A little problem

by TNKT, May 15, 2025, 1:17 PM

Problem. Let a,b,c be three positive real numbers with a+b+c=3. Prove that \color{blue}{\frac{1}{4a^{2}+9}+\frac{1}{4b^{2}+9}+\frac{1}{4c^{2}+9}\le \frac{3}{abc+12}.}
When does equality hold?
P/s: Could someone please convert it to latex help me? Thank you!

inequalities

inequality

by mathematical-forest, May 15, 2025, 12:40 PM

For positive real intengers $x_{1} ,x_{2} ,\cdots,x_{n}$ , such that $\prod_{i=1}^{n} x_{i} =1$
proof:
$\sum_{i=1}^{n} \frac{1}{1+\sum _{j\ne i}x_{j} } \le 1$

inequalities

Nice original fe

by Rayanelba, May 15, 2025, 12:37 PM

Find all functions $f: \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ that verfy the following equation :
$P(x,y):f(x+yf(x))+f(f(x))=f(xy)+2x$

This post has been edited 1 time. Last edited by Rayanelba, 31 minutes ago
Reason: Notation

functional equation

function

Inspired by Baltic Way 2005

by sqing, May 15, 2025, 12:29 PM

Let $a,b,c>0 , a+b+c +ab+bc+ca+abc=7$ . Prove that
$\frac {a}{a^2+2}+\frac {b}{b^2+2}+\frac {c}{c^2+2} \leq 1$ Let $a,b,c>0 , a+b+c +ab+bc+ca=6$ . Prove that
$\frac {a}{a^2+2}+\frac {b}{b^2+2}+\frac {c}{c^2+2} \leq 1$

inequalities proposed

algebra

D1033 : A problem of probability for dominoes 3*1

by Dattier, May 15, 2025, 12:29 PM

Let $G$ a grid of 9*9, we choose a little square in $G$ of this grid three times, we can choose three times the same.

What the probability of cover with 3*1 dominoes this grid removed by theses little squares (one, two or three) ?

probability

dominoes

IMO

Geometry with altitudes and the nine point centre

by Adywastaken, May 14, 2025, 12:47 PM

The foot of the altitude from vertex $A$ of acute triangle $ABC$ is $T_A$ . The ray drawn from $A$ through the circumcenter $O$ intersects $BC$ at $R_A$ . Let the midpoint of $AR_A$ be $F_A$ . Define $T_B$ , $R_B$ , $F_B$ , $T_C$ , $R_C$ , $F_C$ similarly. Prove that $T_AF_A$ , $T_BF_B$ , $T_CF_C$ are concurrent.

geometry

complex

symmetry

Parallel lines in incircle configuration

by GeorgeRP, May 14, 2025, 7:46 AM

Let $I$ be the incenter of triangle $\triangle ABC$ . Let $H_A$ , $H_B$ , and $H_C$ be the orthocenters of triangles $\triangle BCI$ , $\triangle ACI$ , and $\triangle ABI$ , respectively. Prove that the lines through $H_A$ , $H_B$ , and $H_C$ , parallel to $AI$ , $BI$ , and $CI$ , respectively, are concurrent.

This post has been edited 1 time. Last edited by GeorgeRP, Yesterday at 7:48 AM

geometry

Team Selection Test

Good Numbers

by ilovemath04, Sep 22, 2020, 11:39 PM

Let $a$ be a positive integer. We say that a positive integer $b$ is $a$ -good if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$ . Suppose $b$ is a positive integer such that $b$ is $a$ -good, but $b+2$ is not $a$ -good. Prove that $b+1$ is prime.

This post has been edited 1 time. Last edited by ilovemath04, Sep 22, 2020, 11:40 PM

IMO Shortlist

IMO Shortlist 2019

number theory

Geometry from Iran TST 2017

by bgn, Apr 27, 2017, 12:42 PM

In triangle $ABC$ let $O$ and $H$ be the circumcenter and the orthocenter. The point $P$ is the reflection of $A$ with respect to $OH$ . Assume that $P$ is not on the same side of $BC$ as $A$ . Points $E,F$ lie on $AB,AC$ respectively such that $BE=PC \ , CF=PB$ . Let $K$ be the intersection point of $AP,OH$ . Prove that $\angle EKF = 90 ^{\circ}$

Proposed by Iman Maghsoudi

This post has been edited 4 times. Last edited by bgn, Apr 27, 2017, 5:16 PM

Iran

Iranian TST

geometry