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Bulgaria TST 2003-08, 2012-15, 2020 25p

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

(only those not in IMO & BMO Shortlist)

collected inside aops here

2003-08 , 2012-15, 2020


Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its in center on $AC$. Prove that $\angle {APB}=\angle {APD}$ 

Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square.

Let $H$ be the orthocenter of $\triangle ABC$. The points $A_{1} \not= A$, $B_{1} \not= B$ and $C_{1} \not= C$ lie, respectively, on the circumcircles of $\triangle BCH$, $\triangle CAH$ and $\triangle ABH$ and satisfy $A_{1}H=B_{1}H=C_{1}H$. Denote by $H_{1}$, $H_{2}$ and $H_{3}$ the orthocenters of $\triangle A_{1}BC$, $\triangle B_{1}CA$ and $\triangle C_{1}AB$, respectively. Prove that $\triangle A_{1}B_{1}C_{1}$ and $\triangle H_{1}H_{2}H_{3}$ have the same orthocenter

The points $P$ and $Q$ lie on the diagonals $AC$ and $BD$, respectively, of a quadrilateral $ABCD$ such that $\frac{AP}{AC} + \frac{BQ}{BD} =1$. The line $PQ$ meets the sides $AD$ and $BC$ at points $M$ and $N$. Prove that the circumcircles of the triangles $AMP$, $BNQ$, $DMQ$, and $CNP$ are concurrent.

Let $ABC$ be an acute triangle. Find the locus of the points $M$, in the interior of $\bigtriangleup ABC$, such that $AB-FG= \frac{MF.AG+MG.BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to the lines $BC$ and $AC$, respectively.

Let $ABC$, $AC \not= BC$, be an acute triangle with orthocenter $H$ and incenter $I$. The lines $CH$ and $CI$ meet the circumcircle of $\bigtriangleup ABC$ at points $D$ and $L$, respectively. Prove that $\angle CIH = 90^{\circ}$ if and only if $\angle IDL = 90^{\circ}$

Two points $M$ and $N$ are chosen inside a non-equilateral triangle $ABC$ such that $\angle BAM=\angle CAN$, $\angle ABM=\angle CBN$ and\[AM\cdot AN\cdot BC=BM\cdot BN\cdot CA=CM\cdot CN\cdot AB=k\]for some real $k$. Prove that:
a) We have $3k=AB\cdot BC\cdot CA$.
b) The midpoint of $MN$ is the centroid of $\triangle ABC$.
Nikolai Nikolov
Let $k$ be the circumcircle of $\triangle ABC$, and $D$ the point on the arc $AB$, which do not pass through $C$. $I_A$ and $I_B$ are the centers of incircles of $\triangle ADC$ and $\triangle BDC$, respectively. Proove that the circumcircle of $\triangle I_AI_BC$ touches $k$ iff\[ \frac{AD}{BD}=\frac{AC+CD}{BC+CD}.  \]
Stoyan Atanasov
Points $D$ and $E$ are chosen on the sides $AB$ and $AC$, respectively, of a triangle $\triangle ABC$ such that $DE\parallel BC$. The circumcircle $k$ of triangle $\triangle ADE$ intersects the lines $BE$ and $CD$ at the points $M$ and $N$ (different from $E$ and $D$). The lines $AM$ and $AN$ intersect the side $BC$ at points $P$ and $Q$ such that $BC=2\cdot PQ$ and the point $P$ lies between $B$ and $Q$. Prove that the circle $k$ passes through the point of intersection of the side $BC$ and the angle bisector of $\angle BAC$.
Nikolai Nikolov
Let $ABC$ is a triangle with $\angle BAC=\frac{\pi}{6}$ and the circumradius equal to 1. If $X$ is a point inside or in its boundary let $m(X)=\min(AX,BX,CX).$ Find all the angles of this triangle if $\max(m(X))=\frac{\sqrt{3}}{3}.$

Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ is the midpoint of $AB.$ If $CN\parallel IM$ find $\frac{CN}{IM}$.

In isosceles triangle $ABC(AC=BC)$ the point $M$ is in the segment $AB$ such that $AM=2MB,$ $F$ is the midpoint of $BC$ and $H$ is the orthogonal projection of $M$ in $AF.$ Prove that $\angle BHF=\angle ABC.$

The point $P$ lies inside, or on the boundary of, the triangle $ABC$. Denote by $d_{a}$, $d_{b}$ and $d_{c}$ the distances between $P$ and $BC$, $CA$, and $AB$, respectively. Prove that $\max\{AP,BP,CP \} \ge \sqrt{d_{a}^{2}+d_{b}^{2}+d_{c}^{2}}$. When does the equality holds?

In the triangle $ABC$, $AM$ is median, $M \in BC$, $BB_{1}$ and $CC_{1}$ are altitudes, $C_{1} \in AB$, $B_{1} \in AC$. The line through $A$ which is perpendicular to $AM$ cuts the lines $BB_{1}$ and $CC_{1}$ at points $E$ and $F$, respectively. Let $k$ be the circumcircle of $\triangle EFM$. Suppose also that $k_{1}$ and $k_{2}$ are circles touching both $EF$ and the arc $EF$ of $k$ which does not contain $M$. If $P$ and $Q$ are the points at which $k_{1}$ intersects $k_{2}$, prove that $P$, $Q$, and $M$ are collinear.

The quadrilateral $ABCD$ is inscribed in a circle. The point $E$ is symmetric to $B$ wrt the
intersection of $AD$ and $BC$, the point $F$ is symmetric to $B$ wrt the midpoint of the $CD$,
and the point $G$ is symmetrical to $A$ wrt the midpoint of $CE$. Prove that the points $E, F, G$
and $C$ lie on a circle. 2012 Bulgaria TST 2.2
A quadrilateral $ABCD$ is given. A circle $k$ touches $AD$ at point $D$ and $BC$ at point $C$.
The intersections of the side $AB$ and $k$ are denoted by $K$ and $L$, as $DL = CL$. Prove that
the intersection point of $AC$ and $BD$ lies on the line through $K$ and the midpoint of the side $CD$.

Let $A_1, B_1$ and $C_1$ be the feet of the altitudes from vertices $A, B$ and $C$ of acute
$\vartriangle ABC$. Prove that points $A, B$ and the centers of the inscribed circles in $\vartriangle AB_1C_1$
and $\vartriangle BC_1A_1$ lie on one circle .
Given $\vartriangle ABC$, the points $D, E$ and $F$ lie respectively on the sides $BC, CA$ and
$AB$ such that $AF = EF$ and $BF = DF$. Prove that the orthocenter of $\vartriangle ABC$ lies
on the circle circumscribed around $\vartriangle CDE$. 2013 Bulgaria TST 3.1
The circle inscribed in $\vartriangle ABC$ touches the side $BC$ and $AC$ at points $A_1$ and
$B_1$ respectively. The line $B_1A_1$ intersects $AB$ at a point $X$, where $A$ is between $X$
and $B$. Find the angles of $\vartriangle ABC$ if $\angle CXB = 90^o$ and $BC^2 = AB^2 + BC \cdot AC$ 2013 Bulgaria TST 4.2
Given $\vartriangle ABC$ and points $M, N$ and $P$ respectively on the sides $BC, CA$ and $AB$.
The triangles $CNM, APN$ and $BMP$ are acute and let $H_C, H_A$ and $H_B$ be their orthocenters
respectively. Prove that if lines $AH_A, BH_B$ and $CH_C$ intersect at one point, then lines
$MH_A, NH_B$ and $PH_C$ also intersect at one point.

A quadrilateral $ABCD$ is inscribed in a circle with center $O$ and circumscribed around a circle
with center I. Prove that the quadrilateral formed by the lines through the vertices $A, B, C$ and $D$,
perpendicular to $AI, BI, CI$ and $DI$ respectively, is inscribed in a circle whose center lies on line $IO$.
An isosceles acute-angled $\vartriangle ABC$ with orthocenter $H$ and altitudes $AA_1$
($A_1 \in BC$) and $BB_1$ ($B_1 \in AC$). The perpendicular bisectors of $AA_1$ and $BB_1$
intersect the lines $A_1B_1$ at points $P$ and $Q$, respectively. Let $AP$ and $BQ$ intersect at
point $R$. (a) Prove that $RH$ bisects segment $A_1B_1$. b) Find $\angle ACB$ if $HA_1RB_1$ is a parallalogram. 2014 Bulgaria TST 3.2
$\vartriangle ABC$ is given and let the circle $k$ inscribed in it touch the sides $BC$ and $CA$
at points $P$ and $Q$ respectively. Let us denote by $J$ the center of the ex-circle for $\vartriangle ABC$
corresponding to the side $AB$, and with $T$ the second intersection of the circles circumscribed
around $\vartriangle JBP$ and $\vartriangle JAQ$. Prove that the circumscribed circle around
$\vartriangle ABT$ is tangent to $k$.
The center of the circumcircle of $ABC$ is $O, D$ is the midpoint of the arc $BC$, which
does not contain $A$, and $AH$ is the altitude of $\vartriangle ABC$ ($H$ is of line $BC$).
A point $X$ is selected on the ray $AH$ and $M$ is the midpoint of $DX$. Point $N$ of the line
$DX$ is such that $ON \parallel AD$. Prove that $\angle BAM = \angle CAN$. 

Given $\vartriangle ABC$. Point $A'$ is the center of the circle passing through the midpoint
of BC and the intersections of the perpendiculars from $B$ and $C$ with the bisectors of
$\angle ACB$ and $\angle ABC$. Points $B'$ and $C'$ are defined similarly. Prove that the
orthocenter of$ \vartriangle A'B'C'$ is the center of the inscribed circle for $\vartriangle ABC$.

Let ABC be acute triangle. Point M is arbitrary point on the side AB, and N is the midpoint of AC.
Denote by P and Q the feet of the perpendiculars from A to the lines MC and MN, respectively.
Prove that when M vary then the circumcenter of triangle PQN lies on a fixed line.

In acute triangle $\triangle ABC$, $BC>AC$, $\Gamma$ is its circumcircle, $D$ is a point on
segment $AC$ and $E$ is the intersection of the circle with diameter $CD$ and $\Gamma$. $M$
is the midpoint of $AB$ and $CM$ meets $\Gamma$ again at $Q$. The tangents to $\Gamma$ at
$A,B$ meet at $P$, and $H$ is the foot of perpendicular from $P$ to $BQ$. $K$ is a point on line
$HQ$ such that $Q$ lies between $H$ and $K$. Prove that $\angle HKP=\angle ACE$ if and only
if $\frac{KQ}{QH}=\frac{CD}{DA}$. 2020 Bulgaria TST 2.3In triangle $\triangle ABC$, $BC>AC$, $I_B$ is the $B$-excenter, the line through $C$ parallel
to $AB$ meets $BI_B$ at $F$. $M$ is the midpoint of $AI_B$ and the $A$-excircle touches side
$AB$ at $D$. Point $E$ satisfies $\angle BAC=\angle BDE, DE=BC$, and lies on the same side
as $C$ of $AB$. Let $EC$ intersect $AB,FM$ at $P,Q$ respectively. Prove that $P,A,M,Q$ are
concyclic.

random problems from aops

Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$
and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and
$l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and
$l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ is the midpoint of $AB.$
If $CN\parallel IM$ prove that $\frac{CN}{IM} = 2$.

Consider a triangle $ABC$ with circumcircle $\Gamma$. Let $D$, $E$, $F$ be the tangency points
of the excircle corresponding to vertex $A$ with the sides $BC$, $CA$, $AB$, respectively.
Let $K$ be the projection of $D$ on $EF$ and let $M$ be the midpoint of $EF$. Prove that
$K\in \Gamma$ iff $M\in \Gamma$.

Let $\triangle ABC$ be a triangle with circumcircle $\omega$ and circumcenter $O$.
Let $P$ be any point inside $\triangle ABC$ other than $O$. Let $AP$ intersect $\omega$ for the
second time at $A_1$. Let $A_2$ be the reflection of $A_1$ over line $OP$. Let $l_a$ be the line
through the midpoint of $\overline{BC}$ parallel to $AA_2$. Define $l_b$ and $l_c$ similarly.
Prove that $l_a$, $l_b$ and $l_c$ are concurrent.


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