MathDB

2

Part of 2018 IFYM, Sozopol

Problems(5)

Problem 2 of First round

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

9/21/2018
A square is divided into 169 identical small squares and in every small square is written 0 or 1. It isn’t allowed in one row or column to have the following arrangements of adjacent digits in this order: 101, 111 or 1001. What is the the biggest possible number of 1’s in the table?
tablecombinatorics
Problem 2 of Third round

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

9/22/2018
The set of numbers (p,a,b,c)(p, a, b, c) of positive integers is called Sozopolian when:
* p is an odd prime number
* aa, bb and cc are different and
* ab+1ab + 1, bc+1bc + 1 and ca+1ca + 1 are a multiple of pp.
a) Prove that each Sozopolian set satisfies the inequality p+2≤a+b+c3p+2 \leq \frac{a+b+c}{3}
b) Find all numbers pp for which there exist a Sozopolian set for which the equality of the upper inequation is met.
number theory
Problem 2 of Second round

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

9/22/2018
n>1n > 1 is an odd number and a1,a2,...,ana_1, a_2, . . . , a_n are positive integers such that gcd(a1,a2,...,an)=1gcd(a_1, a_2, . . . , a_n) = 1. If
d=gcd(a1n+a1.a2...an,a2n+a1.a2...an,...,ann+a1.a2...an)d = gcd (a_1^n + a_1.a_2. . . a_n, a_2^n + a_1.a_2. . . a_n, . . . , a_n^n + a_1.a_2. . . a_n)
find all possible values of dd.
number theoryDivisorsgreatest common divisor
Problem 2 of Fourth round

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

9/22/2018
xx, yy, and zz are positive real numbers satisfying the equation x+y+z=1x+1y+1zx+y+z=\frac{1}{x} + \frac{1}{y} + \frac{1}{z}.
Prove the following inequality:
xy+yz+zx≥3xy + yz + zx \geq 3.
algebrainequalities
Problem 2 of Finals

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

9/22/2018
a) The real number aa and the continuous function f:[a,∞)→[a,∞)f : [a, \infty) \rightarrow [a, \infty) are such that ∣f(x)−f(y)∣<∣x–y∣|f(x)-f(y)| < |x–y| for every two different x,y∈[a,∞)x, y \in [a, \infty). Is it always true that the equation f(x)=xf(x)=x has only one solution in the interval [a,∞)[a, \infty)?
b) The real numbers aa and bb and the continuous function f:[a,b]→[a,b]f : [a, b] \rightarrow [a, b] are such that ∣f(x)−f(y)∣<∣x–y∣|f(x)-f(y)| < |x–y|, for every two different x,y∈[a,b]x, y \in [a, b]. Is it always true that the equation f(x)=xf(x)=x has only one solution in the interval [a,b][a, b]?
algebrafunctional equationalgebra unsolved