Found problems: 4776
2002 India IMO Training Camp, 10
Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying
\[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
+ f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}
\]
for all nonnegative integers $ p$, $ q$, $ r$.
1998 Tournament Of Towns, 6
In a function $f (x) = (x^2 + ax + b )/ (x^2 + cx + d)$ , the quadratics $x^2 + ax + b$ and $x^2 + cx + d$ have no common roots. Prove that the next two statements are equivalent:
(i) there is a numerical interval without any values of $f(x)$ ,
(ii) $f(x)$ can be represented in the form $f (x) = f_1 (f_2( ... f_{n-1} (f_n (x))... ))$ where each of the functions $f_j$ is o f one of the three forms $k_j x + b_j, 1/x, x^2$ .
(A Kanel)
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 $n \in \mathbb{N}, f(n) < M$.
1990 Putnam, B1
Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, \[ \left( f(x) \right)^2 = \displaystyle\int_0^x \left[ \left( f(t) \right)^2 + \left( f'(t) \right)^2 \right] \, \mathrm{d}t + 1990. \]
1989 IMO Longlists, 54
Let $ n \equal{} 2k \minus{} 1$ where $ k \geq 6$ is an integer. Let $ T$ be the set of all $ n\minus{}$tuples $ (x_1, x_2, \ldots, x_n)$ where $ x_i \in \{0,1\}$ $ \forall i \equal{} \{1,2, \ldots, n\}$ For $ x \equal{} (x_1, x_2, \ldots, x_n) \in T$ and $ y \equal{} (y_1, y_2, \ldots, y_n) \in T$ let $ d(x,y)$ denote the number of integers $ j$ with $ 1 \leq j \leq n$ such that $ x_i \neq x_j$, in particular $ d(x,x) \equal{} 0.$ Suppose that there exists a subset $ S$ of $ T$ with $ 2^k$ elements that has the following property: Given any element $ x \in T,$ there is a unique element $ y \in S$ with $ d(x, y) \leq 3.$ Prove that $ n \equal{} 23.$
1989 National High School Mathematics League, 3
For any function $f(x)$, in the same rectangular coordinates, figures of function $y=f(x-1)$ and $y=f(-x+1)$
$\text{(A)}$ are symmetrical about $x$-axis
$\text{(B)}$ are symmetrical about line $x=1$
$\text{(C)}$ are symmetrical about line $x=-1$
$\text{(D)}$ are symmetrical about $y$-axis
2014 Romania Team Selection Test, 2
Let $m$ be a positive integer and let $A$, respectively $B$, be two alphabets with $m$, respectively $2m$ letters. Let also $n$ be an even integer which is at least $2m$. Let $a_n$ be the number of words of length $n$, formed with letters from $A$, in which appear all the letters from $A$, each an even number of times. Let $b_n$ be the number of words of length $n$, formed with letters from $B$, in which appear all the letters from $B$, each an odd number of times. Compute $\frac{b_n}{a_n}$.
1985 IMO Longlists, 97
In a plane a circle with radius $R$ and center $w$ and a line $\Lambda$ are given. The distance between $w$ and $\Lambda$ is $d, d > R$. The points $M$ and $N$ are chosen on $\Lambda$ in such a way that the circle with diameter $MN$ is externally tangent to the given circle. Show that there exists a point $A$ in the plane such that all the segments $MN$ are seen in a constant angle from $A.$
1991 Putnam, A5
A5) Find the maximum value of $\int_{0}^{y}\sqrt{x^{4}+(y-y^{2})^{2}}dx$ for $0\leq y\leq 1$.
I don't have a solution for this yet. I figure this may be useful: Let the integral be denoted $f(y)$, then according to the [url=http://mathworld.wolfram.com/LeibnizIntegralRule.html]Leibniz Integral Rule[/url] we have
$\frac{df}{dy}=\int_{0}^{y}\frac{y(1-y)(1-2y)}{\sqrt{x^{4}+(y-y^{2})^{2}}}dx+\sqrt{y^{4}+(y-y^{2})^{2}}$
Now what?
1984 Balkan MO, 3
Show that for any positive integer $m$, there exists a positive integer $n$ so that in the decimal representations of the numbers $5^{m}$ and $5^{n}$, the representation of $5^{n}$ ends in the representation of $5^{m}$.
2003 AMC 12-AHSME, 17
If $ \log(xy^3)\equal{}1$ and $ \log(x^2y)\equal{}1$, what is $ \log(xy)$?
$ \textbf{(A)}\ \minus{}\!\frac{1}{2} \qquad
\textbf{(B)}\ 0 \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{3}{5} \qquad
\textbf{(E)}\ 1$
2016 District Olympiad, 3
Find the continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property:
$$ f\left( x+\frac{1}{n}\right) \le f(x) +\frac{1}{n},\quad\forall n\in\mathbb{Z}^* ,\quad\forall x\in\mathbb{R} . $$
2024 USAJMO, 5
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy
\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]
for all $x,y\in\mathbb{R}$.
[i]Proposed by Carl Schildkraut[/i]
2021 Simon Marais Mathematical Competition, B3
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the following two properties.
(i) The Riemann integral $\int_a^b f(t) \mathrm dt$ exists for all real numbers $a < b$.
(ii) For every real number $x$ and every integer $n \ge 1$ we have
\[ f(x) = \frac{n}{2} \int_{x-\frac{1}{n}}^{x+\frac{1}{n}} f(t) \mathrm dt. \]
2019 Teodor Topan, 3
Let be two real numbers $ a<b, $ a natural number $ n\ge 2, $ and a continuous function $ f:[a,b]\longrightarrow (0,\infty ) $ whose image contains $ 1 $ and that admits a primitive $ F:[a,b]\longrightarrow [a,b] . $ Prove that there is a real number $ c\in (a,b) $ such that
$$ (\underbrace{F\circ\cdots\circ F}_{\text{n times}} )(b) -(\underbrace{F\circ\cdots\circ F}_{\text{n times}} )(a) =(f(c))^{n+1} (b-a) $$
[i]Vlad Mihaly[/i]
2002 AIME Problems, 8
Find the least positive integer $k$ for which the equation $\lfloor \frac{2002}{n}\rfloor = k$ has no integer solutions for $n.$ (The notation $\lfloor x \rfloor$ means the greatest integer less than or equal to $x.$)
2009 Stanford Mathematics Tournament, 4
Find all values of $x$ for which $f(x)+xf\left(\frac{1}{x}\right)=x$ for any function $f(x)$
2012 Putnam, 6
Let $f(x,y)$ be a continuous, real-valued function on $\mathbb{R}^2.$ Suppose that, for every rectangular region $R$ of area $1,$ the double integral of $f(x,y)$ over $R$ equals $0.$ Must $f(x,y)$ be identically $0?$
1985 IMO Longlists, 78
The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by
\[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\]
Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$
2005 Grigore Moisil Urziceni, 2
Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R}_{\ge 0} $ that admits primitives and such that $ \lim_{x\to 0 } \frac{f(x)}{x} =0. $ Prove that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} , $ defined as
$$ g(x)=\left\{ \begin{matrix} f(x)/x ,&\quad x\neq 0\\ 0,& \quad x=0 \end{matrix} \right. , $$
is primitivable.
2005 Putnam, B1
Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a\rfloor,\lfloor 2a\rfloor)=0$ for all real numbers $a.$
(Note: $\lfloor v\rfloor$ is the greatest integer less than or equal to $v.$)
1990 AMC 12/AHSME, 12
Let $f$ be the function defined by $f(x)=ax^2-\sqrt2$ for some positive $a$. If $f(f(\sqrt2 ))=-\sqrt 2$, then $a=$
$\text{(A)} \ \frac{2-\sqrt2}{2} \qquad \text{(B)} \ \frac12 \qquad \text{(C)} \ 2-\sqrt2 \qquad \text{(D)} \ \frac{\sqrt{2}}{2} \qquad \text{(E)} \ \frac{2+\sqrt2}{2}$
2007 VJIMC, Problem 3
A function $f:[0,\infty)\to\mathbb R\setminus\{0\}$ is called [i]slowly changing[/i] if for any $t>1$ the limit $\lim_{x\to\infty}\frac{f(tx)}{f(x)}$ exists and is equal to $1$. Is it true that every slowly changing function has for sufficiently large $x$ a constant sign (i.e., is it true that for every slowly changing $f$ there exists an $N$ such that for every $x,y>N$ we have $f(x)f(y)>0$?)
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 $n = m$.
1987 IMO Shortlist, 22
Does there exist a function $f : \mathbb N \to \mathbb N$, such that $f(f(n)) =n + 1987$ for every natural number $n$? [i](IMO Problem 4)[/i]
[i]Proposed by Vietnam.[/i]