Found problems: 4776
Given non-constant linear functions $p_1(x), p_2(x), \dots p_n(x)$. Prove that at least $n-2$ of polynomials $p_1p_2\dots p_{n-1}+p_n, p_1p_2\dots p_{n-2} p_n + p_{n-1},\dots p_2p_3\dots p_n+p_1$ have a real root.
The function $f$, with domain on the set of non-negative integers, is defined by the following :
$\bullet$ $f (0) = 2$
$\bullet$ $(f (n + 1) -1)^2 + (f (n)-1) ^2 = 2f (n) f (n + 1) + 4$, taking $f (n)$ the largest possible value.
Determine $f (n)$.
Let be a differentiable function $ f:[0,1]\longrightarrow\mathbb{R} $ whose derivative has a positive Lipschitz constant $ L. $ Show that
[b]a)[/b] $ x,y\in [0,1]\implies | f(x)-f(y)-f'(y)(x-y) |\le L\cdot (x-y)^2 $
[b]b)[/b] $ \lim_{n\to\infty } \left( n\int_0^1 f(x)dx-\sum_{i=1}^nf\left( \frac{2i-1}{2n} \right) \right) =0. $
Determine the highest power of $1980$ which divides \[\frac{(1980n)!}{(n!)^{1980}}.\]
Let be an odd natural number $ n\ge 3. $ Find all continuous functions $ f:[0,1]\longrightarrow\mathbb{R} $ that satisfy the following equalities.
$$ \int_0^1 \left( f\left(\sqrt[k]{x}\right) \right)^{n-k} dx=k/n,\quad\forall k\in\{ 1,2,\ldots ,n-1\} $$
[i]Titu Andreescu[/i]
For an arbitrary real number $ x$, $ \lfloor x\rfloor$ denotes the greatest integer not exceeding $ x$. Prove that there is exactly one integer $ m$ which satisfy $ \displaystyle m\minus{}\left\lfloor \frac{m}{2005}\right\rfloor\equal{}2005$.
Let $f: \mathbb R \to\mathbb R$ be a function such that $|f(x)| \leq 1$ for all $x \in \mathbb R$ and
\[f \left( x + \frac{13}{42} \right) + f(x) = f \left( x + \frac 17 \right) + f \left( x + \frac 16 \right), \quad \forall x \in \mathbb R.\]
Show that $f$ is a periodic function.
Let $ y$ be the function of $ x$ satisfying the differential equation $ y'' \minus{} y \equal{} 2\sin x$.
(1) Let $ y \equal{} e^xu \minus{} \sin x$, find the differential equation with which the function $ u$ with respect to $ x$ satisfies.
(2) If $ y(0) \equal{} 3,\ y'(0) \equal{} 0$, then determine $ y$.
Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy \[f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1\] for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$.
[i]Proposed by Bojan Bašić, Serbia[/i]
For a non-constant function $f:\mathbb{R}\to \mathbb{R}$ prove that there exist real numbers $x,y$ satisfying $f(x+y)<f(xy)$
Let $a,b$ be two integers and suppose that $n$ is a positive integer for which the set $\mathbb{Z} \backslash \{ax^n + by^n \mid x,y \in \mathbb{Z}\}$ is finite. Prove that $n=1$.
$x_{1}, x_{2}, \cdots, x_{n}$ are positive numbers such that $\sum_{i=1}^{n}x_{i}= 1$. Prove that \[\left( \sum_{i=1}^{n}\sqrt{x_{i}}\right) \left( \sum_{i=1}^{n}\frac{1}{\sqrt{1+x_{i}}}\right) \leq \frac{n^{2}}{\sqrt{n+1}}\]
Let be a natural number $ n, $ denote with $ C $ the square in the complex plane whose vertices are the affixes of $ 2n\pi\left( \pm 1\pm i \right) , $ and consider the set
$$ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} $$
Prove the following implications.
[b]a)[/b] $ \exists \alpha\in\mathbb{R}_{>0}\quad \forall z\in\partial C\quad \left| \cos z \right|\ge\alpha e^{|\text{Im}(z)|} $
[b]b)[/b] $ \forall f\in\Lambda\quad\frac{1}{2\pi i}\int_{\partial C} \frac{f(z)}{z^2\cos z} dz=f'(0)+\frac{4}{\pi^2}\sum_{p=-2n}^{2n-1} \frac{(-1)^{p+1} f(z-p)}{(1+2p)^2} $
[b]c)[/b] $ \forall f\in\Lambda\quad \sum_{p\in\mathbb{Z}}\frac{(-1)^pf\left( \frac{(1+2p)\pi}{2} \right)}{(1+2p)^2} =\frac{\pi^2 f'(0)}{4} $
Let $n$ be an integer greater than $1.$ $n$ pupils are seated around a round table, each having a certain number of candies (it is possible that some pupils don't have a candy) such that the sum of all the candies they possess is a multiple of $n.$ They exchange their candies as follows: For each student's candies at first, there is at least a student who has more candies than the student sitting to his/her right side, in which case, the student on the right side is given a candy by that student. After a round of exchanging, if there is at least a student who has candies greater than the right side student, then he/she will give a candy to the next student sitting to his/her right side. Prove that after the exchange of candies is completed (ie, when it reaches equilibrium), all students have the same number of candies.
Is it possible to colour all positive real numbers by 10 colours so that every two numbers with decimal representations differing in one place only are of different colours? (We suppose that there is no place in a decimal representations such that all digits starting from that place are 9's.)
[i]Proposed by A. Golovanov[/i]
For which $ n\in \mathbb{N}$ do there exist rational numbers $ a,b$ which are not integers such that both $ a \plus{} b$ and $ a^n \plus{} b^n$ are integers?
Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that
\[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \]
Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.
Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant.
[i]Victor Wang.[/i]
Find all positive real numbers \((a,b,c) \leq 1\) which satisfy
\[ \huge \min \Bigg\{ \sqrt{\frac{ab+1}{abc}}\, \sqrt{\frac{bc+1}{abc}}, \sqrt{\frac{ac+1}{abc}} \Bigg \} = \sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\]
A continuous function $ f: \mathbb {R} \to \mathbb {R} $ satisfies $ f (x) f (f (x)) = 1 $ for every real $ x $ and $ f (2020) = 2019 $ . What is the value of $ f (2018) $?
Let $ a\in \mathbb{R} $ and $ f_1(x),f_2(x),\ldots,f_n(x): \mathbb{R} \rightarrow \mathbb{R} $ are the additive functions such that for every $ x\in \mathbb{R} $ we have $ f_1(x)f_2(x) \cdots f_n(x) =ax^n $. Show that there exists $ b\in \mathbb {R} $ and $ i\in {\{1,2,\ldots,n}\} $ such that for every $ x\in \mathbb{R} $ we have $ f_i(x)=bx $.
$x_{1}, x_{2}, \cdots, x_{n}$ are positive numbers such that $\sum_{i=1}^{n}x_{i}= 1$. Prove that \[\left( \sum_{i=1}^{n}\sqrt{x_{i}}\right) \left( \sum_{i=1}^{n}\frac{1}{\sqrt{1+x_{i}}}\right) \leq \frac{n^{2}}{\sqrt{n+1}}\]
Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition :
1) $f(-1)\geq f(1).$
2) $x+f(x)$ is non decreasing function.
3) $\int_{-1}^ 1 f(x)\ dx=0.$
Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$
Calculate the following indefinite integrals.
[1] $\int x(x^2+3)^2 dx$
[2] $\int \ln (x+2) dx$
[3] $\int x\cos x dx$
[4] $\int \frac{dx}{(x+2)^2}dx$
[5] $\int \frac{x-1}{x^2-2x+3}dx$
The function $f: R \longrightarrow R$ is continuous. Prove that if for every real number $x$, there exists a positive integer $n$, such that
$\underbrace{(f \circ f \circ ... \circ f)}_{n}(x) = 1$,
then $f(1) = 1$.