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Bulgaria JBMO TST 2007-22 31p (-09,-15)

geometry problems from Bulgarian Junior Balkan Mathematical Olympiads Team Selection Tests (JBMO TST) with aops links

[not in JBMO Shortlist]
collected inside aops here

2007 - 2022 
2009 and 2015 missing 

A triangle $ABC$ has $\angle ABC = 20^o$ and $\angle ACB = 60^o$ . Let $I$ be its incenter. The ray $(CI$ intersects the side $AB$ at $L$ and the circumcircle of triangle $ABC$ at $M$. The circumcircle of triangle $BIM$ meets $AM$ again at $D$. Determine the size of the angle $\angle ADL$

2007 TST 2 missing ?

Let $O$ be the center of the circle circumscribed to the triangle $ABC$ in which the side $BC$ is parallel to the diameter $AA_1$. If $H$ is the orthocenter of the triangle $ABC$, prove that the line $BC$ passes through the midpoint of the segment $[OH]$.

The bisectors $(BK$ and $(CP$ of the triangle $ABC$, $K \in AC, P \in AB$, intersect the circle circumscribed around the triangle at points $B_1$ and $C_1$, respectively. If  $CC_1 = AB, BB_1 = AC$ and the angle  $\angle ACB$ is obtuse, point to the third bisector of the triangle, $(AM)$, $M \in BC$, is congruent with $KC_1$.

2009 missing
.
Consider the convex quadrilateral $ABCD$ where $\angle ABC  = 90^o$ and  $\angle BAD =  \angle  ADC = 80^o$. On the sides $BC , AD$ of the quadrilateral consider the points $N, M$ respectively  such that $\angle CDN  = \angle ABM = 20^o$. Determine the measure of the angle $\angle MNB$ knowing that $MD = AB$.

Point $O$ is the center of the circle circumscribed by the triangle $ABC$ with $\angle BAC \ne 90^o$.  The circle circumscribed to the triangle $BOC$, denoted $k$, intersects the lines $AB$ and AC at the points $P$ and $Q$, respectively, where $P \ne B, Q \ne C$. The segment $[ON]$ is the diameter of the circle $k$. Prove that the quadrilateral $APNQ$ is a parallelogram.
Let $ADC$ be a triangle with side $AC = 1$ and the circles $k_1, k_2$ which pass through the point $D$, having the centers in $A, C$ respectively. A variable line through $D$ intersects the circles in $M$ and $N$ for the second time.
a) Prove that the perpendicular bisectors of the segments [MN] have one thing in common.
b) Determine the maximum length of the segment $[MN]$.

2011 Bulgaria JBMO TST 2.2
On the sides $[AB]$ and $[CD]$ of the square ABCD of side $6$ cm consider the points $M$ and $N$ respectively. The segments $[AN]$ and $[DM]$ intersect at point $P$, and the segments $[BN]$ and $[CM]$ intersect at point $Q$. Find the smallest possible length of the segment $[PQ]$  and the largest possible area of  $NPMQ$.
In circle $k$ is inscribed the $\vartriangle ABC$. A chord $[AA_1]$ has the midpoint $M$ such that  $CA_1 \parallel BM$. The lines $BM$ and $CM$ intersect the circle $k$ in $B_1$, respectively $C_1$. Show that the point $M$ is the centroid of the triangle $ABC$ if and only if $A_1B_1 = A_1C_1$.

In triangle $ABC$, the bisector $(AA_1$, $A_1 \in  BC$, intersects the circumcircle, $k$, of triangle $ABC$ at point $T$. Perpendicular from $A_1$ on $AC$ intersects the circle $k$ at point $K$. The line $TK$ intersects the side $BC$ at the point $M$. Show that $AM$ is perpendicular to $BK$.

Let $ABC$ be a  triangle with $\angle BAC> 90^o$  and orthocenter $H$ . If $AA', BB', CC'$ are the $3$ altitudes, consider the circle through $H$, tangent to $BC$ at $A'$, the circle through $H$ tangent to $AC$ at $B'$ and the circle through $H$ tangent to $AB$ at $C'$. Prove that $H$ is the center of the circle circumscribed by the triangle formed by the intersections, different from $H$, of two of these circles.

The circle inscribed in the triangle $ABC$ is tangent to the side $AB$ at point $E$, and to the side $AC$ at point $D$. Consider a certain point $P$ on the great arc  $DE$. The points $F$ and $G$ are symmetric of $A$ with respect to $PD$, $PE$ respectively . Determine the locus of the midpoint of the segment $[FG]$.

The circle $k$ is tangent inside the circle circumscribed to the isosceles triangle $ABC$ ($AB = AC$) and is tangent to the sides $[AB], [AC]$ at the points $P, Q$ respectively . Prove that the midpoint, $M$, of the segment $[PQ]$ is the center of the circle inscribed in the triangle $ABC$.

On the sides $AB, BC, CA$ of the triangle $ABC$ are considered the points $P, Q$, respectively $R$, so that $AP = CQ$ and $B, Q, R, P$ are concyclic. Lines $RP$ and $RQ$ intersect the tangents in $A$ and $C$, respectively, at the circle circumscribed to $ABC$ at points $X, Y$. Prove that $RX = RY$.
In triangle $ABC$, on the interior bisector $\ell_B$ of angle $\angle B$ , considers a point $M$  with the property that the perpendicular bisectors of the segments $[MA]$ and$ [MC]$ intersects on $\ell_B$. If the perpendicular bisector of $[AM]$ intersects $AB$ at point $P$, and the perpendicular bisector  of $[CM]$ intersects $BC$ at point $N$, prove that $\frac{BP}{BN}=\frac{BA}{BC}$
Inside the right angle $\angle AOB$  consider a point $P$. Construct two congruent circles $C_1 (O_1, R)$ and $C_2 (O_2, R)$ so that $C_1$ is tangent to the rays $(OA$ and $(OP$ at points $K$ and $M$, respectively and $C_2$ is tangent to rays $(OB$ ¸ and $(OP$ at the points $N$, respectively $Q$. Let $\{T\} = O2_M \cap OA$. If $O_1 \in KQ$ and $\frac{QM}{MO_2}= q$, find the value of the  ratio $\frac{QT}{O_1O_2}$

2014 Bulgaria JBMO TST 1.1 (also)
Points $M$ and $N$ lie on the sides $BC$ and $CD$ of the square $ABCD,$ respectively, and $\angle MAN = 45^{\circ}$. The circle through $A,B,C,D$ intersects $AM$ and $AN$ again at $P$ and $Q$, respectively. Prove that $MN || PQ.$

2014 Bulgaria JBMO TST 2.1 (also)
From the foot  $D$  of the height $CD$ in the triangle $ABC,$  perpendiculars to $BC$ and $AC$ are drawn, which they intersect at points $M$ and $N.$ Let $\angle CAB = 60^{\circ} , \angle CBA = 45^{\circ} ,$  and  $H$ be the orthocentre of $MNC.$ If $O$ is the midpoint of $CD,$ find $\angle COH.$

2015 missing

The quadrilateral $ABCD$, in which $\angle BAC  < \angle DCB$ , is inscribed in a circle $c$, with center $O$. If $\angle BOD  = \angle ADC  = \alpha$. Find out which values of $\alpha$ the inequality $AB <AD + CD$ occurs.
The vertices of the pentagon $ABCDE$ are on a circle, and the points $H_1, H_2, H_3,H_4$ are the orthocenters of the triangles $ABC, ABE, ACD, ADE$ respectively . Prove that the quadrilateral determined by the four orthocenters is square if and only if $BE \parallel CD$ and the distance between them is $\frac{BE + CD}{2}$.

2017 Bulgaria JBMO TST 1.1
Given is a triangle $ABC$ and $ AA_1 $, $ BB_1 $ are angle bisectors. $\angle  AA_1B=24^o$ and $\angle  BB_1A=18^o$ . Find $ \angle BAC:\angle ACB:\angle ABC $.

2017 Bulgaria JBMO TST 2.2
Let $k$ be the circumscribed circle in triangle $ABC$. It touches $AB=c, BC=a, AC=b$ in $ C_1, A_1, B_1 $, respectively. Let $ KC_1 $ be a diameter. $ C_1A_1 $ intersects $ KB_1 $ in N and $ C_1A_1 $ intersects $ KB_1 $ in M. Find $MN$.

2018 Bulgaria JBMO TST 1.1
In the quadrilateral $ABCD$, we have $\measuredangle BAD = 100^{\circ}$, $\measuredangle BCD = 130^{\circ}$, and $AB=AD=1$ centimeter. Find the length of diagonal $AC$.

2018 Bulgaria JBMO TST  2.2
Let $ABC$ be a triangle and $AA_1$ be the angle bisector of $A$ ($A_1 \in BC$). The point $P$ is on the segment $AA_1$ and $M$ is the midpoint of the side $BC$. The point $Q$ is on the line connecting $P$ and $M$ such that $M$ is the midpoint of $PQ$. Define $D$ and $E$ as the intersections of $BQ$, $AC$, and $CQ$, $AB$. Prove that $CD=BE$.

The sides $[AB]$ and $[AC]$ of the acute triangle $ABC$ are chords in the circles $k_1$ and $k_2$, respectively. The circle $k_1$ intersects the segment $[AB]$ at the point $M$, and the circle $k_2$ intersects the segment $[AC]$ at the point $N$. It is known that $AB = AN$ and $BC = BM$. If the circles $k_1$ and $k_2$ intersect for second time at the point $K$ and $\angle ACK = 50^o$ , determine the measure of the angle $\angle BAC$
On the sides of the triangle $ABC$, outside it, are constructed equilateral triangles  $BCA_1, CAB_1$ and $ABC_1$. The line $m$ passes through the midpoint $M$ of the segment $[A_1B_1]$ and is perpendicular to $AB$. The line $n$ passes through the midpoint $N$ of segment $[B_1C_1]$ and is perpendicular to $BC$. The line $p$ passes through the midpoint $P$ of the segment $[C_1A_1]$ and is perpendicular to the $CA$. Prove that the lines $m, n, p$ are concurrent.

Let $k$ with center $O$ be the excircle to triangle $ABC$ and touches the lines $AB$ and $AC$ at point $M$ and $N$ respectively. The points $P,Q$ are the intersection points of the segment $MN$ with $BO, CO$ respectively. The circles $k_1$ and $k_2$ are the circumsircles of $\vartriangle OMQ$ and $\vartriangle ONP$ respectively and intersect at the points $O$ and $D$.
a) Prove that the point $D$ lies on $k$.
b) If $E$ is the intersection point of $CP$ and $BQ$, then prove that the length of the segment $DE$ is the same as the radius of the incircle of triangle of $\vartriangle ABC$.

Let $\vartriangle ABC$ be an acute triangle with orthocenter $H$. The angle bisectors of angles $CAH$ and $CBH$ intersect at point $Q$. The midpoint of the segments $AB$ and $CH$ are $M,N$ respectively.
a) Prove that the point $Q$ lies on the segment $MN$.
b) If the point $Q$ is the midpoint of the segment $MN$, then find $\angle ACB$.

On the sides $BC$ and $CA$ of the acute $\vartriangle ABC$, outside of the triangle, the squares $CBA_1M$ and $ACNB_1$ are constructed. Prove that the intersection point of $AA_1$ and $BB_1$ lies on the height $CC_1$ of $\vartriangle ABC$.

The lines $a$ and $b$ are parallel and the distance between them is $1$. The square $ABCD$ has a side length of $1$. The line $a$ intersects $AB$ and $BC$ at points $M, P$ respectively. The line $b$ intersects $CD$ and $DA$ at $N, Q$ respectively. Find the angle between the lines $MN$ and $PQ$, which intersect inside the square.

Let $ABC (AC < BC)$ be an acute triangle with circumcircle $k$ and midpoint $P$ of $AB$. The altitudes $AM$ and $BN$ ($M\in BC$, $N\in AC$) intersect at $H$. The point $E$ on $k$ is such that the segments $CE$ and $AB$ are perpendicular. The line $EP$ intersects $k$ again at point $K$ and the point $Q$ on $k$ is such that $KQ$ and $AB$ are parallel. The circumcircle of $AHB$ intersects the segment $CP$ at an interior point $R$. Prove that the points $C$, $M$, $R$, $H$, $N$ and $Q$ are concyclic.

Let $ABC$ ($AB < AC$) be a triangle with circumcircle $k$. The tangent to $k$ at $A$ intersects the line $BC$ at $D$ and the point $E\neq A$ on $k$ is such that $DE$ is tangent to $k$. The point $X$ on line $BE$ is such that $B$ is between $E$ and $X$ and $DX = DA$ and the point $Y$ on the line $CX$ is such that $Y$ is between $C$ and $X$ and $DY = DA$. Prove that the lines $BC$ and $YE$ are perpendicular.

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