Found problems: 8
Let $A$ and $B$ be two points inside a circle $C$. Show that there exists a circle that contains $A$ and $B$ and lies completely inside $C$.
Let $ABC$ be a triangle and $D$ a point on its side $BC.$ Points $E, F$ lie on the lines $AB, AC$ beyond vertices $B, C,$ respectively, such that $BE = BD$ and $CF = CD.$ Let $P$ be a point such that $D$ is the incenter of triangle $P EF.$ Prove that $P$ lies inside the circumcircle $\Omega$ of triangle $ABC$ or on it.
A convex quadrangle without parallel sides is given. For each triple of its vertices, a point is constructed that supplements this triple to a parallelogram, one of the diagonals of which coincides with the diagonal of the quadrangle. Prove that of the four points constructed, exactly one lies inside the original quadrangle.
A lattice point is defined as a point on the plane with integer coordinates. Show that for all positive integers $n$, there is a circle on the plane with exactly n lattice points in its interior (not including its boundary).
Given a point $X$ and $n$ vectors $\overrightarrow{x_i}$ with sum zero in the plane. For each permutation of the vectors we form a set of $n$ points, by starting at $X$ and adding the vectors in order. For example, with the original ordering we get $X_1$ such that $XX_1 = \overrightarrow{x_1}, X_2$ such that $X_1X_2 = \overrightarrow{x_2}$ and so on. Show that for some permutation we can find two points $Y, Z$ with angle $\angle YXZ = 60^o $, so that all the points lie inside or on the triangle $XYZ$.
Two equal polygons $F$ and $F'$ are given on the plane. It is known that the vertices of the polygon $F$ belong to $F'$ (may lie inside it or on the border). Is it true that all the vertices of these polygons coincide?
Let $P$ on $AB$, $Q$ on $BC$, $R$ on $CD$ and $S$ on $AD$ be points on the sides of a convex quadrilateral $ABCD$. Show that the following are equivalent:
(1) There is a choice of $P,Q,R,S$, for which all of them are interior points of their side, such that $PQRS$ has minimal perimeter.
(2) $ABCD$ is a cyclic quadrilateral with circumcenter in its interior.
A triangle $ABC$ and a point $P$ inside it are given. $A', B', C'$ are the projections of $P$ onto the straight lines ot the sides $BC,CA,AB$. Prove that the center of the circle circumscribed around the triangle $A'B'C'$ lies inside the triangle $ABC$.