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.

p1. The numerator of the fraction was increased by 20%. By what percentage should its denominator be reduced so that the resulting fraction doubles?
p2. From point

O

on the plane there are four rays

OA

OB

OC

and

OD

(not necessarily in that order). It is known that

\angle AOB =40^o

\angle BOC = 70^o

\angle COD = 80^o

. What values can the angle between rays

OA

and

OD

take? (The angle between the rays is from

0^o

180^o

.)
p3. Three equal circles have a common interior point. Prove that there is a circle of the same radius containing the centers of these three circles.
p4. Two non-leap years are consecutive. The first one has more Mondays than Wednesdays. Which of the seven days of the week will occur most often in the second year?
p5. The difference between two four-digit numbers is

7

. How much can the sums of their digits differ?
p6. The numbers

1, 2, 3, 4, 5, 6, 7, 8, 9

are written on the board. In one move you can increase any of the numbers by

3

5

. What is the minimum number of moves you need to make for all the numbers to become equal?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.

grade6

Problems(2)