MathDB

p1. Can a number ending in

1999

be the square of a natural number?
p2. The Three-Headed Snake Gorynych celebrated his birthday. His heads took turns feasting on birthday cakes and ate two identical cakes in

15

minutes. It is known that each head ate as much time as it would take the other two to eat the same pie together. In how many minutes would the three heads of the Serpent Gorynych eat one pie together?
p3. Find the sum of the coefficients of the polynomial obtained after opening the brackets and bringing similar terms into the expression: a)

(7x - 6)^4 - 1

(7x - 6)^{1999}-1

p4. The general wants to arrange seven anti-aircraft installations so that among any three of them there are two installations, the distance between which is exactly

10

kilometers. Help the general solve this problem.
p5. Gulliver, whose height is

999

millimeters, is building a tower of cubes. The first cube has a height of

1/2

a lilikilometer, the second -

1/4

a lilikilometer, the third -

1/8

a lilikilometer, etc. How many cubes will be in the tower when its height exceeds Gulliver's height. (

1

lilikilometer is equal to

1000

lilimeters).
p6. It is known that in any pentagon you can choose three diagonals from which you can form a triangle. Is there a pentagon in which such diagonals can be chosen in a unique way?
p7. It is known that for natural numbers

a

and

b

the equality

19a = 99b

holds. Can

a + b

be a prime number?
p8. Vitya thought of

5

integers and told Vanya all their pairwise sums:

0, 1, 5, 7, 11, 12, 18, 24, 25, 29.

Help Vanya guess the numbers he has in mind.
p9. In a

3 \times 3

square, numbers are arranged so that the sum of the numbers in each row, in each column and on each major diagonal is equal to

0

. It is known that the sum of the squares of the numbers in the top row is

n

. What can be the sum of the squares of the numbers in the bottom line?
p10.

N

points are marked on a circle. Two players play this game: the first player connects two of these points with a chord, from the end of which the second player draws a chord to one of the remaining points so as not to intersect the already drawn chord. Then the first player makes the same “move” - draws a new chord from the end of the second chord to one of the remaining points so that it does not intersect any of the already drawn ones. The one who cannot make such a “move” loses. Who wins when played correctly? (A chord is a segment whose ends lie on a given circle)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.

Problem Statement