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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
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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IMO ShortList 1998, combinatorics theory problem 5
orl   46
N an hour ago by Maximilian113
Source: IMO ShortList 1998, combinatorics theory problem 5
In a contest, there are $m$ candidates and $n$ judges, where $n\geq 3$ is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most $k$ candidates. Prove that \[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]
46 replies
1 viewing
orl
Oct 22, 2004
Maximilian113
an hour ago
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Abelkonkurransen 2025 3a
Lil_flip38   6
N an hour ago by MathLuis
Source: abelkonkurransen
Let \(ABC\) be a triangle. Let \(E,F\) be the feet of the altitudes from \(B,C\) respectively. Let \(P,Q\) be the projections of \(B,C\) onto line \(EF\). Show that \(PE=QF\).
6 replies
Lil_flip38
Mar 20, 2025
MathLuis
an hour ago
J
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Abelkonkurransen 2025 2b
Lil_flip38   4
N an hour ago by MathLuis
Source: abelkonkurransen
Which positive integers $a$ have the property that \(n!-a\) is a perfect square for infinitely many positive integers \(n\)?
4 replies
Lil_flip38
Mar 20, 2025
MathLuis
an hour ago
J
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Abelkonkurransen 2025 2a
Lil_flip38   2
N an hour ago by MathLuis
Source: Abelkonkurransen
A teacher asks each of eleven pupils to write a positive integer with at most four digits, each on a separate yellow sticky note. Show that if all the numbers are different, the teacher can always submit two or more of the eleven stickers so that the average of the numbers on the selected notes are not an integer.
2 replies
Lil_flip38
Mar 20, 2025
MathLuis
an hour ago
J
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Easy function in turkey TST
egxa   9
N an hour ago by Levieee
Source: 2024 Turkey TST P2
Find all $f:\mathbb{R}\to\mathbb{R}$ functions such that
$$f(x+y)^3=(x+2y)f(x^2)+f(f(y))(x^2+3xy+y^2)$$for all real numbers $x,y$
9 replies
egxa
Mar 18, 2024
Levieee
an hour ago
J
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2023 Iran MO 2nd round P6
Amiralizakeri2007   3
N an hour ago by iliya8788
Source: 2023 Iran MO
6. Circles $W_{1}$ and $W_{2}$ with equal radii are given. Let $P$,$Q$ be the intersection of the circles.
points $B$ and $C$ are on $W_{1}$ and $W_{2}$ such that they are inside $W_{2}$ and $W_{1}$ respectively.
Points $X$,$Y$ $\neq$ $P$ are on $W_{1}$ and $W_{2}$ respectively, such that $\angle{BPQ}=\angle{BYQ}$ and $\angle{CPQ}=\angle{CXQ}$.Denote by $S$ as the other intersection of $(YPB)$ and $(XPC)$. Prove that $QS,BC,XY$ are concurrent.
3 replies
Amiralizakeri2007
May 17, 2023
iliya8788
an hour ago
J
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i love mordell
MR.1   6
N 2 hours ago by epl1
Source: own
find all pairs of $(m,n)$ such that $n^2-79=m^3$
6 replies
MR.1
Apr 10, 2025
epl1
2 hours ago
J
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Abelkonkurransen 2025 1b
Lil_flip38   1
N 2 hours ago by MathLuis
Source: abelkonkurransen
In Duckville there is a perpetual trophy with the words “Best child of Duckville” engraved on it. Each inhabitant of Duckville has a non-empty list (which never changes) of other inhabitants of Duckville. Whoever receives the trophy
gets to keep it for one day, and then passes it on to someone on their list the next day. Gregers has previously received the trophy. It turns out that each time he does receive it, he is guaranteed to receive it again exactly $2025$ days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers $n>0$ for which we can be absolutely certain that she cannot receive the trophy again in $n$ days, given the above information.
1 reply
Lil_flip38
Mar 20, 2025
MathLuis
2 hours ago
J
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Inequality with a,b,c
GeoMorocco   0
2 hours ago
Source: Morocco Training
Let $ a,b,c $ be real numbers such that : $ \sqrt{3a^2+b^2}+\sqrt{3b^2+c^2}+\sqrt{3c^2+a^2} \leq 6$ . Prove that : $$8+abc\geq 3(a+b+c) $$
0 replies
GeoMorocco
2 hours ago
0 replies
J
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Nepal TST DAY 1 Problem 1
Bata325   6
N 3 hours ago by Mathdreams
Source: Nepal TST 2025 p1
Consider a triangle $\triangle ABC$ and some point $X$ on $BC$. The perpendicular from $X$ to $AB$ intersects the circumcircle of $\triangle AXC$ at $P$ and the perpendicular from $X$ to $AC$ intersects the circumcircle of $\triangle AXB$ at $Q$. Show that the line $PQ$ does not depend on the choice of $X$.(Shining Sun, USA)
6 replies
Bata325
Yesterday at 1:21 PM
Mathdreams
3 hours ago
J
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Ihave a minor issue.
CovertQED   1
N 4 hours ago by vanstraelen
The area of triangle ABC is 18，sin2A +sin2B =4sinAsinB.Find the minimum perimeter of triangle ABC.
1 reply
CovertQED
Today at 1:52 PM
vanstraelen
4 hours ago
J
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geometry parabola problem
smalkaram_3549   8
N 4 hours ago by jb2015007
How would you solve this without using calculus?
8 replies
smalkaram_3549
Yesterday at 9:52 PM
jb2015007
4 hours ago
J
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Indonesia Regional MO 2019 Part A
parmenides51   13
N Today at 3:09 PM by SomeonecoolLovesMaths
Indonesia Regional MO
Year 2019 Part A

Time: 90 minutes Rules


p1. In the bag there are $7$ red balls and $8$ white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...


p2. Given a regular hexagon with a side length of $1$ unit. The area of the hexagon is ...


p3. It is known that $r, s$ and $1$ are the roots of the cubic equation $x^3 - 2x + c = 0$. The value of $(r-s)^2$ is ...


p4. The number of pairs of natural numbers $(m, n)$ so that $GCD(n,m) = 2$ and $LCM(m,n) = 1000$ is ...


p5. A data with four real numbers $2n-4$, $2n-6$, $n^2-8$, $3n^2-6$ has an average of $0$ and a median of $9/2$. The largest number of such data is ...


p6. Suppose $a, b, c, d$ are integers greater than $2019$ which are four consecutive quarters of an arithmetic row with $a <b <c <d$. If $a$ and $d$ are squares of two consecutive natural numbers, then the smallest value of $c-b$ is ...


p7. Given a triangle $ABC$, with $AB = 6$, $AC = 8$ and $BC = 10$. The points $D$ and $E$ lies on the line segment $BC$. with $BD = 2$ and $CE = 4$. The measure of the angle $\angle DAE$ is ...


p8. Sequqnce of real numbers $a_1,a_2,a_3,...$ meet $\frac{na_1+(n-1)a_2+...+2a_{n-1}+a_n}{n^2}=1$ for each natural number $n$. The value of $a_1a_2a_3...a_{2019}$ is ....


p9. The number of ways to select four numbers from $\{1,2,3, ..., 15\}$ provided that the difference of any two numbers at least $3$ is ...


p10. Pairs of natural numbers $(m , n)$ which satisfies $$m^2n+mn^2 +m^2+2mn = 2018m + 2019n + 2019$$are as many as ...


p11. Given a triangle $ABC$ with $\angle ABC =135^o$ and $BC> AB$. Point $D$ lies on the side $BC$ so that $AB=CD$. Suppose $F$ is a point on the side extension $AB$ so that $DF$ is perpendicular to $AB$. The point $E$ lies on the ray $DF$ such that $DE> DF$ and $\angle ACE = 45^o$. The large angle $\angle AEC$ is ...


p12. The set of $S$ consists of $n$ integers with the following properties: For every three different members of $S$ there are two of them whose sum is a member of $S$. The largest value of $n$ is ....


p13. The minimum value of $\frac{a^2+2b^2+\sqrt2}{\sqrt{ab}}$ with $a, b$ positive reals is ....


p14. The polynomial P satisfies the equation $P (x^2) = x^{2019} (x+ 1) P (x)$ with $P (1/2)= -1$ is ....


p15. Look at a chessboard measuring $19 \times 19$ square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of $k$ coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of $k$ so that the game never ends for any initial square selection is ....
13 replies
parmenides51
Nov 11, 2021
SomeonecoolLovesMaths
Today at 3:09 PM
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Diophantine Equation in (x^2+4) set
Johann Peter Dirichlet   0
Today at 3:01 PM
Let $S=\{n^2+4 | n \in \mathbb{Z}\}$.

Find all $p,q,r \in S$ so that $pq-r=4$.
0 replies
Johann Peter Dirichlet
Today at 3:01 PM
0 replies
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J
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Geometry: Prove
pythagorazz   5
N Dec 10, 2020 by ATGY
Points A, M, N, and B are collinear, in that order, and AM=4, MN=2, NB=3. If point C is not collinear with these four points and AC=6, prove that CN bisects angle BCM.
5 replies
pythagorazz
Nov 18, 2014
ATGY
Dec 10, 2020
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Geometry: Prove
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Reveal topic tags
geometry
trigonometry
LaTeX
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Law of Cosines
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pythagorazz
373 posts
pythagorazz
#1 h  Nov 18, 2014, 12:07 AM • 2 Y
Y by Adventure10, Mango247
Points A, M, N, and B are collinear, in that order, and AM=4, MN=2, NB=3. If point C is not collinear with these four points and AC=6, prove that CN bisects angle BCM.
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somepersonoverhere
1313 posts
somepersonoverhere
#2 h  Nov 18, 2014, 12:58 AM • 2 Y
Y by pythagorazz, Adventure10
Click to reveal hidden text

PS. Sorry, I don't know LaTeX :(
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wqr
36 posts
wqr
#3 h  Nov 18, 2014, 5:24 AM • 2 Y
Y by Adventure10, Mango247
Are there any alternate solutions? I feel like there might be
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Octophi
377 posts
Octophi
#4 h  Nov 18, 2014, 10:53 PM • 4 Y
Y by pythagorazz, wqr, Adventure10, Mango247
An alternate solution
Click to reveal hidden text
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franzliszt
23531 posts
franzliszt
#5 h  Dec 10, 2020, 1:36 AM
Y by
This happens to be 2014-2015 Philippine MO Areas P2.

Consider a point $P\ne N$ on the line such that $PA=6$. We have $(M,B;N,P)=-1$ and $\angle PCN=90^\circ$ by Thales. The result follows.
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ATGY
2502 posts
ATGY
#6 h  Dec 10, 2020, 3:28 AM
Y by
simpler solution
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