Olympiad problems

1969 Czech and Slovak Olympiad III A, 2

Tags: geometry , Geometric Inequalities , geometrical inequalities , convex , quadrilateral , area

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.

1973 Polish MO Finals, 6

Tags: conics , ellipse , geometry , Geometric Inequalities

Prove that for every centrally symmetric polygon there is at most one ellipse containing the polygon and having the minimal area.

1985 Swedish Mathematical Competition, 5

Tags: geometry , analytic geometry , Geometric Inequalities , inequalities

In a rectangular coordinate system, $O$ is the origin and $A(a,0)$, $B(0,b)$ and $C(c,d)$ the vertices of a triangle. Prove that $AB+BC+CA \ge 2CO$.

2018 Flanders Math Olympiad, 1

Tags: geometry , angles , Geometric Inequalities

In the triangle $\vartriangle ABC$ we have $| AB |^3 = | AC |^3 + | BC |^3$. Prove that $\angle C> 60^o$ .

1994 Swedish Mathematical Competition, 2

Tags: Geometric Inequalities , geometry , Medians

In the triangle $ABC$, the medians from $B$ and $C$ are perpendicular. Show that $\cot B + \cot C \ge \frac23$.

1973 Bulgaria National Olympiad, Problem 6

Tags: geometry , 3D geometry , tetrahedron , inequalities , Geometric Inequalities

In the tetrahedron $ABCD$, $E$ and $F$ are the midpoints of $BC$ and $AD$, $G$ is the midpoint of the segment $EF$. Construct a plane through $G$ intersecting the segments $AB$, $AC$, $AD$ in the points $M,N,P$ respectively in such a way that the sum of the volumes of the tetrahedrons $BMNP$, $CMNP$ and $DMNP$ to be minimal. [i]H. Lesov[/i]

1964 Swedish Mathematical Competition, 4

Tags: min , max , combinatorial geometry , geometry , inequalities , Geometric Inequalities

Points $H_1, H_2, ... , H_n$ are arranged in the plane so that each distance $H_iH_j \le 1$. The point $P$ is chosen to minimise $\max (PH_i)$. Find the largest possible value of $\max (PH_i)$ for $n = 3$. Find the best upper bound you can for $n = 4$.