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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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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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JSMC texas
BossLu99   20
N 4 minutes ago by MC_ADe
who is going to JSMC texas
20 replies
1 viewing
BossLu99
Yesterday at 1:32 PM
MC_ADe
4 minutes ago
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9 Did I make the right choice?
Martin2001   1
N 10 minutes ago by babyzombievillager
If you were in 8th grade, would you rather go to MOP or mc nats? I chose to study the former more and got in so was wondering if that was valid given that I'll never make mc nats.
1 reply
+1 w
Martin2001
13 minutes ago
babyzombievillager
10 minutes ago
J
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Diophantine equation !
ComplexPhi   10
N 33 minutes ago by Namisgood
Determine all triples $(m , n , p)$ satisfying :
\[n^{2p}=m^2+n^2+p+1\]
where $m$ and $n$ are integers and $p$ is a prime number.
10 replies
ComplexPhi
Feb 4, 2015
Namisgood
33 minutes ago
J
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Factorial Equation
Alidq   0
35 minutes ago
Solve in $\mathbb{N}$ $$\frac{x!}{(x-y)!} = 10x+2y-29$$
0 replies
Alidq
35 minutes ago
0 replies
J
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Triangular function and quadrilateral
Kunihiko_Chikaya   1
N 40 minutes ago by Mathzeus1024
Source: National Defence Medical college Entrance exam November 2005
Given a convex quadrilateral such that $AB=AD=\sqrt{5},\ CB=CD=2, \tan \angle{DAB=-2,\ \angle{BCD}=2\alpha\ (0<2\alpha <\pi).}$ Find all the integers $n$ satisfying $\cos n\alpha=\cos \alpha.$
1 reply
Kunihiko_Chikaya
Jan 17, 2006
Mathzeus1024
40 minutes ago
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IMO Shortlist 2011, G5
WakeUp   71
N an hour ago by InterLoop
Source: IMO Shortlist 2011, G5
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $D$ and $E$ be the second intersection points of $\omega$ with $AI$ and $BI$, respectively. The chord $DE$ meets $AC$ at a point $F$, and $BC$ at a point $G$. Let $P$ be the intersection point of the line through $F$ parallel to $AD$ and the line through $G$ parallel to $BE$. Suppose that the tangents to $\omega$ at $A$ and $B$ meet at a point $K$. Prove that the three lines $AE,BD$ and $KP$ are either parallel or concurrent.

Proposed by Irena Majcen and Kris Stopar, Slovenia
71 replies
WakeUp
Jul 13, 2012
InterLoop
an hour ago
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Inspired by old results
sqing   2
N an hour ago by sqing
Source: Own
Let $ a,b>0 , a^2+b^2+ab+a+b=5 . $ Prove that
$$ \frac{ 1 }{a+b+ab+1}+\frac{6}{a^2+b^2+ab+1}\geq \frac{7}{4}$$$$ \frac{ 1 }{a+b+ab+1}+\frac{1}{a^2+b^2+ab+1}\geq \frac{1}{2}$$$$ \frac{41}{a+b+2}+\frac{ab}{a^3+b^3+2} \geq \frac{21}{2}$$
2 replies
sqing
an hour ago
sqing
an hour ago
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If $b^n|a^n-1$ then $a^b >\frac {3^n}{n}$ (China TST 2009)
Fang-jh   15
N an hour ago by ihategeo_1969
Source: Chinese TST 2009 6th P1
Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n-1,$ then $ a^b > \frac {3^n}{n}.$
15 replies
Fang-jh
Apr 4, 2009
ihategeo_1969
an hour ago
J
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inequality (another entrance exam)
nai0610   1
N an hour ago by sqing
Given positive real numbers $a,b,c$ satisfying
$(a+2)b^2+(b+2)c^2+(c+2)a^2\geq 8+abc$
prove that $2(ab+bc+ca)\leq a^2(a+b)+b^2(b+c)+c^2(c+a)$
1 reply
nai0610
Jun 2, 2024
sqing
an hour ago
J
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Inspired by old results
sqing   6
N 2 hours 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
2 hours ago
J
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Integer-Valued FE comes again
lminsl   206
N 2 hours 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
2 hours ago
J
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Integer representation
RL_parkgong_0106   2
N 2 hours 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
2 hours ago
J
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9 Mathpath vs. AMSP
FuturePanda   30
N 3 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
3 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
J
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J
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prime spam
fruitmonster97   28
N Apr 4, 2025 by Apple_maths60
Source: 2024 AMC 10A #3
What is the sum of the digits of the smallest prime that can be written as a sum of $5$ distinct primes?

$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$
28 replies
fruitmonster97
Nov 7, 2024
Apple_maths60
Apr 4, 2025
J
prime spam
G H J
Reveal topic tags
AMC
AMC 10
2024 AMC 10A
2024 AMC
number theory
prime numbers
L
G H BBookmark kLocked kLocked NReply
Source: 2024 AMC 10A #3
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fruitmonster97
2485 posts
fruitmonster97
#1 h  Nov 7, 2024, 4:54 PM
Y by
What is the sum of the digits of the smallest prime that can be written as a sum of $5$ distinct primes?

$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$
This post has been edited 1 time. Last edited by fruitmonster97, Nov 7, 2024, 8:21 PM
Z K Y
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fruitmonster97
2485 posts
fruitmonster97
#2 h  Nov 7, 2024, 4:56 PM
Y by
If $2$ was one of the primes, then the sum would be even, impossible. Otherwise, $3+5+7+11+13=39,$ which isn't prime, but $3+5+7+11+17=43\implies{7}$ is, yielding $\boxed{\textbf{(B)}}.$
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MathRook7817
664 posts
MathRook7817
#3 h  Nov 7, 2024, 4:58 PM • 1 Y
Y by mithu542
The minimum possible sum of 5 distinct primes that are odd (since the number is prime) is
3+5+7+11+13 = 39
41 is impossible since 15 isn't prime, so the number is 43 (3+5+7+11+17).
Therefore, the answer is 4+3 = 7 (B)
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Jonysun
34 posts
Jonysun
#4 h  Nov 7, 2024, 4:59 PM
Y by
The answer is indeed $\fbox{43} \Rightarrow B$
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golden_star_123
206 posts
golden_star_123
#5 h  Nov 7, 2024, 5:16 PM
Y by
All the primes have to be odd for the sum to be odd. We may then just bash out the possibilities to notice that the first prime that works is $43$, so the answer is $\boxed{7}$.
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gracemoon124
872 posts
gracemoon124
#6 h  Nov 7, 2024, 5:35 PM • 2 Y
Y by alexanderhamilton124, songyanxin
sillied ):
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lpieleanu
2973 posts
lpieleanu
#7 h  Nov 7, 2024, 5:38 PM
Y by
Solution
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SomeonecoolLovesMaths
3209 posts
SomeonecoolLovesMaths
#8 h  Nov 7, 2024, 6:06 PM
Y by
$$3+5+7+11+17=43$$7 confirmed
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wangzrpi
159 posts
wangzrpi
#9 h  Nov 7, 2024, 6:10 PM
Y by
4+3=7 confirmed :)

Cannot be 2 because then the sum would be even and therefore not prime
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KenWuMath
58 posts
KenWuMath
#10 h  Nov 7, 2024, 6:16 PM
Y by
apparently 15 isnt a prime :D
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wangzrpi
159 posts
wangzrpi
#11 h  Nov 7, 2024, 6:51 PM
Y by
fruitmonster97 wrote:
If $2$ was one of the primes, then the sum would be even, impossible. Otherwise, $3+5+7+11+13=39,$ which isn't prime, but $3+5+7+11+17=43\implies{7}$ is, yielding $\boxed{\textbf{(B)}}.$

Also sum of primes 5-13 is larger
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dolphinday
1325 posts
dolphinday
#12 h  Nov 7, 2024, 7:19 PM
Y by
I did 11 + 13 = 14 on this one. double checked it too
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GreenBanana666
548 posts
GreenBanana666
#13 h  Nov 7, 2024, 7:21 PM
Y by
Its $43$, $4+3=7$. Its B then
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SkatingKitty
223 posts
SkatingKitty
#14 h  Nov 7, 2024, 7:51 PM
Y by
I miscalculated and and got 53, or 8 chat I’m cooked again
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mhgelgi
689 posts
mhgelgi
#15 h  Nov 7, 2024, 7:53 PM • 1 Y
Y by SkatingKitty
SkatingKitty wrote:
I miscalculated and and got 53, or 8 chat I’m cooked again
yeah, same here, we can weep together
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SkatingKitty
223 posts
SkatingKitty
#16 h  Nov 7, 2024, 8:00 PM
Y by
I literally left like all of them blank except for like 2 that I got wrong
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mhgelgi
689 posts
mhgelgi
#17 h  Nov 7, 2024, 8:01 PM
Y by
127.5 => 103.5 (4 sillies)
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SkatingKitty
223 posts
SkatingKitty
#18 h  Nov 7, 2024, 8:08 PM
Y by
2 sillies- probably bc I don’t know abt my other ones lol
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happypi31415
742 posts
happypi31415
#19 h  Nov 7, 2024, 8:12 PM
Y by
I put 53 like an idiot :(
This post has been edited 2 times. Last edited by happypi31415, Nov 8, 2024, 5:25 PM
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SkatingKitty
223 posts
SkatingKitty
#20 h  Nov 7, 2024, 8:14 PM
Y by
Not the only one here
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SirAppel
877 posts
SirAppel
#21 h  Nov 7, 2024, 8:15 PM
Y by
answer choice E was 11, and I know that because I almost put E before checking my answers oops (47 susus)
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SkatingKitty
223 posts
SkatingKitty
#22 h  Nov 7, 2024, 8:19 PM
Y by
At least you caught your mistake: I forgot to check my answers
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gicyuraok2
1059 posts
gicyuraok2
#23 h  Nov 7, 2024, 11:26 PM
Y by
i for sure would have had to come back to this lol

obv i looked at $5+7+11+13+17=53$ first, but $8$ isn't one of the answers (thank god)
accidentally got a spoiler but probably would have found it anyway
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Existing_Human1
209 posts
Existing_Human1
#24 h  Nov 7, 2024, 11:42 PM
Y by
POV: Your nervousness causes you to think that 2 + 3 + 5 + 7 + 11 is odd
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SkatingKitty
223 posts
SkatingKitty
#25 h  Nov 8, 2024, 1:29 AM
Y by
gicyuraok2 wrote:
i for sure would have had to come back to this lol

obv i looked at $5+7+11+13+17=53$ first, but $8$ isn't one of the answers (thank god)
accidentally got a spoiler but probably would have found it anyway

Rly? Oh whoops I put 8
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mathenrichmentaops
540 posts
mathenrichmentaops
#26 h  Nov 8, 2024, 2:54 AM
Y by
SkatingKitty wrote:
gicyuraok2 wrote:
i for sure would have had to come back to this lol

obv i looked at $5+7+11+13+17=53$ first, but $8$ isn't one of the answers (thank god)
accidentally got a spoiler but probably would have found it anyway

Rly? Oh whoops I put 8

Omg same I think I changed it to B tho
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gicyuraok2
1059 posts
gicyuraok2
#27 h  Nov 9, 2024, 8:48 PM
Y by
mathenrichmentaops wrote:
SkatingKitty wrote:
gicyuraok2 wrote:
i for sure would have had to come back to this lol

obv i looked at $5+7+11+13+17=53$ first, but $8$ isn't one of the answers (thank god)
accidentally got a spoiler but probably would have found it anyway

Rly? Oh whoops I put 8

Omg same I think I changed it to B tho

how do you put 8 if it's not one of the answer choices
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EpicShrek49
17 posts
EpicShrek49
#28 h  Nov 9, 2024, 9:33 PM
Y by
i sillied the one where it was -1, -6, and 10 for the sum and i thought it was -2, -3, and 10..
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Apple_maths60
26 posts
Apple_maths60
#30 h  Apr 4, 2025, 3:28 PM
Y by
Yes we get the prime as 43 so sum is 7
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