MathDB

2

Part of 2016 IFYM, Sozopol

Problems(5)

Problem 2 of First round - Chessboard and lines

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

8/29/2019
A cell is cut from a chessboard 8 x 88\, x\, 8, after which an open broken line was built, which vertices are the centers of the remaining cells. Each segment of the broken line has a length 17\sqrt{17} or 65\sqrt{65}. When is the number of such broken lines bigger – when the cut cell is (1,2)(1,2) or (3,6)(3,6)? (The rows and columns on the board are numerated consecutively from 1 to 8.)
combinatoricsChessboard
Problem 2 of Second round - Polynomial and prime numbers

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

8/31/2019
We are given a polynomial f(x)=x6−11x4+36x2−36f(x)=x^6-11x^4+36x^2-36. Prove that for an arbitrary prime number pp, f(x)≡0(modp)f(x)\equiv 0\pmod{p} has a solution.
algebrapolynomialprime numbersnumber theory
Problem 2 of Third round

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

9/1/2019
Let pp be a prime number and the decimal notation of 1p\frac{1}{p} is periodical with a length of the period 4k4k, 1p=0,a1a2…a4ka1a2…a4k…\frac{1}{p}=0,a_1 a_2…a_{4k} a_1 a_2…a_{4k}… .Prove that a1+a3+...+a4k−1=a2+a4+...+a4ka_1+a_3+...+a_{4k-1}=a_2+a_4+...+a_{4k}.
algebraprime numbers
Problem 2 of Fourth round

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

12/31/2019
Let a0,a1,a2...a_0,a_1,a_2... be a sequence of natural numbers with the following property: an2a_n^2 divides an−1an+1a_{n-1} a_{n+1} for ∀\forall n∈Nn\in \mathbb{N}. Prove that, if for some natural k≥2k\geq 2 the numbers a1a_1 and aka_k are coprime, then a1a_1 divides a0a_0.
number theorySequence
Problem 2 of Finals

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

1/11/2020
On the VI-th International Festival of Young Mathematicians in Sozopol nn teams were participating, each of which was with kk participants (n>k>1n>k>1). The organizers of the competition separated the nknk participants into nn groups, each with kk people, in such way that no two teammates are in the same group. Prove that there can be found nn participants no two of which are in the same team or group.
combinatoricsset theory