We congratulate you on your achievement in reaching the final stage of the Ulpaniada Mathematics Competition and wish you continued success.

Transcription

1 בס ד, כ באייר תש ע Dear Participant, We congratulate you on your achievement in reaching the final stage of the Ulpaniada Mathematics Competition and wish you continued success. Please fill in your personal details on this page before you start answering the questions. Name Grade level address: Name of High School: Town Country/State This question paper consists of two parts. When you have completed the first part and circled your answers, please copy them into this table (a,b,c,d or e) Question number Your answer Wishing you much hatzlacha and bracha, The Ulpaniada Team Mathematics Department, Michlalah College, Jerusalem Ulpaniada Math Department, Michlalah Jerusalem College P.O.B , Bayit Vegan, Jerusalem 91160, Israel Tel: , Fax: ulpaniada@macam.ac.il

2 This question paper consists of two parts, in which you can accumulate a total of 120 points. Part I: 14 multiple choice questions (84 points) Part II: 2 open questions (36 points) You have three and a half hours to complete the whole test. The use of a calculator is permitted B hatzlacha! Part One: There are 5 possible answers given to each question, only one of which is correct. Read the question carefully, solve it, and then encircle the correct answer. Each question is worth 6 points. 1 Which of the following sets of numbers has the highest average? a. The two digit even numbers. b. The two digit numbers which are multiples of 3. c. The two digit numbers which are multiples of 4. d. The two digit numbers which are multiples of 5. e. The two digit numbers which are multiples of 6.

3 3 2 Consider a 770X770 square grid. Its squares are colored in three colors - red, green and yellow in the following order: start from the square at the top left corner, and move towards the right: red, green, yellow, red, green, yellow, etc. At the end of the first row, the order progresses according to the arrows, as is shown in the diagram. What are the colors of the squares in the two bottom corners? a. The right one - red, and the left one - green. b. The right one - yellow, and the left one - red. c. The right one - yellow, and the left one - green. d. The right one - green, and the left one - red. e. The right one - red, and the left one - yellow. 3 Among all the five digit numbers of the form 5a83b, there are some which are divisible by 36. The sum of such numbers is: a b c d e Consider a cube. Every three vertices of the cube form a triangle. How many non congruent triangles can be formed in this way? a. 2 b. 3 c. 4 d. 5 e. 6

4 5 The figure below consists of six small triangles, which have a common vertex in the centre. There are three circles on each side of every triangle. Enter the numbers 1-19 into these circles, so that each number appears in exactly one circle, and so that the sum of the numbers along each side of each triangle is the same. If the numbers 1,2,3 and 6 are located as shown in the diagram, what is the number in the yellow circle? a. 17 b. 4 c. 16 d. 18 e There are odd numbers smaller than 1000, such that the product of each individual number s digits is 180. How many numbers like this are there? a. 2 b. 3 c. 4 d. 5 e. 6

5 7 On the island of knights and Knaves, the knights always tell the truth, whilst the Knaves always lie. The island has 200 inhabitants. Every inhabitant has one of the following jobs: fishing, banana growing or rug weaving. Each one of the inhabitants was asked three questions to which he answered yes or no. The question: Are you a fisherman? was answered yes by 80 people. The question: Do you grow bananas? was answered yes by 95 people. The question: Do you weave rugs? was answered yes by 75 people. How many knights live on the island? a. 50 b. 80 c. 95 d.100 e If the same digit is added before and after a two digit number N, the result is a four digit number which is 36 times the original number. The sum of N s digits is: a. 11 b. 12 c. 13 d. 14 e f(x)is a function which is defined for all x 0 and satisfies: The value of f(-3) is : f 1 1 ( ) f ( x) = x x x 3 a. -6 b. 9 c. -3 d. 1/3 e. -13

6 האולפני א ד ה ה מ מ ט י שלב הגמר - ש ע he 3rd stage t : s l r i g ish Ulpaniada - the math contest for Jew 10 During the archeological excavations in the Jewish Quarter in Jerusalem, a mosaic floor and some vessels were discovered with a six leafed rose on them. This rose is constructed from arcs made out of six circles with the same radius R, passing through one point. 16 The area of the six leafed rose a. R 2 3 b. R 2 6 is c. R 3 3R d. R 2 (2 3 3 ) e. R 2 11 The set {7, 83, 421, 659 } is a set of prime numbers, containing each digit between 1 and 9 exactly once. There are other sets like it. What is the minimum sum of the elements in such a set of prime numbers? a. 193 b. 198 c. 207 d. 225 e. 227

7 האולפני א ד ה ה מ מ ט י שלב הגמר - ש ע he 3rd stage t : s l r i g ish Ulpaniada - the math contest for Jew 12 The following figure shows a large rectangle, with a small rectangle cut off its upper right hand corner. All angles in the diagram are right angles. What is the maximum area of a circle which can be inscribed in this figure? a. 16π b. 145π/4 c. 25π d. 81π/4 e. 20π 13 Consider a 4x4 board. Two different squares on the board (not necessarily adjacent to each other) are called friends if they are in the same row or column. Every square on the board has 6 friends. Each square can be colored in one of the three colors: white, red or blue. When a square is touched, its color and its friends color are changed according to the following rule: a white square turns red, a red square turns blue, and a blue square turns white. For example, if we start with a completely white board, and touch first the top left square and then the square to its right, we get the following board: If we start with a completely white board, and touch exactly four squares one after the other, which one of the following boards can t be obtained? a. b. c. d. e.

8 14 Rachel began working on a mathematical puzzle soon after 6:00 pm, and solved it just before 7:00 pm. She noticed that at both of these times, the angle between the two hands of the clock was 110º. How many minutes did Rachel spend solving the puzzle? a. 35 b. 37 c. 38 d. 40 e. 42

9 Part Two: This section has two questions. Solve them, and include your reasoning in your answer. Provide a proof if it is requested. Each question is worth 18 points - partial answers will also be accepted, and will earn partial credit. 15 a. Prove that for every choice of three different natural numbers, there are always two numbers whose sum is divisible by 2. b. Prove that for every choice of five different natural numbers, there are always three numbers whose sum is divisible by 3. c. Give an example of a set of four natural numbers, so that the sum of any three of them is not divisible by 3. d. Deduce from the previous parts of this question, that for every choice of 11 numbers, there are always six numbers whose sum is divisible by 6. e. Complete the following statement (without proof): For every choice of numbers, there are always n of them whose sum is divisible by n. f. For every natural number n>1, give an example of a set of 2n-2 natural numbers, so that the sum of every n among them is not divisible by n.

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11 האולפני א ד ה ה מ מ ט י שלב הגמר - ש ע he 3rd stage t : s l r i g ish Ulpaniada - the math contest for Jew 16 על חומותייך ירושלים הפקדתי שומרים I have posted watchmen on your walls, O Jerusalem The Old City of Jerusalem has eight gates, which are, in clockwise order: New Gate, Damascus Gate, Flowers Gate, Lions Gate, The Gate of Mercy, Dung Gate, Zion Gate and Jaffa Gate. Eight watchmen are stationed around the wall for an eight hour shift beginning at midnight. At the beginning of the shift, one watchman is stationed at each of the gates. The watchmen patrol around the walls according to the following rules: 1. During each shift, all watchmen patrol at uniform and equal speed, which allows each of them to complete one circuit around the walls in exactly one hour. 2. At the beginning of the shift, each watchman chooses a direction of his choice, and begins patrolling in that direction along the wall. 3. Each time two watchmen meet, they both change direction, thus continuing at the same pace, in the opposite direction from before. Prove: a. If the watchman who left Jaffa Gate is holding the Jaffa Gate flag, and passes it to the first watchman he meets (if there is such a watchman), and this watchman then passes the flag to the next watchman he meets, etc., where will the flag be after exactly one hour? b. Prove that on the hour, every hour, every watchman will be at a gate (not necessarily the gate from which he left).

12 c. At the beginning of the shift, watchman number 1 is at New Gate, watchman number 2 at Damascus Gate, watchman number 3 at Flowers Gate, etc, (watchman number 8 at Jaffa Gate). Between which two watchmen will watchman number 3 be after an hour? After two hours? d. Prove that at the end of the shift, at 8:00a.m., every watchman will be at the gate at which he started the shift.

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