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

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1 Dear Participant, We congratulate you on your achievement in reaching the second 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 is comprised of two parts. When you have completed the first part and encircled 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 The following question paper consists of two parts. The first part contains 14 questions. Each question has 5 possible answers, only one of which is correct. Read the question carefully, solve it and encircle the correct answer. The second part contains 3 questions. Solve them, including all of your reasoning in the solution. Partial answers will also be accepted. You have three and a half hours to complete the whole test. You may use a calculator. B hatzlacha! Part One: 1 G and A are opposite vertices of a cube (see diagram). The shortest path moving along sides from A to G involves traversing three sides. In how many different ways can this be done? a. 3 b. 4 c. 5 d. 6 e. 8 2 In the following sum, digits have been replaced by Hebrew letters. Different letters stand for different digits, and identical letters stand for identical digits.?אדה What is the minimum possible value of the word a. 201 b. 321 c. 102 d. 396 e. 431

3 3 3 Five grandsons, Avi, Baruch, Gavriel, Daniel and Hershey sat around the Seder table. One of them stole the afikoman. When Saba enquired as to its whereabouts, each grandson replied with one sentence: Avi: Daniel took the afikoman. Baruch: I m not the guilty party Gavriel: Hershey is innocent! Daniel: Avi s lying! Hershey: Baruch s telling the truth! Yael, who witnessed the whole scene, told Saba that three of the children were telling the truth and two were lying. Who stole the afikoman? a. Avi b. Baruch c. Gavriel d. Daniel e. Hershey 4 The rectangle below contains 8 inner squares. If the grey area equals 300, then the area of the large, inner square is: a. 400 b. 441 c. 484 d. 676 e The lengths of all three sides of a triangle ABC are integers. If AB=7, and AC=4BC, then the perimeter of the triangle is: a. 12 b. 15 c. 16 d. 17 e. 22

4 6 One side of a square is tangent to a circle of radius 10, and the opposite side is a chord of the circle (see diagram). The square has side of length: a. 10 b. 12 c. 4 d. 16 e. 5 7 How many natural numbers are 7 times the sum of their digits? a. 1 b. 2 c. 3 d. 4 e. 5 8 A polygon is convex if all its interior angles are less than For how many values of n does there exist a convex polygon, such that the ratios between its interior angles are 1:2:3:... n. a. 1 b. 2 c. 3 d. 4 e. There is no such polygon 9 Two distinct four digit numbers each contain all four of the digits 2, 4, 5 and 7. If one of them is an exact integral multiple of the other, then their sum is: a b c d e

5 10 A is a set of 20 consecutive natural numbers. If 8 is not in A, then the maximal number of prime numbers which can be in A is: a. 4 b. 5 c. 6 d. 7 e The Nechamah network consists of five High Schools א, ב, ג, ד, ה which are located in different places. Some of the distances (in km) between respective High Schools are given in the table below: א ב ג ד ה א ב ג The distance between High School ד and High School ה (in km) is: a. 39 b. 42 c. 45 d. 47 e Consider the following equality: =117. All 9 digits on the left hand side appear in order. Besides this sum, how many other sums containing all 9 digits in order, equal 117? a. 1 b. 2 c. 3 d. 4 e. There aren t any other such sums

6 13 Let A be a set containing the number 2, satisfying the following two properties: 1. If n is in A, then n+5 is also in A. 2. If n is in A, then 3n is also in A. Which of the following numbers is not necessarily in A? a. 770 b. 771 c. 772 d. 773 e A 4 digit number is called a Notable Number if it has the following property: Its thousandths digit equals the number of zeros in the number, its hundredths digit equals the number of 1 s in the number, its tenths digit equals the number of 2 s in the number, and its units digit equals the number of 3 s in the number. How many Notable Numbers are there? a. 0 b. 1 c. 2 d. 3 e. 4 16

7 Part Two: In this section there are 3 questions. Solve them, writing down all of your reasoning. Provide a proof when it is requested. Partial solutions are also accepted. Write your answer on the question paper underneath each question. If there is not enough space, continue on the other side of the paper. 15 Yael and Shira play the following game: The game begins with a bag containing 770 coins. The first player removes a number of coins from the bag, not less than 1 and not more than 9, and places them on the table. From that point on, each subsequent player must remove at least one coin from the bag, but not more than the number of coins already on the table, and place them also on the table. The game terminates when all 770 coins have been placed on the table. The player who makes the last move is the winner. Yael is the first player, and Shira is the second. Show that one of them has a way of winning the game regardless of her opponent s moves. Who is she? How should she plan her winning strategy? Outline her moves, explaining your reasoning.

8 16 Consider a regular 8x8 chessboard. Enter one of the numbers 0 or 1 into each of its 64 squares. Now calculate the sums of the numbers in every row, in every column and along the two main diagonals. You will receive a list of 18 numbers. Prove that in this list there is a number that appears at least 3 times.

9 17 a. Consider two parallel lines, one below the other. We wish to place 6 points, some on the upper line, some on the lower line, and then to join all points on the lower line to all points on the upper line by line segments. How should we divide the 6 points between the two lines, so as to maximize the number of such line segments? b. Consider two parallel lines, one below the other. We wish to place 2n points, some on the upper line, some on the lower line, and then join all points on the lower line to all points on the upper line by line segments. How should we divide the 2n points between the two lines, so as to maximize the number of such line segments? What is this maximal number of line segments? Prove your answer. c. Find the 20 th element in the sequence: 0, 1, 9, 36, 100, 225 d. Consider two parallel lines, one below the other, and 2n points, with n of them on the upper line, and the other n on the lower line. Join all points on the lower line to all points on the upper line by line segments. Some of these line segments intersect each other between the two parallel lines. If we are told that no three of these line segments intersect in one point, how many intersection points are there? Prove your answer.

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