Found problems: 1269
1994 Irish Math Olympiad, 1
A sequence $ (x_n)$ is given by $ x_1\equal{}2$ and $ nx_n\equal{}2(2n\minus{}1)x_{n\minus{}1}$ for $ n>1$. Prove that $ x_n$ is an integer for every $ n \in \mathbb{N}$.
2003 All-Russian Olympiad, 3
Let $f(x)$ and $g(x)$ be polynomials with non-negative integer coefficients, and let m be the largest coefficient of $f.$ Suppose that there exist natural numbers $a < b$ such that $f(a) = g(a)$ and $f(b) = g(b)$. Show that if $b > m,$ then $f = g.$
1970 IMO Longlists, 6
There is an equation $\sum_{i=1}^{n}{\frac{b_i}{x-a_i}}=c$ in $x$, where all $b_i >0$ and $\{a_i\}$ is a strictly increasing sequence. Prove that it has $n-1$ roots such that $x_{n-1}\le a_n$, and $a_i \le x_i$ for each $i\in\mathbb{N}, 1\le i\le n-1$.
1985 IMO Longlists, 65
Define the functions $f, F : \mathbb N \to \mathbb N$, by
\[f(n)=\left[ \frac{3-\sqrt 5}{2} n \right] , F(k) =\min \{n \in \mathbb N|f^k(n) > 0 \},\]
where $f^k = f \circ \cdots \circ f$ is $f$ iterated $n$ times. Prove that $F(k + 2) = 3F(k + 1) - F(k)$ for all $k \in \mathbb N.$
1985 IMO Longlists, 88
Determine the range of $w(w + x)(w + y)(w + z)$, where $x, y, z$, and $w$ are real numbers such that
\[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\]
2002 Indonesia MO, 3
Find all solutions (real and complex) for $x,y,z$, given that:
\[ x+y+z = 6 \\
x^2+y^2+z^2 = 12 \\
x^3+y^3+z^3 = 24 \]
1993 USAMO, 1
For each integer $\, n \geq 2, \,$ determine, with proof, which of the two positive real numbers $\, a \,$ and $\, b \,$ satisfying \[ a^n = a + 1, \hspace{.3in} b^{2n} = b + 3a \] is larger.
2012 Postal Coaching, 3
Given an integer $n\ge 2$, prove that
\[\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor\].
[hide="Edit"] Thanks to shivangjindal for pointing out the mistake (and sorry for the late edit)[/hide]
2001 IberoAmerican, 3
Show that it is impossible to cover a unit square with five equal squares with side $s<\frac{1}{2}$.
1989 IMO Longlists, 2
An accurate 12-hour analog clock has an hour hand, a minute hand, and a second hand that are aligned at 12:00 o’clock and make one revolution in 12 hours, 1 hour, and 1 minute, respectively. It is well known, and not difficult to prove, that there is no time when the three hands are equally spaced around the clock, with each separating angle $ \frac{2 \cdot \pi}{3}.$ Let $ f(t), g(t), h(t)$ be the respective absolute deviations of the separating angles from \frac{2 \cdot \pi}{3} at $ t$ hours after 12:00 o’clock. What is the minimum value of $ max\{f(t), g(t), h(t)\}?$
1989 India National Olympiad, 2
Let $ a,b,c$ and $ d$ be any four real numbers, not all equal to zero. Prove that the roots of the polynomial $ f(x) \equal{} x^{6} \plus{} ax^{3} \plus{} bx^{2} \plus{} cx \plus{} d$ can't all be real.
1986 Iran MO (2nd round), 1
Let $f$ be a function such that
\[f(x)=\frac{(x^2-2x+1) \sin \frac{1}{x-1}}{\sin \pi x}.\]
Find the limit of $f$ in the point $x_0=1.$
2003 Irish Math Olympiad, 5
show that thee is no function f definedonthe positive real numbes such that :
$f(y) > (y-x)f(x)^2$
2004 Korea National Olympiad, 1
For arbitrary real number $x$, the function $f : \mathbb R \to \mathbb R$ satisfies $f(f(x))-x^2+x+3=0$. Show that the function $f$ does not exist.
2000 Polish MO Finals, 3
Show that the only polynomial of odd degree satisfying $p(x^2-1) = p(x)^2 - 1$ for all $x$ is $p(x) = x$
2004 Serbia Team Selection Test, 3
Let $P(x)$ be a polynomial of degree $n$ whose roots are $i-1, i-2,\cdot\cdot\cdot, i-n$ (where $i^2=-1$), and let $R(x)$ and $S(x)$ be the polynomials with real coefficients such that $P(x)=R(x)+iS(x)$. Show that the polynomial $R$ has $n$ real roots. (R. Stanojevic)
1998 Austrian-Polish Competition, 3
Find all pairs of real numbers $(x, y)$ satisfying the following system of
equations
$2-x^{3}=y, 2-y^{3}=x$.
1989 IMO Longlists, 5
The sequences $ a_0, a_1, \ldots$ and $ b_0, b_1, \ldots$ are defined for $ n \equal{} 0, 1, 2, \ldots$ by the equalities
\[ a_0 \equal{} \frac {\sqrt {2}}{2}, \quad a_{n \plus{} 1} \equal{} \frac {\sqrt {2}}{2} \cdot \sqrt {1 \minus{} \sqrt {1 \minus{} a^2_n}}
\]
and
\[ b_0 \equal{} 1, \quad b_{n \plus{} 1} \equal{} \frac {\sqrt {1 \plus{} b^2_n} \minus{} 1}{b_n}
\]
Prove the inequalities for every $ n \equal{} 0, 1, 2, \ldots$
\[ 2^{n \plus{} 2} a_n < \pi < 2^{n \plus{} 2} b_n.
\]
2008 USA Team Selection Test, 8
Mr. Fat and Ms. Taf play a game. Mr. Fat chooses a sequence of positive integers $ k_1, k_2, \ldots , k_n$. Ms. Taf must guess this sequence of integers. She is allowed to give Mr. Fat a red card and a blue card, each with an integer written on it. Mr. Fat replaces the number on the red card with $ k_1$ times the number on the red card plus the number on the blue card, and replaces the number on the blue card with the number originally on the red card. He repeats this process with number $ k_2$. (That is, he replaces the number on the red card with $ k_2$ times the number now on the red card plus the number now on the blue card, and replaces the number on the blue card with the number that was just placed on the red card.) He then repeats this process with each of the numbers $ k_3, \ldots k_n$, in this order. After has has gone through the sequence of integers, Mr. Fat then gives the cards back to Ms. Taf. How many times must Ms. Taf submit the red and blue cards in order to be able to determine the sequence of integers $ k_1, k_2, \ldots k_n$?
1999 Vietnam Team Selection Test, 1
Let a sequence of positive reals $\{u_n\}^{\infty}_{n=1}$ be given. For every positive integer $n$, let $k_n$ be the least positive integer satisfying:
\[\sum^{k_n}_{i=1} \frac{1}{i} \geq \sum^n_{i=1} u_i.\]
Show that the sequence $\left\{\frac{k_{n+1}}{k_n}\right\}$ has finite limit if and only if $\{u_n\}$ does.
2011 Croatia Team Selection Test, 1
We define a sequence $a_n$ so that $a_0=1$ and
\[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + d & \textrm{ otherwise. } \end{cases} \]
for all postive integers $n$.
Find all positive integers $d$ such that there is some positive integer $i$ for which $a_i=1$.
1987 USAMO, 3
Construct a set $S$ of polynomials inductively by the rules:
(i) $x\in S$;
(ii) if $f(x)\in S$, then $xf(x)\in S$ and $x+(1-x)f(x)\in S$.
Prove that there are no two distinct polynomials in $S$ whose graphs intersect within the region $\{0 < x < 1\}$.
2006 Bulgaria Team Selection Test, 2
Find all couples of polynomials $(P,Q)$ with real coefficients, such that for infinitely many $x\in\mathbb R$ the condition \[ \frac{P(x)}{Q(x)}-\frac{P(x+1)}{Q(x+1)}=\frac{1}{x(x+2)}\]
Holds.
[i] Nikolai Nikolov, Oleg Mushkarov[/i]
2002 Finnish National High School Mathematics Competition, 1
A function $f$ satisfies $f(\cos x) = \cos (17x)$ for every real $x$. Show that $f(\sin x) =\sin (17x)$ for every $x \in \mathbb{R}.$
2006 MOP Homework, 4
Determine if there exists a strictly increasing sequence of positive integers $a_1$, $a_2$, ... such that $a_n \le n^3$ for every positive integer $n$ and that every positive integer can be written uniquely as the difference of two terms in the sequence.