BG-MO 1972

Prove that there are don't exist integers $a,b,c$ such that for every integer $x$ the number $A=(x+a)(x+b)(x+c)-x^3-1$ is divisible by $9$ .

I. Tonov

Solve the system of equations: $$\begin{cases}\sqrt{\frac{y(t-y)}{t-x}-\frac4x}+\sqrt{\frac{z(t-z)}{t-x}-\frac4x}=\sqrt x\sqrt{\frac{z(t-z)}{t-y}-\frac4y}+\sqrt{\frac{x(t-x)}{t-y}-\frac4y}=\sqrt y\sqrt{\frac{x(t-x)}{t-z}-\frac4z}+\sqrt{\frac{y(t-y)}{t-z}-\frac4z}=\sqrt zx+y+z=2t\end{cases}$$ if the following conditions are satisfied: $0<x<t$ , $0<y<t$ , $0<z<t$ .

H. Lesov

Prove the equality: $$\sum_{k=1}^{n-1}\frac1{\sin^2\frac{(2k+1)\pi}{2n}}=n^2$$ where $n$ is a natural number.

H. Lesov

Find maximal possible number of points lying on or inside a circle with radius $R$ in such a way that the distance between every two points is greater than $R\sqrt2$ .

H. Lesov

In a circle with radius $R$ , there is inscribed a quadrilateral with perpendicular diagonals. From the intersection point of the diagonals, there are perpendiculars drawn to the sides of the quadrilateral.

(a) Prove that the feet of these perpendiculars $P_1,P_2,P_3,P_4$ are vertices of the quadrilateral that is inscribed and circumscribed. (b) Prove the inequalities $2r_1\le\sqrt2 R_1\le R$ where $R_1$ and $r_1$ are radii respectively of the circumcircle and inscircle to the quadrilateral $P_1P_2P_3P_4$ . When does equality hold?

H. Lesov

It is given a tetrahedron $ABCD$ for which two points of opposite edges are mutually perpendicular. Prove that:

(a) the four altitudes of $ABCD$ intersects at a common point $H$ ; (b) $AH+BH+CH+DH<p+2R$ , where $p$ is the sum of the lengths of all edges of $ABCD$ and $R$ is the radii of the sphere circumscribed around $ABCD$ .

H. Lesov