Olympiad problems

1942 Eotvos Mathematical Competition, 1

Prove that in any triangle, at most one side can be shorter than the altitude from the opposite vertex.

1972 Polish MO Finals, 2

Tags: point , geometry , geometric inequalities

On the plane are given $n > 2$ points, no three of which are collinear. Prove that among all closed polygonal lines passing through these points, any one with the minimum length is non-selfintersecting.

VI Soros Olympiad 1999 - 2000 (Russia), 10.4

Tags: geometry , geometric inequality , geometric inequalities , exradius , inequalities

Prove that the inequality $ r^2+r_a^2+r_b^2+ r_c^2 \ge 2S$ holds for an arbitrary triangle, where $r$ is the radius of the circle inscribed in the triangle, $r_a$, $r_b$, $r_c$ are the radii of its three excribed circles, $S$ is the area of the triangle.

1979 Polish MO Finals, 5

Tags: geometry , cyclic quadrilateral , geometric inequalities

Prove that the product of the sides of a quadrilateral inscribed in a circle with radius $1$ does not exceed $4$.

1989 Romania Team Selection Test, 3

Tags: geometry , parallelogram , min , geometric inequalities , inequalities

(a) Find the point $M$ in the plane of triangle $ABC$ for which the sum $MA + MB+ MC$ is minimal. (b) Given a parallelogram $ABCD$ whose angles do not exceed $120^o$, determine $min \{MA+ MB+NC+ND+ MN | M,N$ are in the plane $ABCD\}$ in terms of the sides and angles of the parallelogram.

2018 Czech-Polish-Slovak Match, 4

Tags: geometric inequalities , inequalities , geometry

Let $ABC$ be an acute triangle with the perimeter of $2s$. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers $A, B$, and $C$, respectively. Prove that there exists a circle with the radius of $s$ which contains all the three circles. [i]Proposed by Josef Tkadlec, Czechia[/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 $n = 4$.

1927 Eotvos Mathematical Competition, 3

Tags: geometry , inradius , exradius , geometric inequalities

Consider the four circles tangent to all three lines containing the sides of a triangle $ABC$; let $k$ and $k_c$ be those tangent to side $AB$ between $A$ and $B$. Prove that the geometric mean of the radii of k and $k_c$, does not exceed half the length of $AB$.

1964 Swedish Mathematical Competition, 1

Tags: geometry , min , geometric inequalities

Find the side lengths of the triangle $ABC$ with area $S$ and $\angle BAC = x$ such that the side $BC$ is as short as possible.

1970 Poland - Second Round, 4

Tags: geometry , perimeter , geometric inequalities

Prove that if triangle $T_1$ contains triangle $T_2$, then the perimeter of triangle $T_1$ is not less than the perimeter of triangle $T_2$.

2017 Singapore Junior Math Olympiad, 1

Tags: geometry , rectangle , ratio , geometric inequalities , inequalities , square , sidelenghts

A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.

1953 Poland - Second Round, 3

Tags: geometry , geometric inequalities , area

A triangular piece of sheet metal weighs $900$ g. Prove that by cutting this sheet metal along a straight line passing through the center of gravity of the triangle, it is impossible to cut off a piece weighing less than $400$ g.

1915 Eotvos Mathematical Competition, 2

Tags: geometry , perimeter , geometric inequalities

Triangle $ABC$ lies entirely inside a polygon. Prove that the perimeter of triangle $ABC$ is not greater than that of the polygon.

1988 Poland - Second Round, 5

Tags: geometry , rectangle , geometric inequalities , combinatorial geometry , combinatorics

Decide whether any rectangle that can be covered by 25 circles of radius 2 can also be covered by 100 circles of radius 1.

1992 Poland - Second Round, 5

Tags: geometry , 3d geometry , sphere , geometric inequalities , tetrahedron

Determine the upper limit of the volume of spheres contained in tetrahedra of all heights not longer than $ 1 $.

1950 Polish MO Finals, 4

Tags: geometry , hexagon , geometric inequality , square , geometric inequalities

Someone wants to unscrew a square nut with side $a$, with a wrench whose hole has the form of a regular hexagon with side $b$. What condition should the lengths $a$ and $b$ meet to make this possible?

1999 Singapore MO Open, 4

Tags: cyclic , geometric inequalities , geometry

Let $ABCD$ be a quadrilateral with each interior angle less than $180^o$. Show that if $A, B, C, D$ do not lie on a circle, then $AB \cdot CD + AD\cdot BC > AC \cdot BD$

1998 Yugoslav Team Selection Test, Problem 2

Tags: inequalities , geometric inequality , geometric inequalities , geometry

In a convex quadrilateral $ABCD$, the diagonal $AC$ intersects the diagonal $BD$ at its midpoint $S$. The radii of incircles of triangles $ABS,BCS,CDS,DAS$ are $r_1,r_2,r_3,r_4$, respectively. Prove that $$|r_1-r_2+r_3-r_4|\le\frac18|AB-BC+CD-DA|.$$

2009 Iran MO (2nd Round), 3

Tags: circumcircle , trigonometry , angle bisector , geometric inequalities , geometry , inequalities

Let $ ABC $ be a triangle and the point $ D $ is on the segment $ BC $ such that $ AD $ is the interior bisector of $ \angle A $. We stretch $ AD $ such that it meets the circumcircle of $ \Delta ABC $ at $ M $. We draw a line from $ D $ such that it meets the lines $ MB,MC $ at $ P,Q $, respectively ($ M $ is not between $ B,P $ and also is not between $ C,Q $). Prove that $ \angle PAQ\geq\angle BAC $.