Let $a_1,a_2,\ldots,a_n$ be $n$ real numbers whose squares sum to $1$. Prove that for any integer $k\ge2$, there exist $n$ integers $x_1,x_2,\ldots,x_n$, each with absolute value at most $k-1$ and not all zero, such that
$$\left|\sum_{i=1}^n a_ix_i\right|\le\frac{(k-1)\sqrt n}{k^n-1}.$$
If $a_1,a_2,\ldots,a_n$ are distinct integers, prove that
$$(x-a_1)(x-a_2)\cdots(x-a_n)-1$$
cannot be expressed as a product of two polynomials, each with integer coefficients and of degree at least $1$.
Let $A_1,B_1$ be two points on the base $AB$ of an isosceles triangle $ABC$, with $\angle C\gt 60^\circ$, such that $\angle A_1CB_1=\angle ABC$. A circle externally tangent to the circumcircle of $\triangle A_1B_1C$ is tangent to the rays $CA$ and $CB$ at points $A_2$ and $B_2$, respectively. Prove that $A_2B_2=2AB$.
Let $\mathbb N$ be the set of positive integers, that is, $\mathbb N=\{1,2,\ldots\}$. Find all functions $f:\mathbb N\to\mathbb N$ such that
$$f(f(m)+f(n))=m+n\qquad\text{for all }m,n\in\mathbb N.$$