MathDB

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.

Problem Statement