geometry problems from Stars Of Mathematics (Romanian)
with aops links
Senior part started in 2007, junior part in 2011
junior
2012 Stars of Mathematics Juniors P1
Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ so that $AQ=PQ$ and $\angle PAQ = 45^{\circ}$.
2007-2022 (+2021 winter)
senior
with aops links
Stelele Matematici , ICHB Bucuresti
Senior part started in 2007, junior part in 2011
collected inside aops here
it didn't take place in 2020
2011 - 2022 (+2021 winter)
junior
Let $ABC$ be an acute-angled, not equilateral triangle, where vertex $A$ lies on the perpendicular bisector of the segment $HO$, joining the orthocentre $H$ to the circumcentre $O$. Determine all possible values for the measure of angle $A$.
(U.S.A. - 1989 IMO Shortlist)
Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ so that $AQ=PQ$ and $\angle PAQ = 45^{\circ}$.
(Dan Schwarz)
2013 Stars of Mathematics Juniors P2
Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle. Prove that at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle.
Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle. Prove that at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle.
(Dan Schwarz)
2015 Stars of Mathematics Juniors P3
Let $ABCD$ be cyclic quadrilateral,let $\gamma$ be it's circumscribed circle and let $M$ be the midpoint of arc $AB$ of $\gamma$,which does not contain points $C,D$.The line that passes through $M$ and the intersection point of diagonals $AC,BD$,intersects $\gamma$ in $N\neq M$.
Let $P,Q$ be two points situated on $CD$,such that $\angle{AQD}=\angle{DAP}$ and $\angle{BPC}=\angle{CBQ}$.Prove that circles $\odot(NPQ)$ and $\gamma$ are tangent.
Let $ABCD$ be cyclic quadrilateral,let $\gamma$ be it's circumscribed circle and let $M$ be the midpoint of arc $AB$ of $\gamma$,which does not contain points $C,D$.The line that passes through $M$ and the intersection point of diagonals $AC,BD$,intersects $\gamma$ in $N\neq M$.
Let $P,Q$ be two points situated on $CD$,such that $\angle{AQD}=\angle{DAP}$ and $\angle{BPC}=\angle{CBQ}$.Prove that circles $\odot(NPQ)$ and $\gamma$ are tangent.
(Flavian Georgescu)
2016 Stars of Mathematics Juniors P4
Let $ABC$ be an acute angled triangle with $AB <AC$, $I$ be the center of its incircle, $D, E, F$ touch points of the incircle with the sides $BC, CA$ and $AB$ respectively, $X$ the midpoint of the arc $BAC$ of the circumcircle of the triangle $ABC$. $P$ is the projection of $D$ on $EF, Q$ is the projection of $A$ on $ID$.
a) Prove that that the lines $IX$ and $PQ$ are parallel.
b) If $Y$ is the second point of intersection between the circle of diameter $AI$ and the circumcircle of the triangle $ABC$, prove that the lines $XQ$ and $PI$ intersect at $Y$.
Let $ABC$ be an acute angled triangle with $AB <AC$, $I$ be the center of its incircle, $D, E, F$ touch points of the incircle with the sides $BC, CA$ and $AB$ respectively, $X$ the midpoint of the arc $BAC$ of the circumcircle of the triangle $ABC$. $P$ is the projection of $D$ on $EF, Q$ is the projection of $A$ on $ID$.
a) Prove that that the lines $IX$ and $PQ$ are parallel.
b) If $Y$ is the second point of intersection between the circle of diameter $AI$ and the circumcircle of the triangle $ABC$, prove that the lines $XQ$ and $PI$ intersect at $Y$.
Let $ABC$ be an acute triangle in which $AB < AC$. Let $M$ be the midpoint of the side $BC$ and consider $D$ an arbitrary point of the line segment $AM$. Let $E$ be a point of the line segment $BD$ and consider the point $F$ of the line $AB$ such that lines $EF$ and $BC$ are parallel. If the orthocenter, $H$, of the triangle $ABC$ lies at the intersection point of lines $AE$ and $DF$, prove that the angle bisectors of $\angle BAC$ and $\angle BDC$ meet on the line $BC$.
(Vlad Robu)
2018 Stars of Mathematics Juniors P3
Let be an isosceles trapezoid such that its smaller base is equal to its legs, and a rhombus that has each of its vertexes on a different side of the trapezoid. Prove that the smaller angles of the trapezoid are equal to the smaller ones of the rhombus.
Let be an isosceles trapezoid such that its smaller base is equal to its legs, and a rhombus that has each of its vertexes on a different side of the trapezoid. Prove that the smaller angles of the trapezoid are equal to the smaller ones of the rhombus.
(Vlad Robu)
Let $A$ and $C$ be two points on a circle $X$ so that $AC$ is not diameter and $P$ a segment point on $AC$ different from its middle. The circles $c_1$ and $c_2$, inner tangents in $A$, respectively $C$, to circle $X$, pass through the point $P$ ¸ and intersect a second time at point $Q$. The line $PQ$ intersects the circle $X$ in points $B$ and $D$. The circle $c_1$ intersects the segments $AB$ and $AD$ in $K$, respectively $N$, and circle $c_2$ intersects segments $CB$ and ¸ $CD $ in $L$, respectively $M$. Show that:
a) the $KLMN$ quadrilateral is isosceles trapezoid;
b) $Q$ is the middle of the segment $BD$.
a) the $KLMN$ quadrilateral is isosceles trapezoid;
b) $Q$ is the middle of the segment $BD$.
(Thanos Kalogerakis)
Let $ABC$ be a triangle, let its $A$-symmedian cross the circle $ABC$ again at $D$, and let $Q$ and $R$ be the feet of the perpendiculars from $D$ on the lines $AC$ and $AB$, respectively. Consider a variable point $X$ on the line $QR$, different from both $Q$ and $R$. The line through $X$ and perpendicular to $DX$ crosses the lines $AC$ and $AB$ at $V$ and $W$, respectively. Determine the geometric locus of the midpoint of the segment $VW$.
Adapted from American Mathematical Monthly
Note: this is also the P3 junior level problem, although in the junior competition the locus was already given
Let $ABC$ be a triangle, let $I$ be its incentre and let $D$ be the orthogonal projection of $I$ on $BC.$ The circle $\odot(ABC)$ crosses the line $AI$ again at $M,$ and the line $DM$ again at $N.$ Prove that the lines $AN$ and $IN$ are perpendicular.
(Freddie Illingworth & Dominic Yeo)
Let $ABCD$ be a convex quadrilateral and $P$ be a point in its interior, such that $\angle APB+\angle CPD=\angle BPC+\angle DPA$, $\angle PAD+\angle PCD=\angle PAB+\angle PCB$ and $\angle PDC+ \angle PBC= \angle PDA+\angle PBA$. Prove that the quadrilateral is circumscribed.
senior
2007 Stars of Mathematics Seniors P3
Let $ABC$ be a triangle and $A_1,B_1,C_1$ be the feet of the altitudes from $A,B,C$. Let $A_2$ respectively $A_3$, be the orthogonal projection of $A_1$ onto $AB$, respectively $AC$, points $B_2,B_3$ and $C_2,C_3$ are defined in an analogous way. The lines $B_2B_3$ and $C_2C_3$ meet at $A_4$, the lines $C_2C_3$ and $A_2A_3$ meet at $B_4$, while the lines $A_2A_3$ and $B_2B_3$ meet at $C_4$. Show that the lines $AA_4,BB_4$ and $CC_4$ are concurrent.
2007 Stars of Mathematics Seniors P7
Consider a convex quadrilateral, and the incircles of the triangles determined by one of its diagonals. Prove that the tangency points of the incircles with the diagonal are symmetrical with respect to the midpoint of the diagonal if and only if the line of the incenters passes through the crossing point of the diagonals.
Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Also let $\alpha \in (0, \pi/2)$ be fixed. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ such that $AQ=PQ$ and $\angle PAQ = \alpha$.
Let $ABC$ be a triangle and $A_1,B_1,C_1$ be the feet of the altitudes from $A,B,C$. Let $A_2$ respectively $A_3$, be the orthogonal projection of $A_1$ onto $AB$, respectively $AC$, points $B_2,B_3$ and $C_2,C_3$ are defined in an analogous way. The lines $B_2B_3$ and $C_2C_3$ meet at $A_4$, the lines $C_2C_3$ and $A_2A_3$ meet at $B_4$, while the lines $A_2A_3$ and $B_2B_3$ meet at $C_4$. Show that the lines $AA_4,BB_4$ and $CC_4$ are concurrent.
2007 Stars of Mathematics Seniors P7
Let $A_0A_1...A_{n-1}$ be a regular $n$-gon. For each index $i$, consider a point $B_i$ lying on the side $A_iA_{i+1}$, such that $A_iB_i<A_iA_{i+1}/2$, and a point $C_i$ lying on the segment $A_iB_i$ (indices are reduced modulo $n$). Show that the perimeter of the polygon $C_0C_1...C_{n-1}$ is at least as large as the perimeter of the polygon $B_0B_1...B_{n-1}$.
2008 Stars of Mathematics Seniors P3Consider a convex quadrilateral, and the incircles of the triangles determined by one of its diagonals. Prove that the tangency points of the incircles with the diagonal are symmetrical with respect to the midpoint of the diagonal if and only if the line of the incenters passes through the crossing point of the diagonals.
(Dan Schwarz)
2009 Stars of Mathematics Seniors P2
Let $\omega$ be a circle in the plane and $A,B$ two points lying on it. We denote by $M$ the midpoint of $AB$ and let $P \ne M$ be a new point on $AB$. Build circles $\gamma$ and $\delta$ tangent to $AB$ at $P$ and to $\omega$ at $C$, respectively $D$. Consider $E$ to be the point diametrically opposed to $D$ in $\omega$. Prove that the circumcenter of $\triangle BMC$ lies on the line $BE$.
Let $\omega$ be a circle in the plane and $A,B$ two points lying on it. We denote by $M$ the midpoint of $AB$ and let $P \ne M$ be a new point on $AB$. Build circles $\gamma$ and $\delta$ tangent to $AB$ at $P$ and to $\omega$ at $C$, respectively $D$. Consider $E$ to be the point diametrically opposed to $D$ in $\omega$. Prove that the circumcenter of $\triangle BMC$ lies on the line $BE$.
Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations.
2010 Stars of Mathematics Seniors P2
Let $ABC$ be an acute-angled triangle with $AB \neq BC$, $M$ the midpoint of $AC$, $N$ the point where the median $BM$ meets again the circumcircle of $\triangle ABC$, $H$ the orthocentre of $\triangle ABC$, $D$ the point on the circumcircle for which $\angle BDH = 90^{\circ}$, and $K$ the point that makes $ANCK$ a parallelogram. Prove the lines $AC$, $KH$, $BD$ are concurrent.
Let $ABCD$ be a square and let the points $M$ on $BC$, $N$ on $CD$, $P$ on $DA$, be such that $\angle (AB,AM)=x,\angle (BC,MN)=2x,\angle (CD,NP)=3x$.
1) Show that for any $0\le x\le 22.5$, such a configuration uniquely exists, and that $P$ ranges over the whole segment $DA$;
2) Determine the number of angles $0\le x\le 22.5$ for which$\angle (DA,PB)=4x$.
(Dan Schwarz)
(I. Nagel)
2012 Stars of Mathematics Seniors P2Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Also let $\alpha \in (0, \pi/2)$ be fixed. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ such that $AQ=PQ$ and $\angle PAQ = \alpha$.
(Dan Schwarz)
2013 Stars of Mathematics Seniors P2
Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle.
i) Prove at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle.
ii) Assuming the three points inside the rectangle are three corners of it, prove at least two of the three concave quadrilaterals formed by these four points have perimeters lesser than that of the rectangle.
Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle.
i) Prove at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle.
ii) Assuming the three points inside the rectangle are three corners of it, prove at least two of the three concave quadrilaterals formed by these four points have perimeters lesser than that of the rectangle.
(Dan Schwarz)
Let $\gamma,\gamma_0,\gamma_1,\gamma_2$ be four circles in plane,such that $\gamma_i$ is interiorly tangent to $\gamma$ in point $A_i$,and $\gamma_i$ and $\gamma_{i+1}$ are exteriorly tangent in point $B_{i+2}$,$i=0,1,2$(the indexes are reduced modulo $3$).The tangent in $B_i$,common for circles $\gamma_{i-1}$ and $\gamma_{i+1}$,intersects circle $\gamma$ in point $C_i$,situated in the opposite semiplane of $A_i$ with respect to line $A_{i-1}A_{i+1}$.Prove that the three lines $A_iC_i$ are concurrent.
2016 Stars of Mathematics Seniors P3
Let $ABC$ be a triangle, let $M_A$ be the midpoint of the side $BC$, and let $P_A$ be the orthogonal projection of $A$ on the line $BC$, similarly, define $M_B, P_B$ and $M_C, P_C$. The lines $M_BM_C$ and $P_BP_C$ meet at $S_A$, and the tangent of the circle $ABC$ at $A$ meets the line $BC$ at T$_A$, similarly, define $S_B, T_B$ and $S_C, T_C$. Show that the perpendiculars through $A, B, C$ to the lines $S_AT_A, S_BT_B, S_CT_C$, respectively, are concurrent.
Let $ABC$ be a triangle, let $M_A$ be the midpoint of the side $BC$, and let $P_A$ be the orthogonal projection of $A$ on the line $BC$, similarly, define $M_B, P_B$ and $M_C, P_C$. The lines $M_BM_C$ and $P_BP_C$ meet at $S_A$, and the tangent of the circle $ABC$ at $A$ meets the line $BC$ at T$_A$, similarly, define $S_B, T_B$ and $S_C, T_C$. Show that the perpendiculars through $A, B, C$ to the lines $S_AT_A, S_BT_B, S_CT_C$, respectively, are concurrent.
(Flavian Georgescu)
2017 Stars of Mathematics Seniors P2
Let $ABC$ be a triangle and $O$ be its circumcenter. Let $P,Q$ be two points in the interior of $ABC$, $R$ the reflection of $O$ wrt of the midpoint of $PQ$, and $S$ the reflection of $R$ wrt of the Euler circle center. Let $\omega$ be the circle that passes through $P,Q$ such that $\omega$ and the circumcenter are orthogonal. Let $(OP,(OQ$ intersect $\omega$ in $P,P’$ and $Q,Q’$. Let $P’Q,PQ’$ intersect at $T$. Prove that if $P,Q$ are isogonal conjugates, then $S,T$ are also isogonal conjugates.
2018 Stars of Mathematics Seniors P1
Let $ABC$ be a triangle. Let $M$ be a variable point interior to the segment $AB$, and let $\gamma_B$ be the circle through $M$ and tangent at $B$ to $BC$. Let $P$ and $Q$ be the touch points of $\gamma_B$ and its tangents from $A$, and let $X$ be the midpoint of the segment $PQ$. Similarly, let $N$ be a variable point interior to the segment $AC$, and let $\gamma_C$ be the circle through $M$ and tangent at $C$ to $BC$. Let $R$ and $S$ be the touch points of $\gamma_C$ and its tangents from $A$, and let $Y$ be the midpoint of the segment $RS$. Prove that the line through the centers of the circles $AMN$ and $AXY$ passes through a fixed point.
source: pregatirematematicaolimpiadejuniori.wordpress.com
Let $ABC$ be a triangle and $O$ be its circumcenter. Let $P,Q$ be two points in the interior of $ABC$, $R$ the reflection of $O$ wrt of the midpoint of $PQ$, and $S$ the reflection of $R$ wrt of the Euler circle center. Let $\omega$ be the circle that passes through $P,Q$ such that $\omega$ and the circumcenter are orthogonal. Let $(OP,(OQ$ intersect $\omega$ in $P,P’$ and $Q,Q’$. Let $P’Q,PQ’$ intersect at $T$. Prove that if $P,Q$ are isogonal conjugates, then $S,T$ are also isogonal conjugates.
2018 Stars of Mathematics Seniors P1
Let $ABC$ be a triangle, and let $\ell$ be the line through $A$ and perpendicular to the line $BC$. The reflection of $\ell$ in the line $AB$ crosses the line through $B$ and perpendicular to $AB$ at $P$. The reflection of $\ell$ in the line $AC$ crosses the line through $C$ and perpendicular to $AC$ at $Q$. Show that the line $PQ$ passes through the orthocenter of the triangle $ABC$.
2019 Stars of Mathematics Seniors P3
(Flavian Georgescu)
Let $ABC$ be a triangle. Let $M$ be a variable point interior to the segment $AB$, and let $\gamma_B$ be the circle through $M$ and tangent at $B$ to $BC$. Let $P$ and $Q$ be the touch points of $\gamma_B$ and its tangents from $A$, and let $X$ be the midpoint of the segment $PQ$. Similarly, let $N$ be a variable point interior to the segment $AC$, and let $\gamma_C$ be the circle through $M$ and tangent at $C$ to $BC$. Let $R$ and $S$ be the touch points of $\gamma_C$ and its tangents from $A$, and let $Y$ be the midpoint of the segment $RS$. Prove that the line through the centers of the circles $AMN$ and $AXY$ passes through a fixed point.
Let $ABC$ be a triangle, let its $A$-symmedian cross the circle $ABC$ again at $D$, and let $Q$ and $R$ be the feet of the perpendiculars from $D$ on the lines $AC$ and $AB$, respectively. Consider a variable point $X$ on the line $QR$, different from both $Q$ and $R$. The line through $X$ and perpendicular to $DX$ crosses the lines $AC$ and $AB$ at $V$ and $W$, respectively. Determine the geometric locus of the midpoint of the segment $VW$.
Adapted from American Mathematical Monthly
Note: this is also the P3 junior level problem, although in the junior competition the locus was already given.
Let $ABC$ be a triangle, and let $D, E$ and $F$ be the feet of the altitudes from $A, B$ and $C,$ respectively. A circle $\omega_A$ through $B$ and $C$ crosses the line $EF$ at $X$ and $X'$. Similarly, a circle $\omega_B$ through $C$ and $A$ crosses the line $FD$ at $Y$ and $Y',$ and a circle $\omega_C$ through $A$ and $B$ crosses the line $DE$ at $Z$ and $Z'$. Prove that $X, Y$ and $Z$ are collinear if and only if $X', Y'$ and $Z'$ are collinear.
(Vlad Robu)
Let $ABCD$ be a convex quadrilateral and $P$ be a point in its interior, such that $\angle APB+\angle CPD=\angle BPC+\angle DPA$, $\angle PAD+\angle PCD=\angle PAB+\angle PCB$ and $\angle PDC+ \angle PBC= \angle PDA+\angle PBA$. Prove that the quadrilateral is circumscribed.
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