MathDB

p1. What is the maximum amount of a

12\%

acid solution that can be obtained from

1

liter of

5\%

10\%

and

15\%

solutions?
p2. Which number is greater:

199,719,971,997^2

199,719,971,996 * 19,9719,971,998

?
p3. Is there a convex

1998

-gon whose angles are all integer degrees?
p4. Is there a ten-digit number divisible by

11

that uses all the digits from

0

9

?
p5. There are

20

numbers written in a circle, each of which is equal to the sum of its two neighbors. Prove that the sum of all numbers is

0

.
p6. Is there a convex polygon that has neither an axis of symmetry nor a center of symmetry, but which transforms into itself when rotated around some point through some angle less than

180

degrees?
p7. 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.
p8. Give an example of a natural number that is divisible by

30

and has exactly

105

different natural factors, including

1

and the number itself.
p9. 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?
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.1. There was typo in problem

9

, it asks for

2^3

and not

23

. PS.2. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.

Problem Statement