Found problems: 4776
2007 Moldova National Olympiad, 11.4
The function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(\textrm{cot}x)=\sin2x+\cos2x$, for any $x\in(0,\pi)$. Find the minimum and maximum value of $g: [-1;1]\rightarrow\mathbb{R}$, $g(x)=f(x)\cdot f(1-x)$.
2005 China National Olympiad, 4
The sequence $\{a_n\}$ is defined by: $a_1=\frac{21}{16}$, and for $n\ge2$,\[ 2a_n-3a_{n-1}=\frac{3}{2^{n+1}}. \]Let $m$ be an integer with $m\ge2$. Prove that: for $n\le m$, we have\[ \left(a_n+\frac{3}{2^{n+3}}\right)^{\frac{1}{m}}\left(m-\left(\frac{2}{3}\right)^{{\frac{n(m-1)}{m}}}\right)<\frac{m^2-1}{m-n+1}. \]
2005 District Olympiad, 2
Let $f:[0,1]\to\mathbb{R}$ be a continuous function and let $\{a_n\}_n$, $\{b_n\}_n$ be sequences of reals such that
\[ \lim_{n\to\infty} \int^1_0 | f(x) - a_nx - b_n | dx = 0 . \]
Prove that:
a) The sequences $\{a_n\}_n$, $\{b_n\}_n$ are convergent;
b) The function $f$ is linear.
2016 Serbia National Math Olympiad, 2
Let $n $ be a positive integer. Let $f $ be a function from nonnegative integers to themselves. Let $f (0,i)=f (i,0)=0$, $f (1, 1)=n $, and $ f(i, j)= [\frac {f(i-1,j)}{2}]+ [\frac {f(i, j-1)}{2}] $ for positive integers $i, j$ such that $i*j>1$. Find the number of pairs $(i,j) $ such that $f (i, j) $ is an odd number.( $[x]$ is the floor function).
Dumbest FE I ever created, 6.
Find all non decreasing functions or non increasing function $f \colon \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$
$$ f(x+f(y))=f(x)+f(y) \text{ or } f(f(f(x)))+y$$ .
Taiwan TST 2015 Round 1, 2
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R} \setminus \{ 0 \}$ such that
\[(f(x))^2f(2y)+(f(y))^2f(2x)=2f(x)f(y)f(x+y)\]
for all $x,y\in\mathbb{Q}$
2016 Iran MO (3rd Round), 2
Find all function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for all $a,b\in\mathbb{N}$ ,
$(f(a)+b) f(a+f(b))=(a+f(b))^2$
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]
2014 Putnam, 1
Prove that every nonzero coefficient of the Taylor series of $(1-x+x^2)e^x$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.
2003 Polish MO Finals, 6
Let $n$ be an even positive integer. Show that there exists a permutation $(x_1, x_2, \ldots, x_n)$ of the set $\{1, 2, \ldots, n\}$, such that for each $i \in \{1, 2, \ldots, n\}, x_{i+1}$ is one of the numbers $2x_i, 2x_{i}-1, 2x_i - n, 2x_i - n - 1$, where $x_{n+1} = x_1.$
2016 Romania National Olympiad, 3
Let be a real number $ a, $ and a nondecreasing function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ Prove that $ f $ is continuous in $ a $ if and only if there exists a sequence $ \left( a_n \right)_{n\ge 1} $ of real positive numbers such that
$$ \int_a^{a+a_n} f(x)dx+\int_a^{a-a_n} f(x)dx\le\frac{a_n}{n} , $$
for all natural numbers $ n. $
[i]Dan Marinescu[/i]
2014 Cezar Ivănescu, 3
Find the real numbers $ \lambda $ that have the property that there is a nonconstant, continuous function $ u: [0,1]\longrightarrow\mathbb{R} $ satisfying
$$ u(x)=\lambda\int_0^1 (x-3y)u(y)dy , $$
for any $ x $ in the interval $ [0,1]. $
1952 Miklós Schweitzer, 10
Let $ n$ be a positive integer. Prove that, for $ 0<x<\frac{\pi}{n\plus{}1}$,
$ \sin{x}\minus{}\frac{\sin{2x}}{2}\plus{}\cdots\plus{}(\minus{}1)^{n\plus{}1}\frac{\sin{nx}}{n}\minus{}\frac{x}{2}$
is positive if $ n$ is odd and negative if $ n$ is even.
1998 AMC 12/AHSME, 17
Let $ f(x)$ be a function with the two properties:
[list=a]
[*] for any two real numbers $ x$ and $ y$, $ f(x \plus{} y) \equal{} x \plus{} f(y)$, and
[*] $ f(0) \equal{} 2$
[/list]
What is the value of $ f(1998)$?
$ \textbf{(A)}\ 0\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ 1996\qquad
\textbf{(D)}\ 1998\qquad
\textbf{(E)}\ 2000$
1983 IMO Longlists, 46
Let $f$ be a real-valued function defined on $I = (0,+\infty)$ and having no zeros on $I$. Suppose that
\[\lim_{x \to +\infty} \frac{f'(x)}{f(x)}=+\infty.\]
For the sequence $u_n = \ln \left| \frac{f(n+1)}{f(n)} \right|$, prove that $u_n \to +\infty$ as $n \to +\infty.$
2003 District Olympiad, 4
Let $\displaystyle a,b,c,d \in \mathbb R$ such that $\displaystyle a>c>d>b>1$ and $\displaystyle ab>cd$.
Prove that $\displaystyle f : \left[ 0,\infty \right) \to \mathbb R$, defined through
\[ \displaystyle f(x) = a^x+b^x-c^x-d^x, \, \forall x \geq 0 , \]
is strictly increasing.
1991 Arnold's Trivium, 14
Calculate with at most $10\%$ relative error
\[\int_{-\infty}^{\infty}(x^4+4x+4)^{-100}dx\]
2025 Romania National Olympiad, 3
Define the functions $g_k \colon \mathbb{Z} \to \mathbb{Z}$, $g_k(x) = x^k$, where $k$ is a positive integer.
Find the set $M_k$ of positive integers $n$ for which there exist injective functions $f_1,f_2, \dots ,f_n \colon \mathbb{Z} \to \mathbb{Z}$ such that $g_k=f_1\cdot f_2 \cdot \ldots \cdot f_n$.
(Here, $\cdot$ denotes component-wise function multiplication)
2025 District Olympiad, P3
Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuous and bijective function, such that $$\lim_{x\rightarrow\infty}\frac{f^{-1}(f(x)/x)}{x}=1.$$
[list=a]
[*] Show that $\lim_{x\rightarrow\infty}\frac{f(x)}{x}=\infty$ and $\lim_{x\rightarrow\infty}\frac{f^{-1}(ax)}{f^{-1}(x)}=1$ for any $a>0$.
[*] Give an example of function which satisfies the hypothesis.
1993 Poland - First Round, 6
The function $f: R \longrightarrow R$ is continuous. Prove that if for every real number $x$, there exists a positive integer $n$, such that
$\underbrace{(f \circ f \circ ... \circ f)}_{n}(x) = 1$,
then $f(1) = 1$.
2016 Romania National Olympiad, 4
Determine all functions $f: \mathbb R \to \mathbb R$ which satisfy the inequality
$$f(a^2) - f(b^2) \leq \left( f(a) + b\right)\left( a - f(b)\right),$$
for all $a,b \in \mathbb R$.
2016 Latvia National Olympiad, 4
Two functions are defined by equations: $f(a) = a^2 + 3a + 2$ and $g(b, c) = b^2 - b + 3c^2 + 3c$. Prove that for any positive integer $a$ there exist positive integers $b$ and $c$ such that $f(a) = g(b, c)$.
1992 All Soviet Union Mathematical Olympiad, 574
Let $$f(x) = a \cos(x + 1) + b \cos(x + 2) + c \cos(x + 3)$$, where $a, b, c$ are real. Given that $f(x)$ has at least two zeros in the interval $(0, \pi)$, find all its real zeros.
2012 India IMO Training Camp, 3
Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a function such that $f(x+y+xy)=f(x)+f(y)+f(xy)$ for all $x, y\in\mathbb{R}$. Prove that $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x, y\in\mathbb{R}$.
1963 Miklós Schweitzer, 9
Let $ f(t)$ be a continuous function on the interval $ 0 \leq t \leq 1$, and define the two sets of points \[ A_t\equal{}\{(t,0): t\in[0,1]\} , B_t\equal{}\{(f(t),1): t\in [0,1]\}.\] Show that the union of all segments $ \overline{A_tB_t}$ is Lebesgue-measurable, and find the minimum of its measure with respect to all functions $ f$. [A. Csaszar]