Macedonia North Juniors / JBMO TST 2012-22 17p

geometry problems from North Macedonian Junior Mathematical Olympiads, who also serves as JBMO Team Selection Tests (JBMO TST) with aops links

[not in JBMO Shortlist]

collected inside aops here

2007, 2009, 2012- 2022

2007 Macedonia North Juniors P2
Let $ABCD$ be a parallelogram and let $E$ be a point on the side $AD$, such that $\frac{AE}{ED} = m$. Let $F$ be a point on $CE$, such that $BF \perp CE$, and the point $G$ is symmetrical to $F$ with respect to $AB$. If point $A$ is the circumcenter of triangle $BFG$, find the value of $m$.


2009 Macedonia North Juniors P3
Let $\triangle ABC$ be equilateral. On the side $AB$ points $C_{1}$ and $C_{2}$, on the side $AC$ points $B_{1}$ and $B_{2}$ are chosen, and on the side $BC$ points $A_{1}$ and $A_{2}$ are chosen. The following condition is given : $A_{1}A_{2}$ = $B_{1}B_{2}$ = $C_{1}C_{2}$. Let the intersection lines $A_{2}B_{1}$ and $B_{2}C_{1}$, $B_{2}C_{1}$ and $C_{2}A_{1}$ and $C_{2}A_{1}$ and $A_{2}B_{1}$ are $E$, $F$, and $G$ respectively. Show that the triangle formed by $B_{1}A_{2}$, $A_{1}C_{2}$ and $C_{1}B_{2}$ is similar to $\triangle EFG$.

2012 Macedonia North Juniors P2
Let $ABCD$ be a convex quadrilateral inscribed in a circle of radius $1$. Prove that $0< (AB+BC+CD+AD)-(AC+BD) < 4.$

2013 Macedonia North Juniors P2

A triangle $ABC$ is given, and a segment $PQ=t$ on $BC$ such that $P$ is between $B$ and $Q$ and $Q$ is between $P$ and $C$. Let $PP_1 || AB$, $P_1$ is on $AC$, and $PP_2 || AC$, $P_2$ is on $AB$. Points $Q_1$ and $Q_2$ аrе defined similar. Prove that the sum of the areas of $PQQ_1P_1$ and $PQQ_2P_2$ does not depend from the position of $PQ$ on $BC$.

2014 Macedonia North Juniors P2

Point $M$ is an arbitrary point in the plane and let points $G$ and $H$ be the intersection points of the tangents from point M and the circle $k$. Let $O$ be the center of the circle $k$ and let $K$ be the orthocenter of the triangle $MGH$. Prove that ${\angle}GMH={\angle}OGK$.

2014 Macedonia North Juniors P4

In a convex quadrilateral $ABCD$, $E$ is the intersection of $AB$ and $CD$, $F$ is the intersection of $AD$ and $BC$ and $G$ is the intersection of $AC$ and $EF$. Prove that the following two claims are equivalent:

(i) $BD$ and $EF$ are parallel.

(ii) $G$ is the midpoint of $EF$

2015 Macedonia North Juniors P2

A circle $k$ with center $O$ and radius $r$ and a line $p$ which has no common points with $k$, are given. Let $E$ be the foot of the perpendicular from $O$ to $p$. Let $M$ be an arbitrary point on $p$, distinct from $E$. The tangents from the point $M$ to the circle $k$ are $MA$ and $MB$. If $H$ is the intersection of $AB$ and $OE$, then prove that $OH=\frac{r^2}{OE}$.

2015 Macedonia North Juniors P4

Let $\triangle ABC$ be an acute angled triangle and let $k$ be its circumscribed circle. A point $O$ is given in the interior of the triangle, such that $CE=CF$, where $E$ and $F$ are on $k$ and $E$ lies on $AO$ while $F$ lies on $BO$. Prove that $O$ is on the angle bisector of $\angle ACB$ if and only if $AC=BC$.

2016 Macedonia North Juniors P2

Let $ABCD$ be a parallelogram and let $E$, $F$, $G$, and $H$ be the midpoints of sides $AB$, $BC$, $CD$, and $DA$, respectively. If $BH \cap AC = I$, $BD \cap EC = J$, $AC \cap DF = K$, and $AG \cap BD = L$, prove that the quadrilateral $IJKL$ is a parallelogram.

2017 Macedonia North Juniors P2

In the triangle $ABC$, the medians $AA_1$, $BB_1$, and $CC_1$ are concurrent at a point $T$ such that $BA_1=TA_1$. The points $C_2$ and $B_2$ are chosen on the extensions of $CC_1$ and $BB_2$, respectively, such that $C_1C_2 = \frac{CC_1}{3} \quad \text{and} \quad B_1B_2 = \frac{BB_1}{3}.$ Show that $TB_2AC_2$ is a rectangle.

2017 Macedonia North Juniors P4

In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $BM + CM > AY.$

2018 Macedonia North Juniors P2

We are given a semicircle $k$ with center $O$ and diameter $AB$. Let $C$ be a point on $k$ such that $CO \bot AB$. The bisector of $\angle ABC$ intersects $k$ at point $D$. Let $E$ be a point on $AB$ such that $DE \bot AB$ and let $F$ be the midpoint of $CB$. Prove that the quadrilateral $EFCD$ is cyclic.

2019 Macedonia North Juniors P2

Circles $\omega_{1}$ and $\omega_{2}$ intersect at points $A$ and $B$. Let $t_{1}$ and $t_{2}$ be the tangents to $\omega_{1}$ and $\omega_{2}$, respectively, at point $A$. Let the second intersection of $\omega_{1}$ and $t_{2}$ be $C$, and let the second intersection of $\omega_{2}$ and $t_{1}$ be $D$. Points $P$ and $E$ lie on the ray $AB$, such that $B$ lies between $A$ and $P$, $P$ lies between $A$ and $E$, and $AE = 2 \cdot AP$. The circumcircle to $\bigtriangleup BCE$ intersects $t_{2}$ again at point $Q$, whereas the circumcircle to $\bigtriangleup BDE$ intersects $t_{1}$ again at point $R$. Prove that points $P$, $Q$, and $R$ are collinear.

2020 Macedonia North Juniors P4

Let $ABC$ be an isosceles triangle with base $AC$. Points $D$ and $E$ are chosen on the sides $AC$ and $BC$, respectively, such that $CD = DE$. Let $H, J,$ and $K$ be the midpoints of $DE, AE,$ and $BD$, respectively. The circumcircle of triangle $DHK$ intersects $AD$ at point $F$, whereas the circumcircle of triangle $HEJ$ intersects $BE$ at $G$. The line through $K$ parallel to $AC$ intersects $AB$ at $I$. Let $IH \cap GF =$ {$M$}. Prove that $J, M,$ and $K$ are collinear points.

2020 North Macedonia Juniors p2

Let $ABC$ be an isosceles triangle with base $AC$. Points $D$ and $E$ are chosen on the sides $AC$ and $BC$, respectively, such that $CD = DE$. Let $H, J,$ and $K$ be the midpoints of $DE, AE,$ and $BD$, respectively. The circumcircle of triangle $DHK$ intersects $AD$ at point $F$, whereas the circumcircle of triangle $HEJ$ intersects $BE$ at $G$. The line through $K$ parallel to $AC$ intersects $AB$ at $I$. Let $IH \cap GF =$ {$M$}. Prove that $J, M,$ and $K$ are collinear points.

2021 North Macedonia Juniors p2

Let $ABCD$ be a tangential quadrilateral with inscribed circle $k(O,r)$ which is tangent to the sides $BC$ and $AD$ at $K$ and $L$, respectively. Show that the circle with diameter $OC$ passes through the intersection point of $KL$ and $OD$.

2021 North Macedonia Juniors p5

Let $ABC$ be an acute triangle and let $X$ and $Y$ be points on the segments $AB$ and $AC$ such that $BX = CY$. If $I_{B}$ and $I_{C}$ are centers of inscribed circles in triangles $ABY$ and $ACX$, and $T$ is the second intersection point of the circumcircles of $ABY$ and $ACX$, show that: $\frac{TI_{B}}{TI_{C}} = \frac{BY}{CX}.$ 

2022 North Macedonia Juniors p3

Let $\triangle ABC$ be an acute triangle with orthocenter $H$. The circle $\Gamma$ with center $H$ and radius $AH$ meets the lines $AB$ and $AC$ at the points $E$ and $F$ respectively. Let $E'$, $F'$ and $H'$ be the reflections of the points $E$, $F$ and $H$ with respect to the line $BC$, respectively. Prove that the points $A$, $E'$, $F'$ and $H'$ lie on a circle.

source: https://pregatirematematicaolimpiadejuniori.wordpress.com/