Found problems: 1311
2001 Romania National Olympiad, 4
Let $n\ge 2$ be an even integer and $a,b$ real numbers such that $b^n=3a+1$. Show that the polynomial $P(X)=(X^2+X+1)^n-X^n-a$ is divisible by $Q(X)=X^3+X^2+X+b$ if and only if $b=1$.
1998 Hong kong National Olympiad, 4
Define a function $f$ on positive real numbers to satisfy
\[f(1)=1 , f(x+1)=xf(x) \textrm{ and } f(x)=10^{g(x)},\]
where $g(x) $ is a function defined on real numbers and for all real numbers $y,z$ and $0\leq t \leq 1$, it satisfies
\[g(ty+(1-t)z) \leq tg(y)+(1-t)g(z).\]
(1) Prove: for any integer $n$ and $0 \leq t \leq 1$, we have
\[t[g(n)-g(n-1)] \leq g(n+t)-g(n) \leq t[g(n+1)-g(n)].\]
(2) Prove that \[\frac{4}{3} \leq f(\frac{1}{2}) \leq \frac{4}{3} \sqrt{2}.\]
1951 Miklós Schweitzer, 7
Let $ f(x)$ be a polynomial with the following properties:
(i) $ f(0)\equal{}0$; (ii) $ \frac{f(a)\minus{}f(b)}{a\minus{}b}$ is an integer for any two different integers $ a$ and $ b$. Is there a polynomial which has these properties, although not all of its coefficients are integers?
2021 German National Olympiad, 3
For a fixed $k$ with $4 \le k \le 9$ consider the set of all positive integers with $k$ decimal digits such that each of the digits from $1$ to $k$ occurs exactly once.
Show that it is possible to partition this set into two disjoint subsets such that the sum of the cubes of the numbers in the first set is equal to the sum of the cubes in the second set.
2020 Baltic Way, 5
Find all real numbers $x,y,z$ so that
\begin{align*}
x^2 y + y^2 z + z^2 &= 0 \\
z^3 + z^2 y + z y^3 + x^2 y &= \frac{1}{4}(x^4 + y^4).
\end{align*}
1971 IMO Longlists, 55
Prove that the polynomial $x^4+\lambda x^3+\mu x^2+\nu x+1$ has no real roots if $\lambda, \mu , \nu $ are real numbers satisfying
\[|\lambda |+|\mu |+|\nu |\le \sqrt{2} \]
2005 ITAMO, 2
Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and
\[a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }\\\\a_n+h \text{ otherwise }.\end{array}\]
For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$?
1998 USAMO, 5
Prove that for each $n\geq 2$, there is a set $S$ of $n$ integers such that $(a-b)^2$ divides $ab$ for every distinct $a,b\in S$.
2023 German National Olympiad, 4
Determine all triples $(a,b,c)$ of real numbers with
\[a+\frac{4}{b}=b+\frac{4}{c}=c+\frac{4}{a}.\]
2008 Tournament Of Towns, 5
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 $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, \cdots$ )
2011 Vietnam National Olympiad, 3
Let $n\in\mathbb N$ and define $P(x,y)=x^n+xy+y^n.$
Show that we cannot obtain two non-constant polynomials $G(x,y)$ and $H(x,y)$ with real coefficients such that
$P(x,y)=G(x,y)\cdot H(x,y).$
2014 Brazil National Olympiad, 4
The infinite sequence $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$ is defined as
\[P_0(x)=x,\quad P_n(x) = P_{n-1}(x-1)\cdot P_{n-1}(x+1),\quad n\ge 1.\]
Find the largest $k$ such that $P_{2014}(x)$ is divisible by $x^k$.
1990 Hungary-Israel Binational, 3
Prove that: \[ \frac{1989}{2}\minus{}\frac{1988}{3}\plus{}\frac{1987}{4}\minus{}\cdots\minus{}\frac{2}{1989}\plus{}\frac{1}{1990}\equal{}\frac{1}{996}\plus{}\frac{3}{997}\plus{}\frac{5}{998}\plus{}\cdots\plus{}\frac{1989}{1990}\]
2015 German National Olympiad, 1
Determine all pairs of real numbers $(x,y)$ satisfying
\begin{align*} x^3+9x^2y&=10,\\
y^3+xy^2 &=2.
\end{align*}
2005 IMAR Test, 3
A flea moves in the positive direction on the real Ox axis, starting from the origin. He can only jump over distances equal with $\sqrt 2$ or $\sqrt{2005}$. Prove that there exists $n_0$ such that the flea can reach any interval $[n,n+1]$ with $n\geq n_0$.
1990 IMO Longlists, 67
Let $a + bi$ and $c + di$ be two roots of the equation $x^n = 1990$, where $n \geq 3$ is an integer and $a,b,c,d \in \mathbb R$. Under the linear transformation $f =\left(\begin{array}{cc}a&c\\b &d\end{array}\right)$, we have $(2, 1) \to (1, 2)$. Denote $r$ to be the distance from the image of $(2, 2)$ to the origin. Find the range of $r.$
1951 Miklós Schweitzer, 11
Prove that, for every pair $ n$, $r$ of positive integers, there can be found a polynomial $ f(x)$ of degree $ n$ with integer coefficients, so that every polynomial $ g(x)$ of degree at most $ n$, for which the coefficients of the polynomial $ f(x)\minus{}g(x)$ are integers with absolute value not greater than $ r$, is irreducible over the field of rational numbers.
2003 Czech-Polish-Slovak Match, 3
Numbers $p,q,r$ lies in the interval $(\frac{2}{5},\frac{5}{2})$ nad satisfy $pqr=1$. Prove that there exist two triangles of the same area, one with the sides $a,b,c$ and the other with the sides $pa,qb,rc$.
2007 Iran MO (3rd Round), 6
Scientist have succeeded to find new numbers between real numbers with strong microscopes. Now real numbers are extended in a new larger system we have an order on it (which if induces normal order on $ \mathbb R$), and also 4 operations addition, multiplication,... and these operation have all properties the same as $ \mathbb R$.
[img]http://i14.tinypic.com/4tk6mnr.png[/img]
a) Prove that in this larger system there is a number which is smaller than each positive integer and is larger than zero.
b) Prove that none of these numbers are root of a polynomial in $ \mathbb R[x]$.
2013 China Team Selection Test, 2
Find the greatest positive integer $m$ with the following property:
For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference.
1998 Irish Math Olympiad, 4
A sequence $ (x_n)$ is given as follows: $ x_0,x_1$ are arbitrary positive real numbers, and $ x_{n\plus{}2}\equal{}\frac{1\plus{}x_{n\plus{}1}}{x_n}$ for $ n \ge 0$. Find $ x_{1998}$.
1998 Slovenia National Olympiad, Problem 2
find all functions $f(x)$ satisfying:
$(\forall x\in R) f(x)+xf(1-x)=x^2+1$
2008 IberoAmerican Olympiad For University Students, 2
Prove that for each natural number $n$ there is a polynomial $f$ with real coefficients and degree $n$ such that $ p(x)=f(x^2-1)$ is divisible by $f(x)$ over the ring $\mathbb{R}[x]$.
1986 Federal Competition For Advanced Students, P2, 3
Find all possible values of $ x_0$ and $ x_1$ such that the sequence defined by:
$ x_{n\plus{}1}\equal{}\frac{x_{n\minus{}1} x_n}{3x_{n\minus{}1}\minus{}2x_n}$ for $ n \ge 1$
contains infinitely many natural numbers.
2005 Baltic Way, 4
Find three different polynomials $P(x)$ with real coefficients such that $P\left(x^2 + 1\right) = P(x)^2 + 1$ for all real $x$.