MathDB

p1. The oil pipeline passes by three villages

A

B

C

. In the first village,

30\%

of the initial amount of oil is drained, in the second -

40\%

of the amount that will reach village

B

, and in the third -

50\%

of the amount that will reach village

C

What percentage of the initial amount of oil reaches the end of the pipeline?
p2. There are several ordinary irreducible fractions (not necessarily proper) with natural numerators and denominators (and the denominators are greater than

1

). The product of all fractions is equal to

10

. All numerators and denominators are increased by

1

. Can the product of the resulting fractions be greater than

10

?
p3. The garland consists of

10

light bulbs connected in series. Exactly one of the light bulbs has burned out, but it is not known which one. There is a suitable light bulb available to replace a burnt out one. To unscrew a light bulb, you need

10

seconds, to screw it in - also

10

seconds (the time for other actions can be neglected). Is it possible to be guaranteed to find a burnt out light bulb: a) in

10

minutes, b) in

5

minutes?
p4. When fast and slow athletes run across the stadium in one direction, the fast one overtakes the slow one every

15

minutes, and when they run towards each other, they meet once every

5

minutes. How many times is the speed of a fast runner greater than the speed of a slow runner?
p5. Petya was

35

minutes late for school. Then he decided to run to the kiosk for ice cream. But when he returned, the second lesson had already begun. He immediately ran for ice cream a second time and was gone for the same amount of time. When he returned, it turned out that he was late again, and he had to wait

50

minutes before the start of the fourth lesson. How long does it take to run from school to the ice cream stand and back if each lesson, including recess after it, lasts

55

minutes?
p6. In a convex heptagon, draw as many diagonals as possible so that no three of them are sides of the same triangle, the vertices of which are at the vertices of the original heptagon.
p7. In the writing of the antipodes, numbers are also written with the digits

0, ..., 9

, but each of the numbers has different meanings for them and for us. It turned out that the equalities are also true for the antipodes

5 * 8 + 7 + 1 = 48

2 * 2 * 6 = 24

5* 6 = 30

a) How will the equality

2^3 = ...

in the writing of the antipodes be continued? b) What does the number 9 mean among the Antipodes?
Clarifications: a) It asks to convert

2^3

in antipodes language, and write with what number it is equal and find a valid equality in both numerical systems. b) What does the digit

9

mean among the antipodes, i.e. with which digit is it equal in our number system?
p8. They wrote the numbers

1, 2, 3, 4, ..., 1996, 1997

in a row. Which digits were used more when writing these numbers - ones or twos? How long?
p9. 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.)?
p10. Is there a convex quadrilateral that can be cut along a straight line into two parts of the same size and shape, but neither the diagonal nor the straight line passing through the midpoints of opposite sides divides it into two equal parts?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.

Problem Statement