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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

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[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
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0 replies
jlacosta
Mar 2, 2025
0 replies
J
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P(a+1)=1 for every root a
MTA_2024   1
N 3 minutes ago by pco
Find all real polynomials $P$ of degree $K$ having $k$ distinct real roots ($k \in \mathbb N$), such that for every root $a$ : $P(a+1)=1$.
1 reply
MTA_2024
14 minutes ago
pco
3 minutes ago
J
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Functional equation
socrates   30
N 28 minutes ago by NicoN9
Source: Baltic Way 2014, Problem 4
Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$
30 replies
socrates
Nov 11, 2014
NicoN9
28 minutes ago
J
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Algebra Functions
pear333   1
N 37 minutes ago by whwlqkd
Let $P(z)=z-1/z$. Prove that there does not exist a pair of rational numbers $x,y$ such that $P(x)+P(y)=4$.
1 reply
pear333
Today at 12:20 AM
whwlqkd
37 minutes ago
J
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Polynomial equation with rational numbers
Miquel-point   2
N 42 minutes ago by Assassino9931
Source: Romanian TST 1979 day 2 P1
Determine the polynomial $P\in \mathbb{R}[x]$ for which there exists $n\in \mathbb{Z}_{>0}$ such that for all $x\in \mathbb{Q}$ we have: \[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]
Dumitru Bușneag
2 replies
Miquel-point
Apr 15, 2023
Assassino9931
42 minutes ago
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BdMO National Higher Secondary 2019/2
Olympus_mountaineer   9
N Sep 15, 2021 by jasperE3
Source: BdMO National Higher Secondary 2019/2
Prove that,if $a,b,c$ are positive real numbers,
\[ \dfrac{a}{bc}+ \dfrac{b}{ca}+\dfrac{c}{ab}\geq \dfrac{2}{a}+\dfrac{2}{b}-\dfrac{2}{c}\]
9 replies
Olympus_mountaineer
Mar 4, 2019
jasperE3
Sep 15, 2021
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BdMO National Higher Secondary 2019/2
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Reveal topic tags
algebra
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Source: BdMO National Higher Secondary 2019/2
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Olympus_mountaineer
414 posts
Olympus_mountaineer
#1 h  Mar 4, 2019, 4:01 AM • 3 Y
Y by mathematicsy, Adventure10, Mango247
Prove that,if $a,b,c$ are positive real numbers,
\[ \dfrac{a}{bc}+ \dfrac{b}{ca}+\dfrac{c}{ab}\geq \dfrac{2}{a}+\dfrac{2}{b}-\dfrac{2}{c}\]
This post has been edited 1 time. Last edited by Olympus_mountaineer, Mar 4, 2019, 9:50 AM
Reason: Typo
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Olympus_mountaineer
414 posts
Olympus_mountaineer
#2 h  Mar 4, 2019, 4:07 AM • 3 Y
Y by Mr_pi, Adventure10, Mango247
Hint
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DievilOnlyM
504 posts
DievilOnlyM
#3 h  Mar 4, 2019, 7:38 AM • 2 Y
Y by Adventure10, Mango247
The inequality can be written as:
$\frac{a^2+b^2+c^2}{abc}\geq\frac{2bc+2ca-2ab}{abc}$
$<=>(a+b)^2-2c(b+a)+c^2\geq0$
$<=>(a+b-c)^2\geq0$
This post has been edited 1 time. Last edited by DievilOnlyM, Mar 4, 2019, 7:39 AM
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chem123
2000 posts
chem123
#4 h  Mar 4, 2019, 7:51 AM • 2 Y
Y by Adventure10, Mango247
The hint actually gives way to the answer...
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sqing
41020 posts
sqing
#5 h  Mar 4, 2019, 8:40 AM • 2 Y
Y by Adventure10, Mango247
Olympus_mountaineer wrote:
Prove that,if $a,b,c$ are positive real numbers,
\[ \dfrac{a}{bc}+ \dfrac{b}{ca}+\dfrac{c}{ab}\leq \dfrac{2}{a}+\dfrac{2}{b}-\dfrac{2}{c}\]
Maybe.\[ \dfrac{a}{bc}+ \dfrac{b}{ca}+\dfrac{c}{ab}\geq \dfrac{2}{a}+\dfrac{2}{b}-\dfrac{2}{c}\]
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Olympus_mountaineer
414 posts
Olympus_mountaineer
#6 h  Mar 4, 2019, 9:51 AM • 2 Y
Y by Adventure10, Mango247
Yes,I fixed the problem.
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Steve12345
618 posts
Steve12345
#7 h  Mar 4, 2019, 4:12 PM • 2 Y
Y by Adventure10, Mango247
https://artofproblemsolving.com/community/c6h53823p336839
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Olympus_mountaineer
414 posts
Olympus_mountaineer
#8 h  Mar 4, 2019, 4:15 PM • 4 Y
Y by Steve12345, Sazid, Adventure10, Mango247
Steve12345 wrote:
https://artofproblemsolving.com/community/c6h53823p336839

I am surprized that it is a well known inequality!
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snayer945gmail.com
37 posts
snayer945gmail.com
#9 h  Mar 20, 2020, 4:39 PM
Y by
use AM-GM inequality
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jasperE3
11091 posts
jasperE3
#10 h  Sep 15, 2021, 12:11 AM
Y by
$(a+b-c)^2\ge0\Rightarrow a^2+b^2+c^2\ge2ac+2bc-2ab\Rightarrow\frac a{bc}+\frac b{ca}+\frac c{ab}\ge\frac2a+\frac2b-\frac2c$
It holds for all $a,b,c$ with $a^2+b^2+c^2\ne0$.
This post has been edited 2 times. Last edited by jasperE3, Sep 15, 2021, 12:12 AM
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