Found problems: 4776
2005 Romania National Olympiad, 2
Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which
\[ x(f(x+1)-f(x)) = f(x), \]
for all $x\in\mathbb{R}$ and
\[ | f(x) - f(y) | \leq |x-y| , \]
for all $x,y\in\mathbb{R}$.
[i]Mihai Piticari[/i]
1994 China Team Selection Test, 1
Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i
t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.
2005 Moldova National Olympiad, 10.7
Determine all strictly increasing functions $ f: R\rightarrow R$ satisfying relationship $ f(x\plus{}f(y))\equal{}f(x\plus{}y)\plus{}2005$
for any real values of x and y.
2017 CMIMC Algebra, 2
For nonzero real numbers $x$ and $y$, define $x\circ y = \tfrac{xy}{x+y}$. Compute \[2^1\circ \left(2^2\circ \left(2^3\circ\cdots\circ\left(2^{2016}\circ 2^{2017}\right)\right)\right).\]
2010 Today's Calculation Of Integral, 543
Let $ y$ be the function of $ x$ satisfying the differential equation $ y'' \minus{} y \equal{} 2\sin x$.
(1) Let $ y \equal{} e^xu \minus{} \sin x$, find the differential equation with which the function $ u$ with respect to $ x$ satisfies.
(2) If $ y(0) \equal{} 3,\ y'(0) \equal{} 0$, then determine $ y$.
2018 USA Team Selection Test, 2
Find all functions $f\colon \mathbb{Z}^2 \to [0, 1]$ such that for any integers $x$ and $y$,
\[f(x, y) = \frac{f(x - 1, y) + f(x, y - 1)}{2}.\]
[i]Proposed by Yang Liu and Michael Kural[/i]
1992 India National Olympiad, 10
Determine all functions $f : \mathbb{R} - [0,1] \to \mathbb{R}$ such that \[ f(x) + f \left( \dfrac{1}{1-x} \right) = \dfrac{2(1-2x)}{x(1-x)} . \]
1995 Bulgaria National Olympiad, 5
Let $A = \{1,2,...,m + n\}$, where $m,n$ are positive integers, and let the function f : $A \to A$ be defined by:
$f(m) = 1$, $f(m+n) = m+1$ and $f(i) = i+1$ for all the other $i$.
(a) Prove that if $m$ and $n$ are odd, then there exists a function $g : A \to A$ such that $g(g(a)) = f(a)$ for all $a \in A$.
(b) Prove that if $m$ is even, then there is a function $g : A\to A$ such that $g(g(a))=f(a)$ for all $a \in A$ is and only if $n = m$.
1954 AMC 12/AHSME, 17
The graph of the function $ f(x) \equal{} 2x^3 \minus{} 7$ goes:
$ \textbf{(A)}\ \text{up to the right and down to the left} \\
\textbf{(B)}\ \text{down to the right and up to the left} \\
\textbf{(C)}\ \text{up to the right and up to the left} \\
\textbf{(D)}\ \text{down to the right and down to the left} \\
\textbf{(E)}\ \text{none of these ways.}$
2012 Balkan MO Shortlist, N3
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold:
(i) $f(n!)=f(n)!$ for every positive integer $n$,
(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.
2008 District Olympiad, 2
Consider the positive reals $ x$, $ y$ and $ z$. Prove that:
a) $ \arctan(x) \plus{} \arctan(y) < \frac {\pi}{2}$ iff $ xy < 1$.
b) $ \arctan(x) \plus{} \arctan(y) \plus{} \arctan(z) < \pi$ iff $ xyz < x \plus{} y \plus{} z$.
2009 ISI B.Stat Entrance Exam, 2
Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2\pi]$ and $f''(x) \geq 0$ for all $x$ in $[0,2\pi]$. Show that
\[\int_{0}^{2\pi} f(x)\cos x dx \geq 0\]
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 $1$.
2005 Today's Calculation Of Integral, 39
Find the minimum value of the following function $f(x) $ defined at $0<x<\frac{\pi}{2}$.
\[f(x)=\int_0^x \frac{d\theta}{\cos \theta}+\int_x^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta}\]
2010 Today's Calculation Of Integral, 542
Find continuous functions $ f(x),\ g(x)$ which takes positive value for any real number $ x$, satisfying $ g(x)\equal{}\int_0^x f(t)\ dt$ and $ \{f(x)\}^2\minus{}\{g(x)\}^2\equal{}1$.
2025 Thailand Mathematical Olympiad, 6
Find all function $f: \mathbb{R}^+ \rightarrow \mathbb{R}$,such that the inequality $$f(x) + f\left(\frac{y}{x}\right) \leqslant \frac{x^3}{y^2} + \frac{y}{x^3}$$ holds for all positive reals $x,y$ and for every positive real $x$, there exist positive reals $y$, such that the equality holds.
1997 Pre-Preparation Course Examination, 3
Suppose that $f : \mathbb R^+ \to \mathbb R^+$ is a decreasing function such that
\[f(x+y)+f(f(x)+f(y))=f(f(x+f(y))+f(y+f(x)), \quad \forall x,y \in \mathbb R^+.\]
Prove that $f(x) = f^{-1}(x).$
2009 IMO Shortlist, 5
Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\]
[i]Proposed by Igor Voronovich, Belarus[/i]
1966 IMO Longlists, 46
Let $a,b,c$ be reals and
\[f(a, b, c) = \left| \frac{ |b-a|}{|ab|} +\frac{b+a}{ab} -\frac 2c \right| +\frac{ |b-a|}{|ab|} +\frac{b+a}{ab} +\frac 2c\]
Prove that $f(a, b, c) = 4 \max \{\frac 1a, \frac 1b,\frac 1c \}.$
2015 AMC 10, 11
The ratio of the length to the width of a rectangle is $4:3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$?
$\textbf{(A) }\dfrac27\qquad\textbf{(B) }\dfrac37\qquad\textbf{(C) }\dfrac{12}{25}\qquad\textbf{(D) }\dfrac{16}{25}\qquad\textbf{(E) }\dfrac34$
2011 Uzbekistan National Olympiad, 4
Does existes a function $f:N->N$ and for all positeve integer n
$f(f(n)+2011)=f(n)+f(f(n))$
2007 Singapore Senior Math Olympiad, 2
For any positive integer $n$, let $f(n)$ denote the $n$- th positive nonsquare integer, i.e., $f(1) = 2, f(2) = 3, f(3) = 5, f(4) = 6$, etc. Prove that $f(n)=n +\{\sqrt{n}\}$ where $\{x\}$ denotes the integer closest to $x$.
(For example, $\{\sqrt{1}\} = 1, \{\sqrt{2}\} = 1, \{\sqrt{3}\} = 2, \{\sqrt{4}\} = 2$.)
2004 Moldova Team Selection Test, 7
Let $ABC$ be a triangle, let $O$ be its circumcenter, and let $H$ be its orthocenter.
Let $P$ be a point on the segment $OH$.
Prove that
$6r\leq PA+PB+PC\leq 3R$,
where $r$ is the inradius and $R$ the circumradius of triangle $ABC$.
[b]Moderator edit:[/b] This is true only if the point $P$ lies inside the triangle $ABC$. (Of course, this is always fulfilled if triangle $ABC$ is acute-angled, since in this case the segment $OH$ completely lies inside the triangle $ABC$; but if triangle $ABC$ is obtuse-angled, then the condition about $P$ lying inside the triangle $ABC$ is really necessary.)
2011 South East Mathematical Olympiad, 1
If $\min \left \{ \frac{ax^2+b}{\sqrt{x^2+1}} \mid x \in \mathbb{R}\right \} = 3$, then (1) Find the range of $b$; (2) for every given $b$, find $a$.
2017 South East Mathematical Olympiad, 2
Let $x_i \in \{0,1\}(i=1,2,\cdots ,n)$,if the value of function $f=f(x_1,x_2, \cdots ,x_n)$ can only be $0$ or $1$,then we call $f$ a $n$-var Boole function,and we denote $D_n(f)=\{(x_1,x_2, \cdots ,x_n)|f(x_1,x_2, \cdots ,x_n)=0\}.$
$(1)$ Find the number of $n$-var Boole function;
$(2)$ Let $g$ be a $n$-var Boole function such that $g(x_1,x_2, \cdots ,x_n) \equiv 1+x_1+x_1x_2+x_1x_2x_3 +\cdots +x_1x_2 \cdots x_n \pmod 2$,
Find the number of elements of the set $D_n(g)$,and find the maximum of $n \in \mathbb{N}_+$ such that
$\sum_{(x_1,x_2, \cdots ,x_n) \in D_n(g)}(x_1+x_2+ \cdots +x_n) \le 2017.$