Found problems: 1269
2001 South africa National Olympiad, 6
The unknown real numbers $x_1,x_2,\dots,x_n$ satisfy $x_1 < x_2 < \cdots < x_n,$ where $n \geq 3$. The numbers $s$, $t$ and $d_1,d_2,\dots,d_{n - 2}$ are given, such that \[ \begin{aligned} s & = \sum\limits_{i = 1}^nx_i, \\ t & = \sum\limits_{i = 1}^nx_i^2,\\ d_i & = x_{i + 2} - x_i,\ \ i = 1,2,\dots,n - 2. \end{aligned} \] For which $n$ is this information always sufficient to determine $x_1,x_2,\dots,x_n$ uniquely?
2008 Germany Team Selection Test, 1
Determine $ Q \in \mathbb{R}$ which is so big that a sequence with non-negative reals elements $ a_1 ,a_2, \ldots$ which satisfies the following two conditions:
[b](i)[/b] $ \forall m,n \geq 1$ we have $ a_{m \plus{} n} \leq 2 \left(a_m \plus{} a_n \right)$
[b](ii)[/b] $ \forall k \geq 0$ we have $ a_{2^k} \leq \frac {1}{(k \plus{} 1)^{2008}}$
such that for each sequence element we have the inequality $ a_n \leq Q.$
2005 Junior Balkan Team Selection Tests - Moldova, 4
Let the $A$ be the set of all nonenagative integers.
It is given function such that $f:\mathbb{A}\rightarrow\mathbb{A}$ with $f(1) = 1$ and for every element $n$ od set $A$ following holds:
[b]1)[/b] $3 f(n) \cdot f(2n+1) = f(2n) \cdot (1+3 \cdot f(n))$;
[b]2)[/b] $f(2n) < 6f(n)$,
Find all solutions of $f(k)+f(l) = 293$, $k<l$.
1991 Hungary-Israel Binational, 3
Let $ \mathcal{H}_n$ be the set of all numbers of the form $ 2 \pm\sqrt{2 \pm\sqrt{2 \pm\ldots\pm\sqrt 2}}$ where "root signs" appear $ n$ times.
(a) Prove that all the elements of $ \mathcal{H}_n$ are real.
(b) Computer the product of the elements of $ \mathcal{H}_n$.
(c) The elements of $ \mathcal{H}_{11}$ are arranged in a row, and are sorted by size in an ascending order. Find the position in that row, of the elements of $ \mathcal{H}_{11}$ that corresponds to the following combination of $ \pm$ signs: \[ \plus{}\plus{}\plus{}\plus{}\plus{}\minus{}\plus{}\plus{}\minus{}\plus{}\minus{}\]
1989 IMO Longlists, 98
Let $ A$ be an $ n \times n$ matrix whose elements are non-negative real numbers. Assume that $ A$ is a non-singular matrix and all elements of $ A^{\minus{}1}$ are non-negative real numbers. Prove that every row and every column of $ A$ has exactly one non-zero element.
2004 Kurschak Competition, 2
Find the smallest positive integer $n\neq 2004$ for which there exists a polynomial $f\in\mathbb{Z}[x]$ such that the equation $f(x)=2004$ has at least one, and the equation $f(x)=n$ has at least $2004$ different integer solutions.
2000 Italy TST, 3
Given positive numbers $a_1$ and $b_1$, consider the sequences defined by
\[a_{n+1}=a_n+\frac{1}{b_n},\quad b_{n+1}=b_n+\frac{1}{a_n}\quad (n \ge 1)\]
Prove that $a_{25}+b_{25} \geq 10\sqrt{2}$.
2002 Czech-Polish-Slovak Match, 6
Let $n \ge 2$ be a fixed even integer. We consider polynomials of the form
\[P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + 1\]
with real coefficients, having at least one real roots. Find the least possible value of $a^2_1 + a^2_2 + \cdots + a^2_{n-1}$.
2006 All-Russian Olympiad, 5
Two sequences of positive reals, $ \left(x_n\right)$ and $ \left(y_n\right)$, satisfy the relations $ x_{n \plus{} 2} \equal{} x_n \plus{} x_{n \plus{} 1}^2$ and $ y_{n \plus{} 2} \equal{} y_n^2 \plus{} y_{n \plus{} 1}$ for all natural numbers $ n$. Prove that, if the numbers $ x_1$, $ x_2$, $ y_1$, $ y_2$ are all greater than $ 1$, then there exists a natural number $ k$ such that $ x_k > y_k$.
1990 Dutch Mathematical Olympiad, 3
A polynomial $ f(x)\equal{}ax^4\plus{}bx^3\plus{}cx^2\plus{}dx$ with $ a,b,c,d>0$ is such that $ f(x)$ is an integer for $ x \in \{ \minus{}2,\minus{}1,0,1,2 \}$ and $ f(1)\equal{}1$ and $ f(5)\equal{}70$.
$ (a)$ Show that $ a\equal{}\frac{1}{24}, b\equal{}\frac{1}{4},c\equal{}\frac{11}{24},d\equal{}\frac{1}{4}$.
$ (b)$ Prove that $ f(x)$ is an integer for all $ x \in \mathbb{Z}$.
1983 Bundeswettbewerb Mathematik, 4
Let $f(0), f(1), f(2), \dots$ be a sequence satisfying \[ f(0) = 0 \quad \text{and} \quad f(n) = n - f(f(n-1)) \] for $n=1,2,3,\dots$. Give a formula for $f(n)$ such that its value can be immediately computed using $n$ without having to compute the previous terms.
1986 IMO Longlists, 52
Solve the system of equations
\[\tan x_1 +\cot x_1=3 \tan x_2,\]\[\tan x_2 +\cot x_2=3 \tan x_3,\]\[\vdots\]\[\tan x_n +\cot x_n=3 \tan x_1\]
1993 China Team Selection Test, 2
Let $S = \{(x,y) | x = 1, 2, \ldots, 1993, y = 1, 2, 3, 4\}$. If $T \subset S$ and there aren't any squares in $T.$ Find the maximum possible value of $|T|.$ The squares in T use points in S as vertices.
2000 Vietnam National Olympiad, 3
Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$. Determine the maximum possible number of real roots of $ P(x)$.
2006 Estonia Math Open Senior Contests, 7
A real-valued function $ f$ satisfies for all reals $ x$ and $ y$ the equality
\[ f (xy) \equal{} f (x)y \plus{} x f (y).
\]
Prove that this function satisfies for all reals $ x$ and $ y \ne 0$ the equality
\[ f\left(\frac{x}{y}\right)\equal{}\frac{f (x)y \minus{} x f (y)}{y^2}
\]
2004 Austrian-Polish Competition, 10
For each polynomial $Q(x)$ let $M(Q)$ be the set of non-negative integers $x$ with $0 < Q(x) < 2004.$ We consider polynomials $P_n(x)$ of the form
\[P_n(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_{n-1} \cdot x + 1\]
with coefficients $a_i \in \{ \pm1\}$ for $i = 1, 2, \ldots, n-1.$
For each $n = 3^k, k > 0$ determine:
a.) $m_n$ which represents the maximum of elements in $M(P_n)$ for all such polynomials $P_n(x)$
b.) all polynomials $P_n(x)$ for which $|M(P_n)| = m_n.$
1991 Polish MO Finals, 1
On the Cartesian plane consider the set $V$ of all vectors with integer coordinates. Determine all functions $f : V \rightarrow \mathbb{R}$ satisfying the conditions:
(i) $f(v) = 1$ for each of the four vectors $v \in V$ of unit length.
(ii) $f(v+w) = f(v)+f(w)$ for every two perpendicular vectors $v, w \in V$
(Zero vector is considered to be perpendicular to every vector).
2011 Morocco National Olympiad, 3
Find all functions $f : \mathbb{R} \to \mathbb{R} $ which verify the relation
\[(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.\]
2013 International Zhautykov Olympiad, 1
A quadratic trinomial $p(x)$ with real coefficients is given. Prove that there is a positive integer $n$ such that the equation $p(x) = \frac{1}{n}$ has no rational roots.