Let $a, b, c$ be positive real numbers. Prove that \[1+\frac{3}{ab+bc+ca}\geq\frac{6}{a+b+c}.\]

Let $a, b, c$ be non-negative numbers with $a+b+c = 3$. Prove the inequality \[\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1} \geq \frac 32.\]

Find the largest positive real $ k$, such that for any positive reals $ a,b,c,d$, there is always: \[ (a\plus{}b\plus{}c) \left[ 3^4(a\plus{}b\plus{}c\plus{}d)^5 \plus{} 2^4(a\plus{}b\plus{}c\plus{}2d)^5 \right] \geq kabcd^3\]

Given an arbitrary triangle $ ABC$, denote by $ P,Q,R$ the intersections of the incircle with sides $ BC, CA, AB$ respectively. Let the area of triangle $ ABC$ be $ T$, and its perimeter $ L$. Prove that the inequality \[\left(\frac {AB}{PQ}\right)^3 \plus{}\left(\frac {BC}{QR}\right)^3 \plus{}\left(\frac {CA}{RP}\right)^3 \geq \frac {2}{\sqrt {3}} \cdot \frac {L^2}{T}\] holds.

Let $a,b,c>0$ so that $a+b+c=1$. Then prove that $$(a+2ab+2ac+bc)^a(b+2bc+2ba+ca)^b(c+2ca+2cb+ab)^c\le1.$$ [i]Proposed by Marius Drăgan[/i]

Let $a, b, c$ be positive reals such that their sum is $1$. Prove that \[\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.\]

[u]Round 5[/u] [b]p13.[/b] Find all ordered pairs of real numbers $(x, y)$ satisfying the following equations: $$\begin{cases} \dfrac{1}{xy} + \dfrac{y}{x}= 2 \\ \dfrac{1}{xy^2} + \dfrac{y^2}{x} = 7 \end{cases}$$ [b]p14.[/b] An egg plant is a hollow prism of negligible thickness, with height $2$ and an equilateral triangle base. Inside the egg plant, there is enough space for four spherical eggs of radius $1$. What is the minimum possible volume of the egg plant? [b]p15.[/b] How many ways are there for Farmer James to color each square of a $2\times 6$ grid with one of the three colors eggshell, cream, and cornsilk, so that no two adjacent squares are the same color? [u]Round 6[/u] [b]p16.[/b] In a triangle $ABC$, $\angle A = 45^o$, and let $D$ be the foot of the perpendicular from $A$ to segment $BC$. $BD = 2$ and $DC = 4$. Let $E$ be the intersection of the line $AD$ and the perpendicular line from $B$ to line $AC$. Find the length of $AE$. [b]p17.[/b] Find the largest positive integer $n$ such that there exists a unique positive integer $m$ satisfying $$\frac{1}{10} \le \frac{m}{n} \le \frac19$$ [b]p18.[/b] How many ordered pairs $(A,B)$ of positive integers are there such that $A+B = 10000$ and the number $A^2 + AB + B$ has all distinct digits in base $10$? [u]Round 7[/u] [b]p19.[/b] Pentagon $JAMES$ satisfies $JA = AM = ME = ES = 2$. Find the maximum possible area of $JAMES$. [b]p20.[/b] $P(x)$ is a monic polynomial (a polynomial with leading coecient $1$) of degree $4$, such that $P(2^n+1) =8^n + 1$ when $n = 1, 2, 3, 4$. Find the value of $P(1)$. [b]p21[/b]. PEAcock and Zombie Hen Hao are at the starting point of a circular track, and start running in the same direction at the same time. PEAcock runs at a constant speed that is $2018$ times faster than Zombie Hen Hao's constant speed. At some point in time, Farmer James takes a photograph of his two favorite chickens, and he notes that they are at different points along the track. Later on, Farmer James takes a second photograph, and to his amazement, PEAcock and Zombie Hen Hao have now swapped locations from the first photograph! How many distinct possibilities are there for PEAcock and Zombie Hen Hao's positions in Farmer James's first photograph? (Assume PEAcock and Zombie Hen Hao have negligible size.) [u]Round 8[/u] [b]p22.[/b] How many ways are there to scramble the letters in $EGGSEATER$ such that no two consecutive letters are the same? [b]p23.[/b] Let $JAMES$ be a regular pentagon. Let $X$ be on segment $JA$ such that $\frac{JX}{XA} = \frac{XA}{JA}$ . There exists a unique point $P$ on segment $AE$ such that $XM = XP$. Find the ratio $\frac{AE}{PE}$ . [b]p24.[/b] Find the minimum value of the function $$f(x) = \left|x - \frac{1}{x} \right|+ \left|x - \frac{2}{x} \right| + \left|x - \frac{3}{x} \right|+... + \left|x - \frac{9}{x} \right|+ \left|x - \frac{10}{x} \right|$$ over all nonzero real numbers $x$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949191p26406082]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all of the positive real numbers like $ x,y,z,$ such that : 1.) $ x \plus{} y \plus{} z \equal{} a \plus{} b \plus{} c$ 2.) $ 4xyz \equal{} a^2x \plus{} b^2y \plus{} c^2z \plus{} abc$ Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.

Let $ a$, $ b$, $ c$ be positive real numbers. Prove that \[ \dfrac{(2a \plus{} b \plus{} c)^2}{2a^2 \plus{} (b \plus{} c)^2} \plus{} \dfrac{(2b \plus{} c \plus{} a)^2}{2b^2 \plus{} (c \plus{} a)^2} \plus{} \dfrac{(2c \plus{} a \plus{} b)^2}{2c^2 \plus{} (a \plus{} b)^2} \le 8. \]

Let $a,b,c\in \mathbb{R}_{\ge 0}$ satisfy $a+b+c=1$. Prove the inequality : \[ \frac{3-b}{a+1}+\frac{a+1}{b+1}+\frac{b+1}{c+1}\ge 4 \]

Solve in integers the equation \[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]

Let $ n \geq 3 $ be an integer and let \begin{align*} x_1,x_2, \ldots, x_n \end{align*} be $ n $ distinct integers. Prove that \begin{align*} (x_1 - x_2)^2 + (x_2 - x_3)^2 + \ldots + (x_n - x_1)^2 \geq 4n - 6. \end{align*}

Suppose that $ x, y$ and $ z$ are positive real numbers such that $ xyz \ge 1$. (a) Prove that $ 27 \le (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2$, with equality if and only if $ x \equal{} y \equal{} z \equal{} 1$. (b) Prove that $ (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2$ $ \le 3(x \plus{} y \plus{} z)^2$, with equality if and only if $ x \equal{} y \equal{} z \equal{} 1$.

If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that \[\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).\]

If $x,y,z$ satisfies $x^2+y^2+z^2=1$, find the maximum possible value of $$(x^2-yz)(y^2-zx)(z^2-xy)$$