Found problems: 4776
A sequence $ a_1, a_2, a_3, \ldots$ is defined recursively by $ a_1 \equal{} 1$ and $ a_{2^k\plus{}j} \equal{} \minus{}a_j$ $ (j \equal{} 1, 2, \ldots, 2^k).$ Prove that this sequence is not periodic.
A function $f(\theta)$ satisfies the following conditions $(a),(b)$.
$(a)\ f(\theta)\geq 0$
$(b)\ \int_0^{\pi} f(\theta)\sin \theta d\theta =1$
Prove the following inequality.
\[\int_0^{\pi} f(\theta)\sin n\theta \ d\theta \leq n\ (n=1,2,\cdots)\]
Find all real solutions $x$ of the equation
$\cos\cos\cos\cos x=\sin\sin\sin\sin x$.
(Angles are measured in radians.)
Two circles are tangent to each other internally at a point $\ T $. Let the chord $\ AB $ of the larger circle be tangent to the smaller circle at a point $\ P $. Prove that the line $\ TP $ bisects $\ \angle ATB $.
find all functions from the nonegative integers into themselves, such that: $2f(m^2+n^2)=f^2(m)+f^2(n)$ and for $m\geq n$ $f(m^2)\geq f(n^2)$.
Let $k$ be a fixed positive integer. The $n$th derivative of $\tfrac{1}{x^k-1}$ has the form $\tfrac{P_n(x)}{(x^k-1)^{n+1}}$, where $P_n(x)$ is a polynomial. Find $P_n(1)$.
Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.
[b]a)[/b] Give an example of function $ f:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ that admits a primitive $ F:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ having the property that $ F^e $ is a primitive of $ f^e. $
[b]b)[/b] Prove that there is no derivable function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that has a primitive $ G:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ e^G $ is a primitive of $ e^g. $
Suppose that $f: \mathbb{R}\rightarrow\mathbb{R}$ fulfils $\left|\sum^n_{k=1}3^k\left(f(x+ky)-f(x-ky)\right)\right|\le1$ for all $n\in\mathbb{N},x,y\in\mathbb{R}$. Prove that $f$ is a constant function.
Find the differentiable function $f(x)$ with $f(0)\neq 0$ satisfying $f(x+y)=f(x)f'(y)+f'(x)f(y)$ for all real numbers $x,\ y$.
(a) Show that, if $I \subset R$ is a closed bounded interval, and $f : I \to R$ is a non-constant monic polynomial function such that $max_{x\in I}|f(x)|< 2$, then there exists a non-constant monic polynomial function $g : I \to R$ such that $max_{x\in I} |g(x)| < 1$.
(b) Show that there exists a closed bounded interval $I \subset R$ such that $max_{x\in I}|f(x)| \ge 2$ for every non-constant monic polynomial function $f : I \to R$.
There are $n$ people seated on a circular table that have seats numerated from 1 to $n$ clockwise. Let $k$ be a fix integer with $2 \leq k \leq n$. The people can change their seats. There are two types of moves permitted:
1. Each person moves to the next seat clockwise.
2. Only the ones in seats 1 and $k$ exchange their seats.
Determine, in function of $n$ and $k$, the number of possible configurations of people in the table that can be attain by using a sequence of permitted moves.
Prove that the function $ f:\mathbb{R}\longrightarrow\mathbb{R} , f(x)=\text{arcsin} \frac{2x}{1+x^2} $ admits primitives and describe a primitive of it.
Let be four functions $ f,g,s,i:\mathbb{N}\longrightarrow\mathbb{N} $ such that $ s(x)=\max (f(x),g(x)) $ and $ i(x)=\min (f(x),g(x)) , $ for any natural number $ x. $ Prove that $ f=g $ if $ s $ is surjective and $ i $ injective.
Let $f(x):\mathbb {Q} \rightarrow \mathbb {Q}$ be a function satisfying $f(x+2y)+f(2x-y)=5f(x)+5f(y)$
Find all such functions.
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function having the property that
$$ f(f(x))=f(x)-\frac{1}{4}x +1, $$
for all real numbers $ x. $
[b]a)[/b] Prove that $ f $ is increasing.
[b]b)[/b] Show that the equation $ f(x)=ax $ has at least a real solution in $ x, $ for any real number $ a\ge 1. $
[b]c)[/b] Calculate $ \lim_{x\to\infty } \frac{f(x)}{x} $ supposing that it exists, it's finite, and that $ \lim_{x\to\infty } f(f(x))=\infty . $
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that
$f(f(x)^2+f(y^2))=(x-y)f(x-f(y))$
Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that
\[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}.
\]
[i]Author: Juhan Aru, Estonia[/i]
Functions $f$ and $g$ are defined on the set of all integers in the interval $[-100; 100]$ and take integral values. Prove that for some integral $k$ the number of solutions of the equation $f(x)-g(y)=k$ is odd.\\ ( A. Golovanov)
Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that
\[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]
Let $M$ be an empty set of real numbers. For any $x \in M$ the functions $f: M\to M$ and $g: M\to M$ satisfy the relations $f (g (x)) = g (f (x)) = x$ and $f (x) + g (x) = x$. Show that $- x \in M$ ¸ and $f (-x) = -f (x)$ whatever $x \in M$.
How many positive integer multiples of 1001 can be expressed in the form $10^{j}-10^{i}$, where $i$ and $j$ are integers and $0\leq i < j \leq 99$?
A variable tangent $t$ to the circle $C_1$, of radius $r_1$, intersects the circle $C_2$, of radius $r_2$ in $A$ and $B$. The tangents to $C_2$ through $A$ and $B$ intersect in $P$.
Find, as a function of $r_1$ and $r_2$, the distance between the centers of $C_1$ and $C_2$ such that the locus of $P$ when $t$ varies is contained in an equilateral hyperbola.
[b]Note[/b]: A hyperbola is said to be [i]equilateral[/i] if its asymptotes are perpendicular.
Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that:
(i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and
(ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$,
$$\{ x,f(x),\cdots f^{p-1}(x) \} $$
is a complete residue system modulo $p$. With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$.
[i]Proposed by IndoMathXdZ[/i]