初中二年级奥数题.doc

Individual ContestEnglish VersionInstructions: Do not turn to the first page until you are told to do so. Remember to write down your team name, your name andcontestant number in the spaces indicated on the first page. The Individual Contest is composed of two sections with atotal of 120 points. Section A consists of 12 questions in which blanks are to befilled in and only ARABIC NUMERAL answers arerequired. For problems involving more than one answer,points are given only when ALL answers are correct. Eachquestion is worth 5 points. There is no penalty for a wronganswer. Section B consists of 3 problems of a computational nature,and the solutions should include detailed explanations. Eachproblem is worth 20 points, and partial credit may beawarded. You have a total of 120 minutes to complete thecompetition. No calculator, calculating device, watches or electronicdevices are allowed. Answers must be in pencil or in blue or black ball point pen. All papers shall be collected at the end of this test.Invitational World Youth Mathematics Intercity CompetitionIndividual ContestTime limit: 120 minutes 20th July 2011 Bali, IndonesiaTeam: Name: No.: Score:Section A.In this section, there are 12 questions. Fill in the correct answer in the spaceprovided at the end of each question. Each correct answer is worth 5 points.1. Let a, b, and c be positive integers such that2( ) 8045,2.ab bc ca a b cabc a b c+ + + + + = = Find the value of a+b+c.Answer2. There are two kinds of students in a certain class, those who always lie and thosewho never lie. Each student knows what kind each of the other students is. In ameeting today, each student tells what kind each of the other students is. Theanswer “liar” is given 240 times. Yesterday a similar meeting took place, but oneof the students did not attend. The answer “liar” was given 216 times then. Howmany students are present today?Answer3. The product 1!2! 2011! 2012! is written on the blackboard. Which factor,in term of a factorial of an integer, should be erased so that the product of theremaining factors is the square of an integer? (The factorial sign n! stands for theproduct of all positive integers less than or equal to n.)Answer4. B and E are points on the sides AD and AC of trianglesACD such that BC and DE intersect at F. TrianglesABC and AED are congruent. Moreover, AB=AE=1 andAC=AD=3. Determine the ratio between the areas of thequadrilateral ABFE and the triangle ADC.Answer5. A positive integer n has exactly 4 positive divisors, including 1 and n.Furthermore, n+1 is four times the sum of the other two divisors. Find n.AnswerInvitational World Youth Mathematics Intercity CompetitionE FDCA B6. Jo tells Kate that the product of three positive integers is 36. Jo also tells herwhat the sum of the three numbers is, but Kate still does not know what thethree numbers are. What is the sum of the three numbers?Answer7. Two circles A and B, both with radius 1, touch eachother externally. Four circles P, Q, R and S, all with thesame radius r, are such that P touches A, B, Q and Sexternally; Q touches P, B and R externally; R touchesA, B, Q and S externally; and S touches P, A and Rexternally. Calculate r.Answer8. Find the smallest positive common multiple of 7 and 8 such that each digit iseither 7 or 8, there is at least one 7 and there is at least one 8.Answer9. The side lengths of a triangle are 50 cm, 120 cm and 130 cm. Find the area ofthe region consisting of all the points, inside and outside the triangle, whosedistances from at least one point on the sides of the triangle are 2 cm. Take227 = .Answer10. Find the number of positive integers which satisfy the following conditions:(1) It contains 8 digits each of which is 0 or 1.(2) The first digit is 1.(3) The sum of the digits on the even places equals the sum of the digits onthe odd places.Answer11. A checker is placed on a square of an infinite checkerboard, where each squareis 1 cm by 1 cm. It moves according to the following rules: In the first move, the checker moves 1 square North. All odd numbered moves are North or South and all even numbered movesare East or West. In the n-th move, the checker makes a move of n squares in the samedirection.The checker makes 12 moves so that the distance between the centres of itsinitial and final squares is as small as possible. What is this minimum distance?Answer cm12. Let a, b and c be three real numbers such that( )( )( )( ) 0a b c b c akb c a c b a = = for some constant k. Find the greatest integer less than or equal to k.AnswerS BRQPASection B.Answer the following 3 questions, and show your detailed solution in thespace provided after each question. Each question is worth 20 points.1. The diagonals AC and BD of a quadrilateral ABCD intersect at a point E. IfAE=CE and ABC=ADC, does ABCD have to be a parallelogram?2. When a=1, 2, 3, , 2010, 2011, the roots of the equation x2 2x a2 a = 0 are1 1 2 2 3 3 2010 2010 2011 2011 ( , ), ( , ), ( , ),L, ( , ), ( , ) respectively.Evaluate1 1 2 2 3 3 2010 2010 2011 20111 1 1 1 1 1 1 1 1 1 + + + + + + + + + .3. Consider 15 rays that originate from one point. What is the maximum number ofobtuse angles they can form? (The angle between any two rays is taken to be lessthan or equal to 180)个人赛英文版本说明：不要打开第一页，直到你被告知这样做。请写下您的队名，您的姓名和参赛者在空格中的数字表示在第一页。个人赛两部分组成，与共120分。第一个由12个问题，其中空白填写阿拉伯数字答案是必需的。对于涉及多个答案的问题，点只有当所有的答案是正确的的。每个问题是得5分。有没有一个错误的惩罚答案。B节由3性质的计算问题，和解决方案应包括详细的解释。每个问题是值得20分，部分信贷可能奖励。你有一个120分钟的总完成竞争。没有计算器，计算设备，手表或电子设备是允许的。答案必须用铅笔或蓝色或黑色圆珠笔。所有文件应收集在本次测试结束。邀请赛世界青年数学际比赛个人赛时间限制：120分钟7月20日2011年巴厘岛，印尼团队：名称：编号：分数：A节在本节中，有12个问题。在正确的答案填写在空间提供的每一个问题的结束。每个正确的答案是得5分。1。设a，b和c，这样的正整数2（）8045，2。AB BC CA A B C农行A B C+ + = - - - = - 查找A + B + C的值答2。有两种学生在一定的阶级，这些人总是说谎人永远不会说谎。每个学生都知道什么样的其他学生，每个。在今天的会议，每一位学生告诉什么样的其他学生。 “骗子”的答案是240倍。昨天发生了一次类似的会议，但学生没有出席。 “骗子”的答案是216倍即可。如何今天在座的很多学生都？答3。该产品为1 2！ . 2011！ 2012年！写在黑板上。哪些因素，在长期的一个整数的阶乘，应该清除，使产品的其余的因素是一个整数的平方？ （阶乘符号n!的代表所有的正整数的乘积小于或等于n）答4。 B和E是AD和AC两侧的三角形的点。亚洲合作对话等，BC和DE相交楼三角形ABC和AED是一致的。此外，AB = AE = 1和AC = AD = 3。确定的领域之间的比例ABFE四边形和三角形ADC。答5。正整数n恰好有4个积极的，包括1和n除数，此外，N +1是其他两个约数的总和的四倍。查找N.答邀请赛世界青年数学际比赛E F彗星A B6。乔告诉凯特，三个正整数的乘积是36。曹某还告诉她这三个数字的总和是什么，但凯特仍然不知道什么三个数字。三个数字的总和是什么？答7。 A和B两个圆圈，两个半径为1，相互接触其他外部。四界P，Q，R和S，所有与相同的半径为r，使得P接触，A，B，Q和S外部，Q接触磷，B和R外部，R触及，A，B，Q和S外部和S触及P，A和R外部。计算规则答8。找到7的最小正常见多发和等8个，每个数字7或8，有至少有一个7，并有至少有一个8。答9。三角形的边长50厘米，