2015加拿大滑铁卢大学高斯数学竞赛(CEMC)

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Pascal滑铁卢数学竞赛(Grade 9)-数学Mathematics-2013-试题 exam

1. The value of (4 + 44 + 444) ÷ 4 is
(A) 111
(B) 123
(C) 459
(D) 489
(E) 456
2. Jing purchased eight identical items. If the total cost was $26, then the cost per item, in dollars, was
7. Scoring: Each correct answer is worth 5 in Part A, 6 in Part B, and 8 in Part C. There is no penalty for an incorrect answer. Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
Friday, February 22, 2013
(outside of North America and South America)
Time: 60 minutes Calculators are permitted Instructions
©2012 University of Waterloo
Mass of Jeff s Pet Atlantic Cod
20
Mass in kg
10
00
2
4
6
8
Age in Years
(A) 3
(B) 7
(C) 4
(D) 6
(E) 5
5. What is the value of 13 + 23 + 33 + 43?

2015滑铁卢竞赛试题答案

(b) Points N (5, 3) and P (5, c) lie on the same vertical line. We can consider N P as the base of M N P . Suppose that the length of this base is b. The corresponding height of M N P is the distance from M (1, 4) to the line through N and P . Since M lies on the vertical line x = 1 and N and P lie on the vertical line x = 5, then the height is h = 4.
Therefore x = 0 or x2 − (a + 1)x + 4 = 0. Note that x = 0 is not a solution to x2 − (a + 1)x + 4 = 0, since when x = 0 is substitutedA2AD2 = = = =
B
E
60º 120º
C
2
D
Since AC = CD, then ACD is isosceles with ∠CDA = ∠CAD. 1 Each of these angles equals 2 (180◦ − ∠ACD) = 1 (180◦ − 120◦ ) = 30◦ . 2 √ But DAE is then a 30◦ -60◦ -90◦ triangle, so AD = 2AE = 2 3.
y
P1(5, c1) M (1, 4)

2015年AMC12B竞赛真题及问题详解

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。

canadamo数学竞赛知识点

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

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。

2015加拿大欧几里得数学竞赛真题

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。

加拿大数学竞赛历年试题(滑铁卢大学)

Each question is worth 10 marks
Calculating devices are allowed, provided that they do not have any of the following features: (i) internet access, (ii) the ability to communicate with other devices, (iii) information previously stored by students (such as formulas, programs, notes, etc.), (iv) a computer algebra system, (v) dynamic geometry software.
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
WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED. • Extra paper for your ﬁnished solutions must be supplied by your supervising teacher and inserted into your answer booklet. Write your name, school name, and question number on any inserted pages. • Express answers as simpli√ﬁed exact numbers except where otherwise indicated. For example, π + 1 and 1 − 2 are simpliﬁed exact numbers.

加拿大高斯数学

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。

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2.The value of 10×(5−2)is (A)13
(B)70
(C)7
(D)30
(E)50
3.
The graph shows the total distance that each of ﬁve runners ran during a one-hour training session.Which runner ran the least distance?(A)Phil (B)Tom (C)Pete
(D)Amal
(E)Sanjay
Phil
Tom
4.
The equal-arm scale shown is balanced.One has the same mass as (A)(B)(C)(D)(E)
5.Which of the following is closest to 5cm?(A)The length of a full size school bus (B)The height of a picnic table (C)The height of an elephant (D)The length of your foot (E)The length of your thumb
6.The number of centimetres in 3.5metres is (A)350
(B)30.5
(C)3.05
(D)3.50
(E)305
7.The perimeter of the ﬁgure shown is (A)18(B)17(C)23
(D)20
(E)25
8.
Hannah scored 312points during the basketball season.If her average (mean)was 13points per game,how many games did she play?(A)24
(B)41
(C)17
(D)13
(E)30
9.
The number 6has exactly four positive divisors:1,2,3,and 6.How many positive divisors does 20have?(A)2
(B)6
(C)3
(D)5
(E)8
12.A piece of paper is folded in half,creating two layers of paper.The paper is then folded in half again.This is continued until the paper has been folded in half a total of ﬁve times.The total number of layers of paper in the folded sheet is (A)16(B)32(C)25(D)8(E)6413.How many even whole numbers between 1and 99are multiples of 5?(A)5(B)7(C)9(D)11(E)1314.In the 3×3table shown,the numbers 1,2and 3are placed so that each number occurs only once in each row and only once in each column.The value of X +Y is
(A)3(B)2(C)5
(D)6
(E)4
15.In the rectangle shown,the area of the shaded region is
(A)60cm 2(B)20cm 2(C)30cm 2
(D)
40cm 2
(E)
50cm 2
5 cm
16.You have exactly $4.40(440¢)in quarters (25¢coins),dimes (10¢coins),and nickels
(5¢coins).You have the same number of each type of coin.How many dimes do you have?(A)20(B)11(C)10(D)12(E)417.One corner of a cube is cut oﬀ,creating a new triangular face,as shown.How many edges does this new solid have?
(A)18(B)14(C)24
(D)15
(E)13
18.In the graph shown,which of the following represents the
image of the line segment P Q after a reﬂection across the x -axis?
(A)P S (B)T U (C)MN
(D)W V
(E)F G
decimal point will there also be a 2?
(A)119th (B)121st (C)123rd (D)125th (E)126th
20.In a triangle,the measure of one of the angles is 45◦.The measures of the other two
angles in the triangle are in the ratio 4:5.What is the measure of the largest angle in the triangle?
(A)80◦
(B)90◦
(C)75◦
(D)85◦
(E)100◦
Part C:Each correct answer is worth 8.
21.The numbers 1through 25are arranged into 5rows and 5columns in the table below.
123451098761112131415201918171621
22
23
24
25
What is the largest possible sum that can be made using ﬁve of these numbers such that no two numbers come from the same row and no two numbers come from the same column?(A)75
(B)73
(C)71
(D)70
(E)68
22.The width of a rectangle is doubled and the length is halved.This produces a square
with a perimeter of P .What is the perimeter of the original rectangle?
(A)P
(B)2P
(C)12P
(D)54P (E)5
2P
23.A palindrome is a positive integer that is the same when read forwards or backwards.
The numbers 101and 4554are examples of palindromes.The ratio of the number of 4-digit palindromes to the number of 5-digit palindromes is
(A)4:5
(B)5:2
(C)2:7
(D)4:3
(E)1:10
24.In the diagram,rectangle P QRS is made up of six
identical squares.Points U,V,W,X,Y,and Z are midpoints of sides of the squares,as shown.Which of the following triangles has the greatest area?
(A)P V U (B)P XZ (C)P V X
(D)P Y S
(E)P QW
P
Q
25.Two diﬀerent 2-digit positive integers are called a reversal pair if the position of the
digits in the ﬁrst integer is switched in the second integer.For example,52and 25are a reversal pair.The integer 2015has the property that it is equal to the product of three diﬀerent prime numbers,two of which are a reversal pair.Including 2015,how many positive integers less than 10000have this same property?
(A)18
(B)14
(C)20
(D)17
(E)19。