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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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Theory of Equations
P162008   1
N 37 minutes ago by vanstraelen
Let $a,b,c,d$ and $e\in [-2,2]$ such that $\sum_{cyc} a = 0, \sum_{cyc} a^3 = 0, \sum_{cyc} a^5 = 10.$ Find the value of $\sum_{cyc} a^2.$
1 reply
P162008
Today at 11:27 AM
vanstraelen
37 minutes ago
J
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Divisibility on 101 integers
BR1F1SZ   4
N an hour ago by BR1F1SZ
Source: Argentina Cono Sur TST 2024 P2
There are $101$ positive integers $a_1, a_2, \ldots, a_{101}$ such that for every index $i$, with $1 \leqslant i \leqslant 101$, $a_i+1$ is a multiple of $a_{i+1}$. Determine the greatest possible value of the largest of the $101$ numbers.
4 replies
BR1F1SZ
Aug 9, 2024
BR1F1SZ
an hour ago
J
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inequalities of elements in set
toanrathay   1
N an hour ago by Lankou
Let \( m \) be a positive integer such that \( m \geq 4 \), and let the set
\[ A = \{a_1, a_2, a_3, \ldots, a_m\} \]consist of distinct positive integers not exceeding 2025. Suppose that for every \( a, b \in A \), with \( a \ne b \), if \( a + b \leq 2025 \), then \( a + b \in A \) as well. Prove that

\[ \frac{a_1 + a_2 + a_3 + \cdots + a_m}{m} \geq 1013. \]
1 reply
toanrathay
5 hours ago
Lankou
an hour ago
J
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2^x+3^x = yx^2
truongphatt2668   2
N an hour ago by CM1910
Prove that the following equation has infinite integer solutions:
$$2^x+3^x = yx^2$$
2 replies
truongphatt2668
Yesterday at 3:38 PM
CM1910
an hour ago
J
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Prove perpendicular
shobber   29
N an hour ago by zuat.e
Source: APMO 2000
Let $ABC$ be a triangle. Let $M$ and $N$ be the points in which the median and the angle bisector, respectively, at $A$ meet the side $BC$. Let $Q$ and $P$ be the points in which the perpendicular at $N$ to $NA$ meets $MA$ and $BA$, respectively. And $O$ the point in which the perpendicular at $P$ to $BA$ meets $AN$ produced.

Prove that $QO$ is perpendicular to $BC$.
29 replies
shobber
Apr 1, 2006
zuat.e
an hour ago
J
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The smallest of sum of elements
hlminh   1
N 2 hours ago by nguyenhuybao_06
Let $S=\{1,2,...,2014\}$ and $X\subset S$ such that for all $a,b\in X,$ if $a+b\leq 2014$ then $a+b\in X.$ Find the smallest of sum of all elements of $X.$
1 reply
hlminh
2 hours ago
nguyenhuybao_06
2 hours ago
J
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2024 PUMaC Team Round, Question 14 Inquiry
A22-4   0
2 hours ago
2024 PUMaC Team Round Question 14 reads as follows:

What is the largest value of $m$ for which I can find nonnegative integers $a_1, a_2, \cdots, a_m<2024$ such that for all indices $i>j$, $17$ divides $\binom{a_i}{a_j}$?
(Note: This should say "... nonnegative integers $a_1<a_2<\cdots<a_m<2024$ ...")

I interpreted this correction to mean the following:
What is the largest value of $m$ for which I there exists nonnegative integers $a_1<a_2<\cdots<a_m<2024$ such that for all indices $i>j$, $17$ divides $\binom{a_i}{a_j}$?

The official answer (https://static1.squarespace.com/static/570450471d07c094a39efaed/t/67421bd74806e80a7ab11c7d/1732385751115/PUMaC_2024_Team__Final_.pdf) is 107. However, I believe I have a construction with $108$ integers - take the set of all integers with a digit sum of $19$ in base $17$, then append $2023_{10}=700_{17}$ to the list.

I checked this with Python using the following code:
[code]def digit_sum_base(n, base):
total = 0
while n > 0:
total += n % base
n //= base
return total

target_sum = 19
base = 17
limit = 2024
qualified_numbers = [n for n in range(limit) if digit_sum_base(n, base) == target_sum]

qualified_numbers.append(2023)

from math import comb

all_divisible = True
for i in range(len(qualified_numbers)):
for j in range(i):
a, b = qualified_numbers, qualified_numbers[j]
if comb(a, b) % 17 != 0:
all_divisible = False
break
if not all_divisible:
break

print(len(qualified_numbers), all_divisible)[/code]

Am I wrong or are they wrong? Any insight would be appreciated!
0 replies
A22-4
2 hours ago
0 replies
J
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Inequalities
Scientist10   0
2 hours ago
If $x, y, z \in \mathbb{R}$, then prove that the following inequality holds:
\[ \sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x \]
0 replies
+1 w
Scientist10
2 hours ago
0 replies
J
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NT from ukr contest
mshtand1   3
N 2 hours ago by ravengsd
Source: Ukrainian TST for RMM 2021(2) and EGMO 2022 P2
Find the greatest positive integer $n$ such that there exist positive integers $a_1, a_2, ..., a_n$ for which the following holds $a_{k+2} = \dfrac{(a_{k+1}+a_k)(a_{k+1}+1)}{a_k}$ for all $1 \le k \le n-2$.
Proposed by Mykhailo Shtandenko and Oleksii Masalitin
3 replies
mshtand1
Oct 2, 2021
ravengsd
2 hours ago
J
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Posted before ,but no solution
Nuran2010   1
N 2 hours ago by Nuran2010
Source: 1220 Number Theory Problems
Find all positive integers $n$ where $49n^3+42n^2+11n+1$ is a perfect cube
1 reply
Nuran2010
Apr 11, 2025
Nuran2010
2 hours ago
J
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APMO 2012 #3
syk0526   30
N 2 hours ago by EVKV
Source: APMO 2012 #3
Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.
30 replies
syk0526
Apr 2, 2012
EVKV
2 hours ago
J
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Problem 1
SpectralS   146
N 3 hours ago by YaoAOPS
Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$

(The excircle of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)

Proposed by Evangelos Psychas, Greece
146 replies
SpectralS
Jul 10, 2012
YaoAOPS
3 hours ago
J
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Number theory or function ?
matematikator   15
N 3 hours ago by YaoAOPS
Source: IMO ShortList 2004, algebra problem 3
Does there exist a function $s\colon \mathbb{Q} \rightarrow \{-1,1\}$ such that if $x$ and $y$ are distinct rational numbers satisfying ${xy=1}$ or ${x+y\in \{0,1\}}$, then ${s(x)s(y)=-1}$? Justify your answer.

Proposed by Dan Brown, Canada
15 replies
matematikator
Mar 18, 2005
YaoAOPS
3 hours ago
J
H
How many ways can we indistribute n different marbles into 6 identical boxes
Taiharward   10
N 3 hours ago by MathBot101101
How many ways can we distribute n indifferent marbles into 6 identical boxes and one jar?
10 replies
Taiharward
Today at 2:14 AM
MathBot101101
3 hours ago
J
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J
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(a,b,c) triples
Ecrin_eren   2
N Apr 14, 2025 by pedronis
Let n be a positive integer. How many positive integer triples (a, b, c) are there such that:

a! + b! + c! = 2^n ?
2 replies
Ecrin_eren
Apr 13, 2025
pedronis
Apr 14, 2025
J
(a,b,c) triples
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number theory
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Ecrin_eren
56 posts
Ecrin_eren
#1 h  Apr 13, 2025, 3:22 PM
Y by
Let n be a positive integer. How many positive integer triples (a, b, c) are there such that:

a! + b! + c! = 2^n ?
Z K Y
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alexheinis
10556 posts
alexheinis
#2 h  Apr 13, 2025, 8:20 PM • 1 Y
Y by Ecrin_eren
Let's assume $1\le a\le b\le c$ then $a!|2^n \implies a=1,2$.
If $a=1$ then $b=1, c!+2=2^n$. Then $c!$ has exactly one factor 2, hence $c=2,3$. Both work.
Now suppose $a=2$, then $b!$ has exactly one factor 2, hence $b=2,3$.
If $a=b=2$ we find $c!+4=2^n$ and $c!$ has exactly two factors 2. No solution.
If $a=2,b=3$ then $c!+8=2^n$. Then $c!$ has exactly three factors 2 and $c=4,5$. Both work.
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pedronis
25 posts
pedronis
#3 h  Apr 14, 2025, 1:48 AM
Y by
WLOG $a \geq b \geq c$. If $c \geq 3 \Rightarrow 3 \mid 2^n$ (contradiction), so $c \leq 2$.

Let $c = 1$. By parity, we have $b = 1 \Rightarrow 2^n = a! + 2$.
Clearly, $n \geq 2$.

If $a \geq 4 \Rightarrow 2^n = m(4) + 2$ (contradiction!!!), because $4 \mid 2^n$. So $a \leq 3 \Rightarrow n \leq 3$.

Therefore, the solutions are $(n,a,b,c) = (3,3,1,1)$ and $(2,2,1,1)$.

The other case is $c = 2$. It works!
This post has been edited 1 time. Last edited by pedronis, Apr 14, 2025, 1:49 AM
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