IMC 2015

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

prove that $\det(A) = \det(B)$.

Does the same conclusion follow for matrices with complex entries?

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

When does equality hold?

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 $\sum\limits_{n=0}^{\infty} \dfrac{1}{F(2^n)}$ is a rational number.

Determine whether or not there exist 15 integers $m_1, \ldots, m_{15}$ such that

Let $n \ge 2$, let $A_1, A_2, \ldots, A_{n+1}$ 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 $A_1, A_2, \ldots, A_{n+1}$. Prove that $\angle A_iBA_j > 90^\circ$ holds for at least $n$ pairs $(i, j)$ with $1 \le i < j \le n + 1$.

Prove that

Compute

Consider all $26^{26}$ words of length 26 in the Latin alphabet. Define the weight of a word as $1/(k+1)$, where $k$ is the number of letters not used in this word. Prove that the sum of the weights of all words is $3^{75}$.

An $n \times n$ complex matrix $A$ is called t-normal if $AA^t = A^tA$ where $A^t$ is the transpose of $A$. For each $n$, determine the maximum dimension of a linear space of complex $n \times n$ matrices consisting of t-normal matrices.

Let $n$ be a positive integer, and let $p(x)$ be a polynomial of degree $n$ with integer coefficients. Prove that