IMC 2006

Let $f : \mathbb{R} \to \mathbb{R}$ be a real function. Prove or disprove each of the following statements.

(a) If $f$ is continuous and $\operatorname{range}(f) = \mathbb{R}$ then $f$ is monotonic.

(b) If $f$ is monotonic and $\operatorname{range}(f) = \mathbb{R}$ then $f$ is continuous.

(c) If $f$ is monotonic and $f$ is continuous then $\operatorname{range}(f) = \mathbb{R}$.

Find the number of positive integers $x$ satisfying the following two conditions:

1. $x < 10^{2006}$;

2. $x^2 - x$ is divisible by $10^{2006}$.

Let $A$ be an $n \times n$-matrix with integer entries and $b_1, \dots, b_k$ be integers satisfying $\det A = b_1 \cdot \dots \cdot b_k$. Prove that there exist $n \times n$-matrices $B_1, \dots, B_k$ with integer entries such that $A = B_1 \cdot \dots \cdot B_k$ and $\det B_i = b_i$ for all $i = 1, \dots, k$.

Let $f$ be a rational function (i.e. the quotient of two real polynomials) and suppose that $f(n)$ is an integer for infinitely many integers $n$. Prove that $f$ is a polynomial.

Let $a, b, c, d, e > 0$ be real numbers such that $a^2 + b^2 + c^2 = d^2 + e^2$ and $a^4 + b^4 + c^4 = d^4 + e^4$. Compare the numbers $a^3 + b^3 + c^3$ and $d^3 + e^3$.

Find all sequences $a_0, a_1, \dots, a_n$ of real numbers where $n \ge 1$ and $a_n \ne 0$, for which the following statement is true:

If $f : \mathbb{R} \to \mathbb{R}$ is an $n$ times differentiable function and $x_0 < x_1 < \dots < x_n$ are real numbers such that $f(x_0) = f(x_1) = \dots = f(x_n) = 0$ then there exists an $h \in (x_0, x_n)$ for which

Let $V$ be a convex polygon with $n$ vertices.

(a) Prove that if $n$ is divisible by $3$ then $V$ can be triangulated (i.e. dissected into non-overlapping triangles whose vertices are vertices of $V$) so that each vertex of $V$ is the vertex of an odd number of triangles.

(b) Prove that if $n$ is not divisible by $3$ then $V$ can be triangulated so that there are exactly two vertices that are the vertices of an even number of the triangles.

Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that for any real numbers $a < b$, the image $f\big([a,b]\big)$ is a closed interval of length $b - a$.

Compare $\tan(\sin x)$ and $\sin(\tan x)$ for all $x \in \left(0, \dfrac{\pi}{2}\right)$.

Let $v_0$ be the zero vector in $\mathbb{R}^n$ and let $v_1, v_2, \dots, v_{n+1} \in \mathbb{R}^n$ be such that the Euclidean norm $|v_i - v_j|$ is rational for every $0 \le i, j \le n+1$. Prove that $v_1, \dots, v_{n+1}$ are linearly dependent over the rationals.

Prove that there exists an infinite number of relatively prime pairs $(m,n)$ of positive integers such that the equation

has three distinct integer roots.

Let $A_i$, $B_i$, $S_i$ ($i = 1, 2, 3$) be invertible real $2 \times 2$ matrices such that

(1) not all $A_i$ have a common real eigenvector;

(2) $A_i = S_i^{-1} B_i S_i$ for all $i = 1, 2, 3$;

(3) $A_1 A_2 A_3 = B_1 B_2 B_3 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.

Prove that there is an invertible real $2 \times 2$ matrix $S$ such that $A_i = S^{-1} B_i S$ for all $i = 1, 2, 3$.