Juniors

  1. Let $a,b,c\ge 0$ be three real numbers such that $$ab+bc+ca+2abc=1.$$ Prove that $\sqrt{a}+\sqrt{b}+\sqrt{c}\ge 2$ and determine equality cases.
  2. Prove that there exist an infinite number of odd natural numbers $m_1<m_2<...$ and an infinity of natural numbers $n_1<n_2<...$ such that $(m_k,n_k)=1$ and $m_k^4-2n_k^4$ is a perfect square for all $k\in\mathbb{N}$.
  3. Let $ABCD$ be cyclic quadrilateral,let $\gamma$ be it's circumscribed circle and let $M$ be the midpoint of arc $AB$ of $\gamma$,which does not contain points $C$, $D$. The line that passes through $M$ and the intersection point of diagonals $AC$, $BD$, intersects $\gamma$ in $N\neq M$. Let $P$, $Q$ be two points situated on $CD$ such that $\angle{AQD}=\angle{DAP}$ and $\angle{BPC}=\angle{CBQ}$. Prove that circles $\odot(NPQ)$ and $\gamma$ are tangent.
  4. Let $n\ge 5$ be a positive integer and let $$\{a_1,a_2,...,a_n\}=\{1,2,...,n\}.$$ Prove that at least $\lfloor \sqrt{n}\rfloor +1$ numbers from $a_1,a_1+a_2,...,a_1+a_2+...+a_n$ leave different residues when divided by $n$.

Senior

  1. Prove that there exist an infinite number of odd natural numbers $m_1<m_2<...$ and an infinity of natural numbers $n_1<n_2<...$ such that $(m_k,n_k)=1$ and $m_k^4-2n_k^4$ is a perfect square for all $k\in\mathbb{N}$.
  2. Let $\gamma$, $\gamma_0$, $\gamma_1$, $\gamma_2$ be four circles in plane such that $\gamma_i$ is interiorly tangent to $\gamma$ in point $A_i$, and $\gamma_i$ and $\gamma_{i+1}$ are exteriorly tangent in point $B_{i+2}$ $(i=0,1,2)$ (the indexes are reduced modulo $3$). The tangent in $B_i$, common for circles $\gamma_{i-1}$ and $\gamma_{i+1}$, intersects circle $\gamma$ in point $C_i$, situated in the opposite semiplane of $A_i$ with respect to line $A_{i-1}A_{i+1}$. Prove that the three lines $A_iC_i$ are concurrent.
  3. Let $n$ be a positive integer and let $a_1,a_2,...,a_n$ be non-zero positive integers. Prove that $$\sum_{k=1}^n\frac{\sqrt{a_k}}{1+a_1+a_2+...+a_k}<\sum_{k=1}^{n^2}\frac{1}{k}.$$
  4. Let $S$ be a finite set of points in the plane, situated in general position (any three points in $S$ are not collinear) and let $$D(S,r)=\{\{x,y\}:x,y\in S,\text{dist}(x,y)=r\}$$ where $R$ is a positive real number, and $\text{dist}(x,y)$ is the euclidean distance between points $x$ and $y$. Prove that $$\sum_{r>0}|D(S,r)|^2\le\frac{3|S|^2(|S|-1)}{4}.$$

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