Olympiad problems

1914 Eotvos Mathematical Competition, 1

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 $k$.

1929 Eotvos Mathematical Competition, 3

Tags: geometry , geometric inequalities , concurrency

Let $p, q$ and $r$ be three concurrent lines in the plane such that the angle between any two of them is $60^o$. Let $a$, $b$ and $c$ be real numbers such that $0 < a \le b \le c$. (a) Prove that the set of points whose distances from $p, q$ and $r$ are respectively less than $a, b$ and $c$ consists of the interior of a hexagon if and only if $a + b > c$. (b) Determine the length of the perimeter of this hexagon when $a + b > c$.

1955 Kurschak Competition, 1

Tags: geometry , trapezoid , geometric inequalities

Prove that if the two angles on the base of a trapezoid are different, then the diagonal starting from the smaller angle is longer than the other diagonal. [img]https://cdn.artofproblemsolving.com/attachments/7/1/77cf4958931df1c852c347158ff1e2bbcf45fd.png[/img]

2005 Switzerland - Final Round, 5

Tags: geometry , geometric inequalities , combinatorial geometry , convex

Tweaking a convex $n$-gon means the following: choose two adjacent sides $AB$ and $BC$ and replaces them with the line segment $AM$, $MN$, $NC$, where $M \in AB$ and $N \in BC$ are arbitrary points inside these segments. In other words, you cut off a corner and get an $(n+1)$-corner. Starting from a regular hexagon $P_6$ with area $1$, by continuous Tweaks a sequence $P_6,P_7,P_8, ...$ convex polygons. Show that Area of $P_n$ for all $n\ge 6$ greater than $\frac1 2$ is, regardless of how tweaks takes place.

2017 Singapore Junior Math Olympiad, 1

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

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 $1$.

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 $.

1988 Poland - Second Round, 5

Tags: rectangle , geometric inequalities , combinatorial geometry , combinatorics , geometry

Decide whether any rectangle that can be covered by 25 circles of radius 2 can also be covered by 100 circles of radius 1.

2014 Swedish Mathematical Competition, 4

Tags: geometry , combinatorial geometry , square , geometric inequalities

A square is cut into a finitely number of triangles in an arbitrary way. Show the sum of the diameters of the inscribed circles in these triangles is greater than the side length of the square.

1991 Swedish Mathematical Competition, 6

Tags: geometry , area , geometric inequalities , equilateral

Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original triangle.

Durer Math Competition CD Finals - geometry, 2008.C2

Tags: geometry , geometric inequality , geometric inequalities , median

Given a triangle with sides $a, b, c$ and medians $s_a, s_b, s_c$ respectively. Prove the following inequality: $$a + b + c> s_a + s_b + s_c> \frac34 (a + b + c) $$

2018 Flanders Math Olympiad, 1

Tags: geometry , geometric inequalities , angle

In the triangle $\vartriangle ABC$ we have $| AB |^3 = | AC |^3 + | BC |^3$. Prove that $\angle C> 60^o$ .

1932 Eotvos Mathematical Competition, 2

Tags: geometric inequalities , geometry

In triangle $ABC$, $AB \ne AC$. Let $AF$, $AP$ and $AT$ be the median, angle bisector and altitude from vertex $A$, with $F, P$ and $T$ on $BG$ or its extension. (a) Prove that $P$ always lies between$ F$ and $T$. (b) Prove that $\angle FAP < \angle PAT$ if $ABC$ is an acute triangle.

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} $.

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

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

1986 Poland - Second Round, 6

Tags: geometry , geometric inequalities , area

In the triangle $ ABC $, the point $ A' $ on the side $ BC $, the point $ B' $ on the side $ AC $, the point $ C' $ on the side $ AB $ are chosen so that the straight lines $ AA' $, $ CC' $ intersect at one point, i.e. equivalently $ |BA'| \cdot |CB'| \cdot |AC'| = |CA'| \cdot |AB'| \cdot |BC'| $. Prove that the area of triangle $ A'B'C' $ is not greater than $ 1/4 $ of the area of triangle $ ABC $.

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]

2000 Czech and Slovak Match, 1

Tags: geometry , geometric inequalities , inequalities , algebra

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

1965 German National Olympiad, 6

Tags: trigonometry , inequalities , geometric inequalities , geometry

Let $\alpha,\beta, \gamma$ be the angles of a triangle. Prove that $\cos\alpha, + \cos\beta + \cos\gamma \le \frac{3}{2} $ and find the cases of equality.

1985 Poland - Second Round, 1

Tags: algebra , inequalities , geometric inequalities , geometry

Inside the triangle $ABC$, the point $P$ is chosen. Let $ a, b, c $ be the lengths of the sides $ BC $, $ CA $, $ AB $, respectively, and $ x, y, z $ the distances of the point $ P $ from the vertices $ B, C, A $. Prove that if $$ x^2 + xy + y^2 = a^2 $$ $$y^2 + yz + z^2 = b^2 $$ $$z^2 + zx + x^2 = c^2$$ this $$ a^2 + ab + b^2 > c^2.$$

1974 Czech and Slovak Olympiad III A, 5

Tags: geometry , geometric inequalities , geometrical inequalities , cyclic , hexagon , area , inequalities

Let $ABCDEF$ be a cyclic hexagon such that \[AB=BC,\quad CD=DE,\quad EF=FA.\] Show that \[[ACE]\le[BDF]\] and determine when the equality holds. ($[XYZ]$ denotes the area of the triangle $XYZ.$)

1985 Swedish Mathematical Competition, 5

Tags: geometry , analytic geometry , geometric inequalities , inequalities

In a rectangular coordinate system, $O$ is the origin and $A(a,0)$, $B(0,b)$ and $C(c,d)$ the vertices of a triangle. Prove that $AB+BC+CA \ge 2CO$.

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 $h^2\le ab$.

1973 Bulgaria National Olympiad, Problem 6

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

In the tetrahedron $ABCD$, $E$ and $F$ are the midpoints of $BC$ and $AD$, $G$ is the midpoint of the segment $EF$. Construct a plane through $G$ intersecting the segments $AB$, $AC$, $AD$ in the points $M,N,P$ respectively in such a way that the sum of the volumes of the tetrahedrons $BMNP$, $CMNP$ and $DMNP$ to be minimal. [i]H. Lesov[/i]

1973 Polish MO Finals, 6

Tags: ellipse , geometry , geometric inequalities , conic

Prove that for every centrally symmetric polygon there is at most one ellipse containing the polygon and having the minimal area.