CN-TST 2020

Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$ , and $p$ of them are non-zero. Let $$b_k=\sum_{i=1}^n a_i \omega^{ki}$$ for $k=1,2,\cdots, n$ . Prove that if $p>0$ , then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.

Given an isosceles triangle $\triangle ABC$ , $AB=AC$ . A line passes through $M$ , the midpoint of $BC$ , and intersects segment $AB$ and ray $CA$ at $D$ and $E$ , respectively. Let $F$ be a point of $ME$ such that $EF=DM$ , and $K$ be a point on $MD$ . Let $\Gamma_1$ be the circle passes through $B,D,K$ and $\Gamma_2$ be the circle passes through $C,E,K$ . $\Gamma_1$ and $\Gamma_2$ intersect again at $L \neq K$ . Let $\omega_1$ and $\omega_2$ be the circumcircle of $\triangle LDE$ and $\triangle LKM$ . Prove that, if $\omega_1$ and $\omega_2$ are symmetric wrt $L$ , then $BF$ is perpendicular to $BC$ .

For a non-empty finite set $A$ of positive integers, let $\text{lcm}(A)$ denote the least common multiple of elements in $A$ , and let $d(A)$ denote the number of prime factors of $\text{lcm}(A)$ (counting multiplicity). Given a finite set $S$ of positive integers, and $$f_S(x)=\sum_{\emptyset \neq A \subset S} \frac{(-1)^{|A|} x^{d(A)}}{\text{lcm}(A)}.$$ Prove that, if $0 \le x \le 2$ , then $-1 \le f_S(x) \le 0$ .

Show that the following equation has finitely many solutions $(t,A,x,y,z)$ in positive integers $$\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})$$

Let $a_1,a_2,\cdots,a_n$ be a permutation of $1,2,\cdots,n$ . Among all possible permutations, find the minimum of $$\sum_{i=1}^n \min \{ a_i,2i-1 \}.$$

Given a simple, connected graph with $n$ vertices and $m$ edges. Prove that one can find at least $m$ ways separating the set of vertices into two parts, such that the induced subgraphs on both parts are connected.