Found problems: 1311
2009 IberoAmerican Olympiad For University Students, 6
Let $\alpha_1,\ldots,\alpha_d,\beta_1,\ldots,\beta_e\in\mathbb{C}$ be such that the polynomials
$f_1(x) =\prod_{i=1}^d(x-\alpha_i)$ and $f_2(x) =\prod_{i=1}^e(x-\beta_i)$
have integer coefficients.
Suppose that there exist polynomials $g_1, g_2 \in\mathbb{Z}[x]$ such that $f_1g_1 +f_2g_2 = 1$.
Prove that $\left|\prod_{i=1}^d \prod_{j=1}^e (\alpha_i - \beta_j) \right|=1$
2009 Korea National Olympiad, 4
For a positive integer $n$, define a function $ f_n (x) $ at an interval $ [ 0, n+1 ] $ as
\[ f_n (x) = ( \sum_{i=1} ^ {n} | x-i | )^2 - \sum_{i=1} ^{n} (x-i)^2 . \]
Let $ a_n $ be the minimum value of $f_n (x) $. Find the value of
\[ \sum_{n=1}^{11} (-1)^{n+1} a_n . \]
1999 Turkey Team Selection Test, 3
Determine all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the set
\[\left \{ \frac{f(x)}{x}: x \neq 0 \textnormal{ and } x \in \mathbb{R}\right \}\]
is finite, and for all $x \in \mathbb{R}$
\[f(x-1-f(x)) = f(x) - x - 1\]
1987 IMO Longlists, 65
The [i]runs[/i] of a decimal number are its increasing or decreasing blocks of digits. Thus $024379$ has three [i]runs[/i] : $024, 43$, and $379$. Determine the average number of runs for a decimal number in the set $\{d_1d_2 \cdots d_n | d_k \neq d_{k+1}, k = 1, 2,\cdots, n - 1\}$, where $n \geq 2.$
2008 Serbia National Math Olympiad, 5
The sequence $ (a_n)_{n\ge 1}$ is defined by $ a_1 \equal{} 3$, $ a_2 \equal{} 11$ and $ a_n \equal{} 4a_{n\minus{}1}\minus{}a_{n\minus{}2}$, for $ n \ge 3$. Prove that each term of this sequence is of the form $ a^2 \plus{} 2b^2$ for some natural numbers $ a$ and $ b$.
2006 Singapore MO Open, 2
Show that any representation of 1 as the sum of distinct reciprocals of numbers drawn from the arithmetic progression $\{2,5,8,11,...\}$ such as given in the following example must have at least eight terms: \[1=\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{20}+\frac{1}{41}+\frac{1}{110}+\frac{1}{1640}\]
2008 Tournament Of Towns, 7
In an in finite sequence $a_1, a_2, a_3, \cdots$, the number $a_1$ equals $1$, and each $a_n, n > 1$, is obtained from $a_{n-1}$ as follows:
[list]- if the greatest odd divisor of $n$ has residue $1$ modulo $4$, then $a_n = a_{n-1} + 1,$
- and if this residue equals $3$, then $a_n = a_{n-1} - 1.$[/list]
Prove that in this sequence
[b](a) [/b] the number $1$ occurs infi nitely many times;
[b](b)[/b] each positive integer occurs infi nitely many times.
(The initial terms of this sequence are $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, \cdots$ )
2000 Federal Competition For Advanced Students, Part 2, 1
The sequence an is defined by $a_0 = 4, a_1 = 1$ and the recurrence formula $a_{n+1} = a_n + 6a_{n-1}$. The sequence $b_n$ is given by
\[b_n=\sum_{k=0}^n \binom nk a_k.\]
Find the coefficients $\alpha,\beta$ so that $b_n$ satisfies the recurrence formula $b_{n+1} = \alpha b_n + \beta b_{n-1}$. Find the explicit form of $b_n$.
2008 Moldova National Olympiad, 12.1
Consider the equation $ x^4 \minus{} 4x^3 \plus{} 4x^2 \plus{} ax \plus{} b \equal{} 0$, where $ a,b\in\mathbb{R}$. Determine the largest value $ a \plus{} b$ can take, so that the given equation has two distinct positive roots $ x_1,x_2$ so that $ x_1 \plus{} x_2 \equal{} 2x_1x_2$.
2021 JBMO TST - Turkey, 7
Initially on a blackboard, the equation $a_1x^2+b_1x+c=0$ is written where $a_1, b_1, c_1$ are integers and $(a_1+c_1)b_1 > 0$. At each move, if the equation $ax^2+bx+c=0$ is written on the board and there is a $x \in \mathbb{R}$ satisfying the equation, Alice turns this equation into $(b+c)x^2+(c+a)x+(a+b)=0$. Prove that Alice will stop after a finite number of moves.
2002 Tuymaada Olympiad, 2
Find all the functions $f(x),$ continuous on the whole real axis, such that for every real $x$ \[f(3x-2)\leq f(x)\leq f(2x-1).\]
[i]Proposed by A. Golovanov[/i]
1994 Baltic Way, 5
Let $p(x)$ be a polynomial with integer coefficients such that both equations $p(x)=1$ and $p(x)=3$ have integer solutions. Can the equation $p(x)=2$ have two different integer solutions?
1974 IMO Longlists, 25
Let $f : \mathbb R \to \mathbb R$ be of the form $f(x) = x + \epsilon \sin x,$ where $0 < |\epsilon| \leq 1.$ Define for any $x \in \mathbb R,$
\[x_n=\underbrace{f \ o \ \ldots \ o \ f}_{n \text{ times}} (x).\]
Show that for every $x \in \mathbb R$ there exists an integer $k$ such that $\lim_{n\to \infty } x_n = k\pi.$
1974 IMO Longlists, 21
Let $M$ be a nonempty subset of $\mathbb Z^+$ such that for every element $x$ in $M,$ the numbers $4x$ and $\lfloor \sqrt x \rfloor$ also belong to $M.$ Prove that $M = \mathbb Z^+.$
2011 Romania Team Selection Test, 1
Determine all real-valued functions $f$ on the set of real numbers satisfying
\[2f(x)=f(x+y)+f(x+2y)\]
for all real numbers $x$ and all non-negative real numbers $y$.
2002 India IMO Training Camp, 3
Let $X=\{2^m3^n|0 \le m, \ n \le 9 \}$. How many quadratics are there of the form $ax^2+2bx+c$, with equal roots, and such that $a,b,c$ are distinct elements of $X$?
1999 Romania Team Selection Test, 8
Let $a$ be a positive real number and $\{x_n\}_{n\geq 1}$ a sequence of real numbers such that $x_1=a$ and
\[ x_{n+1} \geq (n+2)x_n - \sum^{n-1}_{k=1}kx_k, \ \forall \ n\geq 1. \]
Prove that there exists a positive integer $n$ such that $x_n > 1999!$.
[i]Ciprian Manolescu[/i]
2007 Moldova National Olympiad, 12.6
Show that the distance between a point on the hyperbola $xy=5$ and a point on the ellipse $x^{2}+6y^{2}=6$ is at least $\frac{9}{7}$.
2012 Turkey MO (2nd round), 3
Find all non-decreasing functions from real numbers to itself such that for all real numbers $x,y$ $f(f(x^2)+y+f(y))=x^2+2f(y)$ holds.
2006 IberoAmerican Olympiad For University Students, 2
Prove that for any positive integer $n$ and any real numbers $a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n$ we have that the equation
\[a_1 \sin(x) + a_2 \sin(2x) +\cdots+a_n\sin(nx)=b_1 \cos(x)+b_2\cos(2x)+\cdots +b_n \cos(nx)\]
has at least one real root.
1985 Federal Competition For Advanced Students, P2, 6
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying: $ x^2 f(x)\plus{}f(1\minus{}x)\equal{}2x\minus{}x^4$ for all $ x \in \mathbb{R}$.
1980 IMO, 2
Let $p: \mathbb C \to \mathbb C$ be a polynomial with degree $n$ and complex coefficients which satisfies
\[x \in \mathbb R \iff p(x) \in \mathbb R.\]
Show that $n=1$
2001 District Olympiad, 1
Let $(a_n)_{n\ge 1}$ be a sequence of real numbers such that
\[a_1\binom{n}{1}+a_2\binom{n}{2}+\ldots+a_n\binom{n}{n}=2^{n-1}a_n,\ (\forall)n\in \mathbb{N}^*\]
Prove that $(a_n)_{n\ge 1}$ is an arithmetical progression.
[i]Lucian Dragomir[/i]
1995 All-Russian Olympiad, 3
Can the equation $f(g(h(x))) = 0$, where $f$, $g$, $h$ are quadratic polynomials, have the solutions $1, 2, 3, 4, 5, 6, 7, 8$?
[i]S. Tokarev[/i]
2006 Pre-Preparation Course Examination, 2
a) Show that you can divide an angle $\theta$ to three equal parts using compass and ruler if and only if the polynomial $4t^3-3t-\cos (\theta)$ is reducible over $\mathbb{Q}(\cos (\theta))$.
b) Is it always possible to divide an angle into five equal parts?