$a,b,c$ are positive reals satisfying $\frac{2}{5} \leq c \leq \min{a,b}$ ; $ac \geq \frac{4}{15}$ and $bc \geq \frac{1}{5}$ Find the maximum value of $\left(\frac{1}{a}+\frac{2}{b}+\frac{3}{c}\right)$.

For any positive real numbers $a, b, c, d$, prove the following inequality: $$(a^2+b^2)(b^2+c^2)(c^2+d^2)(d^2+a^2) \geq 64abcd|(a-b)(b-c)(c-d)(d-a)|$$ [i]Proposed by Anton Trygub[/i]

Let $(a_1, a_2,..., a_n)$ and $(b_1, b_2, ..., b_n)$ be two sequences of positive real numbers. Let $\pi$ be a permutation of the set $\{1, 2, ..., n\}$, for which the sum $a_{\pi(1)}(b_{\pi(1)}+b_{\pi(2)}+...+b_{\pi(n)})+a_{\pi(2)}(b_{\pi(3)}+b_{\pi(3)}+...+b_{\pi(n)})+...+a_{\pi(n)}b_{\pi(n)}$ is minimal. Proce for this permutation $\pi$, that $$ \frac{a_{\pi(1)}}{b_{\pi(1)}}\le \frac{a_{\pi(2})}{b_{\pi(2)}}\le ... \le \frac{a_{\pi(n)}}{b_{\pi(n)}}$$ Application: In an idealized role-playing game you fight against $n$ opponents at the same time. In order to minimize the damage you suffer yourself, you should first take care of your opponent for the ratio of the time it takes to defeat him (if you only focus on him), and the damage it does per second is minimal; next, one should fight the opponent with the second smallest such ratio, and so on.

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

Prove that if $a+b+c=0$ then $a^3+b^3+c^3=3abc$

Let $a,b,c \in \mathbb{R^+}$ such that $abc=1$. Prove that $$\sum_{a,b,c} \sqrt{\frac{a}{a+8}} \ge 1$$

For every positive $a, b, c, d$ such that $a + c\le ac$ and $b + d \le bd$ prove that $ab + cd \ge 8$.

If $a,b,c$ are nonnegative real numbers, then \[{ 3(a^2+b^2+c^2) \geq (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2 \geq (a+b+c)^2.}\]

Let $a, b, c$ be sides of a triangle. Prove that $$\frac{1}{3} \leq \frac{a^2 +b^2 +c^2 }{(a+b+c)^2 } < \frac{1}{2}$$ and show that $\frac{1}{2}$ cannot be replaced with a smaller number.

Show that if a holomorphic function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ has the property that the modulus of any of its derivatives (of any order) is everywhere dominated by $ 1, $ then $ |f(z)|\le e^{|\text{Im} (z)|} , $ for all complex numbers $ z. $

Let $k$ be a positive integer. A sequence $a_0,a_1,...,a_n,n>0$ of positive integers satisfies the following conditions: $(i)$ $a_0=a_n=1$; $(ii)$ $2\leq a_i\leq k$ for each $i=1,2,...,n-1$; $(iii)$For each $j=2,3,...,k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0,a_1,...,a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$; $(iv)$For any $i=1,2,...,n-1$, $\gcd(a_i,a_{i-1})=1=\gcd(a_i,a_{i+1})$, and $a_i$ divides $a_{i-1}+a_{i+1}$. Suppose there is another sequence $b_0,b_1,...,b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}}>\frac{b_i}{a_i}$ for all $i=0,1,...,n-1$. Find the minimum value of $b_n-b_0$.

Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$. Let a line $\ell$ through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$, respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$, respectively such that the equation \[ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} \] holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.

For positive numbers $ a_1,a_2,\dots,a_n$, we define \[ A\equal{}\frac{a_1\plus{}a_2\plus{}\cdots\plus{}a_n}{n}, \quad G\equal{}\sqrt[n]{a_1\cdots a_n}, \quad H\equal{}\frac{n}{a_1^{\minus{}1}\plus{}\cdots\plus{}a_n^{\minus{}1}}\] Prove that (i) $ \frac{A}{H}\leq \minus{}1\plus{}2\left(\frac{A}{G}\right)^n$, for n even (ii) $ \frac{A}{H}\leq \minus{}\frac{n\minus{}2}{n}\plus{}\frac{2(n\minus{}1)}{n}\left(\frac{A}{G}\right)^n$, for $ n$ odd

Let $x, y, z$ be positive real numbers such that $x^2+y^2+z^2=3$. Prove that $$\frac{x+1}{z+x+1}+\frac{y+1}{x+y+1}+\frac{z+1}{y+z+1}\geq\frac{(xy+yz+zx+\sqrt{xyz})^2}{(x+y)(y+z)(z+x)}.$$

$PA_1A_2...A_n$ is a pyramid. The base $A_1A_2...A_n$ is a regular n-gon. The apex $P$ is placed so that the lines $PA_i$ all make an angle $60^{\cdot}$ with the plane of the base. For which $n$ is it possible to find $B_i$ on $PA_i$ for $i = 2, 3, ... , n$ such that $A_1B_2 + B_2B_3 + B_3B_4 + ... + B_{n-1}B_n + B_nA_1 < 2A_1P$?

Let $ r$ and $ n$ be nonnegative integers such that $ r \le n$. $ (a)$ Prove that: $ \frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r}$ is an integer. $ (b)$ Prove that: $ \displaystyle\sum_{r\equal{}0}^{[n/2]}\frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r}<2^{n\minus{}2}$ for all $ n \ge 9$.

The real numbers $a_0, a_1, ... , a_{n+1}$ satisfy $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_k + a_{k+1}| \le 1$ for $k = 1, 2, ... , n$. Show that $|a_k| \le \frac{ k(n + 1 - k)}{2}$ for all $k$.

Let $P(x),\ Q(x)$ be polynomials such that : \[\int_0^2 \{P(x)\}^2dx=14,\ \int_0^2 P(x)dx=4,\ \int_0^2 \{Q(x)\}^2dx=26,\ \int_0^2 Q(x)dx=2.\] Find the maximum and the minimum value of $\int_0^2 P(x)Q(x)dx$.

If $P$ is the area of a triangle $ABC$ with sides $a,b,c$, prove that $\frac{ab+bc+ca}{4P} \ge \sqrt3$

For every positive real pair $(x,y)$ satisfying the equation $x^3+y^4 = x^2y$, if the greatest value of $x$ is $A$, and the greatest value of $y$ is $B$, then $A/B = ?$ $ \textbf{(A)}\ \frac{2}{3} \qquad \textbf{(B)}\ \frac{512}{729} \qquad \textbf{(C)}\ \frac{729}{1024} \qquad \textbf{(D)}\ \frac{3}{4} \qquad \textbf{(E)}\ \frac{243}{256}$

For any $a$, $b$, $c > 0$ and for any $n \in \mathbb{N}^*$, prove the inequality$$(a - b)\left(\frac{a}{b}\right)^n+(b - c)\left(\frac{b}{c}\right)^n+(c - a)\left(\frac{c}{a}\right)^n\geq (a - b)\frac{a}{b}+(b - c)\frac{b}{c}+(c - a)\frac{c}{a}$$

We consider $n$ real variables $x_i(1 \le i \le n)$, where $n$ is an integer and $n \ge 2$. The product of these variables will be denoted by $p$, their sum by $s$, and the sum of their squares by $S$. Furthermore, let $\alpha$ be a positive constant. We now study the inequality $ps \le S\alpha$. Prove that it holds for every $n$-tuple $(x_i)$ if and only if $\alpha=\frac{n+1}{2}$

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be positive real numbers ($ n \ge 2$) whose sum is $ S$. Show that \[ \sum_{i\equal{}1}^n\frac{a_i^{2^{k}}}{\left(S\minus{}a_i\right)^{2^t\minus{}1}}\ge\frac{S^{1\plus{}2^k\minus{}2^t}}{\left(n\minus{}1\right)^{2^t\minus{}1}n^{2^k\minus{}2^t}}\] for any nonnegative integers $ k$, $ t$ with $ k \ge t$. When does equality occur?

Suppose a sequence of reals $\{a_n\}_{n\geq 0}$ satisfies $a_0 = 0$, $\frac{100}{101} <a_{100}<1$, and $$2a_n - a_{n-1} -a_{n+1} \leq 2 (1-a_n )^3$$ for every $n\geq 1$. (1) Define a sequence $b_n = a_n - \frac{n}{n+1}$. Prove that $b_n\leq b_{n+1}$ for any $n\geq 100$. (2) Determine whether infinite series $\sum_{n=1}^\infty \frac{a_n}{n^2}$ converges or diverges.

Let a,b,c be real numbers such that:$a\ge b\ge 1\ge c\ge 0$ and a+b+c=3. a)Prove that $2\le ab +bc+ca\le 3$ b)Prove that $\dfrac{24}{a^3+b^3+c^3}+\dfrac{25}{ab+bc+ca}\ge 14$. Find the equality cases