Olympiad problems

1995 Romania Team Selection Test, 2

Tags: number theory

Find all positive integers $ x,y,z,t$ such that $ x,y,z$ are pairwise coprime and $ (x \plus{} y)(y \plus{} z)(z \plus{} x) \equal{} xyzt$.

2007 USA Team Selection Test, 6

Tags: algebra , polynomial , algorithm , modular arithmetic , counting , distinguishability , induction

For a polynomial $ P(x)$ with integer coefficients, $ r(2i \minus{} 1)$ (for $ i \equal{} 1,2,3,\ldots,512$) is the remainder obtained when $ P(2i \minus{} 1)$ is divided by $ 1024$. The sequence \[ (r(1),r(3),\ldots,r(1023)) \] is called the [i]remainder sequence[/i] of $ P(x)$. A remainder sequence is called [i]complete[/i] if it is a permutation of $ (1,3,5,\ldots,1023)$. Prove that there are no more than $ 2^{35}$ different complete remainder sequences.

2006 Thailand Mathematical Olympiad, 3

Tags: algebra , polynomial , divide

Let $P(x), Q(x)$ and $R(x)$ be polynomials satisfying the equation $2xP(x^3) + Q(-x -x^3) = (1 + x + x^2)R(x)$. Show that $x - 1$ divides $P(x) - Q(x)$.

2007 Moldova Team Selection Test, 4

Tags: prime number , number theory

Show that there are infinitely many prime numbers $p$ having the following property: there exists a natural number $n$, not dividing $p-1$, such that $p|n!+1$.

2007 Danube Mathematical Competition, 3

Tags: induction , number theory

For each positive integer $ n$, define $ f(n)$ as the exponent of the $ 2$ in the decomposition in prime factors of the number $ n!$. Prove that the equation $ n\minus{}f(n)\equal{}a$ has infinitely many solutions for any positive integer $ a$.

2003 AIME Problems, 15

Tags: algebra , polynomial , function

Let \[P(x)=24x^{24}+\sum_{j=1}^{23}(24-j)(x^{24-j}+x^{24+j}). \] Let $z_{1},z_{2},\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2}=a_{k}+b_{k}i$ for $k=1,2,\ldots,r,$ where $i=\sqrt{-1},$ and $a_{k}$ and $b_{k}$ are real numbers. Let \[\sum_{k=1}^{r}|b_{k}|=m+n\sqrt{p}, \] where $m,$ $n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$

2001 India IMO Training Camp, 2

Tags: modular arithmetic , number theory

Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$, define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv 1 \pmod{p}$ and set $kx_k = 1+ pn_k$. Prove that : \[\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}\]