Found problems: 4776
2025 Nordic, 1
Let $n$ be a positive integer greater than $2$. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying:
$(f(x+y))^{n} = f(x^{n})+f(y^{n}),$ for all integers $x,y$
2009 Math Prize For Girls Problems, 2
If $ a$, $ b$, $ c$, $ d$, and $ e$ are constants such that every $ x > 0$ satisfies
\[ \frac{5x^4 \minus{} 8x^3 \plus{} 2x^2 \plus{} 4x \plus{} 7}{(x \plus{} 2)^4}
\equal{} a \plus{} \frac{b}{x \plus{} 2} \plus{} \frac{c}{(x \plus{} 2)^2}
\plus{} \frac{d}{(x \plus{} 2)^3} \plus{} \frac{e}{(x \plus{} 2)^4} \, ,\]
then what is the value of $ a \plus{} b \plus{} c \plus{} d \plus{} e$?
2010 AMC 12/AHSME, 4
If $ x < 0$, then which of the following must be positive?
$ \textbf{(A)}\ \frac{x}{|x|}\qquad \textbf{(B)}\ \minus{}x^2\qquad \textbf{(C)}\ \minus{}2^x\qquad \textbf{(D)}\ \minus{}x^{\minus{}1}\qquad \textbf{(E)}\ \sqrt[3]{x}$
2015 NIMO Summer Contest, 1
For all real numbers $a$ and $b$, let \[a\Join b=\dfrac{a+b}{a-b}.\] Compute $1008\Join 1007$.
[i] Proposed by David Altizio [/i]
2013 Brazil National Olympiad, 3
Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.
2008 ISI B.Stat Entrance Exam, 7
Consider the equation $x^5+x=10$. Show that
(a) the equation has only one real root;
(b) this root lies between $1$ and $2$;
(c) this root must be irrational.
2000 Taiwan National Olympiad, 1
Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.
2013 Romania National Olympiad, 3
Find all injective functions$f:\mathbb{Z}\to \mathbb{Z}$ that satisfy:
$\left| f\left( x \right)-f\left( y \right) \right|\le \left| x-y \right|$ ,for any $x,y\in \mathbb{Z}$.
2014 All-Russian Olympiad, 3
If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x)$, $f(x)g(x)$, $f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3-3x^2+5$ and $x^2-4x$ are written on the blackboard. Can we write a nonzero polynomial of form $x^n-1$ after a finite number of steps?
2003 AIME Problems, 15
Let
\[P(x)=24x^{24}+\sum_{j=1}^{23}(24-j)(x^{24-j}+x^{24+j}). \]
Let $z_{1},z_{2},\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2}=a_{k}+b_{k}i$ for $k=1,2,\ldots,r,$ where $i=\sqrt{-1},$ and $a_{k}$ and $b_{k}$ are real numbers. Let
\[\sum_{k=1}^{r}|b_{k}|=m+n\sqrt{p}, \]
where $m,$ $n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$
2011 Today's Calculation Of Integral, 685
Suppose that a cubic function with respect to $x$, $f(x)=ax^3+bx^2+cx+d$ satisfies all of 3 conditions:
\[f(1)=1,\ f(-1)=-1,\ \int_{-1}^1 (bx^2+cx+d)\ dx=1\].
Find $f(x)$ for which $I=\int_{-1}^{\frac 12} \{f''(x)\}^2\ dx$ is minimized, the find the minimum value.
[i]2011 Tokyo University entrance exam/Humanities, Problem 1[/i]
2005 India IMO Training Camp, 3
For real numbers $a,b,c,d$ not all equal to $0$ , define a real function $f(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}$. Suppose $f(t) = 4a$ for some real $t$. prove that there exist a real number $s$ s.t. $f(s)<0$
2009 Iran MO (3rd Round), 6
Let $z$ be a complex non-zero number such that $Re(z),Im(z)\in \mathbb{Z}$.
Prove that $z$ is uniquely representable as $a_0+a_1(1+i)+a_2(1+i)^2+\dots+a_n(1+i)^n$ where $n\geq 0$ and $a_j \in \{0,1\}$ and $a_n=1$.
Time allowed for this problem was 1 hour.
1991 Irish Math Olympiad, 3
Three operations $f,g$ and $h$ are defined on subsets of the natural numbers $\mathbb{N}$ as follows:
$f(n)=10n$, if $n$ is a positive integer;
$g(n)=10n+4$, if $n$ is a positive integer;
$h(n)=\frac{n}{2}$, if $n$ is an [i]even[/i] positive integer.
Prove that, starting from $4$, every natural number can be constructed by performing a finite number of operations $f$, $g$ and $h$ in some order.
$[$For example: $35=h(f(h(g(h(h(4)))))).]$
2010 Greece Team Selection Test, 4
Find all functions $ f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}$ satisfying $f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))$ for all $x,y\in \mathbb{R^{\ast }}$
and are strictly monotone in $(0,+\infty )$
1990 China Team Selection Test, 3
In set $S$, there is an operation $'' \circ ''$ such that $\forall a,b \in S$, a unique $a \circ b \in S$ exists. And
(i) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ (b \circ c)$.
(ii) $a \circ b \neq b \circ a$ when $a \neq b$.
Prove that:
a.) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ c$.
b.) If $S = \{1,2, \ldots, 1990\}$, try to define an operation $'' \circ ''$ in $S$ with the above properties.
1976 Swedish Mathematical Competition, 5
$f(x)$ is defined for $x \geq 0$ and has a continuous derivative. It satisfies $f(0)=1$, $f'(0)=0$ and $(1+f(x))f''(x)=1+x$. Show that $f$ is increasing and that $f(1) \leq 4/3$.
2012 Centers of Excellency of Suceava, 2
Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify, for any nonzero real number $ x $ the relation
$$ xf(x/a)-f(a/x)=b, $$
where $ a\neq 0,b $ are two real numbers.
[i]Dan Popescu[/i]
2011 Harvard-MIT Mathematics Tournament, 4
For all real numbers $x$, let \[ f(x) = \frac{1}{\sqrt[2011]{1-x^{2011}}}. \] Evaluate $(f(f(\ldots(f(2011))\ldots)))^{2011}$, where $f$ is applied $2010$ times.
1997 National High School Mathematics League, 5
Let $f(x)=x^2-\pi x$, $\alpha=\arcsin\frac{1}{3},\beta=\arctan\frac{5}{4},\gamma=\arccos\left(-\frac{1}{3}\right),\delta=\text{arccot}\left(-\frac{5}{4}\right)$
$\text{(A)}f(\alpha)>f(\beta)>f(\delta)>f(\gamma)$
$\text{(B)}f(\alpha)>f(\delta)>f(\beta)>f(\gamma)$
$\text{(C)}f(\delta)>f(\alpha)>f(\beta)>f(\gamma)$
$\text{(D)}f(\delta)>f(\alpha)>f(\gamma)>f(\beta)$
1989 Brazil National Olympiad, 3
A function $f$, defined for the set of integers, is such that $f(x)=x-10$ if $x>100$ and $f(x)=f(f(x+11))$ if $x \leq 100$.
Determine, justifying your answer, the set of all possible values for $f$.
2005 National High School Mathematics League, 8
$f(x)$ is a decreasing function defined on $(0,+\infty)$, if $f(2a^2+a+1)<f(3a^2-4a+1)$, then the range value of $a$ is________.
2002 National High School Mathematics League, 10
$f(x)$ is a function defined on $\mathbb{R}$. $f(1)=1$, and for all $x\in\mathbb{R}$,
$f(x+5)\geq x+5,f(x+1)\leq f(x)+1$.
If $g(x)=f(x)+1-x$, then $g(2002)=$________.
2020 Brazil Undergrad MO, Problem 6
Let $f(x) = 2x^2 + x - 1, f^{0}(x) = x$, and $f^{n+1}(x) = f(f^{n}(x))$ for all real $x>0$ and $n \ge 0$ integer (that is, $f^{n}$ is $f$ iterated $n$ times).
a) Find the number of distinct real roots of the equation $f^{3}(x) = x$
b) Find, for each $n \ge 0$ integer, the number of distinct real solutions of the equation $f^{n}(x) = 0$
2010 Today's Calculation Of Integral, 547
Find the minimum value of $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$.