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\}}

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