BS D Tevet 5774 /December 2013

Transcription

1 BS D Tevet 5774 /December 2013 Dear Participant, Welcome to the 5774 International Ulpaniada Mathematics Competition. It is our pleasure to welcome you and all your fellow participants-girls from around the world who have risen to the challenge and are taking part in this unique event. We wish you all the very best! Fill in your details before you start answering the questions: Name: Grade level: address: Name of High School: CIty: Country/State: When you have completed the test, return to this page and enter your answers into the following table (a,b,c,d or e) Question number Your answer Ulpaniada Math Department, Michlalah Jerusalem College P.O.B , Bayit Vegan, Jerusalem 91160, Israel Tel: , Fax: ulpaniada@macam.ac.il 1

2 This paper contains 14 questions. Each question shows five possible answers; only one is correct. Read each question carefully, solve it, and encircle the correct answer. You have 90 minutes to complete the test. You may use a calculator. B hatzlacha! 1 The sum of the digits of a six digit number is 8. What is the maximal value of the product of its digits? a. 0 b. 1 c. 2 d. 3 e. 4 2 What is the value of a. 3/2 b. 8/5 c. 5/3 d. 5/8 e. 3/5 3 Rachel drew six lines on a rectangular page, each parallel to one of its sides, dividing the page either lengthwise (from top to bottom) or widthwise (from right to left). As a result, the page was divided into N parts. Which of the following numbers cannot be the value of N? a. 7 b. 12 c. 13 d. 15 e. 16 2

3 3 4 Three circles of radii 2, 3 and 10 are tangential to each other (from the outside). The area of the triangle ABC whose vertices are the centers of the circles is: a. 30 b. 10π c. 60 d. 9π e In every cycle of 19 years, years three, six, eight, eleven, fourteen, seventeen and nineteen are leap years. ( j(5774 תשע ד is the seventeenth year of the cycle, and so it is a leap year. David was born on באדר א תשע א j.ל אדר א occurs only in a leap year and always has thirty days. Also, a regular year has no באדר,ל and so in most years David s birthday will not be celebrated on the date of his birth. This year will be the first time that David will celebrate his birthday on the date of his birth. Which year will be the tenth time that David will celebrate his birthday on the date of his birth? ת ת.e תשפ א.d תשפ ח.c תשצ ח.b תשצ ה.a 6 The diagram below consists of regular hexagons. The number of angles in the diagram is: a. 42 b. 36 c. 30 d. 60 e. 48 3

4 7 Ten cubes have been stuck together to form the tower-like structure shown in the diagram. The exterior sides of the structure (including its base and back sides) are colored in red. How many sides, in all, are red? a. 18 b. 24 c. 34 d. 36 e In the following exercise, different letters stand for different digits, and identical letters stand for identical digits. אולפ X נ יאדה Only one of the following can be true. Which is it? נ= 9 e. נ= 3 d. פ= 5 c. א= b. 1 ה= 2 a. 4

5 9 A magic square of size 4 4 is a table of numbers consisting of four rows and four columns, containing all 16 numbers from 1 to 16, with one number in each square, so that the sum of numbers in each row, column and two main diagonals is equal. The following is a partially completed magic square. What number is supposed to be in the shaded square? a. 2 b. 5 c. 8 d. 9 e In the following diagram, the line segment AB has length 16 cm. The red broken line starts from A, intersecting AB several times, each time at an angle of 60 0 (or ),ending at B. What is the length of the broken line, in cms? a. 32 b. 40 c. 48 d. 64 e. There isn t enough information to calculate the length 5

6 11 The following five numbers are written on five cards (one on each card): 2, 7, 17, 27, and 274. The following five statements are written on the other side of each of the cards, one statement on each card: The number is divisible by 3, the number is prime, the number is odd, the number is a 2 digit number, the number contains the digit 7. If none of the numbers on the cards satisfy the property written on the other side of the card, which statement is on the other side of 274? a. The number is divisible by 3 b. The number is prime c. The number is odd d. The number is a 2 digit number e. The number contains the digit 7 12 The following rectangle is tiled using congruent tiles, where the length of each side of the tiles is a natural number, and all angles of the tiles are right angles. Which of the following numbers can be the area of the rectangle? a. 44 b. 60 c. 132 d. 198 e and 31 are an example of a pair of different two digit prime numbers so that each number is the other one in reverse. The number of such pairs (including the pair in the example) is: a. 2 b. 3 c. 4 d. 5 e. 6 6

7 14 The following diagram shows two regular pentagons with a common vertex, C. The line segments DF and BI intersect at the point J. What is the size of the angle BJF? a. It is impossible to calculate, because we are not given that the pentagons are congruent. b c d e. It is impossible to calculate, because we are not given the angle BCF between the pentagons. 7