Found problems: 4776
2005 Iran MO (2nd round), 3
Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}^{+}$ such that for all positive real numbers $x$ and $y$, the following equation holds:
\[(x+y)f(f(x)y)=x^2f(f(x)+f(y)).\]
2010 Today's Calculation Of Integral, 560
Let $ K$ be the figure bounded by the graph of function $ y \equal{} \frac {x}{\sqrt {1 \minus{} x^2}}$, $ x$ axis and the line $ x \equal{} \frac {1}{2}$.
(1) Find the volume $ V_1$ of the solid generated by rotation of $ K$ around $ x$ axis.
(2) Find the volume $ V_2$ of the solid generated by rotation of $ K$ around $ y$ axis.
Please solve question (2) without using the shell method for Japanese High School Students those who don't learn it.
2017 Pan-African Shortlist, N1
Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.
1991 Romania Team Selection Test, 8
Let $n, a, b$ be integers with $n \geq 2$ and $a \notin \{0, 1\}$ and let $u(x) = ax + b$ be the function defined on integers. Show that there are infinitely many functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)$ for all $x$.
If $a = 1$, show that there is a $b$ for which there is no $f$ with $f_n(x) \equiv u(x)$.
1992 IMO Longlists, 48
Find all the functions $f : \mathbb R^+ \to \mathbb R$ satisfying the identity
\[f(x)f(y)=y^{\alpha}f\left(\frac x2 \right) + x^{\beta} f\left(\frac y2 \right) \qquad \forall x,y \in \mathbb R^+\]
Where $\alpha,\beta$ are given real numbers.
2003 China Girls Math Olympiad, 5
Let $ \{a_n\}^{\infty}_1$ be a sequence of real numbers such that $ a_1 \equal{} 2,$ and \[ a_{n\plus{}1} \equal{} a^2_n \minus{} a_n \plus{} 1, \forall n \in \mathbb{N}.\] Prove that \[ 1 \minus{} \frac{1}{2003^{2003}} < \sum^{2003}_{i\equal{}1} \frac{1}{a_i} < 1.\]
2010 District Olympiad, 1
Prove that any continuos function $ f: \mathbb{R}\rightarrow \mathbb{R}$ with
\[ f(x)\equal{}\left\{ \begin{aligned} a_1x\plus{}b_1\ ,\ \text{for } x\le 1 \\
a_2x\plus{}b_2\ ,\ \text{for } x>1 \end{aligned} \right.\]
where $ a_1,a_2,b_1,b_2\in \mathbb{R}$, can be written as:
\[ f(x)\equal{}m_1x\plus{}n_1\plus{}\epsilon|m_2x\plus{}n_2|\ ,\ \text{for } x\in \mathbb{R}\]
where $ m_1,m_2,n_1,n_2\in \mathbb{R}$ and $ \epsilon\in \{\minus{}1,\plus{}1\}$.
2014 ELMO Shortlist, 4
Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$.
[i]Proposed by Evan Chen[/i]
2008 Putnam, B4
Let $ p$ be a prime number. Let $ h(x)$ be a polynomial with integer coefficients such that $ h(0),h(1),\dots, h(p^2\minus{}1)$ are distinct modulo $ p^2.$ Show that $ h(0),h(1),\dots, h(p^3\minus{}1)$ are distinct modulo $ p^3.$
2006 Tournament of Towns, 2
Do there exist functions $p(x)$ and $q(x)$, such that $p(x)$ is an even function while $p(q(x))$ is an odd function (different from 0)?
[i](3 points)[/i]
1996 China Team Selection Test, 2
$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions:
[b]I.[/b] $f(1) = 2$
[b]II.[/b] $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$
Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.
2006 Taiwan TST Round 1, 3
Every square on a $n\times n$ chessboard is colored with red, blue, or green. Each red square has at least one green square adjacent to it, each green square has at least one blue square adjacent to it, and each blue square has at least one red square adjacent to it. Let $R$ be the number of red squares. Prove that $\displaystyle \frac{n^2}{11} \le R \le \frac{2n^2}{3}$.
2010 Today's Calculation Of Integral, 558
For a positive constant $ t$, let $ \alpha ,\ \beta$ be the roots of the quadratic equation $ x^2 \plus{} t^2x \minus{} 2t \equal{} 0$.
Find the minimum value of $ \int_{ \minus{} 1}^2 \left\{\left(x \plus{} \frac {1}{\alpha ^ 2}\right)\left(x \plus{} \frac {1}{\beta ^ 2}\right) \plus{} \frac {1}{\alpha \beta}\right\}dx.$
2013 Harvard-MIT Mathematics Tournament, 24
Given a point $p$ and a line segment $l$, let $d(p,l)$ be the distance between them. Let $A$, $B$, and $C$ be points in the plane such that $AB=6$, $BC=8$, $AC=10$. What is the area of the region in the $(x,y)$-plane formed by the ordered pairs $(x,y)$ such that there exists a point $P$ inside triangle $ABC$ with $d(P,AB)+x=d(P,BC)+y=d(P,AC)?$
1999 Hungary-Israel Binational, 2
The function $ f(x,y,z)\equal{}\frac{x^2\plus{}y^2\plus{}z^2}{x\plus{}y\plus{}z}$ is defined for every $ x,y,z \in R$ whose sum is not 0. Find a point $ (x_0,y_0,z_0)$ such that $ 0 < x_0^2\plus{}y_0^2\plus{}z_0^2 < \frac{1}{1999}$ and $ 1.999 < f(x_0,y_0,z_0) < 2$.
1977 IMO Longlists, 46
Let $f$ be a strictly increasing function defined on the set of real numbers. For $x$ real and $t$ positive, set\[g(x,t)=\frac{f(x+t)-f(x)}{f(x) - f(x - t)}.\]
Assume that the inequalities\[2^{-1} < g(x, t) < 2\]
hold for all positive t if $x = 0$, and for all $t \leq |x|$ otherwise.
Show that\[ 14^{-1} < g(x, t) < 14\]
for all real $x$ and positive $t.$
1954 AMC 12/AHSME, 16
If $ f(x) \equal{} 5x^2 \minus{} 2x \minus{} 1$, then $ f(x \plus{} h) \minus{} f(x)$ equals:
$ \textbf{(A)}\ 5h^2 \minus{} 2h \qquad \textbf{(B)}\ 10xh \minus{} 4x \plus{} 2 \qquad \textbf{(C)}\ 10xh \minus{} 2x \minus{} 2 \\
\textbf{(D)}\ h(10x \plus{} 5h \minus{} 2) \qquad \textbf{(E)}\ 3h$
1986 Kurschak Competition, 3
A and B plays the following game: they choose randomly $k$ integers from $\{1,2,\dots,100\}$; if their sum is even, A wins, else B wins. For what values of $k$ does A and B have the same chance of winning?
2000 Bulgaria National Olympiad, 3
Let $A$ be the set of all binary sequences of length $n$ and denote $o =(0, 0, \ldots , 0) \in A$. Define the addition on $A$ as $(a_1, \ldots , a_n)+(b_1, \ldots , b_n) =(c_1, \ldots , c_n)$, where $c_i = 0$ when $a_i = b_i$ and $c_i = 1$ otherwise. Suppose that $f\colon A \to A$ is a function such that $f(0) = 0$, and for each $a, b \in A$, the sequences $f(a)$ and $f(b)$ differ in exactly as many places as $a$ and $b$ do. Prove that if $a$ , $b$, $c \in A$ satisfy $a+ b + c = 0$, then $f(a)+ f(b) + f(c) = 0$.
1948 Putnam, B3
Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in N$.
2011 Today's Calculation Of Integral, 769
In $xyz$ space, find the volume of the solid expressed by $x^2+y^2\leq z\le \sqrt{3}y+1.$
1969 IMO Shortlist, 61
$(SWE 4)$ Let $a_0, a_1, a_2, \cdots$ be determined with $a_0 = 0, a_{n+1} = 2a_n + 2^n$. Prove that if $n$ is power of $2$, then so is $a_n$
2021 CIIM, 5
For every positive integer $n$, let $s(n)$ be the sum of the exponents of $71$ and $97$ in the prime factorization of $n$; for example, $s(2021) = s(43 \cdot 47) = 0$ and $s(488977) = s(71^2 \cdot 97) = 3$. If we define $f(n)=(-1)^{s(n)}$, prove that the limit
\[ \lim_{n \to +\infty} \frac{f(1) + f(2) + \cdots+ f(n)}{n} \]
exists and determine its value.
1983 IMO Longlists, 20
Let $f$ and $g$ be functions from the set $A$ to the same set $A$. We define $f$ to be a functional $n$-th root of $g$ ($n$ is a positive integer) if $f^n(x) = g(x)$, where $f^n(x) = f^{n-1}(f(x)).$
(a) Prove that the function $g : \mathbb R \to \mathbb R, g(x) = 1/x$ has an infinite number of $n$-th functional roots for each positive integer $n.$
(b) Prove that there is a bijection from $\mathbb R$ onto $\mathbb R$ that has no nth functional root for each positive integer $n.$
2019 India IMO Training Camp, P3
Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.