geometry problems from North Macedonian Team Selection Tests (TST) , for IMO , Balkan MO and EGMO, with aops links in the names
IMO TST 2021
it started in 2021
Let $ABC$ be an acute triangle such that $AB<AC$. Denote by $A'$ the reflection of $A$ with respect to $BC$. The circumcircle of $A'BC$ meets the rays $AB$ and $AC$ at $D$ and $E$ respectively, such that $B$ is between $A$ and $D$, and $E$ is between $A$ and $C$. Denote by $P$ and $Q$ the midpoints of the segments $CD$ and $BE$, and let $S$ be the midpoint of $BC$. Show that the lines $BC$ and $AA'$ meet on the circumcircle of $PQS$.
by Nikola Velov
Let $ABC$ be an acute triangle such that $AB<AC$ with orthocenter $H$. The altitudes $BH$ and $CH$ intersect $AC$ and $AB$ at $B_{1}$ and $C_{1}$. Denote by $M$ the midpoint of $BC$. Let $l$ be the line parallel to $BC$ passing through $A$. The circle around $ CMC_{1}$ meets the line $l$ at points $X$ and $Y$, such that $X$ is on the same side of the line $AH$ as $B$ and $Y$ is on the same side of $AH$ as $C$. The lines $MX$ and $MY$ intersect $CC_{1}$ at $U$ and $V$ respectively. Show that the circumcircles of $ MUV$ and $ B_{1}C_{1}H$ are tangent.
by Nikola Velov
BMO TST 2021
Let $ABC$ be an acute triangle. Let $D$, $E$ and $F$ be the feet of the altitudes from $A$, $B$ and $C$ respectively and let $H$ be the orthocenter of $\triangle ABC$. Let $X$ be an arbitrary point on the circumcircle of $\triangle DEF$ and let the circumcircles of $\triangle EHX$ and $\triangle FHX$ intersect the second time the lines $CF$ and $BE$ second at $Y$ and $Z$, respectively. Prove that the line $YZ$ passes through the midpoint of $BC$.
EGMO TST 2021
Let $P$ be a point in the interior of $\triangle ABC$. The rays $AP$, $BP$ and $CP$ meet the sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$ respectively. The points $R$, $S$ and $T$ are the images of the point $P$ under reflection over $D$,$E$ and $F$ respectively. Show that:
$$\frac{AR}{PD} \cdot \frac{BS}{PE} \cdot \frac{CT}{PF} \geq 64$$
Let $P$ be a point in the interior of $\triangle ABC$ and let $D,E,F$ be the orthogonal projections of $P$ to $BC$, $CA$ and $AB$ respectively. Denote by $S$ the midpoint of the side $AB$. Show that $FP$ is a bisector in $\triangle DEF$ if and only if $SD = SE$.
Show that for any convex hexagon with area $P$ there exists a diagonal which cuts off a triangle with area $\leq \frac{P}{6}$. Can this bound be improved?
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