MathDB

P1

Part of 2023 4th Memorial "Aleksandar Blazhevski-Cane"

Problems(2)

A bound on perfect squares as values of a polynomial in m

Source: 4th Memorial Mathematical Competition "Aleksandar Blazhevski - Cane"- Junior D1 P2/ Senior D1 P1

12/20/2022
Let a,b,c,da, b, c, d be integers. Prove that for any positive integer nn, there are at least ⌊n4⌋\left \lfloor{\frac{n}{4}}\right \rfloor positive integers m≤nm \leq n such that m5+dm4+cm3+bm2+2023m+am^5 + dm^4 + cm^3 + bm^2 + 2023m + a is not a perfect square.
Proposed by Ilir Snopce
number theorypolynomialfloor functionPerfect Square
Color the lines!

Source: 4th Memorial Mathematical Competition "Aleksandar Blazhevski - Cane"- Junior D1 P1

12/21/2022
Let nn be a fixed positive integer and fix a point OO in the plane. There are nn lines drawn passing through the point OO. Determine the largest kk (depending on nn) such that we can always color kk of the nn lines red in such a way that no two red lines are perpendicular to each other.
Proposed by Nikola Velov
combinatoricsColoringconstructionlines