Found problems: 4776
2001 Hungary-Israel Binational, 3
Find all continuous functions $f : \mathbb{R}\to\mathbb{R}$ such that for all $x \in\mathbb{ R}$,
\[f (f (x)) = f (x)+x.\]
1978 Austrian-Polish Competition, 1
Determine all functions $f:(0;\infty)\to \mathbb{R}$ that satisfy
$$f(x+y)=f(x^2+y^2)\quad \forall x,y\in (0;\infty)$$
PEN K Problems, 4
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+f(f(n))+f(n)=3n.\]
2013 Today's Calculation Of Integral, 893
Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$
2016 Peru IMO TST, 9
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.
[i]Proposed by James Rickards, Canada[/i]
2003 District Olympiad, 2
Let be two distinct continuous functions $ f,g:[0,1]\longrightarrow (0,\infty ) $ corelated by the equality $ \int_0^1 f(x)dx =\int_0^1 g(x)dx , $ and define the sequence $ \left( x_n \right)_{n\ge 0} $ as
$$ x_n=\int_0^1 \frac{\left( f(x) \right)^{n+1}}{\left( g(x) \right)^n} dx . $$
[b]a)[/b] Show that $ \infty =\lim_{n\to\infty} x_n. $
[b]b)[/b] Demonstrate that the sequence $ \left( x_n \right)_{n\ge 0} $ is monotone.
1970 AMC 12/AHSME, 6
The smallest value of $x^2+8x$ for real values of $x$ is:
$\textbf{(A) }-16.25\qquad\textbf{(B) }-16\qquad\textbf{(C) }-15\qquad\textbf{(D) }-8\qquad \textbf{(E) }\text{None of these}$
2016 South East Mathematical Olympiad, 5
Let $n$ is positive integer, $D_n$ is a set of all positive divisor of $n$ and $f(n)=\sum_{d\in D_n}{\frac{1}{1+d}}$
Prove that for all positive integer $m$, $\sum_{i=1}^{m}{f(i)} <m$
2013 Putnam, 2
Let $S$ be the set of all positive integers that are [i]not[/i] perfect squares. For $n$ in $S,$ consider choices of integers $a_1,a_2,\dots, a_r$ such that $n<a_1<a_2<\cdots<a_r$ and $n\cdot a_1\cdot a_2\cdots a_r$ is a perfect square, and let $f(n)$ be the minimum of $a_r$ over all such choices. For example, $2\cdot 3\cdot 6$ is a perfect square, while $2\cdot 3,2\cdot 4, 2\cdot 5, 2\cdot 3\cdot 4,$ $2\cdot 3\cdot 5, 2\cdot 4\cdot 5,$ and $2\cdot 3\cdot 4\cdot 5$ are not, and so $f(2)=6.$ Show that the function $f$ from $S$ to the integers is one-to-one.
2017 VJIMC, 2
Prove or disprove the following statement. If $g:(0,1) \to (0,1)$ is an increasing function and satisfies
$g(x) > x$ for all $x \in (0,1)$, then there exists a continuous function $f:(0,1) \to \mathbb{R}$ satisfying $f(x) < f(g(x)) $ for all $x \in (0,1)$, but $f$ is not an increasing function.
2020 Putnam, B3
Let $x_0=1$, and let $\delta$ be some constant satisfying $0<\delta<1$. Iteratively, for $n=0,1,2,\dots$, a point $x_{n+1}$ is chosen uniformly form the interval $[0,x_n]$. Let $Z$ be the smallest value of $n$ for which $x_n<\delta$. Find the expected value of $Z$, as a function of $\delta$.
2012 Middle European Mathematical Olympiad, 2
Let $ a,b$ and $ c $ be positive real numbers with $ abc = 1 $. Prove that
\[ \sqrt{ 9 + 16a^2}+\sqrt{ 9 + 16b^2}+\sqrt{ 9 + 16c^2} \ge 3 +4(a+b+c)\]
1970 Miklós Schweitzer, 7
Let us use the word $ N$-measure for nonnegative, finitely additive set functions defined on all subsets of the positive integers, equal to $ 0$ on finite sets, and equal to $ 1$ on the whole set. We say that the system $ \Upsilon$ of sets determines the $ N$-measure $ \mu$ if any $ N$-measure coinciding with $ \mu$ on all elements of $ \Upsilon$ is necessarily identical with $ \mu$.
Prove the existence of an $ N$-measure $ \mu$ that cannot be determined by a system of cardinality less than continuum.
[i]I. Juhasz[/i]
2021 Iran MO (3rd Round), 3
Polynomial $P$ with non-negative real coefficients and function $f:\mathbb{R}^+\to \mathbb{R}^+$ are given such that for all $x, y\in \mathbb{R}^+$ we have
$$f(x+P(x)f(y)) = (y+1)f(x)$$
(a) Prove that $P$ has degree at most 1.
(b) Find all function $f$ and non-constant polynomials $P$ satisfying the equality.
PEN K Problems, 31
Find all strictly increasing functions $f: \mathbb{N}\to \mathbb{N}$ such that \[f(f(n))=3n.\]
2015 AMC 12/AHSME, 20
For every positive integer $n$, let $\operatorname{mod_5}(n)$ be the remainder obtained when $n$ is divided by $5$. Define a function $f : \{0, 1, 2, 3, \dots\} \times \{0, 1, 2, 3, 4\} \to \{0, 1, 2, 3, 4\}$ recursively as follows:
\[f(i, j) = \begin{cases}
\operatorname{mod_5}(j+1) & \text{if }i=0\text{ and }0\leq j\leq 4 \\
f(i-1, 1) & \text{if }i\geq 1\text{ and }j=0 \text{, and}\\
f(i-1, f(i, j-1)) & \text{if }i\geq 1\text{ and }1\leq j\leq 4
\end{cases}\]
What is $f(2015, 2)$?
$\textbf{(A) }0 \qquad\textbf{(B) }1 \qquad\textbf{(C) }2 \qquad\textbf{(D) }3 \qquad\textbf{(E) }4$
2010 Today's Calculation Of Integral, 669
Find the differentiable function defined in $x>0$ such that ${\int_1^{f(x)} f^{-1}(t)dt=\frac 13(x^{\frac {3}{2}}-8}).$
2019 CMI B.Sc. Entrance Exam, 1
For a natural number $n$ denote by Map $( n )$ the set of all functions $f : \{ 1 , 2 , 3 , \cdots , n \} \rightarrow \{ 1 , 2 , 3 , \cdots , n \} . $ For $ f , g \in $ Map$( n ) , f \circ g $ denotes the function in Map $( n )$ that sends $x \rightarrow f ( g ( x ) ) . $ \\
\\
$(a)$ Let $ f \in$ Map $( n ) . $ If for all $x \in \{ 1 , 2 , 3 , \cdots , n \} f ( x ) \neq x , $ show that $ f \circ f \neq f $
\\$(b)$ Count the number of functions $ f \in$ Map $( n )$ such that $ f \circ f = f $
2006 USA Team Selection Test, 5
Let $n$ be a given integer with $n$ greater than $7$ , and let $\mathcal{P}$ be a convex polygon with $n$ sides. Any set of $n-3$ diagonals of $\mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $\mathcal{P}$ into $n-2$ triangles. A triangle in the triangulation of $\mathcal{P}$ is an interior triangle if all of its sides are diagonals of $\mathcal{P}$. Express, in terms of $n$, the number of triangulations of $\mathcal{P}$ with exactly two interior triangles, in closed form.
2023 Iran MO (3rd Round), 2
find all $f : \mathbb{C} \to \mathbb{C}$ st:
$$f(f(x)+yf(y))=x+|y|^2$$
for all $x,y \in \mathbb{C}$
1987 Flanders Math Olympiad, 3
Find all continuous functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x)^3 = -\frac x{12}\cdot\left(x^2+7x\cdot f(x)+16\cdot f(x)^2\right),\ \forall x \in \mathbb{R}.\]
1998 IberoAmerican Olympiad For University Students, 1
The definite integrals between $0$ and $1$ of the squares of the continuous real functions $f(x)$ and $g(x)$ are both equal to $1$.
Prove that there is a real number $c$ such that
\[f(c)+g(c)\leq 2\]
2007 Today's Calculation Of Integral, 243
A cubic funtion $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ intersects with the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma\ (\alpha < \beta < \gamma).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\ \gamma$.
2011 Gheorghe Vranceanu, 2
Let $ f:[0,1]\longrightarrow (0,\infty ) $ be a continuous function and $ \left( b_n \right)_{n\ge 1} $ be a sequence of numbers from the interval $ (0,1) $ that converge to $ 0. $
[b]a)[/b] Demonstrate that for any fixed $ n, $ the equation $ F(x)=b_nF(1)+\left( 1-b_n\right) F(0) $ has an unique solution, namely $ x_n, $ where $ F $ is a primitive of $ f. $
[b]b)[/b] Calculate $ \lim_{n\to\infty } \frac{x_n}{b_n} . $
2017 Germany Team Selection Test, 3
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]