ITOT 42 Autumn 2020 - THPT O - Additive practice

Each of the real quadratic polynomials $P(x)$, $Q(x)$ and $P(x)+Q(x)$ has a double root. Must these roots coincide?

Ten points $X_1,\ldots,X_{10}$ lie in that order on a line. On each consecutive segment $X_iX_{i+1}$ an isosceles triangle is constructed, all with the same apex angle $\alpha$. All nine apices lie on a common semicircle with diameter $X_1X_{10}$. Find $\alpha$.

A positive integer $N$ is divisible by 2020. Its decimal digits are all distinct, and swapping any two digits produces a number not divisible by 2020. How many digits can $N$ have?

The angle bisectors of a triangle divide its three sides into six segments. Is it always possible to use all six segments as the sides of two nondegenerate triangles?

There are 101 coins on a circle, each weighing 10g or 11g. Prove that there exists a coin whose nearest $k$ coins on the left have the same total weight as its nearest $k$ coins on the right, for (a) $k=50$; (b) $k=49$. The chosen coin may differ between the two parts.