IMC 2015 for University Students: Day 1

Problem 1. For any integer $n \ge 2$ and two $n \times n$ matrices with real entries $A,B$ that satisfy the equation

$A^{-1}+B^{-1}=(A+B)^{-1}$,

prove that $\det(A)=\det(B)$. Does the same conclusion follow for matrices with complex entries?

Problem 2. For a positive integer $n$, let $f(n)$ be the number obtained by writing $n$ in binary and replacing every $0$ with $1$ and vice versa. For example, $n=23$ is $10111$  in binary, so $f(n)$ is $1000$ in binary, therefore $f(23)=8$. Prove that

$\sum_{k=1}^n f(k) \le \frac{n^2}{4}$.

When does the equality hold?

Problem 3. Let $F(0)=0,F(1)=\frac{3}{2}$, and $F(n)=\frac{5}{2}F(n-1)-F(n-2)$ for $n \ge 2$.

Determine whether or not $\displaystyle \sum_{n=0}^\infty \frac{1}{F(2^n)}$ is a rational number.

Problem 4. Determine whether or not there exist $15$ integers $m_1,m_2,\dots,m_{15}$ such that

$\displaystyle \sum_{k=1}^{15} m_k \cdotp \text{arctan}(k) = \text{arctan}(16).$

Problem 5. Let $n \ge 2$, and let $A_1,A_2,\dots,A_{n+1}$ be $n+1$ points in the $n-$dimensional Euclidean space, not lying on the same hyperplane, and let $B$ be a point strictly inside the convex hull of $A_1,A_2,\dots,A_{n+1}$. Prove that $\angle A_iBA_j > 90^{\circ}$ holds for at least $n$ pairs $(i,j)$ with $1 \le i < j \le n+1$.