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Croatia TST 2001-20 (IMO - MEMO) 28p (-05,-08)

geometry problems from Croatian Team Selection Tests (TST) [for MEMO and IMO] with aops links in the names 
(only those not in Shortlist)

in 2007 started the MEMO TSTs

collected inside aops here


 IMO TST  2001 - 2020 (-05)
Circles $k_1$ and $k_2$ intersect at $P$ and $Q$, and $A$ and $B$ are the tangency points of their common tangent that is closer to $P$ (where $A$ is on $k_1$ and $B$ on $k_2$). The tangent to $k_1$ at $P$ intersects $k_2$ again at $C$. The lines $AP$ and $BC$ meet at $R$. Show that the lines $BP$ and $BC$ are tangent to the circumcircle of triangle $PQR$.

A quadrilateral $ABCD$ is circumscribed about a circle. Lines $AC$ and $DC$ meet at point $E$ and lines $DA$ and $BC$ meet at $F$, where $B$ is between $A$ and $E$ and between $C$ and $F$. Let $I_1, I_2$ and $I_3$ be the incenters of triangles $AFB, BEC$ and $ABC$, respectively. The line $I_1I_3$ intersects $EA$ at $K$ and $ED$ at $L$, whereas the line $I_2I_3$ intersects $FC$ at $M$ and $FD$ at $N$. Prove that $EK = EL$ if and only if $FM = FN$

Let $B$ be a point on a circle $k_1, A \ne B$ be a point on the tangent to the circle at $B$, and $C$ a point not lying on $k_1$ for which the segment $AC$ meets $k_1$ at two distinct points. Circle $k_2$ is tangent to line $AC$ at $C$ and to $k_1$ at point $D$, and does not lie in the same half-plane as $B$. Prove that the circumcenter of triangle $BCD$ lies on the circumcircle of $\vartriangle ABC$

2004 Croatia TST p3 (ARMO 1995)
A line intersects a semicircle with diameter $AB$ and center $O$ at $C$ and $D$, and the line $AB$
at $M$, where $MB < MA$ and $MD < MC.$ If the circumcircles of the triangles $AOC$ and $DOB$
meet again at $K,$ prove that $\angle MKO$ is right.

2004 - 2005 missing

2007 Croatia IMO TST p3 (UK 2006, Round II )
Let $ABC$ be a triangle such that $|AC|>|AB|$. Let $X$ be on line $AB$ (closer to $A$) such that $|BX|=|AC|$ and let $Y$ be on the segment $AC$ such that $|CY|=|AB|$. Intersection of lines $XY$ and bisector of $BC$ is point $P$. Prove that $\angle BPC+\angle BAC = 180^\circ$.

Point $ M$ is taken on side $ BC$ of a triangle $ ABC$ such that the centroid $ T_c$ of triangle $ ABM$ lies on the circumcircle of $ \triangle ACM$ and the centroid $ T_b$ of $ \triangle ACM$ lies on the circumcircle of $ \triangle ABM$. Prove that the medians of the triangles $ ABM$ and $ ACM$ from $ M$ are of the same length.

A triangle $ ABC$ is given with $ \left|AB\right| > \left|AC\right|$. Line $ l$ tangents in a point $ A$ the circumcirle of $ ABC$. A circle centered in $ A$ with radius $ \left|AC\right|$ cuts $ AB$ in the point $ D$ and the line $ l$ in points $ E, F$ (such that $ C$ and $ E$ are in the same halfplane with respect to $ AB$). Prove that the line $ DE$ passes through the incenter of $ ABC$.

Let $ABC$ be a triangle in which $|AB| <|CA| <|BC|$ and let $D$ and $E$ be the points on rays $BA$ and $BC$ such that $|BD| = |BE| = |AC|$. The circumcircle of the triangle $BDE$ intersects the segment $AC$ at the point $P$, and the extension of $BP$ intersects the circumcircle of the triangle $ABC$ at point $Q$ ($Q \ne B$). Prove that $|AQ| + |QC| = |BP|$.

Let $I$ be the incenter of the acute-angled triangle $ABC$ and let $k_c$ be the excircle opposite of the angle $\angle BCA$. If the circle $k_c$ touches the side $AB$ at point $D$ and the line $DI$ meets the circle $k_c$ again in $S$, show that $DI$ bisects the angle $\angle ASB$
A trapezoid $ABCD$ with a longer base $AB$ is inscribed in a circle $k$. Let $A_0, B_0$ be the midpoints of segments $BC, CA$ respectively. Let $N$ be the foot of perpendicular from the vertex $C$ on $AB$, and $G$ the centroid of triangle $ABC$. Circle $k_1$ passes through points $A_0$ and $B_0$ and is tangent to circle $k$ at point $X$, other than $C$. Prove that the points $D, G, N$, and $X$ are collinear.

An isosceles triangle $ABC$ with base $AB$ is given. Point $P$ lies on side $AC$ and point $Q$ on side $BC$ are selected such that $|AP| + |BQ| = |PQ|$. Parallel to the line $BC$ through the midpoint of length $PQ$ intersects the segment $AB$ at point $N$. The circle circumscribed around the triangle $PNQ$ intersects the segment $AC$ at points $P$ and $K$, and the segment $BC$ at points $Q$ and $L$. If the point $R$ is the intersection of the segments $PL$ and $QK$, prove that the line $PQ$ is perpendicular to the segment $CR$.

2014 Croatia IMO TST p3
In an acute-angled triangle $ABC$, in which $|AC| <|BC|$, the points $M$ and $N$ are the feet of the altitudes from vertices $A$ and $B$ respectively. The circumcircle of $ABC$, with center $O$,  intersects the circumcircle of $MNC$, with center $S$,  at points $C$ and $D$. If the point $P$ is the midpoint of the segment $AB$, prove that the points $P, O, S$, and $D$ lie on the same circle.

In the quadrilateral $ABCD$ , $\angle DAB = 110^o$, $\angle ABC = 50^o$, $\angle BCD  = 70^o$. Let  $M, N$ be the midpoints of segments $AB , CD$ respectively. Let $P$ be a point on the segment $MN$  such that $|AM|:|CN|=|MP|:|NP|$ and $|AP|=|CP|$. Determine the angle $\angle APC$.

2016 Croatia IMO TST p3
Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $|BE| = |OF|$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\angle ENO + \angle  OMF = 2 \angle  BAC$.

The inscribed circle of triangle $ABC$ has center $I$ and touches the sides $BC,CA, AB$ respectively at points $D,E,F$. Let $k$ be a circle with center $A$ passing through the point $E$. Second intersection the line $DE$ with the circle $k$  is the point $K$. The parallel with the line $DF$ through the point $I$ intersects the page $AB$ at point $P.$ The point $L$ is the intersection of the line $CP$ and the circle $k$  such that $P$  is located between the points $C$ and $L$. The point $O$ is the center of the circumcircle of triangle $DKL$. Prove that the directions $AI$ and $OD$ are parallel.


MEMO TST 2007 - 2020 (-08)


2007 Croatia  MEMO TST p1
Let there be two circles. Find all points $M$ such that there exist two points, one on each circle such that $M$ is their midpoint.

2008 memo missing

2009 Croatia  MEMO TST 1 p1
On sides $ AB$ and $ AC$ of triangle $ ABC$ there are given points $ D,E$ such that $ DE$ is tangent of circle inscribed in triangle $ ABC$ and $ DE \parallel BC$. Prove $ AB+BC+CA\ge 8DE$

2009 Croatia  MEMO TST 2 p1
It is given a convex quadrilateral $ ABCD$ in which $ \angle B+\angle C < 180^0$.
Lines $ AB$ and $ CD$ intersect in point E. Prove that
$ CD\cdot  CE=AC^2+AB   \cdot AE \Leftrightarrow \angle B= \angle D$

Within the triangle $ABC$, the point $P$ is given such that $\angle ABP = \angle PCA = \frac13 (\angle ABC + \angle BCA)$. Prove that $$ \frac{|AB|}{|AC| + |PB|} =\frac{|AC|}{|AB| + |PC|}$$

Within a acute triangle $ABC$ a point $S$ is given such that $\angle SAB = \angle SBC =\angle SCA$. The lines $AS, BS, CS$ intersect the circles circumscribed around the triangles $SBC, SCA,SAB$ in points $A_1, B_1, C_1$. Prove the inequality for the area of the triangles $P (A_1CB) + P (B_1AC) + P (C_1BA) \ge  3P (ABC)$.
Let $ABC$ be an acute triangle and let $A_1, B_1, C_1$ be points on segments $BC, CA, AB$ respectively. Prove that triangles $ABC$ and $A_1B_1C_1$ are similar ($\angle A =\angle A_1$, $\angle B = \angle B_1$, $\angle C = \angle C_1$) if and only if the orthocenter of triangle $A_1B_1C_1$ coincides with the center of the circumscribed circle of the triangle $ABC$.

The point $N$ is the foot of the altitude at the hypotenuse $AB$ of the right triangle $ABC$. Bisectors of angles $\angle NCA$ and $\angle BCN$ intersect the segment $AB$ respectively at points $K$ and $L$. If $S$ and $T$ are respectively the centers of the circles inscribed inside the triangles $BCN$ and $NCA$, prove that the quadrilateral $KLST$ is  cyclic.

Let $ABC$ be an acute-angled triangle in which $|AC|> |BC|$. Let $H$ be the orthocenter of that triangle, $N$ be the midpoint of the altitide from the vertex $B$, and $P$ be the midpoint of the segment $AB$. Circumcircles of triangles $ABC$ and $CHN$ intersect at points $C$ and $D$. Prove that points $B, D, N$, and $P$ lie on the same circle.

Let $I$ be the center of the inscribed circle of the triangle  $ABC$ and the point $D$ on the side $ AC$ such that  $|AB| = |DB |$. The inscribed circle of a triangle  $BCD$ touches the lines $AC$  and  $BD$  respectively at the points  $E$ and $F$ respectively. Prove that the line $EF$ bisects the segment $DI$.

The cyclic quadrilateral $ABCD$ is given. The rays $AD$ and $BC$ intersect at point $P$. In the interior of the triangle $DCP$,  the point $M$ is given such that the line $PM$ bisects the angle $\angle CMD$. Line $CM$ intersects the circumcircle of triangle $DMP$ again at the point $Q$, and the line $DM$ intersects the circumcircle of triangle $CMP$ again at point $R$.
a) Prove that the segments $CQ$ and $DR$ have equal length.
b) Prove that triangles $PAQ$ and $PBR$ have equal areas.

Let $AD$ be the altitude of the acute triangle $ABC$. Let $E , F$ be different points on ray  $AD$ such that $|DE| = |DF|$ and  point $E$ lies inside the triangle $ABC$. The circle circumscribed around  the triangle $BEF$ intersects the segments $BC , AB$ at points $K , M$ respectively . The circle circumscribed around the triangle $CEF$ intersects the segments $BC , CA$ at points $L , N$ respectively . Prove that the segments $AD, KM$ and $LN$ intersect at one point.

Let $ABCD$ be an isosceles trapezoid with bases $AB$ and $CD$ . The diagonals of the trapezoid intersect at point $S$, and the midpoint of page $AD$ is point $M$. The circle circumscribed around the triangle $BCM$ intersects again line $AD$ at point $K$. Prove that the lines $SK$ and $AB$ are parallel..

The tangents of the incircle of triangle $ABC$ with sides $AB , AC$ are the points $D , E$ respectively. Point of tangency of the excircle opposite the vertex $A$ with the extensions of $AB , AC$ are the points $F , G$ respectively. Let the bisectors of the angles $\angle ABC$ , $\angle ACB$ intersect the line $DE$ at the points $X , Y$ respectively. Let the external bisectors of the angles $\angle ABC$ , $\angle ACB$ intersect the line  $FG$ at the points $Z , W$ respectively. Prove that the quadrilateral $XYZW$ is cyclic.

2020 from IMO ISL

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