MathDB

1

Part of 2019 Iran Team Selection Test

Problems(3)

Iran TST

Source: Iranian TST 2019, first exam day 1, problem 1

6/24/2019
A table consisting of 55 columns and 3232 rows, which are filled with zero and one numbers, are "varied", if no two lines are filled in the same way.\\ On the exterior of a cylinder, a table with 3232 rows and 1616 columns is constructed. Is it possible to fill the numbers cells of the table with numbers zero and one, such that any five consecutive columns, table 32×532\times5 created by these columns, is a varied one?
Proposed by Morteza Saghafian
combinatorics
Iranian TST 2019, second exam, day 1, problem 1

Source: Iranian TST 2019

4/11/2019
SS is a subset of Natural numbers which has infinite members. S’={xy+yx: x,y∈S, x≠y}S’=\left\{x^y+y^x: \, x,y\in S, \, x\neq y\right\} Prove the set of prime divisors of S’S’ has also infinite members Proposed by Yahya Motevassel
Iranian TSTTST
Iran polynomial

Source: Iranian TST 2019, third exam day 1, problem 1

4/15/2019
Find all polynomials P(x,y)P(x,y) with real coefficients such that for all real numbers x,yx,y and zz: P(x,2yz)+P(y,2zx)+P(z,2xy)=P(x+y+z,xy+yz+zx).P(x,2yz)+P(y,2zx)+P(z,2xy)=P(x+y+z,xy+yz+zx).
Proposed by Sina Saleh
algebrapolynomial