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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

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[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
J
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k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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Point satisfies triple property
62861   36
N 4 minutes ago by cursed_tangent1434
Source: USA Winter Team Selection Test #2 for IMO 2018, Problem 2
Let $ABCD$ be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at $H$. Denote by $M$ and $N$ the midpoints of $\overline{BC}$ and $\overline{CD}$. Rays $MH$ and $NH$ meet $\overline{AD}$ and $\overline{AB}$ at $S$ and $T$, respectively. Prove that there exists a point $E$, lying outside quadrilateral $ABCD$, such that
[list]
[*] ray $EH$ bisects both angles $\angle BES$, $\angle TED$, and
[*] $\angle BEN = \angle MED$.
[/list]

Proposed by Evan Chen
36 replies
62861
Jan 22, 2018
cursed_tangent1434
4 minutes ago
J
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Inspired by 2025 Xinjiang
sqing   1
N 10 minutes ago by sqing
Source: Own
Let $ a,b,c >0 . $ Prove that
$$ \left(1+\frac {a} { b}\right)\left(2+\frac {a}{ b}+\frac {b}{ c}\right) \left(1+\frac {a}{b}+\frac {b}{ c}+\frac {c}{ a}\right) \geq 12+8\sqrt 2 $$
1 reply
sqing
Yesterday at 5:32 PM
sqing
10 minutes ago
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Inspired by 2025 Beijing
sqing   4
N 18 minutes ago by pooh123
Source: Own
Let $ a,b,c,d >0 $ and $ (a^2+b^2+c^2)(b^2+c^2+d^2)=36. $ Prove that
$$ab^2c^2d \leq 8$$$$a^2bcd^2 \leq 16$$$$ ab^3c^3d \leq \frac{2187}{128}$$$$ a^3bcd^3 \leq \frac{2187}{32}$$
4 replies
sqing
Yesterday at 4:56 PM
pooh123
18 minutes ago
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Find the minimum
sqing   27
N 27 minutes ago by sqing
Source: Zhangyanzong
Let $a,b$ be positive real numbers such that $a^2b^2+\frac{4a}{a+b}=4.$ Find the minimum value of $a+2b.$
27 replies
sqing
Sep 4, 2018
sqing
27 minutes ago
J
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Sharygin 2025 CR P15
Gengar_in_Galar   7
N 30 minutes ago by Giant_PT
Source: Sharygin 2025
A point $C$ lies on the bisector of an acute angle with vertex $S$. Let $P$, $Q$ be the projections of $C$ to the sidelines of the angle. The circle centered at $C$ with radius $PQ$ meets the sidelines at points $A$ and $B$ such that $SA\ne SB$. Prove that the circle with center $A$ touching $SB$ and the circle with center $B$ touching $SA$ are tangent.
Proposed by: A.Zaslavsky
7 replies
Gengar_in_Galar
Mar 10, 2025
Giant_PT
30 minutes ago
J
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Inequality olympiad algebra
Foxellar   1
N 33 minutes ago by sqing
Given that \( a, b, c \) are nonzero real numbers such that
\[ \frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b}, \]let \( M \) be the maximum value of the expression
\[ \frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}. \]Determine the sum of the numerator and denominator of the simplified fraction representing \( M \).
1 reply
Foxellar
5 hours ago
sqing
33 minutes ago
J
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Inspired by RMO 2006
sqing   2
N 40 minutes ago by sqing
Source: Own
Let $ a,b >0 . $ Prove that
$$ \frac {a^{2}+1}{b+k}+\frac { b^{2}+1}{ka+1}+\frac {2}{a+kb} \geq \frac {6}{k+1} $$Where $k\geq 0.03 $
$$ \frac {a^{2}+1}{b+1}+\frac { b^{2}+1}{a+1}+\frac {2}{a+b} \geq 3 $$
2 replies
sqing
Yesterday at 3:24 PM
sqing
40 minutes ago
J
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Inspired by 2025 Xinjiang
sqing   3
N an hour ago by sqing
Source: Own
Let $ a,b >0 . $ Prove that
$$ \left(1+\frac {a} { b}\right)\left(6+\frac {b}{a}\right) \left(2+\frac {a}{b}+\frac {b}{ a}\right) \geq\frac {625}{ 12}$$$$ \left(1+\frac {a} { b}\right)\left(6+\frac {a}{b}\right) \left(2+\frac {a}{b}+\frac {b}{ a}\right) \geq29+6\sqrt 6$$$$ \left(1+\frac {a} { b}\right)\left(2+\frac {b}{ a}\right) \left(2+\frac {a}{b}+\frac {b}{ a}\right) \geq \frac{3(63+11\sqrt{33})}{16} $$$$ \left(1+\frac {a} { b}\right)\left(2+\frac {a}{ b}\right) \left(2+\frac {a}{b}+\frac {b}{ a}\right) \geq \frac{223+70\sqrt{10}}{27} $$
3 replies
sqing
Yesterday at 5:56 PM
sqing
an hour ago
J
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interesting diophantiic fe in natural numbers
skellyrah   2
N an hour ago by SYBARUPEMULA
Find all functions \( f : \mathbb{N} \to \mathbb{N} \) such that for all \( m, n \in \mathbb{N} \),
\[ mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!). \]
2 replies
skellyrah
Yesterday at 8:01 AM
SYBARUPEMULA
an hour ago
J
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PHP on subsets
SYBARUPEMULA   0
an hour ago
Source: inspired by Romania 2009
Let $X$ be a set of $102$ elements. Let $A_1, A_2, A_3, ..., A_{101}$ be subsets of $X$ such that the union of any $50$ of them has more than $100$ elements. Show that there's a member of $X$ that occurs in at least $7$ different subsets $A_j$.
0 replies
SYBARUPEMULA
an hour ago
0 replies
J
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Moscow Geometry Problems
nataliaonline75   1
N 2 hours ago by MathLuis
Source: MMO 2003 10.4
Let M be the intersection point of the medians of ABC. On the perpendiculars dropped from M to sides BC, AC, AB, points A1, B1, C1 are taken, respectively, with A1B1 perpendicular to MC and A1C1 perpendicular to MB. prove that M is the intersections pf the medians in A1B1C1.
Any solutions without vectors? :)
1 reply
nataliaonline75
Jul 9, 2024
MathLuis
2 hours ago
J
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Serbian selection contest for the IMO 2025 - P4
OgnjenTesic   1
N 2 hours ago by nataliaonline75
Source: Serbian selection contest for the IMO 2025
For a permutation $\pi$ of the set $A = \{1, 2, \ldots, 2025\}$, define its colorfulness as the greatest natural number $k$ such that:
- For all $1 \le i, j \le 2025$, $i \ne j$, if $|i - j| < k$, then $|\pi(i) - \pi(j)| \ge k$.
What is the maximum possible colorfulness of a permutation of the set $A$? Determine how many such permutations have maximal colorfulness.

Proposed by Pavle Martinović
1 reply
1 viewing
OgnjenTesic
May 22, 2025
nataliaonline75
2 hours ago
J
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Another OI geometry problem
chengbilly   7
N 2 hours ago by MathLuis
Source: own
Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$. $H$ the orthocenter of triangle $BIC$ . The incircle of $ABC$ touches side $AC,AB$ at $E,F$ respectively. Suppose that $\odot (AEF)$ and $\odot (AIO)$ intersects $\odot (ABC)$ at $S$ and $T$ respectively (differ from $A$). Prove that $T,H,I,S$ lies on a circle.

($\odot (ABC)$ denote the circumcircle of $ABC$)
7 replies
chengbilly
Apr 14, 2021
MathLuis
2 hours ago
J
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Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   2
N 2 hours ago by nataliaonline75
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
2 replies
OgnjenTesic
May 22, 2025
nataliaonline75
2 hours ago
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J
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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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Reveal topic tags
inequalities
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Cobedangiu
70 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
70 posts
Cobedangiu
#4 h  Apr 3, 2025, 9:03 AM
Y by
...........
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arqady
30258 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 $uvw$.
Z K Y
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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
148 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
211 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 $a+b+c=3u , ab+bc+ca=3v^2 , abc=w^3.$
Z K Y
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Jackson0423
104 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
30258 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
211 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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