英国数学竞赛JMO2016年真题
2016年第57届imo中国国家队选拔考试试题(pdf版)
目录
第 57 届国际数学奥林匹克中国国家队选拔考试一 第一天 2016 年 3 月 15 日上午 8:00-12：30 . . . . . . . . . . . . . . . . . . . . . . . . . 4
第 57 届国际数学奥林匹克中国国家队选拔考试一 第二天 2016 年 3 月 16 日上午 8:00-12：30 . . . . . . . . . . . . . . . . . . . . . . . . . 5
第 57 届国际数学奥林匹克中国国家队选拔考试三 第一天 2016 年 3 月 25 日上午 8:00-12：30 . . . . . . . . . . . . . . . . . . . . . . . . . 8
第 57 届国际数学奥林匹克中国国家队选拔考试三 第二天 2016 年 3 月 26 日上午 8:00-12：30 . . . . . . . . . . . . . . . . . . . . . . . . . 9
第 57 届国际数学奥林匹克中国国家队选拔考试二 第一天 2016 年 3 月 20 日上午 8:00-12：30 . . . . . . . . . . . . . . . . . . . . . . . . . 6
第 57 届国际数学奥林匹克中国国家队选拔考试二 第二天 2016 年 3 月 21 日上午 8:00-12：30 . . . . . . . . . . . . . . . . . . . . . . . . . 7
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2016年高中数学竞赛b试题答案
2016年高中数学竞赛b试题答案2016年高中数学竞赛B试题的答案如下：选择题：1. 答案：A解析：根据题目所给条件，我们可以通过代入验证法或者排除法来确定正确答案。
例如，将选项A代入题目的等式中，如果满足条件，则A为正确答案。
2. 答案：B解析：此题考查了函数的性质，需要利用函数的单调性、奇偶性等性质来求解。
3. 答案：C解析：本题需要运用数列的通项公式和求和公式，通过计算来确定答案。
4. 答案：D解析：考查了几何图形的性质，需要通过几何证明或者代数方法来求解。
5. 答案：E解析：此题涉及到概率统计的知识，需要根据题目所给的条件，运用概率公式来计算。
填空题：1. 答案：3解析：根据题目所给的数列规律，可以推导出答案。
2. 答案：\( \sqrt{2} \)解析：此题考查了二次根式的性质，需要通过化简来求解。
3. 答案：5解析：根据题目所给的几何图形，可以利用面积公式来求解。
4. 答案：\( \frac{\pi}{4} \)解析：此题考查了三角函数的求值，需要运用三角函数的性质和公式。
5. 答案：\( x^2 - 4x + 3 \)解析：本题需要运用因式分解的方法来求解。
解答题：1. 答案：首先设未知数，然后建立方程组，通过解方程组来求解。
2. 答案：根据题目所给的函数表达式，我们可以利用函数的性质来求解。
3. 答案：此题需要运用数列的递推关系，通过递推公式来求解。
4. 答案：本题考查了几何证明，需要运用几何定理和公理来证明。
5. 答案：此题需要运用组合数学的知识，通过组合公式来求解。
请注意，以上答案和解析是根据一般性描述给出的，具体的题目内容和答案可能会有所不同。
如果需要针对具体题目的详细解析，请提供具体的题目内容。
2016年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 the average of the integers 5,15,25,35,45,55?(b)If x 2=2016,what is the value of (x +2)(x −2)?(c)In the diagram,points P (7,5),Q (a,2a ),andR (12,30)lie on a straight line.Determine the value of a .x2.(a)What are all values of n for which n 9=25n?(b)What are all values of x for which (x −3)(x −2)=6?(c)At Willard’s Grocery Store,the cost of 2apples is the same as the cost of3bananas.Ross buys 6apples and 12bananas for a total cost of $6.30.Determinethe cost of 1apple.3.(a)In the diagram,point B is on AC ,point F is on DB ,and point G is on EB .A B CDE F G p ˚q ˚u ˚t ˚r ˚s ˚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 parabolais at V ,which is below the x -axis.If the area of V P Q is 12,determine thecoordinates of V .4.(a)Determine all angles θwith 0◦≤θ≤180◦and sin 2θ+2cos 2θ=74.(b)The sum of the radii of two circles is 10cm.The circumference of the largercircle is 3cm greater than the circumference of the smaller circle.Determine thedifference between the area of the larger circle and the area of the smaller circle.5.(a)Charlotte’s Convenience Centre buys a calculator for $p (where p >0),raises itsprice by n %,then reduces this new price by 20%.If the final price is 20%higherthan $p ,what is the value of n ?(b)A function f is defined so that if n is an odd integer,then f (n )=n −1and if n isan even integer,then f (n )=n 2−1.For example,if n =15,then f (n )=14andif n =−6,then f (n )=35,since 15is an odd integer and −6is an even integer.Determine all integers n for which f (f (n ))=3.6.(a)What is the smallest positive integer x for which 132=x 10yfor some positive integer y ?(b)Determine all possible values for the area of a right-angled triangle with one sidelength equal to 60and with the property that its side lengths form an arithmeticsequence.(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.)7.(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 7km/h and swim at 2km/h.They start from thesame point at the same time with Amrita paddling and Zhang swimming.After awhile,Amrita stops the kayak and immediately starts swimming.Upon reachingthe kayak (which has not moved since Amrita started swimming),Zhang gets inand immediately starts paddling.They arrive on the far side of the lake at thesame time,90minutes after they began.Determine the amount of time duringthese 90minutes that the kayak was not being paddled.(b)Determine all pairs (x,y )of real numbers that satisfy the system of equationsx 12+y −2x 2 =0y 52+x −y =08.(a)In the diagram,ABCD is a parallelogram.Point E is on DC with AE perpendicularto DC ,and point F is on CB with AFperpendicular to CB .If AE =20,AF =32,and cos(∠EAF )=13,determine the exactvalue of the area of quadrilateral AECF .A B C D E F 2032(b)Determine all real numbers x >0for whichlog 4x −log x 16=76−log x 89.(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 justification,the total number of strings of ten letters,each ofwhich is A or B ,that do not include the consecutive letters ABBA .(b)In the diagram,ABCD is a square.PointsE andF are chosen on AC so that∠EDF =45◦.If AE =x ,EF =y ,andF C =z ,prove that y 2=x 2+z 2.A B C DEF x yz45˚10.Let k be a positive integer with k≥2.Two bags each contain k balls,labelled withthe positive integers from1to k.Andr´e removes one ball from each bag.(In eachbag,each ball is equally likely to be chosen.)Define P(k)to be the probability thatthe product of the numbers on the two balls that he chooses is divisible by k.(a)Calculate P(10).(b)Determine,with justification,a polynomial f(n)for which•P(n)≥f(n)n2for all positive integers n with n≥2,and•P(n)=f(n)n2for infinitely many positive integers n with n≥2.(A polynomial f(x)is an algebraic expression of the form f(x)=a m x m+a m−1x m−1+···+a1x+a0for some integer m≥0and for some real numbers a m,a m−1,...,a1,a0.)(c)Prove there exists a positive integer m for which P(m)>2016 m.Euclid Contest(English) 2016。
2016 AMC 8 考题及答案
2016 AMC 8 考题及答案Problem 1The longest professional tennis match lasted a total of 11 hours and 5 minutes. How many minutes was that?Problem 2In rectangle , and . Point is the midpoint of . What is the area of ?Problem 3Four students take an exam. Three of their scores are and . If the average of their four scores is , then what is the remaining score?Problem 4When Cheenu was a boy he could run miles in hours and minutes. As an old man he can now walk miles in hours. How many minutes longer does it take for him to travel a mile now compared to when he was a boy?Problem 5The number is a two-digit number.• When is divided by , the remainder is .• When is divided by , the remainder is .What is the remainder when is divided by ?Problem 6The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names?Problem 7Which of the following numbers is not a perfect square?Problem 8Find the value of the expressionProblem 9What is the sum of the distinct prime integer divisors of ?Problem 10Suppose that means What is the value of ifProblem 11Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum isProblem 12Jefferson Middle School has the same number of boys and girls. of the girls and of the boys went on a field trip. What fraction of the students on the field trip were girls?Problem 13Two different numbers are randomly selected from the setand multiplied together. What is the probability that the product is ?Problem 14Karl's car uses a gallon of gas every miles, and his gas tank holds gallons when it is full. One day, Karl started with a full tank of gas, drove miles, bought gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day?Problem 15What is the largest power of that is a divisor of ?Problem 16Annie and Bonnie are running laps around a -meter oval track. They started together, but Annie has pulled ahead because she runs faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?Problem 17An ATM password at Fred's Bank is composed of four digits from to , with repeated digits allowable. If no password may begin with the sequence then how many passwords are possible?Problem 18In an All-Area track meet, sprinters enter a meter dash competition. The track has lanes, so only sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion sprinter?Problem 19The sum of consecutive even integers is . What is the largest of these consecutive integers?Problem 20The least common multiple of and is , and the least common multiple of and is . What is the least possible value of the least common multiple of and ?Problem 21A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?Problem 22Rectangle below is a rectangle with . What is the area of the "bat wings" (shaded area)?Problem 23Two congruent circles centered at points and each pass through the other circle's center. The line containing both and is extended to intersect the circles at points and . The circles intersect at two points, one of which is . What is the degree measure of ?Problem 24The digits , , , , and are each used once to write a five-digit number . The three-digit number is divisible by , the three-digit number is divisible by , and the three-digit number is divisible by . What is ?Problem 25A semicircle is inscribed in an isosceles triangle with base and heightso that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?2016 AMC 8 Answer Key1.C2.A3.A4.B5.E6.B7.B8.C9.B10.D11.B12.B13.D14.A15.C16.D17.D18.C19.E20.A21.B22.C23.C24.A25.B。
英国数学竞赛JMO2018年真题
The UnLeabharlann ted Kingdom Mathematics Trust is a Registered Charity.
©UKMT 2018
Section A
Try to complete Section A within 30 minutes or so. Only answers are required.
1. Time allowed: 2 hours.
2. The use of calculators, measuring instruments and squared paper is forbidden. 3. All candidates must be in School Year 8 or below (England and Wales), S2 or below
5. Questions A1-A10 are relatively short questions. Try to complete Section A within the first 30 minutes so as to allow well over an hour for Section B.
高中化学国际奥林匹克竞赛样题2016第一轮-参考答案
= 0.0120 g cm−3
2
Give credit if they use 24 dm3 for 1 mol of gas as a known value at STP.
(f) WF6 (g) + 4H2O (l) → H2WO4 (aq) + 6HF (aq)
1
WF6 (g) + 3H2O (l) → WO3 (s) + 6HF (aq) Accept either. State symbols not required.
48th INTERNATIONAL CHEMISTRY OLYMPIAD
2016 UK Round One MARK SCHEME
Although we would encourage students to always quote answers to an appropriate number of significant figures, do not penalise students for significant figure errors. Allow where a student’s answers differ slightly from the mark scheme due to the use of rounded/non-rounded data from an earlier part of the question.
(iii) B SF6 is kinetically stable but thermodynamically unstable
1
(h) Tungsten = 1 atom inside unit cell + 4 × atoms on face + 8 × atoms on corner
jmo-2016-sample-solutions英国小学生数学竞赛试题解答
The Junior Mathematical Olympiad (JMO) has long aimed to help introduce able students to (and to encourage them in) the art of problem solving and proof. The problems are the product of theimaginations of a small number of volunteers writing for the JMO problems group. After the JMO, model solutions for each problem are published in the solutions booklet and the UKMT Yearbook.However, these neatly printed solutions usually represent only one approach (often where several are feasible) and convey none of the sense of investigation, rough working or false starts that usually precede the finished article.In marking JMO scripts, markers want to encourage complete (and concise) proofs in section B but, in doing so, are content to accept less than perfectly written answers, as long as it is clear where the proof is going – we appreciate that there is a limited time available, and that to polish and to present takes the place of considering another problem. Nevertheless, clarity and insight are what are looked for.It is not easy to generalise about what makes a good solution or how a candidate can achieve success in the JMO, but there are a few points to ponder on:∙using trial and improvement to a large degree is generally considered not very mathematical – it will lend little insight into the structure of the problem, and even if you get an answer that works, there should always be a concern that it is, perhaps, not the only answer; ∙if you do have to resort to calculating your way through a large number of cases, then the calculations should be shown as part the solution – the reader should not have to make suppositions about what you have tried (or not tried); ∙diagrams should be large enough to contain all subsequent details – however, the use of just one diagram can convey very little of the order of the proof, especially in geometrical questions; ∙avoid long sentences, particularly those which frequently use if and could be . ∙the use of algebra makes it possible to express connections in simple ways, where a huge number of words would be otherwise necessary; ∙if algebra is used, variables should be fully declared; ∙ attempt as many questions as you can do really well in – it is better, in terms not only of scoring marks but also of honest satisfaction, to spend your time concentrating on a few questions and providing full, clear and accurate solutions, rather than to have a go at everything you can, and to achieve not very much in any of them. Out of the 1000 students or so who take part in the JMO, generally only a handful achieve full marks in all six section B questions.If this all sounds rather negative, it has, in balance, to be said that JMO candidates produce an astonishing amount of work that is worthy of high praise. We hope that the work below will give future candidates a flavour of what is to be encouraged, in style, method and presentation.Junior Mathematical Olympiad 2016We have collected below a small number of solutions submitted for the JMO in 2016, with the aim thatfuture candidates can see what some students in this age group do achieve and that they might aspire toemulate it. In many ways they are ordinary solutions, not brilliantly and startlingly original – butmathematically they are to be commended, in particular, for the logical progression from one point to the next, and for clear presentation.Solution:Solution:Solution:Solution:Solution:Solution:B6 continued:。
2016英国数学奥林匹克竞赛题-初中-第一轮
United Kingdom Mathematics Trust British Mathematical OlympiadRound1:Friday,27November2015Time allowed312hours.Instructions•Full written solutions–not just answers–arerequired,with complete proofs of any assertionsyou may make.Marks awarded will depend on theclarity of your mathematical presentation.Workin roughfirst,and then write up your best attempt.Do not hand in rough work.•One complete solution will gain more credit thanseveral unfinished attempts.It is more importantto complete a small number of questions than totry all the problems.•Each question carries10marks.However,earlierquestions tend to be easier.In general you areadvised to concentrate on these problemsfirst.•The use of rulers,set squares and compassesis allowed,but calculators and protractors areforbidden.•Start each question on a fresh sheet of paper.Writeon one side of the paper only.On each sheet ofworking write the number of the question in thetop left hand corner and your name,initials andschool in the top right hand corner.•Complete the cover sheet provided and attach it tothe front of your script,followed by your solutionsin question number order.•Staple all the pages neatly together in the top lefthand corner.•To accommodate candidates sitting in other timezones,please do not discuss the paper onthe internet until8am GMT on Saturday28November.Do not turn over until toldto do so.United Kingdom Mathematics Trust 2015/16British Mathematical OlympiadRound1:Friday,27November20151.On Thursday1st January2015,Anna buys one book and one shelf.For the next two years,she buys one book every day and one shelf on alternate Thursdays,so she next buys a shelf on15th January2015.On how many days in the period Thursday1st January2015until (and including)Saturday31st December2016is it possible for Anna to put all her books on all her shelves,so that there is an equal number of books on each shelf?2.Let ABCD be a cyclic quadrilateral and let the lines CD and BAmeet at E.The line through D which is tangent to the circle ADE meets the line CB at F.Prove that the triangle CDF is isosceles.3.Suppose that a sequence t0,t1,t2,...is defined by a formula t n=An2+Bn+C for all integers n≥0.Here A,B and C are real constants with A=0.Determine values of A,B and C which give the greatest possible number of successive terms of the sequence which are also successive terms of the Fibonacci sequence.The Fibonacci sequence is defined by F0=0,F1=1and F m=F m−1+F m−2for m≥2.4.James has a red jar,a blue jar and a pile of100pebbles.Initially bothjars are empty.A move consists of moving a pebble from the pile into one of the jars or returning a pebble from one of the jars to the pile.The numbers of pebbles in the red and blue jars determine the state of the game.The following conditions must be satisfied:a)The red jar may never contain fewer pebbles than the blue jar;b)The game may never be returned to a previous state.What is the maximum number of moves that James can make?5.Let ABC be a triangle,and let D,E and F be the feet of theperpendiculars from A,B and C to BC,CA and AB respectively.Let P,Q,R and S be the feet of the perpendiculars from D to BA, BE,CF and CA respectively.Prove that P,Q,R and S are collinear.6.A positive integer is called charming if it is equal to2or is of the form3i5j where i and j are non-negative integers.Prove that every positive integer can be written as a sum of different charming integers.。
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Organised by The United Kingdom Mathematics Trust
Tuesday 14th June 2016
RULES AND GUIDELINES : READ THESE INSTRUCTIONS CAREFULLY BEFORE STARTING
(Scotland), School Year 9 or below (Northern Ireland). 4. Write in blue or black pen or pencil.
For questions in Section A only the answer is required. Enter each answer neatly in the relevant box on the Front Sheet. Do not hand in rough work. For questions in Section B you must give full written solutions, including clear mathematical explanations as to why your method is correct. Solutions must be written neatly on A4 paper. Sheets must be STAPLED together in the top left corner with the Front Sheet on top. Do not hand in rough work. 5. Questions A1-A10 are relatively short questions. Try to complete Section A within the first 30 minutes so as to allow well over an hour for Section B. 6. Questions B1-B6 are longer questions reqeans that each answer must be accompanied by clear explanations and proofs. Work in rough first, then set out your final solution with clear explanations of each step. 7. These problems are meant to be challenging! Do not hurry. Try the earlier questions in each section first (they tend to be easier). Try to finish whole questions even if you are not able to do many. A good candidate will have done most of Section A and given solutions to at least two questions in Section B. 8. Answers must be FULLY SIMPLIFIED, and EXACT using symbols like π, fractions, or square roots if appropriate, but NOT decimal approximations.
10p + 1, where p is the previous term. What is the sum of the first ten terms?
A4. The diagram shows a regular hexagon with area 48 m2.
1. Time allowed: 2 hours. 2. The use of calculators, measuring instruments and squared paper is forbidden. 3. All candidates must be in School Year 8 or below (England and Wales), S2 or below
A1. Roger picks two consecutive integers, one of which ends in a 5. He multiplies the integers
together and then squares the result.
What are the last two digits of his answer?
A2. Three isosceles triangles are put together to
create a larger isosceles triangle, as shown.
What is the value of x?
x° 35°
A3. The first term of a sequence is 0. Each term of the sequence after the first term is equal to
DO NOT OPEN THE PAPER UNTIL INSTRUCTED BY THE INVIGILATOR TO DO SO!
The United Kingdom Mathematics Trust is a Registered Charity.
Section A
Try to complete Section A within 30 minutes or so. Only answers are required.