geometry problems from Belarusian Team Selection Tests (TST)
with aops links in the names
1995, 1998, 2000, 2009-12, 2014-19
1995 Belarus TST 1.2 (British MO 1996 Round 2 p3)
Circles $S,S_1,S_2$ are given in a plane. $S_1$ and $S_2$ touch each other externally, and both touch $S$ internally at $A_1$ and $A_2$ respectively. The common internal tangent to $S_1$ and $S_2$ meets $S$ at $P$ and $Q.$ Let $B_1$ and $B_2$ be the intersections of $PA_1$ and $PA_2$ with $S_1$ and $S_2$, respectively. Prove that $B_1B_2$ is a common tangent to $S_1,S_2$
1995 Belarus TST 2.2
There is a room having a form of right-angled parallelepiped. Four maps of the same scale are hung (generally, on different levels over the floor) on four walls of the room, so that sides of the maps are parallel to sides of the wall. It is known that the four points corresponding to each of Stockholm, Moscow, and Istanbul are coplanar. Prove that the four points coresponding to Hong Kong are coplanar as well.
1996-1997 missing
Let $A,B,C$ denote intersection points of diagonals $A_1A_4$ and $A_2A_5$, $A_1A_6$ and $A_2A_7$, $A_1A_9$ and $A_2A_{10}$ of the regular decagon $A_1A_2...A_{10}$ respectively.
Find the angles of the triangle $ABC$
2017 Belarus TST 1.3
Let $H$ be the orthocenter of an acute triangle $ABC$, $AH=2$, $BH=12$, $CH=9$.
Find the area of the triangle $ABC$.
2019 Belarus TST 1.2
Points $M$ and $N$ are the midpoints of the sides $BC$ and $AD$, respectively, of a convex quadrilateral $ABCD$. Is it possible that $AB+CD>\max(AM+DM,BN+CN)?$
Let $O$ be the circumcenter and $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $T$ is the midpoint of the segment $AO$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$. Prove that the circumcircle of the triangle $AST$ bisects the segment $OH$.
Let $AA_1$ be the bisector of a triangle $ABC$. Points $D$ and $F$ are chosen on the line $BC$ such that $A_1$ is the midpoint of the segment $DF$. A line $l$, different from $BC$, passes through $A_1$ and intersects the lines $AB$ and $AC$ at points $B_1$ and $C_1$, respectively.
Find the locus of the points of intersection of the lines $B_1D$ and $C_1F$ for all possible positions of $l$.
Two circles $\Omega$ and $\Gamma$ are internally tangent at the point $B$. The chord $AC$ of $\Gamma$ is tangent to $\Omega$ at the point $L$, and the segments $AB$ and $BC$ intersect $\Omega$ at the points $M$ and $N$. Let $M_1$ and $N_1$ be the reflections of $M$ and $N$ about the line $BL$; and let $M_2$ and $N_2$ be the reflections of $M$ and $N$ about the line $AC$. The lines $M_1M_2$ and $N_1N_2$ intersect at the point $K$.
Prove that the lines $BK$ and $AC$ are perpendicular.
The internal bisectors of angles $\angle DAB$ and $\angle BCD$ of a quadrilateral $ABCD$ intersect at the point $X_1$, and the external bisectors of these angles intersect at the point $X_2$. The internal bisectors of angles $\angle ABC$ and $\angle CDA$ intersect at the point $Y_1$, and the external bisectors of these angles intersect at the point $Y_2$.
Prove that the angle between the lines $X_1X_2$ and $Y_1Y_2$ equals the angle between the diagonals $AC$ and $BD$.
with aops links in the names
(only those not in IMO Shortlist)
collected inside aops here
1995, 1998, 2000, 2009-12, 2014-19
1995 Belarus TST 1.2 (British MO 1996 Round 2 p3)
Circles $S,S_1,S_2$ are given in a plane. $S_1$ and $S_2$ touch each other externally, and both touch $S$ internally at $A_1$ and $A_2$ respectively. The common internal tangent to $S_1$ and $S_2$ meets $S$ at $P$ and $Q.$ Let $B_1$ and $B_2$ be the intersections of $PA_1$ and $PA_2$ with $S_1$ and $S_2$, respectively. Prove that $B_1B_2$ is a common tangent to $S_1,S_2$
1995 Belarus TST 2.2
There is a room having a form of right-angled parallelepiped. Four maps of the same scale are hung (generally, on different levels over the floor) on four walls of the room, so that sides of the maps are parallel to sides of the wall. It is known that the four points corresponding to each of Stockholm, Moscow, and Istanbul are coplanar. Prove that the four points coresponding to Hong Kong are coplanar as well.
1996-1997 missing
1998 Belarus TST 2.3
2000 Belarus TST 2.1
All vertices of a convex polyhedron are endpoints of exactly four edges. Find the minimal possible number of triangular faces of the polyhedron.
2000 Belarus TST 2.4
In a triangle ABC with $AC = b \ne BC = a$, points $E,F$ are taken on the sides $AC,BC$ respectively such that $AE = BF =\frac{ab}{a+b}$. Let $M$ and $N$ be the midpoints of $AB$ and $EF$ respectively, and $P$ be the intersection point of the segment $EF$ with the bisector of $\angle ACB$. Find the ratio of the area of $CPMN$ to that of $ABC$.
2000 Belarus TST 3.1
In a triangle $ABC$, let $a = BC, b = AC$ and let $m_a,m_b$ be the corresponding medians.
Find all real numbers $k$ for which the equality $m_a+ka = m_b +kb$ implies that $a = b$.
2000 Belarus TST 5.1
Let $AM$ and $AL$ be the median and bisector of a triangle $ABC$ ($M,L \in BC$).
If $BC = a, AM = m_a, AL = l_a$, prove the inequalities:
(a) $a\tan \frac{a}{2} \le 2m_a \le a \cot \frac{a}{2} $ if $a < \frac{\pi}{2}$ and $a\tan \frac{a}{2} \ge 2m_a \ge a \cot \frac{a}{2} $ if $a > \frac{\pi}{2}$
(b) $2l_a \le a\cot \frac{a}{2} $.
2000 Belarus TST 8.1
The diagonals of a convex quadrilateral $ABCD$ with $AB = AC = BD$ intersect at $P$, and $O$ and $I$ are the circumcenter and incenter of $\vartriangle ABP$, respectively. Prove that if $O \ne I$ then $OI$ and $CD$ are perpendicular
2001-2008 missing
2009 Belarus TST 1.1
Two equal circles $S_1$ and $S_2$ meet at two different points. The line $\ell$ intersects $S_1$ at points $A,C$ and $S_2$ at points $B,D$ respectively (the order on $\ell$: $A,B,C,D$) . Define circles $\Gamma_1$ and $\Gamma_2$ as follows: both $\Gamma_1$ and $\Gamma_2$ touch $S_1$ internally and $S_2$ externally, both $\Gamma_1$ and $\Gamma_2$ line $\ell$, $\Gamma_1$ and $\Gamma_2$ lie in the different halfplanes relatively to line $\ell$. Suppose that $\Gamma_1$ and $\Gamma_2$ touch each other. Prove that $AB=CD$.
Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle. Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$
Let $O$ be a point inside an acute angle with the vertex $A$ and $H, N$ be the feet of the perpendiculars drawn from $O$ onto the sides of the angle. Let point $B$ belong to the bisector of the angle, $K$ be the foot of the perpendicular from $B$ onto either side of the angle. Denote by $P,F$ the midpoints of the segments $AK,HN$ respectively. Known that $ON + OH = BK$, prove that $PF$ is perpendicular to $AB$.
I. Voronovich
Ya. Konstantinovski
The incircle of the triangle $ABC$ touches its sides $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. If $r$ is the inradius of $\vartriangle ABC, P,P_1$ are the perimeters of $\vartriangle ABC, \vartriangle A_1B_1C_1$ respectively, prove that $P+P_1 \ge 9 \sqrt3 r$.
I. Voronovich
Two circles $S_1$ and $S_2$ intersect at different points $P,Q$. The arc of $S_1$ lying inside $S_2$ measures $2a$ and the arc of $S_2$ lying inside $S_1$ measures $2b$. Let $T$ be any point on $S_1$. Let $R,S$ be another points of intersection of $S_2$ with $TP$ and $TQ$ respectively. Let $a+2b<\pi$ . Find the locus of the intersection points of $PS$ and $RQ$.
S.Shikh
Let $P$ be a point inside a triangle $ABC$ with $\angle C = 90^o$ such that $AP = AC$, and let $M$ be the midpoint of $AB$ and $CH$ be the altitude. Prove that $PM$ bisects $\angle BPH$ if and only if $\angle A = 60^o$.
All vertices of a convex polyhedron are endpoints of exactly four edges. Find the minimal possible number of triangular faces of the polyhedron.
2000 Belarus TST 2.4
In a triangle ABC with $AC = b \ne BC = a$, points $E,F$ are taken on the sides $AC,BC$ respectively such that $AE = BF =\frac{ab}{a+b}$. Let $M$ and $N$ be the midpoints of $AB$ and $EF$ respectively, and $P$ be the intersection point of the segment $EF$ with the bisector of $\angle ACB$. Find the ratio of the area of $CPMN$ to that of $ABC$.
2000 Belarus TST 3.1
In a triangle $ABC$, let $a = BC, b = AC$ and let $m_a,m_b$ be the corresponding medians.
Find all real numbers $k$ for which the equality $m_a+ka = m_b +kb$ implies that $a = b$.
Let $AM$ and $AL$ be the median and bisector of a triangle $ABC$ ($M,L \in BC$).
If $BC = a, AM = m_a, AL = l_a$, prove the inequalities:
(a) $a\tan \frac{a}{2} \le 2m_a \le a \cot \frac{a}{2} $ if $a < \frac{\pi}{2}$ and $a\tan \frac{a}{2} \ge 2m_a \ge a \cot \frac{a}{2} $ if $a > \frac{\pi}{2}$
(b) $2l_a \le a\cot \frac{a}{2} $.
2000 Belarus TST 8.1
The diagonals of a convex quadrilateral $ABCD$ with $AB = AC = BD$ intersect at $P$, and $O$ and $I$ are the circumcenter and incenter of $\vartriangle ABP$, respectively. Prove that if $O \ne I$ then $OI$ and $CD$ are perpendicular
2001-2008 missing
2009 Belarus TST 1.1
Two equal circles $S_1$ and $S_2$ meet at two different points. The line $\ell$ intersects $S_1$ at points $A,C$ and $S_2$ at points $B,D$ respectively (the order on $\ell$: $A,B,C,D$) . Define circles $\Gamma_1$ and $\Gamma_2$ as follows: both $\Gamma_1$ and $\Gamma_2$ touch $S_1$ internally and $S_2$ externally, both $\Gamma_1$ and $\Gamma_2$ line $\ell$, $\Gamma_1$ and $\Gamma_2$ lie in the different halfplanes relatively to line $\ell$. Suppose that $\Gamma_1$ and $\Gamma_2$ touch each other. Prove that $AB=CD$.
I. Voronovich
Points $T,P,H$ lie on the side $BC,AC,AB$ respectively of triangle $ABC$, so that $BP$ and $AT$ are angle bisectors and $CH$ is an altitude of $ABC$. Given that the midpoint of $CH$ belongs to the segment $PT,$ find the value of $\cos A + \cos B$
I. Voronovich
Given trapezoid $ABCD$ ($AD\parallel BC$) with $AD \perp AB$ and $T=AC\cap BD$.
A circle centered at point $O$ is inscribed in the trapezoid and touches the side $CD$ at point $Q$.
Let $P$ be the intersection point (different from $Q$) of the side $CD$ and the circle passing through $T,Q$ and $O$. Prove that $TP \parallel AD$.
Let $M,N$ be the midpoints of the sides $AD,BC$ respectively of the convex quadrilateral $ABCD$,
$K=AN \cap BM$, $L=CM \cap DN$. Find the smallest possible $c\in R$ such that $S(MKNL)<c \cdot S(ABCD)$ for any convex quadrilateral $ABCD$.
In a triangle $ABC, AM$ is a median, $BK$ is a bisectrix, $L=AM\cap BK$. It is known that $BC=a, AB=c, a>c$. Given that the circumcenter of triangle $ABC$ lies on the line $CL$, find $AC$
A circle centered at point $O$ is inscribed in the trapezoid and touches the side $CD$ at point $Q$.
Let $P$ be the intersection point (different from $Q$) of the side $CD$ and the circle passing through $T,Q$ and $O$. Prove that $TP \parallel AD$.
I. Voronovich
2009 Belarus TST 5.1Let $M,N$ be the midpoints of the sides $AD,BC$ respectively of the convex quadrilateral $ABCD$,
$K=AN \cap BM$, $L=CM \cap DN$. Find the smallest possible $c\in R$ such that $S(MKNL)<c \cdot S(ABCD)$ for any convex quadrilateral $ABCD$.
I. Voronovich
2009 Belarus TST 6.1In a triangle $ABC, AM$ is a median, $BK$ is a bisectrix, $L=AM\cap BK$. It is known that $BC=a, AB=c, a>c$. Given that the circumcenter of triangle $ABC$ lies on the line $CL$, find $AC$
I. Voronovich
Does there exist a convex pentagon $A_1A_2A_3A_4A_5$ and a point $X$ inside it such that $XA_i=A_{i+2}A_{i+3}$ for all $i=1,...,5$ (all indices are considered modulo $5$) ?
I. Voronovich
Points $H$ and $T$ are marked respectively on the sides $BC$ abd $AC$ of triangle $ABC$ so that $AH$ is the altitude and $BT$ is the bisectrix $ABC$. It is known that the gravity center of $ABC$ lies on the line $HT$.
a) Find $AC$ if $BC$=a and $AB$=c.
b) Determine all possible values of $\frac{c}{a}$ for all triangles $ABC$ satisfying the given condition.
I. Voronovich
2010 Belarus TST 2.1
Point $D$ is marked inside a triangle $ABC$ so that $\angle ADC = \angle ABC + 60^o$, $\angle CDB =\angle CAB + 60^o$, $\angle BDA = \angle BCA + 60^o$. Prove that $AB \cdot CD = BC \cdot AD = CA \cdot BD$.
A. Levin
2010 Belarus TST 3.1
Let $I$ be an incenter of a triangle $ABC, A_1,B_1,C_1$ be intersection points of the circumcircle of the triangle $ABC$ and the lines $AI, BI, Cl$ respectively. Prove that
a) $\frac{AI}{IA_1}+ \frac{BI}{IB_1}+ \frac{CI}{IC_1}\ge 3$ b) $AI \cdot BI \cdot CI \le I_1A_1\cdot I_2B_1 \cdot I_1C_1$
D. Pirshtuk
2011 Belarus TST 1.2
Points $L$ and $H$ are marked on the sides $AB$ of an acute-angled triangle ABC so that $CL$ is a bisector and $CH$ is an altitude. Let $P,Q$ be the feet of the perpendiculars from $L$ to $AC$ and $BC$ respectively. Prove that $AP \cdot BH = BQ \cdot AH$.
The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $P$ and $Q$ respectively. $N$ and $M$ are the midpoints of $AC$ and $BC$ respectively. Let $X=AM\cap BP, Y=BN\cap AQ$. Given $C,X,Y$ are collinear, prove that $CX$ is the angle bisector of the angle $ACB$.
2016 Belarus TST 1.2Points $L$ and $H$ are marked on the sides $AB$ of an acute-angled triangle ABC so that $CL$ is a bisector and $CH$ is an altitude. Let $P,Q$ be the feet of the perpendiculars from $L$ to $AC$ and $BC$ respectively. Prove that $AP \cdot BH = BQ \cdot AH$.
I. Gorodnin
2011 Belarus TST 2.2
Two different points $X,Y$ are marked on the side $AB$ of a triangle $ABC$ so that $\frac{AX \cdot BX}{CX^2}=\frac{AY \cdot BY}{CY^2}$ . Prove that $\angle ACX=\angle BCX$.
2015 Belarus TST 1.3Two different points $X,Y$ are marked on the side $AB$ of a triangle $ABC$ so that $\frac{AX \cdot BX}{CX^2}=\frac{AY \cdot BY}{CY^2}$ . Prove that $\angle ACX=\angle BCX$.
I.Zhuk
2011 Belarus TST 3.2
The external angle bisector of the angle $A$ of an acute-angled triangle $ABC$ meets the circumcircle of $\vartriangle ABC$ at point $T$. The perpendicular from the orthocenter $H$ of $\vartriangle ABC$ to the line $TA$ meets the line $BC$ at point $P$. The line $TP$ meets the circumcircce of $\vartriangle ABC$ at point $D$. Prove that $AB^2+DC^2=AC^2+BD^2$
The external angle bisector of the angle $A$ of an acute-angled triangle $ABC$ meets the circumcircle of $\vartriangle ABC$ at point $T$. The perpendicular from the orthocenter $H$ of $\vartriangle ABC$ to the line $TA$ meets the line $BC$ at point $P$. The line $TP$ meets the circumcircce of $\vartriangle ABC$ at point $D$. Prove that $AB^2+DC^2=AC^2+BD^2$
A. Voidelevich
2011 Belarus TST 6.1
$AB$ and $CD$ are two parallel chords of a parabola. Circle $S_1$ passing through points $A,B$ intersects circle $S_2$ passing through $C,D$ at points $E,F$. Prove that if $E$ belongs to the parabola, then $F$ also belongs to the parabola.
$AB$ and $CD$ are two parallel chords of a parabola. Circle $S_1$ passing through points $A,B$ intersects circle $S_2$ passing through $C,D$ at points $E,F$. Prove that if $E$ belongs to the parabola, then $F$ also belongs to the parabola.
I.Voronovich
2011 Belarus TST 7.1
In an acute-angled triangle $ABC$, the orthocenter is $H$. $I_H$ is the incenter of $\vartriangle BHC$. The bisector of $\angle BAC$ intersects the perpendicular from $I_H$ to the side $BC$ at point $K$. Let $F$ be the foot of the perpendicular from $K$ to $AB$. Prove that $2KF+BC=BH +HC$
In an acute-angled triangle $ABC$, the orthocenter is $H$. $I_H$ is the incenter of $\vartriangle BHC$. The bisector of $\angle BAC$ intersects the perpendicular from $I_H$ to the side $BC$ at point $K$. Let $F$ be the foot of the perpendicular from $K$ to $AB$. Prove that $2KF+BC=BH +HC$
A. Voidelevich
$A, B, C, D, E$ are five points on the same circle, so that $ABCDE$ is convex and we have $AB = BC$ and $CD = DE$. Suppose that the lines $(AD)$ and $(BE)$ intersect at $P$, and that the line $(BD)$ meets line $(CA)$ at $Q$ and line $(CE)$ at $T$. Prove that the triangle $PQT$ is isosceles.
I. Voronovich
Let $\Gamma$ be the incircle of an non-isosceles triangle $ABC$, $I$ be it’s center. Let $A_1, B_1, C_1$ be the tangency points of $\Gamma$ with the sides $BC, AC, AB$, respectively. Let $A_2 = \Gamma \cap AA_1, M = C_1B_1 \cup AI$, $P$ and $Q$ be the other (different from $A_1, A_2$) intersection points of $A_1M, A_2M$ and $\Gamma$, respectively. Prove that $A, P, Q$ are collinear.
A. Voidelevich
For any point $X$ inside an acute-angled triangle $ABC$ we define$$f(X)=\frac{AX}{A_1X}\cdot \frac{BX}{B_1X}\cdot \frac{CX}{C_1X}$$where $A_1, B_1$, and $C_1$ are the intersection points of the lines $AX, BX,$ and $CX$ with the sides $BC, AC$, and $AB$, respectively. Let $H, I$, and $G$ be the orthocenter, the incenter, and the centroid of the triangle $ABC$, respectively. Prove that $f(H) \ge f(I) \ge f(G)$ .
D. Bazylev
Two distinct points $A$ and $B$ are marked on the left half of the parabola $y = x^2$. Consider any pair of parallel lines which pass through $A$ and $B$ and intersect the right half of the parabola at points $C$ and $D$. Let $K$ be the intersection point of the diagonals $AC$ and $BD$ of the obtained trapezoid $ABCD$. Let $M, N$ be the midpoints of the bases of $ABCD$. Prove that the difference $KM - KN$ depends only on the choice of points $A$ and $B$ but does not depend on the pair of parallel lines described above
I. Voronovich
Determine the greatest possible value of the constant $c$ that satisfies the following condition: for any convex heptagon the sum of the lengthes of all it’s diagonals is greater than $cP$, where $P$ is the perimeter of the heptagon.
I. Zhuk
2013 missing
All vertices of triangles $ABC$ and $A_1B_1C_1$ lie on the hyperbola $y=1/x$. It is known that $AB \parallel A_1B_1$ and $BC \parallel B_1C_1$. Prove that $AC_1 \parallel A_1C$.
I. Gorodnin
Given a triangle $ABC$. Let $S$ be the circle passing through $C$, centered at $A$. Let $X$ be a variable point on $S$ and let $K$ be the midpoint of the segment $CX$ . Find the locus of the midpoints of $BK$, when $X$ moves along $S$.
I. Gorodnin
Point $L$ is marked on the side $AB$ of a triangle $ABC$. The incircle of the triangle $ABC$ meets the segment $CL$ at points $P$ and $Q$ .Is it possible that the equalities $CP = PQ = QL$ hold if $CL$ is
a) the median?
b) the bisector?
c) the altitude?
d) the segment joining vertex $C$ with the point $L$ of tangency of the excircle of the triangie $ABC$ with $AB$ ?
I. Gorodnin
Circles $\Gamma_1$ and $\Gamma_2$ meet at points $X$ and $Y$. A circle $S_1$ touches internally $\Gamma_1$ at $A$ and $\Gamma_2$ externally at $B$. A circle $S_2$ touches $\Gamma_2$ internally at $C$ and $\Gamma_1$ externally at $D$. Prove that the points $A, B, C, D$ are either collinear or concyclic.
A. Voidelevich
Given triangle $ABC$ with $\angle A = a$. Let $AL$ be the bisector of the triangle $ABC$. Let the incircle of $\vartriangle ABC$ touch the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $X$ be the intersection point of the lines $AQ$ and $LP$. Prove that the lines $BX$ and $AL$ are perpendicular.
V. Karamzin
Let $O$ be the circumcenter of an acute-angled triangle $ABC$. Let $AH$ be the altitude of this triangle, $M,N,P,Q$ be the midpoints of the segments $AB, AC, BH, CH$, respectively. Let $\omega_1$ and $\omega_2$ be the circumferences of the triangles $AMN$ and $POQ$. Prove that one of the intersection points of $\omega_1$ and $\omega_2$ belongs to the altitude $AH$.
A. Voidelevich
Let $I$ be the incenter of a triangle $ABC$. The circle passing through $I$ and centered at $A$ meets the circumference of the triangle $ABC$ at points $M$ and $N$. Prove that the line $MN$ touches the incircle of the triangle $ABC$.
I. Kachan
Let $\Gamma_B$ and $\Gamma_C$ be excircles of an acute-angled triangle $ABC$ opposite to its vertices $B$ and $C$, respectively. Let $C_1$ and $L$ be the tangent points of $\Gamma_C$ and the side $AB$ and the line $BC$ respectively. Let $B_1$ and $M$ be the tangent points of $\Gamma_B$ and the side $AC$ and the line $BC$, respectively. Let $X$ be the point of intersection of the lines $LC_1$ and $MB_1$. Prove that $AX$ is equal to the inradius of the triangle $ABC$.
A. Voidelevich
Let $AA_1, BB_1$ be the altitudes of an acute non-isosceles triangle $ABC$. Circumference of the triangles $ABC$ meets that of the triangle $A_1B_1C$ at point $N$ (different from $C$). Let $M$ be the midpoint of $AB$ and $K$ be the intersection point of $CN$ and $AB$. Prove that the line of centers the circumferences of the triangles $ABC$ and $KMC$ is parallel to the line $AB$.
I. Kachan
The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $P$ and $Q$ respectively. $N$ and $M$ are the midpoints of $AC$ and $BC$ respectively. Let $X=AM\cap BP, Y=BN\cap AQ$. Given $C,X,Y$ are collinear, prove that $CX$ is the angle bisector of the angle $ACB$.
I. Gorodnin
2015 Belarus TST 2.3
Let the incircle of the triangle $ABC$ touch the side $AB$ at point $Q$. The incircles of the triangles $QAC$ and $QBC$ touch $AQ,AC$ and $BQ,BC$ at points $P,T$ and $D,F$ respectively. Prove that $PDFT$ is a cyclic quasrilateral.
Let the incircle of the triangle $ABC$ touch the side $AB$ at point $Q$. The incircles of the triangles $QAC$ and $QBC$ touch $AQ,AC$ and $BQ,BC$ at points $P,T$ and $D,F$ respectively. Prove that $PDFT$ is a cyclic quasrilateral.
I.Gorodnin
2015 Belarus TST 3.2
The medians $AM$ and $BN$ of a triangle $ABC$ are the diameters of the circles $\omega_1$ and $\omega_2$. If $\omega_1$ touches the altitude $CH$, prove that $\omega_2$ also touches $CH$.
The medians $AM$ and $BN$ of a triangle $ABC$ are the diameters of the circles $\omega_1$ and $\omega_2$. If $\omega_1$ touches the altitude $CH$, prove that $\omega_2$ also touches $CH$.
I. Gorodnin
2015 Belarus TST 4.1
A circle intersects a parabola at four distinct points. Let $M$ and $N$ be the midpoints of the arcs of the circle which are outside the parabola. Prove that the line $MN$ is perpendicular to the axis of the parabola.
A circle intersects a parabola at four distinct points. Let $M$ and $N$ be the midpoints of the arcs of the circle which are outside the parabola. Prove that the line $MN$ is perpendicular to the axis of the parabola.
I. Voronovich
2015 Belarus TST 7.2
Given a cyclic $ABCD$ with $AB=AD$. Points $M$ and $N$ are marked on the sides $CD$ and $BC$, respectively, so that $DM+BN=MN$. Prove that the circumcenter of the triangle $AMN$ belongs to the segment $AC$.
Given a cyclic $ABCD$ with $AB=AD$. Points $M$ and $N$ are marked on the sides $CD$ and $BC$, respectively, so that $DM+BN=MN$. Prove that the circumcenter of the triangle $AMN$ belongs to the segment $AC$.
N.Sedrakian
Let $A,B,C$ denote intersection points of diagonals $A_1A_4$ and $A_2A_5$, $A_1A_6$ and $A_2A_7$, $A_1A_9$ and $A_2A_{10}$ of the regular decagon $A_1A_2...A_{10}$ respectively.
Find the angles of the triangle $ABC$
Folkore
Let $a,b,c,d,x,y$ denote the lengths of the sides $AB, BC,CD,DA$ and the diagonals $AC,BD$ of a cyclic quadrilateral $ABCD$ respectively.
Prove that $$(\frac{1}{a}+\frac{1}{c})^2+(\frac{1}{b}+\frac{1}{d})^2 \geq 8 ( \frac{1}{x^2}+\frac{1}{y^2})$$
Prove that $$(\frac{1}{a}+\frac{1}{c})^2+(\frac{1}{b}+\frac{1}{d})^2 \geq 8 ( \frac{1}{x^2}+\frac{1}{y^2})$$
V. Karamzin
2016 Belarus TST 2.3
Point $A,B$ are marked on the right branch of the hyperbola $y=\frac{1}{x},x>0$. The straight line $l$ passing through the origin $O$ is perpendicular to $AB$ and meets $AB$ and given branch of the hyperbola at points $D$ and $C$ respectively. The circle through $A,B,C$ meets $l$ at $F$.
Find $OD:CF$
I. Voronovich
2016 Belarus TST 3.3
Let $D,E,F$ denote the tangent points of the incircle of $ABC$ with sides $BC,AC,AB$ respectively. Let $M$ be the midpoint of the segment $EF$. Let $L$ be the intersection point of the circle passing through $D,M,F$ and the segment $AB$, $K$ be the intersection point of the circle passing through $D,M,E$ and the segment $AC$. Prove that the circle passing through $A,K,L$ touches the line $BC$
Let $D,E,F$ denote the tangent points of the incircle of $ABC$ with sides $BC,AC,AB$ respectively. Let $M$ be the midpoint of the segment $EF$. Let $L$ be the intersection point of the circle passing through $D,M,F$ and the segment $AB$, $K$ be the intersection point of the circle passing through $D,M,E$ and the segment $AC$. Prove that the circle passing through $A,K,L$ touches the line $BC$
V.Voinov
2016 Belarus TST 4.2
A point $A_1$ is marked inside an acute non-isosceles triangle $ABC$ such that $\angle A_1AB = \angle A_1BC$ and $\angle A_1AC=\angle A_1CB$. Points $B_1$ and $C_1$ are defined same way. Let $G$ be the gravity center if the triangle $ABC$. Prove that the points $A_1,B_1,C_1,G$ are concyclic.
D.Voinov
A point $A_1$ is marked inside an acute non-isosceles triangle $ABC$ such that $\angle A_1AB = \angle A_1BC$ and $\angle A_1AC=\angle A_1CB$. Points $B_1$ and $C_1$ are defined same way. Let $G$ be the gravity center if the triangle $ABC$. Prove that the points $A_1,B_1,C_1,G$ are concyclic.
D.Voinov
2016 Belarus TST 6.2
Points $B_1$ and $C_1$ are marked respectively on the sides $AB$ and $AC$ of an acute isosceles triangle $ABC$( $AB=AC$) such that $BB_1=AC_1$. The points $B,C$ and $S$ lie in the same half-plane with respect to the line $B_1C_1$ so that $\angle SB_1C_1=\angle SC_1B_1 = \angle BAC$. Prove that $B,C,S$ are colinear if and only if the triangle $ABC$ is equilateral.
Points $B_1$ and $C_1$ are marked respectively on the sides $AB$ and $AC$ of an acute isosceles triangle $ABC$( $AB=AC$) such that $BB_1=AC_1$. The points $B,C$ and $S$ lie in the same half-plane with respect to the line $B_1C_1$ so that $\angle SB_1C_1=\angle SC_1B_1 = \angle BAC$. Prove that $B,C,S$ are colinear if and only if the triangle $ABC$ is equilateral.
I. Kachan
2016 Belarus TST 8.2
Let $K$ and $L$ be the centers of the excircles of a non-isosceles triangle $ABC$ opposite $B$ and $C$ respectively. Let $B_1$ and $C_1$ be the midpoints of the sides $AC$ and $AB$ respectively Let $M$ and $N$ be symmetric to $B$ and $C$ about $B_1$ and $C_1$ respectively.Prove that the lines $KM$ and $LN$ meet on $BC$.
Let $K$ and $L$ be the centers of the excircles of a non-isosceles triangle $ABC$ opposite $B$ and $C$ respectively. Let $B_1$ and $C_1$ be the midpoints of the sides $AC$ and $AB$ respectively Let $M$ and $N$ be symmetric to $B$ and $C$ about $B_1$ and $C_1$ respectively.Prove that the lines $KM$ and $LN$ meet on $BC$.
D.Voinov
Let $H$ be the orthocenter of an acute triangle $ABC$, $AH=2$, $BH=12$, $CH=9$.
Find the area of the triangle $ABC$.
2017 Belarus TST 1.4
Let four parallel lines $l_1$, $l_2$, $l_3$, and $l_4$ meet the hyperbola $y=1/x$ at points $A_1$ and $B_1$, $A_2$ and $B_2$, $A_3$ and $B_3$, $A_4$ and $B_4$, respectively. Prove that the areas of the quadrilaterals $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ are equal.
2017 Belarus TST 2.4
Given triangle $ABC$, let $D$ be an inner point of the side $BC$. Let $P$ and $Q$ be distinct inner points of the segment $AD$. Let $K=BP\cap AC$, $L=CP\cap AB$, $E=BQ\cap AC$, $F=CQ\cap AB$. Given that $KL\parallel EF$, find all possible values of the ratio $BD:DC$.
2017 Belarus TST 3.1 (also)
Let $I$ be the incenter of a non-isosceles triangle $ABC$. The line $AI$ intersects the circumcircle of the triangle $ABC$ at $A$ and $D$. Let $M$ be the middle point of the arc $BAC$. The line through the point $I$ perpendicular to $AD$ intersects $BC$ at $F$. The line $MI$ intersects the circle $BIC$ at $N$. Prove that the line $FN$ is tangent to the circle $BIC$.
2017 Belarus TST 6.3
Given an isosceles triangle $ABC$ with $AB=AC$. let $\omega(XYZ)$ be the circumcircle of a triangle $XYZ$. Tangents to $\omega(ABC)$ at $B$ and $C$ meet at $D$. Point $F$ is marked on the arc $AB$ (opposite to $C$). Let $K$, $L$ be the intersection points of $AF$ and $BD$, $AB$ and $CF$, respectively.
Prove that if circles $\omega(BTS)$ and $\omega(CFK)$ are tangent to each other, the their tangency point belongs to $AB$. (Here $T$ and $S$ are the centers of the circles $\omega(BLC)$ and $\omega(BLK)$, respectively.)
2018 Belarus TST 1.2
Given the parallelogram $ABCD$. The circle $S_1$ passes through the vertex $C$ and touches the sides $BA$ and $AD$ at points $P_1$ and $Q_1$ respectively. The circle S_2 passes through the vertex B and touches the sides $DC$ and $AD$ at points $P_2$ and $Q_2$ respectively. Let $d_1 , d_2$ be the distances from $C$ and $B$ to the lines $P_1Q_1$ and $P_2Q_2$ respectively. Find all possible values of the ratio $d_1 : d_2$ .
Let four parallel lines $l_1$, $l_2$, $l_3$, and $l_4$ meet the hyperbola $y=1/x$ at points $A_1$ and $B_1$, $A_2$ and $B_2$, $A_3$ and $B_3$, $A_4$ and $B_4$, respectively. Prove that the areas of the quadrilaterals $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ are equal.
Given triangle $ABC$, let $D$ be an inner point of the side $BC$. Let $P$ and $Q$ be distinct inner points of the segment $AD$. Let $K=BP\cap AC$, $L=CP\cap AB$, $E=BQ\cap AC$, $F=CQ\cap AB$. Given that $KL\parallel EF$, find all possible values of the ratio $BD:DC$.
Let $I$ be the incenter of a non-isosceles triangle $ABC$. The line $AI$ intersects the circumcircle of the triangle $ABC$ at $A$ and $D$. Let $M$ be the middle point of the arc $BAC$. The line through the point $I$ perpendicular to $AD$ intersects $BC$ at $F$. The line $MI$ intersects the circle $BIC$ at $N$. Prove that the line $FN$ is tangent to the circle $BIC$.
Given an isosceles triangle $ABC$ with $AB=AC$. let $\omega(XYZ)$ be the circumcircle of a triangle $XYZ$. Tangents to $\omega(ABC)$ at $B$ and $C$ meet at $D$. Point $F$ is marked on the arc $AB$ (opposite to $C$). Let $K$, $L$ be the intersection points of $AF$ and $BD$, $AB$ and $CF$, respectively.
Prove that if circles $\omega(BTS)$ and $\omega(CFK)$ are tangent to each other, the their tangency point belongs to $AB$. (Here $T$ and $S$ are the centers of the circles $\omega(BLC)$ and $\omega(BLK)$, respectively.)
2018 Belarus TST 1.2
Given the parallelogram $ABCD$. The circle $S_1$ passes through the vertex $C$ and touches the sides $BA$ and $AD$ at points $P_1$ and $Q_1$ respectively. The circle S_2 passes through the vertex B and touches the sides $DC$ and $AD$ at points $P_2$ and $Q_2$ respectively. Let $d_1 , d_2$ be the distances from $C$ and $B$ to the lines $P_1Q_1$ and $P_2Q_2$ respectively. Find all possible values of the ratio $d_1 : d_2$ .
I.Voronovich
Let $A_1H_1,A_2H_2,A_3H_3$ be altitudes and $A_1L_1,A_2L_2,A_3L_3$ be bisectors of an acute angled triangle $A_1A_2A_3$ . Prove the inequality $S(L_1L_2L_3) \ge S(H_1H_2H_3)$ where $S$ stands for the area of a triangle.
D. Bazylev
2018 Belarus TST 2.2
Given the isosceles triangle $ABC$ ($CA = CB$). The bisector of the angle $\angle B$ intersects the side $AC$ at point $L$ and the circumcircle of the triangle $ABC$ at point $D$. It is known that $\angle C > 60^o$ . Prove that $DC + DL \le BC$ .
Given the isosceles triangle $ABC$ ($CA = CB$). The bisector of the angle $\angle B$ intersects the side $AC$ at point $L$ and the circumcircle of the triangle $ABC$ at point $D$. It is known that $\angle C > 60^o$ . Prove that $DC + DL \le BC$ .
I.Voronovich
Points $C_1$ and $B_1$ are marked on the sides $AB$ and $AC$ of the triangle $ABC$ respectively. Segments $BB_1$ and $CC_1$ intersect at point $X$ , and segments $B_1C_1$ and $AX$ intersect at point $A_1$ . The circumcircles of the triangles $BXC_1$ and $CXB_1$ intersect the side $BC$ at points $D$ and $E$ respectively. Lines $B_1D$ and $C_1E$ intersect at point $F$ . Prove that the lines $A_1F , B_1E$ and $C_1D$ are either parallel or concurrent.
A.Voidelevich
2018 Belarus TST 5.3
Given a convex quadrilateral $ABCD$. The point $A_1$ is on the border of $ABCD$ such that the segment $AA_1$ divides $ABCD$ into two parts with equal areas. In the same way we define points $B_1 , C_1$ and $D_1$ . It is known that the lengths of all segments $AA_1 , BB_1 , CC_1$ , and $DD_1$ do not exceed $1$. Prove that the area $S(ABCD) < \frac23$ .
Given a convex quadrilateral $ABCD$. The point $A_1$ is on the border of $ABCD$ such that the segment $AA_1$ divides $ABCD$ into two parts with equal areas. In the same way we define points $B_1 , C_1$ and $D_1$ . It is known that the lengths of all segments $AA_1 , BB_1 , CC_1$ , and $DD_1$ do not exceed $1$. Prove that the area $S(ABCD) < \frac23$ .
N. Sedrakian
2018 Belarus TST 7.1
A point $X$ lies inside an isosceles right triangle $ABC$ with $\angle B = 90^o$ .
Prove the inequality $AX + BX + \sqrt2 CX \ge \sqrt5 AB$ and find all points $X$ for which the equality holds.
A point $X$ lies inside an isosceles right triangle $ABC$ with $\angle B = 90^o$ .
Prove the inequality $AX + BX + \sqrt2 CX \ge \sqrt5 AB$ and find all points $X$ for which the equality holds.
M. Karpuk
Given a cyclic quadrilateral $ABCD$ with $AB = AD$. Inner points $M , N$ of the segments $CD$ and $BC$ , respectively, are such that $DM + BN = MN$ . Prove that the circumcenter of $\vartriangle AMN$ belongs to the segment $AC$ .
N. Sedrakian
Points $M$ and $N$ are the midpoints of the sides $BC$ and $AD$, respectively, of a convex quadrilateral $ABCD$. Is it possible that $AB+CD>\max(AM+DM,BN+CN)?$
Folklore
2019 Belarus TST 2.2Let $O$ be the circumcenter and $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $T$ is the midpoint of the segment $AO$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$. Prove that the circumcircle of the triangle $AST$ bisects the segment $OH$.
M. Berindeanu, RMC 2018 book
2019 Belarus TST 5.2Let $AA_1$ be the bisector of a triangle $ABC$. Points $D$ and $F$ are chosen on the line $BC$ such that $A_1$ is the midpoint of the segment $DF$. A line $l$, different from $BC$, passes through $A_1$ and intersects the lines $AB$ and $AC$ at points $B_1$ and $C_1$, respectively.
Find the locus of the points of intersection of the lines $B_1D$ and $C_1F$ for all possible positions of $l$.
M. Karpuk
2019 Belarus TST 6.1 (also)Two circles $\Omega$ and $\Gamma$ are internally tangent at the point $B$. The chord $AC$ of $\Gamma$ is tangent to $\Omega$ at the point $L$, and the segments $AB$ and $BC$ intersect $\Omega$ at the points $M$ and $N$. Let $M_1$ and $N_1$ be the reflections of $M$ and $N$ about the line $BL$; and let $M_2$ and $N_2$ be the reflections of $M$ and $N$ about the line $AC$. The lines $M_1M_2$ and $N_1N_2$ intersect at the point $K$.
Prove that the lines $BK$ and $AC$ are perpendicular.
M. Karpuk
2019 Belarus TST 7.1The internal bisectors of angles $\angle DAB$ and $\angle BCD$ of a quadrilateral $ABCD$ intersect at the point $X_1$, and the external bisectors of these angles intersect at the point $X_2$. The internal bisectors of angles $\angle ABC$ and $\angle CDA$ intersect at the point $Y_1$, and the external bisectors of these angles intersect at the point $Y_2$.
Prove that the angle between the lines $X_1X_2$ and $Y_1Y_2$ equals the angle between the diagonals $AC$ and $BD$.
A. Voidelevich
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