MathDB

6

Part of 2016 IFYM, Sozopol

Problems(5)

Problem 6 of First round

Source: VII International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

8/29/2019
Find all polynomials P∈Q[x]P\in \mathbb{Q}[x], which satisfy the following equation: P2(n)+14=P(n2+14)P^2 (n)+\frac{1}{4}=P(n^2+\frac{1}{4}) for ∀\forall n∈Nn\in \mathbb{N}.
algebraPolynomials
Problem 6 of Second round - Ball and barriers on a chessboard

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

8/31/2019
We are given a chessboard 100 x 100, kk barriers (each with length 1), and one ball. We want to put the barriers between the cells of the board and put the ball in some cell, in such way that the ball can get to each possible cell on the board. The only way that the ball can move is by lifting the board so it can go only forward, backward, to the left or to the right. The ball passes all cells on its way until it reaches a barrier or the edge of the board where it stops. What’s the least number of barriers we need so we can achieve that?
combinatoricsChessboard
Problem 6 of Third round

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

9/1/2019
a,b,m,k∈Za,b,m,k\in \mathbb{Z}, a,b,m>2,k>1a,b,m>2,k>1, for which kna+bk^n a+b is an mm-th power of a natural number for ∀n∈N\forall n\in \mathbb{N}. Prove that bb is an mm-th power of a non-negative integer.
number theory
Problem 6 of Fourth round - Game strategy with a polynomial

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

9/3/2019
Let f(x)f(x) be a polynomial, such that f(x)=x2015+a1x2014+...+a2014x+a2015f(x)=x^{2015}+a_1 x^{2014}+...+a_{2014} x+a_{2015}. Velly and Polly are taking turns, starting from Velly changing the coefficients aia_i with real numbers , where each coefficient is changed exactly once. After 2015 turns they calculate the number of real roots of the created polynomial and if the root is only one, then Velly wins, and if it’s not – Polly wins. Which one has a winning strategy?
Game Theorygame strategypolynomialalgebra
Problem 6 of Finals

Source: VII International Festival of Young Mathematicians Sozopol 2016, Theme for 10-12 grade

9/19/2019
On the sides of a convex, non-regular mm-gon are built externally regular heptagons. It is known that their centers are vertices of a regular mm-gon. What’s the least possible value of mm?
geometryPolygons