Serbia 2007-21 (SMO) 26p

geometry problems from Serbian Mathematical Olympiads
with aops links in the names

Serbian Mathematical Olympiads 2008 - 2017 EN with solutions

                                          

                                               2007 - 2021

2007 Serbian P1
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

2007 Serbian P5
In a scalene triangle $ABC , AD, BE , CF$ are the angle bisectors $(D \in BC , E \in AC , F \in AB)$. Points $K_{a}, K_{b}, K_{c}$ on the incircle of triangle $ABC$ are such that $DK_{a}, EK_{b}, FK_{c}$ are tangent to the incircle and $K_{a}\not\in BC , K_{b}\not\in AC , K_{c}\not\in AB$. Let $A_{1}, B_{1}, C_{1}$ be the midpoints of sides $BC , CA, AB$ , respectively. Prove that the lines $A_{1}K_{a}, B_{1}K_{b}, C_{1}K_{c}$ intersect on the incircle of triangle $ABC$.

2008 Serbian P2
Triangle $\triangle ABC$ is given. Points $D$ i $E$ are on line  $AB$ such that  $D - A - B - E, AD = AC$ and $BE = BC$. Bisector of internal angles at $A$ and $B$ intersect $BC,AC$ at $P$ and $Q$, and circumcircle of  $ABC$ at  $M$ and  $N$. Line which connects  $A$ with center of circumcircle of $BME$ and line which connects $B$ and center of circumcircle of $AND$ intersect at $X$. Prove that $CX \perp PQ$.

2008 Serbian P6
In a convex pentagon $ABCDE$, let $\angle EAB = \angle ABC = 120^{\circ}$, $\angle ADB = 30^{\circ}$ and $\angle CDE = 60^{\circ}$. Let $AB = 1$. Prove that the area of the pentagon is less than $\sqrt {3}$.

2009 Serbian P1
In a convex pentagon $ABCDE$, let $\angle EAB = \angle ABC = 120^{\circ}$, $\angle ADB = 30^{\circ}$ and $\angle CDE = 60^{\circ}$. Let $AB = 1$. Prove that the area of the pentagon is less than $\sqrt {3}$.

2009 Serbian P6
Triangle $ABC$ has incircle $w$ centered as $S$ that touches the sides $BC,CA$ and $AB$ at $P,Q$ and $R$ respectively. $AB$ isn't equal $AC$, the lines $QR$ and BC intersects at point $M$, the circle that passes through points $B$ and $C$ touches the circle $w$ at point N$$, circumcircle of triangle $MNP$ intersects with line $AP$ at $L$ ($L$ isn't equal to $P$). Then prove that $S,L$ and $M$ lie on the same line

2010 Serbian P2
In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$.

2010 Serbian P4
Let $O$ be the circumcenter of triangle $ABC$. A line through $O$ intersects the sides $CA$ and $CB$ at points $D$ and $E$ respectively, and meets the circumcircle of $ABO$ again at point $P \neq O$ inside the triangle. A point $Q$ on side $AB$ is such that $\frac{AQ}{QB}=\frac{DP}{PE}$. Prove that $\angle APQ = 2\angle CAP$.

2011 Serbian P3
Let $H$ be orthocenter and $O$ circumcenter of an acuted angled triangle $ABC$. $D$ and $E$ are feets of perpendiculars from $A$ and $B$ on $BC$ and $AC$ respectively. Let $OD$ and $OE$ intersect $BE$ and $AD$ in $K$ and $L$, respectively. Let $X$ be intersection of circumcircles of $HKD$ and $HLE$ different than $H$, and $M$ is midpoint of $AB$. Prove that $K, L, M$ are collinear iff $X$ is circumcenter of $EOD$.

2011 Serbian P4
On sides $AB, AC, BC$ are points $M, X, Y$, respectively, such that $AX=MX$; $BY=MY$. $K$, $L$ are midpoints of $AY$ and $BX$. $O$ is circumcenter of $ABC$, $O_1$, $O_2$ are symmetric with $O$ with respect to $K$ and $L$. Prove that $X, Y, O_1, O_2$ are concyclic.

2012 Serbian P1
Let $ABCD$ be a parallelogram and $P$ be a point on diagonal $BD$ such that $\angle PCB=\angle ACD$. Circumcircle of triangle $ABD$ intersects line $AC$ at points $A$ and $E$. Prove that $\angle AED=\angle PEB.$

2013 Serbian P3
Let $M$, $N$ and $P$ be midpoints of sides $BC, AC$ and $AB$, respectively, and let $O$ be circumcenter of acute-angled triangle $ABC$. Circumcircles of triangles $BOC$ and $MNP$ intersect at two different points $X$ and $Y$ inside of triangle $ABC$. Prove that $\angle BAX=\angle CAY.$

2013 Serbian P5
Let $A'$ and $B'$ be feet of altitudes from $A$ and $B$, respectively, in acute-angled triangle $ABC$ ($AC\not = BC$). Circle $k$ contains points $A'$ and $B'$ and touches segment $AB$ in $D$. If triangles $ADA'$ and $BDB'$ have the same area, prove that $\angle A'DB'= \angle ACB.$

2014 Serbian P2
On sides $BC$ and $AC$ of $\triangle ABC$ given are $D$ and $E$, respectively. Let $F$ ($F \neq C$) be a point of intersection of circumcircle of $\triangle CED$ and line that is parallel to $AB$ and passing through C. Let $G$ be a point of intersection of line $FD$ and side $AB$, and let $H$ be on line $AB$ such that $\angle HDA = \angle GEB$ and $H-A-B$. If $DG=EH$, prove that point of intersection of $AD$ and $BE$ lie on angle bisector of $\angle ACB$.

2014 Serbian P6 (IMO Shortlist 2013 G3)
In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{  \angle BAC, \angle ABC  \}$

by Dusan Djukic

2015 Serbian P1
Consider circle inscribed quadriateral $ABCD$. Let $M,N,P,Q$ be midpoints of sides $DA,AB,BC,CD$.Let $E$ be the point of intersection of diagonals. Let  $k_1,k_2$ be circles around $EMN$ and $EPQ$ . Let $F$ be point of intersection of $k_1$ and $k_2$ different from $E$. Prove that $EF$ is perpendicular to $AC$.

2016 Serbian P3
Let $ABC$ be a triangle and $O$ its circumcentre. A line tangent to the circumcircle of the triangle $BOC$ intersects sides $AB$ at $D$ and $AC$ at $E$. Let $A'$ be the image of $A$ under $DE$. Prove that the circumcircle of the triangle $A'DE$ is tangent to the circumcircle of triangle $ABC$.

2016 Serbian P4
Let $ABC$be a triangle, and $I$ the incenter, $M$ midpoint of $BC$,  $D$ the touch point of incircle and $BC$. Prove that perpendiculars from $M, D, A$ to $AI, IM, BC$ respectively are concurrent

2017 Serbian P2
Let $ABCD$ be a convex and cyclic quadrilateral. Let $AD\cap BC=\{E\}$, and let $M,N$ be points on $AD,BC$ such that $AM:MD=BN:NC$. Circle around $\triangle EMN$ intersects circle around $ABCD$ at $X,Y$ prove that $AB,CD$ and $XY$ are either parallel or concurrent.

2017 Serbian P6
Let $k$ be the circumcircle of $\triangle ABC$ and let $k_a$ be A-excircle .Let the two common tangents of $k,k_a$ cut $BC$ in $P,Q$.Prove that $\measuredangle PAB=\measuredangle CAQ$.

2018 Serbian P1
Let $\triangle ABC$ be a triangle with incenter $I$. Points $P$ and $Q$ are chosen on segmets $BI$ and $CI$ such that $2\angle  PAQ=\angle BAC$. If $D$ is the touch point of incircle and side $BC$ prove that $\angle PDQ=90$.

2019 Serbian P3
Let $k$ be the circle inscribed in convex quadrilateral $ABCD$. Lines $AD$ and $BC$ meet at $P$ ,and circumcircles of $\triangle PAB$ and $\triangle PCD$ meet in $X$ . Prove that tangents from $X$ to $k$ form equal angles with lines $AX$ and $CX$ .

2019 Serbian P4
For a  $\triangle ABC$ , let $A_1$ be the symmetric point of the intersection of angle bisector of $\angle BAC$ and $BC$ , where center of the symmetry is the midpoint of side $BC$, In the same way we define $B_1$ ( on $AC$ ) and $C_1$ (on $AB$). Intersection of circumcircle of $\triangle A_1B_1C_1$ and line $AB$ is the set $\{Z,C_1 \}$, with $BC$ is the set $\{X,A_1\}$ and with $CA$ is the set $\{Y,B_1\}$. If the perpendicular lines from $X,Y,Z$ on $BC,CA$ and $AB$ , respectively are concurrent , prove that $\triangle ABC$ is isosceles.\

2020 Serbian P3

We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.

2020 Serbian P4

In a trapezoid $ABCD$ such that the internal angles are not equal to $90^{\circ}$, the diagonals $AC$ and $BD$ intersect at the point $E$. Let $P$ and $Q$ be the feet of the altitudes from $A$ and $B$ to the sides $BC$ and $AD$ respectively. Circumscribed circles of the triangles $CEQ$ and $DEP$ intersect at the point $F\neq E$. Prove that the lines $AP$, $BQ$ and $EF$ are either parallel to each other, or they meet at exactly one point.

2021 Serbian P3

In a triangle $ABC$, let $AB$ be the shortest side. Points $X$ and $Y$ are given on the circumcircle of $\triangle ABC$ such that $CX=AX+BX$ and $CY=AY+BY$. Prove that $\measuredangle XCY<60^{o}$.

2021 Serbian P4

A convex quadrilateral $ABCD$ will be called rude if there exists a convex quadrilateral $PQRS$ whose points are all in the interior or on the sides of quadrilateral $ABCD$ such that the sum of diagonals of $PQRS$ is larger than the sum of diagonals of $ABCD$.

Let $r>0$ be a real number. Let us assume that a convex quadrilateral $ABCD$ is not rude, but every quadrilateral $A'BCD$ such that $A'\neq A$ and $A'A\leq r$ is rude. Find all possible values of the largest angle of $ABCD$.

source: srb.imomath.com