Hypatia滑铁卢数学竞赛(Grade 11)-数学Mathematics-2008-试题 exam
2008Hypatia Contest(Grade11)
Wednesday,April16,2008
1.For numbers a and b,the notation a∇b means2a+b2+ab.
For example,1∇2=2(1)+22+(1)(2)=8.
(a)Determine the value of3∇2.
(b)If x∇(−1)=8,determine the value of x.
(c)If4∇y=20,determine the two possible values of y.
(d)If(w−2)∇w=14,determine all possible values of w.
2.(a)Determine the equation of the line through the points A(7,8)and B(9,0).
(b)Determine the coordinates of P,the point of intersection of the line y=2x−10and the
line through A and B.
(c)Is P closer to A or to B?Explain how you obtained your answer.
3.In the diagram,ABCD is a trapezoid with AD parallel to BC and
BC perpendicular to AB.Also,AD=6,AB=20,and BC=30.
(a)Determine the area of trapezoid ABCD.
(b)There is a point K on AB such that the area of KBC
equals the area of quadrilateral KADC.Determine the length of BK.
(c)There is a point M on DC such that the area of MBC
equals the area of quadrilateral MBAD.Determine the length of MC.
C
4.The peizi-sum of a sequence a1,a2,a3,...,a n is formed by adding the products of all of the
pairs of distinct terms in the sequence.For example,the peizi-sum of the sequence a1,a2,a3,a4 is a1a2+a1a3+a1a4+a2a3+a2a4+a3a4.
(a)The peizi-sum of the sequence2,3,x,2x is−7.Determine the possible values of x.
(b)A sequence has100terms.Of these terms,m are equal to1and n are equal to−1.The
rest of the terms are equal to2.Determine,in terms of m and n,the number of pairs of distinct terms that have a product of1.
(c)A sequence has100terms,with each term equal to either2or−1.Determine,with
justification,
the minimum possible peizi-sum of the sequence.。
欧几里得滑铁卢数学竞赛_2010EuclidSolution
Since Bea flies at a constant speed, then the ratio of the two distances equals the ratio of
the corresponding times.
HF 60 minutes 4
Therefore, =
=.
GF 45 minutes 3
(b) Solution 1
Since ∠OP B = 90◦, then OP and P B are perpendicular, so the product of their slopes
is −1.
4−0 4
4−0
4
The slope of OP is
= and the slope of P B is
Since F GH is right-angled at F , then F GH must be similar to a 3-4-5 triangle, and
HG 5
so = .
GF 3
In
particular,
this
means
that
the
ratio
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to
Wednesday, April 7, 2010
Solutions
©2010 Centre for Education in Mathematics and Computing
2010 Euclid Contest Solutions
Page 2
1. (a) Solution 1 Since 3x = 27, then 3x+2 = 3x32 = 27 · 9 = 243.
caribou数学竞赛介绍
caribou数学竞赛介绍
Caribou数学竞赛是一项面向学生的国际性数学竞赛，旨在激发学生对数学的兴趣，并提高他们的数学技能和解决问题的能力。
该竞赛由加拿大Caribou Mathematics Competition组织，每年举办一次。
Caribou数学竞赛分为多个级别，适合不同年级的学生参与，从幼儿园到高中都有相应的竞赛级别。
竞赛题目设计丰富多样，包括选择题、填空题、解答题等形式，涵盖了各个数学领域的知识和技能。
参加Caribou数学竞赛有许多好处。
首先，它可以培养学生的数学思维和解决问题的能力。
通过面对不同类型的数学问题，学生需要思考和分析，找到解题的方法和策略。
其次，竞赛可以激发学生对数学的兴趣和热爱，让他们深入了解数学的魅力和应用。
此外，竞赛还提供了一个与全球各地学生交流和比较的平台，激发学生的竞争意识和合作精神。
参加Caribou数学竞赛需要学校或个人报名，并按照竞赛规定的时间和方式进行答题。
竞赛结果将根据学生的得分和排名进行评定，并颁发相应的证书和奖励。
总而言之，Caribou数学竞赛是一项激发学生数学兴趣和提高数学能力的国际性竞赛，为学生提供了锻炼和展示自己数学才能的机会。
通过参与竞赛，学生可以发展数学思维，提高解决问题的能力，并与来自世界各地的学生交流和比较。
国际中学生数学竞赛含金量排行榜
国际中学生数学竞赛含金量排行榜
国际中学生数学竞赛的含金量因赛事的权威性、参赛选手水平、奖项设置等因素而有所不同。
以下是部分国际中学生数学竞赛及简要介绍：
1. 国际数学奥林匹克（IMO）：这是最具权威性的国际中学生数学竞赛之一，每年有来自世界各地的参赛选手。
IMO的奖项分为金牌、银牌和铜牌，其
中金牌是最高荣誉。
获奖者将获得世界范围内的认可和奖励，对于未来的学术和职业发展具有重要意义。
2. 亚洲太平洋数学奥林匹克（APMO）：这是亚太地区最高水平的数学竞赛，每年有来自多个国家和地区的代表队参赛。
APMO的奖项分为金、银、铜
牌和优秀奖，其中金牌是最高荣誉。
获奖者将获得国际范围内的认可和奖励，对于未来的学术和职业发展具有重要意义。
3. 英国数学奥林匹克（BMO）：这是英国最高水平的数学竞赛，每年有来
自全国各地的参赛选手。
BMO的奖项分为金、银、铜牌和优秀奖，其中金
牌是最高荣誉。
获奖者将获得英国范围内的认可和奖励，对于未来的学术和职业发展具有重要意义。
4. 罗马尼亚数学奥林匹克（RMO）：这是罗马尼亚最高水平的数学竞赛，
每年有来自全国各地的参赛选手。
RMO的奖项分为金、银、铜牌和优秀奖，其中金牌是最高荣誉。
获奖者将获得国际范围内的认可和奖励，对于未来的学术和职业发展具有重要意义。
总的来说，这些国际中学生数学竞赛都具有较高的含金量，获奖者将获得国际范围内的认可和奖励，对于未来的学术和职业发展具有重要意义。
滑铁卢数学竞赛
滑铁卢数学竞赛滑铁卢数学竞赛是加拿大滑铁卢大学举办的一项年度数学竞赛活动。
该竞赛旨在通过一系列难度不断增加的数学问题，考察参赛者的数学思维能力、解题能力以及创造力。
每年都有来自世界各地的学生参加该比赛，其中包括来自中小学的学生以及大学生。
滑铁卢数学竞赛分为两个阶段，第一阶段为全球性选拔赛，任何人都可以参加。
参赛者需要在线完成一套由滑铁卢大学编制的数学测试，题型涵盖代数、几何、组合数学等多个数学领域。
根据第一阶段的成绩，滑铁卢大学将选拔出前几百名成绩优异的参赛者晋级到第二阶段。
第二阶段为面试阶段，只有第一阶段晋级的学生才可以参加。
参赛者需要前往滑铁卢大学进行现场的笔试和面试。
笔试部分主要考察参赛者的数学基础知识和解题能力，而面试部分则更加注重参赛者的思维过程和解题思路。
面试时，学生需要与评委进行面对面的交流，展示自己的数学思考能力。
滑铁卢数学竞赛的题目通常非常有难度，涉及到一些高级数学概念和方法。
参赛者需要具备扎实的数学基础知识，并且具备独立思考和解决问题的能力。
竞赛的目的不仅是测试学生的数学水平，更重要的是培养他们解决问题的能力和数学思维方式。
参加滑铁卢数学竞赛对于学生来说是一次宝贵的经历。
这个竞赛可以提供一个展示自己数学才能的平台，也可以锻炼参赛者的思维能力和团队合作精神。
在竞赛中，学生们可以结识来自不同国家和地区的志同道合的数学爱好者，分享彼此的数学体验和解题方法。
滑铁卢数学竞赛也为参赛者提供了一些奖励和机会。
根据参赛者在竞赛中的表现，滑铁卢大学会为他们颁发证书和奖状，并且可以获得一些奖金和奖品。
此外，优秀的参赛者还有机会获得滑铁卢大学的奖学金和入学机会，为他们的未来发展开启了一扇大门。
总之，滑铁卢数学竞赛是一个非常有挑战性和有意义的数学竞赛活动。
通过参加这个竞赛，学生们可以提升自己的数学能力，拓展自己的数学视野，同时也能够展示自己的才能和潜力。
无论是对于中小学生还是大学生，参加滑铁卢数学竞赛都是一个值得鼓励和支持的选择。
滑铁卢数学竞赛
滑铁卢数学竞赛1、21.|x|>3表示的区间是（）[单选题] *A.（-∞，3）B.(-3,3)C. [-3,3]D. （-∞，-3）∪（3，+ ∞）(正确答案)2、15.下列数中，是无理数的为（）[单选题] *A.-3.14B.6/11C.√3(正确答案)D.03、9.一棵树在离地5米处断裂，树顶落在离树根12米处，问树断之前有多高（）[单选题] *A. 17(正确答案)B. 17.5C. 18D. 204、若tan（π-α）>0且cosα>0，则角α的终边在（）[单选题] *A.第一象限B.第二象限C.第三象限D.第四象限(正确答案)5、下列说法正确的是[单选题] *A.带“+”号和带“-”号的数互为相反数B.数轴上原点两侧的两个点表示的数是相反数C.和一个点距离相等的两个点所表示的数一定互为相反数D.一个数前面添上“-”号即为原数的相反数(正确答案)6、16、在中,则( ). [单选题] *A. AB<2AC (正确答案)B. AB=2ACC. AB>2ACD. AB与2AC关系不确定7、260°是第（）象限角？[单选题] *第一象限第二象限第三象限(正确答案)第四象限8、4.已知两圆的半径分别为3㎝和4㎝，两个圆的圆心距为10㎝,则两圆的位置关系是（）[单选题] *Ａ.内切Ｂ.相交Ｃ.外切Ｄ.外离(正确答案)9、下列各角中，是界限角的是（）[单选题] *A. 1200°B. -1140°C. -1350°(正确答案)D. 1850°10、下列各对象可以组成集合的是（）[单选题] *A、与1非常接近的全体实数B、与2非常接近的全体实数(正确答案)C、高一年级视力比较好的同学D、与无理数相差很小的全体实数11、若2?＝a2＝4 ?，则a?等于( ) [单选题] *A. 43B. 82C. 83(正确答案)D. 4?12、47、若△ABC≌△DEF，AB＝2，AC＝4，且△DEF的周长为奇数，则EF的值为（）[单选题] *A．3B．4C．1或3D．3或5(正确答案)13、8. 下列事件中，不可能发生的事件是(? ? )．[单选题] *A．明天气温为30℃B．学校新调进一位女教师C．大伟身长丈八(正确答案)D．打开电视机，就看到广告14、3．中国是最早采用正负数表示相反意义的量，并进行负数运算的国家．若零上10℃记作+10℃，则零下10℃可记作（）[单选题] *A．10℃B．0℃C.-10 ℃(正确答案)D．-20℃15、19．对于实数a、b、c,“a>b”是“ac2(c平方)>bc2(c平方) ; ”的（）[单选题] * A．充分不必要条件B．必要不充分条件(正确答案)C．充要条件D．既不充分也不必要条件16、22．如图棋盘上有黑、白两色棋子若干，找出所有使三颗颜色相同的棋在同一直线上的直线，满足这种条件的直线共有（）[单选题] *A．5条(正确答案)B．4条C．3条D．2条17、f（x）=-2x+5在x=1处的函数值为（）[单选题] *A、-3B、-4C、5D、3(正确答案)18、10. 如图所示，小明周末到外婆家，走到十字路口处，记不清哪条路通往外婆家，那么他一次选对路的概率是(? ? ?)．[单选题] *A．1/2B．1/3(正确答案)C．1/4D．119、15．如图所示，下列数轴的画法正确的是（）[单选题] *A．B．C．(正确答案)D．20、-950°是（）[单选题] *A. 第一象限角B. 第二象限角(正确答案)C. 第三象限角D. 第四象限角21、1．如图，∠AOB＝120°，∠AOC＝∠BOC，OM平分∠BOC，则∠AOM的度数为（）[单选题] *A．45°B．65°C．75°(正确答案)D．80°22、27．下列计算正确的是（）[单选题] *A．（﹣a3）2＝a6(正确答案)B．3a+2b＝5abC．a6÷a3＝a2D．（a+b）2＝a2+b223、16．“x2（x平方）－4x－5=0”是“x=5”的( ) [单选题] *A．充分不必要条件B．必要不充分条件(正确答案)C．充要条件D．既不充分也不必要条件24、19．下列两个数互为相反数的是（）[单选题] *A．（﹣）和﹣（﹣）B．﹣5和(正确答案)C．π和﹣14D．+20和﹣（﹣20）25、13．在数轴上，下列四个数中离原点最近的数是（）[单选题] *A．﹣4(正确答案)B．3C．﹣2D．626、14．命题“?x∈R，?n∈N*，使得n≥x2（x平方）”的否定形式是（）[单选题] * A．?x∈R，?n∈N*，使得n<x2B．?x∈R，?x∈N*，使得n<x2C．?x∈R，?n∈N*，使得n<x2D．?x∈R，?n∈N*，使得n<x2(正确答案)27、8．(2020·课标Ⅱ)已知集合U={－2，－1,0,1,2,3}，A={－1,0,1}，B={1,2}，则?U(A∪B)=( ) [单选题] *A．{－2,3}(正确答案)B．{－2,2,3}C．{－2，－1,0,3}D．{－2，－1,0,2,3}28、7人小组选出2名同学作正副组长，共有选法（）种。
Fermat滑铁卢数学竞赛(Grade 11)-数学Mathematics-2002-试题 exam
Canadian Instituteof Actuaries Chartered AccountantsSybasei Anywhere SolutionsScoring:There is no penalty for an incorrect answer.Each unanswered question is worth 2, to a maximum of 10 unanswered questions.Part A: Each correct answer is worth 5.1.If x =3, the numerical value of 522–x is(A ) –1(B ) 27(C ) –13(D )–31(E ) 32.332232++ is equal to(A ) 3(B ) 6(C ) 2(D )32(E ) 53.If it is now 9:04 a.m., in 56 hours the time will be(A ) 9:04 a.m.(B ) 5:04 p.m.(C ) 5:04 a.m.(D ) 1:04 p.m.(E ) 1:04 a.m.4.Which one of the following statements is not true?(A ) 25 is a perfect square.(B ) 31 is a prime number.(C ) 3 is the smallest prime number.(D ) 8 is a perfect cube.(E ) 15 is the product of two prime numbers.5. A rectangular picture of Pierre de Fermat, measuring 20 cmby 40 cm, is positioned as shown on a rectangular postermeasuring 50 cm by 100 cm. What percentage of the areaof the poster is covered by the picture?(A ) 24%(B ) 16%(C ) 20%(D ) 25%(E ) 40%6.Gisa is taller than Henry but shorter than Justina. Ivan is taller than Katie but shorter than Gisa. Thetallest of these five people is(A ) Gisa (B ) Henry (C ) Ivan (D ) Justina (E ) Katie7. A rectangle is divided into four smaller rectangles. Theareas of three of these rectangles are 6, 15 and 25, as shown.The area of the shaded rectangle is(A ) 7(B ) 15(C ) 12(D ) 16(E) 108.In the diagram, ABCD and DEFG are squares with equal side lengths, and ∠=°DCE 70. The value of y is (A ) 120(B ) 160(C ) 130(D ) 110(E ) 1409.The numbers 1 through 20 are written on twenty golf balls, with one number on each ball. The golfballs are placed in a box, and one ball is drawn at random. If each ball is equally likely to be drawn,what is the probability that the number on the golf ball drawn is a multiple of 3?(A )320(B )620(C )1020(D )520(E )12010.ABCD is a square with AB x =+16 and BC x =3, as shown.The perimeter of ABCD is(A ) 16(B ) 32(C ) 96(D ) 48(E ) 24Part B: Each correct answer is worth 6.11. A line passing through the points 02,−() and 10,() also passes through the point 7,b (). The numericalvalue of b is(A ) 12(B )92(C ) 10(D ) 5(E ) 1412.How many three-digit positive integers are perfect squares?(A ) 23(B ) 22(C ) 21(D ) 20(E ) 1913. A “double-single” number is a three-digit number made up of two identical digits followed by adifferent digit. For example, 553 is a double-single number. How many double-single numbers are there between 100 and 1000?(A ) 81(B ) 18(C ) 72(D ) 64(E ) 9014.The natural numbers from 1 to 2100 are entered sequentially in 7 columns, with the first 3 rows asshown. The number 2002 occurs in column m and row n . The value of m n + isColumn 1Column 2Column 3Column 4Column 5Column 6Column 7Row 1 1 2 3 4 5 6 7Row 2 8 91011121314Row 315161718192021M M M M M M M M(A ) 290(B ) 291(C ) 292(D ) 293(E ) 294x + 163xA BD C15.In a sequence of positive numbers, each term after the first two terms is the sum of all of the previousterms . If the first term is a ,the second term is 2, and the sixth term is 56, then the value of a is(A ) 1(B ) 2(C ) 3(D ) 4(E ) 516.If ac ad bc bd +++=68 and c d +=4, what is the value of a b c d +++?(A ) 17(B ) 85(C ) 4(D ) 21(E ) 6417.The average age of a group of 140 people is 24. If the average age of the males in the group is 21 andthe average age of the females is 28, how many females are in the group?(A ) 90(B ) 80(C ) 70(D ) 60(E ) 5018. A rectangular piece of paper AECD has dimensions 8 cm by 11 cm. Corner E is folded onto point F , which lies on DC ,as shown. The perimeter of trapezoid ABCD is closest to (A ) 33.3 cm (B ) 30.3 cm (C ) 30.0 cm(D ) 41.3 cm (E ) 35.6 cm 19.If 238610a b =(), where a and b are integers, then b a − equals(A ) 0(B ) 23(C )−13(D )−7(E )−320.In the diagram, YQZC is a rectangle with YC =8 and CZ = 15. Equilateral triangles ABC and PQR , each withside length 9, are positioned as shown with R and B on sidesYQ and CZ , respectively. The length of AP is (A ) 10(B )117(C ) 9(D ) 8(E )72Part C: Each correct answer is worth 8.21.If 31537521219⋅⋅⋅⋅+−=L n n , then the value of n is(A ) 38(B ) 1(C ) 40(D ) 4(E ) 3922.The function f x () has the property that f x y f x f y xy +()=()+()+2, for all positive integers x and y .If f 14()=, then the numerical value of f 8() is(A ) 72(B ) 84(C ) 88(D ) 64(E ) 80continued ...Figure 1Figure 223.The integers from 1 to 9 are listed on a blackboard. If an additional m eights and k nines are added tothe list, the average of all of the numbers in the list is 7.3. The value of k m + is(A ) 24(B ) 21(C ) 11(D ) 31(E ) 8924. A student has two open-topped cylindrical containers. (Thewalls of the two containers are thin enough so that theirwidth can be ignored.) The larger container has a height of20 cm, a radius of 6 cm and contains water to a depth of 17cm. The smaller container has a height of 18 cm, a radius of5 cm and is empty. The student slowly lowers the smallercontainer into the larger container, as shown in the cross-section of the cylinders in Figure 1. As the smaller container is lowered, the water first overflows out of the larger container (Figure 2) and then eventually pours into thesmaller container. When the smaller container is resting onthe bottom of the larger container, the depth of the water in the smaller container will be closest to(A ) 2.82 cm (B ) 2.84 cm (C ) 2.86 cm(D ) 2.88 cm (E ) 2.90 cm25.The lengths of all six edges of a tetrahedron are integers. The lengths of five of the edges are 14, 20,40, 52, and 70. The number of possible lengths for the sixth edge is(A ) 9(B ) 3(C ) 4(D ) 5(E ) 6。
Cayley滑铁卢数学竞赛(Grade 10)-数学Mathematics-1998-试题 exam
Chartered Accountants SybaseInc. (Waterloo) IBMCanada Ltd.Canadian Institute of ActuariesDo not open the contest booklet until you are told to do so.You may use rulers, compasses and paper for rough work.Calculators are permitted, providing they are non-programmable and without graphic displays.Part A: Each question is worth 5 credits.1.The value of 03012..()+ is(A ) 0.7(B ) 1(C ) 0.1(D ) 0.19(E ) 0.1092.The pie chart shows a percentage breakdown of 1000 votesin a student election. How many votes did Sue receive?(A ) 550(B ) 350(C ) 330(D ) 450(E ) 9353.The expression a a a 9153× is equal to(A ) a 45(B ) a 8(C ) a 18(D ) a 14(E ) a 214.The product of two positive integers p and q is 100. What is the largest possible value of p q +?(A ) 52(B ) 101(C ) 20(D ) 29(E ) 255.In the diagram, ABCD is a rectangle with DC =12. If the area of triangle BDC is 30, what is the perimeter ofrectangle ABCD ?(A ) 34(B ) 44(C ) 30(D ) 29(E ) 606.If x =2 is a solution of the equation qx –311=, the value of q is (A ) 4(B ) 7(C ) 14(D ) –7(E ) –47.In the diagram, AB is parallel to CD . What is the value ofy ?(A ) 75(B ) 40(C ) 35(D ) 55(E ) 508.The vertices of a triangle have coordinates 11,(), 71,() and 53,(). What is the area of this triangle?(A ) 12(B ) 8(C ) 6(D ) 7(E ) 99.The number in an unshaded square is obtained by adding thenumbers connected to it from the row above. (The ‘11’ is one such number.) The value of x must be (A ) 4(B ) 6(C ) 9(D ) 15(E) 10Scoring:There is no penalty for an incorrect answer.Each unanswered question is worth 2 credits, to a maximum of 20 credits.A BCD DAC B10.The sum of the digits of a five-digit positive integer is 2. (A five-digit integer cannot start with zero.)The number of such integers is(A ) 1(B ) 2(C ) 3(D ) 4(E ) 5Part B: Each question is worth 6 credits.11.If x y z ++=25, x y +=19 and y z +=18, then y equals(A ) 13(B ) 17(C ) 12(D ) 6(E ) –612. A regular pentagon with centre C is shown. The value of xis(A ) 144(B ) 150(C ) 120(D ) 108(E ) 7213.If the surface area of a cube is 54, what is its volume?(A ) 36(B ) 9(C ) 8138(D ) 27(E ) 162614.The number of solutions x y ,() of the equation 3100x y +=, where x and y are positive integers, is(A ) 33(B ) 35(C ) 100(D ) 101(E ) 9715.If y –55= and 28x =, then x y + equals(A ) 13(B ) 28(C ) 3316.Rectangle ABCDhas length 9 and width 5. Diagonal is divided into 5 equal parts at W , X , Y , and Z area of the shaded region.(A ) 36(B ) 365(C ) 18(D ) 41065(E ) 2106517.If N p q =()()()+75243 is a perfect cube, where p and q are positive integers, the smallest possible valueof p q + is(A ) 5(B ) 2(C ) 8(D ) 6(E ) 1218.Q is the point of intersection of the diagonals of one face ofa cube whose edges have length 2 units. The length of QRis(A ) 2(B ) 8(C ) 5(D ) 12(E ) 619.Mr. Anderson has more than 25 students in his class. He has more than 2 but fewer than 10 boys andmore than 14 but fewer than 23 girls in his class. How many different class sizes would satisfy these conditions?(A ) 5(B ) 6(C ) 7(D ) 3(E ) 420.Each side of square ABCD is 8. A circle is drawn through A and D so that it is tangent to BC . What is the radius of thiscircle?(A ) 4(B ) 5(C ) 6(D ) 42(E ) 5.25Part C: Each question is worth 8 credits.21.When Betty substitutes x =1 into the expression ax x c 32–+ its value is –5. When she substitutesx =4 the expression has value 52. One value of x that makes the expression equal to zero is(A ) 2(B ) 52(C ) 3(D ) 72(E ) 422. A wheel of radius 8 rolls along the diameter of a semicircleof radius 25 until it bumps into this semicircle. What is thelength of the portion of the diameter that cannot be touchedby the wheel?(A ) 8(B ) 12(C ) 15(D ) 17(E ) 2023.There are four unequal, positive integers a , b , c , and N such that N a b c =++535. It is also true thatN a b c =++454 and N is between 131 and 150. What is the value of a b c ++?(A ) 13(B ) 17(C ) 22(D ) 33(E ) 3624.Three rugs have a combined area of 2002m . By overlapping the rugs to cover a floor area of 1402m ,the area which is covered by exactly two layers of rug is 242m . What area of floor is covered by three layers of rug?(A ) 122m (B ) 182m (C ) 242m (D) 362m (E ) 422m 25.One way to pack a 100 by 100 square with 10000 circles, each of diameter 1, is to put them in 100rows with 100 circles in each row. If the circles are repacked so that the centres of any three tangent circles form an equilateral triangle, what is the maximum number of additional circles that can be packed?(A ) 647(B ) 1442(C ) 1343(D) 1443(E ) 1344。
欧几里得滑铁卢数学竞赛_2011EuclidSolution
=
6,
then
∠BDA
=
∠BAD
=
1 2
(180◦
− ∠DBA)
=
1 2
(180◦
−
30◦)
=
75◦.
We calculate the length of AD.
Method 1
AD
BA
By the Sine Law in DBA, we have
=
.
sin(∠DBA) sin(∠BDA)
6 sin(30◦)
4. (a) We consider choosing the three numbers all at once. We list the possible sets of three numbers that can be chosen:
{1, 2, 3} {1, 2, 4} {1, 2, 5} {1, 3, 4} {1, 3, 5} {1, 4, 5} {2, 3, 4} {2, 3, 5} {2, 4, 5} {3, 4, 5}
∠B C D)
=
1 2
(180◦
−
60◦)
=
60◦.
Therefore, BCD is equilateral, and so BD = BC = CD = 6.
Consider DBA.
Note that ∠DBA = 90◦ − ∠CBD = 90◦ − 60◦ = 30◦.
Since
BD
=
BA
∠ADC = 360◦ − ∠ADB − ∠CDB = 360◦ − 130◦ − 150◦ = 80◦ .
(c) By the Pythag√orean Theorem in EAD, we have EA2 +AD2 = ED2 or 122 +AD2 = 132, and so AD = 169 − 144 = 5, since AD > 0. By the Pythag√orean Theorem in ACD, we have AC2 + CD2 = AD2 or AC2 + 42 = 52, and so AC = 25 − 16 = 3, since AC > 0. (We could also have determined the lengths of AD and AC by recognizing 3-4-5 and 5-12-13 right-angled triangles.) By the Pythag√orean The√orem in ABC, we have AB2 + BC2 = AC2 or AB2 + 22 = 32, and so AB = 9 − 4 = 5, since AB > 0.
欧几里得数学竞赛奖项设置
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。
1999滑铁卢竞赛试题答案
( x – 2)( x + 1) = 0
∴ x = 2 or x = –1
Solution 2 1 6 x2 – x – 6 1– – 2 = x x x2 x 2 – x – 2) – 4 ( = x2 –4 2 2 (since x – x – 2 = 0 ) x 2 But x – x – 2 = ( x – 2)( x + 1) = 0 ∴ x = 2 or x = –1. x = –1, Substituting x = 2, or –4 –4 = 4 1 = –1. = –4 =
y
6
–3
–1 O
x
Solution 3 Let the equation of the parabola be y = a( x + 2)2 + c . Since (0, 6) is on parabola, 6 = 4 a + c , and ( –1, 0) is on parabola, 0 = a + c . Solving, a = 2, c = – 2. ∴ Equation is y = 2( x + 2)2 – 2 . 3. (a) How many equilateral triangles of side 1 cm, placed as shown in the diagram, are needed to completely cover the interior of an equilateral triangle of side 10 cm?
(b)
If the point P( – 3, 2) is on the line 3 x + 7ky = 5 , what is the value of k? Solution Since P is on the line, its coordinates must satisfy the equation of the line. Thus, 3( – 3) + 7k (2) = 5 14 k = 14 k =1