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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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0 replies
jlacosta
Mar 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
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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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Don Sirloin Bowel's Algorithm
BadAtCompetitionMath21420   1
N 6 minutes ago by ohiorizzler1434
Can you sum
\begin{align*} \sum_{k=1}^n F^m_k? \end{align*}
I coined a strategy Don Sirloin Bowel's algorithm because I found it and didn't see it anywhere else. Please tell me if this looks familiar because it can have absolutely scrumptious applications. (I hope that this stuff can help you sum that.) Starting with the "golden quadratic" or whatever the sigma it's called $x^2=x+1$. Multiplying by $x$ on both sides and resubstituting $x^2$, we find $x^3=x^2+x=(x+1)+x=2x+1$. Continuing this process, we find $x^4=3x+2$, $x^5=5x+3$, and so forth. We claim that $x^n=F_nx+F_{n-1}.$ We already have our base case. $x^n=F_nx+F_{n-1}$. Multiplying by $x$, $x^{n+1}=F_nx^2+F_{n-1}x=F_n(x+1)+F_{n-1}x=F_{n+1}x+F_{n}$. Hence, we have proved our claim by induction. Now, raising both sides to the $m$th power,
\begin{align*} x^{mn}&=\sum_{k=0}^m \binom{m}{k}F_{n}^{m-k}F_{n-1}^kx^m\\ &=\sum_{k=0}^m \binom{m}{k}F_{n}^{m-k}F_{n-1}^k(F_mx+F_{m-1})\\ &=x\sum_{k=0}^m \binom{m}{k}F_{n}^{m-k}F_{n-1}^kF_m+\sum_{k=0}^mF_{n}^{m-k}F_{n-1}^kF_{m-1}\\ &=F_{mn}x+F_{mn-1} \end{align*}So,
\begin{align*} F_{mn}=\sum_{k=0}^m \binom{m}{k}F_{n}^{m-k}F_{n-1}^kF_m \end{align*}and
\begin{align*} F_{mn-1}=\sum_{k=0}^mF_{n}^{m-k}F_{n-1}^kF_{m-1} \end{align*}Are these identities useful? And can you use them to compute the sum?
1 reply
+1 w
BadAtCompetitionMath21420
23 minutes ago
ohiorizzler1434
6 minutes ago
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Incenter geometry with parallel lines
nAalniaOMliO   2
N an hour ago by nAalniaOMliO
Source: Belarusian MO 2023
Let $\omega$ be the incircle of triangle $ABC$. Line $l_b$ is parallel to side $AC$ and tangent to $\omega$. Line $l_c$ is parallel to side $AB$ and tangent to $\omega$. It turned out that the intersection point of $l_b$ and $l_c$ lies on circumcircle of $ABC$
Find all possible values of $\frac{AB+AC}{BC}$
2 replies
nAalniaOMliO
Apr 16, 2024
nAalniaOMliO
an hour ago
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Chessboard
Ecrin_eren   5
N an hour ago by TrendCrusher
On an 8×8 checkerboard, what is the minimum number of squares that must be marked (including the marked ones) so that every square has exactly one marked neighbor? (We define neighbors as squares that share a common edge, and a square is not considered a neighbor of itself.)

5 replies
Ecrin_eren
Yesterday at 8:55 PM
TrendCrusher
an hour ago
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Problem about Euler's function
luutrongphuc   3
N 2 hours ago by ishan.panpaliya
Prove that for every integer $n \ge 5$, we have:
$$ 2^{n^2+3n-13} \mid \phi \left(2^{2^{n}}-1 \right)$$
3 replies
luutrongphuc
6 hours ago
ishan.panpaliya
2 hours ago
J
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Function equation
Dynic   3
N 2 hours ago by Filipjack
Find all function $f:\mathbb{Z}\to\mathbb{Z}$ satisfy all conditions below:
i) $f(n+1)>f(n)$ for all $n\in \mathbb{Z}$
ii) $f(-n)=-f(n)$ for all $n\in \mathbb{Z}$
iii) $f(a^3+b^3+c^3+d^3)=f^3(a)+f^3(b)+f^3(c)+f^3(d)$ for all $n\in \mathbb{Z}$
3 replies
Dynic
5 hours ago
Filipjack
2 hours ago
J
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solve in positive integers: 3 \cdot 2^x +4 =n^2
parmenides51   3
N 3 hours ago by ali123456
Source: Greece JBMO TST 2019 p2
Find all pairs of positive integers $(x,n) $ that are solutions of the equation $3 \cdot 2^x +4 =n^2$.
3 replies
parmenides51
Apr 29, 2019
ali123456
3 hours ago
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Why does the combined equation have two negative solutions?
Luking   1
N 3 hours ago by vanstraelen
It is known that the moving point $G(x,y)$ is on the curve $C_1: y^2 - x^2 = 1$. There is a parabola $C_2: x^2 = 4y$ with focus $F$. Two tangent lines to$C_2$ are drawn through a point $P$ on $C1$, and the tangent points are $A$ and $B$ respectively. The line $l$ parallel to the line $AB$ is tangent to $C_2$ at point $Q$. Question: When the line $l$ and $C_1$ have two intersection points, find the range of $|QF|$.\
This may make some of the information in the question useless, because I deleted the first two questions of this big question in order to avoid making the question too long and get straight to the point.\
According to the calculation, I get the analytical expression of the line $l$ as $y=\frac{x_0}{2} x - \frac{x_0^2}{4}$.\
At first, I thought it only needed to be not parallel to the parabola asymptotes.\
That is, the slope of the straight line $k \neq \frac{a}{b}$ , then $\frac{x_0}{2} \neq \frac{a}{b}$ , so $x_0 \neq \pm \frac{2a}{b} = \pm2$ ,and $x_0^2 \neq 4$.\
$|QF| = y_0 +1 \neq 5$.\
But when I checked the answer, it was wrong.\
It combines the straight line $l$ with the curve $C_1$ to get an equation and then uses Vieta's theorem.
\begin{cases}
y=\frac{x_0}{2} x - \frac{x_0^2}{4}
x^2=4y
\end{cases}
$$( 4 x_{0}^{2}-1 6 ) y^{2}-8 x_{0}^{2} y-x_{0}^{4}-4 x_{0}^{2}=0 $$The answer is that according to the question, the equation has two negative roots. I can't understand this, and this is exactly where my problem lies.\
Then it gets the following system of equations:
\begin{cases}
4x_0-16 \neq 0
\Delta > 0
x_1+x_2 <0
x_1\cdot x_2>0
\end{cases}
Solve, $2\sqrt{5}-2<x_0<4$.
So we get $|QF|=\frac {x_0^2}{4} + 1 \in (\frac {\sqrt{5}+1}{2},2)$.
I hope you can help me figure out why both roots of that equation are negative.
IMAGE
IMAGE
1 reply
Luking
Today at 4:02 PM
vanstraelen
3 hours ago
J
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2^n = p+3^p
reeh_haan   7
N 3 hours ago by MITDragon
Find all pairs $(p, n)$ of positive integers which satisfy the equation $$2^n = p+3^p$$
7 replies
reeh_haan
Dec 28, 2021
MITDragon
3 hours ago
J
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2^a + 3^b + 1 = 6^c
togrulhamidli2011   3
N 3 hours ago by Tamam
Find all positive integers (a, b, c) such that:

\[ 2^a + 3^b + 1 = 6^c \]
3 replies
togrulhamidli2011
Mar 16, 2025
Tamam
3 hours ago
J
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min A=x+1/x+y+1/y if 2(x+y)=1+xy for x,y>0 , 2020 ISL A3 for juniors
parmenides51   12
N 3 hours ago by mathmax001
Source: 2021 Greece JMO p1 (serves also as JBMO TST) / based on 2020 IMO ISL A3
If positive reals $x,y$ are such that $2(x+y)=1+xy$, find the minimum value of expression $$A=x+\frac{1}{x}+y+\frac{1}{y}$$
12 replies
parmenides51
Jul 21, 2021
mathmax001
3 hours ago
J
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3a^2b+16ab^2 is perfect square for primes a,b >0
parmenides51   5
N 4 hours ago by ali123456
Source: 2020 Greek JBMO TST p3
Find all pairs $(a,b)$ of prime positive integers $a,b$ such that number $A=3a^2b+16ab^2$ equals to a square of an integer.
5 replies
parmenides51
Nov 14, 2020
ali123456
4 hours ago
J
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minimum value of S, ISI 2013
Sayan   13
N 4 hours ago by Apple_maths60
Let $a,b,c$ be real number greater than $1$. Let
\[S=\log_a {bc}+\log_b {ca}+\log_c {ab}\]
Find the minimum possible value of $S$.
13 replies
Sayan
May 12, 2013
Apple_maths60
4 hours ago
J
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classical R+ FE
jasperE3   2
N 4 hours ago by jasperE3
Source: kent2207, based on 2019 Slovenia TST
wanted to post this problem in its own thread: https://artofproblemsolving.com/community/c6h1784825p34307772
Find all functions $f:\mathbb R^+\to\mathbb R^+$ for which:
$$f(f(x)+f(y))=yf(1+yf(x))$$for all $x,y\in\mathbb R^+$.
2 replies
jasperE3
Yesterday at 3:55 PM
jasperE3
4 hours ago
J
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Geometry
srnjbr   0
4 hours ago
in triangle abc, we know that bac=60. the circumcircle of the center i is tangent to the sides ab and ac at points e and f respectively. the midpoint of side bc is called m. if lines bi and ci intersect line ef at points p and q respectively, show that pmq is equilateral.
0 replies
srnjbr
4 hours ago
0 replies
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J
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Classic geometry problem
Raul_S_Baz   2
N Mar 17, 2025 by Raul_S_Baz
IMAGE
2 replies
Raul_S_Baz
Mar 4, 2025
Raul_S_Baz
Mar 17, 2025
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Classic geometry problem
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Reveal topic tags
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Raul_S_Baz
10 posts
Raul_S_Baz
#1 h  Mar 4, 2025, 3:49 AM
Y by
https://imgur.com/a/Oi4SbIj
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sunken rock
4372 posts
sunken rock
#2 h  Mar 4, 2025, 11:42 AM • 2 Y
Y by Raul_S_Baz, sangriabeaver5
$\triangle ADC\sim\triangle BNC$; as $AD=2CD$, it follows that $BN=2CN$, done.

Best regards,
sunken rock
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Raul_S_Baz
10 posts
Raul_S_Baz
#3 h  Mar 17, 2025, 4:10 PM
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Thanx a lot! Very fine solution!
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