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 Swedish Mathematical Competition, 6
Tags: geometric inequalities , geometry , centroid
Find the sharpest inequalities of the form $a\cdot AB < AG < b\cdot AB$ and $c\cdot AB < BG < d\cdot AB$ for all triangles $ABC$ with centroid $G$ such that $GA > GB > GC$.

2001 Estonia Team Selection Test, 2
Tags: regular polygon , geometric inequalities , 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 $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$

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?

2009 Postal Coaching, 1
Tags: inequalities , geometric inequalities , right triangle
Two circles $\Gamma_a$ and $\Gamma_b$ with their centres lying on the legs $BC$ and $CA$ of a right triangle, both touching the hypotenuse $AB$, and both passing through the vertex $C$ are given. Let the radii of these circles be denoted by $\gamma_a$ and $\gamma_b$. Find the greatest real number $p$ such that the inequality $\frac{1}{\gamma_a}+\frac{1}{\gamma_b}\ge p \left(\frac{1}{a}+\frac{1}{b}\right)$ ($BC = a,CA = b$) holds for all right triangles $ABC$.

1963 Polish MO Finals, 3
Tags: geometry , rectangle , geometric inequalities
From a given triangle, cut out the rectangle with the largest area.

2020 Jozsef Wildt International Math Competition, W58
Tags: geometric inequalities , geometry , inequalities
In all triangles $ABC$ does it hold that: $$\sum\sqrt{\frac{a(h_a-2r)}{(3a+b+c)(h_a+2r)}}\le\frac34$$ [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

1970 Polish MO Finals, 1
Tags: geometry , geometric inequalities
Diameter $AB$ divides a circle into two semicircles. Points $P_1$ , $P_2$, $...$, $P_n$ are given on one of the semicircles in this order. How should a point C be chosen on the other semicircle in order to maximize the sum of the areas of triangles $CP_1P_2$, $CP_2P_3$, $...$,$CP_{n-1}P_n$?

2009 Postal Coaching, 4
Tags: triangle , geometry , circumcircle , 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?

1955 Poland - Second Round, 5
Tags: rectangle , geometry , geometric inequalities , area
Given a triangle $ ABC $. Find the rectangle of smallest area containing the triangle.

1978 Bulgaria National Olympiad, Problem 6
Tags: geometry , 3d geometry , pyramid , inequalities , geometric inequalities
The base of the pyramid with vertex $S$ is a pentagon $ABCDE$ for which $BC>DE$ and $AB>CD$. If $AS$ is the longest edge of the pyramid prove that $BS>CS$. [i]Jordan Tabov[/i]

1968 Poland - Second Round, 4
Tags: algebra , inequalities , geometric inequalities , geometry
Prove that if the numbers $ a, b, c $, are the lengths of the sides of a triangle and the sum of the numbers $x,y,z$ is zero, then $$a^2yz + b^2zx + c^2xy \leq 0.$$

1963 Kurschak Competition, 3
Tags: geometric inequalities , circumcircle , geometry
A triangle has no angle greater than $90^o$. Show that the sum of the medians is greater than four times the circumradius.

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?

1986 French Mathematical Olympiad, Problem 2
Tags: inequalities , geometric inequalities , geometry
Points $A,B,C$, and $M$ are given in the plane. (a) Let $D$ be the point in the plane such that $DA\le CA$ and $DB\le CB$. Prove that there exists point $N$ satisfying $NA\le MA,NB\le MB$, and $ND\le MC$. (b) Let $A',B',C'$ be the points in the plane such that $A'B'\le AB,A'C'\le AC,B'C'\le BC$. Does there exist a point $M'$ which satisfies the inequalities $M'A'\le MA,M'B'\le MB,M'C'\le MC$?

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.

1987 Swedish Mathematical Competition, 2
Tags: arc , geometric inequalities , equal area , geometry
A circle of radius $R$ is divided into two parts of equal area by an arc of another circle. Prove that the length of this arc is greater than $2R$.

1965 Swedish Mathematical Competition, 1
Tags: altitude , geometric inequalities , geometry , max , angle
The feet of the altitudes in the triangle $ABC$ are $A', B', C'$. Find the angles of $A'B'C'$ in terms of the angles $A, B, C$. Show that the largest angle in $A'B'C'$ is at least as big as the largest angle in $ABC$. When is it equal?

1958 Polish MO Finals, 6
Tags: perimeter , geometric inequalities , tangential , geometry
Prove that of all the quadrilaterals circuscribed around a given circle, the square has the smallest perimeter.

1972 Spain Mathematical Olympiad, 2
Tags: geometry , geometric inequalities , analytic geometry , inequalities
A point moves on the sides of the triangle $ABC$, defined by the vertices $A(-1.8, 0)$, $B(3.2, 0)$, $C(0, 2.4)$ . Determine the positions of said point, in which the sum of their distance to the three vertices is absolute maximum or minimum. [img]https://cdn.artofproblemsolving.com/attachments/2/5/9e5bb48cbeefaa5f4c069532bf5605b9c1f5ea.png[/img]

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]

2010 Junior Balkan Team Selection Tests - Moldova, 7
Tags: geometric inequalities , isosceles
In the triangle $ABC$ with $| AB | = c, | BC | = a, | CA | = b$ the relations hold simultaneously $$a \ge max \{ b, c, \sqrt{bc}\}, \sqrt{(a - b) (a + c)} + \sqrt{(a - c) (a + b) } \ge 2\sqrt{a^2-bc}$$ Prove that the triangle $ABC$ is isosceles.

1981 Kurschak Competition, 1
Tags: geometry , geometric inequalities
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.

2021 Yasinsky Geometry Olympiad, 3
Tags: altitude , geometry , geometric inequality , geometric inequalities
In the triangle $ABC$, $h_a, h_b, h_c$ are the altitudes and $p$ is its half-perimeter. Compare $p^2$ with $h_ah_b + h_bh_c + h_ch_a$. (Gregory Filippovsky)

1956 Polish MO Finals, 5
Tags: geometry , geometric inequalities
Prove that every polygon with perimeter $ 2a $ can be covered by a disk with diameter $ a $.

1996 Romania National Olympiad, 4
Tags: inequalities , geometric inequality , geometric inequalities , 3d geometry , geometry , algebra
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 $x, y, z, t, u, v \in (0, 1)$ : $$xyz + uv(1- x) + (1- y)(1- v)t + (1- z)(1- w)(1- t) < 1.$$