‫האולפניאדה‬
‫ה‬
‫מ‬
‫ת‬
ewish Girls
‫מטית‬
‫ה‬
‫ע‬
st for J
‫ו‬
e
t
‫ל‬
n
‫מ‬
‫י‬
o
‫ שלב הגמר‬- ‫ ת תשע“ו‬The Math C
a-
‫ה‬
‫ מכללה ירושלים‬,‫חוג למתמטיקה‬
Ulpaniad
‫בס“ד‬
June 8, 2016, ‫ב‘ בסיון תשע“ו‬
Dear Participant,
We congratulate you on your participation in the final round 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:
Email address:
Name of High School:
CIty:
Country/State:
This question paper is comprised of two parts.
When you have completed the first part and then circled your answers, please copy them into this table
(a,b,c,d or e):
Question
number
1
2
3
4
5
6
7
8
9
10
Your answer
Please write your answers to Section B in the spaces provided after each question.
Wishing you much hatzlacha,
The Ulpaniada Team
Ulpaniada Math Department, Michlalah – Jerusalem College
P.O.B. 16078, Bayit Vegan, Jerusalem 91160, Israel
Tel: +972-2-6750931, Fax: +972-2-6750718
http://ulpaniada.macam.ac.il/english
[email protected]
1
‫האולפניאדה‬
‫ה‬
‫מ‬
‫ת‬
ewish Girls
‫מטית‬
‫ה‬
‫ע‬
st for J
‫ו‬
e
t
‫ל‬
n
‫מ‬
‫י‬
o
‫ שלב הגמר‬- ‫ ת תשע“ו‬The Math C
a-
‫ה‬
‫ מכללה ירושלים‬,‫חוג למתמטיקה‬
Ulpaniad
This question paper consists of two parts, in which you can accumulate a total of 130
points. You have four hours to complete both parts of the paper.
You may use a calculator.
B’hatzlacha!
Part One:
This section consists of 10 questions.
Each question has five possible answers, only one of which is correct.
Read the question carefully, solve it, and then circle the correct answer.
Each question is worth 7 points (70 points in total).
1
The sum of four consecutive prime numbers is 272.
The sum of the digits of the largest of these four numbers is:
a. 7
b. 8
c. 10
d. 13
e. 16
2 Consider a square. Mark the following eight points on its perimeter in red; its four vertices and the
midpoints of each of its four sides. A point P inside the square is chosen at random.
What is the probability that the red point nearest to P is a vertex (and not a midpoint)?
a. 1/16
b. 1/8
2
c. 4/12
d. 1/4
e. 1/2
‫האולפניאדה‬
‫ה‬
‫מ‬
‫ת‬
ewish Girls
‫מטית‬
‫ה‬
‫ע‬
st for J
‫ו‬
e
t
‫ל‬
n
‫מ‬
‫י‬
o
‫ שלב הגמר‬- ‫ ת תשע“ו‬The Math C
a-
‫ה‬
‫ מכללה ירושלים‬,‫חוג למתמטיקה‬
Ulpaniad
3 The following 3X3 grid is filled with three circles, three squares and three triangles colored in the three3
colors red, green and yellow.
The following three illustrations alongside it provide partial information as to how it is filled.
Insert three circles, three squares and three triangles, one of each color, into the empty 3X3 grid below,
according to the eight hints shown:
Note: When a square is completely colored in a solid color, it means that square contains a shape of
that color, and when a square contains a black and white shape, the square contains that shape in one
of the three colors.
The square in the center of a grid filled according to these hints contains:
a. A green circle b. A yellow square c. A yellow triangle d. A red square e. A green square
3
‫האולפניאדה‬
‫ה‬
‫מ‬
‫ת‬
ewish Girls
‫מטית‬
‫ה‬
‫ע‬
st for J
‫ו‬
e
t
‫ל‬
n
‫מ‬
‫י‬
o
‫ שלב הגמר‬- ‫ ת תשע“ו‬The Math C
a-
‫ה‬
‫ מכללה ירושלים‬,‫חוג למתמטיקה‬
Ulpaniad
4 In the following addition exercise, distinct letters stand for distinct digits, and identical letters stand
for the same digit.
If the answer ‫ הגבלה‬is maximal, then what is the sum of digits of ‫?הגבלה‬
a. 17
5
b. 19
c. 21
d. 29
e. 35
The Hebrew calendar has six types of years, three of which are ordinary years, and the other three are
leap years. A leap year has an additional month of 30 days, Adar I.
The three types of ordinary years are:
A regular ordinary year in which Tishrei has 30 days, Cheshvan has 29 days, and Kislev has 30 days,
and so on, alternately 29 and 30, so that in total an ordinary regular year has six months of 30 days and
six months of 29 days, a deficient ordinary year which has one day less than a regular ordinary year
(Kislev has 29 days), and a complete ordinary year which has one day more than a regular ordinary year
(Cheshvan has 30 days).
In a similar fashion, the three types of leap years are regular leap years, deficient leap years and
complete leap years, each containing an extra month of 30 days over their corresponding ordinary
years.
About a thousand years ago, in both of the two consecutive years 4775 and 4776, the first day of Rosh
Hashanah fell on Shabbat.
The year 4775 was :
a. A deficient ordinary year b. A regular ordinary year c. A deficient leap year
d. A regular leap year e. A complete leap year
4
‫האולפניאדה‬
‫ה‬
‫מ‬
‫ת‬
ewish Girls
‫מטית‬
‫ה‬
‫ע‬
st for J
‫ו‬
e
t
‫ל‬
n
‫מ‬
‫י‬
o
‫ שלב הגמר‬- ‫ ת תשע“ו‬The Math C
a-
‫ה‬
‫ מכללה ירושלים‬,‫חוג למתמטיקה‬
Ulpaniad
6
A setting is an assignment of the digits
example, the following setting:
1,2,3,4 to a 2X2 table, with each digit in a different cell. For
A cut is an operation where 1 is subtracted simultaneously from two numbers in the same row or
column.
Here’s the table from the above example after a cut:
(We subtracted 1 from each number in the left column).
A setting is called successful if it can be reduced to a table of zeros using a series of cuts.
How many successful settings are there?
a. 4
7
b. 6
c. 8
d. 12
e. 16
Shira and Tamar, two contestants in the Ulpaniada, each separately switched the letters in the word
‫“( מתמודדת‬contestant”) to digits, so that identical letters were replaced by identical digits, and different
letters by different digits. Both of these seven figure numbers are divisible by 11.
Shira’s number is the biggest possible number satisfying these conditions, and Tamar’s number is the
smallest possible number. The two numbers have one digit in common.
What is it?
a. 0
b. 1
5
c. 3
d. 6
e. 8
‫האולפניאדה‬
‫ה‬
‫מ‬
‫ת‬
ewish Girls
‫מטית‬
‫ה‬
‫ע‬
st for J
‫ו‬
e
t
‫ל‬
n
‫מ‬
‫י‬
o
‫ שלב הגמר‬- ‫ ת תשע“ו‬The Math C
a-
‫ה‬
‫ מכללה ירושלים‬,‫חוג למתמטיקה‬
Ulpaniad
8
Consider 8 cards in a circular arrangement, as in the illustration. Each card has a number on it, which
is not visible because the cards are placed face down. The number next to each card shows the sum
of the numbers of the five cards furthest away from that card (i.e. of all numbers except for its own
number and the numbers of its two neighbors).
The number on the red colored card is:
a. 10
9
b. 7
c. 6
d. 3
e. 1
Consider an equiangular hexagon ABCDEF, where we are given the lengths of four of its sides:
AB=5, BC=7, CD=9, DE=11.
What is the product of the lengths of its remaining two sides?
a. 15
b. 18
6
c. 21
d. 30
e. there isn’t enough information to calculate this product.
‫האולפניאדה‬
‫ה‬
‫מ‬
‫ת‬
ewish Girls
‫מטית‬
‫ה‬
‫ע‬
st for J
‫ו‬
e
t
‫ל‬
n
‫מ‬
‫י‬
o
‫ שלב הגמר‬- ‫ ת תשע“ו‬The Math C
a-
‫ה‬
‫ מכללה ירושלים‬,‫חוג למתמטיקה‬
Ulpaniad
10 Consider the following cube ABCDEFGH divided into 27 smaller cubes of side 1 –see diagram.
The plane containing the vertices B,E,G also contains additional vertices of the smaller cubes.
How many different vertices of smaller cubes lie on this plane altogether?
a. 9
b. 10
7
c. 11
d. 12
e. 14
‫האולפניאדה‬
‫ה‬
‫מ‬
‫ת‬
ewish Girls
‫מטית‬
‫ה‬
‫ע‬
st for J
‫ו‬
e
t
‫ל‬
n
‫מ‬
‫י‬
o
‫ שלב הגמר‬- ‫ ת תשע“ו‬The Math C
a-
‫ה‬
‫ מכללה ירושלים‬,‫חוג למתמטיקה‬
Ulpaniad
Part Two:
This section has two questions. Solve them, and include your reasoning in your answer. Provide
a proof when it is requested. Each question is worth 30 points, 60 points in total. Partial answers
will also be accepted, and will earn partial credit.
11 Consider a rectangular board, and 2X1 domino tiles, each one of which exactly fits two adjacent squares
of the board.
A coverage of the board is a placement of dominos so that each square of the board is covered by exactly
one domino, with no part of any domino lying outside the confines of the board.
Here is an example of a coverage of a 3X4 board-and a shorter way of displaying it using line segments.
This board can be separated into two smaller rectangular boards without cutting any domino.
Such a separation is called a split.
For example, the following diagram shows a split of the 3X4 board above.
8
‫האולפניאדה‬
‫ה‬
‫מ‬
‫ת‬
ewish Girls
‫מטית‬
‫ה‬
‫ע‬
st for J
‫ו‬
e
t
‫ל‬
n
‫מ‬
‫י‬
o
‫ שלב הגמר‬- ‫ ת תשע“ו‬The Math C
a-
‫ה‬
‫ מכללה ירושלים‬,‫חוג למתמטיקה‬
Ulpaniad
a) Prove that any coverage of a 3X4 board has a split.
b) Prove that any coverage of a 6X6 board has a split.
c) Fill in a coverage of the following 8X6 board that does not have a split.
The next page contains various boards for your convenience, and the page after, has been left blank
for you to write your proofs of sections a and b.
9
‫האולפניאדה‬
‫ה‬
‫מ‬
‫ת‬
‫‪ewish Girls‬‬
‫מטית‬
‫ה‬
‫ע‬
‫‪st for J‬‬
‫ו‬
‫‪e‬‬
‫‪t‬‬
‫ל‬
‫‪n‬‬
‫מ‬
‫י‬
‫‪o‬‬
‫‪ The Math C‬ת תשע“ו ‪ -‬שלב הגמר‬
‫‪a-‬‬
‫ה‬
‫חוג למתמטיקה‪ ,‬מכללה ירושלים‬
‫‪Ulpaniad‬‬
‫‪Practicle boards‬‬
‫‪10‬‬
‫האולפניאדה‬
‫ה‬
‫מ‬
‫ת‬
‫‪ewish Girls‬‬
‫מטית‬
‫ה‬
‫ע‬
‫‪st for J‬‬
‫ו‬
‫‪e‬‬
‫‪t‬‬
‫ל‬
‫‪n‬‬
‫מ‬
‫י‬
‫‪o‬‬
‫‪ The Math C‬ת תשע“ו ‪ -‬שלב הגמר‬
‫‪a-‬‬
‫ה‬
‫חוג למתמטיקה‪ ,‬מכללה ירושלים‬
‫‪Ulpaniad‬‬
‫‪11‬‬
‫האולפניאדה‬
‫ה‬
‫מ‬
‫ת‬
ewish Girls
‫מטית‬
‫ה‬
‫ע‬
st for J
‫ו‬
e
t
‫ל‬
n
‫מ‬
‫י‬
o
‫ שלב הגמר‬- ‫ ת תשע“ו‬The Math C
a-
‫ה‬
‫ מכללה ירושלים‬,‫חוג למתמטיקה‬
Ulpaniad
12 Define a function f(n) of non negative integers as follows:
f(1)=0, and for every natural number n,
f(2n)=2f(n); f(2n+1)=n+2f(n)
a) Fill in the following table:
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
f(n)
b) Find all natural numbers n for which f(n)=0. Prove your answer!
c) What is (2n+2k), for k,n natural numbers and k<n? Prove your answer!
d) Find all natural numbers n for which f(n) is an odd number. Prove your answer!
e) Find all natural numbers n for which f(n+1)=f(n). Prove your answer!
f) Find all natural numbers n for which f(n+1)<f(n). Prove your answer!
Note: For section (f) you may use the following inequality (without proving it):
f(n+1)<f(n)+n/2
12
‫האולפניאדה‬
‫ה‬
‫מ‬
‫ת‬
‫‪ewish Girls‬‬
‫מטית‬
‫ה‬
‫ע‬
‫‪st for J‬‬
‫ו‬
‫‪e‬‬
‫‪t‬‬
‫ל‬
‫‪n‬‬
‫מ‬
‫י‬
‫‪o‬‬
‫‪ The Math C‬ת תשע“ו ‪ -‬שלב הגמר‬
‫‪a-‬‬
‫ה‬
‫חוג למתמטיקה‪ ,‬מכללה ירושלים‬
‫‪Ulpaniad‬‬
‫‪13‬‬
‫האולפניאדה‬
‫ה‬
‫מ‬
‫ת‬
‫‪ewish Girls‬‬
‫מטית‬
‫ה‬
‫ע‬
‫‪st for J‬‬
‫ו‬
‫‪e‬‬
‫‪t‬‬
‫ל‬
‫‪n‬‬
‫מ‬
‫י‬
‫‪o‬‬
‫‪ The Math C‬ת תשע“ו ‪ -‬שלב הגמר‬
‫‪a-‬‬
‫ה‬
‫חוג למתמטיקה‪ ,‬מכללה ירושלים‬
‫‪Ulpaniad‬‬
‫‪14‬‬