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

2019 Durer Math Competition Finals, 3
Tags: geometry , perimeter , geometric inequalities
Let $P$ be an interior point of triangle $ABC$. The lines $AP$, $BP$ and $CP$ divide each of the three sides into two segments. If the so-obtained six segments all have distinct integer lengths, what is the minimum possible perimeter of $ABC$?

1937 Eotvos Mathematical Competition, 3
Tags: geometric inequalities , geometry
Let $n$ be a positive integer. Let $P,Q,A_1,A_2,...,A_n$ be distinct points such that $A_1,A_2,...,A_n$ are not collinear. Suppose that $PA_1 + PA_2 + ...+PA_n$, and $QA_1 + QA_2 +...+ QA_n$, have a common value $s$ for some real number $s$. Prove that there exists a point $R$ such that $$RA_1 + RA_2 +... + RA_n < s.$$

1972 Polish MO Finals, 4
Tags: perimeter , geometric inequalities , 3d geometry , sphere , geometry
Points $A$ and $B$ are given on a line having no common points with a sphere $K$. The feet $P$ of the perpendicular from the center of $K$ to the line $AB$ is positioned between $A$ and $B$, and the lengths of segments $AP$ and $BP$ both exceed the radius of $K$. Consider the set $Z$ of all triangles $ABC$ whose sides $AC$ and $BC$ are tangent to $K$. Prove that among all triangles in $Z$, a triangle $T$ with a maximum perimeter also has a maximum area.

1930 Eotvos Mathematical Competition, 3
Tags: geometry , circumradius , geometric inequalities
Inside an acute triangle $ABC$ is a point $P$ that is not the circumcenter. Prove that among the segments $AP$, $BP$ and $CP$, at least one is longer and at least one is shorter than the circumradius of $ABC$.

1973 Spain Mathematical Olympiad, 7
Tags: analytic geometry , inequalities , geometric inequalities , geometry
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.

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

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 $ABC$.

1993 Abels Math Contest (Norwegian MO), 1b
Tags: inequalities , geometric inequalities
Given a triangle with sides of lengths $a,b,c$, prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}< 2$.

1991 Poland - Second Round, 6
Tags: geometry , 3d geometry , sphere , geometric inequalities , parallelepiped
The parallelepiped contains a sphere of radius $r$ and is contained within a sphere of radius $R$. Prove that $ \frac{R}{r} \geq \sqrt{3} $.

1965 Kurschak Competition, 3
Tags: 3d geometry , geometry , geometric inequalities , pyramid
A pyramid has square base and equal sides. It is cut into two parts by a plane parallel to the base. The lower part (which has square top and square base) is such that the circumcircle of the base is smaller than the circumcircles of the lateral faces. Show that the shortest path on the surface joining the two endpoints of a spatial diagonal lies entirely on the lateral faces. [img]https://cdn.artofproblemsolving.com/attachments/c/8/170bec826d5e40308cfd7360725d2aba250bf6.png[/img]

1972 Polish MO Finals, 2
Tags: geometry , geometric inequalities , point
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.

2000 Swedish Mathematical Competition, 4
Tags: geometry , lattice , area , geometric inequalities , 3d geometry , combinatorial geometry
The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.

1990 Swedish Mathematical Competition, 2
Tags: sum , geometric inequalities , geometry , collinear
The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$. $P$ is chosen to minimise $\sum PA_i$. Find the minimum.

1953 Poland - Second Round, 3
Tags: geometry , geometric inequalities , area
A triangular piece of sheet metal weighs $900$ g. Prove that by cutting this sheet metal along a straight line passing through the center of gravity of the triangle, it is impossible to cut off a piece weighing less than $400$ g.

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

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 $$\frac32 \le \frac{b + c}{b + c + 2a}+ \frac{a + c}{a + c + 2b}+ \frac{a + b}{a + b + 2c}\le \frac53.$$

1960 Polish MO Finals, 6
Tags: geometry , rectangle , geometric inequalities
On the perimeter of a rectangle, point $ M $ is chosen. Find the shortest path whose beginning and end are point $ M $ and which has a point in common with each side of the rectangle.

1999 German National Olympiad, 4
Tags: geometry , perimeter , geometric inequalities , inequalities , convex , polygon
A convex polygon $P$ is placed inside a unit square $Q$. Prove that the perimeter of $P$ does not exceed $4$.

1996 North Macedonia National Olympiad, 3
Tags: trigonometry , inequalities , geometric inequalities , angle
Prove that if $\alpha, \beta, \gamma$ are angles of a triangle, then $\frac{1}{\sin \alpha}+ \frac{1}{\sin \beta} \ge \frac{8}{ 3+2 \ cos\gamma}$ .

1986 Swedish Mathematical Competition, 2
Tags: geometry , geometric inequalities , triangle area
The diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect at $O$. If $S_1$ and $S_2$ are the areas of triangles $AOB$ and $COD$ and S that of $ABCD$, show that $\sqrt{S_1} + \sqrt{S_2} \le \sqrt{S}$. Prove that equality holds if and only if $AB$ and $CD$ are parallel.

2016 Flanders Math Olympiad, 3
Tags: geometry , geometric inequalities , 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]

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?

1971 Bulgaria National Olympiad, Problem 4
Tags: geometry , inequalities , geometric inequalities , circumcircle , triangle
It is given a triangle $ABC$. Let $R$ be the radius of the circumcircle of the triangle and $O_1,O_2,O_3$ be the centers of excircles of the triangle $ABC$ and $q$ is the perimeter of the triangle $O_1O_2O_3$. Prove that $q\le6R\sqrt3$. When does equality hold?

1992 Czech And Slovak Olympiad IIIA, 2
Tags: inequalities , geometric inequalities , area , 3d geometry , tetrahedron , geometry
Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$

1995 North Macedonia National Olympiad, 2
Tags: geometric inequalities , inequalities , altitude
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?