Found problems: 4776
1996 IMO Shortlist, 7
Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1$ and
\[ f \left( x \plus{} \frac{13}{42} \right) \plus{} f(x) \equal{} f \left( x \plus{} \frac{1}{6} \right) \plus{} f \left( x \plus{} \frac{1}{7} \right).\]
Prove that $ f$ is a periodic function (that is, there exists a non-zero real number $ c$ such $ f(x\plus{}c) \equal{} f(x)$ for all $ x \in \mathbb{R}$).
2015 Canada National Olympiad, 1
Let $\mathbb{N} = \{1, 2, 3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^2< f(n)f(f(n)) < n^2+n$ for every positive integer $n$.
2013 IFYM, Sozopol, 4
Let $k<<n$ denote that $k<n$ and $k\mid n$. Let $f:\{1,2,...,2013\}\rightarrow \{1,2,...,M\}$ be such that, if $n\leq 2013$ and $k<<n$, then $f(k)<<f(n)$. What’s the least possible value of $M$?
2003 SNSB Admission, 3
Let be the set $ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} . $ Show that:
$ \text{(1)}\sin\in\Lambda $
$ \text{(2)}\sum_{p\in\mathbb{Z}}\frac{1}{(1+2p)^2} =\frac{\pi^2}{4} $
$ \text{(3)} f\in\Lambda\implies \left| f'(0) \right|\le 1 $
1988 IMO Longlists, 77
A function $ f$ defined on the positive integers (and taking positive integers values) is given by:
$ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\
f(2 \cdot n) \equal{} f(n) \\
f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\
f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$
for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$
2014 Iran Team Selection Test, 3
let $m,n\in \mathbb{N}$ and $p(x),q(x),h(x)$ are polynomials with real Coefficients such that $p(x)$ is Descending.
and for all $x\in \mathbb{R}$
$p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m$ .
prove that dont exist function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x\in \mathbb{R}$
$f(q(p(x))+h(x))=f(x)^{2}+1$
1983 AMC 12/AHSME, 21
Find the smallest positive number from the numbers below
$\text{(A)} \ 10-3\sqrt{11} \qquad \text{(B)} \ 3\sqrt{11}-10 \qquad \text{(C)} \ 18-5\sqrt{13} \qquad \text{(D)} \ 51-10\sqrt{26} \qquad \text{(E)} \ 10\sqrt{26}-51$
1986 IMO Longlists, 19
Let $f : [0, 1] \to [0, 1]$ satisfy $f(0) = 0, f(1) = 1$ and
\[f(x + y) - f(x) = f(x) - f(x - y)\]
for all $x, y \geq 0$ with $x - y, x + y \in [0, 1].$ Prove that $f(x) = x$ for all $x \in [0, 1].$
2014 Iran MO (3rd Round), 2
Find all continuous function $f:\mathbb{R}^{\geq 0}\rightarrow \mathbb{R}^{\geq 0}$ such that :
\[f(xf(y))+f(f(y)) = f(x)f(y)+2 \: \: \forall x,y\in \mathbb{R}^{\geq 0}\]
[i]Proposed by Mohammad Ahmadi[/i]
2011 India IMO Training Camp, 3
Let $\{a_0,a_1,\ldots\}$ and $\{b_0,b_1,\ldots\}$ be two infinite sequences of integers such that
\[(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0\]
for all integers $n\geq 2$. Prove that there exists a positive integer $k$ such that
\[a_{k+2011}=a_{k+2011^{2011}}.\]
2006 Flanders Math Olympiad, 4
Find all functions $f: \mathbb{R}\backslash\{0,1\} \rightarrow \mathbb{R}$ such that
\[ f(x)+f\left(\frac{1}{1-x}\right) = 1+\frac{1}{x(1-x)}. \]
2001 Tuymaada Olympiad, 4
Is it possible to colour all positive real numbers by 10 colours so that every two numbers with decimal representations differing in one place only are of different colours? (We suppose that there is no place in a decimal representations such that all digits starting from that place are 9's.)
[i]Proposed by A. Golovanov[/i]
2005 Italy TST, 1
Suppose that $f:\{1, 2,\ldots ,1600\}\rightarrow\{1, 2,\ldots ,1600\}$ satisfies $f(1)=1$ and
\[f^{2005}(x)=x\quad\text{for}\ x=1,2,\ldots ,1600. \]
$(a)$ Prove that $f$ has a fixed point different from $1$.
$(b)$ Find all $n>1600$ such that any $f:\{1,\ldots ,n\}\rightarrow\{1,\ldots ,n\}$ satisfying the above condition has at least two fixed points.
2003 District Olympiad, 4
Consider the continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R}, g: [0,1]\longrightarrow\mathbb{R} , $ where $
f $ has a finite limit at $ \infty . $ Show that:
$$ \lim_{n \to \infty} \frac{1}{n}\int_0^n f(x) g\left( \frac{x}{n} \right) dx =\int_0^1 g(x)dx\cdot\lim_{x\to\infty} f(x) . $$
1994 Irish Math Olympiad, 5
Let $ f(n)$ be defined for $ n \in \mathbb{N}$ by $ f(1)\equal{}2$ and $ f(n\plus{}1)\equal{}f(n)^2\minus{}f(n)\plus{}1$ for $ n \ge 1$. Prove that for all $ n >1:$
$ 1\minus{}\frac{1}{2^{2^{n\minus{}1}}}<\frac{1}{f(1)}\plus{}\frac{1}{f(2)}\plus{}...\plus{}\frac{1}{f(n)}<1\minus{}\frac{1}{2^{2^n}}$
2013 Today's Calculation Of Integral, 886
Find the functions $f(x),\ g(x)$ such that
$f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$
$g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$
2021 CCA Math Bonanza, I15
Let $N$ be the number of functions $f$ from $\{1,2,\ldots, 8\}$ to $\{1,2,3,\ldots, 255\}$ with the property that:
[list]
[*] $f(k)=1$ for some $k \in \{1,2,3,4,5,6,7,8\}$
[*] If $f(a) =f(b)$, then $a=b$.
[*] For all $n \in \{1,2,3,4,5,6,7,8\}$, if $f(n) \neq 1$, then $f(k)+1>\frac{f(n)}{2} \geq f(k)$ for some $k \in \{1,2,\ldots, 7,8\}$.
[*] For all $k,n \in \{1,2,3,4,5,6,7,8\}$, if $f(n)=2f(k)+1$, then $k<n$.
[/list]
Compute the number of positive integer divisors of $N$.
[i]2021 CCA Math Bonanza Individual Round #15[/i]
2007 Purple Comet Problems, 7
Allowing $x$ to be a real number, what is the largest value that can be obtained by the function $25\sin(4x)-60\cos(4x)?$
2016 Uzbekistan National Olympiad, 4
$a,b,c,x,y,z$ are positive real numbers and $bz+cy=a$, $az+cx=b$, $ay+bx=c$. Find the least value of following function
$f(x,y,z)=\frac{x^2}{1+x}+\frac{y^2}{1+y}+\frac{z^2}{1+z}$
2009 IMO Shortlist, 7
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
2011 National Olympiad First Round, 14
What is the remainder when $2011^{(2011^{(2011^{(2011^{2011})})})}$ is divided by $19$ ?
$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1$
2018 India IMO Training Camp, 2
Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.
1988 Federal Competition For Advanced Students, P2, 3
Show that there is precisely one sequence $ a_1,a_2,...$ of integers which satisfies $ a_1\equal{}1, a_2>1,$ and $ a_{n\plus{}1}^3\plus{}1\equal{}a_n a_{n\plus{}2}$ for $ n \ge 1$.
1982 Putnam, B5
For each $x>e^e$ define a sequence $S_x=u_0,u_1,\ldots$ recursively as follows: $u_0=e$, and for $n\ge0$, $u_{n+1}=\log_{u_n}x$. Prove that $S_x$ converges to a number $g(x)$ and that the function $g$ defined in this way is continuous for $x>e^e$.
1976 Euclid, 6
Source: 1976 Euclid Part A Problem 6
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The $y$-intercept of the graph of the function defined by $y=\frac{4(x+3)(x-2)-24}{(x+4)}$ is
$\textbf{(A) } -24 \qquad \textbf{(B) } -12 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } -4 \qquad \textbf{(E) } -48$