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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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Be sure to mark your calendars for the following events:
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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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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Existence of perfect squares
egxa   0
a few seconds ago
Source: All Russian 2025 10.3
Find all natural numbers \(n\) for which there exists an even natural number \(a\) such that the number
\[ (a - 1)(a^2 - 1)\cdots(a^n - 1) \]is a perfect square.
0 replies
egxa
a few seconds ago
0 replies
J
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Weighted graph problem
egxa   0
2 minutes ago
Source: All Russian 2025 10.4
In the plane, $106$ points are marked, no three of which are collinear. All possible segments between them are drawn. Grisha assigned to each drawn segment a real number with absolute value no greater than 1. For every group of 6 marked points, he calculated the sum of the numbers on all 15 connecting segments. It turned out that the absolute value of each such sum is at least \(C\), and there are both positive and negative such sums. What is the maximum possible value of \(C\)?
0 replies
egxa
2 minutes ago
0 replies
J
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Board problem with complex numbers
egxa   0
4 minutes ago
Source: All Russian 2025 11.1
$777$ pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers \(a\) and \(b\) from the board such that:
\[ a^2 + b^2 + 1 = 2ab \]Ways that differ by the order of selection are considered the same. Prove that there exist two numbers \(c\) and \(d\) from the board such that:
\[ c^2 + d^2 + 2025 = 2cd \]
0 replies
egxa
4 minutes ago
0 replies
J
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two 3D problems in one day
egxa   0
5 minutes ago
Source: All Russian 2025 11.2
A right prism \(ABCA_1B_1C_1\) is given. It is known that triangles \(A_1BC\), \(AB_1C\), \(ABC_1\), and \(ABC\) are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle \(ABC\), lie on the same sphere.
0 replies
egxa
5 minutes ago
0 replies
J
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Important pairs of polynomials
egxa   0
7 minutes ago
Source: All Russian 2025 11.3
A pair of polynomials \(F(x, y)\) and \(G(x, y)\) with integer coefficients is called \emph{important} if from the divisibility of both differences \(F(a, b) - F(c, d)\) and \(G(a, b) - G(c, d)\) by $100$, it follows that both \(a - c\) and \(b - d\) are divisible by 100. Does there exist such an important pair of polynomials \(P(x, y)\), \(Q(x, y)\), such that the pair \(P(x, y) - xy\) and \(Q(x, y) + xy\) is also important?
0 replies
egxa
7 minutes ago
0 replies
J
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3D russian combo
egxa   0
8 minutes ago
Source: All Russian 2025 11.4
A natural number \(N\) is given. A cube with side length \(2N + 1\) is made up of \((2N + 1)^3\) unit cubes, each of which is either black or white. It turns out that among any $8$ cubes that share a common vertex and form a \(2 \times 2 \times 2\) cube, there are at most $4$ black cubes. What is the maximum number of black cubes that could have been used?
0 replies
egxa
8 minutes ago
0 replies
J
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cubefree divisibility
DottedCaculator   57
N 32 minutes ago by tastyl
Source: 2021 ISL N1
Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$
57 replies
DottedCaculator
Jul 12, 2022
tastyl
32 minutes ago
J
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ARMO 2025
Oksutok   0
37 minutes ago
All-Russian MO 2025 Day 1
0 replies
Oksutok
37 minutes ago
0 replies
J
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general form
pennypc123456789   1
N 39 minutes ago by sqing
If $a,b,c$ are positive real numbers, $k \ge 3$ then
$$ \frac{a + b}{a + kb + c} + \dfrac{b + c}{b + kc + a}+\dfrac{c + a}{c + ka + b} \geq \dfrac{6}{k+2}$$
1 reply
pennypc123456789
4 hours ago
sqing
39 minutes ago
J
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JBMO Shortlist 2023 A4
Orestis_Lignos   6
N 42 minutes ago by Novmath
Source: JBMO Shortlist 2023, A4
Let $a,b,c,d$ be positive real numbers with $abcd=1$. Prove that

$$\sqrt{\frac{a}{b+c+d^2+a^3}}+\sqrt{\frac{b}{c+d+a^2+b^3}}+\sqrt{\frac{c}{d+a+b^2+c^3}}+\sqrt{\frac{d}{a+b+c^2+d^3}} \leq 2$$
6 replies
Orestis_Lignos
Jun 28, 2024
Novmath
42 minutes ago
J
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2 var inquality
sqing   5
N an hour ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $ a+ b+2ab=4 . $ Prove that
$$ 3(a+b-2)(2 - ab) \ge (a-1)(b-1)(a-b)$$$$ 9 (a+b-2)(3 - 2ab) \ge 2\sqrt 5(a-1)(b-1)(a-b)$$$$9(a+b-2)(6 - 5ab) \ge2\sqrt {14} (a-1)(b-1)(a-b)$$
5 replies
sqing
Today at 1:50 AM
sqing
an hour ago
J
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Extent of Group Theory needed for NT
Math-Problem-Solving   0
an hour ago
How much of group theory, knowledge of rings and fields is required for doing number theory in full fledge. Given I am a class 11 high schooler student with a little background in math olympiad, so please mention some resources for learning these things which I can understand.
0 replies
Math-Problem-Solving
an hour ago
0 replies
J
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No perfect squares in A-A
JustPostChinaTST   7
N an hour ago by kes0716
Source: 2022 China TST, Test 3 P5
Show that there exist constants $c$ and $\alpha > \frac{1}{2}$, such that for any positive integer $n$, there is a subset $A$ of $\{1,2,\ldots,n\}$ with cardinality $|A| \ge c \cdot n^\alpha$, and for any $x,y \in A$ with $x \neq y$, the difference $x-y$ is not a perfect square.
7 replies
JustPostChinaTST
Apr 30, 2022
kes0716
an hour ago
J
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Quadric function
soryn   2
N 2 hours ago by soryn
If f(x)=ax^2+bx+c, a,b,c integers, |a|>=3, and M îs the set of integers x for which f(x) is a prime number and f has exactly one integer solution,prove that M has at most three elements.
2 replies
soryn
Today at 2:47 AM
soryn
2 hours ago
J
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J
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Bosnia and Herzegovina TST 2001 Day 1 Problem 1
gobathegreat   1
N Sep 20, 2018 by Tintarn
Source: Bosnia and Herzegovina Team Selection Test 2001
On circle there are points $A$, $B$ and $C$ such that they divide circle in ratio $3:5:7$. Find angles of triangle $ABC$
1 reply
gobathegreat
Sep 19, 2018
Tintarn
Sep 20, 2018
J
Bosnia and Herzegovina TST 2001 Day 1 Problem 1
G H J
Reveal topic tags
ratio
geometry
circle
angles
arc
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G H BBookmark kLocked kLocked NReply
Source: Bosnia and Herzegovina Team Selection Test 2001
The post below has been deleted. Click to close.
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gobathegreat
741 posts
gobathegreat
#1 h  Sep 19, 2018, 10:45 PM • 1 Y
Y by Adventure10
On circle there are points $A$, $B$ and $C$ such that they divide circle in ratio $3:5:7$. Find angles of triangle $ABC$
Z K Y
The post below has been deleted. Click to close.
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Tintarn
9037 posts
Tintarn
#2 h  Sep 20, 2018, 11:00 AM • 1 Y
Y by Adventure10
If a segment $XY$ of a circle centered at $O$ cuts a portion of $x$ of its circumference, then we have $\angle XOY=x \cdot 360^\circ$ and hence for any point $Z$ on the other side of the circle we have $\angle XZY=\frac{1}{2} \angle XOY=x \cdot 180^\circ$.
So the ratios of the circle divison correspond proportionally to the ratios of the angles. So the angles of $ABC$ are also in ratio $3:5:7$. So they are $\frac{3}{15} \cdot 180^\circ=36^\circ, \frac{5}{15} \cdot 180^\circ=60^\circ$ and $\frac{7}{15} \cdot 180^\circ=84^\circ$.
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