MathDB

3

Part of 2016 IFYM, Sozopol

Problems(5)

Problem 3 of First round - inequalities

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

4/21/2022
Let x≤y≤zx\leq y\leq z be real numbers such that x+y+z=12x+y+z=12, x2+y2+z2=54x^2+y^2+z^2=54. Prove that: a) x≤3x\leq 3 and z≥5z\geq 5 b) xyxy, yzyz, zx∈[9,25]zx\in [9,25]
algebrainequalities
Problem 3 of Second round

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

8/31/2019
Let f:R2→Rf: \mathbb{R}^2\rightarrow \mathbb{R} be a function for which for arbitrary x,y,z∈Rx,y,z\in \mathbb{R} we have that f(x,y)+f(y,z)+f(z,x)=0f(x,y)+f(y,z)+f(z,x)=0. Prove that there exist function g:R→Rg:\mathbb{R}\rightarrow \mathbb{R} for which: f(x,y)=g(x)−g(y), ∀x,y∈Rf(x,y)=g(x)-g(y),\, \forall x,y\in \mathbb{R}.
algebraFunctional Equationsfunctions
Problem 3 of Third round

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

9/1/2019
Let A1A2…A66A_1 A_2…A_{66} be a convex 66-gon. What’s the greatest number of pentagons AiAi+1Ai+2Ai+3Ai+4,1≤i≤66,A_i A_{i+1} A_{i+2} A_{i+3} A_{i+4},1\leq i\leq 66, which have an inscribed circle? (A66+i≡AiA_{66+i}\equiv A_i).
combinatorial geometrycombinatorics
Problem 3 of Fourth round - 2 circles with radius r and 2r and a convex polygon

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

1/11/2020
The angle of a rotation ρ\rho is α<180∘\alpha <180^\circ and ρ\rho maps the convex polygon MM in itself. Prove that there exist two circles c1c_1 and c2c_2 with radius rr and 2r2r, so that c1c_1 is inner for MM and MM is inner for c2c_2.
geometryrotationconvex polygon
Problem 3 of Finals

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

9/19/2019
Find the least natural number n≥5n\geq 5, for which xn≡16 (mod p)x^n\equiv 16\, (mod\, p) has a solution for any prime number pp.
number theoryprime numbers