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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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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H
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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A Characterization of Rectangles
buratinogigle   1
N 35 minutes ago by lbh_qys
Source: VN Math Olympiad For High School Students P8 - 2025
Prove that if a convex quadrilateral $ABCD$ satisfies the equation
\[ (AB + CD)^2 + (AD + BC)^2 = (AC + BD)^2, \]then $ABCD$ must be a rectangle.
1 reply
buratinogigle
Today at 1:35 AM
lbh_qys
35 minutes ago
J
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A Segment Bisection Problem
buratinogigle   1
N an hour ago by Giabach298
Source: VN Math Olympiad For High School Students P9 - 2025
In triangle $ABC$, let the incircle $\omega$ touch sides $BC, CA, AB$ at $D, E, F$, respectively. Let $P$ lie on the line through $D$ perpendicular to $BC$. Let $Q, R$ be the intersections of $PC, PB$ with $EF$, respectively. Let $K, L$ be the projections of $R, Q$ onto line $BC$. Let $M, N$ be the second intersections of $DQ, DR$ with the incircle $\omega$. Let $S$ be the intersection of $KM$ and $LN$. Prove that the line $DS$ bisects segment $QR$.
1 reply
buratinogigle
Today at 1:36 AM
Giabach298
an hour ago
J
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2017 PAMO Shortlsit: Power of a prime is a sum of cubes
DylanN   3
N an hour ago by AshAuktober
Source: 2017 Pan-African Shortlist - N2
For which prime numbers $p$ can we find three positive integers $n$, $x$ and $y$ such that $p^n = x^3 + y^3$?
3 replies
DylanN
May 5, 2019
AshAuktober
an hour ago
J
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Hard number theory
Hip1zzzil   14
N an hour ago by bonmath
Source: FKMO 2025 P6
Positive integers $a, b$ satisfy both of the following conditions.
For a positive integer $m$, if $m^2 \mid ab$, then $m = 1$.
There exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 = z^2 + w^2$ and $z^2 + w^2 > 0$.
Prove that there exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 + n = z^2 + w^2$, for each integer $n$.
14 replies
Hip1zzzil
Mar 30, 2025
bonmath
an hour ago
J
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Constant Angle Sum
i3435   6
N 2 hours ago by bin_sherlo
Source: AMASCIWLOFRIAA1PD (mock oly geo contest) P3
Let $ABC$ be a triangle with circumcircle $\Omega$, $A$-angle bisector $l_A$, and $A$-median $m_A$. Suppose that $l_A$ meets $\overline{BC}$ at $D$ and meets $\Omega$ again at $M$. A line $l$ parallel to $\overline{BC}$ meets $l_A$, $m_A$ at $G$, $N$ respectively, so that $G$ is between $A$ and $D$. The circle with diameter $\overline{AG}$ meets $\Omega$ again at $J$.

As $l$ varies, show that $\angle AMN + \angle DJG$ is constant.

MP8148
6 replies
i3435
May 11, 2021
bin_sherlo
2 hours ago
J
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NEPAL TST 2025 DAY 2
Tony_stark0094   8
N 2 hours ago by cursed_tangent1434
Consider an acute triangle $\Delta ABC$. Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.

Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$, respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$. Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$, and by $Q$ the intersection of ray $EB$ with $\Gamma$.

If $O$ is the circumcenter of $\Delta ABC$, prove that $O$, $D$, and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.

$\textbf{Proposed by Kritesh Dhakal, Nepal.}$
8 replies
Tony_stark0094
Apr 12, 2025
cursed_tangent1434
2 hours ago
J
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Interesting inequalities
sqing   4
N 2 hours ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ ab+bc+ca+abc=4$ . Prove that
$$k(a+b+c) -ab-bc\geq 4\sqrt{k(k+1)}-(k+4)$$Where $ k\geq \frac{16}{9}. $
$$ \frac{16}{9}(a+b+c) -ab-bc\geq \frac{28}{9}$$
4 replies
sqing
5 hours ago
sqing
2 hours ago
J
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NEPAL TST DAY 2 PROBLEM 2
Tony_stark0094   6
N 2 hours ago by cursed_tangent1434
Kritesh manages traffic on a $45 \times 45$ grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the $45 \times 45$ grid.

Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove $k$ cars from the $45 \times 45$ grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.

What is the minimum value of $k$ that guarantees that Kritesh's job is possible?

$\textbf{Proposed by Shining Sun. USA}$
6 replies
Tony_stark0094
Apr 12, 2025
cursed_tangent1434
2 hours ago
J
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State target p8 sol
EaZ_Shadow   101
N 2 hours ago by Craftybutterfly
This solution was so educational! It helped me understand how to solve this problem in many ways I couldn't, and I feel so enlightened!
101 replies
EaZ_Shadow
Apr 6, 2025
Craftybutterfly
2 hours ago
J
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NEPAL TST DAY-2 PROBLEM 1
Tony_stark0094   9
N 3 hours ago by cursed_tangent1434
Let the sequence $\{a_n\}_{n \geq 1}$ be defined by
\[ a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N} \]Prove that
\[ a_n^{2025} >n^{2024} \]for all positive integers $n \geq 2$.

$\textbf{Proposed by Prajit Adhikari, Nepal.}$
9 replies
Tony_stark0094
Apr 12, 2025
cursed_tangent1434
3 hours ago
J
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Math and AI 4 Girls
mkwhe   18
N 3 hours ago by EvaLin
Hey everyone!

The 2025 MA4G competition is now open!

Apply Here: https://xmathandai4girls.submittable.com/submit


Visit https://www.mathandai4girls.org/ to get started!

Feel free to PM or email mathandai4girls@yahoo.com if you have any questions!
18 replies
mkwhe
Apr 5, 2025
EvaLin
3 hours ago
J
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Inspired by Omerking
sqing   1
N 3 hours ago by lbh_qys
Source: Own
Let $ a,b,c>0 $ and $ ab+bc+ca\geq \dfrac{1}{3}. $ Prove that
$$ ka+ b+kc\geq \sqrt{\frac{4k-1}{3}}$$Where $ k\geq 1.$$$ 4a+ b+4c\geq \sqrt{5}$$
1 reply
sqing
3 hours ago
lbh_qys
3 hours ago
J
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The daily problem!
Leeoz   106
N 3 hours ago by Leeoz
Every day, I will try to post a new problem for you all to solve! If you want to post a daily problem, you can! :)

Please hide solutions and answers, hints are fine though! :)

Problems usually get harder throughout the week, so Sunday is the easiest and Saturday is the hardest!

Past Problems!
106 replies
Leeoz
Mar 21, 2025
Leeoz
3 hours ago
J
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2025 MATHCOUNTS State Hub
SirAppel   551
N 5 hours ago by DhruvJha
Previous Years' "Hubs": (2022) (2023) (2024)Please Read

Now that it's April and we're allowed to discuss ...
[list=disc]
[*] CA: 43 (45 44 43 43 43 42 42 41 41 41)
[*] NJ: 43 (45 44 44 43 39 42 40 40 39 38) *
[*] NY: 42 (43 42 42 42 41 40)
[*] TX: 42 (43 43 43 42 42 40 40 38 38 38)
[*] MA: 41 (45 43 42 41)
[*] WA: 41 (41 45 42 41 41 41 41 41 41 40) *
[*]VA: 40 (41 40 40 40)
[*] FL: 39 (42 41 40 39 38 37 37)
[*] IN: 39 (41 40 40 39 36 35 35 35 34 34)
[*] NC: 39 (42 42 41 39)
[*] IL: 38 (41 40 39 38 38 38)
[*] OR: 38 (44 40? 38 38)
[*] PA: 38 (41 40 40 38 38 37 36 36 34 34) *
[*] MD: 37 (43 39 39 37 37 37)
[*] AZ: 36 (40? 39? 39 36)
[*] CT: 36 (44 38 38 36 35 35 34 34 34 33 33)
[*] MI: 36 (39 41 41 36 37 37 36 36 36 36) *
[*] MN: 36 (40 36 36 36 35 35 35 34)
[*] CO: 35 (41 37 37 35 35 35 ?? 31 31 30) *
[*] GA: 35 (38 37 36 35 34 34 34 34 34 33)
[*] OH: 35 (41 37 36 35)
[*] AR: 34 (46 45 35 34 33 31 31 31 29 29)
[*] NV: 34 (41 38 ?? 34)
[*] TN: 34 (38 ?? ?? 34)
[*] WI: 34 (40 37 37 34 35 30 28 29 29 29) *
[*] HI: 32 (35 34 32 32)
[*] NH: 31 (42 35 33 31 30)
[*] DE: 30 (34 33 32 30 30 29 28 27 26? 24)
[*] SC: 30 (33 33 31 30)
[*] IA: 29 (33 30 31 29 29 29 29 29 29 29 29 29) *
[*] NE: 28 (34 30 28 28 27 27 26 26 25 25)
[*] SD: 22 (30 29 24 22 22 22 21 21 20 20)
[/list]
Cutoffs Unknown

* means that CDR is official in that state.

Notes

For those asking about the removal of the tiers, I'd like to quote Jason himself:
[quote=peace09]
learn from my mistakes
[/quote]

Help contribute by sharing your state's cutoffs!
551 replies
SirAppel
Apr 1, 2025
DhruvJha
5 hours ago
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J
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mathcounts state discussion
Soupboy0   64
N Apr 5, 2025 by Andyluo
les goo its finally april
64 replies
Soupboy0
Apr 1, 2025
Andyluo
Apr 5, 2025
J
mathcounts state discussion
G H J
Reveal topic tags
MATHCOUNTS
L
G H BBookmark kLocked kLocked NReply
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Elodie-AW
238 posts
Elodie-AW
#51 h  Apr 3, 2025, 6:52 PM
Y by
I got pretty nervous and messed up kind of badly :oops:
Z K Y
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iwastedmyusername
78 posts
iwastedmyusername
#52 h  Apr 3, 2025, 7:22 PM
Y by
was there a non bashy way to do target #4
Z K Y
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jb2015007
1862 posts
jb2015007
#53 h  Apr 3, 2025, 7:23 PM
Y by
yes js let the side lengths be a and b, and the hypotneuse is 40-a-b, and then solve
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iwastedmyusername
78 posts
iwastedmyusername
#54 h  Apr 3, 2025, 7:26 PM
Y by
wouldnt you have to square 40-a-b though
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jb2015007
1862 posts
jb2015007
#55 h  Apr 3, 2025, 7:28 PM
Y by
i js timed myself and did it in 15 secs
its not that big of a deal
Z K Y
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Bummer12345
132 posts
Bummer12345
#56 h  Apr 3, 2025, 8:28 PM
Y by
cheese sol
Z K Y
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gamma_pi
153 posts
gamma_pi
#57 h  Apr 3, 2025, 9:14 PM • 1 Y
Y by Eddie_tiger
jb2015007 wrote:
nah it was 27/2 if i remember correctly

yeah i think it is
Z K Y
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EightYearLetterman
14 posts
EightYearLetterman
#58 h  Apr 3, 2025, 9:17 PM
Y by
iwastedmyusername wrote:
wouldnt you have to square 40-a-b though

a^2 + b^2 = 1600 - 80(a+b) + (a+b)^2
0 = 1600 - 80(40-c) + 2ab
-80c = 1600 - 80*40 + 80
c = -20 + 40 - 1
Z K Y
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Pi_isCool31415
25 posts
Pi_isCool31415
#59 h  Apr 4, 2025, 12:30 AM
Y by
target 6 u could just induct and not prove it lol
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Pi_isCool31415
25 posts
Pi_isCool31415
#60 h  Apr 4, 2025, 12:31 AM
Y by
Bummer12345 wrote:
cheese sol

bruh even worse sol, calc bash and guess based on answer form
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mathprodigy2011
301 posts
mathprodigy2011
#61 h  Apr 4, 2025, 11:24 PM
Y by
iwastedmyusername wrote:
was there a non bashy way to do target #4

Click to reveal hidden text
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chesswillow
205 posts
chesswillow
#62 h  Apr 5, 2025, 3:00 AM
Y by
How were you supposed to do target 6 though

I randomly added up all the squares from 1 to 2025 and divided that by the sum of all integers 1-2025 (I don't even know why) and somehow got the right answer.
Z K Y
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Soupboy0
322 posts
Soupboy0
#63 h  Apr 5, 2025, 3:01 AM
Y by
i just used arithmetico geometric sequence
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sadas123
1214 posts
sadas123
#64 h  Apr 5, 2025, 11:09 AM
Y by
chesswillow wrote:
How were you supposed to do target 6 though

I randomly added up all the squares from 1 to 2025 and divided that by the sum of all integers 1-2025 (I don't even know why) and somehow got the right answer.

I used a method kind of guessing but we can see the number of subsets that have 2025 as the greateset is (Not Blank)

2^2025-1

Then the number of subsets that have 2024 as the greatest is:

2^2024 -1

Then the numbers of subsets that have 2023 as the greatest is:

2^2023-1

We establish a pattern similar to an geometric sequence but not exact.... 1/2, 1/4, 1/8, 1/16, 1/32,1/64,1/128,1/256 So with this pattern we already know that the numbers that count the most are the first 3 numbers that I listed but if you look at it carefully you can see that when you subtract some integer n from n_1+n_2 we see what it is always 1/2^n so we can use this finding to see that is has to be somewhere between 2024>=x>=2023
but we again see that 1/4-1/8=1/8 so we are done and our final answer is $2024$
This post has been edited 1 time. Last edited by sadas123, Apr 5, 2025, 11:09 AM
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Andyluo
920 posts
Andyluo
#65 h  Apr 5, 2025, 11:41 AM
Y by
you can also try cases with n=3-6 which is what I did to convince myself that $n-1$ was the answer
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