IMC 2002

A standard parabola is the graph of a quadratic polynomial $y = x^2 + ax + b$ with leading co"efficient 1. Three standard parabolas with vertices $V1, V2, V3$ intersect pairwise at points $A1, A2, A3$ . Let $A \mapsto s(A)$ be the reflection of the plane with respect to the $x$ -axis.

Prove that standard parabolas with vertices $s (A1), s (A2), s (A3)$ intersect pairwise at the points $s (V1), s (V2), s (V3)$ .

Does there exist a continuously differentiable function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that for every $x \in \mathbb{R}$ we have $f(x) > 0$ and $f'(x) = f(f(x))$ ?

Let $n$ be a positive integer and let $a_k = \dfrac{1}{\binom{n}{k}}, b_k = 2^{k-n},\ (k=1..n)$ .

Show that $\sum_{k=1}^n \dfrac{a_k-b_k}{k} = 0$ .

Let $f : [a, b] \rightarrow [a, b]$ be a continuous function and let $p \in [a, b]$ . Define $p_0 = p$ and $p_{n+1} = f(p_n)$ for $n = 0, 1, 2,...$ . Suppose that the set $T_p = \{p_n : n = 0, 1, 2,...\}$ is closed, i.e., if $x \not\in T_p$ then $\exists \delta > 0$ such that for all $x' \in T_p$ we have $|x'-x|\ge\delta$ .

Show that $T_p$ has finitely many elements.

Prove or disprove the following statements: (a) There exists a monotone function $f : [0, 1] \rightarrow [0, 1]$ such that for each $y \in [0, 1]$ the equation $f(x) = y$ has uncountably many solutions $x$ . (b) There exists a continuously differentiable function $f : [0, 1] \rightarrow [0, 1]$ such that for each $y \in [0, 1]$ the equation $f(x) = y$ has uncountably many solutions $x$ .

For an $n\times n$ matrix with real entries let $||M||=\sup_{x\in \mathbb{R}^{n}\setminus\{0\}}\frac{||Mx||_{2}}{||x||_{2}}$ , where $||\cdot||_{2}$ denotes the Euclidean norm on $\mathbb{R}^{n}$ . Assume that an $n\times n$ matrxi $A$ with real entries satisfies $||A^{k}-A^{k-1}||\leq\frac{1}{2002k}$ for all positive integers $k$ . Prove that $||A^{k}||\leq 2002$ for all positive integers $k$ .

Compute the determinant of the $n\times n$ matrix $A=(a_{ij})_{ij}$ , $$a_{ij}=\begin{cases} (-1)^{|i-j|} & \text{if} i\ne j, 2 & \text{if} i= j. \end{cases}$$

200 students participated in a math contest. They had 6 problems to solve. Each problem was correctly solved by at least 120 participants. Prove that there must be 2 participants such that every problem was solved by at least one of these two students.

For each $n\geq 1$ let $$a_{n}=\sum_{k=0}^{\infty}\frac{k^{n}}{k!}, \; b_{n}=\sum_{k=0}^{\infty}(-1)^{k}\frac{k^{n}}{k!}.$$ Show that $a_{n}\cdot b_{n}$ is an integer.