There are $100$ positive numbers $a_1$, $a_2$, $...$, $a_{100}$ such that $$\frac{1}{a_1+1}+\frac{1}{a_2+1}+...+\frac{1}{a_{100}+1} \le 1.$$ Prove that $$a_1 \cdot a_2\cdot ... \cdot a_{100} \ge 99^{100}.$$

If $ a,b,c $ represent the lengths of the sides of a triangle, prove the inequality: $$ 3\le\sum_{\text{cyc}}\sqrt{\frac{a}{-a+b+c}} . $$

Prove the following statement: If a polynomial $f(x)$ with real coefficients takes only nonnegative values, then there exists a positive integer $n$ and polynomials $g_1(x), g_2(x),\cdots, g_n(x)$ such that \[f(x) = g_1(x)^2 + g_2(x)^2 +\cdots+ g_n(x)^2\]

Prove that if $n$ is large enough, among any $n$ points of plane we can find $1000$ points such that these $1000$ points have pairwise distinct distances. Can you prove the assertion for $n^{\alpha}$ where $\alpha$ is a positive real number instead of $1000$?

Let $a=\lfloor \sqrt{n} \rfloor$ for given positive integer $n.$ Express the summation $\sum_{k=1}^{n}\lfloor \sqrt{k} \rfloor$ in terms of $n$ and $a.$

Zeros of a fourth-degree polynomial $f (x)$ form an arithmetic progression. Prove that the zeros of $f '(x)$ also form an arithmetic progression.

Find all possible real values of $a$ for which the system of equations \[\{\begin{array}{cc}x +y +z=0\\\text{ } \\ xy+yz+azx=0\end{array}\] has exactly one solution.

Let $a,b,c$ be positive real numbers. Prove that $\frac{a^3+3b^3}{5a+b}+\frac{b^3+3c^3}{5b+c}+\frac{c^3+3a^3}{5c+a} \geq \frac{2}{3}(a^2+b^2+c^2)$.

Let $a$ be a positive real number. Collinear points $Z_1, Z_2, Z_3, Z_4$ (in that order) are plotted on the $(x, y)$ Cartesian plane. Suppose that the graph of the equation \[ x^2 + (y+a)^2 + x^2 + (y-a)^2 = 4a^2 + \sqrt{(x^2 + (y+a)^2)(x^2 + (y-a)^2)} \] passes through points $Z_1$ and $Z_4$, and the graph of the equation \[ x^2 + (y+a)^2 + x^2 + (y-a)^2 = 4a^2 - \sqrt{(x^2 + (y+a)^2)(x^2 + (y-a)^2)} \] passes through points $Z_2$ and $Z_3$. If $Z_1Z_2 = 5$, $Z_2Z_3 = 1$, and $Z_3Z_4 = 3$, then $a^2$ can be written as $\frac{m + n\sqrt{p}}{q}$, where $m$, $n$, $p$, and $q$ are positive integers, $m$, $n$, and $q$ are relatively prime, and $p$ is squarefree. Find $m + n + p + q$.

Prove that if $x$, $y$, and $z$ are non-negative numbers and $x^2+y^2+z^2=1$, then the following inequality is true: $\frac{x}{1-x^2}+\frac{y}{1-y^2}+\frac{z}{1-z^2 }\geq \frac{3\sqrt{3}}{2}$

Consider the sequence of Fibonacci numbers $F_0, F_1, F_2, ... $, given by $F_0 = F_1= 1$ and $F_{n+1} =F_n + F_{n-1}$ for $n\ge 1$. Define the sequence $x_0, x_1, x_2, ....$ by $x_0 = 1$ and $x_{k+1} = x^2_k + F^2_{2^k}$ for $k \ge 0$. Define the sequence $y_0, y_1, y_2, ...$ by $y_0 = 1$ and $y_{k+1} = 2x_ky_k - y^2_k$ for $k \ge 0$. If $$\sum^{\infty}_{k=0} \frac{1}{y_k}= \frac{a -\sqrt{b}}{c}$$ for positive integers a$, b, c$ with $gcd (a, c) = 1$, find $a + b + c$.

Are there integers $m, n \geq 2$ such that the following property is always true? $$``\text{For any real numbers } x, y, \text{ if } x^m + y^m \text{ and } x^n + y^n \text{ are integers, then } x + y \text{ is an integer}".$$

Let $(a_n)_{n\geq 1}$ be a sequence for real numbers given by $a_1=1/2$ and for each positive integer $n$ \[ a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}. \] Prove that for every positive integer $n$ we have $a_1+a_2+\cdots + a_n<1$.

Let \(c\) be a real number. For all \(x\) and \(y\) real numbers we have, \[f(x-f(y))=f(x-y)+c(f(x)-f(y))\] and \(f(x)\) is not constant. \(a)\) Find all possible values of \(c\). \(b)\) Can \(f\) be periodic?

Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$ [i]Proposed by Carl Schildkraut[/i]

Find all $g:\mathbb{R}\to\mathbb{R}$ so that there exists a unique $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(0)=g(0)$ and \[f(x+g(y))+f(-x-g(-y))=g(x+f(y))+g(-x-f(-y))\] for all $x,y\in\mathbb{R}$. [i] Proposed by usjl[/i]

An infinite sequence of integers $a_1,a_2,a_3, ...$ is given with $a_1 = 0$ and further holds for every natural number $n$ that $a_{n+1} = a_n - n$ if $a_n \ge n$ and $a_{n+1} = a_n + n$ if $a_n < n$ . (a) Prove that there are infinitely many numbers in the sequence equal to $0$. (b) Express in terms of $k$ the ordinal number of the $k^e$ number from the sequence, which is equal to $0$.

Show, for all positive integers $n = 1,2 , \dots ,$ that $14$ divides $ 3 ^ { 4 n + 2 } + 5 ^ { 2 n + 1 }$.

Solve the system $\begin{cases} 3xyz -x^3 - y^3-z^3 = b^3 \\ x + y+ z = 2b \\ x^2 + y^2-z^2 = b^2 \end{cases}$ in $C$

[b]p1.[/b] Suppose that $x$ and $y$ are positive real numbers such that $\log_2 x = \log_x y = \log_y 256$. Find $xy$. [b]p2.[/b] Let the roots of $x^2 + 7x + 11$ be $r$ and $s$. If f(x) is the monic polynomial with roots $rs + r + s$ and $r^2 + s^2$, what is $f(3)$? [b]p3.[/b] Call a positive three digit integer $\overline{ABC}$ fancy if $\overline{ABC} = (\overline{AB})^2 - 11 \cdot \overline{C}$. Find the sum of all fancy integers. [b]p4.[/b] In triangle $ABC$, points $D$ and $E$ are on line segments $BC$ and $AC$, respectively, such that $AD$ and $BE$ intersect at $H$. Suppose that $AC = 12$, $BC = 30$, and $EC = 6$. Triangle $BEC$ has area $45$ and triangle $ADC$ has area $72$, and lines $CH$ and $AB$ meet at $F$. If $BF^2$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ with $c$ squarefree and $gcd(a, b, d) = 1$, then find $a + b + c + d$. [b]p5.[/b] Find the minimum possible integer $y$ such that $y > 100$ and there exists a positive integer $x$ such that $x^2 + 18x + y$ is a perfect fourth power. [b]p6.[/b] Let $ABCD$ be a quadrilateral such that $AB = 2$, $CD = 4$, $BC = AD$, and $\angle ADC + \angle BCD = 120^o$. If the sum of the maximum and minimum possible areas of quadrilateral $ABCD$ can be expressed as $a\sqrt{b}$ for positive integers $a, b$ with $b$ squarefree, then find $a + b$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $N = \{1, 2, \ldots, n\}$, $n \geq 3$. To each pair $i \neq j $ of elements of $N$ there is assigned a number $f_{ij} \in \{0, 1\}$ such that $f_{ij} + f_{ji} = 1$. Let $r(i)=\sum_{i \neq j} f_{ij}$, and write $M = \max_{i\in N} r(i)$, $m = \min_{i\in N} r(i)$. Prove that for any $w \in N$ with $r(w) = m$ there exist $u, v \in N$ such that $r(u) = M$ and $f_{uv}f_{vw} = 1$.

If $a$ and $b$ are the roots of $x^2 - 2x + 5$, what is $|a^8 + b^8|$?

Two positive real numbers $\alpha, \beta$ satisfies that for any positive integers $k_1,k_2$, it holds that $\lfloor k_1 \alpha \rfloor \neq \lfloor k_2 \beta \rfloor$, where $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$. Prove that there exist positive integers $m_1,m_2$ such that $\frac{m_1}{\alpha}+\frac{m_2}{\beta}=1$.

With the conditions $a,b,c\in\mathbb{R^+}$ and $a+b+c=1$, prove that \[\frac{7+2b}{1+a}+\frac{7+2c}{1+b}+\frac{7+2a}{1+c}\geq\frac{69}{4}\]

We say that a positive integer $n$ is [i]memorable[/i] if it has a binary representation with strictly more $1$'s than $0$'s (for example $25$ is memorable because $25=(11001)_{2}$ has more $1$'s than $0$'s). Are there infinitely many memorable perfect squares? [i]Proposed by Nikola Velov[/i]