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 $a$, $b$, $c$ be the sides of a triangle, and $S$ its area. Prove: $$a^{2} + b^{2} + c^{2}\geq 4S \sqrt {3}$$ In what case does equality hold?

%V0 Let $ABC$ be a triangle, and let $P$, $Q$, $R$ be three points in the interiors of the sides $BC$, $CA$, $AB$ of this triangle. Prove that the area of at least one of the three triangles $AQR$, $BRP$, $CPQ$ is less than or equal to one quarter of the area of triangle $ABC$. Alternative formulation: Let $ABC$ be a triangle, and let $P$, $Q$, $R$ be three points on the segments $BC$, $CA$, $AB$, respectively. Prove that $\min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$, where the abbreviation $\left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $P_1P_2P_3$.

%V0 We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots,5$ is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.

%V0 Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.

%V0 Is it possible to put $1987$ points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a non-degenerate triangle with rational area? (IMO Problem 5) Proposed by Germany, DR