Found problems: 4776
2003 China Team Selection Test, 3
Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define
\[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.
2014 India IMO Training Camp, 3
For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$.
Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that $f^{k}((n,m))=(m,n)$,where
$f^{k}(x)=f(f(f(...f(x))))$,$f$ being composed with itself $k$ times.
2007 Romania Team Selection Test, 1
Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find
\[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]
2005 Romania National Olympiad, 3
a) Prove that there are no one-to-one (injective) functions $f: \mathbb{N} \to \mathbb{N}\cup \{0\}$ such that
\[ f(mn) = f(m)+f(n) , \ \forall \ m,n \in \mathbb{N}. \]
b) Prove that for all positive integers $k$ there exist one-to-one functions $f: \{1,2,\ldots,k\}\to\mathbb{N}\cup \{0\}$ such that $f(mn) = f(m)+f(n)$ for all $m,n\in \{1,2,\ldots,k\}$ with $mn\leq k$.
[i]Mihai Baluna[/i]
2003 AMC 12-AHSME, 24
Positive integers $ a$, $ b$, and $ c$ are chosen so that $ a<b<c$, and the system of equations
\[ 2x\plus{}y\equal{}2003\text{ and }y\equal{}|x\minus{}a|\plus{}|x\minus{}b|\plus{}|x\minus{}c|
\]has exactly one solution. What is the minimum value of $ c$?
$ \textbf{(A)}\ 668 \qquad
\textbf{(B)}\ 669 \qquad
\textbf{(C)}\ 1002 \qquad
\textbf{(D)}\ 2003 \qquad
\textbf{(E)}\ 2004$
2012 Macedonia National Olympiad, 3
Find all functions $f : \mathbb{R} \to \mathbb{Z}$ which satisfy the conditions:
$f(x+y) < f(x) + f(y)$
$f(f(x)) = \lfloor {x} \rfloor + 2$
2013 Iran Team Selection Test, 7
Nonnegative real numbers $p_{1},\ldots,p_{n}$ and $q_{1},\ldots,q_{n}$ are such that $p_{1}+\cdots+p_{n}=q_{1}+\cdots+q_{n}$
Among all the matrices with nonnegative entries having $p_i$ as sum of the $i$-th row's entries and $q_j$ as sum of the $j$-th column's entries, find the maximum sum of the entries on the main diagonal.
1998 VJIMC, Problem 3
Give an example of a sequence of continuous functions on $\mathbb R$ converging pointwise to $0$ which is not uniformly convergent on any nonempty open set.
2010 Postal Coaching, 3
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$\boxed{1} \ f(1) = 1$
$\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}$
2015 Indonesia MO Shortlist, A1
Function $f: R\to R$ is said periodic , if $f$ is not a constant function and there is a number real positive $p$ with the property of $f (x) = f (x + p)$ for every $x \in R$. The smallest positive real number p which satisfies the condition $f (x) = f (x + p)$ for each $x \in R$ is named period of $f$. Given $a$ and $b$ real positive numbers, show that there are periodic functions $f_1$ and $f_2$, with periods $a$ and $b$ respectively, so that $f_1 (x)\cdot f_2 (x)$ is also a periodic function.
2013 Romanian Master of Mathematics, 2
Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?
KoMaL A Problems 2021/2022, A. 821
[b]a)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ and positive integer $M$ there exists $n\in\mathbb N$ such that set $\left\{k\in \mathbb N : f(n,k)=g(k)\right\}$ has at least $M$ elements?
[b]b)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ there exists $n\in \mathbb N$ such that set $\left\{k\in\mathbb N : f(n,k)=g(k)\right\}$ has an infinite number of elements?
2003 Gheorghe Vranceanu, 2
Let be a real number $ a $ and a function $ f:[a,\infty )\longrightarrow\mathbb{R} $ that is continuous at $ a. $ Prove that $ f $ is primitivable on $ (a,\infty ) $ if and only if $ f $ is primitivable on $ [a,\infty ) . $
2008 Harvard-MIT Mathematics Tournament, 5
Let $ f(x) \equal{} x^3 \plus{} x \plus{} 1$. Suppose $ g$ is a cubic polynomial such that $ g(0) \equal{} \minus{} 1$, and the roots of $ g$ are the squares of the roots of $ f$. Find $ g(9)$.
1967 Putnam, A4
Show that if $\lambda > \frac{1}{2}$ there does not exist a real-valued function $u(x)$ such that for all $x$ in the closed interval $[0,1]$ the following holds:
$$u(x)= 1+ \lambda \int_{x}^{1} u(y) u(y-x) \; dy.$$
2002 AMC 10, 1
The ratio $ \dfrac{10^{2000}\plus{}10^{2002}}{10^{2001}\plus{}10^{2001}}$ is closest to which of the following numbers?
$ \text{(A)}\ 0.1\qquad
\text{(B)}\ 0.2\qquad
\text{(C)}\ 1\qquad
\text{(D)}\ 5\qquad
\text{(E)}\ 10$
2012 Indonesia TST, 4
Let $\mathbb{N}$ be the set of positive integers. For every $n \in \mathbb{N}$, define $d(n)$ as the number of positive divisors of $n$. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that:
a) $d(f(x)) = x$ for all $x \in \mathbb{N}$
b) $f(xy)$ divides $(x-1)y^{xy-1}f(x)$ for all $x,y \in \mathbb{N}$
2012 Spain Mathematical Olympiad, 2
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[(x-2)f(y)+f(y+2f(x))=f(x+yf(x))\]
for all $x,y\in\mathbb{R}$.
1984 AMC 12/AHSME, 16
The function $f(x)$ satisfies $f(2+x) = f(2-x)$ for all real numbers $x$. If the equation $f(x) = 0$ has exactly four distinct real roots, then the sum of these roots is
A. 0
B. 2
C. 4
D. 6
E. 8
1986 Iran MO (2nd round), 2
[b](a)[/b] Sketch the diagram of the function $f$ if
\[f(x)=4x(1-|x|) , \quad |x| \leq 1.\]
[b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$
[b](c)[/b] Let $g$ be a function such that
\[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\right.\]
Is the function $g$ continuous in the point $x=0 \ ?$
[b](d)[/b] Sketch the diagram of $g.$
2007 Today's Calculation Of Integral, 225
2 Points $ P\left(a,\ \frac{1}{a}\right),\ Q\left(2a,\ \frac{1}{2a}\right)\ (a > 0)$ are on the curve $ C: y \equal{}\frac{1}{x}$. Let $ l,\ m$ be the tangent lines at $ P,\ Q$ respectively. Find the area of the figure surrounded by $ l,\ m$ and $ C$.
2004 AMC 12/AHSME, 16
A function $ f$ is defined by $ f(z) \equal{} i\bar z$, where $ i \equal{}\sqrt{\minus{}\!1}$ and $ \bar z$ is the complex conjugate of $ z$. How many values of $ z$ satisfy both $ |z| \equal{} 5$ and $ f (z) \equal{} z$?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 8$
2010 Today's Calculation Of Integral, 570
Let $ f(x) \equal{} 1 \minus{} \cos x \minus{} x\sin x$.
(1) Show that $ f(x) \equal{} 0$ has a unique solution in $ 0 < x < \pi$.
(2) Let $ J \equal{} \int_0^{\pi} |f(x)|dx$. Denote by $ \alpha$ the solution in (1), express $ J$ in terms of $ \sin \alpha$.
(3) Compare the size of $ J$ defined in (2) with $ \sqrt {2}$.
2010 Today's Calculation Of Integral, 666
Let $f(x)$ be a function defined in $0<x<\frac{\pi}{2}$ satisfying:
(i) $f\left(\frac{\pi}{6}\right)=0$
(ii) $f'(x)\tan x=\int_{\frac{\pi}{6}}^x \frac{2\cos t}{\sin t}dt$.
Find $f(x)$.
[i]1987 Sapporo Medical University entrance exam[/i]
Today's calculation of integrals, 893
Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$