Found problems: 15925
2021 Final Mathematical Cup, 3
For a positive integer $n$ we define $f (n) = \max X_1^{X_2^{...^{X_k}}}$ where the maximum is taken over all possible decompositions of natural numbers $n = X_1X_2...X_k$. Determine $f(n)$.
I Soros Olympiad 1994-95 (Rus + Ukr), 10.4
It is known that for some $a$ and $b$ the equation $$\frac{x-3}{(x-6)^2} -\frac{x-6}{(x-3)^2} =a(b-9x+x^2)$$ has as its largest root the number $1995$. Find the smallest root of this equation for the same $a$ and $b$.
2022 Saint Petersburg Mathematical Olympiad, 4
We will say that a point of the plane $(u, v)$ lies between the parabolas $y = f(x)$ and $y = g(x)$ if $f(u) \leq v \leq g(u)$. Find the smallest real $p$ for which the following statement is true: for any segment, the ends and the midpoint of which
lie between the parabolas $y = x^2$ and $y=x^2+1$, then they lie entirely between the parabolas $y=x^2$ and $y=x^2+p$.
1991 Bundeswettbewerb Mathematik, 4
Given wo non-negative integers $a$ and $b$, one of them is odd and the other one even. By the following rule we define two sequences $(a_n),(b_n)$:
\[ a_0 = a, \quad a_1 = b, \quad a_{n+1} = 2a_n - a_{n-1} + 2 \quad (n = 1,2,3, \ldots)\]
\[ b_0 = b, \quad b_1 = a, \quad b_{n+1} = 2a_n - b_{n-1} + 2 \quad (n = 1,2,3, \ldots)\]
Prove that none of these two sequences contain a negative element if and only if we have $|\sqrt{a} - \sqrt{b}| \leq 1$.
2010 Indonesia TST, 1
Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.
1995 All-Russian Olympiad Regional Round, 11.6
The sequence $ a_n$ satisfies $ a_{m\plus{}n}\plus{} a_{m\minus{}n}\equal{}\frac12(a_{2m}\plus{}a_{2n})$ for all $ m\geq n\geq 0$. If $ a_1\equal{}1$, find $ a_{1995}$.
2018 Austria Beginners' Competition, 1
Let $a, b$ and $c$ denote positive real numbers. Prove that $\frac{a}{c}+\frac{c}{b}\ge \frac{4a}{a + b}$ .
When does equality hold?
(Walther Janous)
1987 Dutch Mathematical Olympiad, 2
For $x >0$ , prove that $$\frac{1}{2\sqrt{x+1}}<\sqrt{x+1}-\sqrt{x}<\frac{1}{2\sqrt{x}}$$
and for all $n \ge 2$ prove that $$1 <2\sqrt{n} - \sum_{k=1}^n\frac{1}{\sqrt{k}}<2$$
2017 EGMO, 2
Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties:
$(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$
$(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$
[i]In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.[/i]
1975 Spain Mathematical Olympiad, 1
Calculate the limit
$$\lim_{n \to \infty} \frac{1}{n} \left(\frac{1}{n^k} +\frac{2^k}{n^k} +....+\frac{(n-1)^k}{n^k} +\frac{n^k}{n^k}\right).$$
(For the calculation of the limit, the integral construction procedure can be followed).
2020 LIMIT Category 1, 17
The sum of $k$ consecutive integers is $90$. Then the sum of all possible values of $k$ is?
(A)$89$
(B)$179$
(C)$168$
(D)$119$
2010 Purple Comet Problems, 25
Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$. There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$. Find $m+n$.
1996 Iran MO (3rd Round), 3
Find all sets of real numbers $\{a_1,a_2,\ldots, _{1375}\}$ such that
\[2 \left( \sqrt{a_n - (n-1)}\right) \geq a_{n+1} - (n-1), \quad \forall n \in \{1,2,\ldots,1374\},\]
and
\[2 \left( \sqrt{a_{1375}-1374} \right) \geq a_1 +1.\]
2013 Peru MO (ONEM), 1
We define the polynomial $$P (x) = 2014x^{2013} + 2013x^{2012} +... + 4x^3 + 3x^2 + 2x.$$ Find the largest prime divisor of $P (2)$.
2003 Junior Tuymaada Olympiad, 4
The natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_n $ satisfy the condition $ 1 / a_1 + 1 / a_2 + \ldots + 1 / a_n = 1 $. Prove that all these numbers do not exceed $$ n ^ {2 ^ n} $$
2005 France Team Selection Test, 6
Let $P$ be a polynom of degree $n \geq 5$ with integer coefficients given by $P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_0 \quad$ with $a_i \in \mathbb{Z}$, $a_n \neq 0$.
Suppose that $P$ has $n$ different integer roots (elements of $\mathbb{Z}$) : $0,\alpha_2,\ldots,\alpha_n$. Find all integers $k \in \mathbb{Z}$ such that $P(P(k))=0$.
1985 Vietnam Team Selection Test, 2
Find all real values of a for which the equation $ (a \minus{} 3x^2 \plus{} \cos \frac {9\pi x}{2})\sqrt {3 \minus{} ax} \equal{} 0$ has an odd number of solutions in the interval $ [ \minus{} 1,5]$
1996 Austrian-Polish Competition, 8
Show that there is no polynomial $P(x)$ of degree $998$ with real coefficients which satisfies $P(x^2 + 1) = P(x)^2 - 1$ for all $x$.
2010 AMC 12/AHSME, 6
A [i]palindrome[/i], such as $ 83438$, is a number that remains the same when its digits are reversed. The numbers $ x$ and $ x \plus{} 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x?
$ \textbf{(A)}\ 20\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 23\qquad \textbf{(E)}\ 24$
1990 IMO Longlists, 86
Given function $f(x) = \sin x + \sin \pi x$ and positive number $d$. Prove that there exists real number $p$ such that $|f(x + p) - f(x)| < d$ holds for all real numbers $x$, and the value of $p$ can be arbitrarily large.
2013 Baltic Way, 19
Let $a_0$ be a positive integer and $a_n=5a_{n-1}+4$ for all $n\ge 1$. Can $a_0$ be chosen so that $a_{54}$ is a multiple of $2013$?
2008 Bulgaria Team Selection Test, 3
Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.
1995 Balkan MO, 1
For all real numbers $x,y$ define $x\star y = \frac{ x+y}{ 1+xy}$. Evaluate the expression \[ ( \cdots (((2 \star 3) \star 4) \star 5) \star \cdots ) \star 1995. \]
[i]Macedonia[/i]
2002 IMO Shortlist, 6
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer.
[i]Laurentiu Panaitopol, Romania[/i]
2013 Baltic Way, 5
Numbers $0$ and $2013$ are written at two opposite vertices of a cube. Some real numbers are to be written at the remaining $6$ vertices of the cube. On each edge of the cube the difference between the numbers at its endpoints is written. When is the sum of squares of the numbers written on the edges minimal?