geometry problems from Armenia Republican Mathematical Olympiad from with aops links in the names
collected inside aops here
2009 - 2021
when a problem was proposed in 2 grades,
only the younger is mentioned here
Points $E, F, M, K$ are taken on sides $AB$, $BC$, $CD$, $DA$ of rectangle $ABCD$, respectively. Draw parallels to the sides of $ABCD$ passing through $E,F,M,K$ and create the rectangle $A_1B_1C_1D_1$ as shown in the figure. Find the area of $EFMK$ given that $S_{ABCD}=24$ cm$^2$ and $S_{A_1B_1C_1D_1}=2$ cm$^2$.
Given a cyclic quadrilateral $ABCD$ with $BC = CD$. The points $E , F$ are taken on the line $AC$ , such that $\angle EBA = \angle FDA = 90 ^o$. Prove that $EC = FC$.
Points $E, F, M, K$ are taken on sides $AB$, $BC$, $CD$, $DA$ of rectangle $ABCD$, respectively, such that the area of $EFMK$ is equal to half the area of $ABCD$. Describe all possible cases.
Two congruent equilateral triangles intersect at the points $A,B,C,D,E,F$ (as shown in the figure). Prove that $$AB^2+CD^2+EF^2=BC^2+DE^2+FA^2$$
Points $C_1$, $A_1$, $B_1$ are taken on the sides $AB$, $BC$, $CA$ of triangle $ABC$ respectively, such that the segments $AA_1$, $BB_1$, $CC_1$ pass through point $M$. Is is also known that the triangles $AMC_1$, $BMA_1$ and $CMB_1$ have equal areas. Prove that the triangles $AMB_1$, $BMC_1$ and $CMA_1$ have equal areas.
$SABC$ is a regular pyramid, which is inscribed in the sphere with center $O$. Prove that for the angles $\alpha = \angle AOS$, $\beta = \angle AOB$, holds the inequality $\cos\alpha + \cos \beta \ge - \frac23$.
Two congruent rectangles intersect at the points $A,B,C,D,E,F,P,K$ (as shown in the figure). Prove that
$$AB+CD+EF+PK=BC+DE+FP+KA$$ 
In triangle $ABC$, $AB = 9\, m$, $AC = 19\, m$, $\angle A = 59 ^ o$. What is that triangle, acute , right or obtuse?
From the vertex of the right angle of the right triangle $ABC$ are drawn the median $CM$, the altitude $CH$ and the angle bisector $CE$. It is known that two of the segments $MH$, $ME$ and $HE$ are such that the length of one of them is twice the length of the other. Find the angles of triangle $ABC$.
$M$ is the midpoint of the side $BC$ in quadrilateral $ABCD$. If $S_ {ADM} = \frac12 S_ {ABCD}$ prove that $AB\parallel CD$.
The points $A_1, B_1, C_1$ are taken on the sides $BC$, $CA$, $AB$ of triangle $ABC$ such that $AA_1$, $BB_1$, $CC_1$ intersect at the same point. Prove that $S_ {A_1B_1C_1} \le \frac14 S_ {ABC}$.
Note. Grade XI problem asked for the maximum value of area of triangle $A_1 B_1,C_1$ in terms of area of $ABC$.
Segments $OA,OB,OC$ are drawn in space not lying on the same plane such that $\angle AOB+ \angle AOC=180^o$. Find the angle between $OA$ and the angle bisector of $\angle BOC$.
$ABCA_1B_1C_1$ is a triangular prism. Is it possible that through that prism the segments $AB_1$, $BC_1$, $CA_1$ may be parallel to the same plane? Justify the answer.
Prove that $\vartriangle ABC=\vartriangle A_1B_1C_1$, if it is known that $AB=A_1B_1$, $BC=B_1C_1$ and $\angle A-\angle C= \angle A_1 -\angle C_1 > 0$.
It is known that the point $M$ lies inside the isosceles triangle $ABC$ with base $AC$. Also $\angle MBA=10^o$, $\angle MBC=30^o$ and $BM=AC$. Find the angle $\angle MCA$.
It is known that the point of intersection of the angle bisectors is equidistant from the midpoints of all its sides. Prove that this triangle is equilateral:
In a cyclic quadrilateral $ABCD,$ $AB=AD$. $M$ and $N$ are points on the sides $CD$ and $CB$ respectively such that $DM+BN=MN.$ Prove that the circumcenter of the triangle $AMN$ is on the segment $AC.$
In a triangle $ABC$ $\angle B= 120^o$ and the incircle touches the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $K$ be the reflection of the midpoint of $AC$ with respect to the line $PQ$. Find $\angle AKC.$
Let a hexagone with a diameter $D$ be given and let $d>\frac D 2.$ On each side of the hexagon one constructs a isosceles triangle with two equal sides of length $d$. Prove that the sum of the areas of those isoscele triangles is greater than the area of a rhombus with side lengths $d$ and a diagonal of length $D$.
(The diameter of a polygon is the maximum of the lengths of all its sides and diagonals.)
A quadrilateral $ABCD$ is such that $\angle A= \angle C=60^o$ and $\angle B=100^o$. Let $O_1$ and $O_2$ be the centers of the incircles of triangles $ABD$ and $CBD$ respectively. Find the angle between the lines $AO_2$ and $CO_1$.
On the sides $AB $ and $BC$ of an abtuse triangle $ABC$ $(\angle B> {{90} ^{0}})$ are taken points $M$ and $N$ respectively . Prove that $AN + CM> AM + MN + NC$
The interior points $M,N,P,Q$ are taken on the sides $AB$,$BC$,$CD$,$DA$ of the rectangle $ABCD$, respectively. It is known that the four circles circumscribed around triangles $AMQ$,$BMN$, $CNP$ and $DQP$ have a common point. Prove that the sum of the squares of two radii of these four circles is equal to the sum of the squares of the other two radii.
On the small arcs $AC $ , $BC $ of the circumscribed circle of the triangle $ABC$ are given the points $M$ , $N $respectively such that $\angle MAI = \angle NBI $, where $I$ is the center of the circle inscribed in the triangle $ABC $. Prove that the lines $CI$ and $MN$ are perpendicular
Circle ${{\omega} _ {1}}$ passing through vertices $B$ and $C$ of trapezoid $ABCD$ ($BC\parallel AD$) is tangent to line $AB$. Circle ${{\omega} _ {2}}$ passing through points $A$ and $D$ is tangent to line $CD$. Circle ${{\omega} _ {3}}$, passing through $A$ and $B$ intersects the circles ${{\omega} _{1}}$ and ${{\omega} _ {2}} $ for the second time at the points $P$ and $Q$ respectively. Prove that the points $C, P, Q, D $ are on the same circle.
$I$ is the center of the circle inscribed in an not isosceles triangle $ABC$, and $\omega$ is the circle circumscribed around that triangle. It is known that a circle with diameter $AI$ and the ray $AI $ intersect circle $\omega$ at points ${{A} _ {1}}$ and ${{A} _ {2}}$ (different from $A$) respectively . Similarly are defined ${{B} _ {1}}, {{B} _ {2}} $ and ${{C} _ {1}}, {{C} _ {2}} $ . Prove that lines ${{A} _ {1}} {{A} _ {2}}$, ${{B} _ {1}} {{B} _ {2}} $ and ${{C } _ {1}} {{C} _ {2}} $ intersect at a point.
The following conditions are true for a convex octagon $ABCDEFGH $: $AB\parallel EF$, $BC\parallel GF$, $AH\parallel DE$, $\angle AHG = \angle BCD$, $AB = EF$, $BC-GF = AH-DE$. Prove that $CD = GH$
From a point $M$ of the altitude $BB_1$ of the triangle $ABC$ are drawn to the sides $AB$, $BC$, the perpendiculars $ME$ and $MF$ respectively . It is known that the circumscribed circle of the triangle $EF {{B} _ {1}}$ is tangent to the line $AC$. Prove that $AB = BC$.
The sides of two given parallelograms intersect at eight points. Prove that the area of this constructed $16$-gon is greater than half the area of any of those parallelograms.
The circle inscribed in triangle $ABC$ with right angle $B$, touches sides $AB$, $BC$ and $AC$ at points $P$, $Q$ and $R$, respectively. The line passing point $Q$ perpendicular on PR intersects the line $AB$ at point $X$ . The line passing point $P$ perpendicular on $QR$ intersects the line $BC$ at point $Y$ . Prove that $AX = CY$
Points $P$ and $Q$ are taken on different sides of the angle with vertex $S$. such that the segments $SP$ and $SQ$ are not equal. Through the midpoint $S$ of the segment $PQ$ is drawn a perpendixular to the angle bisector of the angle $S$ intersecting the line $SP$ at point $T$. Prove that the line perpendicular to $SP$ passing through point $T$ and he perpendicular bisector of $PQ$ passing through point $M$, intersect at the angle bisector of the angle $S$.
The altitudes $AD$ and $CE$ are drawn in the triangle $ABC$. Let $M, N$ be the the feet of the perpendiculars drawn from the points $A, C$ on the line $DE$ respectively. Prove that $ME = DN$.
Let $P$ be an arbitrary point on the side $AC$ of triangle $ABC$. The points $M , N$ are marked on the sides $AB , BC$ such that $AM = AP$ and $CN = CP$. The perpendicular on sides $AB , BC$ at the points $M , N$ respectively intersect at point $Q$. Prove that $\angle QIB = 90^o$, where $I$ is the center of the circle inscribed in the triangle $ABC$.
In triangle $ABC$, $AC$ is the side with the smallest length. On the sides $AB$ , $BC$ are taken respectively points $K , L$ , so that $KA = AC = CL$. Let M be the intersection point of the segments $AL$ and $KC$ . Let $I$ be the center of the inscribed circle of the triangle $ABC$. Prove that $MI$ is perpendicular to $AC$.
From point $A$ outside a circle, the tangents $AB , AC$ are drawn to that circle ($B , C$ are the touchpoints). Let $DC$ be a diameter of the circle. Let $H$ be the feet of the perpendicular drawn from $B$ on $CD$. Prove that the line $AD$ bisects the segment $BH$.
$AB$ and $CD$ are chords intersecting inside the circle. Take a point $M$ on segment $AB$ , such that $AM = AC$, and take a point $N$ on the segment $CD$ such that that $DN = DB$. Prove that if the points $M , N$ do not coincide, then $MN$ is parallel to $AD$.
The inscribed circle of the tangential quadrilateral $ABCD$, with center $O$, touches its non-parallel sides $BC$, $AD$ at points $E , F$ respectively. Lines $AO$ and $EF$ intersect at point $K$. Lines $DO$ and $EF$ intersect at point $N$. Lines $BK$ and $CN$ intersect at point $M$. Prove that the points $O, K, M$ and $N$ are on the same circle.
Let $a, b$ and $c$ be the sidelengths of a triangle and $S$ be the area of that triangle. Prove that $ab+bc+ac \ge 4\sqrt3 S$.
The circle inscribed in triangle $ABC$ touches the sides $BC$ , $CA$ , $AB$ at points $K$, $L$ ,$M$ respectively. $Q$ is the other point of intersection of the incircle with $AK$. The line passing through point $A$ parallel to $BC$ intersects $KL$ and $KM$ at points $R$ and $P$ respectively. Prove that $\angle PQR = \angle MQL$.
It is known for the convex quadrilateral $ABCD$ that $\angle C = 70 ^ \circ$ , and the bisectors of angles of $ A$ and $ B$ intersect at the midpoint of the side $CD$. Find all possible values of the angle $\angle D$ .
In triangle $ABC$, $\angle ABC =120^\circ$ and $BC = 2AB$. Calculate $\angle ABF$ where $F$ is the midpoint of the side $AC$ .
Given a right triangle $ABC$ ( $\angle C = 90 ^ \circ$). The altitude $CH$ intersect angle bisectors $AK$ and $BL$ at the points $ P$ and $Q$, respectively. Let $F$ and $E$ be the midpoints of the segments $PK$ and $QL$ respectively. Prove that $EF$ is parallel to $AB$.
In triangle $ABC$, $\angle AMB = 60^\circ$ and $BC = 2AB$, where $M$ is the centroid of the triangle $ABC$ . Calculate $\angle ABC$
Given the quadrilateral $ABCD$ with $\angle CAB = 72 ^ \circ $, $\angle CAD = \angle BDC =36^ \circ$, $\angle DBC =18^\circ$. Prove that $ AB = AD $.
The circles ${{\omega} _ {1}}$ and ${{\omega} _ {2}} $ intersect at the points $A$ and $B$. The circle with diameter of $AP$ and $AQ $ intersect ${{\omega} _ {2}} $ at the point $P '$ and the circle with diameter of $AQ$ intersects ${{\omega } _ {1}}$ at point $Q '$. Prove that the point of intersection of the circles circumscribed around the triangles $PP'B $ and $QQ'B $ is on the line $AB $.
Let the altitudes $A {{A} _ {1}}$ and $C {{C} _ {1}} $ of the acute triangle $ABC$ (${{A} _ {1 }} \in BC, {{C} _ {1}} \in AB $) intersect at the point $H$. The segments $BH$ and ${{A} _ {1}} {{C} _ {1}} $ intersect at the point $D $ . Let $P$ be is the midpoint of the segment $BH$. Prove that the symmetric point of $D$ wrt line $AC$ lies on the circle circumscribed around the triangle $APC$.
Circles $\Omega$ and $\omega$ are tangent at point $P$ and $\omega $ is inside $\Omega$. The chord $AB$ of the circle $\Omega$ is tangent to the circle $\omega$ at the point $C $, and the line $PC$ intersects $\Omega$ for the second time at point $Q$. The chords $QR$ and $QS$ of the circle $\Omega$ are tangent to $\omega$. Let $I,X$ and $Y$ be the centers of the circles inscribed in the triangles $APB$, $ARB$ and $ASB$, respectively. Prove that $\angle PXI + \angle PYI = 90 ^\circ $.
Let $\omega_1$ and $\omega_2$ be two circles intersecting at points $A,B$, and $EF$ be their common tangent ($E \in \omega_1$ , $F \in \omega_2$, and $A$ is closer to $EF$ than $ B$). Let the perpendicular from point $A$ on the line $EB$ and the line $AB$ intersects the circumcircle of the triangle $EFB$ at points $P,Q$ respectively ($P,Q$ lie on the same side of the plane wrt $EF$). Prove that $EQ=PF$.
The angle bisector $BD$ of the triangle $ABC$ ($AB <BC$) and it's circumscribed circle $\omega$ intersect at point $E$. The perpendicular from point $A$ on $BE$ intersects the segments $BE$, $BC$ and the circle $\omega$ at points $L, M$ and $P$ respectively. Segments $EP$ and $BC$ intersect at point $T$. Prove that $LT = \frac12 AC$.
In the quadrilateral $ABCD$,$ \angle ABC = 30^o$, $\angle BAC = 90^o$,$\angle ACD = 85^o$, $\angle ADC = 20^o$. Prove that $AB + BC> CD$.
In the triangle $ABC$, select the point $M$ such that $\angle MAB = 20^o$, $\angle MBA = 40^o$, $\angle MBC = 70^o$, $\angle MCB = 30^o$. Find $\angle MAC$
The circle with center $O$ inscribed in the triangle $ABC$ intersects the sides $BC$ and $AC$ at points $M$ and $N$, respectively. Let $E$ and $F$ be the midpoints of the sides $AB$ and $AC$, respectively. The lines $BO$ and $EF$ intersect at point $D$. Prove that the points $M$, $N$ and $D$ lie on one line.
The angle bisector $BD$ of the triangle $ABC$ ($AB <BC$) and it's circumscribed circle intersect at point $E$. The point $M$ lies on the segment $BC$ is chosen so that $EC = EM$. Prove that the lines $AB$, $MD$ and $EC$ intersect at one point if and only if $\angle BAC = 90^o$.
From the point $P$ outside the circle, the tangents $PA$ and $PB$ of the circle are drawn, as well as the secant $PD$, which intersects the circle also at the point $C$ ($D$ lies on the circle, $C$ les on the segment $PD$). The line parallel to $PA$ passing through point $B$ intersects the lines $AC$ and $AD$ at points $E$ and $F$, respectively. Prove that $BE = BF$.
Points $E, F, K$ are taken on the semicircle with diameter $AB$ (points $A, E, F, K, B$ are in the specified order), and points $C, D$ on the diameter $AB$ ($C$ is on the segment $AD$) such that $\angle ACE = \angle BCF$ and $\angle
CDF = \angle BDK$. Prove that $\angle AEC + \angle BKD = 90^o$.
In quadrilateral $ABCD$ , $AB\parallel CD$, $\angle DBC = 10^o$, $\angle BCD=130^o$ and $AB = AD$. Find the measure of the angle $\angle CAD$ .
Let $BH$ be the altitude of the triangle $ABC$ and $O$ be the center of the circumscribed circle . Let the points $ P$ and $Q$ be the symmetric of $H$ wrt the sides $BA$ and $BC$, respectively. Let $M$ be the intersection point of the lines $BO$ and $AC$. Prove that $\angle APM = \angle CQM$.
In quadrilateral $ABCD$ , $AB\parallel CD$, $\angle DBC = 10^o$, $\angle BCD=130^o$ and $AB = AD$. Prove that $BO = AO + OD$, where $O$ is the intersection point of the diagonals of $ABCD$.
The altitudes $CF$ and $AE$ of the triangle ABC intersect at point $H$. Let $P$ and $Q$ be the symmetric points of point $ B$ wrt the points $F$ and $E$ (points $P , Q$ lie on the segments $AB , BC$), and $K$ is the symmetric point of $ B$ wrt line $AC$. Prove that the points $K, P, H, Q$ are on a circle.
Let $ABCD$ be a quadrilateral inscribed in a circle with $BC> AD$ and $CD> AB$. The points $E , F$ are marked on the sides $BC$ and $CD$ of the quadrilateral such that $BE = AD$ and $DF = AB$. Prove that $BM \perp DM$, where $M$ is the midpoint of the segment $EF$.
Let the altitudes of the acute triangle $ABC$ intersect at point $H$. Point $D$ is marked such that $HABD$ is a parallelogram. The point $E$ is marked on the line $DH$ so that the line $AC$ passes through the midpoint of the segment $EH$. Let $F$ be the second intersection point of the circumcircle of the triangle $DCE$ with line $AC$ (the first point is $C$). Prove that $EF = AH$.
In right triangle $ABC$ ($\angle C=90^o$) , take a point $K$ on side $AC$ such that $CK=AB-AC$. Prove that $\angle A = 2\angle CBK$ .
$CD$ is an altitude of right triangle $ABC$ ($\angle C=90^o$) . The circle with center $D$ and radius $CD$ intersects the line $AB$ at points $E$ and $F$ ($A$ is between the points $E$ and $F$ ) and intersects the segment $BC$ at point $Q$. Let the segments $EQ$ and $AC$ intersect at point $P$. Prove that $EP = QF$.
The point $P$ is taken outside the isosceles right triangle $ABC$ ($\angle B=90^o$) such that $\angle PAC = 15^o$ and $\angle PCA = 30^o$. Find the angles $\angle APB$ and $\angle BPC$.
The tangents at points $A$ and $C$ of the circumscribed circle of triangle $ABC$ intersect at point $K$. The perpendicular bisector of side $AB$ intersects side $BC$ at point $Q$, and the perpendicular bisector of side $BC$ intersects side $AB$ at point $P$. Let the segments $BK$ and $PQ$ intersect at point $M$. Prove that $PM=QM$.
Let $ABC$ be a triangle right at $C$. Point $K$ lies on the side $AC$ and point $E$ lies on the side $CB$ such that $CK = AB - AC$ and $CE = AB - BC$. Prove that $\angle CBK + \angle CAE = 45^o$.
Segments $AA_1$, $BB_1$, $CC_1$ are the medians of the triangle $ABC$. A circle with diameter $AA_1$ intersects the circumscribed circle of the triangle $ABC$ at point $A_2$. The points $B_2$,$C_2$ are defined similarly. Prove that lines $A_1A_2$, $B_1B_2$, $C_1C_2$ intersect at one point.
Suppose that the incircle of the triangle $ABC$ has center $I$ and is tangent to the sides $AB,BC$ at points $E,F$, respectively. Line $CI$ intersects the circumcircle of the triangle $ABC$ at point $P$. Lines $EF$ and $CP$ intersect at point $T$. It is known that $PT=TI$.. Find $\angle ABC$.
Let $E$ be point on the side $BC$ of the triangle $ABC$ such that $AC=EC$. On a line parallel to the line $AE$ passing through point $C$ , take a point $F$ such that $\angle BAE= \angle EAF$. Prove that the line $EF$ intersects the segment $AB$ at it's midpoint.
Suppose the circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A line passing through point $B$ intersects the circle $\omega_1$ at point C and the circle $\omega_2$ at point $D$ (point $B$ lies between points $C$ and $D$). The line $AD$ intersects $\omega_1$ at point $E$ and the line $AC$ intersects $\omega_2$ at point $F$ (point $A$ lies between points $E, D$ and $C, F$). Let $O$ be the center of the circumscribed circle of the triangle $AEF$. Prove that $OB \perp CD$.
Suppose $I$ is the center of the circle inscribed in the isosceles triangle $ABC$ ($AB=BC$).
a) If $AC+AI=BC$, find the angles of triangle $ABC$.
b) If $\angle ABC=36^o$. prove that $AC+AI=BC$.
Note. Grade 8 had (a), grade 9 had (b)
Let $O$ be the center of the circumscribed circle of $ABC$ ($AB<BC$) and $BD$ be the altitude. Let $DR$ be the altitude of triangle $BDC$ . Let the lines $BO$ and $DR$ intersect at point $ P$. Prove that $DM=MP$, where $M$ is the midpoint of side $AC$ .
Point $E $is marked on the side $AB$ of the parallelogram $ABCD$. The points $F$ and $G$ are the centers of the circles circumscribed around the triangles $BCE$ and $ADE$ respectively. Prove that the length of the segment $FG$ does not depend on the choice of point $E$.
Suppose the circles $\omega_1$ , $\omega_2$ intersect at points $B,C$, with $BC$ diameter of the circle $\omega_1$. The tangent drawn on the circle $\omega_1$ at point $C$ intersects the circle $\omega_2$ at point $A$. Segment $AB$ intersects with circle $\omega_1$ at point $E$, the segment $CE$ intersects the circle $\omega_2$ at point $F$. The line passing through point $H$, taken on the segment $AF$, and $E$ intersects circle $\omega_1$ at point $G$. The lines $BG$,$AC$ intersect at point $D$. Prove that $\frac{HF}{AH}=\frac{CD}{AC}$.
Suppose $X$ is an arbitrary point on the base $AB$ of the equilateral triangle $ABC$. Let point $E$ be such that $ACEX$ is a parallelogram and the point $F$ lie on the ray $XE$ such that $BE=EF$. The line $BE$ intersects the circle circumscribed around the triangle $ABF$ at point $T$. Prove that $BT=2 BC$.
Draw a line containing the angle bisector $BD$ of the right triangle $ABC$, with right angle at C . The tangent drawn at point $A$ to the circle circumcscribed of the triangle $ABC$ intersects that line at point $E$, and $F$ is the symmetric point of $E$ wrt point $A$. Let $P$ be the intersection point of the lines $FD$ and $AB$. Prove that the lines $EP$ and $BF$ are perpendicular.
On the sides $AB$ , $DC$ of the square $ABCD$ are taken points $E , N$, respectively and on the side $BC$ are taken points $M , F$ such that the triangles $AMN$ and $DEF$ are equilateral. Let $P$ be the intersection point of the segments $AN$ and $DE$ . Let Q be the intersection point of the segments $AM$ and $FE$. Prove that $PQ=FM$.
Let$ I$ be the center of the circle inscribed in the acute triangle $ABC$. The symmetric point of $I$ wrt side $AC$ lies on the circumcscribed circle of the triangle $ABC$. Prove that $\angle IHB=\frac32 \angle BAC$, where $H$ is the intersection point of the altitudes of triangle $ABC$.
In triangle $ABC$, $\angle ABC=30^o$, $\angle ACB=15^o$. Let points $M , N$ lie on side $BC$ such that $BM=MC$ and $CN=AB$. Prove that $\angle MAN=\angle CAN$
Let $P$ be the intersection point of the external angle bisector of $A$ and the circumscribed circle of triangle $ABC$ ($AB>BC$). Suppose a circle passing through points $A$ and $P$ intersects the segments $BP$ and $CP$ at points $E$ and $F$, respectively. Suppose $AD$ is the angle bisector of the triangle $ABC$. Prove that $\angle PED = \angle PFD$.
The symmetric of the intersection of the medians of the triangle $ABC$ wrt side $BC$ lies on the circle circumscribed around the triangle $ABC$. Prove that $\frac{AB + AC}{2} \le BC$. Find out when the equality occurs.
Suppose $E$ is any point on the side $CD$ of the right trapezoid $ABCD$ ($\angle A = \angle B = 90^o$). Let the circumscribed circle $\omega$ of the triangle $ABE$ touch the sides $AB, AE, BE$ at points $P, F, K$ respectively. Let the line $KF$ intersects the segments $BC$ and $AD$ at the points $M$ and $N$, respectively. Let $PM$ and $PN$ intersect $\omega$ at the points $H$ and $T$, respectively. Prove that $PH = PT$.
The $C$-excircle and $B$-excircle of triangle $ABC$ ($AB \ne AC$) touch the sides of triangle $AB$ and $AC$ at points $F$ and $E$, respectively. Let the lines $BE$ and $CF$ intersect at point $N$. It is known that the symmetric of the point $N$ wrt line $BC$ lies on the circle circumscribed around the triangle $ABC$. Prove that $AB + AC = 2BC$.
Let the extension of angle bisector $BL$ of the triangle $ABC$ intersect the circumscribed circle of the triangle $ABC$ at point $P$. Let $\ell$ be a line parallel to the line $AB$ passing through point $P$. Let the circle passing through points $A,P$ tangent to $\ell$ intersects line $AB$ at point $Q$, which is different from $A$. It is known that the points $Q,P,C$ lie on one line. Prove that $QL \perp BC$.
Let the diagonals of parallelogram $ABCD$ intersect at point $O$, and the perpendicular bisector of the side $CD$ intersects the line $AB$ at point $M$, with point $B$ lying between points $A$ and $M$. Let line $MO$ intersect the line $AD$ at point $E$. Prove that $AE = EM$.
Let $P$ be the midpoint of the arc $AC$, that does not contain the vertex $B$, of the circle circumscribed around triangle $ABC$. Suppose $\ell$ is a line parallel to the line $AB$ passing through the point $P$. Let the circle passing through point $A$ and point $P$, tangent to $\ell$, intersect the line $AB$ at point $Q$, different than point $A$. Prove that $BQ = BC$.
Let the diagonals of parallelogram $ABCD$ intersect at point $O$, and the perpendicular bisector of the side $CD$ intersects the line $AB$ at point $M$, with point $ B$ lying between points $A$ and $M$. Let line $MO$ intersect the line $AD$ at point $E$. Let the circumscribed circle of the triangle $BME$ intersect $BC$ at point $T$. Prove that $ME = BT$.
Let $P$ be the midpoint of the arc $AC$, that does not contain the vertex $B$, of the circle circumscribed around triangle $ABC$. Suppose $\ell_1$ is a line parallel to the line $AB$ passing through the point $P$, and $\ell_2$ is a line parallel to $BC$ passing through the point $P$. Let the circle passing through point $A$ and point $P$, tangent to $\ell_1$, intersect the line $AB$ at point $Q$, different than point $A$. Let a circle passing through point $C$ and point $P$, tangent to $\ell_2$, intersect the line $BC$ at point $K$, different from point $C$.
a) Prove that $AK \parallel QC$
b) Prove that the points $A, Q, C$ and $K$ lie on a circle.
Note: Grade 10 had (a), grades 11-12 had (b)
Let $H$ be the point of intersection of the altitudes of the triangle $ABC$. Let the circle passing through points $A, B, H$ have center $R$ and intersect the segment $BC$ at the point $D$ ($D\ne B$). Let $P$ be the intersection point of the line $DH$ with segment $AC$. Let $Q$ be the center of the circumscribed circle of the triangle $ADP$. Prove that the points $B, D, Q, R$ lie on a circle.
Let $AD$, $BF$, $CE$ be the altitudes of the triangle $ABC$, $M$ be the midpoint of the side $BC$, $H$ be the point of intersection of the altitudes, $S$ be the midpoint of the segment $AH$, $G$ be the intersection point of the segments $FE$ and $AH$, and $N$ be the intersection point of line $AM$ with the circumscribed circle of triangle $BCH$ . Prove $\angle HMA = \angle GNS$.
Suppose the points $Q ,T$ lie on the side $AB$ and the points $P , M$ lie on the side $BC$ of the right triangle $ABC$ ($\angle ABC=90^o$) such that $\angle BAP =\angle PAM = \angle MAC$ and $\angle BCQ = \angle QCT = \angle TCA$. Let the lines $CT$ and $AM$ intersect at point $R$. Find $\angle BAC$ if it is known that $\angle QRC = 142^o$
The points $X.Y$ are taken on the sides $AD , BC$ of the square $ABCD$ respectively such that $AY = BX$. Let $DZ$ be the altitude of triangle $DAY$. Prove that the measure of the angle $XZC$ does not depend on the position of the points $X, Y.$
Suppose $H$ is the foot of the altitude $BH$ of the triangle $ABC$ and $O$ is the center of it's circumcscribed circle. Let a line parallel to the line $OC$ passing through point $H$ intersect the line $BO$ at point $P$. Prove that the midpoints of the segments $AB$, $AC$ and point $P$ lie on one line.
Let $ABCD$ be a parallelogram with $\angle ABC = 40^o$.Let the angle bisector of $ADC$ intersect the segment $AB$ at the point $H$. Let point $K$ lie on segment $DC$, such that $HK \parallel AD$. Let $E$ be the midpoint of the segment $HK$ , and $F$ be the foot of the perpendicular from point $A$ on the segment $DE$. Find the angle $\angle
DFK$.
In the acute triangle $ABC$, $I$ is the center of the inscribed circle. The ray $BI$ intersects intersects the circumscribed circle $\omega$ at point $M$. A circle passing through points $M$ and $I$ intersects the segment $AC$ at points $E$ and $F$. Let the lines $ME$ and $MF$ intersect $\omega$ at the points $P$ and $Q$, respectively. Prove that the points $P, I, Q$ lie on a line.
Let $AH$ , $AM$ be the altitude and the median , respectively, of the right-angled triangle $ABC$, with right angle at $A$. Suppose sides $AB , AC$ are bases of the isosceles triangles $ABP$ , $ACQ$ so that those triangles have no common interior points with triangle $ABC$ and $\angle APB = \angle AQC$. Let the line $MA$ intersect the segment $PQ$ at point $N$. Prove that $\angle NHP =\angle AHQ$.
Suppose the altitudes $BE$ and $CF$ of the triangle $ABC$ intersect at point $H$ and $M$ is the midpoints of the side $BC$. Let $D$ be the foot ot the perpendicular drawn from point $A$ on line $MH$. Prove that the bisectors of the angles $\angle DBH$ , $\angle DCH$ and the line $MH$ intersect at one point.
source: www.olymp.am/
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