BG-MO 1965

Prove the inequality: $$(1+\sin^2\alpha)^n+(1+\cos^2\alpha)^n\ge2\left(\frac32\right)^n$$ is true for every natural number $n$ . When does equality hold?

In the triangle $ABC$ , angle bisector $CD$ intersects the circumcircle of $ABC$ at the point $K$ .

(a) Prove the equalities: $$\frac1{ID}-\frac1{IK}=\frac1{CI},\enspace\frac{CI}{ID}-\frac{ID}{DK}=1$$ where $I$ is the center of the inscribed circle of triangle $ABC$ . (b) On the segment $CK$ some point $P$ is chosen whose projections on $AC,BC,AB$ respectively are $P_1,P_2,P_3$ . The lines $PP_3$ and $P_1P_2$ intersect at a point $M$ . Find the locus of $M$ when $P$ moves around segment $CK$ .

In the space there are given crossed lines $s$ and $t$ such that $\angle(s,t)=60^\circ$ and a segment $AB$ perpendicular to them. On $AB$ it is chosen a point $C$ for which $AC:CB=2:1$ and the points $M$ and $N$ are moving on the lines $s$ and $t$ in such a way that $AM=2BN$ . The angle between vectors $\overrightarrow{AM}$ and $\overrightarrow{BM}$ is $60^\circ$ . Prove that:

(a) the segment $MN$ is perpendicular to $t$ ; (b) the plane $\alpha$ , perpendicular to $AB$ in point $C$ , intersects the plane $CMN$ on fixed line $\ell$ with given direction in respect to $s$ ; (c) all planes passing by $ell$ and perpendicular to $AB$ intersect the lines $s$ and $t$ respectively at points $M$ and $N$ for which $AM=2BN$ and $MN\perp t$ .