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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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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
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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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help me ..
exoticc   2
N 16 minutes ago by exoticc
Find all pairs of functions (f,g) : R->R that satisfy:
f(1)=2025;
g(f(x+y))+2x+y-1=f(x)+(2x+y)g(y) , ∀x,y∈R
2 replies
exoticc
6 hours ago
exoticc
16 minutes ago
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trigonometric inequality
MATH1945   11
N 35 minutes ago by sqing
Source: ?
In triangle $ABC$, prove that $$sin^2(A)+sin^2(B)+sin^2(C) \leq \frac{9}{4}$$
11 replies
1 viewing
MATH1945
May 26, 2016
sqing
35 minutes ago
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Centrally symmetric polyhedron
genius_007   0
37 minutes ago
Source: unknown
Does there exist a convex polyhedron with an odd number of sides, where each side is centrally symmetric?
0 replies
genius_007
37 minutes ago
0 replies
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Combinatorial Game
Cats_on_a_computer   2
N an hour ago by Cats_on_a_computer

Let n>1 be odd. A row of n spaces is initially empty. Alice and Bob alternate moves (Alice first); on each turn a player may either
1. Place a stone in any empty space, or
2. Remove a stone from a non-empty space S, then (if they exist) place stones in the nearest empty spaces immediately to the left and to the right of S.

Furthermore, no move may produce a position that has appeared earlier. The player loses when they cannot make a legal move.
Assuming optimal play, which move(s) can Alice make on her first turn?
2 replies
Cats_on_a_computer
an hour ago
Cats_on_a_computer
an hour ago
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Inequalities
sqing   0
an hour ago
Let $ a,b> 0 $ and $2a+2b+ab=5. $ Prove that
$$ \frac{a^4}{b^4}+\frac{1}{a^4}+42ab-a^4\geq 43$$$$ \frac{a^5}{b^5}+\frac{1}{a^5}+64ab-a^5\geq 65$$$$ \frac{a^6}{b^6}+\frac{1}{a^6}+90ab-a^6\geq 91$$$$ \frac{a^7}{b^7}+\frac{1}{a^7}+121ab-a^7\geq 122$$
0 replies
sqing
an hour ago
0 replies
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2-var inequality
sqing   4
N an hour ago by sqing
Source: Own
Let $ a,b> 0 $ and $2a+2b+ab=5. $ Prove that
$$ \frac{a^2}{b^2}+\frac{1}{a^2}-a^2\geq 1$$$$ \frac{a^3}{b^3}+\frac{1}{a^3}-a^3\geq 1$$
4 replies
sqing
2 hours ago
sqing
an hour ago
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A game of digits and seventh powers
v_Enhance   28
N an hour ago by blueprimes
Source: Taiwan 2014 TST3 Quiz 1, P2
Alice and Bob play the following game. They alternate selecting distinct nonzero digits (from $1$ to $9$) until they have chosen seven such digits, and then consider the resulting seven-digit number by concatenating the digits in the order selected, with the seventh digit appearing last (i.e. $\overline{A_1B_2A_3B_4A_6B_6A_7}$). Alice wins if and only if the resulting number is the last seven decimal digits of some perfect seventh power. Please determine which player has the winning strategy.
28 replies
v_Enhance
Jul 18, 2014
blueprimes
an hour ago
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n containers distribute gas to a car for a single loop along the track
parmenides51   16
N an hour ago by v_Enhance
Source: Spanish Mathematical Olympiad 1997 P6
The exact quantity of gas needed for a car to complete a single loop around a track is distributed among $n$ containers placed along the track. Prove that there exists a position starting at which the car, beginning with an empty tank of gas, can complete a loop around the track without running out of gas. The tank of gas is assumed to be large enough.
16 replies
1 viewing
parmenides51
Jul 31, 2018
v_Enhance
an hour ago
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Factorial: n!|a^n+1
Nima Ahmadi Pour   68
N an hour ago by blueprimes
Source: IMO Shortlist 2005 N4, Iran preparation exam
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property:
\[ n!\mid a^n + 1 \]

Proposed by Carlos Caicedo, Colombia
68 replies
Nima Ahmadi Pour
Apr 24, 2006
blueprimes
an hour ago
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Geometry problem
Whatisthepurposeoflife   1
N an hour ago by Royal_mhyasd
Source: Derived from MEMO 2024 I3
Triangle ∆ABC is scalene the circle w that goes through the points A and B intersects AC at E BC at D let the Lines BE and AD intersect at point F. And let the tangents A and B of circle w Intersect at point G.
Prove that C F and G are collinear
1 reply
1 viewing
Whatisthepurposeoflife
2 hours ago
Royal_mhyasd
an hour ago
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\frac{x^2}{x+y}+\frac{y^2}{1-x}+\frac{(1-x-y)^2}{1-y}\geq\frac{1}{2} if 0<x,y<1
parmenides51   6
N 2 hours ago by JH_K2IMO
Source: KJMO 2009 p3
For two arbitrary reals $x, y$ which are larger than $0$ and less than $1.$ Prove that$$\frac{x^2}{x+y}+\frac{y^2}{1-x}+\frac{(1-x-y)^2}{1-y}\geq\frac{1}{2}.$$
6 replies
parmenides51
May 2, 2019
JH_K2IMO
2 hours ago
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Original Problem (qrxz17)
qrxz17   0
2 hours ago
Problem. Suppose that
\begin{align*} (a^2+b^2)^2 + (b^2+c^2)^2 +(c^2+a^2)^2 &= 138 \\ (a^2+b^2+c^2)^2 &=100. \end{align*}Find \(a^2(a^2-1) + b^2(b^2-1)+c^2(c^2-1)\).
Answer: Click to reveal hidden text
Solution. Subtracting the two given equations, we get
\begin{align*} a^4+b^4+c^4=38. \end{align*}Taking the square root of the second equation, we get

\begin{align*} a^2+b^2+c^2 = 10. \end{align*}
Then,
\begin{align*} a^4+b^4+c^4-(a^2+b^2+c^2) &= a^4-a^2+b^4-b^2+c^4-c^2 \\ &= a^2(a^2-1)+b^2(b^2-1)+c^2(c^2-1) = \boxed{28}. \end{align*}
0 replies
qrxz17
2 hours ago
0 replies
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17th PMO Area Stage #5
yes45   0
2 hours ago
Triangle \(ABC\) has a right angle at \(B\), with \(AB = 3\) and \(BC = 4\). If \(D\) and \(E\) are points
on \(AC\) and \(BC\), respectively, such that \(CD = DE = \frac{5}{3}\), find the perimeter of quadrilateral
\(ABED\).
Answer Confirmation
Solution
0 replies
yes45
2 hours ago
0 replies
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2-var inequality
sqing   6
N 2 hours ago by sqing
Source: Own
Let $ a,b>0 , a^2+b^2-ab\leq 1 . $ Prove that
$$a^3+b^3 -\frac{a^4}{b+1} -\frac{b^4}{a+1} \leq 1 $$
6 replies
sqing
Yesterday at 2:15 AM
sqing
2 hours ago
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geometry question
kjhgyuio   5
N Apr 26, 2025 by valisaxieamc
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5 replies
kjhgyuio
Apr 8, 2025
valisaxieamc
Apr 26, 2025
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geometry question
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Reveal topic tags
geometry
right angled triangles
Triangle
move to msm
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kjhgyuio
69 posts
kjhgyuio
#1 h  Apr 8, 2025, 9:04 AM
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kjhgyuio
69 posts
kjhgyuio
#2 h  Apr 8, 2025, 9:16 AM
Y by
someone pls help
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alexheinis
10631 posts
alexheinis
#3 h  Apr 8, 2025, 10:22 AM
Y by
Denote the height difference of $A,P$ by $k$ and the height difference of $Q,B$ by $m$. Then $(k:4:m)=(AP:PQ:QB)=(22\cdot 11: 26\cdot 11: 26\cdot 22)=(11:13:26)$. This gives us $k,m$ and $OA=k+m+4$.
Then we have $OAB$, the total area. Finally $OPQ={4\over {k+m+4}} \cdot OAB$. I leave the details to you.
This post has been edited 2 times. Last edited by alexheinis, Apr 8, 2025, 10:25 AM
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vanstraelen
9062 posts
vanstraelen
#4 h  Apr 8, 2025, 3:21 PM
Y by
Given $26 \cdot AP=22 \cdot PQ=11 \cdot QB$ or $26(y_{P}-y_{A})=22(y_{Q}-y_{P})=11(y_{B}-y_{Q})$,
$26(y_{P}-y_{A})=22(-4)=11(-y_{Q})$, with solutions $y_{Q}=8,y_{P}=12$ and $y_{A}=\frac{200}{13}$.
Area $\triangle OPQ\ =\ 26$.
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undefined-NaN
11 posts
undefined-NaN
#5 h  Apr 25, 2025, 5:54 PM
Y by
AREA OF NEW OPQ TRIANGLE = S
ALSO NEW OQB TRIANGLES AREA = 2S
BECAUSE THEY HAVE SAME HEIGH
BUT HAVE DIFFERENT BASE => 22PQ=11QB SO 2PQ=1QB

ALSO 2 TIMES PQ VERTICAL DISTANCE = 1 TIME QB VERTICAL DISTANCE SO PB VERTICAL DISTANCE = 12

AFTER OPB TRIANGEL ARE = 12 x 13 / 2 =78

SO THIS WAY WE CAN FIND S = AREA OF OPQ

3S = 78 => S 78/3 = 26
This post has been edited 1 time. Last edited by undefined-NaN, Apr 25, 2025, 6:04 PM
Reason: NO NEED TO PHOTO
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valisaxieamc
506 posts
valisaxieamc
#7 h  Apr 26, 2025, 4:52 AM
Y by
@above R u ok, why is this all caps or did you acidentally do all caps and then was too lazy to rewrite it
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