MathDB

4

Part of 2020 Tournament Of Towns

Problems(3)

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

Source: Tournament of Towns, Junior O-Level Paper, Spring 2020 , p4

6/3/2020
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
Diophantine equationdiophantinenumber theory
real numbers into the cells of a square of size N x N

Source: Tournament of Towns, Junior A-Level Paper, Spring 2020 , p4

6/10/2020
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
combinatoricssquare gridtable
p(x)=a(x) +b(x), squares of polynomials

Source: Tournament of Towns, Senior Ο-Level Paper, Spring 2020 , p4

6/4/2020
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
algebrapolynomial