Find all pairs of integers $x,y$ for which \[x^3+x^2+x=y^2+y.\]

When a company releases a new social media software, the marketing development of the company researches and analyses the characteristics of the customer group apart from paying attention to the active customer depending on the change of the time. We use $n(t, x)$ to express the customer density (which will be abbreviated as density). Here $t$ is the time and $x$ is the time of the customer spent on the social media software. In the instant time $t$, for $0<x_1<x_2$, the number of customers of spending time between $x_1$ and $x_2$ is $\int_{x_1}^{x_2}n(t,x)dx$. We assume the density $n(t,x)$ depends on the time and the following factors: Assumption 1. When the customer keeps using that social media software, their time spent on social media increases linearly. Assumption 2. During the time that the customer uses the social media software, they may stop using it. We assumption the speed of stopping using it $d(x)>0$ only depends on $x$. Assumption 3. There are two sources of new customer. (i) The promotion from the company: A function of time that expresses the increase of number of people in a time unit, expressed by $c(t)$. (ii) The promotion from previous customer: Previous customer actively promotes this social media software to their colleagues and friends actively. The speed of promoting sucessfully depends on $x$, denoted as $b(x)$. Assume if in an instant time, denoted as $t=0$, the density function is known and $n(0,x)=n_0(x)$. We can derive. The change of time $n(t,x)$ can satisfy the equation: $\begin{cases} \frac{\partial}{\partial t}n(t,x)+\frac{\partial}{\partial x}n(t,x)+d(x)n(t,x)=0, t\ge 0, x\ge 0 \\ N(t):=n(t,x=0)=c(t)+\int_{0}^{\infty}b(y)n(t,y)dy \end{cases}\,$ where $N(t)$ iis the speed of the increase of new customers. We assume $b, d \in L^\infty_-(0, \infty)$. $b(x)$ and $d(x)$ is bounded in essence. The following, we first make a simplified assumption: $c(t)\equiv 0$, i.e. the increase of new customer depends only on the promotion of previous customer. (a) According to assumption 1 and 2, formally derive the PDE that $n(t, x)$ satisfies in the two simtaneous equation above. You are required to show the assumption of model and the relationship between the Maths expression. Furthermore, according to assumption 3, explain the definition and meaning of $N(t)$ in the simtaneous equation above. (b) We want to research the relationship of the speed of the increase of the new customers $N(t)$ and the speed of promoting sucessfully $b(x)$. Derive an equation that $N(t)$ satisfies in terms of $N(t), n_0(x), b(x), d(x)$ only and does not include $n(t, x)$. Prove that $N(t)$ satifies the estimation $|N(t)|\le ||b||_\infty e^{||b||_\infty t}\int_{0}^{\infty}|n_0(x)|dx$, where $||\cdot||_\infty$ is the norm of $L^\infty$. (c) Finally, we want to research, after sufficiently long time, what trend of number density function $n(t, x) $\frac{d} has. As the total number of customers may keep increasing so it is not comfortable for us to research the number density function $n(t, x)$. We should try to find a density function which is renormalized. Hence, we first assume there is one only solution $(\lambda_0,\varphi(x))$ of the following eigenvalue problem: $\begin{cases} \varphi'(x)+(\lambda_0+d(x))\varphi(x)=0, x\ge 0 \\ \varphi(x)>0,\varphi(0)=\int_{0}^{\infty}b(x)\varphi(x)dx=1 \end{cases}\, $ and its dual problem has only solution $\psi(x)$: $\begin{cases} -\varphi'(x)+(\lambda_0+d(x))\psi(x)=\psi(0)b(x), x\ge 0 \\ \psi(x)>0,\int_{0}^{\infty}\psi(x)\varphi(x)dx=1 \end{cases}\,$ Prove that for any convex function $H:\mathbb{R}^+\to \mathbb{R}^+$ which satisfies $H(0)=0$. We have $\frac{d}{dx}\int_{0}^{\infty}\psi(x)\varphi(x)H(\frac{\tilde{n}(t,x)}{\varphi(x)})dx\le 0, \forall t\ge 0$. Furthermore, prove that $\int_{0}^{\infty}\psi(x)n(t,x)dx=e^{\lambda_0t}\int_{0}^{\infty}\psi(x)n_0(x)dx$ To simplify the proof, the contribution of boundary terms in $\infty$ is negligible.

Solve logarithmical equation $x^{\log _{3} {x-1}} + 2(x-1)^{\log _{3} {x}}=3x^2$

We call a polynomial $P(x)$ intresting if there are $1398$ distinct positive integers $n_1,...,n_{1398}$ such that $$P(x)=\sum_{}{x^{n_i}}+1$$ Does there exist infinitly many polynomials $P_1(x),P_2(x),...$ such that for each distinct $i,j$ the polynomial $P_i(x)P_j(x)$ is interesting.

Let the positive integers $x_1$, $x_2$, $...$, $x_{100}$ satisfy the equation \[\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+...+\frac{1}{\sqrt{x_{100}}}=20.\] Show that at least two of these integers are equal to each other.

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

Find the real roots of the equation $$(5-x)^4+ (x-2)^ 4 = 17$$ and the real roots of a more general equation $$(a - x) ^4+ (x - b)^4 = c$$

An integer $n$ will be called [i]good[/i] if we can write \[n=a_1+a_2+\cdots+a_k,\] where $a_1,a_2, \ldots, a_k$ are positive integers (not necessarily distinct) satisfying \[\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.\] Given the information that the integers 33 through 73 are good, prove that every integer $\ge 33$ is good.

Find the roots of the equation $(x-a)(x-b)=(x-c)(x-d)$, if you know that $a+d=b+c=2015$ and $a \ne c$ (numbers $a, b, c, d$ are not given).

Find all triplets of integers $(a, b, c)$, that verify the equation $$|a+3|+b^2+4\cdot c^2-14\cdot b-12\cdot c+55=0.$$

For positive integers $x$, $y$ and $z$ holds $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$. Prove that $xyz\geq 3600$

Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

Let $n$ be a positive integer. Determine all positive real numbers $x$ satisfying $nx^2 +\frac{2^2}{x + 1}+\frac{3^2}{x + 2}+...+\frac{(n + 1)^2}{x + n}= nx + \frac{n(n + 3)}{2}$

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?

Let $P$ be a cubic polynomial given by $P(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are integers and $a\ne0$. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.

In how many ways you can pay $2015\$$ using bills of $1\$$, $10\$$, $100\$$ and $200\$$

Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.

An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$, we have: $$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$$ [i]Proposed by John Berman, United States[/i]

An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$, we have: $$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$$ [i]Proposed by John Berman, United States[/i]

Find all triples $(x, y,z)$ such that $x, y, z \in (0,1)$ and $$\left(x+\frac{1}{2x}-1\right) \left(y+\frac{1}{2y}-1\right) \left(z+\frac{1}{2z}-1\right) = \left(1-\frac{xy}{z}\right)\left(1-\frac{yz}{x}\right)\left(1-\frac{zx}{y}\right)$$ (D. Bazylev)

Having three sets $ A,B\subset C, $ solve the set equation $ (X\cup (C\setminus A))\cap ((C\setminus X)\cup A)=B. $

Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.

Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that \[\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}\]

$(x_n)$ be a sequence with $x_1=0$, $$x_{n+1}=5x_n + \sqrt{24x_n^2+1}$$. Prove that for $k \geq 2$ $x_k$ is a natural number.

For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?