geometry problems from Junior Round of Clock-Tower School (Romanian) - Școala cu ceas
with aops linksJunior Round lasted only years 2007-14
collected inside aops here
2007 - 2014
Consider a triangle $ ABC $ with $ AB=AC, $ and a point $ M $ on the segment $ BC. $ Let $ N,P $ be the projections of $ M $ on $ AB, $ respectively, $ AC, $ let $ Q $ be the intersection of $ BP,CN, $ and $ R $ be the intersection of $ AQ $ with $ NP. $ Show that $ MR $ bisects $ \angle NMP . $
Francisc Bozgan
Consider $ 1210 $ points inside a circle of radius $ 1. $ Show that there are at least $ 11 $ of these points that are inside a circle of radius $ 1/10. $
Vasile Pop
Consider a circle of center $O$ and a chord $AB$ of it (not a diameter). Take a point $T$ on the ray $OB$. The perpendicular at $T$ onto $OB$ meets the chord $AB$ at $C$ and the circle at $D$ and $E$. Denote by $S$ the orthogonal projection of $T$ onto the chord $AB$. Show that $AS \cdot BC = TE \cdot TD$
Consider any $25$ points, three by three non-collinear, in the interior of a square of side length $3$. Show that there exist, four among them that form a quadrilateral perimeter less than $5$.
Let $P$ be the set of all points of the plane, and $O \in P$ fixed. The function $f: P - \{O\} \to R$ has the property:
For any four distinct points $A,B,C,D \in P - \{O\}$ with $\vartriangle AOB \sim \vartriangle COD$, $f(A)-f(B)+f(C) - f(D) =0$ occurs.
Prove the function $f$ is constant.
Consider a triangle $ABC$ with the property that there is a point $D$ inside it so that $\angle DAC = \angle DCA = 30^o$ and $\angle DBA = 60^o$. Let $E$ be the midpoint of $[BC]$ and $F \in [AC]$ such that $CF = \frac13 AC$ . Show that $DE \perp EF$.
Let be a regular polygon $A_1A_2...A_{2010}$ having the center at the point $O$. On each of the segments $OA_k$, with $k = 1, 2, ..., 2010$, the point $B_k$ is considered such that $\frac{OB_k}{OA_k}=\frac{1}{k}$. Determine the ratio between the area of the polygon $B_1B_2... B_{2010}$ and that of the polygon $A_1A_2...A_{2010}$ .
Consider a triangle $ABC$ and the points $M \in (BC), N \in (AC), P \in (AB)$ such that $\angle BMP = \angle CNM =\angle APN$ and $BM = CN = AP$. Prove that the triangle $ABC$ is equilateral.
Let $ABC$ be a right-angled triangle, where$ A$ is the right angle. On the catheti $(AB)$ and $(AC)$ we consider points $D$ and $E$, respectively, such us the perpendicular lines through $D$ and $A$ to $(BE)$ intersect $(BC)$ in $F$ and $G$, respectively and $[FG]=[GC]$.
a) Prove that $BD\cdot GC = AD \cdot BF$ .
b) If $EG \perp BC$ , prove that $AE < EC$ .
Ştefan SMĂRĂNDOIU – Rm. Vâlcea
Let $ABC$ be a triangle with $\angle B=\angle C$ . On $(AB)$ we consider a point $D$ and on $(AC - [AC]$ we consider a point $E$ , such as $D$ is symmetric of $E$ with respect to $F$ , where $\{F\}= DE \cap BC$ . Prove that $[BD]=[CE$].
Ştefan SMĂRĂNDOIU – Rm. Vâlcea
In the acute triangle $ABC$, let $O$ and $I$ be the centers of the circumscribed circles and inscribed respectively , and $D$ and $E$ the feet of the heights of $A$ and $B$. respectively. Line $CI$ cuts the circle around the triangle $ABC$ in $K$, and $\{P\}=OK\cap AB$ . If $OK =IK$ , show that the triangle $DEP$ is equilateral.
Dorian CROITORU, Chisinau
Let $\angle XOY$ be a right angle and $[OP$ its bisector. On the right half $(OX$ take a point $A$ so that $\angle OAP$ is an obtuse angle, and on the half-ray $(OY$ a point$ B$ is taken so that $\angle OBP$ is a acute angle and $[PA]=[PB]$. Find the measure of the angle $\angle APB$.
Ştefan SMĂRĂNDOIU, Rm. Vâlcea
Let $ABCD$ be a convex quadrilateral with $AB = AD$. Let $E$ be the midpoint of the arc $BC$ of the circle $\odot (ABC)$ which does not contain point $A$, and $F$ be the midpoint of the arc $CD$ of the circle $\odot (ACD)$ which does not contain point $A$. Prove that $EF \perp AC$.
L. Ploscaru
The triangle $ABC$ is right with $D$ in the midpoint of the hypotenuse $[AC]$. $E$ is a point such that so that $D\in (BE)$ and $[AC]= [DE]$ . Knowing that $[AB]=[AE]$ , prove that $\angle BAE = 120^o$.
Ștefan SMĂRĂNDOIU, Râmnicu Vâlcea
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