Found problems: 4776
2010 Today's Calculation Of Integral, 669
Find the differentiable function defined in $x>0$ such that ${\int_1^{f(x)} f^{-1}(t)dt=\frac 13(x^{\frac {3}{2}}-8}).$
1970 Putnam, A1
Show that the power series for the function
$$e^{ax} \cos bx,$$
where $a,b >0$, has either no zero coefficients or infinitely many zero coefficients.
2007 Grigore Moisil Intercounty, 3
Let be two functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ g $ has infinite limit at $ \infty . $
[b]a)[/b] Prove that if $ g $ continuous then $ \lim_{x\to\infty } f(x) =\lim_{x\to\infty } f(g(x)) $
[b]b)[/b] Provide an example of what $ f,g $ could be if $ f $ has no limit at $ \infty $ and $ \lim_{x\to\infty } f(g(x)) =0. $
The Golden Digits 2024, P1
Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ with the following properties:
1) For every natural number $n\geq 3$, $\gcd(f(n),n)\neq 1$.
2) For every natural number $n\geq 3$, there exists $i_n\in\mathbb{Z}_{>0}$, $1\leq i_n\leq n-1$, such that $f(n)=f(i_n)+f(n-i_n)$.
[i]Proposed by Pavel Ciurea[/i]
2011 IMO, 3
Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies
\[f(x + y) \leq yf(x) + f(f(x))\]
for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x \leq 0$.
[i]Proposed by Igor Voronovich, Belarus[/i]
2017 Romania National Olympiad, 4
Let be two natural numbers $ b>a>0 $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property.
$$ f\left( x^2+ay\right)\ge f\left( x^2+by\right) ,\quad\forall x,y\in\mathbb{R} $$
[b]a)[/b] Show that $ f(s)\le f(0)\le f(t) , $ for any real numbers $ s<0<t. $
[b]b)[/b] Prove that $ f $ is constant on the interval $ (0,\infty ) . $
[b]c)[/b] Give an example of a non-monotone such function.
2018 Mathematical Talent Reward Programme, SAQ: P 3
Does there exist any continuous function $ f$ such that $ f(f(x))=-x^{2019}\ \forall\ x \in \mathbb{R}$
1998 Korea - Final Round, 3
Let $F_n$ be the set of all bijective functions $f:\left\{1,2,\ldots,n\right\}\rightarrow\left\{1,2,\ldots,n\right\}$ that satisfy the conditions:
1. $f(k)\leq k+1$ for all $k\in\left\{1,2,\ldots,n\right\}$
2. $f(k)\neq k$ for all $k\in\left\{2,3,\ldots,n\right\}$
Find the probability that $f(1)\neq1$ for $f$ randomly chosen from $F_n$.
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}$).
1981 IMO, 3
The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.
1987 IMO, 1
Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n))=n+1987$ for all $n$.
1994 IMO, 3
For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k \plus{} 1, k \plus{} 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s.
(a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) \equal{} m$.
(b) Determine all positive integers $ m$ for which there exists [i]exactly one[/i] $ k$ with $ f(k) \equal{} m$.
2006 China Northern MO, 3
$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$.
[i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.
2010 ELMO Shortlist, 1
For a positive integer $n$, let $\mu(n) = 0$ if $n$ is not squarefree and $(-1)^k$ if $n$ is a product of $k$ primes, and let $\sigma(n)$ be the sum of the divisors of $n$. Prove that for all $n$ we have
\[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \]
and determine when equality holds.
[i]Wenyu Cao.[/i]
2011 USA TSTST, 1
Find all real-valued functions $f$ defined on pairs of real numbers, having the following property: for all real numbers $a, b, c$, the median of $f(a,b), f(b,c), f(c,a)$ equals the median of $a, b, c$.
(The [i]median[/i] of three real numbers, not necessarily distinct, is the number that is in the middle when the three numbers are arranged in nondecreasing order.)
2003 Baltic Way, 1
Find all functions $f:\mathbb{Q}^{+}\rightarrow \mathbb{Q}^{+}$ which for all $x \in \mathbb{Q}^{+}$ fulfil
\[f\left(\frac{1}{x}\right)=f(x) \ \ \text{and} \ \ \left(1+\frac{1}{x}\right)f(x)=f(x+1). \]
2007 Nicolae Coculescu, 1
Let be the set $ G=\{ (u,v)\in \mathbb{C}^2| u\neq 0 \} $ and a function $ \varphi :\mathbb{C}\setminus\{ 0\}\longrightarrow\mathbb{C}\setminus\{ 0\} $ having the property that the operation $ *:G^2\longrightarrow G $ defined as
$$ (a,b)*(c,d)=(ac,bc+d\varphi (a)) $$
is associative.
[b]a)[/b] Show that $ (G,*) $ is a group.
[b]b)[/b] Describe $ \varphi , $ knowing that $(G,*) $ is a commutative group.
[i]Marius Perianu[/i]
2014 District Olympiad, 1
For each positive integer $n$ we consider the function $f_{n}:[0,n]\rightarrow{\mathbb{R}}$ defined by $f_{n}(x)=\arctan{\left(\left\lfloor x\right\rfloor \right)} $, where $\left\lfloor x\right\rfloor $ denotes the floor of the real number $x$. Prove that $f_{n}$ is a Riemann Integrable function and find $\underset{n\rightarrow\infty}{\lim}\frac{1}{n}\int_{0}^{n}f_{n}(x)\mathrm{d}x.$
1968 Vietnam National Olympiad, 1
Let $a$ and $b$ satisfy $a \ge b >0, a + b = 1$.
i) Prove that if $m$ and $n$ are positive integers with $m < n$, then $a^m - a^n \ge b^m- b^n > 0$.
ii) For each positive integer $n$, consider a quadratic function $f_n(x) = x^2 - b^nx- a^n$.
Show that $f(x)$ has two roots that are in between $-1$ and $1$.
1972 Miklós Schweitzer, 5
We say that the real-valued function $ f(x)$ defined on the interval $ (0,1)$ is approximately continuous on $ (0,1)$ if for any $ x_0 \in (0,1)$ and $ \varepsilon >0$ the point $ x_0$ is a point of interior density $ 1$ of the set \[ H\equal{} \{x : \;|f(x)\minus{}f(x_0)|< \varepsilon \ \}.\] Let $ F \subset (0,1)$ be a countable closed set, and $ g(x)$ a real-valued function defined on $ F$. Prove the existence of an approximately continuous function $ f(x)$ defined on $ (0,1)$ such that \[ f(x)\equal{}g(x) \;\textrm{for all}\ \;x \in F\ .\]
[i]M. Laczkovich, Gy. Petruska[/i]
2012 Brazil National Olympiad, 1
In a culturing of bacteria, there are two species of them: red and blue bacteria.
When two red bacteria meet, they transform into one blue bacterium.
When two blue bacteria meet, they transform into four red bacteria.
When a red and a blue bacteria meet, they transform into three red bacteria.
Find, in function of the amount of blue bacteria and the red bacteria initially in the culturing,
all possible amounts of bacteria, and for every possible amount, the possible amounts of red and blue bacteria.
2008 Middle European Mathematical Olympiad, 1
Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that
\[ x f(x \plus{} xy) \equal{} x f(x) \plus{} f \left( x^2 \right) f(y) \quad \forall x,y \in \mathbb{R}.\]
2003 National Olympiad First Round, 16
For which of the following values of real number $t$, the equation $x^4-tx+\dfrac 1t = 0$ has no root on the interval $[1,2]$?
$
\textbf{(A)}\ 6
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 8
\qquad\textbf{(D)}\ 9
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2019 PUMaC Algebra A, 4
Let $\mathbb N_0$ be the set of non-negative integers. There is a triple $(f,a,b)$, where $f$ is a function from $\mathbb N_0$ to $\mathbb N_0$ and $a,b\in\mathbb N_0$ that satisfies the following conditions:
[list]
[*]$f(1)=2$
[*]$f(a)+f(b)\leq 2\sqrt{f(a)}$
[*]For all $n>0$, we have $f(n)=f(n-1)f(b)+2n-f(b)$
[/list]
Find the sum of all possible values of $f(b+100)$.
2000 USAMO, 1
Call a real-valued function $ f$ [i]very convex[/i] if
\[ \frac {f(x) \plus{} f(y)}{2} \ge f\left(\frac {x \plus{} y}{2}\right) \plus{} |x \minus{} y|
\]
holds for all real numbers $ x$ and $ y$. Prove that no very convex function exists.