BG-MO 1973

Let the sequence $a_1,a_2,\ldots,a_n,\ldots$ is defined by the conditions: $a_1=2$ and $a_{n+1}=a_n^2-a_n+1$ $(n=1,2,\ldots)$ . Prove that:

(a) $a_m$ and $a_n$ are relatively prime numbers when $m\ne n$ . (b) $\lim_{n\to\infty}\sum_{k=1}^n\frac1{a_k}=1$ I. Tonov

Let the numbers $a_1,a_2,a_3,a_4$ form an arithmetic progression with difference $d\ne0$ . Prove that there are no exists geometric progressions $b_1,b_2,b_3,b_4$ and $c_1,c_2,c_3,c_4$ such that: $$a_1=b_1+c_1,a_2=b_2+c_2,a_3=b_3+c_3,a_4=b_4+c_4.$$

Let $a_1,a_2,\ldots,a_n$ are different integer numbers in the range $[100,200]$ for which: $a_1+a_2+\ldots+a_n\ge11100$ . Prove that it can be found at least number from the given in the representation of decimal system on which there are at least two equal digits.

L. Davidov

Find all functions $f(x)$ defined in the range $\left(-\frac\pi2,\frac\pi2\right)$ that are differentiable at $0$ and satisfy $$f(x)=\frac12\left(1+\frac1{\cos x}\right)f\left(\frac x2\right)$$ for every $x$ in the range $\left(-\frac\pi2,\frac\pi2\right)$ .

L. Davidov

Let the line $\ell$ intersects the sides $AC,BC$ of the triangle $ABC$ respectively at the points $E$ and $F$ . Prove that the line $\ell$ is passing through the incenter of the triangle $ABC$ if and only if the following equality is true: $$BC\cdot\frac{AE}{CE}+AC\cdot\frac{BF}{CF}=AB.$$ H. Lesov

In the tetrahedron $ABCD$ , $E$ and $F$ are the midpoints of $BC$ and $AD$ , $G$ is the midpoint of the segment $EF$ . Construct a plane through $G$ intersecting the segments $AB$ , $AC$ , $AD$ in the points $M,N,P$ respectively in such a way that the sum of the volumes of the tetrahedrons $BMNP$ , $CMNP$ and $DMNP$ to be minimal.

H. Lesov