Very interesting inequality

Let $x,y$ and $z$ be positive real numbers such that $xyz=1$ . Prove that

$\frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)} \geq \frac{3}{4}.$

69 replies

orl

Oct 22, 2004

Marcus_Zhang

13 minutes ago

IMO ShortList 2001, algebra problem 6

orl 137

N 35 minutes ago by Levieee

Source: IMO ShortList 2001, algebra problem 6

Prove that for all positive real numbers $a,b,c$ , $\frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1.$

137 replies

2 viewing

orl

Sep 30, 2004

Levieee

35 minutes ago

9 Three concurrent chords

v_Enhance 1

N an hour ago by YaoAOPS

Three distinct circles $\Omega_1$ , $\Omega_2$ , $\Omega_3$ cut three common chords concurrent at $X$ . Consider two distinct circles $\Gamma_1$ , $\Gamma_2$ which are internally tangent to all $\Omega_i$ . Determine, with proof, which of the following two statements is true.

(1) $X$ is the insimilicenter of $\Gamma_1$ and $\Gamma_2$
(2) $X$ is the exsimilicenter of $\Gamma_1$ and $\Gamma_2$ .

1 reply

+2 w

v_Enhance

an hour ago

YaoAOPS

an hour ago

Simple vector geometry existence

AndreiVila 3

N 2 hours ago by Ianis

Source: Romanian District Olympiad 2025 9.1

Let $ABCD$ be a parallelogram of center $O$ . Prove that for any point $M\in (AB)$ , there exist unique points $N\in (OC)$ and $P\in (OD)$ such that $O$ is the center of mass of $\triangle MNP$ .

3 replies

AndreiVila

Mar 8, 2025

Ianis

2 hours ago

BD tangent to (MDE) , rhombus ABCD with <DCB=60^o

parmenides51 1

N 2 hours ago by vanstraelen

Source: 2021 Germany R4 10.6 https://artofproblemsolving.com/community/c3208025_

Let a rhombus $ABCD$ with $|\angle DCB| = 60^o$ be given . On the extension of the segment $\overline{CD}$ beyond $D$ , a point $E$ is chosen arbitrarily. Let the line through $E$ and $A$ intersect the line $BC$ at the point $F$ . Let $M$ be the intersection of the lines $BE$ and $DF$ . Prove that the line $BD$ is tangent to the circumcircle of the triangle $MDE$ .

1 reply

parmenides51

Oct 6, 2024

vanstraelen

2 hours ago

Geometry Problem #42

vankhea 2

N 3 hours ago by kaede_Arcadia

Source: Van Khea

Let $P$ be any point. Let $D, E, F$ be projection point from $P$ to $BC, CA, AB$ . Circumcircle $(ABC)$ cuts circumcircle $(AEF), (BFD), (CDE)$ at $A_1, B_1, C_1$ . Let $A_2, B_2, C_2$ be antipode of $A_1, B_1, C_1$ wrt $(AEF), (BFD), (CDE)$ . Prove that $A_2, B_2, C_2, P$ are cyclic.

2 replies

vankhea

Sep 6, 2023

kaede_Arcadia

3 hours ago

divisibility

srnjbr 3

N 3 hours ago by srnjbr

Find all natural numbers n such that there exists a natural number l such that for every m members of the natural numbers the number m+m^2+...m^l is divisible by n.

3 replies

srnjbr

5 hours ago

srnjbr

3 hours ago

Very easy inequality

pggp 5

N 3 hours ago by ionbursuc

Source: Polish Junior MO Second Round 2019

Let $x$ , $y$ be real numbers, such that $x^2 + x \leq y$ . Prove that $y^2 + y \geq x$ .

5 replies

pggp

Oct 26, 2020

ionbursuc

3 hours ago

Solve in gaussian integers

CHESSR1DER 0

3 hours ago

Solve in gaussian integers.
$\sin\left(\ln\left(x^{x^{x^2}}\right)\right) = x^4$

0 replies

1 viewing

CHESSR1DER

3 hours ago

0 replies

Inequality and function

srnjbr 4

N 3 hours ago by srnjbr

Find all f:R--R such that for all x,y, yf(x)+f(y)>=f(xy)

4 replies

srnjbr

5 hours ago

srnjbr

3 hours ago

Problem 4

blug 3

N 3 hours ago by sunken rock

Source: Polish Junior Math Olympiad Finals 2025

In a rhombus $ABCD$ , angle $\angle ABC=100^{\circ}$ . Point $P$ lies on $CD$ such that $\angle PBC=20^{\circ}$ . Line parallel to $AD$ passing trough $P$ intersects $AC$ at $Q$ . Prove that $BP=AQ$ .

3 replies

blug

Mar 15, 2025

sunken rock

3 hours ago

CMI Entrance 19#6

bubu_2001 5

N 4 hours ago by quasar_lord

$(a)$ Compute -
$\begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg] \end{align*}$ $(b)$ For $x > 0$ define $F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t .$

$1.$ Determine the open interval(s) (if any) where $F ( x )$ is decreasing and all the open interval(s) (if any) where $F ( x )$ is increasing.

$2.$ Determine all the local minima of $F ( x )$ (if any) and all the local maxima of $F ( x )$ (if any) $.$

5 replies

bubu_2001

Nov 1, 2019

quasar_lord

4 hours ago

Very interesting inequality

sqing 0

Mar 19, 2025

Source: Own

Let $a,b,c\geq 2 .$ Prove that
$(a-1)(b^2-2)(c^3-3)- \frac{5}{2}abc\geq -10$ $(a-\frac{3}{2})(b^2-2)(c^3-3)- \frac{5}{2}abc\geq -15$ $(a-\frac{3}{2})(b^2-\frac{3}{2})(c^3-3)- \frac{25}{8}abc\geq - \frac{155}{8}$ $(a-\frac{3}{2})(b^2-\frac{3}{2})(c^3-3)- 3abc\geq - \frac{363}{20}$ $(a-\frac{3}{2})(b^2-\frac{3}{2})(c^3-\frac{5}{2})- \frac{55}{16}abc\geq - \frac{341}{16}$