Found problems: 4776
Let $f$ be a function $f: \mathbb{N} \cup \{0\} \mapsto \mathbb{N},$ and satisfies the following conditions:
(1) $f(0) = 0, f(1) = 1,$
(2) $f(n+2) = 23 \cdot f(n+1) + f(n), n = 0,1, \ldots.$
Prove that for any $m \in \mathbb{N}$, there exist a $d \in \mathbb{N}$ such that $m | f(f(n)) \Leftrightarrow d | n.$
The game of $pebbles$ is played on an infinite board of lattice points $(i,j)$. Initially there is a $pebble$ at $(0,0)$. A move consists of removing a $pebble$ from point $(i,j)$and placing a $pebble$ at each of the points $(i+1,j)$ and $(i,j+1)$ provided both are vacant. Show taht at any stage of the game there is a $pebble$ at some lattice point $(a,b)$ with $0 \leq a+b \leq 3$
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 $5^{m}$.
Consider the function $f(x)=5x^4-12x^3+30x^2-12x+5$. Let $f(x_1)=p$, wher $x_1$ and $p$ are non-negative integers, and $p$ is prime. Find with proof the largest possible value of $p$.
[i]Proposed by tkhalid[/i]
Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$. A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called [i]nice[/i] if there exists an index $k$ such that \[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\] Prove that the number of nice functions is at least $n^n$.
Consider a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admits bounded primitives. Prove that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as
$$ f(x)=\left\{ \begin{matrix} x, & \quad x\le 0 \\ f(1/x)\cdot\ln x ,& \quad x>0 \end{matrix}\right. $$
admits primitives.
[i]Florian Dumitrel[/i]
Find all functions $\displaystyle f : \mathbb N^\ast \to \mathbb N^\ast$ ($\displaystyle N^\ast = \{ 1,2,3,\ldots \}$) with the property that, for all $\displaystyle n \geq 1$, \[ f(1) + f(2) + \ldots + f(n) \] is a perfect cube $\leq n^3$.
[i]Dinu Teodorescu[/i]
Two functions are defined by equations: $f(a) = a^2 + 3a + 2$ and $g(b, c) = b^2 - b + 3c^2 + 3c$. Prove that for any positive integer $a$ there exist positive integers $b$ and $c$ such that $f(a) = g(b, c)$.
Suppose that $f$ is a function such that $3f(x)- 5xf \left(\frac{1}{x}\right)= x - 7$ for all non-zero real numbers $x.$ Find $f(2010).$
In a triangle $ABC$, let $a,b,c$ be its sides and $m,n,p$ be the corresponding medians. For every $\alpha>0$, let $\lambda(\alpha)$ be the real number such that
$$a^\alpha+b^\alpha+c^\alpha=\lambda(\alpha)^\alpha\left(m^\alpha+n^\alpha+p^\alpha\right)^\alpha.$$
(a) Compute $\lambda(2)$.
(b) Find the limit of $\lambda(\alpha)$ as $\alpha$ approaches $0$.
(c) For which triangles $ABC$ is $\lambda(\alpha)$ independent of $\alpha$?
Let $\{a_0,a_1,\ldots\}$ and $\{b_0,b_1,\ldots\}$ be two infinite sequences of integers such that
\[(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0\]
for all integers $n\geq 2$. Prove that there exists a positive integer $k$ such that
\[a_{k+2011}=a_{k+2011^{2011}}.\]
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function with the property that $ (f\circ f) (x) =[x], $ for any real number $ x. $ Show that there exist two distinct real numbers $ a,b $ so that $ |f(a)-f(b)|\ge |a-b|. $
$ [] $ denotes the integer part.
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 $f(a)+ f(b) + f(c) = 0$.
Let $f:[0,1] \to [0,1]$ a increasing continuous function, diferentiable in $(0,1)$ and with derivative smaller than 1 in every point. The sequence of sets $A_1,A_2,A_3,\dots$ is define as: $A_1 = f([0,1])$, and for $n \geq 2, A_n = f(A_{n-1}).$ Prove that $\displaystyle \lim_{n\to+\infty} d(A_n) = 0$, where $d(A)$ is the diameter of the set $A$.
Note: The diameter of a set $X$ is define as $d(X) = \sup_{x,y\in X} |x-y|.$
Let $ g,f: \mathbb C\longrightarrow\mathbb C$ be two continuous functions such that for each $ z\neq 0$, $ g(z)\equal{}f(\frac1z)$. Prove that there is a $ z\in\mathbb C$ such that $ f(\frac1z)\equal{}f(\minus{}\bar z)$
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 $\sum\limits_{n=1}^{+\infty}\frac{1}{\binom{2n}{n}} = \frac{1}{3} + \frac{2\pi\sqrt{3}}{27}$.
Prove that there are infinitely many positive integers $n$ such that $2^{2^n+1}+1$ is divisible by $n$ but $2^n+1$ is not.
[i](Russia) Valery Senderov[/i]
A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Let $n \geq 3$ be an integer and let $a,b\in\mathbb{R}$ such that $nb\geq a^2$. We consider the set \[ X = \left\{ (x_1,x_2,\ldots,x_n)\in\mathbb{R}^n \mid \sum_{k=1}^n x_k = a, \ \sum_{k=1}^n x_k^2 = b \right\} . \] Find the image of the function $M: X\to \mathbb{R}$ given by \[ M(x_1,x_2,\ldots,x_n) = \max_{1\leq k\leq n} x_k . \]
[i]Dan Schwarz[/i]
Let $a$ and $b$ be real numbers satisfying $a>b>0$. Evaluate \[\int_0^{2\pi}\dfrac{1}{a+b\cos(\theta)}d\theta.\] Express your answer in terms of $a$ and $b$.
Let $ f(x)$ be a nonnegative, integrable function on $ (0,2\pi)$ whose Fourier series is $ f(x)\equal{}a_0\plus{}\sum_{k\equal{}1}^{\infty} a_k \cos (n_k x)$, where none of the positive integers $ n_k$ divides another. Prove that $ |a_k| \leq a_0$.
[i]G. Halasz[/i]
Let be a natural number $ n, $ two $ \text{n-tuplets} $ of real numbers $ a:=\left( a_1,a_2,\ldots, a_n \right) , b:=\left( b_1,b_2,\ldots, b_n \right) , $ and the function $ f:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=\sum_{i=1}^na_i\cos \left( b_ix \right) $. Prove that if the numbers of $ b $ are all positive and pairwise distinct,
[b]a)[/b] then, $ f\ge 0 $ implies that the numbers of $ a $ are all equal.
[b]b)[/b] if the numbers of $ a $ are all nonzero and $ f $ is periodic, then the ratio of any two numbers of $ b $ is rational.
[i]Marin Tolosi[/i]
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $|f(x)-f(y)| \leq |x-y|$, for all $x,y \in \mathbb{R}$.
Prove that if for any real $x$, the sequence $x,f(x),f(f(x)),\ldots$ is an arithmetic progression, then there is $a \in \mathbb{R}$ such that $f(x)=x+a$, for all $x \in \mathbb R$.
Let us use the word $ N$-measure for nonnegative, finitely additive set functions defined on all subsets of the positive integers, equal to $ 0$ on finite sets, and equal to $ 1$ on the whole set. We say that the system $ \Upsilon$ of sets determines the $ N$-measure $ \mu$ if any $ N$-measure coinciding with $ \mu$ on all elements of $ \Upsilon$ is necessarily identical with $ \mu$.
Prove the existence of an $ N$-measure $ \mu$ that cannot be determined by a system of cardinality less than continuum.
[i]I. Juhasz[/i]
Find all functions $f: \mathbb{R}\to\mathbb{R}$ that satidfy :
$$2f(x+y+xy)= a f(x)+ bf(y)+f(xy)$$ for any $x,y \in\mathbb{R}$ όπου $a,b\in\mathbb{R}$ with $a^2-a\ne b^2-b$