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MR - geometry olympiad

olympiad level geometry problems from Mathematical Reflections

A circle centered at $O$ is tangent to all sides of the convex quadrilateral $ABCD$ . The rays $BA$ and $CD$ intersect at $K$ , the rays $AD$ and $BC$ intersect at $L$ . The points $X, Y$ are considered on the line segments $OA,OC$ , respectively. Prove that $\angle XKY = \frac{1}{2} \angle AKC$ if and only if $\angle XLY = \frac{1}{2} \angle ALC$ .

Proposed by Pavlo Pylyavskyy, MIT

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Let $AB$ be a diameter of the circle $\Gamma$ and let $C$ be a point on the circle, different from $A$ and $B$ . Denote by $D$ the projection of $C$ on $AB$ and let $\omega$ be a circle tangent to $AD, CD$ , and $\Gamma$ , touching $\Gamma$ at $X$ . Prove that the angle bisectors of $\angle AXB$ and $\angle ACD$ meet on $AB$ .

Proposed by Liubomir Chiriac, Princeton

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In the convex hexagon $ABCDEF$ the following equalities hold: $AD = BC + EF, BE = AF + CD, CF = AB + DE$ .
Prove that $\frac{AB}{DE}=\frac{CD}{AF}=\frac{EF}{BC}$ .

Proposed by Nairi Sedrakyan, Armenia

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O13

Let $ABC$ be a triangle and $P$ be an arbitrary point inside the triangle. Let $A',B',C'$ , respectively, be the intersections of $AP, BP$ , and $CP$ with the triangles sides. Through $P$ we draw a line perpendicular to $PA$ that intersects $BC$ at $A_1$ . We define $B_1$ and $C_1$ analogously. Let $P'$ be the isogonal conjugate of the point $P$ with respect to triangle $A'B'C'$ . Show that $A_1,B_1$ , and $C_1$ lie on a line $\ell$ that is perpendicular to $PP'$ .

Proposed by Khoa Lu Nguyen, Sam Houston High School, Houston, Texas.

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O16

Let $ABC$ be an acute-angled triangle. Let $\omega$ be the center of the nine point circle and let $G$ be its centroid. Let $A',B',C',A'',B'',C''$ be the projections of $\omega$ and $G$ on the corresponding sides. Prove that the perimeter of $A''B''C''$ is not less than the perimeter of $A'B'C'$ .

Proposed by Iurie Boreico, student, Chișinău, Moldova

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O20

The incircle of triangle $ABC$ touches $AC$ at $E$ and $BC$ at $D$ . The excircle corresponding to $A$ touches the side $BC$ at $A_1$ and the extensions of $AB, AC$ at $C_1$ and $B_1$ , respectively. Let $DE \cap A_1B_1 = L$ . Prove that $L$ lies on the circumcircle of triangle $A_1BC_1$ .

Liubomir Chiriac, Princeton University

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O22

Consider a triangle $ABC$ and points $P,Q$ in its plane. Let $A_1,B_1,C_1$ and $A_2,B_2,C_2$ be cevians in this triangle. Denote by $U, V,W$ the second intersections of circles $(AA_1A_2), (BB_1B_2), (CC_1C_2)$ with circle $(ABC)$ , respectively. Let $X$ be the point of intersection of $AU$ with $BC$ . Similarly define $Y$ and $Z$ . Prove that $X, Y,Z$ are collinear.

Khoa Lu Nguyen, M.I.T and Ivan Borsenco, University of Texas at Dallas

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O23

Let $ABC$ be a triangle and let $A_1,B_1,C_1$ be the points where the angle bisectors of $A,B$ and $C$ meet the circumcircle of triangle $ABC$ , respectively. Let $M_a$ be the midpoint of the segment connecting the intersections of segments $A_1B_1$ and $A_1C_1$ with segment $BC$ . Define $M_b$ and $M_c$ analogously. Prove that $AM_a,BM_b$ , and $CM_c$ are concurrent if and only if $ABC$ is isosceles.

Dr. Zuming Feng, Phillips Exeter Academy, New Hampshire

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O26

Consider a triangle $ABC$ and let $O$ be its circumcenter. Denote by $D$ the foot of the altitude from $A$ and by $E$ the intersection of $AO$ and $BC$ . Suppose tangents to the circumcircle of triangle $ABC$ at $B$ and $C$ intersect at $T$ and that $AT$ intersects this circumcircle at $F$ . Prove that the circumcircles of triangles $DEF$ and $ABC$ are tangent.

Proposed by Ivan Borsenco, University of Texas at Dallas

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O33

Let $ABC$ be a triangle with cicrumcenter $O$ and incenter $I$ . Consider a point $M$ lying on the small arc $BC$ . Prove that $AM + 2OI \ge MB +MC \ge MA - 2OI$

Proposed by Hung Quang Tran, Ha Noi University, Vietnam

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