Školjka

%V0 In a given right triangle $ABC$, the hypotenuse $BC$, of length $a$, is divided into $n$ equal parts ($n$ and odd integer). Let $\alpha$ be the acute angel subtending, from $A$, that segment which contains the mdipoint of the hypotenuse. Let $h$ be the length of the altitude to the hypotenuse fo the triangle. Prove that: $$\tan{\alpha}=\dfrac{4nh}{(n^2-1)a}.$$

%V0 Suppose medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that: a.) Using medians of that triangle it is possible to construct a rectangular triangle. b.) The following inequality: $$5(a^2+b^2-c^2) \geq 8ab,$$ is valid, where $a,b$ and $c$ are side length of the given triangle.

%V0 Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality $$abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.$$ Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.

%V0 Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$. CommentAlternative formulation, from IMO ShortList 1974, Finland 2: We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.

%V0 Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then $$\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N$$ Proposed by Greece.

%V0 In an acute-angled triangle $ABC,$ let $AD,BE$ be altitudes and $AP,BQ$ internal bisectors. Denote by $I$ and $O$ the incenter and the circumcentre of the triangle, respectively. Prove that the points $D, E,$ and $I$ are collinear if and only if the points $P, Q,$ and $O$ are collinear.