Ukraine TST 2007-20 28p

geometry problems from Ukrainian IMO Team Selection Tests (TST) with aops links in the names 

(only those not in IMO Shortlist)

collected inside aops here

                                                 2007 -2020

2007 Ukraine TST p2

$ABCD$ is convex $AD\parallel BC$, $AC\perp BD$. $M$ is interior point of $ABCD$ which is not a intersection of diagonals $AC$ and $BD$ such that $\angle AMB =\angle CMD =\frac{\pi}{2}$ .$P$ is intersection of angel bisectors of $\angle A$ and $\angle C$. $Q$ is intersection of angel bisectors of $\angle B$ and $\angle D$. Prove that $\angle PMB =\angle QMC$.

2007 Ukraine TST p5

$AA_{3}$ and $BB_{3}$ are altitudes of acute-angled $\triangle ABC$. Points $A_{1}$ and $B_{1}$ are second points of intersection lines $AA_{3}$ and $BB_{3}$ with circumcircle of $\triangle ABC$ respectively. $A_{2}$ and $B_{2}$ are points on $BC$ and $AC$ respectively. $A_{1}A_{2}\parallel AC$, $B_{1}B_{2}\parallel BC$. Point $M$ is midpoint of $A_{2}B_{2}$. $\angle BCA = x$. Find $\angle A_{3}MB_{3}$.

2008 Ukraine TST p4

Two circles $\omega_1$ and $\omega_2$ tangents internally in point $P$. On their common tangent points $A$, $B$ are chosen such that $P$ lies between $A$ and $B$. Let $C$ and $D$ be the intersection points of tangent from $A$ to $\omega_1$, tangent from $B$ to $\omega_2$ and tangent from $A$ to $\omega_2$, tangent from $B$ to $\omega_1$, respectively. Prove that $CA + CB = DA + DB$.

2008 Ukraine TST p9

Given $\triangle ABC$ with point $D$ inside. Let $A_0=AD\cap BC$, $B_0=BD\cap AC$, $C_0 =CD\cap AB$ and $A_1$, $B_1$, $C_1$, $A_2$, $B_2$, $C_2$ are midpoints of $BC$, $AC$, $AB$, $AD$, $BD$, $CD$ respectively. Two lines parallel to $A_1A_2$ and $C_1C_2$ and passes through point $B_0$ intersects $B_1B_2$ in points $A_3$ and $C_3$respectively. Prove that $\frac{A_3B_1}{A_3B_2}=\frac{C_3B_1}{C_3B_2}$.

2008 Ukraine TST p11

Let $ABCDE$ be convex pentagon such that $S(ABC) = S(BCD) = S(CDE) = S(DEA) = S(EAB)$. Prove that there is a point $M$ inside pentagon such that $S(MAB) = S(MBC) = S(MCD) = S(MDE) = S(MEA)$.

2009 Ukraine TST p4
Let $A,B,C,D,E$ be consecutive points on a circle with center $O$ such that $AC=BD=CE=DO$. Let $H_1,H_2,H_3$ be the orthocenters  triangles $ACD,BCD,BCE$  respectively. Prove that the triangle $H_1H_2H_3$  is right.

2009 Ukraine TST p8
Two circles $\gamma_1, \gamma_2$ are given, with centers at points $O_1, O_2$ respectively. Select a point $K$ on circle  $\gamma_2$ and construct two circles,  one $\gamma_3$ that touches circle $\gamma_2$ at   point $K$ and circle $\gamma_1$ at a point $A$, and the other $\gamma_4$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $B$. Prove that, regardless of the choice of  point K on circle $\gamma_2$, all lines $AB$ pass through a fixed point of the plane.

2010 Ukraine TST p2
Let $ABCD$ be a quadrilateral inscribled in a circle with the center $O, P$ be the point of intersection of the diagonals $AC$ and $BD$, $BC\nparallel AD$. Rays $AB$ and $DC$ intersect at the point $E$. The circle with center $I$ inscribed in the triangle $EBC$ touches $BC$ at point $T_1$. The $E$-excircle with center $J$ in the triangle $EAD$ touches the side $AD$ at the point T$_2$. Line $IT_1$ and $JT_2$ intersect at $Q$. Prove that the points $O, P$, and $Q$ lie on a straight line.

2010 Ukraine TST p7
Denote in the triangle $ABC$ by $h$ the length of the height drawn from vertex $A$, and by $\alpha = \angle BAC$. Prove that the inequality $AB + AC \ge  BC \cdot \cos \alpha   + 2h \cdot \sin \alpha$ . Are there triangles for which this inequality turns into equality?

2010 Ukraine TST p11
Let $ABC$ be the triangle in which $AB> AC$. Circle $\omega_a$ touches the segment of the $BC$ at point $D$, the extension of the segment $AB$ towards point $B$ at the point $F$, and the extension of the segment $AC$ towards point $C$ at the point $E$. The ray $AD$ intersects circle $\omega_a$ for second time at point $M$. Denote the circle circumscribed around the triangle $CDM$ by $\omega$. Circle $\omega$ intersects the segment $DF$ at N. Prove that $FN > ND$.

2011 Ukraine TST p6
The circle $\omega$ inscribed in triangle $ABC$ touches its sides $AB, BC, CA$ at points $K, L, M$ respectively. In the arc $KL$ of the circle $\omega$ that does not contain the point $M$, we select point $S$. Denote by $P, Q, R, T$ the intersection points of straight $AS$ and $KM, ML$ and $SC, LP$ and $KQ, AQ$ and $PC$ respectively. It turned out that the points $R, S$ and $M$ are collinear. Prove that the point $T$ also lies on the line $SM$.

2011 Ukraine TST p9
Inside the inscribed quadrilateral $ABCD$, a point $P$ is marked such that $\angle PBC = \angle PDA$, $\angle PCB = \angle PAD$. Prove that there exists a circle that touches the straight lines $AB$ and $CD$, as well as the circles circumscribed by the triangles $ABP$ and $CDP$.

2011 Ukraine TST p10
Let $H$ be the point of intersection of the altitudes $AP$ and $CQ$ of the acute-angled triangle $ABC$. The points $E$ and $F$ are marked on the median $BM$ such that $\angle APE = \angle BAC$, $\angle CQF = \angle BCA$, with point $E$ lying inside the triangle $APB$ and point $F$ is inside the triangle $CQB$. Prove that the lines $AE, CF$, and $BH$ intersect at one point.

2012 Ukraine TST p2

$E$ is the intersection point of the diagonals of the cyclic quadrilateral, $ABCD, F$ is the intersection point of the lines $AB$ and $CD, M$ is the midpoint of the side $AB$, and $N$ is the midpoint of the side $CD$. The circles circumscribed around the triangles $ABE$ and $ACN$ intersect for the second time at point $K$. Prove that the points $F, K, M$ and $N$ lie on one circle.

2012 Ukraine TST p4

Given an isosceles triangle $ABC$ ($AB = AC$), the inscribed circle $\omega$ touches its sides $AB$ and $AC$ at points $K$ and $L$, respectively. On the extension of the side of the base $BC$, towards $B$, an arbitrary point $M$. is chosen. Line $M$ intersects $\omega$ at the point $N$ for the second time, line $BN$ intersects the second point $\omega$ at the point $P$. On the line $PK$, there is a point $X$ such that $K$ lies between $P$ and $X$ and $KX = KM$. Determine the locus of the point $X$.

2012 Ukraine TST p9

The inscribed circle $\omega$ of the triangle $ABC$ touches its sides $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$, respectively. Let $S$ be the intersection point of lines passing through points $B$ and $C$ and parallel to $A_1C_1$ and $A_1B_1$ respectively, $A_0$ be the foot of the perpendicular drawn from point $A_1$ on $B_1C_1$, $G_1$ be the centroid of triangle $A_1B_1C_1$, $P$ be the intersection point of the ray $G_1A_0$ with $\omega$. Prove that points $S, A_1$, and $P$ lie on a straight line.

2013 Ukraine TST p1
Let $ABC$ be an isosceles triangle $ABC$ with base $BC$ insribed in a circle. The segment $AD$ is the diameter of the circle, and point $P$ lies on the smaller arc $BD$. Line $DP$ intersects rays $AB$ and $AC$ at points $M$ and $N$, and the lines $BP$ and $CP$ intersects the line $AD$ at points $Q$ and $R$. Prove that the midpoint of the segment $MN$ lies on the circumscribed circle of triangle $PQR$.

2013 Ukraine TST p6
Six different points $A, B, C, D, E, F$ are marked on the plane, lie on one circle and no two segments with ends at these points lie on parallel lines. Let $P, Q,R$ be the points of intersection of the perpendicular bisectors to pairs of segments $(AD, BE)$, $(BE, CF)$ ,$(CF, DA)$ respectively, and $P', Q' ,R'$ are points the intersection of the perpendicular bisectors to the pairs of segments $(AE, BD)$, $(BF, CE)$ , $(CA, DF)$ respectively. Show that $P \ne  P', Q \ne Q', R  \ne R'$, and prove that the lines $PP', QQ'$ and $RR'$ intersect at one point or are parallel.

2014 Ukraine TST p4
The $A$-excircle of the triangle $ABC$ touches the side $BC$ at point $K$. The circumcircles of triangles $AKB$ and $AKC$ intersect for the second time with the bisector of angle $A$ at points $X$ and $Y$ respectively. Let $M$ be the midpoint of $BC$. Prove that the circumcenter of triangle $XYM$ lies on $BC$.

2014 Ukraine TST p8
The quadrilateral $ABCD$ is inscribed in the circle $\omega$ with the center $O$. Suppose that the angles $B$ and $C$ are obtuse and lines $AD$ and $BC$ are not parallel. Lines $AB$ and $CD$ intersect at point $E$. Let $P$ and $R$ be the feet of the perpendiculars from the point $E$ on the lines $BC$ and $AD$ respectively. $Q$ is the intersection point of $EP$ and $AD, S$ is the intersection point of $ER$ and $BC$. Let K be the midpoint of the segment $QS$ . Prove that the points $E, K$, and $O$ are collinear

2015 Ukraine TST p1
Let $O$ be the circumcenter of the triangle $ABC, A'$ be a point symmetric of $A$ wrt line $BC, X$ is an arbitrary point on the ray $AA'$ ($X \ne A$). Angle bisector of angle $BAC$ intersects the circumcircle of triangle $ABC$ at point $D$ ($D \ne A$). Let $M$ be the midpoint of the segment $DX$. A line passing through point $O$ parallel to $AD$, intersects $DX$ at point $N$. Prove that angles $BAM$ and $CAN$ angles are equal.

2015 Ukraine TST p6

Given an acute triangle $ABC, H$ is the foot of the altitude drawn from the point $A$ on the line $BC, P$ and $K \ne H$ are arbitrary points on the segments $AH$ and$BC$ respectively. Segments $AC$ and $BP$ intersect at point $B_1$, lines $AB$ and $CP$ at point $C_1$. Let $X$ and $Y$ be the projections of point $H$ on the lines $KB_1$ and $KC_1$, respectively. Prove that points $A, P, X$ and $Y$ lie on one circle.

2016 Ukraine TST p8

Let $ABC$ be an acute triangle with $AB<BC$. Let $I$ be the incenter of $ABC$, and let $\omega$ be the circumcircle of $ABC$. The incircle of $ABC$ is tangent to the side $BC$ at $K$. The line $AK$ meets $\omega$ again at $T$. Let $M$ be the midpoint of the side $BC$, and let $N$ be the midpoint of the arc $BAC$ of $\omega$. The segment $NT$ intersects the circumcircle of $BIC$ at $P$. Prove that $PM\parallel AK$.

2017 all from Shortlist

2018 Ukraine TST p9
Let $AA_1, BB_1, CC_1$ be the heights of triangle $ABC$ and $H$ be its orthocenter. Liune $\ell$ parallel to $AC$, intersects straight lines $AA_1$ and $CC_1$ at points $A_2$ and $C_2$, respectively. Suppose that point $B_1$ lies outside the circumscribed circle of triangle $A_2 HC_2$. Let $B_1P$ and $B_1T$ be tangent to of this circle. Prove that points $A_1, C_1, P$, and $T$ are cyclic.

2018 Ukraine TST p10
Let $ABC$ be a triangle with $AH$ altitude. The point $K$ is chosen on the segment $AH$ as follows such that $AH =3KH$. Let $O$ be the center of the circle circumscribed around by triangle $ABC, M$ and $N$ be the midpoints of $AC$ and AB respectively. Lines $KO$ and $MN$ intersect at the point $Z$, a perpendicular to $OK$ passing through point $Z$ intersects lines $AC$ and $AB$ at points $X$ and $Y$ respectively. Prove that $\angle XKY =\angle CKB$.

2019 Ukraine TST p1

In a triangle $ABC$, $\angle ABC= 60^o$, point $I$ is the incenter. Let the points $P$ and $T$ on the sides $AB$ and $BC$ respectively such that $PI \parallel BC$ and $TI \parallel   AB$ , and points $P_1$ and $T_1$ on the sides $AB$ and $BC$ respectively such that $AP_1 = BP$ and $CT_1 = BT$. Prove that point $I$ lies on segment $P_1T_1$.

(Anton Trygub)

2019 Ukraine TST p16

Given an acute triangle $ABC$ . It's altitudes $AA_1 , BB_1$ and $CC_1$ intersect at a point $H$ , the orthocenter of $\vartriangle ABC$. Let the lines $B_1C_1$ and $AA_1$ intersect at a point $K$, point $M$ be the midpoint of the segment $AH$. Prove that the circumscribed circle of $\vartriangle MKB_1$ touches the circumscribed circle of $\vartriangle ABC$ if and only if $BA1 = 3A1C$.

(Bondarenko Mykhailo)

2020 Ukraine TST p1

Let $BH$ be an altitude of acute scalene triangle $ABC$. Points $K$ and $L$ are chosen on sides

$AB$, $BC$ respectively such that $BK = BH = BL$. The circumcircles of $\triangle AKH$ and

$\triangle CLH$ intersect for the second time at point $P$ and intersect ray $BH$ for the second time

at points $Q$ and $R$ respectively. Prove that the circumcenter of $\triangle PQR$ lies on the angle

bisector of $\angle ABC$.

(Mykhailo Shtandenko)

2020 Ukraine TST p2

Altitudes $AH1$ and $BH2$ of acute triangle $ABC$ intersect at $H$. Let $w1$ be the circle that

goes through $H2$ and touches the line $BC$ at $H1$, and let $w2$ be the circle that goes through

$H1$ and touches the line $AC$ at $H2$. Prove, that the intersection point of two other tangent lines

$BX$ and $AY$( $X$ and $Y$ are different from $H1$ and $H2$) to circles $w1$ and $w2$

respectively, lies on the circumcircle of triangle $HXY$.

(Danilo Khilko)