Olympiad problems

1965 Polish MO Finals, 1

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 $$\frac{\pi}{3}\leq \frac{a \alpha + b \beta +c \gamma}{a+b+c}<\frac{\pi }{ 2}.$$

1964 Swedish Mathematical Competition, 4

Tags: combinatorial geometry , min , max , geometric inequalities , inequalities , geometry

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 $n = 4$.

1999 German National Olympiad, 4

Tags: perimeter , geometric inequalities , convex , polygon , inequalities , geometry

A convex polygon $P$ is placed inside a unit square $Q$. Prove that the perimeter of $P$ does not exceed $4$.

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.

1955 Poland - Second Round, 5

Tags: rectangle , geometry , geometric inequalities , area

Given a triangle $ ABC $. Find the rectangle of smallest area containing the triangle.

2016 Sharygin Geometry Olympiad, 7

Tags: geometric inequalities , geometry , 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.

1994 French Mathematical Olympiad, Problem 2

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

Let be given a semi-sphere $\Sigma$ whose base-circle lies on plane $p$. A variable plane $Q$, parallel to a fixed plane non-perpendicular to $P$, cuts $\Sigma$ at a circle $C$. We denote by $C'$ the orthogonal projection of $C$ onto $P$. Find the position of $Q$ for which the cylinder with bases $C$ and $C'$ has the maximum volume.

Cono Sur Shortlist - geometry, 2018.G5

Tags: geometry , square , inscribed , circumscribed , geometric inequalities , ratio

We say that a polygon $P$ is inscribed in another polygon $Q$ when all the vertices of $P$ belong to the perimeter of $Q$. We also say in this case that $Q$ is circumscribed to $P$. Given a triangle $T$, let $\ell$ be the largest side of a square inscribed in $T$ and $L$ is the shortest side of a square circumscribed to $T$ . Find the smallest possible value of the ratio $L/\ell$ .

2017 Singapore Junior Math Olympiad, 1

Tags: geometry , rectangle , ratio , geometric inequalities , square , sidelenghts , inequalities

A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.

2017 Hanoi Open Mathematics Competitions, 13

Tags: distance , geometric inequalities , inequalities , geometry

Let $ABC$ be a triangle. For some $d>0$ let $P$ stand for a point inside the triangle such that $|AB| - |P B| \ge d$, and $|AC | - |P C | \ge d$. Is the following inequality true $|AM | - |P M | \ge d$, for any position of $M \in BC $?

1963 Kurschak Competition, 3

Tags: circumcircle , geometric inequalities , geometry

A triangle has no angle greater than $90^o$. Show that the sum of the medians is greater than four times the circumradius.

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

1992 Poland - Second Round, 5

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

Determine the upper limit of the volume of spheres contained in tetrahedra of all heights not longer than $ 1 $.

1993 Abels Math Contest (Norwegian MO), 1b

Tags: inequalities , geometric inequalities

Given a triangle with sides of lengths $a,b,c$, prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}< 2$.

2009 Postal Coaching, 1

Tags: circumcircle , geometric inequalities , geometry

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?

1956 Polish MO Finals, 5

Tags: geometric inequalities , geometry

Prove that every polygon with perimeter $ 2a $ can be covered by a disk with diameter $ a $.

2001 Estonia Team Selection Test, 2

Tags: geometry , distance , regular polygon , geometric inequalities

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 $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$

2000 Belarus Team Selection Test, 5.1

Tags: inequalities , angle bisector , geometric inequalities , median

Let $AM$ and $AL$ be the median and bisector of a triangle $ABC$ ($M,L \in BC$). If $BC = a, AM = m_a, AL = l_a$, prove the inequalities: (a) $a\tan \frac{a}{2} \le 2m_a \le a \cot \frac{a}{2} $ if $a < \frac{\pi}{2}$ and $a\tan \frac{a}{2} \ge 2m_a \ge a \cot \frac{a}{2} $ if $a > \frac{\pi}{2}$ (b) $2l_a \le a\cot \frac{a}{2} $.

1972 Polish MO Finals, 4

Tags: sphere , geometric inequalities , 3d geometry , perimeter , 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.

2000 Czech and Slovak Match, 1

Tags: geometry , algebra , inequalities , geometric inequalities

$a,b,c$ are positive real numbers which satisfy $5abc>a^3+b^3+c^3$. Prove that $a,b,c$ can form a triangle.

2019 LIMIT Category A, Problem 3

Tags: inequalities , geometry , geometric inequalities

In $\triangle ABC$, $\left|\overline{AB}\right|=\left|\overline{AC}\right|$, $D$ is the foot of the perpendicular from $C$ to $AB$ and $E$ the foot of the perpendicular from $B$ to $AC$, then $\textbf{(A)}~\left|\overline{BC}\right|^3>\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(B)}~\left|\overline{BC}\right|^3<\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(C)}~\left|\overline{BC}\right|^3=\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(D)}~\text{None of the above}$

2017 Swedish Mathematical Competition, 5

Tags: algebra , inequalities , geometric inequalities

Find a costant $C$, such that $$ \frac{S}{ab+bc+ca}\le C$$ where $a,b,c$ are the side lengths of an arbitrary triangle, and $S$ is the area of the triangle. (The maximal number of points is given for the best possible constant, with proof.)

1985 Polish MO Finals, 4

Tags: geometry , inequalities , inradius , geometric inequalities

$P$ is a point inside the triangle $ABC$ is a triangle. The distance of $P$ from the lines $BC, CA, AB$ is $d_a, d_b, d_c$ respectively. If $r$ is the inradius, show that $$\frac{2}{ \frac{1}{d_a} + \frac{1}{d_b} + \frac{1}{d_c}} < r < \frac{d_a + d_b + d_c}{2}$$

2009 Thailand Mathematical Olympiad, 8

Tags: inequalities , geometric inequalities , geometry

Let $a, b, c$ be side lengths of a triangle, and define $s =\frac{a+b+c}{2}$. Prove that $$\frac{2a(2a-s)}{b + c}+\frac{2b(2b - s)}{c + a}+\frac{2c(2c - s)}{a + b}\ge s.$$

1968 Swedish Mathematical Competition, 3

Tags: geometric inequalities , geometry

Show that the sum of the squares of the sides of a quadrilateral is at least the sum of the squares of the diagonals. When does equality hold?