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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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Darboux cubic
srirampanchapakesan   1
N 14 minutes ago by srirampanchapakesan
Source: Own
Let P be a point on the Darboux cubic (or the McCay Cubic ) of triangle ABC.

P1P2P3 is the circumcevian or pedal triangle of P wrt ABC.

Prove that P also lie on the Darboux cubic ( or the McCay Cubic) of P1P2P3 .
1 reply
1 viewing
srirampanchapakesan
May 7, 2025
srirampanchapakesan
14 minutes ago
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IMO Shortlist 2011, Algebra 2
orl   43
N 17 minutes ago by ezpotd
Source: IMO Shortlist 2011, Algebra 2
Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\]

Proposed by Warut Suksompong, Thailand
43 replies
orl
Jul 11, 2012
ezpotd
17 minutes ago
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Sequence inequality
BR1F1SZ   1
N 24 minutes ago by IndoMathXdZ
Source: 2025 Francophone MO Seniors P1
Let $a_1, a_2, a_3, \ldots$ be a sequence of positive integers satisfying the following property: for all positive integers $k < \ell$, for all distinct integers $m_1, m_2, \ldots, m_k$ and for all distinct integers $n_1, n_2, \ldots, n_\ell$,
\[ a_{m_1} + a_{m_2} + \cdots + a_{m_k} \leqslant a_{n_1} + a_{n_2} + \cdots + a_{n_\ell}. \]Prove that there exist two integers $N$ and $b$ such that $a_n = b$ for all $n \geqslant N$.
1 reply
BR1F1SZ
2 hours ago
IndoMathXdZ
24 minutes ago
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Hard geometry
Lukariman   1
N 31 minutes ago by Lukariman
Given triangle ABC, a line d intersects the sides AB, AC and the line BC at D, E, F respectively.

(a) Prove that the circles circumscribing triangles ADE, BDF and CEF pass through a point P and P belongs to the circumcircle of triangle ABC.

(b) Prove that the centers of the circles circumscribing triangles ADE, BDF, CEF and ABC are all on the circle.

(c) Let $O_a$,$ O_b$, $O_c$ be the centers of the circles circumscribing triangles ADE, BDF, CEF. Prove that the orthocenter of triangle $O_a$$O_b$$O_c$ belongs to d.

(d) Prove that the orthocenters of triangles ADE, ABC, BDF, CEF are collinear.
1 reply
Lukariman
Yesterday at 12:53 PM
Lukariman
31 minutes ago
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Anything real in this system must be integer
Assassino9931   1
N 33 minutes ago by lksb
Source: Al-Khwarizmi International Junior Olympiad 2025 P1
Determine the largest integer $c$ for which the following statement holds: there exists at least one triple $(x,y,z)$ of integers such that
\begin{align*} x^2 + 4(y + z) = y^2 + 4(z + x) = z^2 + 4(x + y) = c \end{align*}and all triples $(x,y,z)$ of real numbers, satisfying the equations, are such that $x,y,z$ are integers.

Marek Maruin, Slovakia
1 reply
Assassino9931
Friday at 9:26 AM
lksb
33 minutes ago
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geometry problem
kjhgyuio   0
an hour ago
........
0 replies
kjhgyuio
an hour ago
0 replies
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Concurrency of two lines and a circumcircle
BR1F1SZ   1
N an hour ago by MathLuis
Source: 2025 Francophone MO Juniors P3
Let $\triangle{ABC}$ be a triangle, $\omega$ its circumcircle and $O$ the center of $\omega$. Let $P$ be a point on the segment $BC$. We denote by $Q$ the second intersection point of the circumcircles of triangles $\triangle{AOB}$ and $\triangle{APC}$. Prove that the line $PQ$ and the tangent to $\omega$ at point $A$ intersect on the circumcircle of triangle $\triangle AOB$.
1 reply
BR1F1SZ
2 hours ago
MathLuis
an hour ago
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IMO Shortlist 2009 - Problem A2
April   93
N an hour ago by ezpotd
Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that:
\[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\]
Proposed by Juhan Aru, Estonia
93 replies
April
Jul 5, 2010
ezpotd
an hour ago
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Product of consecutive terms divisible by a prime number
BR1F1SZ   0
an hour ago
Source: 2025 Francophone MO Seniors P4
Determine all sequences of strictly positive integers $a_1, a_2, a_3, \ldots$ satisfying the following two conditions:
[list]
[*]There exists an integer $M > 0$ such that, for all indices $n \geqslant 1$, $0 < a_n \leqslant M$.
[*]For any prime number $p$ and for any index $n \geqslant 1$, the number
\[ a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p} \]is a multiple of $p$.
[/list]


0 replies
BR1F1SZ
an hour ago
0 replies
J
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Fixed and variable points
BR1F1SZ   0
an hour ago
Source: 2025 Francophone MO Seniors P3
Let $\omega$ be a circle with center $O$. Let $B$ and $C$ be two fixed points on the circle $\omega$ and let $A$ be a variable point on $\omega$. We denote by $X$ the intersection point of lines $OB$ and $AC$, assuming $X \neq O$. Let $\gamma$ be the circumcircle of triangle $\triangle AOX$. Let $Y$ be the second intersection point of $\gamma$ with $\omega$. The tangent to $\gamma$ at $Y$ intersects $\omega$ at $I$. The line $OI$ intersects $\omega$ at $J$. The perpendicular bisector of segment $OY$ intersects line $YI$ at $T$, and line $AJ$ intersects $\gamma$ at $P$. We denote by $Z$ the second intersection point of the circumcircle of triangle $\triangle PYT$ with $\omega$. Prove that, as point $A$ varies, points $Y$ and $Z$ remain fixed.
0 replies
BR1F1SZ
an hour ago
0 replies
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Use 3d paper
YaoAOPS   7
N an hour ago by EGMO
Source: 2025 CTST p4
Recall that a plane divides $\mathbb{R}^3$ into two regions, two parallel planes divide it into three regions, and two intersecting planes divide space into four regions. Consider the six planes which the faces of the cube $ABCD-A_1B_1C_1D_1$ lie on, and the four planes that the tetrahedron $ACB_1D_1$ lie on. How many regions do these ten planes split the space into?
7 replies
YaoAOPS
Mar 6, 2025
EGMO
an hour ago
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Cyclic ine
m4thbl3nd3r   2
N 2 hours ago by m4thbl3nd3r
Let $a,b,c>0$ such that $a^2+b^2+c^2=3$. Prove that $$\sum \frac{a^2}{b}+abc \ge 4$$
2 replies
m4thbl3nd3r
Yesterday at 3:34 PM
m4thbl3nd3r
2 hours ago
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GCD and LCM operations
BR1F1SZ   0
2 hours ago
Source: 2025 Francophone MO Juniors P4
Charlotte writes the integers $1,2,3,\ldots,2025$ on the board. Charlotte has two operations available: the GCD operation and the LCM operation.
[list]
[*]The GCD operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{gcd}(a, b)$.
[*]The LCM operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{lcm}(a, b)$.
[/list]
An integer $N$ is called a winning number if there exists a sequence of operations such that, at the end, the only integer left on the board is $N$. Find all winning integers among $\{1,2,3,\ldots,2025\}$ and, for each of them, determine the minimum number of GCD operations Charlotte must use.

Note: The number $\operatorname{gcd}(a, b)$ denotes the greatest common divisor of $a$ and $b$, while the number $\operatorname{lcm}(a, b)$ denotes the least common multiple of $a$ and $b$.
0 replies
BR1F1SZ
2 hours ago
0 replies
J
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Balanced grids
BR1F1SZ   0
2 hours ago
Source: 2025 Francophone MO Juniors/Seniors P2
Let $n \geqslant 2$ be an integer. We consider a square grid of size $2n \times 2n$ divided into $4n^2$ unit squares. The grid is called balanced if:
[list]
[*]Each cell contains a number equal to $-1$, $0$ or $1$.
[*]The absolute value of the sum of the numbers in the grid does not exceed $4n$.
[/list]
Determine, as a function of $n$, the smallest integer $k \geqslant 1$ such that any balanced grid always contains an $n \times n$ square whose absolute sum of the $n^2$ cells is less than or equal to $k$.
0 replies
BR1F1SZ
2 hours ago
0 replies
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Three vertices consecutives and its circumcircle
Johann Peter Dirichlet   5
N Nov 29, 2013 by GlassBead
Source: Problem 2 - Brazil Math Olympiad 1994
Given any convex polygon, show that there are three consecutive vertices such that the polygon lies inside the circle through them.
5 replies
Johann Peter Dirichlet
Mar 17, 2006
GlassBead
Nov 29, 2013
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Three vertices consecutives and its circumcircle
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Reveal topic tags
geometry
circumcircle
trigonometry
combinatorial geometry
trig identities
Law of Sines
geometry proposed
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G H BBookmark kLocked kLocked NReply
Source: Problem 2 - Brazil Math Olympiad 1994
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Johann Peter Dirichlet
376 posts
Johann Peter Dirichlet
#1 h  Mar 17, 2006, 5:00 AM • 2 Y
Y by Adventure10, Mango247
Given any convex polygon, show that there are three consecutive vertices such that the polygon lies inside the circle through them.
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Atlas_ha78
69 posts
Atlas_ha78
#2 h  Nov 19, 2013, 6:08 PM • 2 Y
Y by Adventure10, Mango247
I think we can solve it with extremal.
Any ideas?
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fmasroor
779 posts
fmasroor
#3 h  Nov 27, 2013, 8:04 PM • 2 Y
Y by Adventure10, Mango247
Well look at the convex hull, make a really really really really big circle around it and gradually shrink the circle down. Would that work?
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GlassBead
1583 posts
GlassBead
#4 h  Nov 29, 2013, 7:45 PM • 1 Y
Y by Adventure10
This is a consequence of Bose's theorem for polygons (a somewhat stronger version of the discrete four vertex theorem). The idea is very similar to the idea that every finite set of points has a Delaunay triangulation (i.e. can be triangulated by triangles whose circumcircles do not contain any other vertices).
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infiniteturtle
1131 posts
infiniteturtle
#5 h  Nov 29, 2013, 10:22 PM • 1 Y
Y by Adventure10
Also posted here. However, I believe the last sentence in the proof was never actually resolved...
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GlassBead
1583 posts
GlassBead
#6 h  Nov 29, 2013, 10:58 PM • 1 Y
Y by Adventure10
The last statement (of each solution) seems to be inspired by extended law of sines, though I'm not sure if the arguments are quite correct.
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