A sequence $\{a_n\}$ is defined by $a_n=\int_0^1 x^3(1-x)^n dx\ (n=1,2,3.\cdots)$ Find the constant number $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}$

Let $x_0(t)=1$, $x_{k+1}(t)=(1+t^{k+1})x_k(t)$ for all $k\geq 0$; $y_{n,0}(t)=1$, $y_{n,k}(t)=\frac{t^{n-k+1}-1}{t^k-1}y_{n,k-1}(t)$ for all $n\geq 0$, $1\leq k \leq n$. Prove that $\sum_{j=0}^{n-1}(-1)^j x_{n-j-1}(t)y_{n,j}(t)=\frac{1-(-1)^n}{2}$ for all $n\geq 1$.

Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.

The sequence ${a_0, a_1, a_2, ...}$ of real numbers satisfies the recursive relation $$n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n$$ for every positive integer $n$, where $a_0 = a_1 = 1$. Calculate the sum $$\frac{a_0}{a_1} + \frac{a_1}{a_2} + ... + \frac{a_{2008}}{a_{2009}}$$.

Given the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...,$ find $n$ such that the sum of the first $n$ terms is $2008$ or $2009$.

Consider a sequence of polynomials such that $P_0(x)=2,P_1(x)=x$ and for all $n\ge1$ \[P_{n+1}(x)+P_{n-1}(x)=xP_n(x).\] a) Determine the polynomial \[Q_n(x)=P^2_n(x)-xP_n(x)P_{n-1}(x)+P^2_{n-1}(x)\] for $n=1972.$ b) Express the polynomial \[\bigl(P_{n+1}(x)-P_{n-1}(x)\bigr)^2\] in terms of $P_n(x),Q_n(x).$

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ a function having the property that $$ \left| f(x+y)+\sin x+\sin y \right|\le 2, $$ for all real numbers $ x,y. $ [b]a)[/b] Prove that $ \left| f(x) \right|\le 1+\cos x, $ for all real numbers $ x. $ [b]b)[/b] Give an example of what $ f $ may be, if the interval $ \left( -\pi ,\pi \right) $ is included in its [url=https://en.wikipedia.org/wiki/Support_(mathematics)]support.[/url]

Find the sum of all integer values of $n$ such that the equation $\frac{x}{(yz)^2} + \frac{y}{(zx)^2} + \frac{z}{(xy)^2} = n$ has a solution in positive integers.

Let $S = \{2, 3, 4, \ldots\}$ denote the set of integers that are greater than or equal to $2$. Does there exist a function $f : S \to S$ such that \[f (a)f (b) = f (a^2 b^2 )\text{ for all }a, b \in S\text{ with }a \ne b?\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

Find the sum of all the real values of x satisfying $(x+\frac{1}{x}-17)^2$ $= x + \frac{1}{x} + 17.$

Let $n$ be a positive integer. Show that there exists a one-to-one function $\sigma : \{1,2,...,n\} \to \{1,2,...,n\}$ such that $$\sum_{k=1}^{n} \frac{k}{(k+\sigma(k))^2} < \frac{1}{2}.$$

A little boy takes a $ 12$ in long strip of paper and makes a Mobius strip out of it by tapping the ends together after adding a half twist. He then takes a $ 1$ inch long train model and runs it along the center of the strip at a speed of $ 12$ inches per minute. How long does it take the train model to make two full complete loops around the Mobius strip? A complete loop is one that results in the train returning to its starting point.

The sequence $a_1,a_2,\dots$ of integers satisfies the conditions: (i) $1\le a_j\le2015$ for all $j\ge1$, (ii) $k+a_k\neq \ell+a_\ell$ for all $1\le k<\ell$. Prove that there exist two positive integers $b$ and $N$ for which\[\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2\]for all integers $m$ and $n$ such that $n>m\ge N$. [i]Proposed by Ivan Guo and Ross Atkins, Australia[/i]

(1) Let $x>0,\ y$ be real numbers. For variable $t$, find the difference of Maximum and minimum value of the quadratic function $f(t)=xt^2+yt$ in $0\leq t\leq 1$. (2) Let $S$ be the domain of the points $(x,\ y)$ in the coordinate plane forming the following condition: For $x>0$ and all real numbers $t$ with $0\leq t\leq 1$ , there exists real number $z$ for which $0\leq xt^2+yt+z\leq 1$ . Sketch the outline of $S$. (3) Let $V$ be the domain of the points $(x,\ y,\ z) $ in the coordinate space forming the following condition: For $0\leq x\leq 1$ and for all real numbers $t$ with $0\leq t\leq 1$, $0\leq xt^2+yt+z\leq 1$ holds. Find the volume of $V$. [i]2011 Tokyo University entrance exam/Science, Problem 6[/i]

$3.$ Positive integers $a$ and $b$ are such that $a+b=\frac{a}{b}+\frac{b}{a}.$ What is the value of $a^2+b^2 ?$

We say that $1\leq a\leq101$ is a quadratic polynomial residue modulo $101$ with respect to a quadratic polynomial $f(x)$ with integer coefficients if there exists an integer $b$ such that $101 \mid a-f(b)$. For a quadratic polynomial $f$, we define its quadratic residue set as the set of quadratic residues modulo $101$ with respect to $f(x)$. Compute the number of quadratic residue sets. [i]Proposed by Michael Ren[/i]

Prove that there exists an infinity of irrational numbers $x,y$ such that the number $x+y=xy$ is a nonnegative integer.

$(USS 1)$ Prove that for a natural number $n > 2, (n!)! > n[(n - 1)!]^{n!}.$

Let $n\geq 2$ be some fixed positive integer and suppose that $a_1, a_2,\dots,a_n$ are positive real numbers satisfying $a_1+a_2+\cdots+a_n=2^n-1$. Find the minimum possible value of $$\frac{a_1}{1}+\frac{a_2}{1+a_1}+\frac{a_3}{1+a_1+a_2}+\cdots+\frac{a_n}{1+a_1+a_2+\cdots+a_{n-1}}$$

Let $f$ be a function defined on $(0, \infty )$ as follows: \[f(x)=x+\frac1x\] Let $h$ be a function defined for all $x \in (0,1)$ as \[h(x)=\frac{x^4}{(1-x)^6}\] Suppose that $g(x)=f(h(x))$ for all $x \in (0,1)$. (a) Show that $h$ is a strictly increasing function. (b) Show that there exists a real number $x_0 \in (0,1)$ such that $g$ is strictly decreasing in the interval $(0,x_0]$ and strictly increasing in the interval $[x_0,1)$.

Find all triples of reals $(a, b, c)$ such that $$a - \frac{1}{b}=b - \frac{1}{c}=c - \frac{1}{a}.$$

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\] [i]Proposed by Igor Voronovich, Belarus[/i]

Find the number of pairs of positive integers $a$ and $b$ such that $a\leq 100\,000$, $b\leq 100\,000$, and $$ \frac{a^3-b}{a^3+b}=\frac{b^2-a^2}{b^2+a^2}. $$

[b]a)[/b] Show that there exists exactly a sequence $ \left( x_n,y_n \right)_{n\ge 0} $ of pairs of nonnegative integers, that satisfy the property that $ \left( 1+\sqrt 33 \right)^n=x_n+y_n\sqrt 33, $ for all nonegative integers $ n. $ [b]b)[/b] Having in mind the sequence from [b]a),[/b] prove that, for any natural prime $ p, $ at least one of the numbers $ y_{p-1} ,y_p $ and $ y_{p+1} $ are divisible by $ p. $

Suppose the function $f:[1,+\infty)\to[1,+\infty)$ satisfies the following two conditions: (i) $f(f(x))=x^2$ for any $x\geq 1$; (ii) $f(x)\leq x^2+2021x$ for any $x\geq 1$. 1. Prove that $x<f(x)<x^2$ for any $x\geq 1$. 2. Prove that there exists a function $f$ satisfies the above two conditions and the following one: (iii) There are no real constants $c$ and $A$, such that $0<c<1$, and $\frac{f(x)}{x^2}<c$ for any $x>A$.