TH-TMO 2014

Find all functions $f : R \to R$ satisfying $f(xy - 1) + f(x)f(y) = 2xy - 1$ for all real numbers $x, y$

Let $M$ and $N$ be positive integers. Pisut walks from point $(0, N)$ to point $(M, 0)$ in steps so that $\bullet$ each step has unit length and is parallel to either the horizontal or the vertical axis, and $\bullet$ each point ( $x, y)$ on the path has nonnegative coordinates, i.e. $x, y > 0$ . During each step, Pisut measures his distance from the axis parallel to the direction of his step, if after the step he ends up closer from the origin (compared to before the step) he records the distance as a positive number, else he records it as a negative number. Prove that, after Pisut completes his walk, the sum of the signed distances Pisut measured is zero.

Find $P(x)\in Z[x]$ st : $P(n)|2557^{n}+213.2014$ with any $n\in N^{*}$

Determine the maximal value of $k$ such that the inequality $$\left(k +\frac{a}{b}\right) \left(k + \frac{b}{c}\right)\left(k + \frac{c}{a}\right) \le \left( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\right) \left( \frac{b}{a}+ \frac{c}{b}+ \frac{a}{c}\right)$$ holds for all positive reals $a, b, c$ .

Find all primes $p$ such that $2p^2 - 3p - 1$ is a positive perfect cube

Let $n$ be a positive integer. We want to make up a collection of cards with the following properties:

1. each card has a number of the form $m!$ written on it, where $m$ is a positive integer.

2. for any positive integer $t \le n!$ , we can select some card(s) from this collection such that the sum of the number(s) on the selected card(s) is $t$ .

Determine the smallest possible number of cards needed in this collection.