Found problems: 4776
Let $\mathbb{R^+}$ denote the set of all positive real numbers. Find all functions $f: \mathbb{R^+}\rightarrow \mathbb{R^+}$ such that
$$xf(x + y) + f(xf(y) + 1) = f(xf(x))$$
for all $x, y \in\mathbb{R^+}.$
[i]Proposed by Amadej Kristjan Kocbek, Jakob Jurij Snoj[/i]
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function, derivable on $R\backslash\{x_0\}$, having finite side derivatives in $x_0$. Show that there exists a derivable function $g:\mathbb{R}\rightarrow\mathbb{R}$, a linear function $h:\mathbb{R}\rightarrow\mathbb{R}$ and $\alpha\in\{-1,0,1\}$ such that:
\[ f(x)=g(x)+\alpha |h(x)|,\ \forall x\in\mathbb{R} \]
An arithmetic function is a real-valued function whose domain is the set of positive integers. Define the convolution product of two arithmetic functions $ f$ and $ g$ to be the arithmetic function $ f * g$, where \[ (f * g)(n) \equal{} \sum_{ij\equal{}n} f(i) \cdot g(j),\] and $ f^{*k} \equal{} f * f * \ldots * f$ ($ k$ times) We say that two arithmetic functions $ f$ and $ g$ are dependent if there exists a nontrivial polynomial of two variables $ P(x, y) \equal{} \sum_{i,j} a_{ij} x^i y^j$ with real coefficients such that
\[ P(f,g) \equal{} \sum_{i,j} a_{ij} f^{*i} * g^{*j} \equal{} 0,\]
and say that they are independent if they are not dependent. Let $ p$ and $ q$ be two distinct primes and set
\[ f_1(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} p, \\
0 & \text{ otherwise}. \end{cases}\]
\[ f_2(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} q, \\
0 & \text{ otherwise}. \end{cases}\]
Prove that $ f_1$ and $ f_2$ are independent.
Given the function $f(x) = x^2$, the sector of $f$ from $a$ to $b$ is defined as the limited region between the
graph of $y = f(x)$ and the straight line segment that joins the points $(a, f(a))$ and $(b, f(b))$. Define the
increasing sequence $x_0$, $x_1, \cdots$ with $x_0 = 0$ and $x_1 = 1$, such that the area of the sector of $f$ from $x_n$ to $x_{n+1}$ is constant for $n \geq 0$. Determine the value of $x_n$ in function of $n$.
The following functions are written on the board, $$F(x) = x^2 + \frac{12}{x^2}, G(x) = \sin(\pi x^2), H(x) = 1.$$ If functions $f,g$ are currently on the board, we may write on the board the functions $$f(x) + g(x), f(x) - g(x), f(x)g(x), cf(x)$$ (the last for any real number $c$). Can a function $h(x)$ appear on the board such that $$|h(x) - x| < \frac{1}{3}$$ for all $x \in [1,10]$ ?
For each integer $\, n \geq 2, \,$ determine, with proof, which of the two positive real numbers $\, a \,$ and $\, b \,$ satisfying \[ a^n = a + 1, \hspace{.3in} b^{2n} = b + 3a \] is larger.
Consider all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying $f(1-f(x))=x$ for all $x \in \mathbb{R}$.
a) By giving a concrete example, show that such a function exists.
b) For each such function define the sum
\[S_f=f(-2017)+f(-2016)+\dots+f(-1)+f(0)+f(1)+\dots+f(2017)+f(2018).\]
Determine all possible values of $S_f$.
Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.
Let $f:[0,1]\rightarrow \mathbb{R}$ a differentiable function such that $f(0)=f(1)=0$ and $|f'(x)|\le 1,\ \forall x\in [0,1]$. Prove that:
\[\left|\int_0 ^1f(t)dt\right|<\frac{1}{4}\]
Determine all functions $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ satisfying
$f(x+y+f(y))=4030x-f(x)+f(2016y), \forall x,y \in \mathbb{R}^+$.
Let $ S$ be the set of nonnegative real numbers. Find all functions $ f: S\rightarrow S$ which satisfy $ f(x\plus{}y\minus{}z)\plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz})\equal{}f(x\plus{}y\plus{}z)$ for all nonnegative $ x,y,z$ with $ x\plus{}y\ge z$.
Let $n\ge 4$ be a positive integer and let $M$ be a set of $n$ points in the plane, where no three points are collinear and not all of the $n$ points being concyclic. Find all real functions $f:M\to\mathbb{R}$ such that for any circle $\mathcal{C}$ containing at least three points from $M$, the following equality holds:
\[\sum_{P\in\mathcal{C}\cap M} f(P)=0\]
[i]Dorel Mihet[/i]
Suppose that a cubic function with respect to $x$, $f(x)=ax^3+bx^2+cx+d$ satisfies all of 3 conditions:
\[f(1)=1,\ f(-1)=-1,\ \int_{-1}^1 (bx^2+cx+d)\ dx=1\].
Find $f(x)$ for which $I=\int_{-1}^{\frac 12} \{f''(x)\}^2\ dx$ is minimized, the find the minimum value.
[i]2011 Tokyo University entrance exam/Humanities, Problem 1[/i]
Prove the following inequality:
$x_1 + 2x_2 + 3x_3 + ... + nx_n \leq \frac{n(n-1)}{2} + x_1 + x_2 ^2 + x_3 ^3 + ... + x_n ^n$
where $\forall _{x_i} x_i > 0$
If $ A$ and $ B$ are fixed points on a given circle and $ XY$ is a variable diameter of the same circle, determine the locus of the point of intersection of lines $ AX$ and $ BY$. You may assume that $ AB$ is not a diameter.
Assume that the system of differential equations $y'=-z^3$, $z'=y^3$ with the initial conditions $y(0)=1$, $z(0)=0$ has a unique solution $y=f(x)$, $z=g(x)$ defined for real $x$. Prove that there exists a positive constant $L$ such that for all real $x$,
$$f(x+L)=f(x),\enspace g(x+L)=g(x).$$
The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations
\[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.
Triangle $ABC$ and a function $f:\mathbb{R}^+\to\mathbb{R}$ have the following property: for every line segment $DE$ from the interior of the triangle with midpoint $M$, the inequality $f(d(D))+f(d(E))\le 2f(d(M))$, where $d(X)$ is the distance from point $X$ to the nearest side of the triangle ($X$ is in the interior of $\triangle ABC$). Prove that for each line segment $PQ$ and each point interior point $N$ the inequality $|QN|f(d(P))+|PN|f(d(Q))\le |PQ|f(d(N))$ holds.
Find the strictly increasing functions $f : \{1,2,\ldots,10\} \to \{ 1,2,\ldots,100 \}$ such that $x+y$ divides $x f(x) + y f(y)$ for all $x,y \in \{ 1,2,\ldots,10 \}$.
[i]Cristinel Mortici[/i]
Consider the function $f : N \to N$ given by
(i) $f(1) = 1$,
(ii) $f(2n) = f(n)$ for any $n \in N$,
(iii) $f(2n+1) = f(2n)+1$ for any $n \in N$.
(a) Find the maximum value of $f(n)$ for $1 \le n \le 1995$;
(b) Find all values of $f$ on this interval.
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)
Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial
\[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n.
\]
For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where
\[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}. \]
Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$?
[i]Proposed by David Yang[/i]
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 $f \equiv g$.