geometry problems from Aops Mocks for Junior Math Olympiads with aops links in the names
collected inside aops here
2010-21
unofficial mock contests for JMO started in 2010 in aops
\
Let $ABC$ be a triangle with $\angle ABC = 90^o$ and $AB> BC$. Let $D$ be a point on side $AB$ such that $BD = BC$. Let $E$ be the foot of the perpendicular from $D$ on $AC$, and $F$ the reflection of $B$ wrt $CD$. Show that $EC$ is the bisector of angle $\angle BEF$.
Given two fixed, distinct points $B$ and $C$ on plane $P$, find the locus of all points $A$ belonging to $P$ such that the quadrilateral formed by point $A$, the midpoint of $AB$, the centroid of $\vartriangle ABC$, and the midpoint of$ AC$ (in that order) can be inscribed in a circle.
Proposed by Ray Li
In convex, tangential quadrilateral $ABCD$, let the incircle touch the four sides $AB, BC, CD, DA$ at points $E, F, G, H$, respectively. Prove that$$\frac{[EFGH]}{[ABCD]} \le \frac12.$$
Proposed by Victor Wang
In $\vartriangle ABC$, $\angle A=\frac{4\pi}{7}$ and $\angle B =\frac{\pi}{7}$. Let I be the incenter of $\vartriangle ABC$ so that $CI$ meets $AB$ at $D$. Also let the circumcircles of $\vartriangle AID$ and $\vartriangle BDC$ meet at $Q$. If $BQ$ intersects $CD$ at $P$, and $QI$ intersects $BC$ at $R$, prove that $\frac{DQ}{DP}=\frac{IC}{IR}$.
Let $ABC$ be a triangle with incenter $I$. Let $P$ be a point in the plane of $ABC$ other than $I$. Let the reflections of $P$ across lines $BI$ and $CI$ be $B'$ and $C'$ respectively. Show that $B'C'$ is parallel to $BC$ if and only if P lies on line $AI$.
Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$, where $A$ and $C$ are diametrically opposite on $\omega$,. Also, $AD > AB$. Let the reflection of $B$ over $AC$ be $E$. Prove that $ED\cdot BD = AD^2 - AB^2.$
Let $ABCD$ be a cyclic quadrilateral, with $\angle ABD = \angle ADC$. Denote the incenters of $\vartriangle ABC$ and $\vartriangle ABD$ by $I_1,I_2$ respectively. Show that lines $AB$ and $I_1I_2$ meet on the circumcircle of $\vartriangle BI_1C$.
Let $ABCD$ be a convex quadrilateral inscribed in circle $\omega$. Let diagonals $AC$ and $BD$ meet at $P$. Construct circle $\omega_1$ tangent to circle $\omega$, to segment $AP$ at $Q$, and to segment $DP$. Construct circle $\omega_2$ tangent to circle $\omega$, to segment $BP$ at $R$, and to segment $CP$. Prove that $AB \parallel CD$ if and only if $AB \parallel QR$.
Proposed by Kevin Ren
Let $I$ be the incenter of the orthic triangle $DEF$ of acute triangle $ABC$ with $D \in BC$, $E \in AC$, $F \in AB$. Furthermore, let $X$ be the intersection of the $A$-symmedian with $BC$ and $L$ be the point on line segment $EF$ such that $\frac{CD}{CX} = \frac{BD}{BX} = \frac{LE}{LF}$ . Prove that $A, L$, and $I$ are collinear.
Note: the $A$-symmedian is defined to be the reflection of the $A$-median over the $A$-angle bisector.
Proposed by Tristan Shin
Four points $A, B, C, D$ lie on a circle with center $O$. Points $X$ and $Y$ are chosen on $AB$ and $AD$ respectively so that $CX\perp CD$ and $CY\perp BC$. Prove that points $O, X, Y$ are collinear.
Proposed by Tan
Let $ABC$ be a triangle, and $X, Y$ be points on the circumcircle of $ABC$ such that $AX = AY$ . Let $B'$ and $C'$ be the intersections of $XY$ with $AB$ and $AC$, respectively. Let $P$ be an arbitrary point in the plane, and let $Q$ be the second intersection of the circumcircles of $BB'P$ and $CC'P$. Prove that $PQ$ passes through $A$.
Proposed by Chezbgone
Let $\Gamma$ be the circumcircle to $\vartriangle ABC$ and let $D$ be the antipode of $A$ w.r.t $\Gamma$. A line $\ell$ parallel to $BC$ passes through $\Gamma$ at $E, F$ such that $B, E, F, C$ lie on $\Gamma$ in that order. Let $P$ be the intersection of $BD$ with $AE$ and let $Q$ be the intersection of $CD$ with $AF$; finally, let $X$ be the intersection of $CP$ with $BQ$. Prove that $AX$ is perpendicular to $BC$.
On $\vartriangle ABC$, let $D, E, F$ be the tangency points of the incircle $\omega$ of $\vartriangle ABC$ with sides $BC, CA, AB$ respectively. Let $\omega_0$ be a circle concentric to $\omega$ such that it is tangent to $EF$, and let $M, N$ be the intersections of $EF$ with the tangents from $D$ to $\omega_0$. Prove that $D, M, N$, and the midpoint of $BC$ are concyclic.
Let $ABC$ be a triangle and let $D$ be a point on arc $AB$ of the circumcircle of $ABC$ not containing $C$ and let $E$ be a point on arc $AC$ of the circumcircle of $ABC$ not containing $B$. Suppose that line $DE$ intersects $AB$ at $M$ and $AC$ at $N$, and let the circumcircles of $BDN$ and $CEM$ intersect at $X$ and $Y$. Prove that $A$ lies on $XY$ if and only if $DE$ is parallel to $BC$.
Let $WXYZ$ be a quadrilateral that is not self-intersecting, and let $A = WX \cap YZ$ and $B = W Z \cap XY$ . Then, let the second intersection (the point that is not $W$) of the circumcircle of $\vartriangle AWZ$ and the circumcircle of $\vartriangle$ $B WX$ be $C$. You are given that $A, B$ and $C$ are collinear. If points $W, X, Y$ are fixed and $Z$ is allowed to vary in a way that preserves the collinearity of $A, B, C$, determine the locus of the circumcenter of $\vartriangle WPZ$, where $P = W Y \cap XZ$.
In $\vartriangle ABC$ define points $D, E$ to be the intersections of the circle with diameter $BC$ with the nine-point circle of $ABC$, with $BD > BE$. Let $M$ be the midpoint of segment $BC$, and let line $\ell $ be the line through $A$ parallel to $BC$. Let $MD \cap \ell$ be $S$ and $ME \cap \ell$ be $T$. Prove that $ET$ and $DS$ intersect on the $A$-median of triangle $ABC$.
In acute $\vartriangle ABC$ the feet of the perpendiculars from $A, B$, and $C$ to the sides $BC, CA$, and $AB$ are $D, E$, and $F$, respectively. Assume $\angle E > \angle F$. Define $U$ and $V$ to be the foot of the perpendicular from $A$ to line $DE$ and $DF$, respectively. The circumcircles of $\vartriangle UEA$ and $\vartriangle V F A$ intersect at a point $X$ other than $A$. Let $S$ be on segment $DF$ such that $DE = DS$, and let $T$ be the foot of the perpendicular from $D$ to $EF$. Define the midpoint of segment $DT$ as $M$. The intersection of the segment $MX$ with the circumcircle of $\vartriangle ESF$ is $H$. Prove that $H$ is the orthocenter of $\vartriangle ABC$.
In acute $\vartriangle ABC$, let $P$ be a point on $BC$, and let $\omega$ be the circle with diameter $AP$. Let $X$ be the reflection of the orthocenter across the side $BC$, and let $E$ and $F$ be the intersections of the tangents from $X$ to $\omega$. Show that as $P$ varies on $BC, E$ and $F$ vary on a fixed circle.
In a triangle $\vartriangle ABC$, let $\omega$ be the circle with diameter $AD$, where $D$ is the foot of the altitude from $A$ to $BC$. This circle intersects the sides $AB$ and $AC$ at the points $E$ and $F$, respectively. Show that the tangents at $E$ and $F$ to $\omega$ concur on the $A$-median of $\vartriangle ABC$
In $\vartriangle ABC$ let $D, E$, and $F$ be the tangency points of the incircle on $BC, CA$, and $AB$, respectively. Let $Q$ be the second intersection of the incircle and the circle with diameter $AD$, and let $P$ be the second intersection point of line $AQ$ with the incircle. Let $M$ be the midpoint of $BC, X$ be the intersection of $P Q$ and $EF$, and $Z$ be the second intersection of the circle with diameter $AP$ and the incircle. The line parallel to $BC$ through $P$ meets line $MQ$ at $R$, and $L$ is the foot of the perpendicular from $R$ to line $P Q$. Show that $L, X, Z, D$ lie on a circle.
Let $\triangle ABC$ be a triangle with centroid $G$ and incenter $I$, and let $B', C'$ be the intouch points of the incircle with sides $AC$ and $AB$, respectively. Let $M, N$ be the midpoints of $AB$ and $AC$, respectively, and let $B''$ be the reflection of $B'$ across $N$ and $C''$ be the reflection of $C'$ across $M$. Let $X, Y, Z$ be the intersections of lines $BB''$ and $CC''$, $NI$ and $CC''$, $MI$ and $BB''$ respectively. Show that:
(i) $IYXZ$ is a parallelogram
(ii) $[BIG] + [CIG] = [MGX] + [NGX]$
Proposed by FedeX333X
Let $I$ be the incenter of $\triangle ABC$, and $M$ be the midpoint of $\overline{BC}$. Let $\Omega$ be the nine-point circle of $\triangle BIC$. Suppose that $\overline{BC}$ intersects $\Omega$ at a point $D\ne M$. If $Y$ is the intersection of $\overline{BC}$ and the $A$-intouch chord, and $X$ is the projection of $Y$ onto $\overline{AM}$, prove that $X$ lies on $\Omega$, and the intersection of the tangents to $\Omega$ at $D$ and $X$ lies on the $A$-intouch chord of $\triangle ABC$.
Note. The nine-point circle of $\triangle ABC$ is the circumcircle of its medial triangle, and if the incircle touches $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$, respectively, then $\overline{EF}$ is the $A$-intouch chord.
Proposed by TheUltimate123
Let $ABC$ be a triangle with orthocenter $H$, and define $E$ and $F$ as the intersections of $\overline{AH}$ with the perpendicular bisectors of $\overline{AB}$ and $\overline{AC}$ respectively. Furthermore, let $D$ be the intersection of $\overline{BE}$ and $\overline{CF}$. Suppose that $X$ and $Y$ lie on $\overline{AB}$ and $\overline{AC}$ respectively such that $\overline{FX}$, $\overline{EY}$, and $\overline{BC}$ are all parallel. Prove that $X$ and $Y$ lie on the exterior angle bisector of $\angle BDC$.
Proposed by Blast_S1
Determine all positive reals $r$ such that, for any triangle $ ABC$, we can choose points $D,E,F$ trisecting the perimeter of the triangle into three equal-length sections so that the area of $\triangle DEF$ is exactly $r$ times that of $\triangle ABC$.
Proposed by talkon
Consider $\triangle ABC$ with point $D$ on $CA$ and $E$ on $AB.$ Let $BD$ intersect $CE$ at $P.$ Denote the incircle of $\triangle BPC$ as $\omega.$ Let the tangent to $\omega$ through $D$ intersect $\omega$ at point $X$ not on line $BD$ and let the tangent to $\omega$ through $E$ intersect $\omega$ at point $Y$ not on line $CE.$ Let $DX$ and $EY$ intersect at $Z.$ Prove that $DY,$ $EX,$ and $PZ$ concur.
Proposed by Dennis Chen
Let $ABC$ be an acute triangle with circumcenter $O$, orthocenter $H$, and $\angle A=45^\circ$. Denote by $M$ the midpoint of $\overline{BC}$, and let $P$ be a point such that $\overline{AP}$ is parallel to $\overline{BC}$ and $\angle HMB=\angle PMC$. Show that if segment $OP$ intersects the circle with diameter $\overline{AH}$ at $Q$, then $\overline{OA}$ is tangent to the circumcircle of $\triangle APQ$.
Proposed by Blast_S1
Let $ABC$ be a triangle, and $D$ be a point on the internal angle bisector of $\angle BAC$ but not on the circumcircle of $\triangle ABC$. Suppose that the circumcircle of $\triangle ABD$ intersects $\overline{AC}$ again at $P$ and the circumcircle of $\triangle ACD$ intersects $\overline{AB}$ again at $Q$. Denote by $O_1$ and $O_2$ the circumcenters of $\triangle ABD$ and $\triangle ACD$, respectively. Prove that the circumcenters of $\triangle ABC$, $\triangle APQ$, and $\triangle AO_1O_2$ are collinear.
Proposed by TheUltimate123
Let $\triangle ABC$ with $BC$ being the minimum length, and $D$ as the intersection of the tangents of $(ABC)$ at points $B$ and $C$. Internal angle bisectors of $\angle B, \angle C$ intersects circle with radius $DB$ at $E$ and $F$ respectively. Define $I$ as its incenter.
Suppose $M,N$ are the circumcenters of $AIE, AIF$, and $BI, CI$ intersects $(ABC)$ at $X,Y$. Prove that
\[ \frac{YM}{XN} = \frac{CH - AH}{BH - AH} \]where $H$ is the orthocenter of $\triangle ABC$.
Proposed by Jonathan Christian, Indonesia
Let $ABC$ be an acute triangle such that $D,E,F$ lies on $BC, CA, AB$ respectively and $AD,BE,CF$ be its altitudes. Let $P$ be the common point of $EF$ with the circumcircle of $\triangle ABC$, with $P$ on the minor arc of $AC$. Define $H' \not= B$ as the common point of $BE$ with the circumcircle of $\triangle ABC$. Prove that $\angle ADH' = \angle APF$ if and only if $\angle ABP = \angle CBM$ where $M$ denotes the midpoint of $AC$.
Proposed by Orestis Lignos, Greece
Let $\triangle ABC$ be an acute, scalene triangle with incenter $I$, and circumcenter $O$. Assume that the lines $OI$ and $BC$ meet at point $X$ and $M$ is the midpoint of the arc $BC$ in the circumcircle of $\triangle ABC$ not containing $A$. Suppose that the four points $A,O,M$, and $X$ lie on a circle. Prove that $\angle BAC=60^{\circ}$.
Proposed by psi241.
Let $\triangle ABC$ be a triangle. Points $D$, $E$, and $F$ are placed on sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively such that $EF\parallel BC$. The line $DE$ meets the circumcircle of $\triangle ADC$ again at $X\ne D$. Similarly, the line $DF$ meets the circumcircle of $\triangle ADB$ again at $Y\ne D$. If $D_1$ is the reflection of $D$ across the midpoint of $\overline{BC}$, prove that the four points $D$, $D_1$, $X$, and $Y$ lie on a circle.
Proposed by MarkBcc168.
Given a segment $AB$ and a circle $\Omega$ passing through points $A$ and $B$, let $C$ be an arbitrary point distinct from $A$ and $B$ on $\Omega$. Let the external angle bisectors of $\angle BAC$ and $\angle ABC$ meet at a point $I_C$. Let $D$, $E$, and $F$ be the feet of the perpendiculars from $I_C$ onto lines $BC$, $CA$, and $AB$, respectively. Let $H$ be the orthocenter of $\triangle DEF$. As $C$ varies on $\Omega$, let $P$ be the fixed point on line $HI_C$.
Next, let $\ell$ be an arbitrary line which cuts segments $BC$ and $CA$ at points $M$ and $N$, respectively. Let $X$, $Y$, and $Z$ be the midpoints of segments $AM$, $BN$, and $MN$, respectively. Show that the foot of the perpendicular from $P$ onto $\ell$ lies on the circumcircle of $\triangle XYZ$.
Proposed by NJOY
In acute $\triangle ABC$ with circumcenter $O$ and circumcircle $\Omega$, line $OB$ meets $\overline{AC}$ at $E$ and meets $\Omega$ again at $B'$. Similarly, line $OC$ meets $\overline{AB}$ at $F$ and meets $\Omega$ again at $C'$. Let lines $B'F$ and $C'E$ meet at a point $X$. Line $AX$ meets the altitude from $B$ to $\overline{AC}$ at a point $Y$. The circles with diameters $\overline{BY}$ and $\overline{CY}$ meet $\Omega$ again at points $B_1$ and $C_1$, respectively. Prove that if line $BC_1$ meets line $CB_1$ at a point $T$, then points $T$, $O$, $Y$ are collinear.
Proposed by kevinmathz; Modified by NJOY and tastymath75025
In triangle $ABC$ with circumcircle $\Gamma$, let $\ell_1$, $\ell_2$, and $\ell_3$ be the tangents to $\Gamma$ at points $A$, $B$, and $C$, respectively. Choose a variable point $P$ on side $\overline{BC}$. Let the lines parallel to $\ell_2$ and $\ell_3$, passing through $P$, meet $\ell_1$ at points $C_1$ and $B_1$, respectively. Let the circumcircles of $\triangle PBB_1$ and $\triangle PCC_1$ meet each other again at a point $Q\neq P$. Let lines $\ell_1$ and $BC$ meet at a point $R$, and let lines $\ell_2$ and $\ell_3$ meet at a point $X$. Prove that, as $P$ varies on side $\overline{BC}$, lines $PQ$ and $RX$ meet at a fixed point.
Proposed by NJOY & Orestis_Lignos
Let $ABC$ be a triangle with circumcenter $O$, incenter $I$, and circumcircle $\Gamma$. Let there be a circle touching $\overline{AB}$ and $\overline{AC}$, and tangent to $\Gamma$ internally at a point $X$. The perpendicular bisector of $\overline{BC}$ meets line $AX$ at a point $S$. Additionally, let $K$ be the point on the circumcircle of $\triangle AIX$, distinct from $I$, such that $\overline{KI} \parallel \overline{BC}$. Line $KS$ meets the circumcircle of $\triangle AIX$ again at $T$. Prove that the tangent at $T$ to the circumcircle of $\triangle TBC$ passes through the circumcenter of $\triangle TAO$.
Proposed by Orestis_Lignos
Let $O$ and $\Omega$ denote the circumcenter and circumcircle, respectively, of scalene triangle $\triangle ABC$. Furthermore, let $M$ be the midpoint of side $BC$. The tangent to $\Omega$ at $A$ intersects $BC$ and $OM$ at points $X$ and $Y$, respectively. If the circumcircle of triangle $\triangle OXY$ intersects $\Omega$ at two distinct points $P$ and $Q$, prove that $PQ$ bisects $\overline{AM}$.
Proposed by Andrew Wen
Let $B$ and $C$ be points on a semicircle with diameter $AD$ such that $B$ is closer to $A$ than $C$. Diagonals $AC$ and $BD$ intersect at point $E$. Let $P$ and $Q$ be points such that $\overline{PE} \perp \overline{BD}$ and $\overline{PB} \perp \overline{AD}$, while $\overline{QE} \perp \overline{AC}$ and $\overline{QC} \perp \overline{AD}$. If $BQ$ and $CP$ intersect at point $T$, prove that $\overline{TE} \perp \overline{BC}$.
Proposed by Andrew Wen
Let $H$ be the orthocenter of acute triangle $\triangle ABC$. $X$ and $Y$ are points on the circumcircle of triangle $\triangle ABC$ such that $H$ lies on chord $XY$. Then, let $P$ and $Q$ be the feet of the altitudes from $H$ onto $AX$ and $AY$, respectively, and let line $PQ$ intersect line $XY$ at $T$.
(i) Prove that as the chord $XY$ containing $H$ varies, point $T$ traces out part of a circle $\Omega$.
(ii) Prove that the center of $\Omega$ lies on line $EF$, where $E$ and $F$ are the feet of the altitudes from $B$ and $C$ to $AC$ and $AB$, respectively.
Proposed by William Yue
Let $\triangle ABC$ be a triangle with $A$-excenter $I_A$. Let $X$ and $Y$ be the feet of the perpendiculars from $B$ and $C$ to the angle bisectors of $\angle ACB$ and $\angle ABC$, respectively. If the circumcircles of $I_ABX$ and $I_ACY$ meet again at $P$, show that $\angle BJC = 90^{\circ}$, where $J$ is the incenter of triangle $\triangle PXY$.
Proposed by Andrew Wen
In triangle $ABC$, let $\omega$ be a circle on the opposite side of $\overline{BC}$ as $A$ and tangent to lines $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively. Line $AD$ intersects $\omega$ at $D$ and $K$, such that segment $\overline{DE}$ bisects $\angle AEK$. Let $L$ be the reflection of $E$ over point $K$. Let $P$ be any point on the circumcircle of $\triangle DEL$ such that $AK=AP$. Prove that line $AP$ is tangent to the circumcircle of $\triangle DEL$.
Proposed by Awesome_guy
Let $ABC$ be a right scalene triangle with right angle at $A$. The incircle $\omega$ of $\triangle ABC$ is tangent to sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at points $A_1$, $B_1$, and $C_1$, respectively. Let $A_1$ and $D$ be the intersections of line $AA_1$ and $\omega$, and let $E$ and $F$ be the reflections of $A_1$ over points $B_1$ and $C_1$, respectively. Let $K$ be the foot of the perpendicular from $E$ to $\overline{A_1F}$, and let $L$ be the foot of the perpendicular from $F$ to $\overline{A_1E}$. Prove that lines $B_1C_1$, $KL$, and the line through $D$ perpendicular to $\overline{AA_1}$ are concurrent.
Proposed by Awesome_guy & magicarrow
Let $ABC$ be an acute triangle, and let the feet of the altitudes from $A$, $B$, $C$ to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ be $D$, $E$, $F$, respectively. Points $X\ne F$ and $Y\ne E$ lie on lines $CF$ and $BE$ respectively such that $\angle XAD=\angle DAB$ and $\angle YAD=\angle DAC$. Prove that $X$, $D$, $Y$ are collinear.
Proposed by TheUltimate123
Let $ABC$ be a triangle and let $M$ be the midpoint of $\overline{BC}$. The circumcircle of $\triangle ABM$ intersects $\overline{AC}$ again at $P$, and the circumcircle of $\triangle ACM$ intersects $\overline{AB}$ again at $Q$. Select point $T$ on the circumcircle of $\triangle MPQ$ such that $\overline{MT}\parallel\overline{PQ}$, and let $\omega$ be the circle through $T$ tangent to $\overline{BC}$ at $M$. The circumcircles of $\triangle ABM$ and $\triangle ACM$ intersect $\omega$ again at $X$ and $Y$. Prove that line $XY$ is tangent to the circumcircle of $\triangle ABC$.
Proposed by TheUltimate123
In a $\triangle ABC$, let $K$ be the intersection of the $A$-angle bisector and $\overline{BC}$. Let $H$ be the orthocenter of $\triangle ABC$. If the line through $K$ perpendicular to $\overline{AK}$ meets $\overline{AH}$ at $P$, and the line through $H$ parallel to $\overline{AK}$ meets the $A$-tangent of $(ABC)$ at $Q$, then prove that $\overline{PQ}$ is parallel to the $A$-symmedian
Note: The $A$-symmedian is the reflection of the $A$-median over the $A$-angle bisector).
Proposed by i3435
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