1. What is the greatest value that the sum of the cosines of all angles of an isosceles triangle can take?
Solution. Let $\alpha$ be the acute angle of an isosceles triangle, $0 < \alpha < \pi / 2$.

$2 \cos \alpha + \cos (\pi - 2 \alpha) = -2 \cos^2 \alpha + 2 \cos \alpha + 1$. When $\cos \alpha = 1 / 2$, this quadratic trinomial takes the maximum value, which is 1.5.

Comment. An unjustified answer is scored 0 points. The analysis of the quadratic trinomial should include the indication of the x-coordinate of the vertex of the parabola.