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

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 $4$.

1980 Swedish Mathematical Competition, 6
Tags: combinatorial geometry , combinatorics , Geometric Inequalities
Find the smallest constant $c$ such that for every $4$ points in a unit square there are two a distance $\leq c$ apart.

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.

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 $.

2009 Postal Coaching, 1
Tags: geometry , Geometric Inequalities , circumcircle
In a triangle $ABC$, let $D,E, F$ be interior points of sides $BC,CA,AB$ respectively. Let $AD,BE,CF$ meet the circumcircle of triangle $ABC$ in $K, L,M$ respectively. Prove that $\frac{AD}{DK} + \frac{BE}{EL} + \frac{CF}{FM} \ge 9$. When does the equality hold?

2020 Jozsef Wildt International Math Competition, W57
Tags: inequalities , Geometric Inequalities , geometry , Triangles
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]

2009 Postal Coaching, 3
Tags: inequalities , polygon , Cyclic , Geometric Inequalities
Let $\Omega$ be an $n$-gon inscribed in the unit circle, with vertices $P_1, P_2, ..., P_n$. (a) Show that there exists a point $P$ on the unit circle such that $PP_1 \cdot PP_2\cdot ... \cdot PP_n \ge 2$. (b) Suppose for each $P$ on the unit circle, the inequality $PP_1 \cdot PP_2\cdot ... \cdot PP_n \le 2$ holds. Prove that $\Omega$ is regular.

1965 Polish MO Finals, 1
Tags: algebra , inequalities , geometry , Geometric Inequalities
Prove the theorem: the lengths $ a$, $ b $, $ c $ of the sides of a triangle and the arc measures $ \alpha $, $ \beta $, $ \gamma $of its opposite angles satisfy the inequalities $$\frac{\pi}{3}\leq \frac{a \alpha + b \beta +c \gamma}{a+b+c}<\frac{\pi }{ 2}.$$

1990 Romania Team Selection Test, 5
Tags: geometry , circumcircle , circles , area , Geometric Inequalities , inequalities
Let $O$ be the circumcenter of an acute triangle $ABC$ and $R$ be its circumcenter. Consider the disks having $OA,OB,OC$ as diameters, and let $\Delta$ be the set of points in the plane belonging to at least two of the disks. Prove that the area of $\Delta$ is greater than $R^2/8$.

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.

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.

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.

2022 Durer Math Competition Finals, 9
Tags: geometry , perimeter , Geometric Inequalities
Every side of a right triangle is an integer when measured in cm, and the difference between the hypotenuse and one of the legs is $75$ cm. What is the smallest possible value of its perimeter?

1914 Eotvos Mathematical Competition, 1
Tags: geometry , arcs , arc , Geometric Inequalities
Let $A$ and $B$ be points on a circle $k$. Suppose that an arc $k'$ of another circle, $\ell$, connects $A$ with $B$ and divides the area inside the circle $k$ into two equal parts. Prove that arc $k'$ is longer than the diameter of $k$.

2016 Romania National Olympiad, 3
Tags: geometry , geometric inequality , Geometric Inequalities
If $a, b$ and $c$ are the length of the sides of a triangle, show that $$\frac32 \le \frac{b + c}{b + c + 2a}+ \frac{a + c}{a + c + 2b}+ \frac{a + b}{a + b + 2c}\le \frac53.$$

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

2015 Regional Olympiad of Mexico Center Zone, 6
Tags: geometry , Geometric Inequalities
We have $3$ circles such that any $2$ of them are externally tangent. Let $a$ be length of the outer tangent common to a pair of them. The lengths $b$ and $c$ are defined similarly. If $T$ is the sum of the areas of such circles, show that $\pi (a + b + c)^2 \le 12T $. Note: In In the case of externally tangent circles, the common external tangent is the segment tangent to them that touches them at different points.

1934 Eotvos Mathematical Competition, 2
Tags: geometry , polynomial , Cyclic , Geometric Inequalities
Which polygon inscribed in a given circle has the property that the sum of the squares of the lengths of its sides is maximum?

2009 Postal Coaching, 4
Tags: geometry , circumcircle , Triangles , perimeter , Geometric Inequalities
Let $ABC$ be a triangle, and let $DEF$ be another triangle inscribed in the incircle of $ABC$. If $s$ and $s_1$ denote the semiperimeters of $ABC$ and $DEF$ respectively, prove that $2s_1 \le s$. When does equality hold?

2000 Belarus Team Selection Test, 1.4
Tags: trigonometry , inequalities , Geometric Inequalities , sphere , geometry , 3D geometry
A closed pentagonal line is inscribed in a sphere of the diameter $1$, and has all edges of length $\ell$. Prove that $\ell \le \sin \frac{2\pi}{5}$ .

1961 Polish MO Finals, 4
Tags: geometry , Geometric Inequalities
Prove that if every side of a triangle is less than $ 1 $, then its area is less than $ \frac{\sqrt{3}}{4} $.

1981 Swedish Mathematical Competition, 5
Tags: geometry , Geometric Inequalities , areas
$ABC$ 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 $CXY$.

1915 Eotvos Mathematical Competition, 3
Tags: geometry , parallelogram , inscribed , areas , Geometric Inequalities
Prove that a triangle inscribed in a parallelogram has at most half the area of the parallelogram.

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?

2001 Switzerland Team Selection Test, 2
Tags: inequalities , Geometric Inequalities
If $a,b$, and $c$ are the sides of a triangle, prove the inequality $\sqrt{a+b-c}+\sqrt{c+a-b}+\sqrt{b+c-a } \le \sqrt{a}+\sqrt{b}+\sqrt{c}$. When does equality occur?