Školjka

%V0 For what real values of $x$ is $$\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A$$ given a) $A=\sqrt{2}$; b) $A=1$; c) $A=2$, where only non-negative real numbers are admitted for square roots?

%V0 Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

%V0 Each of the numbers in the set $N = \{1, 2, 3, \cdots, n - 1\}$, where $n \geq 3$, is colored with one of two colors, say red or black, so that: (i) $i$ and $n - i$ always receive the same color, and (ii) for some $j \in N$, relatively prime to $n$, $i$ and $|j - i|$ receive the same color for all $i \in N, i \neq j.$ Prove that all numbers in $N$ must receive the same color.

%V0 Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$. (IMO Problem 3) Proposed by Germany, FR

%V0 Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.(IMO Problem 6) Original Formulation Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes. Proposed by Soviet Union.

%V0 Let $a$ and $b$ be two positive integers such that $a \cdot b + 1$ divides $a^{2} + b^{2}$. Show that $\frac {a^{2} + b^{2}}{a \cdot b + 1}$ is a perfect square.