Prove that $$\frac{x}{\sqrt{1+x^2}}+\frac{y}{\sqrt{1+y^2}}+\frac{z}{\sqrt{1+z^2}} \leq\frac{3\sqrt{3}}{2}$$ for any positive real numbers $x, y,z$ such that $x+y+z = xyz.$ [url=https://artofproblemsolving.com/community/c7h236610p10925499]2008 VTRMC #1[/url] [url=http://www.math.vt.edu/people/plinnell/Vtregional/solutions.pdf]here[/url]

Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.

The sides of a triangle have lengths $a,b,c$. Prove that: \begin{align*}(-a+b+c)(a-b+c)\, +\, & (a-b+c)(a+b-c)+(a+b-c)(-a+b+c)\\ &\le\sqrt{abc}(\sqrt{a}+\sqrt{b}+\sqrt{c})\end{align*}

Let $x\geq 5, y\geq 6, z\geq 7$ such that $x^2+y^2+z^2\geq 125$. Find the minimum value of $x+y+z$.

Given positive reals $a,b,c,p,q$ satisfying $abc=1$ and $p \geq q$, prove that \[ p \left(a^2+b^2+c^2\right) + q\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geq (p+q) (a+b+c). \][i]Proposed by AJ Dennis[/i]

A function \[\text{f:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] is called contract if, for every numbers $x,y\in \text{(0,}\infty \text{)}$ we have, $\underset{n\to \infty }{\mathop{\lim }}\,\left( {{f}^{n}}\left( x \right)-{{f}^{n}}\left( y \right) \right)=0$ where ${{f}^{n}}=\underbrace{f\circ f\circ ...\circ f}_{n\ f\text{'s}}$ a) Consider \[f:\text{(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] a function contract, continue with the property that has a fixed point, that existing ${{x}_{0}}\in \text{(0,}\infty \text{) }$ there so that $f\left( {{x}_{0}} \right)={{x}_{0}}.$ Show that $f\left( x \right)>x,$ for every $x\in \text{(0,}{{x}_{0}}\text{)}\,$ and $f\left( x \right)<x$, for every $x\in \text{(}{{x}_{0}}\text{,}\infty \text{)}\,$. b) Show that the given function \[f\text{:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] given by $f\left( x \right)=x+\frac{1}{x}$ is contracted but has no fix number.

For each side of a given pentagon, divide its length by the total length of all other sides. Prove that the sum of all the fractions obtained is less than 2.

Let $ a, b, c$ be positive reals. Find the smallest value of \[ \frac {a \plus{} 3c}{a \plus{} 2b \plus{} c} \plus{} \frac {4b}{a \plus{} b \plus{} 2c} \minus{} \frac {8c}{a \plus{} b \plus{} 3c}. \]

Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal, \[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4\] and $ac=bd$. Find the maximum possible value of \[\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.\]

Let $ ABC$ be a triangle with $ AB \equal{} 5$, $ BC \equal{} 4$ and $ AC \equal{} 3$. Let $ \mathcal P$ and $ \mathcal Q$ be squares inside $ ABC$ with disjoint interiors such that they both have one side lying on $ AB$. Also, the two squares each have an edge lying on a common line perpendicular to $ AB$, and $ \mathcal P$ has one vertex on $ AC$ and $ \mathcal Q$ has one vertex on $ BC$. Determine the minimum value of the sum of the areas of the two squares. [asy]import olympiad; import math; import graph; unitsize(1.5cm); pair A, B, C; A = origin; B = A + 5 * right; C = (9/5, 12/5); pair X = .7 * A + .3 * B; pair Xa = X + dir(135); pair Xb = X + dir(45); pair Ya = extension(X, Xa, A, C); pair Yb = extension(X, Xb, B, C); pair Oa = (X + Ya)/2; pair Ob = (X + Yb)/2; pair Ya1 = (X.x, Ya.y); pair Ya2 = (Ya.x, X.y); pair Yb1 = (Yb.x, X.y); pair Yb2 = (X.x, Yb.y); draw(A--B--C--cycle); draw(Ya--Ya1--X--Ya2--cycle); draw(Yb--Yb1--X--Yb2--cycle); label("$A$", A, W); label("$B$", B, E); label("$C$", C, N); label("$\mathcal P$", Oa, origin); label("$\mathcal Q$", Ob, origin);[/asy]

Prove that for every natural $n$ the following inequality is valid $$|\sin 1| + |\sin 2| + |\sin (3n-1)| + |\sin 3n| > \frac{8n}{5}$$

Let $a, b$, and $c$ be positive real numbers. Find the largest integer $n$ such that $$\frac{1}{ax + b + c} +\frac{1}{a + bx + c}+\frac{1}{a + b + cx} \ge \frac{n}{a + b + c},$$ for all $ x \in [0, 1]$ .

Find the greatest and smallest value of $f(x, y) = y-2x$, if x, y are distinct non-negative real numbers with $\frac{x^2+y^2}{x+y}\leq 4$.

How many different integer solutions to the inequality $|x| + |y| < 100$ are there?

Show that the system of equations \begin{align*} \lfloor x\rfloor^2+\lfloor y\rfloor &=0, \\ 3x+y &=2, \end{align*} has infinitely many solutions and all these solutions satisfy bounds \begin{align*} 0<\ &x <4, \\ -9\le\ &y\le 1. \end{align*}

Prove that for the arbitrary numbers $x_1, x_2, ... , x_n$ from the $[0,1]$ segment $$(x_1 + x_2 + ...+ x_n + 1)^2 \ge 4(x_1^2 + x_2^2 + ... + x_n^2)$$

Let $a,b,c,d \in [0,1] .$ Prove that$$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+d}+\frac{1}{1+d+a}\leq \frac{4}{1+2\sqrt[4]{abcd}}$$

Prove that for any $ p, q\in\mathbb{N}$ with $ q > 1$ the following inequality holds: \[ \left\vert\pi\minus{}\frac{p}{q}\right\vert\ge q^{\minus{}42}.\]

Show: For all real numbers $ a,b,c$ with $ 0<a,b,c<1$ is: \[ \sqrt{a^2bc\plus{}ab^2c\plus{}abc^2}\plus{}\sqrt{(1\minus{}a)^2(1\minus{}b)(1\minus{}c)\plus{}(1\minus{}a)(1\minus{}b)^2(1\minus{}c)\plus{}(1\minus{}a)(1\minus{}b)(1\minus{}c)^2}<\sqrt{3}.\]

Let $a,b,c$ be non-negative real numbers such that $a^2+b^2+c^2 \le 3$ then prove that; $$(a+b+c)(a+b+c-abc)\ge2(a^2b+b^2c+c^2a)$$

(a) Let there be two given points $A,B$ in the plane. i. Find the triangles $MAB$ with the given area and the minimal perimeter. ii. Find the triangles $MAB$ with a given perimeter and the maximal area. (b) In a tetrahedron of volume $V$, let $a,b,c,d$ be the lengths of its four edges, no three of which are coplanar, and let $L=a+b+c+d$. Determine the maximum value of $\frac V{L^3}$.

Find all triplets $(x,y,n)$ of positive integers which satisfy: \[ (x-y)^n=xy \]

Kathryn, for a history project on sports, chronicled the history of college football. When she mentioned that Auburn got cheated out of the NCAA Football championship in the $2004-05$ season due to the many flaws in the BCS system, her teacher just couldn’t contain her applause, and awarded an automatic A to her for the rest of the year. The lecture was so popular, in fact, that many students pressed Kathryn to record the lecture on video and sell DVDs of it. If the function for Kathryn’s profit for selling DVDs of her college football presentation is $y = -x^2 + 14x + 251$, where $y$ is Kathryn’s profit and $x$ is the price per DVD, what price (in dollars) will maximize her profit?

Prove that for any triangle $ABC$, the inequality $\displaystyle\sum_{\text{cyclic}}\cos A\le\sum_{\text{cyclic}}\sin (A/2)$ holds.

Prove that $(a+b+c)^5 \ge 81 (a^2+b^2+c^2)abc$ for any positive real numbers $a,b,c$ I.Gorodnin