PlaneGeometry IMO

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2147   
Let $ABCD$ be a cyclic quadrilateral. Let $P, Q, R$ be the feet of the perpendiculars from $D$ to the lines $BC, CA$ and $AB$ respectively. Show that $PQ = QR$ iff the bisectors of $\angle ABC$ and $\angle ADC$ meet on $AC$.

2149   
Let $ABCD$ be a convex quadrilateral for which $AC = BD$. Equilateral triangles are constructed on the sides of the quadrilateral and pointing outward. Let $O_1, O_2, O_3, O_4$ be the centres of the triangles constructed on $AB, BC, CD,$ and $DA$ respectively. Prove that lines $O_1O_3$ and $O_2O_4$ are perpendicular.

2438   
Given an acute $\triangle{ABC}$ with side length $BC > CA$.

2455   
Let $A$, $B$, $C$ and $D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at the points $X$ and $Y$. The line $XY$ meets $BC$ at the point $Z$. Let $P$ be a point on the line $XY$ different from $Z$. The line $CP$ intersects the circle with diameter $AC$ at the points $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at the points $B$ and $N$. Prove that the lines $AM$, $DN$ and $XY$ are concurrent


3027   
(Weitzenbock's Inequality) Let $a, b, c$, and $S$ be a triangle's three sides' lengths and its area, respectively. Show that $$a^2 + b^2 + c^2 \ge 4\sqrt{3}\cdot S$$

3044   
The diagonals $AC$ and $CD$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ such that $AM:AC = CN:CE=r$. Determine $r$ if $B, M,$ and $N$ are collinear.

3106   
Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.

4095   

Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.


4100   

A convex quadrilateral $ABCD$ satisfies $AB\cdot CD=BC \cdot DA.$ Point $X$ lies inside $ABCD$ so that $\angle XAB = \angle XCD$ and $\angle XBC = \angle XDA.$ Prove that $\angle BXA + \angle DXC = 180^{\circ}$ .


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