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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 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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Simultaneous System of Equations
djmathman   3
N 20 minutes ago by Tetra_scheme
Find the real number $k$ such that $a$, $b$, $c$, and $d$ are real numbers that satisfy the system of equations
\begin{align*} abcd &= 2007,\\ a &= \sqrt{55 + \sqrt{k+a}},\\ b &= \sqrt{55 - \sqrt{k+b}},\\ c &= \sqrt{55 + \sqrt{k-c}},\\ d &= \sqrt{55 - \sqrt{k-d}}. \end{align*}
3 replies
djmathman
Sep 17, 2018
Tetra_scheme
20 minutes ago
J
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prove that at least one of them is divisible by some other member of the set.
Martin.s   0
an hour ago
Given \( n + 1 \) integers \( a_1, a_2, \ldots, a_{n+1} \), each less than or equal to \( 2n \), prove that at least one of them is divisible by some other member of the set.
0 replies
Martin.s
an hour ago
0 replies
J
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estimate for \( a_1 \) is the best possible
Martin.s   0
an hour ago
Let \( a_1 < a_2 < \cdots < a_n < 2n \) be positive integers such that no one of them is divisible by any other member of the sequence. Then
\[ a_1 \geq 2^k, \]where \( k \) is defined by the inequalities
\[ 3^k < 2n < 3^{k+1}. \]This estimate for \( a_1 \) is the best possible.
0 replies
Martin.s
an hour ago
0 replies
J
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best possible estimate.
Martin.s   0
an hour ago
Let \( a_1 < a_2 < \cdots < a_n < 2n \) be a sequence of positive integers. Then
\[ \max \left( (a_i, a_j) \right) > \frac{38n}{147} - c, \]where \( c \) is a constant independent of \( n \), and \( (a_i, a_j) \) denotes the greatest common divisor of \( a_i \) and \( a_j \). This is the best possible estimate.
0 replies
Martin.s
an hour ago
0 replies
J
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Polynomial
Fang-jh   6
N an hour ago by yofro
Source: Chinese TST 2007 1st quiz P3
Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) = 1$ and $ (x + 1)[f(x)]^2 - 1$ is an odd function.
6 replies
Fang-jh
Jan 3, 2009
yofro
an hour ago
J
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collinear wanted, toucpoints of incircle related
parmenides51   2
N an hour ago by Tamam
Source: 2018 Thailand October Camp 1.2
Let $\Omega$ be the inscribed circle of a triangle $\vartriangle ABC$. Let $D, E$ and $F$ be the tangency points of $\Omega$ and the sides $BC, CA$ and $AB$, respectively, and let $AD, BE$ and $CF$ intersect $\Omega$ at $K, L$ and $M$, respectively, such that $D, E, F, K, L$ and $M$ are all distinct. The tangent line of $\Omega$ at $K$ intersects $EF$ at $X$, the tangent line of $\Omega$ at $L$ intersects $DE$ at $Y$ , and the tangent line of $\Omega$ at M intersects $DF$ at $Z$. Prove that $X,Y$ and $Z$ are collinear.
2 replies
parmenides51
Oct 15, 2020
Tamam
an hour ago
J
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show that there are at least eight vertices where exactly three edges meet.
Martin.s   0
an hour ago
If all the faces of a convex polyhedron have central symmetry, show that there are at least eight vertices where exactly three edges meet. (The cube has exactly eight such vertices.)
0 replies
Martin.s
an hour ago
0 replies
J
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Nice problem
Martin.s   0
an hour ago
If \(p\) is a prime and \(n \geq p\), then
\[ n! \sum_{pi+j=n} \frac{1}{p^i i! j!} \equiv 0 \pmod{p}. \]
0 replies
Martin.s
an hour ago
0 replies
J
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2-var inequality
sqing   12
N an hour ago by ytChen
Source: Own
Let $ a,b> 0 ,a^3+ab+b^3=3.$ Prove that
$$ (a+b)(a+1)(b+1) \leq 8$$$$ (a^2+b^2)(a+1)(b+1) \leq 8$$Let $ a,b> 0 ,a^3+ab(a+b)+b^3=3.$ Prove that
$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$
12 replies
1 viewing
sqing
Yesterday at 1:35 PM
ytChen
an hour ago
J
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[15th PMO] National Orals, Part 1, #9
LilKirb   6
N an hour ago by iwastedmyusername
If $x^2+2x+5$ is a factor of $x^4 +ax^2 + b$, find the sum of $a+b.$
6 replies
LilKirb
Yesterday at 4:04 PM
iwastedmyusername
an hour ago
J
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How many friends can sit in that circle at most?
Arytva   4
N 2 hours ago by JohannIsBach

A group of friends sits in a ring. Each friend picks a different whole number and holds a stone marked with it. Then they pass their stone one seat to the right so everyone ends up with two stones: one they made and one they received. Now they notice something odd: if your original number is $x$, your right-neighbor’s is $y$, and the next person over is $z$, then for every trio in the circle they see

$$ x + z = (2 - x)\,y. $$
They want as many friends as possible before this breaks (since all stones must stay distinct).

How many friends can sit in that circle at most?
4 replies
Arytva
Today at 10:00 AM
JohannIsBach
2 hours ago
J
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Bisectors, perpendicularity and circles
JuanDelPan   15
N 2 hours ago by zuat.e
Source: Pan-American Girls’ Mathematical Olympiad 2022, Problem 3
Let $ABC$ be an acute triangle with $AB< AC$. Denote by $P$ and $Q$ points on the segment $BC$ such that $\angle BAP = \angle CAQ < \frac{\angle BAC}{2}$. $B_1$ is a point on segment $AC$. $BB_1$ intersects $AP$ and $AQ$ at $P_1$ and $Q_1$, respectively. The angle bisectors of $\angle BAC$ and $\angle CBB_1$ intersect at $M$. If $PQ_1\perp AC$ and $QP_1\perp AB$, prove that $AQ_1MPB$ is cyclic.
15 replies
JuanDelPan
Oct 27, 2022
zuat.e
2 hours ago
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[SHS Sipnayan 2023] Series of Drama F-E
Magdalo   6
N 2 hours ago by trangbui
Find the units digit of
\[\sum_{n=1}^{2025}n^5\]
6 replies
Magdalo
2 hours ago
trangbui
2 hours ago
J
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Decreasing Digits of Increasing Bases
Magdalo   2
N 2 hours ago by trangbui
Some base $10$ numbers can be expressed in $n+2$ digits in base $k$, $n+1$ digits in base $k+1$, and $n$ digits in base $k+2$ for some positive integers $n,k$. How many such two-digit base $10$ numbers are there?
2 replies
Magdalo
2 hours ago
trangbui
2 hours ago
J
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J
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Solve an equation
lgx57   4
N Apr 17, 2025 by lgx57
Find all positive integers $n$ and $x$ such that:
$$2^{2n+1}-7=x^2$$
4 replies
lgx57
Mar 12, 2025
lgx57
Apr 17, 2025
J
Solve an equation
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Reveal topic tags
equation
number theory
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lgx57
53 posts
lgx57
#1 h  Mar 12, 2025, 10:50 PM • 1 Y
Y by Kizaruno
Find all positive integers $n$ and $x$ such that:
$$2^{2n+1}-7=x^2$$
Z K Y
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Kempu33334
643 posts
Kempu33334
#2 h  Mar 12, 2025, 11:06 PM • 1 Y
Y by Kizaruno
lgx57 wrote:
Find all positive integers $n$ and $x$ such that:
$$2^{2n+1}-7=x^2$$

https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Nagell_equation

The solutions for $n$ are just the odd ones given.
Z K Y
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lgx57
53 posts
lgx57
#3 h  Apr 16, 2025, 2:56 PM
Y by
Kempu33334 wrote:
lgx57 wrote:
Find all positive integers $n$ and $x$ such that:
$$2^{2n+1}-7=x^2$$

https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Nagell_equation

The solutions for $n$ are just the odd ones given.

I can't go to this website in China. Can you give me a brief introduction of it?
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Kempu33334
643 posts
Kempu33334
#5 h  Apr 17, 2025, 12:41 AM • 1 Y
Y by Kizaruno
lgx57 wrote:
I can't go to this website in China. Can you give me a brief introduction of it?

Sure.

The Ramanujan-Nagell equation is \[\displaystyle 2^{n}-7=x^{2}\,\]and solutions in natural numbers n and x exist just when $n = 3, 4, 5, 7$ and $15$ (sequence A060728 in the OEIS).

The values of n correspond to the values of x as: $x = 1, 3, 5, 11$ and $181$ (sequence A038198 in the OEIS).
Wikipedia wrote:
This was conjectured in 1913 by Indian mathematician Srinivasa Ramanujan, proposed independently in 1943 by the Norwegian mathematician Wilhelm Ljunggren, and proved in 1948 by the Norwegian mathematician Trygve Nagell. [...]

If you want, a proof is given here, however, it involves abstract algebra.
Z K Y
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lgx57
53 posts
lgx57
#6 h  Apr 17, 2025, 1:32 PM
Y by
Kempu33334 wrote:
lgx57 wrote:
I can't go to this website in China. Can you give me a brief introduction of it?

Sure.

The Ramanujan-Nagell equation is \[\displaystyle 2^{n}-7=x^{2}\,\]and solutions in natural numbers n and x exist just when $n = 3, 4, 5, 7$ and $15$ (sequence A060728 in the OEIS).

The values of n correspond to the values of x as: $x = 1, 3, 5, 11$ and $181$ (sequence A038198 in the OEIS).
Wikipedia wrote:
This was conjectured in 1913 by Indian mathematician Srinivasa Ramanujan, proposed independently in 1943 by the Norwegian mathematician Wilhelm Ljunggren, and proved in 1948 by the Norwegian mathematician Trygve Nagell. [...]

If you want, a proof is given here, however, it involves abstract algebra.

Thank you very much!
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