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Maths Beyond Limits 2017-22 (Poland) 47p

geometry problems from Maths Beyond Limits, an international mathematical camp fot talented youth in Poland,  with aops links in the names

it started in 2016
collected inside aops here

Qualifying Quiz 2017-22

Let $\omega$ be the incircle of $\vartriangle ABC$. Denote by $I$ the center of $\omega$. The circle with radius $AI$ and center $A$ intersects circumcircle of $\vartriangle ABC$ at points $P$ and $Q$. Prove that line $PQ$ is tangent to $\omega$.

Denote by $\omega$ the incircle of $\vartriangle ABC$. Suppose $\omega$ touches the sides $BC, AC, AB$ at the points $D, E, F$ respectively. Let $I$ be the center of $\omega$, the segment $BI$ intersects $\omega$ at point $M$. The line tangent to $\omega$ at the point $M$ intersects the line $AC$ at the point $R$. Let the incircle of $\vartriangle DEF$ touches the segments $EF, FD$ at points $D', E'$ respectively. The line going through $A$ and perpendicular to $ID'$ intersects line going through $B$ and perpendicular to $IE'$ at the point $T$. Let $J$ be the $A$-excenter of $\vartriangle ABC$. Prove that the points $R, T, J$ are collinear.

Let $H_A, H_B, H_C$ be feet of altitudes from vertices $A, B, C$ of triangle $ABC$, respectively. Line parallel to $CA$ passing through $B$ intersects line $H_BH_C$ at point $X$. Point $M$ is the middle of segment $AB$. Show that $\angle ACM = \angle XH_AB$.

Let $P$ be a point in the plane of triangle $ABC$ such that $\frac{AP}{BC} =\frac{ BP}{CA} = \frac{CP}{AB}$. Prove that $P$ lies on the Euler line of triangle $ABC$.

Let $ABC$ be a triangle such that $\angle ACB = 30^o$. Let $K$ and L be points on arcs $BC$ and $CA$ (whose interiors do not contain vertices of $ABC$) of the circumcircle of $ABC$, respectively, such that $CK = CL = AB$. enote by $H, P, Q$ the orthocenters of triangles $ABC, BKC, CLA$, respectively. Prove that $\angle PHQ = 90^o$

Point $P$ lies in the interior of triangle $ABC$. Lines $AP, BP, CP$ intersect its sides $BC, CA, AB$ at points $D, E, F$, respectively. Point $H$ is the foot of the altitude of $ABC$ from vertex $A$. Prove that point $A$ and the reflections of $H$ in lines $DE, DF$ are collinear.

Let $ABC$ be an acute triangle such that $\angle ACB = 60^{\circ}$. Point $M$ is the middle of shorter arc $AB $ of the circumcircle of $ABC$. Point $H$ is the orthocenter of $ABC$. Line $MH$ intersects $AB$ at $S$. Points $P$, $Q$ lie on $BC$, $CA$, respectively, in such a way that $H$ lies on $PQ$ and $PQ$ is perpendicular to $MH$. Prove that the circumcircle of $PQS$ passes through the foot of altitude of $ABC$ from $C$.

Radek challenges you to a duel. He draws a triangle $ABC$ with a [i]small circle[/i] inside. Afterwards he marks the incenter of $ABC$ and connects it with the vertices of $ABC$. This results in a split of $ABC$ into three triangles. Now, in each move you may choose one of triangles, mark its incenter and connect it with vertices of chosen triangle (which would result in exchanging chosen triangle with three new ones). Your task is to, after a finite number of moves, mark some incenter inside the [i]small circle[/i]. Can you always do that, regardless of what triangle and circle was drawn by Radek?

A triangle $ABC$ is given with $\angle BAC = 60^o$. Let $E$ and $F$ be feet of angle bisectors from $B$ and $C$ respectively, and $I$ be the intersection of $BE$ and $CF$. $M$ and $N$ are midpoints of $AE$ and $AF$, while $P$ and $Q$ are midpoints of $IE$ and $IF$ respectively. Prove that $I$ lies on a line through circumcenters of triangles $CMQ$ and $BPN$.

In a convex $n$-gon $A_1A_2 . . . A_n$ equalities
$A_1A_3 = A_2A_4 = . . . = A_{n-1}A_1 = A_nA_2$ and$ A_1A_4 = A_2A_5 = . . . = A_{n-1}A_2 = A_nA_3$
hold.
For which n do they imply that this n-gon is regular? Investigate this question in cases:
a) $n = 6$,
b) $n = 7$,
c) $n = 11$.

$ABC$ is an acute triangle with $AB < AC$. Let $O$ be its circumcenter, $D$ be the foot of the altitude from $A$, $M$ be the midpoint of $BC$ and $S$ be the projection of $B$ onto $AO$. Prove that the circumcircle of the triangle $DMS$ is tangent to the circle with diameter $AC$.

Triangle $ABC$ satisfies the property that $\angle ABC = 2 \angle ACB < 60^o$. $X$ and $Y$ are the intersections of the perpendicular bisector of $BC$ with a circle centered at $A$ with radius $AB$. $X$ is closer to $BC$ than $Y$ . Show that $3\angle XAC = \angle BAC$.


Matches 2017 - 18


Let $ABCD$ be a convex quadrilateral satisfying $AC = BD = AB$. Let $M$ and $N$ be the midpoints of the segments $AD$ and $BC$ respectively. Denote by $T$ the common point of the diagonals. Prove that the line $MN$ passes through the touching points of the incircle of $\vartriangle ATB$ with the sides $AT$ and $TB$.

Equilateral triangle $ABC$ is given. Let $o$ be the circumcircle and $\omega$ be the incircle of $\vartriangle ABC$ with common center $O$. Let P and Q be the points lying on the sides $AC$ and $AB$ respectively, such that $O$ lies on $PQ$. Let $\gamma_b$ and $\gamma_c$ be the circles with diameters $BP$ and $CQ$ respectively. Show that one of the intersection points of $\gamma_b$ and $\gamma_c$ lies on $o$ and the other lies on $\omega$.

Let $H$ be the ortocenter of $\vartriangle ABC$. The point $P$ lies on circumcircle of $\vartriangle ABC$ and the point $M$ is the midpoint of the side $BC$. Let $T$ be the projection of $H$ onto line $AP$. Prove that $MT = MP$.

Given is an acute $\vartriangle ABC$. Choose $T$ - an arbitrary point at the side $AB$. Let $N$ be the midpoint of the segment $AC$. The foots of the perpendiculars from the point $A$ to the segments $TC$ and $TN$ are the points $P$ and $Q$ respectively. Prove that center of the circumcircle of $\vartriangle PQN$ lies on a fixed line for all the points $T$ from the side $AB$.

Let $\omega_1, \omega_2$ be the intersecting circles with centers $O_1, O_2$ respectively. Point $A$ lies on $\omega_1$, point $B$ lies on $\omega_2$ and $AB$ is an ex-tangent line of $\omega_1$ and $\omega_2$. Suppose $C$ is one of the intersection points of $\omega_1$ and $\omega_2$ nearer to $AB$. Let $D$ be the point laying on $O_1O_2$ such that the line $BD$ is perpendicular to $AC$. Prove that $\angle BCD = 90^o$.

Let $ABC$ be a triangle with incenter $I$. Let $D$ be a point on side BC and let $\omega_B$ and $\omega_C$ be the incircles of $\vartriangle ABD$ and $\vartriangle  ACD$ respectively. Suppose that $\omega_B$ and $\omega_C$ are tangent to segment $BC $at points $E$ and $F$ respectively. Let $P$ be the intersection of segment $AD$ with the line joining the centers of $\omega_B$ and $\omega_C$. Let $X$ be the intersection point of the lines $BI$ and $CP$ and let $Y$ be the intersection point of the lines $CI$ and $BP$. Prove that lines $EX$ and $FY$ meet on the incircle of $\vartriangle ABC$.

Let $H$ be the orthocenter of $\vartriangle ABC$. The lines $\ell_1$ and $\ell_2$ pass through $H$ and $\ell_1 \perp \ell_2$. The line $\ell_1$ intersects the lines $AB$ and $BC$ at the points $K$ and $P$ respectively. The line $\ell_2$ intersects the lines $AC$ and $BC$ at the points $L$ and $Q$ respectively. The line passing through the point $Q$ and parallel to the line $AB$ intersects the line passing through $P$ and parallel to the line $AC$ at the point $T$. Prove that the points $K, T$ and $L$ are collinear

Let $ABC$ be an acute triangle with the orthocenter $H$. On the segments $HA, HB$ and $HC$, we choose three points $A_1, B_1$ and $C_1$ respectively such that $\angle BA_1C = \angle  CB_1A = \angle AC_1B = 90^o$. Denote by $A_2 = BC_1 \cap CB_1$, $B_2 = CA_1 \cap  AC_1$ and $C_2 = AB_1 \cap  BA_1$.
a) Prove that the sides of the hexagon $A_1B_2C_1A_2B_1C_2$ are tangent to the one circle.
b) Let $J$ be a incenter of the hexagon $A_1B_2C_1A_2B_1C_2$. Prove that the lines $A_1A_2, B_1B_2, C_1C_2$ and $HJ$ are concurrent.

Given a $\vartriangle ABC$ has incenter at the point $I$. A line passes through $I$ and intersects circumcircles of $\vartriangle IBC$, $\vartriangle  ICA$ and $\vartriangle IAB$ again at the points $D, E$ and $F$ respectively. Prove that the Euler lines of $\vartriangle  DBC$, $\vartriangle ECA$ and $\vartriangle FAB$ are either concurrent or parallel.


Let $I$ be the incenter of a triangle $ABC, M$ - the midpoint of the side $AB$ and $W$ - the midpoint of the arc $BC$ of the circumcircle of $ABC$ not containing $A$. It is known that $\angle BIM =90^o$. Find ratio $AI:IW$

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. Let $K$ be the midpoint of the segment $AH$. The line perpendicular to $BK$ passing through $K$ intersects the line $AC$ at $P$. Prove that $OP \parallel BC$.

Let $M$ be the midpoint of the hypotenuse $BC$ of a right triangle ABC. Let $\omega_1$ be the circle passing through $B, M$ and touching $AC$ at $X$. Similarly, let $\omega_2$ be the circle passing through $C, M$ and touching $AB$ at $Y$ , where $X$and $Y$ lie on the same side of the line $BC$. Prove that the line $XY$ passes through the midpoint of arc $BC$ of the circumcircle of $ABC$.

Let $X$ be a point inside a triangle ABC such that $XA \cdot BC =XB\cdot AC =XC \cdot AB$. Let $I_1, I_2, I_3$ be the incenters of the triangles $XBC, XCA, XAB$, respectively. Prove that the lines $AI_1, BI_2$ and $CI_3$ are concurrent.

Let $ABC$ be a triangle with $AB =AC$. Let $P, Q$ be points inside the triangle such that $\angle BAP+ \angle CAQ =1/2 \angle BAC$.  Moreover, it is known that $BP=PQ=CQ$. Let $AP$ intersect $BQ$ at  $X$ and $AQ$ intersect $CP$ at $Y$ . Prove that the quadrilateral $PQYX$ is cyclic.

Let the incircle of a triangle $ABC$ with $AB <AC$ be tangent to the sides $BC, CA, AB$ at points $D, E, F$, respectively. Let $A$ and $T$ be the intersection points of the circumcircles of triangles $AEF$ and $ABC$. The line perpendicular to the line $EF$ passing through $D$ meets side $AB$ at $X$. Prove that $TX \perp TF$.

Let $M$ be the midpoint of the side $BC$ of a triangle $ABC$. Points $E, F$ lie on the sides $AB, AC$, respectively, in such a way that $ME=MF$. Let the circumcircles of the triangles $ABC$ and $AEF$ intersect at $A$ and $P$. The tangents at $E, F$ to the circumcircle of $AEF$ intersect each other at $K$. Prove that $\angle KPA=90^o$.

Let $ABC$ be a triangle and $A', B', C'$ be the midpoints of the sides $BC, CA, AB$, respectively. Let $P$ and $P'$ be points such that $PA=P'A', PB =P'B', PC =P'C'$. Prove that all lines $PP'$ pass through a fixed point.

Rectangles $ABA_1B_2$, $BCB_1C_2$ and $CAC_1A_2$ lie outside triangle $ABC$. Let $C'$ be a point such that $C'A_1 \perp A_1C_2$ and $C'B_2 \perp B_2C_1$. Points $A'$ and $B'$ are defined analogously. Prove that the lines $AA', BB'$ and $CC'$ concur.


Olympic Challenge 2017


Denote by $O$ the circumcenter of $\vartriangle ABC$. The circumcircle of $\vartriangle AOC$ and $\vartriangle AOB$ intersects the lines $AB$ and $AC$ at points $AB, AC$ respectively. Similarly define points $B_A, B_C, C_A, C_B$. Prove that lines $A_BA_C, B_AB_C, C_AC_B$ have a common point.

Let $ABC$ be an acute scalene triangle with $O$ as its circumcenter. Point $P$ lies inside triangle $ABC$ and satisfies $\angle   PAB = \angle  PBC$ and $\angle  PAC = \angle  PCB$. Point $Q$ lies on line $BC$ and satisfies $QA = QP$. Show that $\angle  AQP = 2\angle  OQB$.

Let $\omega$ be the incircle of the $\vartriangle ABC$ with center $I$. The line which contains $I$ and is parallel to the line $BC$ intersects the sides $AC$ and $AB$ at the points $E$ and $F$ respectively. Let $E'$ be the symmetric point of $E$ through $BI$. Let $F'$ be the symmetric point of $F$ through $CI$. Prove that the line $E'F'$ is tangent to the circle $\omega$.

Two distinct circle $\omega_1$ and $\omega_2$ meet at two distinct points $P$ and $Q$. A line $\ell$ intersects $\omega_1$ at the points $A, C$ and intersects $\omega_2$ at the points $B, D$ such that the points $A, B, C$ and $D$ are all distinct and lie on $\ell$ in this order. Let $T$ be a point on the line $PQ$ such that $P$ lies between $T$ and $Q$. Lines $AT, BP$ intersect at $X$ and lines $DT, CP$ intersect at $Y$ . Finally, let $M$ and $N$ be the midpoints of the segments $AD$ and $BC$ respectively. Prove that $TM, PN$ and $XY$ are concurrent.

Let $I_b$ and $I_c$ be the $B$-excenter and $C$-excenter of $\vartriangle ABC$ respectively. Consider a chord $PQ$ of circumcircle of $\vartriangle ABC$ which is parallel to $BC$ and intersects the segments $AB$ and $AC$. Suppose the lines $BC$ and $AP$ intersect at $R$, prove that $\angle I_bQI_c + \angle  I_bRI_c = 180^o$.

On the sides $AB$ and $AC$ of $\vartriangle ABC$ choose points $K$ and $L$ respectively. Let $P$ and $Q$ be the points laying on the side $BC$ such that $KP \parallel  AC$ and $LQ \parallel  AB$. The incircle of $\vartriangle BKP$ touches side $AB$ at the point $X$. The incircle of $\vartriangle CLQ$ touches side $AC$ at the point $Y$ . The lines $QY$ and $AB$ intersect at the point $M$, the lines $XP$ and $AC$ intersect at the point $N$. Show that if $AX = AY$ then $MN \parallel BC$.

Triangle $ABC$ is given. The line perpendicular to $AC$ and going and through $B$ intersects the circle with diameter $AC$ at the points $X$ and $K$ where $X$ lies closer to $B$ than $K$. Analogously the line perpendicular to $AB$ and going through $C$ intersects the circle with diameter $AB$ at the points $Y$ and $L$ where $Y$ lies closer to $C$ than $L$. Prove that the intersection point of $XY$ and $KL$ lies on the line $BC$.

Let $\omega$ be the incircle of $\vartriangle ABC$ such that $AB = AC$. The circle $\omega$ touches sides $BC, CA$ and $AB$ at the points $D, E$ and $F$ respectively. The line $BE$ intersects $\omega$ again at the point $X$, the line $CF$ intersects $\omega$ again at the point $Y$ . Line $XY$ intersects segments $AB, AC, DF$ and $DE$ at the points $K, L, P$ and $Q$ respectively. Show that $KX = XP = PQ = QY = YL$.

Let $ABC$ be an acute scalene triangle inscribed in the circle $\omega$. Circle $\omega$, centred at the point $O$, passes through the points $B$ and $C$ and intersects the sides $AB$ and $AC$ at the points $E$ and $D$ respectively. The point $P$ lies on the arc $BC$ of circumcircle of $\vartriangle ABC$ with the point $A$. Prove that the lines $BD,CE$ and $OP$ are concurrent if and only if $\vartriangle PBD$ and $\vartriangle PCE$ have the same incenter.

In acute triangle $ABC$, let $\odot O$ be its circumcircle, $\odot I$ be its incircle. Tangents at $B,C$ to $\odot O$ meet at $L$, $\odot I$ touches $BC$ at $D$. $AY$ is perpendicular to $BC$ at $Y$, $AO$ meets $BC$ at $X$, and $OI$ meets $\odot O$ at $P,Q$. Prove that $P,Q,X,Y$ are concyclic if and only if $A,D,L$ are collinear.

Let $O$ and $I$ be the circumcenter and incenter of triangle $ABC$. The perpendicular from $I$ to $OI$ meets $AB$ and the external bisector of angle $C$ at points $X$ and $Y$ respectively. In what ratio does $I$ divide the segment $XY$?

Let $\omega$ be the circumcircle of $\vartriangle ABC$. Let the points $D, E$ lie on the sides $AC, AB$ respectively. Let $K = BD\cap \omega$ , $L = CE \cap \omega$ . Tangents to $\omega$ at the points $B, C$ intersect the line $DE$ at the points $P, Q$ respectively. Prove that $PL$ and $QK$ intersect on $\omega$ .

Given a right-angled triangle $ABC$ and two perpendicular lines $x$ and $y$ passing through the vertex $A$ of its right angle. For an arbitrary point $X$ on $x$ define $y_B$ and $y_C$ as the reflections of $y$ about $XB$ and $ XC $ respectively. Let $Y$ be the common point of $y_b$ and $y_c$. Find the locus of $Y$ (when $y_b$ and $y_c$ do not coincide).

Let $ABCD$ be the quadrilateral inscribed in the circle $\omega$. Line $\ell$ is tangent to the circle $\omega$ at the point $T$. The lines $AB, BC, CD$ and $AD$ intersect the line l at the points $Q, X, P$ and $Y$ respectively. Show that if $TY = TX$ then $TP = TQ$.

Let $H_B$ and $H_C$ be the foots of the altitudes from the vertexes $B$ and $C$ in $\vartriangle ABC$ respectively. Denote by $\Gamma$ the circumcircle of $\vartriangle ABC$. Prove that circumcircle of triangle bounded by the lines tangent to $\Gamma$ at the points $B$ and $C$ and the line $H_BH_C$ is tangent to $\Gamma$

Given triangle $ABC$ and its incircle $\omega$ prove you can use just a ruler and drawing at most 8 lines to construct points$A',B',C'$ on $\omega$ such that $A,B',C'$ and $B,C',A'$ and $C,A',B'$ are collinear.

Let $\omega$ be the incircle of $\vartriangle ABC$. The circle $\omega$ touches sides $BC$ and $AB$ at the points $D$ and $F$ respectively. Denote by $D'$ the reflection of the point $D$ with respect to the point $C$. The circle passing through the points $B$ and $C$ is tangent to $\omega$ at the point $T$. Let $J$ be the $A$-excenter of $\vartriangle ABC$. Prove that the points $T, F, B, J$ and $D'$ lie on the one circle.

Let $P$ be any point inside $\vartriangle ABC$. Denote by $H_C$ and $H_B$ the orthocenters of $\vartriangle PAB$ and $\vartriangle PAC$ respectively. The line $PA$ cuts circumcircle of $\vartriangle BPC$ again at the point $Q$. The point $H$ is the projection of $Q$ onto the line $BC$. Prove that $AH \perp H_BH_C$.

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