Not so beautiful

by m4thbl3nd3r, May 29, 2025, 3:23 AM

Let $a, b,c>0$ such that $b+c>a$ . Prove that $2 \sqrt[4]{\frac{a}{b+c-a}}\ge 2 +\frac{2a^2-b^2-c^2}{(a+b)(a+c)}.$

Inspired by a9opsow_

by sqing, May 29, 2025, 1:49 AM

Let $a,b > 0 .$ Prove that
$\frac{(ka^2 - kab-b)^2 + (kb^2 - kab-a)^2 + (ab-ka-kb )^2}{ (ka+b)^2 + (kb+a)^2+(a - b)^2 }\geq \frac {1}{(k+1)^2}$ Where $k\geq 0.37088 .$
$\frac{(a^2 - ab-b)^2 + (b^2 - ab-a)^2 + ( ab-a-b)^2}{a^2 +b^2+(a - b)^2 } \geq 1$ $\frac{(2a^2 - 2ab-b)^2 + (2b^2 - 2ab-a)^2 + (ab-2a-2b )^2}{ (2a+b)^2 + (2b+a)^2+(a - b)^2 }\geq \frac 19$

This post has been edited 2 times. Last edited by sqing, 2 hours ago

inequalities proposed

algebra

inequalities

Iran TST Starter

by M11100111001Y1R, May 27, 2025, 7:36 AM

Let $a_n$ be a sequence of positive real numbers such that for every $n > 2025$ , we have:
$a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}$ Prove that there exists a natural number $M$ such that for all $n > M$ , the following holds:
$a_n < \frac{1}{1404}$

Sequence

Cute NT Problem

by M11100111001Y1R, May 27, 2025, 7:20 AM

A number $n$ is called lucky if it has at least two distinct prime divisors and can be written in the form:
$n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}$ where $p_1, \dots, p_k$ are distinct prime numbers that divide $n$ . (Note: it is possible that $n$ has other prime divisors not among $p_1, \dots, p_k$ .) Prove that for every prime number $p$ , there exists a lucky number $n$ such that $p \mid n$ .

This post has been edited 1 time. Last edited by M11100111001Y1R, Tuesday at 6:24 PM

residue

legandre symbol

number theory

3 var inequality

by SunnyEvan, May 17, 2025, 1:05 PM

Let $a,b,c \in R$ ,such that $a^2+b^2+c^2=4(ab+bc+ca)$ Prove that : $\frac{7-2\sqrt{14}}{48} \leq \frac{a^3b+b^3c+c^3a}{(a^2+b^2+c^2)^2} \leq \frac{7+2\sqrt{14}}{48}$

This post has been edited 3 times. Last edited by SunnyEvan, May 18, 2025, 6:04 AM

inequalities

inequalities proposed

Twin Prime Diophantine

by awesomeming327., Mar 7, 2025, 8:28 PM

Determine all positive integers $a$ , $b$ , $c$ , $p$ , where $p$ and $p+2$ are odd primes and
$2^ap^b=(p+2)^c-1.$

number theory

Troublesome median in a difficult inequality

by JG666, May 22, 2022, 2:35 AM

Determine the minimum value of $\lambda\in\mathbb{R}$ , such that for any positive integer $n$ and non-negative reals $x_1, x_2, \cdots, x_n$ , the following inequality always holds:
$\sum_{i=1}^n(m_i-a_i)^2\leqslant \lambda\cdot\sum_{i=1}^nx_i^2,$ Here $m_i$ and $a_i$ denote the median and arithmetic mean of $x_1, x_2, \cdots, x_i$ , respectively.

Duanyang ZHANG, High School Affiliated to Renmin University of China

This post has been edited 1 time. Last edited by JG666, May 22, 2022, 2:37 AM

Inequality

by Amin12, Sep 2, 2017, 9:18 AM

Let $a,b$ and $c$ be positive real numbers. Prove that
$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3}$

trigonometric inequality

by MATH1945, May 26, 2016, 7:23 AM

In triangle $ABC$ , prove that $sin^2(A)+sin^2(B)+sin^2(C) \leq \frac{9}{4}$

This post has been edited 1 time. Last edited by MATH1945, May 26, 2016, 7:25 AM
Reason: 2

inequalities

algebra

trigonometry

Every subset of size k has sum at most N/2

by orl, Apr 20, 2006, 5:58 PM

For a given positive integer $k$ find, in terms of $k$ , the minimum value of $N$ for which there is a set of $2k + 1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $\tfrac{N}{2}.$