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IMO Shortlist 2003 problem C5

Regard a plane with a Cartesian coordinate system; for each point with integer coordinates, draw a circular disk centered at this point and having the radius $\frac{1}{1000}$ .

a) Prove the existence of an equilateral triangle whose vertices lie in the interior of different disks;

b) Show that every equilateral triangle whose vertices lie in the interior of different disks has a sidelength 96.

Radu Gologan, Romania
Remark
[The " 96" in (b) can be strengthened to " 124". By the way, part (a) of this problem is the place where I used the well-known "Dedekind" theorem.]

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Zadaci

IMO Shortlist 2001 problem G6

Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$ , $BP$ , $CP$ meet the sides $BC$ , $CA$ , $AB$ (or extensions thereof) in $D$ , $E$ , $F$ , respectively. Suppose further that the areas of triangles $PBD$ , $PCE$ , $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

IMO Shortlist 2002 problem G5

For any set $S$ of five points in the plane, no three of which are collinear, let $M(S)$ and $m(S)$ denote the greatest and smallest areas, respectively, of triangles determined by three points from $S$ . What is the minimum possible value of $M(S)/m(S)$ ?

IMO Shortlist 2003 problem G3

Let $ABC$ be a triangle, and $P$ a point in the interior of this triangle. Let $D$ , $E$ , $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$ , $CA$ , $AB$ , respectively. Assume that

$AP^{2}+PD^{2}=BP^{2}+PE^{2}=CP^{2}+PF^{2}$ .

Furthermore, let $I_{a}$ , $I_{b}$ , $I_{c}$ be the excenters of triangle $ABC$ . Show that the point $P$ is the circumcenter of triangle $I_{a}I_{b}I_{c}$ .

IMO Shortlist 2006 problem A5

If $a$ , $b$ , $c$ are the sides of a triangle, prove that
$\sum_{\mbox{cyc}}\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\leq 3$

IMO Shortlist 2007 problem G6

Determine the smallest positive real number $k$ with the following property. Let $ABCD$ be a convex quadrilateral, and let points $A_1$ , $B_1$ , $C_1$ , and $D_1$ lie on sides $AB$ , $BC$ , $CD$ , and $DA$ , respectively. Consider the areas of triangles $AA_1D_1$ , $BB_1A_1$ , $CC_1B_1$ and $DD_1C_1$ ; let $S$ be the sum of the two smallest ones, and let $S_1$ be the area of quadrilateral $A_1B_1C_1D_1$ . Then we always have $kS_1\ge S$ .

Author: unknown author, USA

IMO Shortlist 2009 problem G6

Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$ ) intersect at point $P$ . Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$ , respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$ , respectively. Prove that the perpendicular from $E_1$ on $CD$ , the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent.

Proposed by Ukraine