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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
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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Inspired by old results
sqing   6
N an hour ago by sqing
Source: Own
Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that
$$ \frac{2}{a}+\frac {2}{ab}+\frac{1}{abc}\geq 4$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{2}{abc}\geq 2+\sqrt 3$$$$ \frac{3}{a}+\frac {3}{ab}+\frac{1}{abc}\geq\frac {7+\sqrt {13}}{2}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{3}{abc}\geq\frac {5+\sqrt {21}}{2}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{4}{abc}\geq 3+2\sqrt 2$$
6 replies
sqing
Apr 26, 2025
sqing
an hour ago
J
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Integer-Valued FE comes again
lminsl   206
N an hour ago by anudeep
Source: IMO 2019 Problem 1
Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$Proposed by Liam Baker, South Africa
206 replies
lminsl
Jul 16, 2019
anudeep
an hour ago
J
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Integer representation
RL_parkgong_0106   2
N an hour ago by maromex
Source: Own
Show that for any positive integer $n$, there exists some positive integer $k$ that makes the following equation have no integer root $(x_1, x_2, x_3, \dots, x_n)$.

$$x_1^{2^1}+x_2^{2^2}+x_3^{2^3}+\dots+x_n^{2^n}=k$$
2 replies
RL_parkgong_0106
Apr 22, 2025
maromex
an hour ago
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geometry+algebra(ver beatiful)
ehsan2004   7
N an hour ago by NicoN9
Source: Serbia and Montenegro 2004
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.
7 replies
ehsan2004
Aug 11, 2005
NicoN9
an hour ago
J
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Japan Mathematical Olympiad Preliminary 2007 Problem 3
Kunihiko_Chikaya   1
N an hour ago by Mathzeus1024
On a plane given the line segment with length 7. The distance of a point $P$ and the segment is 3. Find the possible minimum value of $AP*BP.$
1 reply
Kunihiko_Chikaya
Jan 18, 2007
Mathzeus1024
an hour ago
J
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Polynomial approximation and intersections
egxa   2
N an hour ago by iliya8788
Source: All Russian 2025 10.6
What is the smallest value of \( k \) such that for any polynomial \( f(x) \) of degree $100$ with real coefficients, there exists a polynomial \( g(x) \) of degree at most \( k \) with real coefficients such that the graphs of \( y = f(x) \) and \( y = g(x) \) intersect at exactly $100$ points?
2 replies
egxa
Apr 18, 2025
iliya8788
an hour ago
J
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Quadrilateral geo
a_507_bc   16
N 2 hours ago by zuat.e
Source: Mexico 2023/3
Let $ABCD$ be a convex quadrilateral. If $M, N, K$ are the midpoints of the segments $AB, BC$, and $CD$, respectively, and there is also a point $P$ inside the quadrilateral $ABCD$ such that, $\angle BPN= \angle PAD$ and $\angle CPN=\angle PDA$. Show that $AB \cdot CD=4PM\cdot PK$.
16 replies
a_507_bc
Nov 8, 2023
zuat.e
2 hours ago
J
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Very easy number theory
darij grinberg   102
N 2 hours ago by ND_
Source: IMO Shortlist 2000, N1, 6th Kolmogorov Cup, 1-8 December 2002, 1st round, 1st league,
Determine all positive integers $ n\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]
102 replies
darij grinberg
Aug 6, 2004
ND_
2 hours ago
J
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9 Mathpath vs. AMSP
FuturePanda   30
N 2 hours ago by Pengu14
Hi everyone,

For an AIME score of 7-11, would you recommend MathPath or AMSP Level 2/3?

Thanks in advance!
Also people who have gone to them, please tell me more about the programs!
30 replies
FuturePanda
Jan 30, 2025
Pengu14
2 hours ago
J
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Positive integers and a_n
Kunihiko_Chikaya   1
N 2 hours ago by Mathzeus1024
Source: 2018 The University entrance of exam / Humanities, Problem 2
Define a sequence $a_1,\ a_2\cdots$ by the expression

$$a_n=\frac{_{2n}C_n}{n!}\ \ (n=1,\ 2,\ \cdots\cdots).$$
(1) Compare the magnitudes of the values $a_7$ and 1.

(2) Let $n\geq 2.$ Find the range of $n$ such that $\frac{a_n}{a_{n-1}}<1.$

(3) Determine all of the integers $n\geq 1$ such that $a_n$ is an integer.
1 reply
1 viewing
Kunihiko_Chikaya
Feb 25, 2018
Mathzeus1024
2 hours ago
J
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Killer NT that nobody solved (also my hardest NT ever created)
mshtand1   13
N 2 hours ago by mshtand1
Source: Ukraine IMO 2025 TST P8
A positive integer number \( a \) is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence \( \{b_k\}_{k=1}^{\infty} \), where
\[ b_k = a^{k^k} \cdot 2^{2^k - k} + 1. \]
Proposed by Arsenii Nikolaev and Mykhailo Shtandenko
13 replies
mshtand1
Apr 19, 2025
mshtand1
2 hours ago
J
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Jumping on Lily Pads to Avoid a Snake
brandbest1   53
N Today at 5:14 AM by ESAOPS
Source: 2014 AMC 10B #25 & 2014 AMC 12B #22
In a small pond there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake?

$ \textbf {(A) } \frac{32}{79} \qquad \textbf {(B) } \frac{161}{384} \qquad \textbf {(C) } \frac{63}{146} \qquad \textbf {(D) } \frac{7}{16} \qquad \textbf {(E) } \frac{1}{2} $
53 replies
brandbest1
Feb 20, 2014
ESAOPS
Today at 5:14 AM
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JSMC texas
BossLu99   19
N Today at 4:24 AM by jkim0656
who is going to JSMC texas
19 replies
BossLu99
Yesterday at 1:32 PM
jkim0656
Today at 4:24 AM
J
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sussy baka stop intersecting in my lattice points
Spectator   22
N Today at 3:55 AM by xHypotenuse
Source: 2022 AMC 10A #25
Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \cup S$, and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S$. See the figure (not drawn to scale).

IMAGE

The fraction of lattice points in $S$ that are in $S \cap T$ is 27 times the fraction of lattice points in $R$ that are in $R \cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$?

$\textbf{(A) }336\qquad\textbf{(B) }337\qquad\textbf{(C) }338\qquad\textbf{(D) }339\qquad\textbf{(E) }340$
22 replies
Spectator
Nov 11, 2022
xHypotenuse
Today at 3:55 AM
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Colored Pencils for Math Competitions
Owinner   17
N Apr 1, 2025 by lord_of_the_rook
I've heard using colored pencils is really useful for geometry problems. Is this only for very hard problems, or can it be used in MATHCOUNTS/AMC 8/10? An example problem would be much appreciated.
17 replies
Owinner
Mar 29, 2025
lord_of_the_rook
Apr 1, 2025
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Colored Pencils for Math Competitions
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Reveal topic tags
geometry
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Owinner
53 posts
Owinner
#1 h  Mar 29, 2025, 5:56 PM
Y by
I've heard using colored pencils is really useful for geometry problems. Is this only for very hard problems, or can it be used in MATHCOUNTS/AMC 8/10? An example problem would be much appreciated.
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mathprodigy2011
324 posts
mathprodigy2011
#2 h  Mar 29, 2025, 6:49 PM
Y by
Owinner wrote:
I've heard using colored pencils is really useful for geometry problems. Is this only for very hard problems, or can it be used in MATHCOUNTS/AMC 8/10? An example problem would be much appreciated.

i think colored pencils are more useful for really complicated and involved geometry problems. So I would think they are useful for AMC 10 but mathcounts and amc8 usually have geometry problems with simple-ish solutions
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LearnMath_105
149 posts
LearnMath_105
#3 h  Mar 29, 2025, 7:19 PM
Y by
I would say that they are most helpful for marking concyclic/colinear/concurrent objects so later aime - olyis where its most commonly used
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Andyluo
945 posts
Andyluo
#4 h  Mar 29, 2025, 7:43 PM
Y by
they're useless for the competitions listed, it's way too fast-paced to spend time and color a diagram, maybe on olympiads but you're losing precious time if you do it on those. Maybe the AIME would utilize this? But I highly doubt that it would give any benefits

@below coloring lines is basically the same thing c'mon
This post has been edited 1 time. Last edited by Andyluo, Mar 29, 2025, 7:49 PM
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cheltstudent
617 posts
cheltstudent
#5 h  Mar 29, 2025, 7:48 PM
Y by
Bruh, andyluo, literally not colouring just marking different lines they talked about this in my AMC 10 seminar

@above nah like drawing lines with a colour pencil bro
This post has been edited 1 time. Last edited by cheltstudent, Mar 29, 2025, 7:51 PM
Reason: gg
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ChaitraliKA
1004 posts
ChaitraliKA
#6 h  Mar 29, 2025, 7:57 PM
Y by
It can be useful, but I feel like in the actual competition you would end up not using it because you would be too invested in thinking about how to solve the problem instead of "oh, I should mark this with a different color", and would just end up drawing over your diagram with your regular pencil again. Drawing a specific part of your diagram with harder pressure compared to other areas of your diagram can also help you too, especially if it's just mathcounts or AMC 8.
I brought a ruler, a pen, and a compass to Aime, but didn't use any of those materials. Didn't touch them at all. No time to think ¯⁠\⁠_⁠(⁠ツ⁠)⁠_⁠/⁠¯
The only time I've ever used different colors is when I draw my diagrams on my chromebook canvas, and I'm lazy to get paper and pencil.
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mathprodigy2011
324 posts
mathprodigy2011
#7 h  Mar 29, 2025, 10:45 PM
Y by
Andyluo wrote:
they're useless for the competitions listed, it's way too fast-paced to spend time and color a diagram, maybe on olympiads but you're losing precious time if you do it on those. Maybe the AIME would utilize this? But I highly doubt that it would give any benefits

@below coloring lines is basically the same thing c'mon

i mean i used different colors to get aime p14 correct so they are useful just not for faster paced competitions. It really depends on how you think.
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mdk2013
637 posts
mdk2013
#8 h  Mar 29, 2025, 11:38 PM • 1 Y
Y by Exponent11
colored pencils way better for olympiads so you can show your steps, for example, there were a few problems on the BAMO and SDMO that i took that i could have used colored pencils to more accurately show my proofs. however
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Jack_w
109 posts
Jack_w
#9 h  Mar 30, 2025, 5:52 PM
Y by
it’s not very helpful
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resources
748 posts
resources
#10 h  Mar 30, 2025, 5:56 PM
Y by
they can be extremely helpful if used correctly.
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vincentwant
1354 posts
vincentwant
#11 h  Mar 30, 2025, 6:31 PM
Y by
I used colored pencils on jmo p3
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EaZ_Shadow
1235 posts
EaZ_Shadow
#12 h  Mar 30, 2025, 6:37 PM
Y by
Owinner wrote:
I've heard using colored pencils is really useful for geometry problems. Is this only for very hard problems, or can it be used in MATHCOUNTS/AMC 8/10? An example problem would be much appreciated.

Ofc u can use it for those comp if u want
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ChaitraliKA
1004 posts
ChaitraliKA
#13 h  Mar 30, 2025, 7:13 PM
Y by
@op, you're getting a bunch of different responses, but in the end, it's your decision obv, whether you wanna use it or not
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Andyluo
945 posts
Andyluo
#14 h  Mar 30, 2025, 7:23 PM
Y by
the general consensus is using colored pencils is a waste of time unless you're doing AIME/Olympiads, at that point, you choose if you should or should not.
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BS2012
1025 posts
BS2012
#15 h  Mar 30, 2025, 7:28 PM
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for contests AIME difficulty or easier the geo diagrams are usually simple enough that using colored pencils would be useless, for oly could be useful though
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Mr.Sharkman
498 posts
Mr.Sharkman
#16 h  Mar 31, 2025, 8:16 PM
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vincentwant wrote:
I used colored pencils on jmo p3

Lol same; it kinda helped solve it
I also did on #2 of last year but still got it wrong anyway
This post has been edited 1 time. Last edited by Mr.Sharkman, Mar 31, 2025, 8:17 PM
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deduck
214 posts
deduck
#17 h  Mar 31, 2025, 9:48 PM • 1 Y
Y by fake123
colored pencils make me confused but i think on olympiads it's personal preference

i think on mathcounts/amc8/10/12 and other fast paced comeptition it's not worth it. if u cant solve the problem in pencil black/white it probably wont get solved just bc u colored it xd because usually the solutions arent that invovled about concyclic/collinear/etc

just my opinion xd
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lord_of_the_rook
146 posts
lord_of_the_rook
#18 h  Apr 1, 2025, 1:33 AM
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@op I feel like you should try them out at home, and see if you find them useful.
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