Olympiad problems

1973 Polish MO Finals, 6

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

1995 North Macedonia National Olympiad, 2

Tags: geometric inequalities , altitude , inequalities

Let $ a, $ $ b $, and $ c $ be sides in a triangle, a $ h_a, $ $ h_b $, and $ h_c $ are the corresponding altitudes. Prove that $h ^ 2_a + h ^ 2_b + h ^ 2_c \leq \frac{3}{4} (a ^ 2 + b ^ 2 + c ^ 2). $ When is the equation valid?

2001 Estonia Team Selection Test, 2

Tags: geometric inequalities , regular polygon , geometry , distance

Point $X$ is taken inside a regular $n$-gon of side length $a$. Let $h_1,h_2,...,h_n$ be the distances from $X$ to the lines defined by the sides of the $n$-gon. Prove that $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$

1973 Bulgaria National Olympiad, Problem 6

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

2003 Singapore MO Open, 4

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

The pentagon $ABCDE$ which is inscribed in a circle with $AB < DE$ is the base of a pyramid with apex $S$. If the longest side from $S$ is $SA$, prove that $BS > CS$.

1975 Bulgaria National Olympiad, Problem 4

Tags: geometric inequalities , inequalities , geometrical inequalities , circles , geometry

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]

1961 Polish MO Finals, 3

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

Prove that if a plane section of a tetrahedron is a parallelogram, then half of its perimeter is contained between the length of the smallest and the length of the largest edge of the tetrahedron.

1983 Swedish Mathematical Competition, 5

Tags: geometric inequalities , min , geometry , disc , combinatorics , disks , combinatorial geometry

Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$. What is the smallest possible radius?

2016 Romania National Olympiad, 3

Tags: geometric inequalities , geometric inequality , geometry

If $a, b$ and $c$ are the length of the sides of a triangle, show that $$\frac32 \le \frac{b + c}{b + c + 2a}+ \frac{a + c}{a + c + 2b}+ \frac{a + b}{a + b + 2c}\le \frac53.$$

1954 Kurschak Competition, 1

Tags: geometric inequalities , geometry

$ABCD$ is a convex quadrilateral with $AB + BD = AC + CD$. Prove that $AB < AC$.

2002 Estonia National Olympiad, 3

Tags: geometric inequalities , sidelenghts , inequalities

Prove that for positive real numbers $a, b$ and $c$ the inequality $2(a^4+b^4+c^4) < (a^2+b^2+c^2)^2$ holds if and only if $a,b,c$ are the sides of a triangle.

1999 French Mathematical Olympiad, Problem 3

Tags: inradius , geometric inequalities , triangle , ratio , geometry

For which acute-angled triangles is the ratio of the smallest side to the inradius the maximum?

1985 Austrian-Polish Competition, 3

Tags: quadrilateral , geometric inequalities , sum , area , geometry , inequalities

In a convex quadrilateral of area $1$, the sum of the lengths of all sides and diagonals is not less than $4+\sqrt 8$. Prove this.

2002 Junior Balkan Team Selection Tests - Romania, 4

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

Five points are given in the plane that each of $10$ triangles they define has area greater than $2$. Prove that there exists a triangle of area greater than $3$.

2016 Flanders Math Olympiad, 3

Tags: geometric inequalities , geometry , area

Three line segments divide a triangle into five triangles. The area of these triangles is called $u, v, x,$ yand $z$, as in the figure. (a) Prove that $uv = yz$. (b) Prove that the area of the great triangle is at most $ \frac{xz}{y}$ [img]https://cdn.artofproblemsolving.com/attachments/9/4/2041d62d014cf742876e01dd8c604c4d38a167.png[/img]

1975 Polish MO Finals, 5

Tags: geometric inequalities , trigonometry , geometry

Show that it is possible to circumscribe a circle of radius $R$ about, and inscribe a circle of radius $r$ in some triangle with one angle equal to $a$, if and only if $$\frac{2R}{r} \ge \dfrac{1}{ \sin \frac{a}{2} \left(1- \sin \frac{a}{2} \right)}$$

2009 Iran MO (2nd Round), 3

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

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 $.

1916 Eotvos Mathematical Competition, 2

Tags: geometric inequalities , geometry

Let the bisector of the angle at $C$ of triangle $ABC$ intersect side $AB$ in point $D$. Show that the segment $CD$ is shorter than the geometric mean of the sides $CA$ and $CB$. (The geometric mean of two positive numbers is the square root of their product; the geometric mean of $n$ numbers is the $n$-th root of their product.

1972 Polish MO Finals, 2

Tags: point , geometric inequalities , geometry

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.

2015 Sharygin Geometry Olympiad, P15

Tags: geometry , geometric inequalities , circumradius , inradius

The sidelengths of a triangle $ABC$ are not greater than $1$. Prove that $p(1 -2Rr)$ is not greater than $1$, where $p$ is the semiperimeter, $R$ and $r$ are the circumradius and the inradius of $ABC$.

2012 Bogdan Stan, 4

Tags: geometric inequalities , angle inequalities , rational geometry , inequalities , geometry , ratio

Let $ D $ be a point on the side $ BC $ (excluding its endpoints) of a triangle $ ABC $ with $ AB>AC, $ such that $ \frac{\angle BAD}{\angle BAC} $ is a rational number. Prove the following: $$ \frac{\angle BAD}{\angle BAC} < \frac{AB\cdot AC - AC\cdot AD}{AB\cdot AD - AC\cdot AD} $$