MathDB

p1. For

25

bagels they paid as many rubles as the number of bagels you can buy with a ruble. How much does one bagel cost?
p2. Cut the square into the figure into

4

parts of the same shape and size so that each part contains exactly one shaded square. https://cdn.artofproblemsolving.com/attachments/a/2/14f0d435b063bcbc55d3dbdb0a24545af1defb.png
p3. The numerator and denominator of the fraction are positive numbers. The numerator is increased by

1

, and the denominator is increased by

10

. Can this increase the fraction?
p4. The brother left the house

5

minutes later than his sister, following her, but walked one and a half times faster than her. How many minutes after leaving will the brother catch up with his sister?
p5. Three apples are worth more than five pears. Can five apples be cheaper than seven pears? Can seven apples be cheaper than thirteen pears? (All apples cost the same, all pears too.)
p6. Give an example of a natural number divisible by

6

and having exactly

15

different natural divisors (counting

1

and the number itself).
p7. In a round dance,

30

children stand in a circle. Every girl's right neighbor is a boy. Half of the boys have a boy on their right, and all the other boys have a girl on their right. How many boys and girls are there in a round dance?
p8. A sheet of paper was bent in half in a straight line and pierced with a needle in two places, and then unfolded and got

4

holes. The positions of three of them are marked in figure Where might the fourth hole be? https://cdn.artofproblemsolving.com/attachments/c/8/53b14ddbac4d588827291b27c40e3f59eabc24.png
p9 The numbers 1

, 2, 3, 4, 5, _, 2000

are written in a row. First, third, fifth, etc. crossed out in order. Of the remaining

1000

numbers, the first, third, fifth, etc. are again crossed out. They do this until one number remains. What is this number?
p10. On the number axis there lives a grasshopper who can jump

1

and

4

to the right and left. Can he get from point

1

to point

2

of the numerical axis in

1996

jumps if he must not get to points with coordinates divisible by

4

(points

0

\pm 4

\pm 8

etc.)?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.

Problem Statement