Compute the sum of roots of $(2 - x)^{2005} + x^{2005} = 0$.

For an integer $m\geq 4,$ let $T_{m}$ denote the number of sequences $a_{1},\dots,a_{m}$ such that the following conditions hold: (1) For all $i=1,2,\dots,m$ we have $a_{i}\in \{1,2,3,4\}$ (2) $a_{1} = a_{m} = 1$ and $a_{2}\neq 1$ (3) For all $i=3,4\cdots, m, a_{i}\neq a_{i-1}, a_{i}\neq a_{i-2}.$ Prove that there exists a geometric sequence of positive integers $\{g_{n}\}$ such that for $n\geq 4$ we have that \[ g_{n} - 2\sqrt{g_{n}} < T_{n} < g_{n} + 2\sqrt{g_{n}}.\]

A positive integer $d$ and a permutation of positive integers $a_1,a_2,a_3,\dots$ is given such that for all indices $i\geq 10^{100}$, $|a_{i+1}-a_{i}|\leq 2d$ holds. Prove that there exists infinity many indices $j$ such that $|a_j -j|< d$.

The sequence $(a_n)_{n>=1}$ satisfies that : $a_1=a_2=1$ $a_n=7a_{n-1}-a_{n-2}$ ($n>=3$) , prove that : for all positive integer n , number $a_n+2+a_{n+1}$ is a perfect square .

While waiting for their next class on Killian Court, Alesha and Belinda both write the same sequence $S$ on a piece of paper, where $S$ is a $2020$-term strictly increasing geometric sequence with an integer common ratio . Every second, Alesha erases the two smallest terms on her paper and replaces them with their geometric mean, while Belinda erases the two largest terms in her paper and replaces them with their geometric mean. They continue this process until Alesha is left with a single value $A$ and Belinda is left with a single value $B$. Let $r_0$ be the minimal value of $r$ such that $\frac{A}{B}$ is an integer. If $d$ is the number of positive factors of $r_0$, what is the closest integer to $\log_{2} d$?

A bottle contains $5$ gallons of a $10\%$ solution of oil. How many gallons of pure oil must be added to make a $30\%$ oil solution? (round to the nearest hundredth)

Find all the functions $f:\mathbb{N}\to\mathbb{R}$ that satisfy \[ f(x+y)=f(x)+f(y) \] for all $x,y\in\mathbb{N}$ satisfying $10^6-\frac{1}{10^6} < \frac{x}{y} < 10^6+\frac{1}{10^6}$. Note: $\mathbb{N}$ denotes the set of positive integers and $\mathbb{R}$ denotes the set of real numbers.

Given non-constant linear functions $p(x), q(x), r(x)$. Prove that at least one of three trinomials $pq+r, pr+q, qr+p$ has a real root. [b]proposed by S. Berlov[/b]

Let $a_0, a_1, \ldots , a_n$ and $b_0, b_1, \ldots , b_n$ be sequences of real numbers such that $a_0 = b_0 \geqslant 0$, $a_n = b_n > 0$ and \[a_i=\sqrt{\frac{a_{i+1}+a_{i-1}}{2}},\quad b_i=\sqrt{\frac{b_{i+1}+b_{i-1}}{2}},\]for all $i=1,\ldots,n-1$. Prove that $a_1 = b_1$.

Find a polynomial with integer coefficients which has $\sqrt2 + \sqrt3$ and $\sqrt2 + \sqrt[3]{3}$ as roots.

Does there exist an increasing sequence of positive integers for which any large enough integer can be expressed uniquely as the sum of two (possibly equal) terms of the sequence? [i]Proposed by Vlad Spătaru and David Anghel[/i]

a) Factorize $xy - x - y + 1$. b) Prove that if integers $a$ and $b$ satisfy $ |a + b| > |1 + ab|$, then $ab = 0$.

A sequence $(a_{n})$ is defined by $a_{1}= 1$ and $a_{n}= a_{1}a_{2}...a_{n-1}+1$ for $n \geq 2.$ Find the smallest real number $M$ such that $\sum_{n=1}^{m}\frac{1}{a_{n}}<M\; \forall m\in\mathbb{N}$.

Find all functions $f: \mathbb R^+ \rightarrow \mathbb R^+$ satisfying the following condition: for any three distinct real numbers $a,b,c$, a triangle can be formed with side lengths $a,b,c$, if and only if a triangle can be formed with side lengths $f(a),f(b),f(c)$.

Find four consecutive terms $a, b, c, d$ of an arithmetic progression and four consecutive terms $a_1, b_1, c_1, d_1$ of a geometric progression such that $$\begin{cases}a + a_1 = 27 \\\ b + b_1 = 27 \\ c + c_1 = 39 \\ d + d_1 = 87\end{cases}$$.

Find all real $\alpha$ s.t. \[ [ \sqrt{n + \alpha} + \sqrt{n} ] = [ \sqrt{4n+1} ] \] holds for all natural numbers $n$

Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.

(10.4) A set of numbers $a_0, a_1,..., a_n$ satisfies the conditions: $a_0 = 0$, $0 \le a_{k+1}- a_k \le 1$ for $k = 0, 1, .. , n -1$. Prove the inequality $$\sum_{k=1}^n a^3_k \le \left(\sum_{k=1}^n a_k \right)^2$$ (11.3) A set of numbers $a_0, a_1,..., a_n$ satisfies the conditions: $a_0 = 0$, $a_{k+1} \ge a_k + 1$ for $k = 0, 1, .. , n -1$. Prove the inequality $$\sum_{k=1}^n a^3_k \ge \left(\sum_{k=1}^n a_k \right)^2$$

Find all functions $f : \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that : $f(x)f(y)f(z)=9f(z+xyf(z))$, where $x$, $y$, $z$, are three positive real numbers.

Prove that if $a, b, c$ are positive numbers and $ab + bc + ca > a+ b + c$, then $a + b + c > 3$.

$n\geq 3$ is a integer. $\alpha,\beta,\gamma \in (0,1)$. For every $a_k,b_k,c_k\geq0(k=1,2,\dotsc,n)$ with $\sum\limits_{k=1}^n(k+\alpha)a_k\leq \alpha, \sum\limits_{k=1}^n(k+\beta)b_k\leq \beta, \sum\limits_{k=1}^n(k+\gamma)c_k\leq \gamma$, we always have $\sum\limits_{k=1}^n(k+\lambda)a_kb_kc_k\leq \lambda$. Find the minimum of $\lambda$

Show that when the product of three conscutive numbers we add arithmetic mean of them it is a perfect cube.

Let $x, y, z$ be nonzero real numbers with $$\frac{x + y}{z}=\frac{y + z}{x}=\frac{z + x}{y}.$$ Determine all possible values of $$\frac{(x + y)(y + z)(z + x)}{xyz}.$$ [i](Walther Janous)[/i]

Given positive reals $x,y,z$ satisfying $x+y+z=3$, prove that \[\sum_{cyc}\left( x^2+y^2+x^2y^2+\frac{y^2}{x^2}\right)\geq 4\sum_{cyc}\frac{y}{x}.\] [i]Proposed by chengbilly.[/i]

Fix a real polynomial $P$ with degree at least $1$, and a real number $c$. Prove that there exist a real number $k$ such that for all reals $a$ and $b$, $$P(a)+P(b)=c \quad \Rightarrow \quad |a+b|<k$$ [i]Proposed by Wong Jer Ren[/i]