Let $a,b,c>0$ such that $a+b+c=3$. Prove that :$$\frac{ab}{ab+a+b}+\frac{bc}{bc+b+c}+\frac{ca}{ca+c+a}+\frac{1}{9}\left(\frac{(a-b)^2}{ab+a+b}+\frac{(b-c)^2}{bc+b+c}+\frac{(c-a)^2}{ca+c+a}\right)\leq1.$$

Let $f, g:\mathbb{Z}\rightarrow [0,\infty )$ be two functions such that $f(n)=g(n)=0$ with the exception of finitely many integers $n$. Define $h:\mathbb{Z}\rightarrow [0,\infty )$ by \[h(n)=\max \{f(n-k)g(k): k\in\mathbb{Z}\}.\] Let $p$ and $q$ be two positive reals such that $1/p+1/q=1$. Prove that \[ \sum_{n\in\mathbb{Z}}h(n)\geq \Bigg(\sum_{n\in\mathbb{Z}}f(n)^p\Bigg)^{1/p}\Bigg(\sum_{n\in\mathbb{Z}}g(n)^q\Bigg)^{1/q}.\]

Prove that for any natural number $n$ the following inequality holds $$4^n < (2n+1)C_{2n}^n$$

Let $x_1 \le x_2 \le \ldots \le x_{2n-1}$ be real numbers whose arithmetic mean equals $A$. Prove that $$2\sum_{i=1}^{2n-1}\left( x_{i}-A\right)^2 \ge \sum_{i=1}^{2n-1}\left( x_{i}-x_{n}\right)^2.$$

Prove or disprove that $\forall a,b,c,d \in \mathbb{R}^+$ we have the following inequality: \[3 \leq \frac{4a+b}{a+4b} + \frac{4b+c}{b+4c} + \frac{4c+a}{c+4a} < \frac{33}{4}\]

Let $x>1$ ,$n$ be positive integer. Prove that$$\sum_{k=1}^{n}\frac{\{kx \}}{[kx]}<\sum_{k=1}^{n}\frac{1}{2k-1}$$ Where $[kx ]$ be the integer part of $kx$ ,$\{kx \}$ be the decimal part of $kx$.

Prove that if the real numbers $ a $, $ b $, $ c $ satisfy the inequalities $$a + b + c> 0,$$ $$ ab + bc + ca > 0$$ $$ abc > 0$$ then $a > 0, b > 0, c > 0$.

A function $f : [0,1] \to R$ has the following properties: (a) $f(x) \ge 0$ for $0 < x < 1$, (b) $f(1) = 1$, (c) $f(x+y) \ge f(x)+ f(y) $ whenever $x,y,x+y \in [0,1]$. Prove that $f(x) \le 2x$ for all $x \in [0,1]$.

Let $a,b,c,d$ real numbers such that $a+b+c+d=19$ and $a^2+b^2+c^2+d^2=91$. Find the maximum value of \[ \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d} \]

Given $a_1\ge 1$ and $a_{k+1}\ge a_k+1$ for all $k\ge 1,2,\dots,n$, show that $a_1^3+a_2^3+\dots+a_n^3\ge (a_1+a_2+\dots+a_n)^2$

Let $a,b,c$ be positive real numbers such that $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}=3$. Prove that $$\frac{a+b}{a^2+ab+b^2}+ \frac{b+c}{b^2+bc+c^2}+ \frac{c+a}{c^2+ca+a^2}\le 2$$ When is the equality valid?

Let $a$, $b$, $c$, $d$, $e$ be real strictly positive real numbers such that $abcde = 1$. Then is true the following inequality:$$\frac{de}{a(b+1)}+\frac{ea}{b(c+1)}+\frac{ab}{c(d+1)}+\frac{bc}{d(e+1)}+\frac{cd}{e(a+1)}\geq \frac{5}{2}$$

Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?

Let $a,b,c \in \mathbb{C}$ such that $a|bc| + b|ca| + c|ab| = 0$. Prove that $|(a-b)(b-c)(c-a)| \ge 3\sqrt{3}|abc|$.

Suppose the numbers $a_0, a_1, a_2, ... , a_n$ satisfy the following conditions: $a_0 =\frac12$, $a_{k+1} = a_k +\frac{1}{n}a_k^2$ for $k = 0, 1, ... , n - 1$. Prove that $1 - \frac{1}{n}< a_n < 1$

If $a,b,c,x$ are positive reals, show that $$\frac{a^{x+2}+1}{a^xbc+1}+\frac{b^{x+2}+1}{b^xac+1}+\frac{c^{x+2}+1}{c^xab+1}\geq 3$$

Hi guys, I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this: 1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though. 2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary. 3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions: A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh? B. Do NOT go back to the previous problem(s). This causes too much confusion. C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for. 4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving! Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D

Let $ x_1,x_2,\ldots,x_n$ be real numbers greater than 1. Show that \[ \frac{x_1x_2}{x_3}\plus{}\frac{x_2x_3}{x_4}\plus{}\cdots\plus{}\frac{x_nx_1}{x_2}\ge4n\] and determine when the equality holds.

How many non-congruent triangles with perimeter $ 7$ have integer side lengths? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

A function $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, where $\mathbb{Z}_+$ is the set of positive integers, is non-decreasing and satisfies $f(mn) = f(m)f(n)$ for all relatively prime positive integers $m$ and $n$. Prove that $f(8)f(13) \ge (f(10))^2$.

Find the number of $16$-tuples $(x_1, x_2,..., x_{16})$ such that (i) $x_i = \pm 1$ for $i = 1,..., 16$, (ii) $0 \le x_1 + x_2 +... + x_r < 4$, for $r = 1, 2,... , 15$, (iii) $x_1 + x_2 +...+ x_{10} = 4$

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

Given integer $n\geq 2$ and real numbers $x_1,x_2,\cdots, x_n$ in the interval $[0,1]$. Prove that there exist real numbers $a_0,a_1,\cdots,a_n$ satisfying the following conditions: (1) $a_0+a_n=0$; (2) $|a_i|\leq 1$, for $i=0,1,\cdots,n$; (3) $|a_i-a_{i-1}|=x_i$, for $i=1,2,\cdots,n$.

For real numbers, $a,b,c$ with $bc \ne 0$ we have to $\frac{1-c^2}{bc} \ge 0$. Prove that $$5( a^2+b^2+c^2 -bc^3) \ge ab.$$

Let the real numbers $a, b, c$ be such that $a \ge b \ge c > 0$. Show that $$\frac{a^2-b^2}{c}+ \frac{c^2-b^2}{a}+ \frac{a^2-c^2}{b}\ge 3a - 4b + c.$$ When does equality hold?