Found problems: 1311
2001 Irish Math Olympiad, 4
Find all nonnegative real numbers $ x$ for which $ \sqrt[3]{13\plus{}\sqrt{x}}\plus{}\sqrt[3]{13\minus{}\sqrt{x}}$ is an integer.
2008 Mongolia Team Selection Test, 3
Given positive integers $ m,n > 1$. Prove that the equation
$ (x \plus{} 1)^n \plus{} (x \plus{} 2)^n \plus{} ... \plus{} (x \plus{} m)^n \equal{} (y \plus{} 1)^{2n} \plus{} (y \plus{} 2)^{2n} \plus{} ... \plus{} (y \plus{} m)^{2n}$ has finitely number of solutions $ x,y \in N$
2012 Waseda University Entrance Examination, 1
Answer the following questions:
(1) For complex numbers $\alpha ,\ \beta$, if $\alpha \beta =0$, then prove that $\alpha =0$ or $\beta =0$.
(2) For complex number $\alpha$, if $\alpha^2$ is a positive real number, then prove that $\alpha$ is a real number.
(3) For complex numbers $\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_{2n+1}\ (n=1,\ 2,\ \cdots)$, assume that $\alpha_1\alpha_2,\ \cdots ,\ \alpha_k\alpha_{k+1},\ \cdots,\ \alpha_{2n}\alpha_{2n+1}$ and $\alpha_{2n+1}\alpha_1$ are all positive real numbers. Prove that $\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_{2n+1}$ are all real numbers.
2012 ELMO Shortlist, 7
Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant.
[i]Victor Wang.[/i]
2012 Romania Team Selection Test, 5
Let $p$ and $q$ be two given positive integers. A set of $p+q$ real numbers $a_1<a_2<\cdots <a_{p+q}$ is said to be balanced iff $a_1,\ldots,a_p$ were an arithmetic progression with common difference $q$ and $a_p,\ldots,a_{p+q}$ where an arithmetic progression with common difference $p$. Find the maximum possible number of balanced sets, so that any two of them have nonempty intersection.
Comment: The intended problem also had "$p$ and $q$ are coprime" in the hypothesis. A typo when the problems where written made it appear like that in the exam (as if it were the only typo in the olympiad). Fortunately, the problem can be solved even if we didn't suppose that and it can be further generalized: we may suppose that a balanced set has $m+n$ reals $a_1<\cdots <a_{m+n-1}$ so that $a_1,\ldots,a_m$ is an arithmetic progression with common difference $p$ and $a_m,\ldots,a_{m+n-1}$ is an arithmetic progression with common difference $q$.
1986 India National Olympiad, 7
If $ a$, $ b$, $ x$, $ y$ are integers greater than 1 such that $ a$ and $ b$ have no common factor except 1 and $ x^a \equal{} y^b$ show that $ x \equal{} n^b$, $ y \equal{} n^a$ for some integer $ n$ greater than 1.
2021 German National Olympiad, 1
Determine all real numbers $a,b,c$ and $d$ with the following property: The numbers $a$ and $b$ are distinct roots of $2x^2-3cx+8d$ and the numbers $c$ and $d$ are distinct roots of $2x^2-3ax+8b$.
2007 Indonesia TST, 2
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying
\[ f(f(x \plus{} y)) \equal{} f(x \plus{} y) \plus{} f(x)f(y) \minus{} xy\]
for all real numbers $x$ and $y$.
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]
2013 National Olympiad First Round, 35
What is the least positive integer $n$ such that $\overbrace{f(f(\dots f}^{21 \text{ times}}(n)))=2013$ where $f(x)=x+1+\lfloor \sqrt x \rfloor$? ($\lfloor a \rfloor$ denotes the greatest integer not exceeding the real number $a$.)
$
\textbf{(A)}\ 1214
\qquad\textbf{(B)}\ 1202
\qquad\textbf{(C)}\ 1186
\qquad\textbf{(D)}\ 1178
\qquad\textbf{(E)}\ \text{None of above}
$
2005 Greece National Olympiad, 2
The sequence $(a_n)$ is defined by $a_1=1$ and $a_n=a_{n-1}+\frac{1}{n^3}$ for $n>1.$
(a) Prove that $a_n<\frac{5}{4}$ for all $n.$
(b) Given $\epsilon>0$, find the smallest natural number $n_0$ such that ${\mid a_{n+1}-a_n}\mid<\epsilon$ for all $n>n_0.$
2007 Baltic Way, 5
A function $f$ is defined on the set of all real numbers except $0$ and takes all real values except $1$. It is also known that
$\color{white}\ . \ \color{black}\ \quad f(xy)=f(x)f(-y)-f(x)+f(y)$
for any $x,y\not= 0$ and that
$\color{white}\ . \ \color{black}\ \quad f(f(x))=\frac{1}{f(\frac{1}{x})}$
for any $x\not\in\{ 0,1\}$. Determine all such functions $f$.
2001 Brazil National Olympiad, 4
A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.)
1982 IMO Longlists, 34
Let $M$ be the set of all functions $f$ with the following properties:
[b](i)[/b] $f$ is defined for all real numbers and takes only real values.
[b](ii)[/b] For all $x, y \in \mathbb R$ the following equality holds: $f(x)f(y) = f(x + y) + f(x - y).$
[b](iii)[/b] $f(0) \neq 0.$
Determine all functions $f \in M$ such that
[b](a)[/b] $f(1)=\frac 52$,
[b](b)[/b] $f(1)= \sqrt 3$.
2018 Bundeswettbewerb Mathematik, 2
Consider all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying $f(1-f(x))=x$ for all $x \in \mathbb{R}$.
a) By giving a concrete example, show that such a function exists.
b) For each such function define the sum
\[S_f=f(-2017)+f(-2016)+\dots+f(-1)+f(0)+f(1)+\dots+f(2017)+f(2018).\]
Determine all possible values of $S_f$.
1993 Korea - Final Round, 5
Given $n \in\mathbb{N}$, find all continuous functions $f : \mathbb{R}\to \mathbb{R}$ such that for all $x\in\mathbb{R},$
\[\sum_{k=0}^{n}\binom{n}{k}f(x^{2^{k}})=0. \]
2010 Iran Team Selection Test, 3
Find all two-variable polynomials $p(x,y)$ such that for each $a,b,c\in\mathbb R$:
\[p(ab,c^2+1)+p(bc,a^2+1)+p(ca,b^2+1)=0\]
2007 All-Russian Olympiad, 6
Do there exist non-zero reals $a$, $b$, $c$ such that, for any $n>3$, there exists a polynomial $P_{n}(x) = x^{n}+\dots+a x^{2}+bx+c$, which has exactly $n$ (not necessary distinct) integral roots?
[i]N. Agakhanov, I. Bogdanov[/i]
1988 Romania Team Selection Test, 15
Let $[a,b]$ be a given interval of real numbers not containing integers. Prove that there exists $N>0$ such that $[Na,Nb]$ does not contain integer numbers and the length of the interval $[Na,Nb]$ exceedes $\dfrac 16$.
2012 Vietnam National Olympiad, 2
Let $\langle a_n\rangle $ and $ \langle b_n\rangle$ be two arithmetic sequences of numbers, and let $m$ be an integer greater than $2.$ Define $P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m.$ Prove that if the quadratic expressions $P_1(x), P_m(x)$ do not have any real roots, then all the remaining polynomials also don't have real roots.
2007 ISI B.Stat Entrance Exam, 1
Suppose $a$ is a complex number such that
\[a^2+a+\frac{1}{a}+\frac{1}{a^2}+1=0\]
If $m$ is a positive integer, find the value of
\[a^{2m}+a^m+\frac{1}{a^m}+\frac{1}{a^{2m}}\]
2001 Irish Math Olympiad, 5
Determine all functions $ f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfy:
$ f(x\plus{}f(y))\equal{}f(x)\plus{}y$ for all $ x,y \in \mathbb{N}$.
2009 Finnish National High School Mathematics Competition, 2
A polynomial $P$ has integer coefficients and $P(3)=4$ and $P(4)=3$. For how many $x$ we might have $P(x)=x$?
2008 Tournament Of Towns, 6
Let $P(x)$ be a polynomial with real coefficients so that equation $P(m) + P(n) = 0$ has infi nitely many pairs of integer solutions $(m,n)$. Prove that graph of $y = P(x)$ has a center of symmetry.
1950 Miklós Schweitzer, 7
Let $ x$ be an arbitrary real number in $ (0,1)$. For every positive integer $ k$, let $ f_k(x)$ be the number of points $ mx\in [k,k \plus{} 1)$ $ m \equal{} 1,2,...$
Show that the sequence $ \sqrt [n]{f_1(x)f_2(x)\cdots f_n(x)}$ is convergent and find its limit.