5. [8] The three medians of a triangle divide its angle into six angles, of which exactly $k$ are greater than $30^{\circ}$. What is the greatest possible value of $k$?

(N. Sedrakyan)
Answer. $k=3$.

Evaluation. First method. Let $h_{1} \leq h_{2} \leq h_{3}$ be the heights of the triangle, and $m_{1}, m_{2}, m_{3}$ be the medians from the corresponding vertices (in fact, $m_{1} \leq m_{2} \leq m_{3}$, but this will not be used). Then we have $m_{3} \geq h_{i}$ for $i=1,2$, 3. Dropping perpendiculars from the end of $m_{3}$ to the adjacent sides, each of them equals half of the corresponding height and, therefore, is not greater than $m_{3} / 2$. Thus, the angles adjacent to $m_{3}$ are not greater than $30^{\circ}$. Similarly, one of the angles at $m_{2}$ is not greater than $30^{\circ}$.

Second method. Let there be a triangle $A B C$ with medians $A X, B Y, C Z$. Both angles $B A X$ and $B C Z$ cannot be greater than $30^{\circ}$: otherwise, points $A$ and $C$ lie inside a circle with chord $Z X$ and a diameter twice as large as $Z X$, but then $A C=2 Z X$, meaning a segment equal to the diameter lies strictly inside the circle - a contradiction. Similarly, in each of the other two pairs, at most one angle is greater than $30^{\circ}$.

Example 1. Consider first a triangle $A B C$ where $\angle C=90^{\circ}, \angle A=30^{\circ}$. Let $B K$ and $C L$ be its medians. Then $\angle A C L=30^{\circ}, \angle B C L=60^{\circ}, \angle C B K>30^{\circ}$ (since the angle bisector of angle $B$ lies between the leg and the median). Now slightly decrease the leg $A C$. In this case, the angle $A C L$ will slightly increase (i.e., become greater than $30^{\circ}$), while the angles $B C L$ and $C B K$ will slightly decrease but remain greater than $30^{\circ}$.

Example 2. Consider a right triangle with legs 4 and 6 (see figure). We have $\operatorname{tg} 30^{\circ}<\operatorname{tg} \alpha<\operatorname{tg} \beta<\operatorname{tg} \gamma$, since $\frac{1}{\sqrt{3}}<\frac{2}{3}<\frac{3}{4}<\frac{3}{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_034089706a976fe8a7b8g-3.jpg?height=294&width=411&top_left_y=1275&top_left_x=1482)