This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 185

1985 Austrian-Polish Competition, 3
Tags: quadrilateral , geometric inequalities , geometry , inequalities , sum , area
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.

1973 Spain Mathematical Olympiad, 7
Tags: analytic geometry , geometry , inequalities , geometric inequalities
The two points $P(8, 2)$ and $Q(5, 11)$ are considered in the plane. A mobile moves from $P$ to $Q$ according to a path that has to fulfill the following conditions: The moving part of $ P$ and arrives at a point on the $x$-axis, along which it travels a segment of length $1$, then it departs from this axis and goes towards a point on the $y$ axis, on which travels a segment of length $2$, separates from the $y$ axis finally and goes towards the point $Q$. Among all the possible paths, determine the one with the minimum length, thus like this same length.

2011 Indonesia TST, 1
Tags: inequalities , geometric inequalities , algebra
Let $a, b, c$ be the sides of a triangle with $abc = 1$. Prove that $$\frac{\sqrt{b + c -a}}{a}+\frac{\sqrt{c + a - b}}{b}+\frac{\sqrt{a + b - c}}{c} \ge a + b + c$$

1981 Swedish Mathematical Competition, 5
Tags: geometry , geometric inequalities , area
$ABC$ is a triangle. $X$, $Y$, $Z$ lie on $BC$, $CA$, $AB$ respectively. Show that area $XYZ$ cannot be smaller than each of area $AYZ$, area $BZX$, area $CXY$.

2020 Jozsef Wildt International Math Competition, W57
Tags: triangle , inequalities , geometric inequalities , geometry
In all triangles $ABC$ does it hold that: $$\sum\sin^2\frac A2\cos^2A\ge\frac{3\left(s^2-(2R+r)^2\right)}{8R^2}$$ [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

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 $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$

1981 Polish MO Finals, 6
Tags: geometry , 3d geometry , tetrahedron , geometric inequalities
In a tetrahedron of volume $V$ the sum of the squares of the lengths of its edges equals $S$. Prove that $$V \le \frac{S\sqrt{S}}{72\sqrt{3}}$$

Indonesia Regional MO OSP SMA - geometry, 2012.4
Tags: geometry , geometric inequality , geometric inequalities , trigonometric inequality , trigonometry
Given an acute triangle $ABC$. Point $H$ denotes the foot of the altitude drawn from $A$. Prove that $$AB + AC \ge BC cos \angle BAC + 2AH sin \angle BAC$$

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 $MA \cdot AN + NB \cdot BL + LC \cdot CM < BC^2$.

2016 Sharygin Geometry Olympiad, 7
Tags: geometry , geometric inequalities , trigonometry
From the altitudes of an acute-angled triangle, a triangle can be composed. Prove that a triangle can be composed from the bisectors of this triangle.