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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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geometry
srnjbr   3
N 7 minutes ago by maromex
the points f,n,o, t a lie in the plane such that the triangles tfo ton are similar, preserving direction and order, and fano is a parallelogram. show that of×on=oa×ot.
3 replies
srnjbr
Mar 15, 2025
maromex
7 minutes ago
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Good Permutations in Modulo n
swynca   6
N 11 minutes ago by dangerousliri
Source: BMO 2025 P1
An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$, such that:
$(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$;
$(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$.
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
6 replies
+1 w
swynca
5 hours ago
dangerousliri
11 minutes ago
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all functions satisfying f(x+yf(x))+y = xy + f(x+y)
falantrng   24
N 11 minutes ago by Springles
Source: Balkan MO 2025 P3
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[f(x+yf(x))+y = xy + f(x+y).\]
Proposed by Giannis Galamatis, Greece
24 replies
falantrng
Today at 11:52 AM
Springles
11 minutes ago
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Sum of angles are equal
mofumofu   18
N 32 minutes ago by zuat.e
Source: China Mathematical Olympiad 2021 P4
In acute triangle $ABC (AB>AC)$, $M$ is the midpoint of minor arc $BC$, $O$ is the circumcenter of $(ABC)$ and $AK$ is its diameter. The line parallel to $AM$ through $O$ meets segment $AB$ at $D$, and $CA$ extended at $E$. Lines $BM$ and $CK$ meet at $P$, lines $BK$ and $CM$ meet at $Q$. Prove that $\angle OPB+\angle OEB =\angle OQC+\angle ODC$.
18 replies
mofumofu
Nov 25, 2020
zuat.e
32 minutes ago
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No more topics!
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hard problem
Cobedangiu   8
N Apr 23, 2025 by IceyCold
Let $x,y,z>0$ and $xy+yz+zx=3$ : Prove that :
$\sum \ \frac{x}{y+z}\ge\sum \frac{1}{\sqrt{x+3}}$
8 replies
Cobedangiu
Apr 2, 2025
IceyCold
Apr 23, 2025
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hard problem
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inequalities
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Cobedangiu
56 posts
Cobedangiu
#1 h  Apr 2, 2025, 6:11 PM
Y by
Let $x,y,z>0$ and $xy+yz+zx=3$ : Prove that :
$\sum \ \frac{x}{y+z}\ge\sum \frac{1}{\sqrt{x+3}}$
Z K Y
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Cobedangiu
56 posts
Cobedangiu
#4 h  Apr 3, 2025, 9:03 AM
Y by
...........
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arqady
30216 posts
arqady
#5 h  Apr 3, 2025, 10:37 AM
Y by
Cobedangiu wrote:
Let $x,y,z>0$ and $xy+yz+zx=3$ : Prove that :
$\sum \ \frac{x}{y+z}\ge\sum \frac{1}{\sqrt{x+3}}$
Use Jensen and $uvw$.
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Math-Problem-Solving
66 posts
Math-Problem-Solving
#6 h  Apr 22, 2025, 3:32 PM
Y by
arqady wrote:
Cobedangiu wrote:
Let $x,y,z>0$ and $xy+yz+zx=3$ : Prove that :
$\sum \ \frac{x}{y+z}\ge\sum \frac{1}{\sqrt{x+3}}$
Use Jensen and $uvw$.

What does this $uvw$ stand for?
This post has been edited 1 time. Last edited by Math-Problem-Solving, Apr 22, 2025, 3:33 PM
Reason: Err
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giangtruong13
133 posts
giangtruong13
#7 h  Apr 22, 2025, 3:39 PM
Y by
$x+y+z=u$,$xy+yz+zx=v$,$xyz=w$
This post has been edited 1 time. Last edited by giangtruong13, Apr 22, 2025, 3:39 PM
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IceyCold
202 posts
IceyCold
#8 h  Apr 22, 2025, 3:54 PM
Y by
giangtruong13 wrote:
$x+y+z=u$,$xy+yz+zx=v$,$xyz=w$

That's $pqr$,not $uvw$.

$uvw$ should be $a+b+c=3u , ab+bc+ca=3v^2 , abc=w^3.$
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Jackson0423
48 posts
Jackson0423
#9 h  Apr 22, 2025, 4:01 PM
Y by
pqr LMAO
I prefer pqr
This post has been edited 1 time. Last edited by Jackson0423, Apr 22, 2025, 4:01 PM
Reason: d
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arqady
30216 posts
arqady
#10 h  Apr 22, 2025, 7:48 PM
Y by
Math-Problem-Solving wrote:

What does this $uvw$ stand for?
$x+y+z=3u$, $xy+xz+yz=3v^2$ and $xyz=w^3$.
Z K Y
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IceyCold
202 posts
IceyCold
#11 h  Apr 23, 2025, 1:52 PM
Y by
Jackson0423 wrote:
$pqr$ LMAO
I prefer $pqr$

I also prefer $pqr$,but $uvw$ just seems stronger-
Why is this though?
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