Let there be a triangle $\triangle ABC$ with $BC=a$, $CA=b$, $AB=c$. Let $T$ be a point not inside $\triangle ABC$ and on the same side of $A$ with respect to $BC$, such that $BT-CT=c-b$. Let $n=BT$ and $m=CT$. Find the point $P$ that minimizes $f(P)=-a \cdot AP + m \cdot BP + n \cdot CP$.

Given a convex pentagon $ ABCDE. $ Let $ A_1 $ be the intersection of $ BD $ with $ CE $ and define $ B_1, C_1, D_1, E_1 $ similarly, $ A_2 $ be the second intersection of $ \odot (ABD_1),\odot (AEC_1) $ and define $ B_2, C_2, D_2, E_2 $ similarly. Prove that $ AA_2, BB_2, CC_2, DD_2, EE_2 $ are concurrent. [i]Proposed by Telv Cohl[/i]

$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$

Circle of radius $R_1$ is inscribed in an acute angle $\alpha$. Second circle with radius $R_2$ touches one of the sides forming the angle $\alpha$ in same point as first circle and intersects the second side in points $A$ and $B$, such that centers of both circles lie inside angle $\alpha$. Prove that $$AB=4\cos{\frac{\alpha}{2}}\sqrt{(R_2-R_1)\left(R_1 \cos^2 \frac{\alpha}{2}+R_2 \sin^2 \frac{\alpha}{2}\right)}$$

In a circle with centre at $O$ and diameter $AB$, two chords $BD$ and $AC$ intersect at $E$. $F$ is a point on $AB$ such that $EF \perp AB$. $FC$ intersects $BD$ in $G$. If $DE = 5$ and $EG =3$, determine $BG$.

From the point $P$ outside a circle $\omega$ with center $O$ draw the tangents $PA$ and $PB$ where $A$ and $B$ belong to $\omega$.In a random point $M$ in the chord $AB$ we draw the perpendicular to $OM$, which intersects $PA$ and $PB$ in $C$ and $D$. Prove that $M$ is the midpoint $CD$.

Let $ABC$ be an acute-angled triangle, let $M$ be the midpoint of $BC$ and let $H$ be the foot of the $B$-altitude. Let $Q$ be the circumcenter of $ABM$ and let $X$ be the intersection point between $BH$ and the axis of $BC$. Show that the circumcircles of the two triangles $ACM$, $AXH$ and the line $CQ$ pass through a same point if and only if $BQ$ is perpendicular to $CQ$.

$ ABCD$ is a rectangle (see the accompanying diagram) with $ P$ any point on $ \overline{AB}$. $ \overline{PS} \perp \overline{BD}$ and $ \overline{PR} \perp \overline{AC}$. $ \overline{AF} \perp \overline{BD}$ and $ \overline{PQ} \perp \overline{AF}$. Then $ PR \plus{} PS$ is equal to: [asy]defaultpen(linewidth(.8pt)); unitsize(3cm); pair D = origin; pair C = (2,0); pair B = (2,1); pair A = (0,1); pair P = waypoint(B--A,0.2); pair S = foot(P,D,B); pair R = foot(P,A,C); pair F = foot(A,D,B); pair Q = foot(P,A,F); pair T = intersectionpoint(P--Q,A--C); pair X = intersectionpoint(A--C,B--D); draw(A--B--C--D--cycle); draw(A--C); draw(B--D); draw(P--S); draw(A--F); draw(P--R); draw(P--Q); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$P$",P,N); label("$S$",S,SE); label("$T$",T,N); label("$E$",X,SW+SE); label("$R$",R,SW); label("$F$",F,SE); label("$Q$",Q,SW);[/asy] $ \textbf{(A)}\ PQ\qquad \textbf{(B)}\ AE\qquad \textbf{(C)}\ PT \plus{} AT\qquad \textbf{(D)}\ AF\qquad \textbf{(E)}\ EF$

For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below. (a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$. (b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.

Circle $\omega$ is inscribed in the $\vartriangle ABC$, with center $I$. Using only a ruler, divide segment $AI$ in half. (Grigory Filippovsky)

Given a circle (O,R). A point P lies inside the circle, OP=d, d<R. We consider quadrilaterals ABCD, inscribed in (O), such that AC is perp to BD at point P. Evaluate the maximum and minimum values of the perimeter of ABCD in terms of R and d.

Let $ABCD$ be a square and $P$ be a point on the shorter arc $AB$ of the circumcircle of the square. Which values can the expression $\frac{AP+BP}{CP+DP}$ take?

Let $ABC$ be a triangle and $H$ its orthocenter. Let $D$ be a point lying on the segment $AC$ and let $E$ be the point on the line $BC$ such that $BC\perp DE$. Prove that $EH\perp BD$ if and only if $BD$ bisects $AE$.

An angle and a point $A$ inside it is given. Is it possible to draw through $A$ three straight lines so that on either side of the angle one of three points of intersection of these lines be the midpoint of two other points of intersection with that side?

Call the ordered pair of distinct circles $(\omega, \gamma)$ scribable if there exists a triangle with circumcircle $\omega$ and incircle $\gamma$. Prove that among $n$ distinct circles there are at most $(n/2)^2$ scribable pairs. [i]Proposed by Daniel Liu

Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$. (Vietnam)

In a triangle $ABC$, $\angle BAC = 100^{\circ}, AB = AC$. A point $D$ is chosen on the side $AC$ such that $\angle ABD = \angle CBD$. Prove that $AD + DB = BC.$

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 $m$.

Let $ABC$ be a triangle with angles $\angle A = 80^\circ, \angle B = 70^\circ, \angle C = 30^\circ$. Let $P$ be a point on the bisector of $\angle BAC$ satisfying $\angle BPC =130^\circ$. Let $PX, PY, PZ$ be the perpendiculars drawn from $P$ to the sides $BC, AC, AB$, respectively. Prove that the following equation with segment lengths is satisfied $$AY^3+BZ^3+CX^3=AZ^3+BX^3+CY^3.$$

Altitudes $AH1$ and $BH2$ of acute triangle $ABC$ intersect at $H$. Let $w1$ be the circle that goes through $H2$ and touches the line $BC$ at $H1$, and let $w2$ be the circle that goes through $H1$ and touches the line $AC$ at $H2$. Prove, that the intersection point of two other tangent lines $BX$ and $AY$( $X$ and $Y$ are different from $H1$ and $H2$) to circles $w1$ and $w2$ respectively, lies on the circumcircle of triangle $HXY$. Proposed by [i]Danilo Khilko[/i]

Let $\alpha, \beta$ and $\gamma$ denote the angles of the triangle $ABC$. The perpendicular bisector of $AB$ intersects $BC$ at the point $X$, the perpendicular bisector of $AC$ intersects it at $Y$. Prove that $\tan(\beta) \cdot \tan(\gamma) = 3$ implies $BC= XY$ (or in other words: Prove that a sufficient condition for $BC = XY$ is $\tan(\beta) \cdot \tan(\gamma) = 3$). Show that this condition is not necessary, and give a necessary and sufficient condition for $BC = XY$.

$(NET 1)$ The vertices of an $(n + 1)-$gon are placed on the edges of a regular $n-$gon so that the perimeter of the $n-$gon is divided into equal parts. How does one choose these $n + 1$ points in order to obtain the $(n + 1)-$gon with $(a)$ maximal area; $(b)$ minimal area?

A special kind of parallelogram tile is made up by attaching the legs of two right isosceles triangles of side length $1$. We want to put a number of these tiles on the floor of an $n\times n$ room such that the distance from each vertex of each tile to the sides of the room is an integer and also no two tiles overlap. Prove that at least an area $n$ of the room will not be covered by the tiles. [i]Proposed by Ali Khezeli[/i]

Let $n$ be an integer, $n \ge 3$. Select $n$ points on the plane, none of which are three on the same line. Consider all triangles with vertices at selected points, denote the smallest of all the interior angles of these triangles by the variable $\alpha$. Find the largest possible value of $\alpha$ and identify all the selected $n$ point placements for which the max occurs.

Let $ABCD$ be a quadrilateral with an inscribed circle, centre $O$. Let \[AO = 5, BO =6, CO = 7, DO = 8.\] If $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$, determine $\frac{OM}{ON}$ .