Found problems: 4776
2013 NIMO Problems, 2
Let $f$ be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers $x$. Find the sum of all possible values of $f(1)$.
[i]Proposed by Ahaan S. Rungta[/i]
2009 BMO TST, 3
For the give functions in $\mathbb{N}$:
[b](a)[/b] Euler's $\phi$ function ($\phi(n)$- the number of natural numbers smaller than $n$ and coprime with $n$);
[b](b)[/b] the $\sigma$ function such that the $\sigma(n)$ is the sum of natural divisors of $n$.
solve the equation $\phi(\sigma(2^x))=2^x$.
2008 Philippine MO, 4
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{2008^{2x}}{2008+2008^{2x}}$. Prove that
\[\begin{aligned}
f\left(\frac{1}{2007}\right)+f\left(\frac{2}{2007}\right)+\cdots+f\left(\frac{2005}{2007}\right)+f\left(\frac{2006}{2007}\right)=1003.
\end{aligned}\]
2007 South East Mathematical Olympiad, 3
Let $a_i=min\{ k+\dfrac{i}{k}|k \in N^*\}$, determine the value of $S_{n^2}=[a_1]+[a_2]+\cdots +[a_{n^2}]$, where $n\ge 2$ . ($[x]$ denotes the greatest integer not exceeding x)
2022 Miklós Schweitzer, 10
Is there a continuous function $f : \mathbb R \backslash \mathbb Q \to \mathbb R \backslash \mathbb Q$ for which the archetype of every irrational number has a positive Hausdorff dimension?
2016 USAMTS Problems, 4:
Find all functions $f(x)$ from nonnegative reals to nonnegative reals such that $f(f(x))=x^4$ and $f(x)\leq Cx^2$ for some constant $C$.
1987 Iran MO (2nd round), 2
Find all continuous functions $f: \mathbb R \to \mathbb R$ such that
\[f(x^2-y^2)=f(x)^2 + f(y)^2, \quad \forall x,y \in \mathbb R.\]
2022 Iran MO (3rd Round), 1
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for all $x,y,z\in\mathbb{R}^+$
$$f(x+f(y)+f(f(z)))=z+f(y+f(x))$$
1977 Polish MO Finals, 1
A function $h : \mathbb{R} \rightarrow \mathbb{R}$ is differentiable and satisfies $h(ax) = bh(x)$ for all $x$, where $a$ and $b$ are given positive numbers and $0 \not = |a| \not = 1$. Suppose that $h'(0) \not = 0$ and the function $h'$ is continuous at $x = 0$. Prove that $a = b$ and that there is a real number $c$ such that $h(x) = cx$ for all $x$.
2010 Today's Calculation Of Integral, 576
For a function $ f(x)\equal{}(\ln x)^2\plus{}2\ln x$, let $ C$ be the curve $ y\equal{}f(x)$. Denote $ A(a,\ f(a)),\ B(b,\ f(b))\ (a<b)$ the points of tangency of two tangents drawn from the origin $ O$ to $ C$ and the curve $ C$. Answer the following questions.
(1) Examine the increase and decrease, extremal value and inflection point , then draw the approximate garph of the curve $ C$.
(2) Find the values of $ a,\ b$.
(3) Find the volume by a rotation of the figure bounded by the part from the point $ A$ to the point $ B$ and line segments $ OA,\ OB$ around the $ y$-axis.
2024 IRN-SGP-TWN Friendly Math Competition, 4
Consider the function $f_k:\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}$ satisfying
\[f_k(x)=x+k\varphi(x)\]
where $\varphi(x)$ is Euler's totient function, that is, the number of positive integers up to $x$ coprime to $x$. We define a sequence $a_1,a_2,...,a_{10}$ with
[list]
[*] $a_1=c$, and
[*] $a_n=f_k(a_{n-1}) \text{ }\forall \text{ } 2\le n\le 10$
[/list]
Is it possible to choose the initial value $c\ne 1$ such that each term is a multiple of the previous, if
(a) $k=2025$ ?
(b) $k=2065$ ?
[i]Proposed by chorn[/i]
2005 Harvard-MIT Mathematics Tournament, 8
If $f$ is a continuous real function such that $ f(x-1) + f(x+1) \ge x + f(x) $ for all $x$, what is the minimum possible value of $ \displaystyle\int_{1}^{2005} f(x) \, \mathrm{d}x $?
1995 Grosman Memorial Mathematical Olympiad, 7
For a given positive integer $n$, let $A_n$ be the set of all points $(x,y)$ in the coordinate plane with $x,y \in \{0,1,...,n\}$. A point $(i, j)$ is called internal if $0 < i, j < n$. A real function $f$ , defined on $A_n$, is called [i]good [/i] if it has the following property: For every internal point $x$, the value of $f(x)$ is the arithmetic mean of its values on the four neighboring points (i.e. the points at the distance $1$ from $x$). Prove that if $f$ and $g$ are good functions that coincide at the non-internal points of $A_n$, then $f \equiv g$.
2013 Math Prize For Girls Problems, 17
Let $f$ be the function defined by $f(x) = -2 \sin(\pi x)$. How many values of $x$ such that $-2 \le x \le 2$ satisfy the equation $f(f(f(x))) = f(x)$?
2010 India IMO Training Camp, 11
Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$
1985 Iran MO (2nd round), 3
Let $f: \mathbb R \to \mathbb R,g: \mathbb R \to \mathbb R$ and $\varphi: \mathbb R \to \mathbb R$ be three ascendant functions such that
\[f(x) \leq g(x) \leq \varphi(x) \qquad \forall x \in \mathbb R.\]
Prove that
\[f(f(x)) \leq g(g(x)) \leq \varphi(\varphi(x)) \qquad \forall x \in \mathbb R.\]
[i]Note. The function is $k(x)$ ascendant if for every $ x,y \in D_k, x \leq {y}$ we have $g(x)\leq{g(y)}$.[/i]
2011 AMC 12/AHSME, 21
Let $f_1(x)=\sqrt{1-x}$, and for integers $n \ge 2$, let $f_n(x)=f_{n-1}(\sqrt{n^2-x})$. If $N$ is the largest value of $n$ for which the domain of $f_n$ is nonempty, the domain of $f_N$ is ${c}$. What is $N+c$?
$ \textbf{(A)}\ -226 \qquad
\textbf{(B)}\ -144 \qquad
\textbf{(C)}\ -20 \qquad
\textbf{(D)}\ 20 \qquad
\textbf{(E)}\ 144$
1993 IMO Shortlist, 4
Let $n \geq 2, n \in \mathbb{N}$ and $A_0 = (a_{01},a_{02}, \ldots, a_{0n})$ be any $n-$tuple of natural numbers, such that $0 \leq a_{0i} \leq i-1,$ for $i = 1, \ldots, n.$
$n-$tuples $A_1= (a_{11},a_{12}, \ldots, a_{1n}), A_2 = (a_{21},a_{22}, \ldots, a_{2n}), \ldots$ are defined by: $a_{i+1,j} = Card \{a_{i,l}| 1 \leq l \leq j-1, a_{i,l} \geq a_{i,j}\},$ for $i \in \mathbb{N}$ and $j = 1, \ldots, n.$ Prove that there exists $k \in \mathbb{N},$ such that $A_{k+2} = A_{k}.$
2008 Middle European Mathematical Olympiad, 2
On a blackboard there are $ n \geq 2, n \in \mathbb{Z}^{\plus{}}$ numbers. In each step we select two numbers from the blackboard and replace both of them by their sum. Determine all numbers $ n$ for which it is possible to yield $ n$ identical number after a finite number of steps.
2005 Iran Team Selection Test, 2
Suppose there are $n$ distinct points on plane. There is circle with radius $r$ and center $O$ on the plane. At least one of the points are in the circle. We do the following instructions. At each step we move $O$ to the baricenter of the point in the circle. Prove that location of $O$ is constant after some steps.
1993 India National Olympiad, 8
Let $f$ be a bijective function from $A = \{ 1, 2, \ldots, n \}$ to itself. Show that there is a positive integer $M$ such that $f^{M}(i) = f(i)$ for each $i$ in $A$, where $f^{M}$ denotes the composition $f \circ f \circ \cdots \circ f$ $M$ times.
2010 Today's Calculation Of Integral, 563
Determine the pair of constant numbers $ a,\ b,\ c$ such that for a quadratic function $ f(x) \equal{} x^2 \plus{} ax \plus{} b$, the following equation is identity with respect to $ x$.
\[ f(x \plus{} 1) \equal{} c\int_0^1 (3x^2 \plus{} 4xt)f'(t)dt\]
.
2023 Brazil Undergrad MO, 2
Let $a_n = \frac{1}{\binom{2n}{n}}, \forall n \leq 1$.
a) Show that $\sum\limits_{n=1}^{+\infty}a_nx^n$ converges for all $x \in (-4, 4)$ and that the function $f(x) = \sum\limits_{n=1}^{+\infty}a_nx^n$ satisfies the differential equation $x(x - 4)f'(x) + (x + 2)f(x) = -x$.
b) Prove that $\sum\limits_{n=1}^{+\infty}\frac{1}{\binom{2n}{n}} = \frac{1}{3} + \frac{2\pi\sqrt{3}}{27}$.
2009 Iran Team Selection Test, 6
We have a closed path on a vertices of a $ n$×$ n$ square which pass from each vertice exactly once . prove that we have two adjacent vertices such that if we cut the path from these points then length of each pieces is not less than quarter of total path .
2006 Federal Competition For Advanced Students, Part 1, 4
Given is the function $ f\equal{} \lfloor x^2 \rfloor \plus{} \{ x \}$ for all positive reals $ x$. ( $ \lfloor x \rfloor$ denotes the largest integer less than or equal $ x$ and $ \{ x \} \equal{} x \minus{} \lfloor x \rfloor$.)
Show that there exists an arithmetic sequence of different positive rational numbers, which all have the denominator $ 3$, if they are a reduced fraction, and don’t lie in the range of the function $ f$.