Serbia Yugoslavia TST 1968 - 2019 (IMO - EGMO - RMM) 52p

geometry problems from Serbian Team Selection Tests (TST), 
with aops links in the names

(only those not in IMO Shortlist)

collected inside aops here

IMO TST 1968 - 2019

(only years found)

1968 Yugoslavia TST p3

Each side of a triangle $ABC$ is divided into three equal parts, and the middle segment in each of the sides is painted green. In the exterior of $\triangle ABC$ three equilateral triangles are constructed, in such a way that the three green segments are sides of these triangles. Denote by $A',B',C'$ the vertices of these new equilateral triangles that don’t belong to the edges of $\triangle ABC$, respectively. Let $A'',B'',C''$ be the points symmetric to $A',B',C'$ with respect to $BC,CA,AB$.

(a) Prove that $\triangle A'B'C'$ and $\triangle A''B''C''$ are equilateral.

(b) Prove that $ABC,A'B'C'$, and $A''B''C''$ have a common centroid.

1968 Yugoslavia TST p6

Prove that the incenter coincides with the circumcenter of a tetrahedron if and only if each pair of opposite edges are of equal length.

1969 Yugoslavia TST p3

Points $A$ and $B$ move with a constant speed along lines $a$ and $b$. Two corresponding positions of these points $A_1,B_1$, and $A_2,B_2$ are known. Find the position of $A$ and $B$ for which the length of $AB$ is minimal.

1969 Yugoslavia TST p5

Prove that the product of the sines of two opposite dihedrals in a tetrahedron is proportional to the product of the lengths of the edges of these dihedrals.

1970 Yugoslavia TST p2

Describe how to place the vertices of a triangle in the faces of a cube in such a way that the shortest side of the triangle is the biggest possible.

1970 Yugoslavia TST p3

If all edges of a non-planar quadrilateral tangent the faces of a sphere, prove that all of the points of tangency belong to a plane.

1972 Yugoslavia TST p2

If a convex set of points in the line has at least two diameters, say $AB$ and $CD$, prove that $AB$ and $CD$ have a common point.

1973 Yugoslavia TST p1

All sides of a rectangle are odd positive integers. Prove that there does not exist a point inside the rectangle whose distance to each of the vertices is an integer.

1973 Yugoslavia TST p2

A circle $k$ is drawn using a given disc (e.g. a coin). A point $A$ is chosen on $k$. Using just the given disc, determine the point $B$ on $k$ so that $AB$ is a diameter of $k$. (You are allowed to choose an arbitrary point in one of the drawn circles, and using the given disc it is possible to construct either of the two circles that passes through the points at a distance that is smaller than the radius of the circle.)

1974 Yugoslavia TST p2

Given two directly congruent triangles $ABC$ and $A'B'C'$ in a plane, assume that the circles with centers $C$ and $C'$ and radii $CA$ and $C'A'$ intersect. Denote by $\mathcal M$ the transformation that maps $\triangle ABC$ to $\triangle A'B'C'$. Prove that $\mathcal M$ can be expressed as a composition of at most three rotations in the following way: The first rotation has the center in one of $A,B,C$ and maps $\triangle ABC$ to $\triangle A_1B_1C_1$; The second rotation has the center in one of $A_1,B_1,C_1$, and maps $\triangle A_1B_1C_1$ to $\triangle A_2B_2C_2$; The third rotation has the center in one of $A_2,B_2,C_2$ and maps $\triangle A_2B_2C_2$ to $\triangle A'B'C'$.

1974 Yugoslavia TST p3

Let $S$ be a set of $n$ points $P_1,P_2,\ldots,P_n$ in a plane such that no three of the

points are collinear. Let $\alpha$ be the smallest of the angles $\angle P_iP_jP_k$ ($i\ne j\ne k\ne i,i,j,k\in\{1,2,\ldots,n\}$). Find $\max_S\alpha$ and determine those sets $S$ for which this maximal value is attained.

1976 Yugoslavia TST p1

Prove that for a given convex polygon of area $A$ and perimeter $P$ there exists a circle of radius $\frac AP$ that is contained in the interior of the polygon.

1977 Yugoslavia TST p3

Assume that the equality $2BC=AB+AC$ holds in $\triangle ABC$. Prove that:

(a) The vertex $A$, the midpoints $M$ and $N$ of $AB$ and $AC$ respectively, the incenter $I$, and the circumcenter $O$ belong to a circle $k$.

(b) The line $GI$, where $G$ is the centroid of $\triangle ABC$ is a tangent to $k$.

1978 Yugoslavia TST p2

Let $k_0$ be a unit semi-circle with diameter $AB$. Assume that $k_1$ is a circle of radius $r_1=\frac12$ that is tangent to both $k_0$ and $AB$. The circle $k_{n+1}$ of radius $r_{n+1}$ touches $k_n,k_0$, and $AB$. Prove that:

(a) For each $n\in\{2,3,\ldots\}$ it holds that $\frac1{r_{n+1}}+\frac1{r_{n-1}}=\frac6{r_n}-4$.

(b) $\frac1{r_n}$ is either a square of an even integer, or twice a square of an odd integer.

1981 Yugoslavia TST p1

Circles $k$ and $l$ intersect at points $P$ and $Q$. Let $A$ be an arbitrary point on $k$ distinct from $P$ and $Q$. Lines $AP$ and $AQ$ meet $l$ again at $B$ and $C$. Prove that the altitude from $A$ in triangle $ABC$ passes through a point that does not depend on $A$.

1985 Yugoslavia TST p2

Let $ABCD$ be a parallelogram and let $E$ be a point in the plane such that $AE\perp AB$ and $BC\perp EC$. Show that either $\angle AED=\angle BEC$ or $\angle AED+\angle BEC=180^\circ$.

1987 Yugoslavia TST p3

Let there be given lines $a,b,c$ in the space, no two of which are parallel. Suppose that there exist planes $\alpha,\beta,\gamma$ which contain $a,b,c$ respectively, which are perpendicular to each other. Construct the intersection point of these three planes. (A space construction permits drawing lines, planes and spheres and translating objects for any vector.)

1992 Yugoslavia TST p2

Three squares $BCDE,CAFG$ and $ABHI$ are constructed outside the triangle $ABC$. Let $GCDQ$ and $EBHP$ be parallelograms. Prove that $APQ$ is an isosceles right triangle.

1995 Yugoslavia TST p3

Let $SABCD$ be a pyramid with the vertex $S$ whose all edges are equal. Points $M$ and $N$ on the edges $SA$ and $BC$ respectively are such that $MN$ is perpendicular to both $SA$ and $BC$. Find the ratios $SM:MA$ and $BN:NC$.

1997 Yugoslavia TST p1

Consider a regular $n$-gon $A_1A_2\ldots A_n$ with area $S$. Let us draw the lines $l_1,l_2,\ldots,l_n$ perpendicular to the plane of the $n$-gon at $A_1,A_2,\ldots,A_n$ respectively. Points $B_1,B_2,\ldots,B_n$ are selected on lines $l_1,l_2,\ldots,l_n$ respectively so that:

(i) $B_1,B_2,\ldots,B_n$ are all on the same side of the plane of the $n$-gon;

(ii) Points $B_1,B_2,\ldots,B_n$ lie on a single plane;

(iii) $A_1B_1=h_1,A_2B_2=h_2,\ldots,A_nB_n=h_n$.

Express the volume of polyhedron $A_1A_2\ldots A_nB_1B_2\ldots B_n$ as a function in $S,h_1,\ldots,h_n$.

1998 Yugoslavia TST p1

In a convex quadrilateral $ABCD$, the diagonal $AC$ intersects the diagonal $BD$ at its midpoint $S$. The radii of incircles of triangles $ABS,BCS,CDS,DAS$ are $r_1,r_2,r_3,r_4$, respectively. Prove that

$$|r_1-r_2+r_3-r_4|\le\frac18|AB-BC+CD-DA|.$$

1999 Yugoslavia TST p2

Let $ABC$ be a triangle such that $\angle A=90^{\circ }$ and $\angle B<\angle C$. The tangent at $A$ to the circumcircle $\omega$ of triangle $ABC$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ in the line $BC$, let $X$ be the foot of the perpendicular from $A$ to $BE$, and let $Y$ be the midpoint of the segment $AX$. Let the line $BY$ intersect the circle $\omega$ again at $Z$. Prove that the line $BD$ is tangent to the circumcircle of triangle $ADZ$.

2002 Serbia TST P2
Let $ABC$ be a triangle with semiperimeter $s$.  Let $E$ and $F$ be the points on the line $AB$, such that $CE = CF = s$. Prove that the circumcircle of triangle $EFC$ and the $C$-excircle  of triangle $ABC$ are tangent.

2002 Serbia TST P5
Let $ABCD$ is convex quadrilateral such that $\angle DAB=\angle ABC=\angle BCD$. Prove that Euler line of $\triangle ABC$ passes through $D$.

2003 Serbia TST P2
Let $M$ and $N$ be the distinct points in the plane of the triangle $ABC$ such that $AM : BM : CM = AN : BN : CN$. Prove that the line $MN$ contains the circumcenter of $\triangle ABC$.

2004 Serbia TST P1
Let $ABCD$ be a square and $K$ be a circle with diameter $AB$. For an arbitrary point $P$ on side $CD$, segments $AP$ and $BP$ meet $K$ again at points $M$ and $N$, respectively, and lines $DM$ and $CN$ meet at point $Q$. Prove that $Q$ lies on the circle $K$ and that $AQ : QB = DP : PC$.

2006 Serbia TST P2
A point $P$ is taken in the interior of a right triangle$ABC$ with $\angle C = 90$ such hat
$AP = 4, BP = 2$, and$CP = 1$. Point $Q$ symmetric to $P$ with respect to $AC$ lies on
the circumcircle of triangle $ABC$. Find the angles of triangle $ABC$.

2009 Serbia TST P1
Let $\alpha$ and $\beta$ be the angles of a non-isosceles triangle $ABC$ at points $A$ and $B$, respectively. Let the bisectors of these angles intersect opposing sides of the triangle in $D$ and $E$, respectively. Prove that the acute angle between the lines $DE$ and $AB$ isn't greater than $\frac{|\alpha-\beta|}3$.

2009 Serbia TST P6
Let $k$ be the inscribed circle of non-isosceles triangle $\triangle ABC$, which center is $S$. Circle $k$ touches sides $BC,CA,AB$ in points $P,Q,R$ respectively. Line $QR$ intersects $BC$ in point $M$. Let a circle which contains points $B$ and $C$ touch $k$ in point $N$. Circumscribed circle of $\triangle MNP$ intersects line $AP$ in point $L$, different from $P$. Prove that points $S,L$ and $M$ are collinear.

2012 Serbia TST P3 
Let $P$ and $Q$ be points inside triangle $ABC$ satisfying $\angle PAC=\angle QAB$ and $\angle PBC=\angle QBA$.
a) Prove that feet of perpendiculars from $P$ and $Q$ on the sides of triangle $ABC$ are concyclic.
b) Let $D$ and $E$ be feet of perpendiculars from $P$ on the lines $BC$ and $AC$ and $F$ foot of perpendicular from $Q$ on $AB$. Let $M$ be intersection point of $DE$ and $AB$. Prove that $MP\bot CF$.

2013 Serbia TST P2
In an acute $\triangle ABC$ ($AB \neq AC$) with angle $\alpha$ at the vertex $A$, point $E$ is the nine-point center, and $P$ a point on the segment $AE$. If $\angle ABP = \angle ACP = x$, prove that $x = 90$° $-2 \alpha$.

 by Dusan Djukic

2017 Serbia TST P1
Let $ABC$ be a triangle and $D$ the midpoint of the side $BC$. Define points $E$ and $F$ on $AC$ and $B$, respectively, such that $DE=DF$ and $\angle EDF =\angle BAC$. Prove that $$DE\geq \frac {AB+AC} 4.$

2018 Serbia TST P5

Let $H$ be the orthocenter of $ABC$ ,$AB\neq AC$ ,and let $F$ be a point on circumcircle of $ABC$ such that $\angle AFH=90^{\circ}$.$K$ is the symmetric point of $H$ wrt $B$.Let $P$ be a point such that $\angle PHB=\angle PBC=90^{\circ}$,and $Q$ is the foot of $B$ to $CP$.Prove that $HQ$ is tangent to tge circumcircle of $FHK$.

2019 Serbia TST P2

Given triangle $\triangle ABC$ with $AC\neq BC$,and let $D$ be a point inside triangle such that $\measuredangle ADB=90^{\circ} + \frac {1}{2}\measuredangle ACB$.Tangents from $C$ to the circumcircles of $\triangle ABC$ and $\triangle ADC$ intersect $AB$ and $AD$ at $P$ and $Q$ , respectively.Prove that $PQ$ bisects the angle $\measuredangle BPC$.

EGMO TST 2012-19

2012 Serbia EGMO TST P3
Let $\omega$ be the circumcircle of an acute angled triangle $ABC$. On sides $AB$ and $AC$ of this triangle were select  $E$ and $D$, respectively, so that $\angle ABD = \angle ACE$. Lines $BD$ and $CE$ intersect the circle $\omega$ at $M$ and $N$ ($M \ne B$ and $N \ne C$), respectively. The tangents at the points $B$ and $C$ on the circle $\omega$, intersect the line  $DE$ at  $P$ and $Q$ respectively.  Prove that the intersection point of lines $PN$ and $QM$ lies on the circle $\omega$.

2012 Serbia EGMO TST P7
Let the $r$ of the radius of the incircle of the triangle $ABC$. Let  $D, E$, and $F$ be points on the sides $BC, CA$, and $AB$  respectively, such that the  incircles of the triangles $AEF, BFD$ and $CDE$ have equal radius  $\rho$. If $r '$ is the radius of the incircle of the triangle $DEF$, prove that it is  $\rho = r - r'$.

2013 Serbia EGMO TST P2

Let $O$ be the center of the circumcircle , and $AD$ ($D \in BC$) be the interior angle bisector of a triangle ABC. Let $\ell$ be a line passing through $O$ parallel to  $AD$. Prove that $\ell$  passes through the orthocenter of the triangle $ABC$ if and only if $\angle BAC = 120^o$.

2013 Serbia EGMO TST P4
Let $ABCD$ be a square of the plane $P$. Define the minimum and the maximum the value of the function $f: P \to R$ is given by $f (P) =\frac{PA + PB}{PC + PD}$

2014 Serbia EGMO TST P1
Let $ABC$ be a triangle. Circle $k_1$ passes through $A$ and $B$ and touches the line $AC$, and circle $k_2$ passes through $C$ and $A$ and touches the line $AB$. Circle $k_1$ intersects line $BC$ at point  $D$ ($D \ne B$) and also  intersects the circle $k_2$ at point $E$ ($E \ne A$). Prove that the line $DE$ bisects segment $AC$.

2015 Serbia EGMO TST P3
The incircle of the triangle $ABC$ touches the side $BC$ at point $D$. Let $I$ be the center of the incircle $k$, $M$ the midpoint of side $BC$, and $K$ the  orthocenter of triangle $AIB$. Prove that $KD$ is perpendicular to $IM$.

2016 Serbia EGMO TST P3
Let $ABCD$ be cyclic quadriateral and let $AC$ and $BD$ intersect at $E$ and $AB$ and $CD$ at $F$. Let $K$ be point in plane such that $ABKC$ is parallelogram. Prove $\angle AFE=\angle CDF$.

2017 Serbia EGMO TST P3
In $\triangle ABC$ the incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$, respectively. Let $G$ be the foot of the altitude from $D$ in $\triangle DEF$. Prove that one of intersections of line $IG$ and $\odot ABC$ is the antipode of $A$ in $\odot ABC$.

2018 Serbia EGMO TST P4
A parallelogram $ABCD$  is given. On sides $AB,  BC$ and the extension of side $CD$ behind the vertice $D$ , lie the points $K, L$ and $M$  respectively, such that $KL = BC, LM = CA$ and $MK = AB$. $KM$ intersects $AD$ at point $N$ .Prove that $LN  // AB$.

2018 Serbia EGMO TST P6

Let $ABCD$ be an isosceles trapezoid ($AB // CD$) . Let $E$ be  a point of that arc of $AB$ of the circumcircle of the trapezoid,  that does not contain the trapezoid. From each of $A$ and $B$, we drop perpendicular on $EC$ and $ED$. Prove that the those four projections of $A$ and $B$ on $EC$ and $ED$ are concyclic.

2019 Serbia EGMO TST P2

On the sides $BC, CA$ and $AB$ of triangle $ABC$ are given the points $P, Q$ and $R$ respectively so the quadrilateral $AQPR$ is cyclic and $BR = CQ$. Tangents to the circumscribed the circle of the triangle $ABC$ at points $B$ and $C$ intersect the $PR$ and $PQ$ at points $X$ and $Y$, in a row. Prove that $PX = PY$.

2020 Serbia EGMO TST P2

Let $ABC$ be an acute triangle  with  $\angle BAC = \alpha$. Let point $O$ be the circumcenter of this triangle, $H$ the orthocenter, and $F$ the midpoint of $AB$. Point $P$ lies inside the triangle such that $\angle APF = \alpha$ and $\angle APB = 180^o -\alpha$ . Prove that $\angle HPO = 180^o -\alpha$

RMM TST 2012-19

2012 Serbia RMM TST P2
Inside the convex quadrilateral $ABCD$ (not a trapezoid)  given is a point $X$ such that $\angle ADX =  \angle BCX <90^o$ and $\angle DAX =\angle CBX <90^o$. If the perpendicular bisectors of $AB$ and $CD$ intersect at point $Y$, prove that $\angle AYB = 2 \angle ADX$

2015 Serbia RMM TST P2
Given an acute isosceles triangle $ABC$ with  $AB = AC$. Point $M$ is the midpoint of the arc $BC$ (not containing $A$) of the circumcscribed circle of triangle $ABC$. The line passing through $M$ parallel to $AC$ intersects $BC$ and $AB$ at points $D$ and $E$, respectively. The line passing through $D$ parallel to $AB$ intersects AC at point F. Prove that $\angle M EF = 90^o$

2016 Serbia RMM TST P2

Given a pentagon with a broken line connecting two vertices and dividing into two congruent pentagons. Prove that the starting pentagon has a pair of equal angles, and that the pentagons intto which it is divided have two pairs of parallel sides.

2017 Serbia RMM TST P4

Given a circle $k$ with center at point $O$ and point $A \in  k$. Let $C$ be the symmetric point of $O$ wrt $A$ and $B$ be the midpoint of $AC$. A point $Q \in  k$ such that $\angle AOQ$ is obtuse. Let  the line $QO$ and the perpendicular bisector of $CQ$ intersect at point $P$. Prove $\angle P OB = 2\angle P BO$.

2018 Serbia RMM TST P2
Let $P$ be a point inside triangle $ABC$ such that $\angle AP B \ne 90^o$.  Lines $AP$ and $BP$ intersect the sides $BC$ and $AC$ respectively at points $D$ and $E$. Let $K$ and $L$ be the orthocentres of triangles $AP E$ and $BPD$. Prove that point $C$ lies on the line $KL$ if and only if the angle $ACB$ is right.

2019 Serbia RMM TST P4 (Brazil MO 2015 P6)

In a scalene triangle $ABC$, $H$ is the orthocenter and $G$ is the centroid. The points $X, Y$ and $Z$ are selected respectively on sides $BC, CA$ and $AB$ so that $\angle AXB =\angle BY C = \angle CZA$.  The circumscribed circles of triangles $BXZ$ and $CXY$ intersect at $P \ne  X$. Prove that $\angle GP H = 90^o$.