KURSCHAK 1999

For any positive integer $m$ , denote by $d_i(m)$ the number of positive divisors of $m$ that are congruent to $i$ modulo $2$ . Prove that if $n$ is a positive integer, then $$\left|\sum_{k=1}^n \left(d_0(k)-d_1(k)\right)\right|\le n.$$

Given a triangle on the plane, construct inside the triangle the point $P$ for which the centroid of the triangle formed by the three projections of $P$ onto the sides of the triangle happens to be $P$ .

We are given more than $2^k$ integers, where $k\in\mathbb{N}$ . Prove that we can choose $k+2$ of them such that if some of our selected numbers satisfy $$x_1+x_2+\dots+x_m=y_1+y_2+\dots+y_m$$ where $x_1<\dots<x_m$ and $y_1<\dots<y_m$ , then $x_i=y_i$ for any $1\le i\le m$ .