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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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0 replies
jlacosta
Apr 2, 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
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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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KSEA NMSC Mock Contest Group B (Last Problem)
Shiyul   2
N 14 minutes ago by Shiyul
Let $a_n$ be a sequence defined by $a_n = a^2 + 1$. Then the product of four consecutive terms in $a_n$ can be written as the product of two terms in $a_n$. Find $p + q$ if $(a_(11))(a_(12))(a_(13))(a_(14)) = (a_p)(a_q)$.
2 replies
1 viewing
Shiyul
4 hours ago
Shiyul
14 minutes ago
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Problem 2
SlovEcience   1
N an hour ago by Primeniyazidayi
Let \( a, n \) be positive integers and \( p \) be an odd prime such that:
\[ a^p \equiv 1 \pmod{p^n}. \]Prove that:
\[ a \equiv 1 \pmod{p^{n-1}}. \]
1 reply
SlovEcience
3 hours ago
Primeniyazidayi
an hour ago
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H not needed
dchenmathcounts   45
N an hour ago by EpicBird08
Source: USEMO 2019/1
Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar.

Robin Son
45 replies
dchenmathcounts
May 23, 2020
EpicBird08
an hour ago
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Problem 1
blug   4
N 2 hours ago by grupyorum
Source: Polish Math Olympiad 2025 Finals P1
Find all $(a, b, c, d)\in \mathbb{R}$ satisfying
\[\begin{aligned} \begin{cases} a+b+c+d=0,\\ a^2+b^2+c^2+d^2=12,\\ abcd=-3.\\ \end{cases} \end{aligned}\]
4 replies
blug
Today at 11:46 AM
grupyorum
2 hours ago
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A board with crosses that we color
nAalniaOMliO   3
N 2 hours ago by nAalniaOMliO
Source: Belarusian National Olympiad 2025
In some cells of the table $2025 \times 2025$ crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least $k$ crosses.
Find the minimal $k$ for which it is always possible to color crosses in two colors such that any balanced set has crosses of both colors.
3 replies
nAalniaOMliO
Mar 28, 2025
nAalniaOMliO
2 hours ago
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April Fools Geometry
awesomeming327.   6
N 2 hours ago by GreekIdiot
Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ be the projection from $A$ onto $BC$. Let $E$ be a point on the extension of $AD$ past $D$ such that $\angle BAC+\angle BEC=90^\circ$. Let $L$ be on the perpendicular bisector of $AE$ such that $L$ and $C$ are on the same side of $AE$ and
\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]Let the reflection of $D$ across $AB$ and $AC$ be $W$ and $Y$, respectively. Let $X\in AW$ and $Z\in AY$ such that $\angle XBE=\angle ZCE=90^\circ$. Let $EX$ and $EZ$ intersect the circumcircles of $EBD$ and $ECD$ at $J$ and $K$, respectively. Let $LB$ and $LC$ intersect $WJ$ and $YK$ at $P$ and $Q$. Let $PQ$ intersect $BC$ at $F$. Prove that $FB/FC=DB/DC$.
6 replies
awesomeming327.
Apr 1, 2025
GreekIdiot
2 hours ago
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Functional equations
hanzo.ei   14
N 2 hours ago by jasperE3
Source: Greekldiot
Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$
14 replies
hanzo.ei
Mar 29, 2025
jasperE3
2 hours ago
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Problem 1
SlovEcience   2
N 2 hours ago by Raven_of_the_old
Prove that
\[ C(p-1, k-1) \equiv (-1)^{k-1} \pmod{p} \]for \( 1 \leq k \leq p-1 \), where \( C(n, m) \) is the binomial coefficient \( n \) choose \( m \).
2 replies
SlovEcience
4 hours ago
Raven_of_the_old
2 hours ago
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Conditional maximum
giangtruong13   1
N 2 hours ago by giangtruong13
Source: Specialized Math
Let $a,b$ satisfy that: $1 \leq a \leq2$ and $1 \leq b \leq 2$. Find the maximum: $$A=(a+b^2+\frac{4}{a^2}+\frac{2}{b})(b+a^2+\frac{4}{b^2}+\frac{2}{a})$$
1 reply
giangtruong13
Mar 22, 2025
giangtruong13
2 hours ago
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four variables inequality
JK1603JK   0
2 hours ago
Source: unknown?
Prove that $$27(a^4+b^4+c^4+d^4)+148abcd\ge (a+b+c+d)^4,\ \ \forall a,b,c,d\ge 0.$$
0 replies
JK1603JK
2 hours ago
0 replies
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a hard geometry problen
Tuguldur   0
3 hours ago
Let $ABCD$ be a convex quadrilateral. Suppose that the circles with diameters $AB$ and $CD$ intersect at points $X$ and $Y$. Let $P=AC\cap BD$ and $Q=AD\cap BC$. Prove that the points $P$, $Q$, $X$ and $Y$ are concyclic.
( $AB$ and $CD$ are not the diagnols)
0 replies
Tuguldur
3 hours ago
0 replies
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Inequalities
sqing   4
N 3 hours ago by sqing
Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=11.$ Prove that
$$a+ab+abc\leq\frac{49}{6}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=10.$ Prove that
$$a+ab+abc\leq\frac{169}{24}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=14.$ Prove that
$$a+ab+abc\leq\frac{63+5\sqrt 5}{6}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=32.$ Prove that
$$a+ab+abc\leq48+\frac{64\sqrt{2}}{3}$$
4 replies
sqing
Apr 1, 2025
sqing
3 hours ago
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Geo Mock #9
Bluesoul   1
N 3 hours ago by vanstraelen
Consider $\triangle{ABC}$ with $AB=12, AC=22$. The points $D,E$ lie on $AB,AC$ respectively, such that $\frac{AD}{BD}=\frac{AE}{CE}=3$. Extend $CD,BE$ to meet the circumcircle of $\triangle{ABC}$ at $P,Q$ respectively. Let the circumcircles of $\triangle{ADP}, \triangle{AEQ}$ meet at points $A,T$. Extend $AT$ to $BC$ at $R$, given $AR=16$, find $[ABC]$.
1 reply
Bluesoul
Apr 1, 2025
vanstraelen
3 hours ago
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Regarding IMO prepartion
omega2007   0
3 hours ago
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compilled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your prespective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
0 replies
omega2007
3 hours ago
0 replies
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An inequality
JK1603JK   3
N Apr 1, 2025 by lbh_qys
Let a,b,c\ge 0: ab+bc+ca>0 then prove \frac{4ab+5c^2}{a+b}+\frac{4bc+5a^2}{b+c}+\frac{4ca+5b^2}{c+a}\ge \frac{3}{2}\cdot\frac{(a+b+c)^3}{ab+bc+ca}.
3 replies
JK1603JK
Mar 31, 2025
lbh_qys
Apr 1, 2025
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An inequality
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Reveal topic tags
inequalities
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JK1603JK
42 posts
JK1603JK
#1 h  Mar 31, 2025, 7:11 AM
Y by
Let a,b,c\ge 0: ab+bc+ca>0 then prove \frac{4ab+5c^2}{a+b}+\frac{4bc+5a^2}{b+c}+\frac{4ca+5b^2}{c+a}\ge \frac{3}{2}\cdot\frac{(a+b+c)^3}{ab+bc+ca}.
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lbh_qys
476 posts
lbh_qys
#2 h  Mar 31, 2025, 9:27 AM
Y by
JK1603JK wrote:
Let $a,b,c\ge 0$: $ab+bc+ca>0$ then prove $\frac{4ab+5c^2}{a+b}+\frac{4bc+5a^2}{b+c}+\frac{4ca+5b^2}{c+a}\ge \frac{3}{2}\cdot\frac{(a+b+c)^3}{ab+bc+ca}.$

It appears that this inequality can be proven using the following generalization of the Iran 96 inequality

\[ \frac{15}{16} \cdot \frac{\sum a(a-b)(a-c)}{\prod (a+b)} \leq \left(\sum ab\right)\left(\sum \frac{1}{(a+b)^2}\right) -\frac 94\leq \frac{\sum a(a-b)(a-c)}{\prod (a+b)} \]
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JK1603JK
42 posts
JK1603JK
#3 h  Apr 1, 2025, 8:14 AM
Y by
Can you explain more detail?
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lbh_qys
476 posts
lbh_qys
#4 h  Apr 1, 2025, 9:22 AM
Y by
We have
\[ \frac{4ab+5c^2}{a+b}=\frac{4(a+c)(b+c)}{a+b}+\frac{c^2}{a+b}-4c. \]According to the above inequality, we have
\[ 4\Bigl(\sum ab\Bigr) \sum \frac{(a+c)(b+c)}{a+b}=4\Bigl(\sum ab\Bigr)\prod (a+b)\sum\frac{1}{(a+b)^2}\ge 9\prod (a+b)+\frac{15}{4}\sum a(a-b)(a-c)\ge 9\prod (a+b)+\frac{1}{2}\sum a(a-b)(a-c), \]and
\[ \Bigl(\sum ab\Bigr)\sum\frac{c^2}{a+b}=\sum c^3+abc\sum\frac{c}{a+b}\ge \sum c^3+\frac{3}{2}abc. \]Thus, it suffices to prove
\[ 9\prod (a+b)+\frac{1}{2}\sum a(a-b)(a-c)+\sum c^3+\frac{3}{2}abc\ge \frac{3}{2}(a+b+c)^3+4(ab+bc+ca)(a+b+c). \]In fact, this is an identity; hence, it is only necessary to prove the generalized form of the Iran 96 inequality as stated above.
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