Found problems: 70
1972 Canada National Olympiad, 10
What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?
1991 Flanders Math Olympiad, 4
A word of length $n$ that consists only of the digits $0$ and $1$, is called a bit-string of length $n$. (For example, $000$ and $01101$ are bit-strings of length 3 and 5.) Consider the sequence $s(1), s(2), ...$ of bit-strings of length $n > 1$ which is obtained as follows :
(1) $s(1)$ is the bit-string $00...01$, consisting of $n - 1$ zeros and a $1$ ;
(2) $s(k+1)$ is obtained as follows :
(a) Remove the digit on the left of $s(k)$. This gives a bit-string $t$ of length $n - 1$.
(b) Examine whether the bit-string $t1$ (length $n$, adding a $1$ after $t$) is already in $\{s(1), s(2), ..., s(k)\}$. If this is the not case, then $s(k+1) = t1$. If this is the case then $s(k+1) = t0$.
For example, if $n = 3$ we get :
$s(1) = 001 \rightarrow s(2) = 011 \rightarrow s(3) = 111 \rightarrow s(4) = 110 \rightarrow s(5) = 101$
$\rightarrow s(6) = 010 \rightarrow s(7) = 100 \rightarrow s(8) = 000 \rightarrow s(9) = 001 \rightarrow ...$
Suppose $N = 2^n$.
Prove that the bit-strings $s(1), s(2), ..., s(N)$ of length $n$ are all different.
2014 Lithuania Team Selection Test, 3
Given such positive real numbers $a, b$ and $c$, that the system of equations:
$ \{\begin{matrix}a^2x+b^2y+c^2z=1&&\\xy+yz+zx=1&&\end{matrix} $
has exactly one solution of real numbers $(x, y, z)$. Prove, that there is a triangle, which borders lengths are equal to $a, b$ and $c$.
2007 Bosnia Herzegovina Team Selection Test, 3
Find all $ x\in \mathbb{Z} $ and $ a\in \mathbb{R} $ satisfying
\[\sqrt{x^2-4}+\sqrt{x+2} = \sqrt{x-a}+a \]
1989 IMO Longlists, 38
A sequence of real numbers $ x_0, x_1, x_2, \ldots$ is defined as follows: $ x_0 \equal{} 1989$ and for each $ n \geq 1$
\[ x_n \equal{} \minus{} \frac{1989}{n} \sum^{n\minus{}1}_{k\equal{}0} x_k.\]
Calculate the value of $ \sum^{1989}_{n\equal{}0} 2^n x_n.$
2002 Tournament Of Towns, 2
Does there exist points $A,B$ on the curve $y=x^3$ and on $y=x^3+|x|+1$ respectively such that distance between $A,B$ is less than $\frac{1}{100}$ ?
2003 Bundeswettbewerb Mathematik, 2
The sequence $\{a_1,a_2,\ldots\}$ is recursively defined by $a_1 = 1$, $a_2 = 1$, $a_3 = 2$, and \[ a_{n+3} = \frac 1{a_n}\cdot (a_{n+1}a_{n+2}+7), \ \forall \ n > 0. \] Prove that all elements of the sequence are integers.
2014 Tuymaada Olympiad, 5
For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$. How many such linear functions $\ell(x)$ can exist?
[i](A. Golovanov)[/i]
2004 Baltic Way, 2
Let $ P(x)$ be a polynomial with a non-negative coefficients. Prove that if the inequality $ P\left(\frac {1}{x}\right)P(x)\geq 1$ holds for $ x \equal{} 1$, then this inequality holds for each positive $ x$.
2002 Canada National Olympiad, 5
Let $\mathbb N = \{0,1,2,\ldots\}$. Determine all functions $f: \mathbb N \to \mathbb N$ such that
\[ xf(y) + yf(x) = (x+y) f(x^2+y^2) \]
for all $x$ and $y$ in $\mathbb N$.
2004 Tournament Of Towns, 3
P(x) and Q(x) are polynomials of positive degree such that for all x P(P(x))=Q(Q(x)) and P(P(P(x)))=Q(Q(Q(x))). Does this necessarily mean that P(x)=Q(x)?
1990 IMO Longlists, 66
Find all the continuous bounded functions $f: \mathbb R \to \mathbb R$ such that
\[(f(x))^2 -(f(y))^2 = f(x + y)f(x - y) \text{ for all } x, y \in \mathbb R.\]
2005 MOP Homework, 3
Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that
(a) $f(1)=1$
(b) $f(n+2)+(n^2+4n+3)f(n)=(2n+5)f(n+1)$ for all $n \in \mathbb{N}$.
(c) $f(n)$ divides $f(m)$ if $m>n$.
2004 Baltic Way, 1
Given a sequence $a_1,a_2,\ldots $ of non-negative real numbers satisfying the conditions:
1. $a_n + a_{2n} \geq 3n$;
2. $a_{n+1}+n \leq 2\sqrt{a_n \left(n+1\right)}$
for all $n\in\mathbb N$ (where $\mathbb N=\left\{1,2,3,...\right\}$).
(1) Prove that the inequality $a_n \geq n$ holds for every $n \in \mathbb N$.
(2) Give an example of such a sequence.
2014 BMO TST, 4
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$.
2014 Kazakhstan National Olympiad, 2
$\mathbb{Q}$ is set of all rational numbers. Find all functions $f:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{Q}$ such that for all $x$, $y$, $z$ $\in\mathbb{Q}$ satisfy
$f(x,y)+f(y,z)+f(z,x)=f(0,x+y+z)$
2005 MOP Homework, 4
Deos there exist a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$,
$f(x^2y+f(x+y^2))=x^3+y^3+f(xy)$
2014 Bosnia Herzegovina Team Selection Test, 1
Sequence $a_n$ is defined by $a_1=\frac{1}{2}$, $a_m=\frac{a_{m-1}}{2m \cdot a_{m-1} + 1}$ for $m>1$. Determine value of $a_1+a_2+...+a_k$ in terms of $k$, where $k$ is positive integer.
2006 MOP Homework, 7
Let $S$ denote the set of rational numbers in the interval $(0,1)$. Determine, with proof, if there exists a subset $T$ of $S$ such that every element in $S$ can be uniquely written as the sum of finitely many distinct elements in $T$.
2013 Middle European Mathematical Olympiad, 1
Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that
\[ f( xf(x) + 2y) = f(x^2)+f(y)+x+y-1 \]
holds for all $ x, y \in \mathbb{R}$.
2019 Canadian Mathematical Olympiad Qualification, 1
A function $f$ is called injective if when $f(n) = f(m)$, then $n = m$.
Suppose that $f$ is injective and $\frac{1}{f(n)}+\frac{1}{f(m)}=\frac{4}{f(n) + f(m)}$. Prove $m = n$
2005 MOP Homework, 5
Let $S$ be a finite set of positive integers such that none of them has a prime factor greater than three. Show that the sum of the reciprocals of the elements in $S$ is smaller than three.
2014 Contests, 4
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$.
2010 Middle European Mathematical Olympiad, 1
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x, y\in\mathbb{R}$, we have
\[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]
2004 Tournament Of Towns, 1
Functions f and g are defined on the whole real line and are mutually inverse: g(f(x))=x, f(g(y))=y for all x, y. It is known that f can be written as a sum of periodic and linear functions: f(x)=kx+h(x) for some number k and a periodic function h(x). Show that g can also be written as a sum of periodic and linear functions. (A functions h(x) is called periodic if there exists a non-zero number d such that h(x+d)=h(x) for any x.)