MathDB

2020 Tournament Of Towns

Part of Tournament Of Towns

Subcontests

(7)
4
3

x^2 + y^2 + z^2 − xy − yz − zx = n , x^2 + y^2 − xy = n, integer solution

For some integer n the equation x2+y2+z2−xy−yz−zx=nx^2 + y^2 + z^2 -xy -yz - zx = n has an integer solution x,y,zx, y, z. Prove that the equationx2+y2−xy=n x^2 + y^2 - xy = n also has an integer solution x,yx, y.
Alexandr Yuran

real numbers into the cells of a square of size N x N

For which integers NN it is possible to write real numbers into the cells of a square of size N×NN \times N so that among the sums of each pair of adjacent cells there are all integers from 11 to 2(N−1)N2(N-1)N (each integer once)?
Maxim Didin

p(x)=a(x) +b(x), squares of polynomials

We say that a nonconstant polynomial p(x)p(x) with real coefficients is split into two squares if it is represented as a(x)+b(x)a(x) +b(x) where a(x)a(x) and b(x)b(x) are squares of polynomials with real coefficients. Is there such a polynomial p(x)p(x) that it may be split into two squares: a) in exactly one way; b) in exactly two ways?
Note: two splittings that differ only in the order of summands are considered to be the same.
Sergey Markelov
7
1
1
5
3
4
6
3

Alice has a deck of 36 cards, 4 suits of 9 cards each, game with Bob

Alice has a deck of 3636 cards, 44 suits of 99 cards each. She picks any 1818 cards and gives the rest to Bob. Now each turn Alice picks any of her cards and lays it face-up onto the table, then Bob similarly picks any of his cards and lays it face-up onto the table. If this pair of cards has the same suit or the same value, Bob gains a point. What is the maximum number of points he can guarantee regardless of Alice’s actions?
Mikhail Evdokimov

2n consecutive integers on a board, replace pairs by their difference and sum

There are 2n2n consecutive integers on a board. It is permitted to split them into pairs and simultaneously replace each pair by their difference (not necessarily positive) and their sum. Prove that it is impossible to obtain any 2n2n consecutive integers again.
Alexandr Gribalko

N white, blue and red cubes around a circle, a robot destroys 2, puts 1 back

Given an endless supply of white, blue and red cubes. In a circle arrange any NN of them. The robot, standing in any place of the circle, goes clockwise and, until one cube remains, constantly repeats this operation: destroys the two closest cubes in front of him and puts a new one behind him a cube of the same color if the destroyed ones are the same, and the third color if the destroyed two are different colors. We will call the arrangement of the cubes good if the color of the cube remaining at the very end does not depends on where the robot started. We call NN successful if for any choice of NN cubes all their arrangements are good. Find all successful NN.
I. Bogdanov
5
4
2
5