多人有机会参加和获得成功。

例如，北美驯鹿国际数学竞赛题中常常有许多有趣的趣味性质问题。

这些问题不需要使用过多的抽象数学知识，但需要学生思考、分析和推理，从而帮助培养学生的数学思维能力。

四、意义北美驯鹿国际数学竞赛作为中小学生国际数学竞赛，一方面可以激发学生学习数学的兴趣和热情，另一方面也可以帮助学生深入了解数学，并解决实际问题。

此外，竞赛还有以下几个方面的意义：1.推动数学教育的发展。

通过北美驯鹿国际数学竞赛的组织和开展，不仅能够推广数学竞赛活动，还能够推动数学的教育方式和方法的发展。

它可以在全世界范围内分享数学教学的最佳实践，促进数学教育的向高质量的方向发展。

2.培养高素质人才。

高素质的数学人才是社会发展的不可或缺的力量。

竞赛可以帮助学生发掘自己的数学潜力和兴趣，促进人才培养和流动。

加拿大滑铁卢大学举办费马国际数学竞赛及欧几里得数学竞赛。

滑铁卢大学是加拿大综合排名第三的大学，其创新精神在加拿大排名第一，该大学面向青少年举办的数学竞赛，在全球具有影响力。

费马国际数学竞赛面向高一学生的费马数学竞赛，考试时间60为分钟，满分为150分，其间学生必须面对全英语试卷解答问题。

欧几里得数学竞赛据360教育集团介绍，面向高二学生的欧几里得数学竞赛，考试时间150为分钟，满分为100分，其间学生必须面对全英语试卷解答问题。

最初的数学考试是由安大略省西南部的几个高中老师联合创办的，从六十年代初年每年300人参加考试到今天，累计已经有21万名学生参加了这个考试。

根据滑铁卢大学的校方统计资料：21万名学生中有40%是来自安大略省的学生，20%是来自英属哥伦比亚省的学生，35%是来自加拿大其他省份的，还有5%是来自国际学生，包括美国、英国、中国等世界各国的学生。

2003年因为安大略省取消13年级，部分涉及微积分的试题不再使用，于是将迪卡尔（法国著名数学家）数学竞赛（DescartesContest）更名为（欧几里德数学竞赛）。

现在，欧几里德数学竞赛的分数已经成为Waterloo数学学院各专业以及“软件工程”专业入学录取的重要指标，更成为学生申请该学院奖学金的重要考核标准。

欧几里德数学竞赛（EuclidContest）主要是为高二年级（加拿大11年级）的高中学生提供的考试，考试内容主要包括：代数（函数、三角、排列、组合）、平面组合、解析几何等，他不仅仅看的是结果，更看重的是学生的解题思路和技巧。

考试的及格分数每年大概在40分左右。

因滑铁卢大学在数学领域的优良声誉及传统，以及欧几里德数学竞赛考察标准的严格性和专业性，该竞赛成绩在加拿大大学中已经得到广泛认可，被誉为类似加拿大“数学托福”的考试。

滑铁卢大学数学竞赛。

欧几里得数学竞赛_（实用版）目录1.欧几里得数学竞赛简介2.竞赛的含金量和认可度3.竞赛的难度和价值4.如何准备和报名欧几里得数学竞赛5.竞赛对留学申请的影响正文欧几里得数学竞赛是全球三大数学竞赛之一，由加拿大滑铁卢大学与计算教育中心（CEMC）主办，面向全球高中生举办。

该竞赛自 1963 年创办以来，每年都有来自 10 多个国家和地区、1850 多所学校的 2 万多名学生报名参加。

竞赛成绩在数学界中已经得到广泛认可，被誉为数学届托福。

欧几里得数学竞赛的含金量和认可度非常高，被誉为全球高含金量高认可度的数学竞赛之一。

在北美地区，该竞赛的知名度与 AMC 和 BMO 齐名。

每年成绩排名前 25% 的学生会获得滑铁卢大学颁发的证书，将为留学申请履历添上丰富的一笔。

虽然欧几里得数学竞赛的含金量和认可度非常高，但是参赛者需要面对一定的难度。

竞赛的知识点考察非常平稳，主要集中在几何和代数模块。

在几何题中，参赛者可能需要运用、探索较为抽象和复杂的几何关系，从切入解决问题。

对于代数题，参赛者要求理解和的问题。

因此，参赛者需要具备较强的数学基础和解题能力。

对于如何准备和报名欧几里得数学竞赛，首先可以找负责数学部门的老师进行咨询。

此外，可以参加专门的欧几里得数学竞赛培训班，了解更多关于竞赛的详细信息和解题技巧。

在准备过程中，可以多做历年真题，提高自己的应试能力。

参加欧几里得数学竞赛对于留学申请有着非常积极的影响。

竞赛成绩可以为留学申请履历增色，提高被名校录取的几率。

此外，在竞赛中获得优异成绩的学生，更容易获得国外大学奖学金和助学金。