Olympiad problems

1994 Swedish Mathematical Competition, 2

In the triangle $ABC$, the medians from $B$ and $C$ are perpendicular. Show that $\cot B + \cot C \ge \frac23$.

1972 Kurschak Competition, 1

Tags: inequalities , geometric inequalities , geometry

A triangle has side lengths $a, b, c$. Prove that $$a(b -c)^2 + b(c - a)^2 + c(a - b)^2 + 4abc > a^3 + b^3 + c^3$$

1955 Polish MO Finals, 5

Tags: construction , geometric inequalities , geometry

In the plane, a straight line $ m $ is given and points $ A $ and $ B $ lie on opposite sides of the straight line $ m $. Find a point $ M $ on the line $ m $ such that the difference in distances of this point from points $ A $ and $ B $ is as large as possible.

1970 Poland - Second Round, 2

Tags: area , geometry , geometric inequalities

On the sides of the regular $ n $-gon, $ n + 1 $ points are taken dividing the perimeter into equal parts. At what position of the selected points is the area of the convex polygon with these $ n + 1 $ vertices a) the largest, b) the smallest?

2004 Germany Team Selection Test, 2

Tags: locus problems , geometric inequalities , geometry

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 $N$).

1997 Brazil Team Selection Test, Problem 5

Tags: inequalities , geometry , geometric inequality , geometric inequalities , triangle , area

Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly. (a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$. (b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.

2004 Swedish Mathematical Competition, 6

Tags: area , geometry , geometric inequalities , convex

Prove that every convex $n$-gon of area $1$ contains a quadrilateral of area at least $\frac12 $. .

1995 Chile National Olympiad, 7

Tags: circles , geometric inequalities , radius , semicircle

In a semicircle of radius $4$ three circles are inscribed, as indicated in the figure. Larger circles have radii $ R_1 $ and $ R_2 $, and the larger circle has radius $ r $. a) Prove that $ \dfrac {1} {\sqrt{r}} = \dfrac {1} {\sqrt{R_1}} + \dfrac {1} {\sqrt{R_2}} $ b) Prove that $ R_1 + R_2 \le 8 (\sqrt{2} -1) $ c) Prove that $ r \le \sqrt{2} -1 $ [img]https://cdn.artofproblemsolving.com/attachments/0/9/aaaa65d1f4da4883973751e1363df804b9944c.jpg[/img]

1988 Poland - Second Round, 5

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

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

2008 Postal Coaching, 2

Tags: geometry , perpendicular , distance , geometric inequalities

Let $ABC$ be a triangle, $AD$ be the altitude from $A$ on to $BC$. Draw perpendiculars $DD_1$ and $DD_2$ from $D$ on to $AB$ and $AC$ respectively and let $p(A)$ be the length of the segment $D_1D_2$. Similarly define $p(B)$ and $p(C)$. Prove that $\frac{p(A)p(B)p(C)}{s^3}\le \frac18$ , where s is the semi-perimeter of the triangle $ABC$.