Science ON 2021 (Romania) 8p

  geometry problems from Science ON   in Romania , with aops links in the names

2021

2021 Science ON 6.1

Triangle $ABC$ is such that $\angle BAC>\angle ABC>60^o$. The perpendicular bisector of $\overline{AB}$ intersects the segment $\overline {BC}$ at $O$. Suppose there exists a point $D$ on the segment $\overline{AC}$ such that $OD=AB$ and $\angle ODA=30^o$. Find $\angle BAC$.

(Vlad Robu)

2021 Science ON 6.2

Is it possible for an isosceles triangle with all its sides of positive integer lengths to have an angle of $36^o$?

(Adapted from Archimedes 2011, Traian Preda)

2021 Science ON 7.2

In triangle $ABC$, we have $\angle ABC=\angle ACB=44^o$. Point $M$ is in its interior such that $\angle MBC=16^o$ and $\angle MCB=30^o$. Prove that $\angle MAC=\angle MBC$.

(Andra Elena Mircea)

2021 Science ON 8.3

$ABCD$ is a scalene tetrahedron and let $G$ be its baricentre. A plane $\alpha$ passes through $G$ such that it intersects neither the interior of $\Delta BCD$ nor its perimeter. Prove that

$$dist(A,\alpha)=dist(B,\alpha)+dist(C,\alpha)+dist(D,\alpha).$$

(Adapted from folklore)

2021 Science ON 9.2

Consider the acute-angled triangle $ABC$, with orthocentre $H$ and circumcentre $O$. $D$ is the intersection point of lines $AH$ and $BC$ and $E$ lies on $\overline{AH}$ such that $AE=DH$. Suppose $EO$ and $BC$ meet at $F$. Prove that $BD=CF$.

(Călin Pop & Vlad Robu)

2021 Science ON 9.4

(a) On the sides of triangle $ABC$ we consider the points $M\in \overline{BC}$, $N\in \overline{AC}$ and $P\in \overline{AB}$ such that the quadrilateral $MNAP$ with right angles $\angle MNA$ and $\angle  MPA$ has an inscribed circle. Prove that $MNAP$ has to be a kite.

(b) Is it possible for an isosceles trapezoid to be orthodiagonal and circumscribed too?

(Călin Udrea)

2021 Science ON Juniors p3

Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $P$. A random line $\ell$ cuts $\omega_1$ at $A$ and $C$ and $\omega_2$ at $B$ and $D$ (points $A,C,B,D$ are in this order on $\ell$). Line $AP$ meets $\omega_2$ again at $E$ and line $BP$ meets $\omega_1$ again at $F$. Prove that the radical axis of circles $(PCD)$ and $(PEF)$ is parallel to $\ell$.

(Vlad Robu)

2021 Science ON  Seniors p4

$ABCD$ is a cyclic convex quadrilateral whose diagonals meet at $X$. The circle $(AXD)$ cuts $CD$ again at $V$ and the circle $(BXC)$ cuts $AB$ again at $U$, such that $D$ lies strictly between $C$ and $V$ and $B$ lies strictly between $A$ and $U$. Let $P\in AB\cap CD$. If $M$ is the intersection point of the tangents to $U$ and $V$ at $(UPV)$ and $T$ is the second intersection of circles $(UPV)$ and $(PAC)$, prove that $\angle PTM=90^o$.

(Vlad Robu)