## Solution of problem T10/363 in M&Y

$\left(a-\dfrac{1}{b}\right)\left(b-\dfrac{1}{c}\right)\left(c-\dfrac{1}{a}\right)\geq \left(a-\dfrac{1}{a}\right)\left(b-\dfrac{1}{b}\right)\left(c-\dfrac{1}{c}\right)\\ \forall a,b,c\in [1,+\infty).$

Solution of a my student.

## Solution of Problem 2, Grade 9, RNO 2007

Let $ABC$ be an acute angled triangle and point $M$ chosen differently from $A,B,C$. Prove that $M$ is the orthocenter of triangle $ABC$ if and only if

$\dfrac{BC}{MA}\cdot\overrightarrow{MA}+\dfrac{CA}{MB}\cdot\overrightarrow{MB}+\dfrac{AB}{MC}\cdot\overrightarrow{MC}= \overrightarrow{0}.(*)$

Solution of my students.

## Solution of problem 11306 in AMM

Let $a,b,$ and $c$ be the lengths of the sides of a nondegenerate triangle, let $p=(a+b+c)/2$, and let $r$ and $R$ be the inradius and circumradius of the triangle, respectively. Show that $\dfrac{a}{2}\cdot \dfrac{4r-R}{R}\leq\sqrt{(p-b)(p-c)}\leq \dfrac{a}{2},$

and determine the cases of equality.

My solution.