Olympiad problems

1981 Kurschak Competition, 1

Prove that $$AB + PQ + QR + RP \le AP + AQ + AR + BP + BQ + BR$$ where $A, B, P, Q$ and $R $ are any five points in a plane.

1999 Romania National Olympiad, 2

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

On the sides $(AB)$, $(BC)$, $(CD)$ and $(DA)$ of the regular tetrahedron $ABCD$, one considers the points $M$, $N$, $P$, $Q$, respectively Prove that $$MN \cdot NP \cdot PQ \cdot QM \ge AM \cdot BN \cdot CP \cdot DQ.$$

1990 Bulgaria National Olympiad, Problem 6

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

The base $ABC$ of a tetrahedron $MABC$ is an equilateral triangle, and the lateral edges $MA,MB,MC$ are sides of a triangle of the area $S$. If $R$ is the circumradius and $V$ the volume of the tetrahedron, prove that $RS\ge2V$. When does equality hold?

2011 Indonesia TST, 1

Tags: inequalities , geometric inequalities , algebra

Let $a, b, c$ be the sides of a triangle with $abc = 1$. Prove that $$\frac{\sqrt{b + c -a}}{a}+\frac{\sqrt{c + a - b}}{b}+\frac{\sqrt{a + b - c}}{c} \ge a + b + c$$

2010 Estonia Team Selection Test, 3

Tags: inequalities , geometric inequalities , trigonometry , circumradius , geometry

Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?

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 $$\frac{2R}{r} \ge \dfrac{1}{ \sin \frac{a}{2} \left(1- \sin \frac{a}{2} \right)}$$

1984 Poland - Second Round, 5

Tags: geometry , analytic geometry , lattice , geometric inequalities

Calculate the lower bound of the areas of convex hexagons whose vertices all have integer coordinates.

2007 Estonia Team Selection Test, 2

Tags: circumradius , inradius , altitude , geometry , geometric inequalities , circumcircle , incircle

Let $D$ be the foot of the altitude of triangle $ABC$ drawn from vertex $A$. Let $E$ and $F$ be points symmetric to $D$ w.r.t. lines $AB$ and $AC$, respectively. Let $R_1$ and $R_2$ be the circumradii of triangles $BDE$ and $CDF$, respectively, and let $r_1$ and $r_2$ be the inradii of the same triangles. Prove that $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$

2000 Singapore Team Selection Test, 3

Tags: combinatorial geometry , combinatorics , geometric inequalities , inequalities

There are $n$ blue points and $n$ red points on a straight line. Prove that the sum of all distances between pairs of points of the same colour is less than or equal to the sum of all distances between pairs of points of different colours

2018 Czech-Polish-Slovak Match, 4

Tags: geometric inequalities , geometry , inequalities

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]

1997 Singapore MO Open, 1

Tags: equilateral , geometry , geometric inequalities

$\vartriangle ABC$ is an equilateral triangle. $L, M$ and $N$ are points on $BC, CA$ and $AB$ respectively. Prove that $MA \cdot AN + NB \cdot BL + LC \cdot CM < BC^2$.

1924 Eotvos Mathematical Competition, 1

Tags: number theory , geometry , triangle inequality , inequalities , geometric inequalities

Let $a, b, c$ be fìxed natural numbers. Suppose that, for every positive integer n, there is a triangle whose sides have lengths $a^n$, $b^n$, and $c^n$ respectively. Prove that these triangles are isosceles.

1973 Poland - Second Round, 1

Tags: algebra , inequalities , geometric inequalities , geometry

Prove that if positive numbers $ x, y, z $ satisfy the inequality $$ \frac{x^2+y^2-z^2}{2xy} + \frac{y^2+z^2-x^2}{2yz} + \frac{z^2+x^2-y ^2}{2xz} > 1,$$ then they are the lengths of the sides of a certain triangle.

1952 Kurschak Competition, 1

Tags: geometry , geometric inequalities , tangent circle

A circle $C$ touches three pairwise disjoint circles whose centers are collinear and none of which contains any of the others. Show that its radius must be larger than the radius of the middle of the three circles.

1961 Polish MO Finals, 3

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

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.

1963 Polish MO Finals, 3

Tags: geometry , rectangle , geometric inequalities

From a given triangle, cut out the rectangle with the largest area.

2009 Estonia Team Selection Test, 4

Tags: geometry , geometric inequalities , inequalities , ratio

Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$

2020 Jozsef Wildt International Math Competition, W35

Tags: inequalities , geometry , geometric inequalities

In all triangles $ABC$ does it hold: $$(b^n+c^p)\tan^{n+p}\frac A2+(c^n+a^p)\tan^{n+p}\frac B2+(a^n+b^p)\tan^{n+p}\frac C2\ge6\sqrt{\left(\frac{4r^2}{R\sqrt3}\right)^{n+p}}$$ where $n,p\in(0,\infty)$. [i]Proposed by Nicolae Papacu[/i]

2002 Junior Balkan Team Selection Tests - Romania, 4

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

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 $3$.

1955 Polish MO Finals, 5

Tags: geometry , construction , geometric inequalities

In the plane, a straight line $ m $ is given and points $ A $ and $ B $ lie on opposite sides of the straight line $ m $. Find a point $ M $ on the line $ m $ such that the difference in distances of this point from points $ A $ and $ B $ is as large as possible.

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 $n = 4$.

1996 Romania National Olympiad, 4

Tags: algebra , geometric inequality , geometric inequalities , 3d geometry , geometry , inequalities

a) Let $AB CD$ be a regular tetrahedron. On the sides $AB$, $AC$ and $AD$, the points $M$, $N$ and $P$, are considered. Determine the volume of the tetrahedron $AMNP$ in terms of $x, y, z$, where $x=AM$, $y=AN$, $z=AP$. b) Show that for any real numbers $x, y, z, t, u, v \in (0, 1)$ : $$xyz + uv(1- x) + (1- y)(1- v)t + (1- z)(1- w)(1- t) < 1.$$

1972 Bulgaria National Olympiad, Problem 5

Tags: geometry , cyclic quadrilateral , perpendicular , inequalities , geometric inequalities

In a circle with radius $R$, there is inscribed a quadrilateral with perpendicular diagonals. From the intersection point of the diagonals, there are perpendiculars drawn to the sides of the quadrilateral. (a) Prove that the feet of these perpendiculars $P_1,P_2,P_3,P_4$ are vertices of the quadrilateral that is inscribed and circumscribed. (b) Prove the inequalities $2r_1\le\sqrt2 R_1\le R$ where $R_1$ and $r_1$ are radii respectively of the circumcircle and inscircle to the quadrilateral $P_1P_2P_3P_4$. When does equality hold? [i]H. Lesov[/i]

Indonesia MO Shortlist - geometry, g9

Tags: geometry , geometric inequalities , perimeter , semiperimeter

Given a triangle $ABC$, the points $D$, $E$, and $F$ lie on the sides $BC$, $CA$, and $AB$, respectively, are such that $$DC + CE = EA + AF = FB + BD.$$ Prove that $$DE + EF + FD \ge \frac12 (AB + BC + CA).$$

1979 Polish MO Finals, 5

Tags: cyclic quadrilateral , geometric inequalities , geometry

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