Home > Olympiad > Vojtech Jarnik Competition 2015 – Problems Category 2

## Vojtech Jarnik Competition 2015 – Problems Category 2

Problem 1. Let ${A}$ and ${B}$ be two ${3 \times 3}$ matrices with real entries. Prove that

$\displaystyle A - (A^{-1}+(B^{-1}-A)^{-1})^{-1} = ABA,$

provided all the inverses appearing on the left-hand side of the equality exist.

Problem 2. Determine all pairs ${(n,m)}$ of positive integers satisfying the equation

$\displaystyle 5^n = 6m^2+1.$

Problem 3. Determine the set of real values ${x}$ for which the following series converges, and find its sum:

$\displaystyle \sum_{n=1}^\infty \left( \sum_{k_i \geq 0, k_1+2k_2+...+nk_n = n} \frac{(k_1+...+k_n)!}{k_1!...k_n!} x^{k_1+...+k_n}\right).$

Problem 4. Find all continuously differentiable functions ${f : \Bbb{R} \rightarrow \Bbb{R}}$, such that for every ${a \geq 0}$ the following relation holds:

$\displaystyle \int_{D(a)} xf\left( \frac{ay}{\sqrt{x^2+y^2}}\right) dxdydz = \frac{\pi a^3}{8}(f(a)+\sin a -1),$

where ${D(a) = \left\{ (x,y,z) : x^2+y^2+z^2 \leq a^2,\ |y| \leq \frac{x}{\sqrt{3}}\right\}}$