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Problem 3 IMO 2005 (Day 1)
Valentin Vornicu   121
N an hour ago by Rayvhs
Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that
\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]
Hojoo Lee, Korea
121 replies
Valentin Vornicu
Jul 13, 2005
Rayvhs
an hour ago
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one nice!
MihaiT   3
N Today at 12:53 PM by Pin123
Find positiv integer numbers $(a,b) $ s.t. $\frac{a}{b-2} $ and $\frac{3b-6}{a-3}$ be positiv integer numbers.
3 replies
MihaiT
Jan 14, 2025
Pin123
Today at 12:53 PM
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Algebra problem
Deomad123   1
N Today at 8:28 AM by lbh_qys
Let $n$ be a positive integer.Prove that there is a polynomial $P$ with integer coefficients so that $a+b+c=0$,then$$a^{2n+1}+b^{2n+1}+c^{2n+1}=abc[P(a,b)+P(b,c)+P(a,c)]$$.
1 reply
Deomad123
May 3, 2025
lbh_qys
Today at 8:28 AM
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GCD of consecutive terms
nsato   39
N Yesterday at 8:19 PM by Shan3t
The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.
39 replies
nsato
Mar 14, 2006
Shan3t
Yesterday at 8:19 PM
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Inequality
nhhlqd   27
N Yesterday at 12:18 PM by sqing
Given that $a,b$ are positive real number such that $a\geq 1$. Prove that
$$\dfrac{b^2}{a+b}+\dfrac{a}{b^2+b}+\dfrac{1}{a+1}\geq \dfrac{3}{2}$$
27 replies
nhhlqd
Feb 20, 2020
sqing
Yesterday at 12:18 PM
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Polynomial Minimization
ReticulatedPython   2
N Yesterday at 7:46 AM by lgx57
Consider the polynomial $$p(x)=x^{n+1}-x^{n}$$, where $x, n \in \mathbb{R+}.$

(a) Prove that the minimum value of $p(x)$ always occurs at $x=\frac{n}{n+1}.$
2 replies
ReticulatedPython
May 6, 2025
lgx57
Yesterday at 7:46 AM
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Inequalities
sqing   12
N Yesterday at 4:08 AM by sqing
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$ (1-abc) (1-a)(1-b)(1-c) \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c) \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840 $$
12 replies
sqing
Jul 12, 2024
sqing
Yesterday at 4:08 AM
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find number of elements in H
Darealzolt   1
N May 5, 2025 by alexheinis
If \( H \) is the set of positive real solutions to the system
\[ x^3 + y^3 + z^3 = x + y + z \]\[ x^2 + y^2 + z^2 = xyz \]then find the number of elements in \( H \).
1 reply
Darealzolt
May 5, 2025
alexheinis
May 5, 2025
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Polynomial
kellyelliee   1
N May 5, 2025 by Jackson0423
Let the polynomial $f(x)=x^2+ax+b$, where $a,b$ integers and $k$ is a positive integer. Suppose that the integers
$m,n,p$ satisfy: $f(m), f(n), f(p)$ are divisible by k. Prove that:
$(m-n)(n-p)(p-m)$ is divisible by k
1 reply
kellyelliee
May 5, 2025
Jackson0423
May 5, 2025
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Largest Divisor
4everwise   19
N May 4, 2025 by reni_wee
What is that largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$?
19 replies
4everwise
Dec 22, 2005
reni_wee
May 4, 2025
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Sequences and GCD problem
BBNoDollar   0
May 4, 2025
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
0 replies
BBNoDollar
May 4, 2025
0 replies
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confusing inequality
giangtruong13   5
N Apr 20, 2025 by arqady
Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{b+c}{a} \geq 2\sqrt{3(ab+bc+ca)}$$
5 replies
giangtruong13
Apr 18, 2025
arqady
Apr 20, 2025
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confusing inequality
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giangtruong13
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giangtruong13
#1 h  Apr 18, 2025, 2:07 PM
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Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{b+c}{a} \geq 2\sqrt{3(ab+bc+ca)}$$
This post has been edited 3 times. Last edited by giangtruong13, Apr 20, 2025, 3:02 PM
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arqady
30240 posts
arqady
#2 h  Apr 18, 2025, 4:14 PM
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giangtruong13 wrote:
Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{a}{b+c} \geq 2\sqrt{3(ab+bc+ca)}$$
It's $$\sum_{cyc}\frac{a}{b+c}\geq2\sqrt{\frac{(ab+ac+bc)(a^2b^2+a^2c^2+b^2c^2)}{a^2b^2c^2}}.$$Are you sure that it's true?
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giangtruong13
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giangtruong13
#3 h  Apr 19, 2025, 3:03 PM
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Oh sorry, i write wrongly, i will fix it here :oops_sign:
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giangtruong13
145 posts
giangtruong13
#4 h  Apr 20, 2025, 2:41 AM
Y by
Bummppppp
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giangtruong13
145 posts
giangtruong13
#5 h  Apr 20, 2025, 2:57 PM
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This inequality was from a book by an inactive user $toanmuonmau$
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arqady
30240 posts
arqady
#6 h  Apr 20, 2025, 7:57 PM • 1 Y
Y by kiyoras_2001
giangtruong13 wrote:
Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{b+c}{a} \geq 2\sqrt{3(ab+bc+ca)}$$
Because by C-S we obtain:
$$\sum_{cyc}\frac{b+c}{a}=\frac{\sum\limits_{cyc}(a^2b+a^2c)}{abc}=\frac{\sum\limits_{cyc}a^2\sum\limits_{cyc}a-\sum\limits_{cyc}a^3}{abc}=$$$$=\frac{\sqrt{\left(\sum\limits_{cyc}a^2\right)^2\left(\sum\limits_{cyc}a\right)^2}-\sum\limits_{cyc}a^3}{abc}=\frac{\sqrt{\sum\limits_{cyc}(a^4+2a^2b^2)\sum\limits_{cyc}(a^2+2ab)}-\sum\limits_{cyc}a^3}{abc}\geq$$$$\geq\frac{\sqrt{\sum\limits_{cyc}a^4\sum\limits_{cyc}a^2}+2\sqrt{\sum\limits_{cyc}a^2b^2\sum\limits_{cyc}ab}-\sum\limits_{cyc}a^3}{abc}\geq\frac{2\sqrt{\sum\limits_{cyc}a^2b^2\sum\limits_{cyc}ab}}{abc}=2\sqrt{3(ab+ac+bc)}.$$
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