Olympiad problems

1975 Bulgaria National Olympiad, Problem 4

In the plane are given a circle $k$ with radii $R$ and the points $A_1,A_2,\ldots,A_n$, lying on $k$ or outside $k$. Prove that there exist infinitely many points $X$ from the given circumference for which $$\sum_{i=1}^n A_iX^2\ge2nR^2.$$ Does there exist a pair of points on different sides of some diameter, $X$ and $Y$ from $k$, such that $$\sum_{i=1}^n A_iX^2\ge2nR^2\text{ and }\sum_{i=1}^n A_iY^2\ge2nR^2?$$ [i]H. Lesov[/i]

1969 Czech and Slovak Olympiad III A, 2

Tags: area , geometric inequalities , quadrilateral , geometrical inequalities , convex , geometry

Five different points $O,A,B,C,D$ are given in plane such that \[OA\le OB\le OC\le OD.\] Show that for area $P$ of any convex quadrilateral with vertices $A,B,C,D$ (not necessarily in this order) the inequality \[P\le \frac12(OA+OD)(OB+OC)\] holds and determine when equality occurs.

1974 Bulgaria National Olympiad, Problem 6

Tags: geometry , 3d geometry , geometrical inequalities , geometric inequalities , inequalities , pyramid

In triangle pyramid $MABC$ at least two of the plane angles next to the edge $M$ are not equal to each other. Prove that if the bisectors of these angles form the same angle with the angle bisector of the third plane angle, the following inequality is true $$8a_1b_1c_1\le a^2a_1+b^2b_1+c^2c_1$$ where $a,b,c$ are sides of triangle $ABC$ and $a_1,b_1,c_1$ are edges crossed respectively with $a,b,c$. [i]V. Petkov[/i]

2019 Polish Junior MO Finals, 4.

Tags: geometrical inequalities , geometric inequality , inequalities , geometry

The point $D$ lies on the side $AB$ of the triangle $ABC$. Assume that there exists such a point $E$ on the side $CD$, that $$ \sphericalangle EAD = \sphericalangle AED \quad \text{and} \quad \sphericalangle ECB = \sphericalangle CEB. $$ Show that $AC + BC > AB + CE$.

1972 Bulgaria National Olympiad, Problem 6

Tags: tetrahedron , geometry , inequalities , 3d geometry , geometrical inequalities

It is given a tetrahedron $ABCD$ for which two points of opposite edges are mutually perpendicular. Prove that: (a) the four altitudes of $ABCD$ intersects at a common point $H$; (b) $AH+BH+CH+DH<p+2R$, where $p$ is the sum of the lengths of all edges of $ABCD$ and $R$ is the radii of the sphere circumscribed around $ABCD$. [i]H. Lesov[/i]

1974 Czech and Slovak Olympiad III A, 5

Tags: geometry , inequalities , geometric inequalities , geometrical inequalities , cyclic , hexagon , area

Let $ABCDEF$ be a cyclic hexagon such that \[AB=BC,\quad CD=DE,\quad EF=FA.\] Show that \[[ACE]\le[BDF]\] and determine when the equality holds. ($[XYZ]$ denotes the area of the triangle $XYZ.$)