Problems

The point $A$ is fixed on a circle. Find the locus of the point $X$ which divides the chords that end at point $A$ in a $1:2$ ratio, starting from the point $A$.

Let $O$ be the center of the rectangle $ABCD$. Find the geometric points of $M$ for which $AM\ge OM,BM\ge OM$, $CM\ge OM$, and $DM\ge OM$.

Construct the triangle $ABC$ along the side $a$, the height $h_a$ and the angle $A$.

Construct a right-angled triangle along the leg and the hypotenuse.

Construct a circle with a given centre, tangent to a given circle.

Construct a straight line passing through a given point and tangent to a given circle.

Three segments whose lengths are equal to $a,b$ and $c$ are given. Construct a segment of length: a) $ab/c$; b) $\sqrt{ab}$.

Construct the triangle ABC by the medians $m_a,m_b$ and $m_c$.

Solve the equations $x^2=14+y^2$ in integers.

Solve the equations in integers:

a) $3x^2+5y^2=345$;

b) $1+x+x^2+x^3=2^y$.