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IMO Shortlist 1994 problem G5

A circle $C$ with center $O.$ and a line $L$ which does not touch circle $C.$ $OQ$ is perpendicular to $L,$ $Q$ is on $L.$ $P$ is on $L,$ draw two tangents $L_1, L_2$ to circle $C.$ $QA, QB$ are perpendicular to $L_1, L_2$ respectively. ( $A$ on $L_1,$ $B$ on $L_2$ ). Prove that, line $AB$ intersect $QO$ at a fixed point.

Original formulation:

A line $l$ does not meet a circle $\omega$ with center $O.$ $E$ is the point on $l$ such that $OE$ is perpendicular to $l.$ $M$ is any point on $l$ other than $E.$ The tangents from $M$ to $\omega$ touch it at $A$ and $B.$ $C$ is the point on $MA$ such that $EC$ is perpendicular to $MA.$ $D$ is the point on $MB$ such that $ED$ is perpendicular to $MB.$ The line $CD$ cuts $OE$ at $F.$ Prove that the location of $F$ is independent of that of $M.$

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IMO Shortlist 2008 problem G6

There is given a convex quadrilateral $ABCD$ . Prove that there exists a point $P$ inside the quadrilateral such that
$\angle PAB + \angle PDC = \angle PBC + \angle PAD = \angle PCD + \angle PBA = \angle PDA + \angle PCB = 90^{\circ}$
if and only if the diagonals $AC$ and $BD$ are perpendicular.

Proposed by Dukan Dukic, Serbia

IMO Shortlist 2006 problem G6

Circles $w_{1}$ and $w_{2}$ with centres $O_{1}$ and $O_{2}$ are externally tangent at point $D$ and internally tangent to a circle $w$ at points $E$ and $F$ respectively. Line $t$ is the common tangent of $w_{1}$ and $w_{2}$ at $D$ . Let $AB$ be the diameter of $w$ perpendicular to $t$ , so that $A, E, O_{1}$ are on the same side of $t$ . Prove that lines $AO_{1}$ , $BO_{2}$ , $EF$ and $t$ are concurrent.

IMO Shortlist 2004 problem G5

Let $A_1A_2A_3...A_n$ be a regular n-gon. Let $B_1$ and $B_n$ be the midpoints of its sides $A_1A_2$ and $A_{n-1}A_n$ . Also, for every $i\in\left\{2;\;3;\;4;\;...;\;n-1\right\}$ , let $S$ be the point of intersection of the lines $A_1A_{i+1}$ and $A_nA_i$ , and let $B_i$ be the point of intersection of the angle bisector bisector of the angle $\measuredangle A_iSA_{i+1}$ with the segment $A_iA_{i+1}$ .

Prove that: $\sum_{i=1}^{n-1} \measuredangle A_1B_iA_n=180^{\circ}$ .

IMO Shortlist 2000 problem G6

Let $ABCD$ be a convex quadrilateral. The perpendicular bisectors of its sides $AB$ and $CD$ meet at $Y$ . Denote by $X$ a point inside the quadrilateral $ABCD$ such that $\measuredangle ADX = \measuredangle BCX < 90^{\circ}$ and $\measuredangle DAX = \measuredangle CBX < 90^{\circ}$ . Show that $\measuredangle AYB = 2\cdot\measuredangle ADX$ .

IMO Shortlist 1996 problem G6

Let the sides of two rectangles be $\{a,b\}$ and $\{c,d\},$ respectively, with $a < c \leq d < b$ and $ab < cd.$ Prove that the first rectangle can be placed within the second one if and only if

$\left(b^2 - a^2\right)^2 \leq \left(bc - ad \right)^2 + \left(bd - ac \right)^2.$

IMO Shortlist 1994 problem G3

A circle $C$ has two parallel tangents $L'$ and $L"$ . A circle $C'$ touches $L'$ at $A$ and $C$ at $X$ . A circle $C"$ touches $L"$ at $B$ , $C$ at $Y$ and $C'$ at $Z$ . The lines $AY$ and $BX$ meet at $Q$ . Show that $Q$ is the circumcenter of $XYZ$