Olympiad problems

1955 Poland - Second Round, 5

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

2016 Flanders Math Olympiad, 3

Tags: geometry , areas , Geometric Inequalities

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]

1993 Abels Math Contest (Norwegian MO), 1b

Tags: inequalities , Geometric Inequalities

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

2004 Germany Team Selection Test, 2

Tags: geometry proposed , geometry , Locus problems , Geometric Inequalities , Germany , TST , Team Selection Test

Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$. Find all points $B$ on the diameter $d$ in the interior of $k$ such that \[\measuredangle MPA = \measuredangle BPN \quad \text{and} \quad PA \leq PB.\] (i. e. give an explicit description of these points without using the points $M$ and $N$).

1970 Bulgaria National Olympiad, Problem 5

Tags: inequalities , Geometric Inequalities , geometry , polygon

Prove that for $n\ge5$ the side of regular inscribable $n$-gon is bigger than the side of regular $n+1$-gon circumscribed around the same circle and if $n\le4$ the opposite statement is true.

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 $GA > GB > GC$.

1962 Kurschak Competition, 3

Tags: geometry , 3D geometry , tetrahedron , Geometric Inequalities

$P$ is any point of the tetrahedron $ABCD$ except $D$. Show that at least one of the three distances $DA$, $DB$, $DC$ exceeds at least one of the distances $PA$, $PB$ and $PC$.

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

1924 Eotvos Mathematical Competition, 1

Tags: number theory , geometry , triangle inequality , inequalities , Geometric Inequalities

Let $a, b, c$ be fìxed natural numbers. Suppose that, for every positive integer n, there is a triangle whose sides have lengths $a^n$, $b^n$, and $c^n$ respectively. Prove that these triangles are isosceles.

2002 Estonia National Olympiad, 3

Tags: inequalities , Geometric Inequalities , sidelenghts

Prove that for positive real numbers $a, b$ and $c$ the inequality $2(a^4+b^4+c^4) < (a^2+b^2+c^2)^2$ holds if and only if $a,b,c$ are the sides of a triangle.

1983 Swedish Mathematical Competition, 5

Tags: geometry , combinatorial geometry , combinatorics , disks , discs , Geometric Inequalities , min

Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$. What is the smallest possible radius?

1972 Bulgaria National Olympiad, Problem 5

Tags: geometry , cyclic quadrilateral , perpendicular , inequalities , Geometric Inequalities

In a circle with radius $R$, there is inscribed a quadrilateral with perpendicular diagonals. From the intersection point of the diagonals, there are perpendiculars drawn to the sides of the quadrilateral. (a) Prove that the feet of these perpendiculars $P_1,P_2,P_3,P_4$ are vertices of the quadrilateral that is inscribed and circumscribed. (b) Prove the inequalities $2r_1\le\sqrt2 R_1\le R$ where $R_1$ and $r_1$ are radii respectively of the circumcircle and inscircle to the quadrilateral $P_1P_2P_3P_4$. When does equality hold? [i]H. Lesov[/i]

1990 Bulgaria National Olympiad, Problem 6

Tags: geometry , 3D geometry , tetrahedron , Geometric Inequalities , inequalities

The base $ABC$ of a tetrahedron $MABC$ is an equilateral triangle, and the lateral edges $MA,MB,MC$ are sides of a triangle of the area $S$. If $R$ is the circumradius and $V$ the volume of the tetrahedron, prove that $RS\ge2V$. When does equality hold?

Cono Sur Shortlist - geometry, 2018.G5

Tags: geometry , ratio , inscribed , square , circumscribed , Geometric Inequalities

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 $L/\ell$ .

1970 Bulgaria National Olympiad, Problem 6

Tags: geometry , inequalities , Geometric Inequalities

In space, we are given the points $A,B,C$ and a sphere with center $O$ and radius $1$. Find the point $X$ from the sphere for which the sum $f(X)=|XA|^2+|XB|^2+|XC|^2$ attains its maximal and minimal value. Prove that if the segments $OA,OB,OC$ are pairwise perpendicular and $d$ is the distance from the center $O$ to the centroid of the triangle $ABC$ then: (a) the maximum of $f(X)$ is equal to $9d^2+3+6d$; (b) the minimum of $f(X)$ is equal to $9d^2+3-6d$. [i]K. Dochev and I. Dimovski[/i]

1964 Polish MO Finals, 5

Tags: geometry , Geometric Inequalities , construction

Given an acute angle and a circle inside the angle. Find a point $ M $ on the circle such that the sum of the distances of the point $ M $ from the sides of the angle is a minimum.

1989 Greece National Olympiad, 4

Tags: geometry , trapezoid , tangential , geometric inequality , Geometric Inequalities

A trapezoid with bases $a,b$ and altitude $h$ is circumscribed around a circl.. Prove that $h^2\le ab$.

1975 Bulgaria National Olympiad, Problem 4

Tags: inequalities , circles , geometry , geometrical inequalities , Geometric Inequalities

In the plane are given a circle $k$ with radii $R$ and the points $A_1,A_2,\ldots,A_n$, lying on $k$ or outside $k$. Prove that there exist infinitely many points $X$ from the given circumference for which $$\sum_{i=1}^n A_iX^2\ge2nR^2.$$ Does there exist a pair of points on different sides of some diameter, $X$ and $Y$ from $k$, such that $$\sum_{i=1}^n A_iX^2\ge2nR^2\text{ and }\sum_{i=1}^n A_iY^2\ge2nR^2?$$ [i]H. Lesov[/i]