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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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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
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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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logarithm rational problem
smalkaram_3549   0
40 minutes ago
I keep getting 94 as my answer, but the correct answer is 81.
0 replies
smalkaram_3549
40 minutes ago
0 replies
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Number Theory Chain!
JetFire008   28
N 42 minutes ago by Sedro
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
28 replies
JetFire008
Apr 7, 2025
Sedro
42 minutes ago
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Classic Invariant
Mathdreams   2
N an hour ago by Tony_stark0094
Source: 2025 Nepal Mock TST Day 1 Problem 1

Prajit and Kritesh challenge each other with a marble game. In a bag, there are initially $2024$ red marbles and $2025$ blue marbles. The rules of the game are as follows:

Move: In each turn, a player (either Prajit or Kritesh) removes two marbles from the bag.

If the two marbles are of the same color, they are both discarded and a red marble is added to the bag.
If the two marbles are of different colors, they are both discarded and a blue marble is added to the bag.

The game continues by repeating the above move.

Prove that no matter what sequence of moves is made, the process always terminates with exactly one marble left. In addition, find the possible colors of the marble remaining.
2 replies
Mathdreams
Thursday at 1:28 PM
Tony_stark0094
an hour ago
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geometry parabola problem
smalkaram_3549   6
N an hour ago by ReticulatedPython
How would you solve this without using calculus?
6 replies
smalkaram_3549
4 hours ago
ReticulatedPython
an hour ago
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ineq.trig.
wer   18
N 2 hours ago by anduran
If a, b, c are the sides of a triangle, show that: $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\frac{r}{R}\le2$
18 replies
wer
Jul 5, 2014
anduran
2 hours ago
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Assam Mathematics Olympiad 2023 Category III Q1
SomeonecoolLovesMaths   4
N 2 hours ago by kjhgyuio
What is the $288$th term of the sequence $a,b,b,c,c,c,d,d,d,d,e,e,e,e,e,f,f,f,f,f,f,...?$
4 replies
SomeonecoolLovesMaths
Sep 11, 2024
kjhgyuio
2 hours ago
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P2 Cono Sur 2021
Leo890   9
N 2 hours ago by jordiejoh
Source: Cono Sur 2021 P2
Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$, respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$, respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$, and the circle $C_2$ intersects $AC$ at $R$ and $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.
9 replies
1 viewing
Leo890
Nov 30, 2021
jordiejoh
2 hours ago
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collinearity eanted, line tangent to 3 incircles ABP, ACP, BCP related
parmenides51   5
N 2 hours ago by leon.tyumen
Source: MGO p6 https://artofproblemsolving.com/community/c594864h3379839p31486784
Let $P$ be a point inside $\vartriangle ABC$. It is known that there exists a line tangent to the incircles of $\vartriangle ABP$, $\vartriangle ACP$ and $\vartriangle BCP$. Prove that if $X$ is the intersection point of the common external tangents of a random pair of these incircles and Y is the intersection point of common external tangents of some other pair of these three incircles, then $XY$ passes through either $A$, $B$ or $C$.
5 replies
parmenides51
Sep 2, 2024
leon.tyumen
2 hours ago
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Inequality with a,b,c
GeoMorocco   0
4 hours ago
Source: Morocco Training
Let $ a,b,c $ be positive real numbers such that : $ ab+bc+ca=3 $ . Prove that : $$\frac{\sqrt{1+a^2}}{1+ab}+\frac{\sqrt{1+b^2}}{1+bc}+\frac{\sqrt{1+c^2}}{1+ca}\ge \sqrt{\frac{3(a+b+c)}{2}}$$
0 replies
GeoMorocco
4 hours ago
0 replies
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Y2K Game
MithsApprentice   13
N 4 hours ago by zuat.e
Source: USAMO 1999 Problem 5
The Y2K Game is played on a $1 \times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.
13 replies
MithsApprentice
Oct 3, 2005
zuat.e
4 hours ago
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Config geo with the Euler line
a_507_bc   11
N 4 hours ago by falantrng
Source: BMO SL 2023 G4
Let $O$ and $H$ be the circumcenter and orthocenter of a scalene triangle $ABC$, respectively. Let $D$ be the intersection point of the lines $AH$ and $BC$. Suppose the line $OH$ meets the side $BC$ at $X$. Let $P$ and $Q$ be the second intersection points of the circumcircles of $\triangle BDH$ and $\triangle CDH$ with the circumcircle of $\triangle ABC$, respectively. Show that the four points $P, D, Q$ and $X$ lie on a circle.
11 replies
a_507_bc
May 3, 2024
falantrng
4 hours ago
J
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Grasshoppers facing in four directions
Stuttgarden   2
N 4 hours ago by biomathematics
Source: Spain MO 2025 P5
Let $S$ be a finite set of cells in a square grid. On each cell of $S$ we place a grasshopper. Each grasshopper can face up, down, left or right. A grasshopper arrangement is Asturian if, when each grasshopper moves one cell forward in the direction in which it faces, each cell of $S$ still contains one grasshopper.
[list]
[*] Prove that, for every set $S$, the number of Asturian arrangements is a perfect square.
[*] Compute the number of Asturian arrangements if $S$ is the following set:
2 replies
Stuttgarden
Mar 31, 2025
biomathematics
4 hours ago
J
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the triangles have the same centroid
V-217   2
N 4 hours ago by V-217
Source: Own
Let $ABC$ an acute triangle inscribed in a circle with its center $O$. Let the lines $B_{a}C_{a}, C_{b}A_{b}$ and $A_{c}B_{c}$ the perpendiculars in $O$ to $AO, BO, CO$ respectively, where $A_{b},A_{c}\in BC$, $B_{a},B_{c}\in AC$ and $C_{a},C_{b}\in AB$. Let $O_{a}, O_{b}, O_{c}$ be the circumcircles of triangles $AC_{a}B_{a}, BA_{b}C_{b}, CA_{c}B_{c}$.
Prove that the triangles $O_{a}O_{b}O_{c}$ and $ABC$ have the same centroid.
2 replies
V-217
Jul 20, 2023
V-217
4 hours ago
J
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uncanny one
Valentin Vornicu   5
N 4 hours ago by DensSv
Source: Romanian Junior BkMO TST 2004, problem 12
One considers the positive integers $a < b \leq c < d $ such that $ad=bc$ and $\sqrt d - \sqrt a \leq 1 $.

Prove that $a$ is a perfect square.
5 replies
Valentin Vornicu
May 3, 2004
DensSv
4 hours ago
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Congruence
Ecrin_eren   3
N Apr 7, 2025 by lbh_qys
Find the number of integer pairs (x, y) satisfying the congruence equation:

3y² + 3x²y + y³ ≡ 3x² (mod 41)

for 0 ≤ x, y < 41.

3 replies
Ecrin_eren
Apr 3, 2025
lbh_qys
Apr 7, 2025
J
Congruence
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Reveal topic tags
number theory
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Ecrin_eren
42 posts
Ecrin_eren
#1 h  Apr 3, 2025, 10:34 AM
Y by
Find the number of integer pairs (x, y) satisfying the congruence equation:

3y² + 3x²y + y³ ≡ 3x² (mod 41)

for 0 ≤ x, y < 41.
Z K Y
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Ecrin_eren
42 posts
Ecrin_eren
#2 h  Apr 4, 2025, 1:39 PM
Y by
Bump bump
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Ecrin_eren
42 posts
Ecrin_eren
#3 h  Apr 6, 2025, 8:42 AM
Y by
Any ideas?
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lbh_qys
507 posts
lbh_qys
#4 h  Apr 7, 2025, 3:39 AM
Y by
If $y\neq0,1$, then the original equation asserts that

\[ \left( 3x(1-y) y^{-1} \right)^2 = 3(1-y)(y+3) \]
The number of solutions satisfying this equation is given by

\[ \left(\frac{3 (1-y)(y+3)}{p}\right) + 1, \]
where $\left(\frac{\cdot}{p}\right)$ denotes the Legendre symbol.

Moreover, since

\[ \sum_{y=0}^{p-1} \left(\frac{(y-1)(y+3)}{p}\right) = -1, \]
it follows that the summation for this part is

\[ \sum_{y=2}^{p-1}\left( \left(\frac{3 (1-y)(y+3)}{p}\right) + 1 \right) = p-2+\left(\frac{-3}{p}\right)\left(\sum_{y=0}^{p-1} \left(\frac{(y-1)(y+3)}{p}\right) - \left(\frac{-3}{p}\right) - \left(\frac{0}{p}\right) \right) = p - 2 - \left(\frac{-3}{p}\right) - 1. \]
When $y = 0$, it is evident that $x = 0$ is the unique solution; and when $y = 1$, there is evidently no solution. Therefore, the total number of solutions is

\[ p - 2 - \left(\frac{-3}{p}\right) - 1 + 1 = p - 2 - \left(\frac{-3}{p}\right) = p - 2 - \left(\frac{p}{3}\right). \]
Since $p = 41 \equiv 2 \pmod{3}$, it follows that $\left(\frac{p}{3}\right) = -1$. Consequently, the number of solutions is

\[ p - 2 - (-1) = p - 1 = 40. \]
This post has been edited 1 time. Last edited by lbh_qys, Apr 7, 2025, 3:41 AM
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