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.

p1. a) There are barrels weighing

1, 2, 3, 4, ..., 19, 20

pounds. Is it possible to distribute them equally (by weight) into three trucks?
b) The same question for barrels weighing

1, 2, 3, 4, ..., 9, 10

pounds.
p2. There are apples and pears in the basket. If you add the same number of apples there as there are now pears (in pieces), then the percentage of apples will be twice as large as what you get if you add as many pears to the basket as there are now apples. What percentage of apples are in the basket now?
p3. What is the smallest number of integers from

1000

1500

that must be marked so that any number

x

from

1000

1500

differs from one of the marked numbers by no more than

10\%

of the value of

x

?
p4. Draw a perpendicular from a given point to a given straight line, having a compass and a short ruler (the length of the ruler is significantly less than the distance from the point to the straight line; the compass reaches from the point to the straight line “with a margin”).
p5. There is a triangle on the chessboard (left figure). It is allowed to roll it around the sides (in this case, the triangle is symmetrically reflected relative to the side around which it is rolled). Can he, after a few steps, take the position shown in right figure? https://cdn.artofproblemsolving.com/attachments/f/5/eeb96c92f30b837e7ed2cdf7cf77b0fbb8ceda.png
p6. The natural number

a

is less than the natural number

b

. In this case, the sum of the digits of number

a

100

less than the sum of the digits of number

b

. Prove that between the numbers

a

and

b

there is a number whose sum of digits is

43

more than the sum of the digits of

a

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

grade8

Problems(2)