MathDB

6

Part of 2022 IFYM, Sozopol

Problems(5)

Problem 6 of First round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022
Let kk be a fixed circle in a given plane and a point CC out of the plane. Let AA be a random point from kk and BB be its diametrically opposite one in kk. Find the geometric place of the center of the circumscribed circle of ABCABC.
geometrygeometric place
Problem 6 of Second round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022
Let DD be an infinite in both sides sequence of 00s and 11s. For each positive integer nn we denote with ana_n the number of different subsequences of 00s and 11s in DD of length nn. Does there exist a sequence DD for which for each n≥22n\geq 22 the number ana_n is equal to the nn-th prime number?
number theory
Problem 6 of Third round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022
A mixing of the sequence a1,a2,…,a3na_1,a_2,\dots ,a_{3n} is called the following sequence: a3,a6,…,a3n,a2,a5,…,a3n−1,a1,a4,…,a3n−2a_3,a_6,\dots ,a_{3n},a_2,a_5,\dots ,a_{3n-1},a_1,a_4,\dots ,a_{3n-2}. Is it possible after finite amount of mixings to reach the sequence 192,191,…,1192,191,\dots ,1 from 1,2,…,1921,2,\dots ,192?
combinatorics
\sum f(i, j) <= 2 (sum^n_{k=0} 1/k! )(\sum^n_{p=1}p!)+ 3

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade, 4th round p6

11/12/2022
For the function f:Z≥02→Z≥0f : Z^2_{\ge0} \to Z_{\ge 0} it is known that f(0,j)=f(i,0)=1,     ∀i,j∈N0f(0, j) = f(i, 0) = 1, \,\,\,\,\, \forall i, j \in N_0 f(i,j)=if(i,j−1)+jf(i−1,j),     ∀i,j∈Nf(i, j) = if (i, j - 1) + jf(i - 1, j),\,\,\,\,\, \forall i, j \in N Prove that for every natural number nn the following inequality holds: ∑0≤i+j≤n+1f(i,j)≤2(∑k=0n1k!)(∑p=1np!)+3\sum_{0\le i+j\le n+1} f(i, j) \le 2 \left(\sum^n_{k=0}\frac{1}{k!}\right)\left(\sum^n_{p=1}p!\right)+ 3
inequalitiesalgebraFunctional inequalityfunctional equation
P_i(a) = N, infinitely many primes

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade,finals p6

11/13/2022
Let nn be a natural number and P1,P2,...,PnP_1, P_2, ... , P_n are polynomials with integer coefficients, each of degree at least 22. Let SS be the set of all natural numbers NN for which there exists a natural number aa and an index 1≤i≤n1 \le i \le n such that Pi(a)=NP_i(a) = N. Prove, that there are infinitely many primes that do not belong to SS.
polynomialalgebra