$(BEL 4)$ Let $O$ be a point on a nondegenerate conic. A right angle with vertex $O$ intersects the conic at points $A$ and $B$. Prove that the line $AB$ passes through a fixed point located on the normal to the conic through the point $O.$

Determine the locus of orthocenters of triangles, given the midpoint of a side and the feet of the altitudes drawn on two other sides.

$\Phi$ is the union of all triangles that are symmetric of the triangle $ABC$ wrt a point $O$, as point $O$ moves along the triangle's sides. If the area of the triangle is $E$, find the area of $\Phi$.

Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios. [b]a.)[/b] Prove that the quadrilateral $MNPQ$ is a parallelogram. [b]b.)[/b] What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ? (Consecutive vertices of the parallelograms are labelled in alphabetical order.

Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$

Let $O$ be the center of the regular triangle $ABC$. Find the set of all points $M$ such that any line containing the point $M$ intersects one of the segments $AB, OC$.

Given a circle and a point $K$ inside it. An arbitrary circle equal to the given one and passing through the point $K$ has a common chord with the given circle. Find the geometric locus of the midpoints of these chords.

A line $MN$ is given in the plane. Consider circles $k_1$, $k_2$ tangent to the line at points $M$, $N$, respectively, while touching each other externally. Let $X$ be the midpoint of the segment $PQ$, where $P$, $Q$ are in this order tangent points of the second common external tangent of the circles $k_1$, $k_2$. Find the locus of the points $X$ for all pairs of circles of the specified properties.

Given are a circle and an ellipse lying inside it with focus $C$. Find the locus of the circumcenters of triangles $ABC$, where $AB$ is a chord of the circle touching the ellipse.

Let a semicircle $AB$ be given and let $X$ be an inner point of the arc. Consider a point $Y$ on ray $XA$ such that $XY=XB$. Find the locus of all points $Y$ when $X$ moves on the arc $AB$ (excluding the endpoints).

Let $ABC$ be a triangle. A point $P$ varies inside $BC$. Let $Q, R$ be the points on $AC, AB$ in that order, such that $PQ\parallel AB, PR\parallel AC$. 1. Prove that, when $P$ varies, the circumcircle of triangle $AQR$ always passes through a fixed point $X$ other than $A$. 2. Extend $AX$ so that it cuts the circumcircle of $ABC$ a second time at point $K$. Prove that $AX=XK$.

Find locus of points $ M $ inside an equilateral triangle $ ABC $ such that $$ \angle MBC+\angle MCA +\angle MAB={\pi}/{2}. $$

Find the locus of the orthocentres (i.e. the point where three altitudes meet) of the triangles inscribed in a given circle . (A. Andjans, Riga)

Let $ABCD$ be a square with side length $1$. Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.

Let $ABCD$ be a trapezium (trapezoid) with $AD$ parallel to $BC$ and $J$ be the intersection of the diagonals $AC$ and $BD$. Point $P$ a chosen on the side $BC$ such that the distance from $C$ to the line $AP$ is equal to the distance from $B$ to the line $DP$. [i]The following three questions 1, 2 and 3 are independent, so that a condition in one question does not apply in another question.[/i] 1.Suppose that $Area( \vartriangle AJB) =6$ and that $Area(\vartriangle BJC) = 9$. Determine $Area(\vartriangle APD)$. 2. Find all points $Q$ on the plane of the trapezium such that $Area(\vartriangle AQB) = Area(\vartriangle DQC)$. 3. Prove that $PJ$ is the angle bisector of $\angle APD$.

Two perpendicular lines $p,q$ and a point $A\notin p\cup q$ are given in plane. Find locus of all points $X$ such that \[XA=\sqrt{|Xp|\cdot|Xq|\,},\] where $|Xp|$ denotes the distance of $X$ from $p.$

Three cockroaches run along a circle in the same direction. They start simultaneously from a point $S$. Cockroach $A$ runs twice as slow than $B$, and thee times as slow than $C$. Points $X, Y$ on segment $SC$ are such that $SX = XY =YC$. The lines $AX$ and $BY$ meet at point $Z$. Find the locus of centroids of triangles $ZAB$.

Let $T_1$ and $T_2$ internally tangent circumferences at $P$, with radius $R$ and $2R$, respectively. Find the locus traced by $P$ as $T_1$ rolls tangentially along the entire perimeter of $T_2$

Consider a quadrilateral $ABCD$ and points $K, L, M, N$ on sides $AB, BC, CD$ and $AD$, respectively, such that $KB = BL = a, MD = DN = b$ and $KL \nparallel MN$. Find the set of all the intersection points of $KL$ with $MN$ as $a$ and $b$ vary.

A segment $AB$ is given in (Euclidean) plane. Consider all triangles $XYZ$ such, that $X$ is an inner point of $AB$, triangles $XBY$ and $XZA$ are similar (in this order of vertices), and points $A, B, Y, Z$ lie on a circle in this order. Find the locus of midpoints of all such segments $YZ$. (Day 1, 2nd problem authors: Michal Rolínek, Jaroslav Švrček)

Around triangle $ABC$ with acute angle C is circumscribed a circle. On the arc $AB$, which does not contain point $C$, point $D$ is chosen. Point $D'$ is symmetric on point $D$ with respect to line $AB$. Straight lines $AD'$ and $BD'$ intersect segments $BC$ and $AC$ at points $E$ and $F$. Let point $C$ move along its arc $AB$. Prove that the center of the circumscribed circle of a triangle $CEF$ moves on a straight line.

$(BEL 4)$ Let $O$ be a point on a nondegenerate conic. A right angle with vertex $O$ intersects the conic at points $A$ and $B$. Prove that the line $AB$ passes through a fixed point located on the normal to the conic through the point $O.$

Consider two lines $\ell$ and $\ell ' $ and a fixed point $P$ equidistant from these lines. What is the locus of projections $M$ of $P$ on $AB$, where $A$ is on $\ell $, $B$ on $\ell ' $, and angle $\angle APB$ is right?

Find the locus of centroids of equilateral triangles whose vertices lie on sides of a given square $ABCD$.

In a parallelogram $ABCD$, triangle $ABD$ is acute-angled and $\angle BAD = \pi /4$. Consider all possible choices of points $K,L,M,N$ on sides $AB,BC, CD,DA$ respectively, such that $KLMN$ is a cyclic quadrilateral whose circumradius equals those of triangles $ANK$ and $CLM$. Find the locus of the intersection of the diagonals of $KLMN$