A [i]triangulation[/i] $ \mathcal{T}$ of a polygon $ P$ is a finite collection of triangles whose union is $ P,$ and such that the intersection of any two triangles is either empty, or a shared vertex, or a shared side. Moreover, each side of $ P$ is a side of exactly one triangle in $ \mathcal{T}.$ Say that $ \mathcal{T}$ is [i]admissible[/i] if every internal vertex is shared by $ 6$ or more triangles. For example [asy] size(100); dot(dir(-100)^^dir(230)^^dir(160)^^dir(100)^^dir(50)^^dir(5)^^dir(-55)); draw(dir(-100)--dir(230)--dir(160)--dir(100)--dir(50)--dir(5)--dir(-55)--cycle); pair A = (0,-0.25); dot(A); draw(A--dir(-100)^^A--dir(230)^^A--dir(160)^^A--dir(100)^^A--dir(5)^^A--dir(-55)^^dir(5)--dir(100)); [/asy] Prove that there is an integer $ M_n,$ depending only on $ n,$ such that any admissible triangulation of a polygon $ P$ with $ n$ sides has at most $ M_n$ triangles.

Given $a_1, a_2, ... , a_n$. Define $$b_k = \frac{a_1 + a_2 + ... + a_k}{k}$$ for $1 \le k\le n.$ Let $$C = (a_1 - b_1)^2 + (a_2 - b_2)^2 + ... + (a_n - b_n)^2, D = (a_1 - b_n)^2 + (a_2 - b_n)^2 + ... + (a_n - b_n)^2$$ Prove that $C \le D \le 2C$.

Let $x, y$ be real numbers such that $1\le x^2-xy+y^2\le2$. Show that: a) $\dfrac{2}{9}\le x^4+y^4\le 8$; b) $x^{2n}+y^{2n}\ge\dfrac{2}{3^n}$, for all $n\ge3$. [i]Laurențiu Panaitopol[/i] and [i]Ioan Tomescu[/i]

Show that for all reals $x,y,z$, we have $$\left(x^2+3\right)\left(y^2+3\right)\left(z^2+3\right)\ge(xyz+x+y+z+4)^2.$$

If $a,b,c>0$, what is the smallest possible value of $ \left\lfloor \dfrac {a+b}{c} \right\rfloor + \left\lfloor \dfrac {b+c}{a} \right\rfloor + \left\lfloor \dfrac {c+a}{b} \right\rfloor $? (Note that $ \lfloor x \rfloor $ denotes the greatest integer less than or equal to $x$.)

Let $a>0$ and $x_1,x_2,x_3$ be real numbers with $x_1+x_2+x_3=0$. Prove that $$\log_2\left(1+a^{x_1}\right)+\log_2\left(1+a^{x_2}\right)+\log_2\left(1+a^{x_3}\right)\ge3.$$

The numbers $x, y, z, t, a$ and $b$ are positive integers, so that $xt-yz = 1$ and $$\frac{x}{y} \ge \frac{a}{b} \ge \frac{z}{t} .$$Prove that $$ab \le (x + z) (y +t)$$

Let $\{a_n\}$ be a sequence of positive terms such that $a_{n+1}=a_n+ \frac{n^2}{a_n}$ . Let $b_n =a_n-n$ . (1) Are there infinitely many $n$ such that $b_n \ge 0$ ? (2) Prove that there is a positive number $M$ such that $\sum^{\infty}_{n=3} \frac{b_n}{n+1}<M$.

Find the smallest real $ K$ such that for each $ x,y,z\in\mathbb R^{ \plus{} }$: \[ x\sqrt y \plus{} y\sqrt z \plus{} z\sqrt x\leq K\sqrt {(x \plus{} y)(y \plus{} z)(z \plus{} x)} \]

The number $a$ is such that both fractions $$(1-3a)/(2a + 3) \,\,\, and \,\,\, (17 + 4a)/(7 + a)$$ are positive. Which one is closer to $\sqrt5$?

Solve for real $x$: \[ \frac{1}{[x]} + \frac{1}{[2x]} = x - [x] + \frac{1}{3}. \]

Let be $ a,b,c\in (0,\infty)$ such that $ 3abc\equal{}1$ . Show that : $ \frac{3a^5}{3a^5\plus{}2bc}\plus{}\frac{3b^5}{3b^5\plus{}2ca}\plus{}\frac{3c^5}{3c^5\plus{}2ab}\ge 1$

a,b,c are positive & abc=1, then show that a^(b+c) .b^(c+a) .c^(a+b) ≤ 1

Let \(a,b,c\) be three distinct positive real numbers such that \(abc=1\). Prove that $$\dfrac{a^3}{(a-b)(a-c)}+\dfrac{b^3}{(b-c)(b-a)}+\dfrac{c^3}{(c-a)(c-b)} \ge 3$$

Prove that the product of two sides of a triangle is always greater than the product of the diameters of the inscribed circle and the circumscribed circle.

Guess I should stop proposing problems at 2:00 AM, as this can lead to ones like this here: Let $a$, $b$, $c$ be three positive reals. Prove the inequality $\frac{a^2+2b^2}{b+c}+\frac{b^2+2c^2}{c+a}+\frac{c^2+2a^2}{a+b}\geq\frac32\left(a+b+c\right)$.

Let $ABC$ be a triangle, let $H$ be the orthocentre and $L,M,N$ the midpoints of the sides $AB, BC, CA$ respectively. Prove that \[HL^{2} + HM^{2} + HN^{2} < AL^{2} + BM^{2} + CN^{2}\] if and only if $ABC$ is acute-angled.

Let $ X_1, \ldots ,X_n$ be $ n$ points in the unit square ($ n>1$). Let $ r_i$ be the distance of $ X_i$ from the nearest point (other than $ X_i$). Prove that the inequality \[ r_1^2\plus{} \ldots \plus{}r_n^2 \leq 4.\] [i]L. Fejes-Toth, E. Szemeredi[/i]

A sequence of integers $1 = x_1 < x_2 < x_3 <...$ satisfies $x_{n+1} \le 2n$ for all $n$. Show that every positive integer $k$ can be written as $x_j -x_i$ for some $i, j$.

Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

Let $n$ be a positive integer. Prove that \[-1<\sum_{k=1}^{n}\frac{k}{k^2+1}-\ln n\le\frac{1}{2}\]

$(CZS 4)$ Let $K_1,\cdots , K_n$ be nonnegative integers. Prove that $K_1!K_2!\cdots K_n! \ge \left[\frac{K}{n}\right]!^n$, where $K = K_1 + \cdots + K_n$

For a positive integer $n \ge 2 $, define set $ T = \{ (i,j) | 1 \le i < j \le n , i | j \} $. For nonnegative real numbers $ x_1 , x_2 , \cdots , x_n $ with $ x_1 + x_2 + \cdots + x_n = 1 $, find the maximum value of \[ \sum_{(i,j) \in T} x_i x_j \] in terms of $n$.

For which real numbers $c$ is $$\frac{e^x +e^{-x} }{2} \leq e^{c x^2 }$$ for all real $x?$

For which real values of $x,y$ the expression$\frac{2-\left(\dfrac{x+y}{3}-1\right)^2}{\left(\dfrac{x-3}{2}+\dfrac{2y-x}{3}\right)^2+4}$ becomes maximum? Which is that maximum value?