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‫בס"ד‬
Mathematics Challenge for Girls, 5770
Sample Questions
These questions appeared in the first stage of the Mathematics Ulpaniada
5768
Each question has only one correct answer.. The use of a calculator is
permitted.
1. Yael took apart these 2 two towers and used
the bricks to build a single tower of the same
shape in their place, so that the top row
contained one brick, and each row contained one
more brick than the row above. After Yael had
built the tallest tower that she could, how many
bricks were left over?
8 .‫ה‬
6 .‫ד‬
4 .‫ג‬
2 .‫ב‬
0 .‫א‬
2. Each one of the letters of the heading "‫"התחרות המתמטית תשס"ח‬
(Mathematics Competition 5768) is represented by a digit so that the
same letters are represented by the same digits and different letters by
different digits. The values of the letters are chosen so that the total sum
of all the letters of the 3 words is as large as possible. What is this sum?
4 .‫ה‬
75 .‫ד‬
100 .‫ג‬
101 .‫ב‬
102 .‫א‬
3. A rectangle is divided into 6 squares
(as illustrated). Two of the squares are
identical, and the smallest square's side
is 1 unit long.
What is the length of the largest
square's side?
7.5 .‫ה‬
7 .‫ד‬
6.5 .‫ג‬
6 .‫ב‬
5.5 .‫א‬
3, a, b,8, c,22, x... … is constructed such that every
4. The sequence:
term, starting from the fourth is equal to the sum of the three
previous terms. The value of x is:
43 .‫ה‬
42 .‫ד‬
41 .‫ג‬
40 .‫ב‬
.‫א‬
34
5. In the following multiplication exercise there are three digits missing.
2 7 x
5 = 1 2
The sum of the missing digits is:
20.‫ה‬
15.‫ד‬
13.‫ג‬
12 .‫ב‬
10 .‫א‬
A
6. ABC is an isosceles triangle (AB=AC). D is a
point on AC such that BC=CD and BD=DA. The
size of angle C is:
0
1
77 .‫ג‬
7
80 .‫ב‬
0
0
1
66 .‫ה‬
6
D
60 .‫א‬
0
0
1
75 .‫ד‬
2
C
B
A, B and C are 3 different natural prime numbers which are smaller than
20. (A natural number is prime if it is not equal to 1 and is only divisible
by itself and one).
 A is a 2 digit number and the sum of its digits is also prime.
 C is the average of A and B.
What is the sum of the 3 numbers?
39 .‫ה‬
33 .‫ד‬
25 .‫ג‬
21 .‫ב‬
19 .‫א‬
8. Insert all of the digits from 1 to 9 into these 9 circles so that
3
the four digits on all three sides of the triangle have the same
sum.
Three of the numbers have already been inserted.
1
2
The sum of the digits on each side is:
19 .‫ה‬
18. ‫ד‬
17.‫ג‬
16. ‫ב‬
15 .‫א‬
9. The numbers a, b and c are three distinct prime numbers (remember
that 1 is not a prime number) whose sum is 40. If we multiply them
together the result is:
690 .‫ה‬
76 .‫ד‬
722 .‫ג‬
434 .‫ב‬
330 .‫א‬
10. Exactly one of three friends Gila, Rina and Ditza is taking part in
the "Ulpaniada". Of the following statements only one is true.
Which one is it?
‫ ) א‬Rina is taking part in the Ulpaniada.
‫ ) ב‬Gila is not taking part but Rina is.
‫ ) ג‬Ditza is not taking part but Gila is.
‫ ) ד‬Ditza is taking part.
‫ ) ה‬Rina is not taking part but Gila is.
11. In every cycle of 19 years, the years numbering 3,6,8,11,14,17 and
19 are Jewish leap years . The year ‫( תשס"ח‬5768) was a leap year,
as the 11th year in the cycle, and it was also a shmittah year. When
prior to the year ‫תשס"ח‬, was there a shmittah year that was also a
leap year?
‫ תשמ"ז‬.‫ה‬
‫ תשמ"א‬.‫ד‬
‫ תש"ם‬.‫ג‬
‫ תשל"ד‬.‫ב‬
‫ תשל"ג‬.‫א‬
12. How many different triangles can you find with both of the following
properties?
a. The length of each one of the triangle's sides is a whole number.
b. The triangle's perimeter is 8.
4 .‫ה‬
3 .‫ד‬
2 .‫ג‬
1 .‫ב‬
0 .‫א‬
13. If the digits of a certain 4 digit number are multiplied together, the
result is 810. If all the digits are different from each other, what is
their sum?
25 .‫ה‬
23 .‫ד‬
22 .‫ג‬
19 .‫ב‬
18
.‫א‬
14. If you fold this figure to form a cube,
what is the product of the 4 neighbors of face number 1?
360 .‫ה‬
240 .‫ד‬
180 .‫ג‬
144 .‫ב‬
120 ..‫א‬
15. In the country of Strudel, they use addition and multiplication just like
us, but they also have an operation @. For example;
0@0=0
m@n=n@m
(m+1)@n=(m@n)+n+1
Calculate the value of 7@5:
47 .‫ה‬
35 .‫ד‬
30 .‫ג‬
36 .‫ב‬
42 .‫א‬
16. A convex polygon (a polygon with all
of its diagonals inside) has 50 sides.
What is the largest number of
diagonals that can be drawn inside the
polygon, such that none of them
intersect each other? (They may meet
at a vertex).
50 .‫ה‬
49 .‫ד‬
48 .‫ג‬
This diagram is for illustration only.
47 .‫ב‬
46
.‫א‬
17. In the diagram below there are 8x8 squares.
Some of the squares are colored black. Add black squares where
necessary, until the pattern of squares is symmetrical with respect to
both the axes.
What is the smallest total number of black squares needed in order to
obtain such symmetry?
24 .‫ה‬
20.‫ד‬
16.‫ג‬
8.‫ב‬
32.‫א‬
18. 100 Ulpaniada graduates had a reunion. In every grouping of 4 girls
there was always one girl who knew the other three. The number of girls
at the reunion who knew everybody was at least:
99 .‫ה‬
98 .‫ד‬
97 .‫ג‬
3 .‫ב‬
4 .‫א‬
19. a  b  c are 3 positive numbers (not necessarily whole) whose
sum is 10. Which one of the following statements is definitely true?
b  4 .‫ג‬
a  b  20 .‫ב‬
a  b  1 .‫א‬
a  b  5 .‫ה‬
b  c  5 .‫ד‬
20. For how many natural numbers m does there exist a natural number
n such that:
nm
 10 ?
nm
12 .‫ה‬
9 .‫ד‬
3 .‫ג‬
!‫בהצלחה‬
2 .‫ב‬
0 .‫א‬
‫‪Answers‬‬
‫‪10 11 12 13 14 15 16 17 18 19 20‬‬
‫ד‬
‫ד‬
‫ג‬
‫ה‬
‫ב‬
‫ה‬
‫ב‬
‫ד ב‬
‫א‬
‫ד‬
‫‪9‬‬
‫‪8‬‬
‫‪7‬‬
‫‪6‬‬
‫‪5‬‬
‫‪4‬‬
‫‪3‬‬
‫‪2‬‬
‫‪1‬‬
‫ב ג ב ג א ה ד א ד‬