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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
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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2024 Putnam A1
KevinYang2.71   21
N 37 minutes ago by KAME06
Determine all positive integers $n$ for which there exists positive integers $a$, $b$, and $c$ satisfying
\[ 2a^n+3b^n=4c^n. \]
21 replies
KevinYang2.71
Dec 10, 2024
KAME06
37 minutes ago
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weird conditions in geo
Davdav1232   1
N an hour ago by NO_SQUARES
Source: Israel TST 7 2025 p1
Let \( \triangle ABC \) be an isosceles triangle with \( AB = AC \). Let \( D \) be a point on \( AC \). Let \( L \) be a point inside the triangle such that \( \angle CLD = 90^\circ \) and
\[ CL \cdot BD = BL \cdot CD. \]Prove that the circumcenter of triangle \( \triangle BDL \) lies on line \( AB \).
1 reply
Davdav1232
2 hours ago
NO_SQUARES
an hour ago
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Functional equation on R
rope0811   15
N an hour ago by ezpotd
Source: IMO ShortList 2003, algebra problem 2
Find all nondecreasing functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that
(i) $f(0) = 0, f(1) = 1;$
(ii) $f(a) + f(b) = f(a)f(b) + f(a + b - ab)$ for all real numbers $a, b$ such that $a < 1 < b$.

Proposed by A. Di Pisquale & D. Matthews, Australia
15 replies
rope0811
Sep 30, 2004
ezpotd
an hour ago
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all functions satisfying f(x+yf(x))+y = xy + f(x+y)
falantrng   34
N 2 hours ago by LenaEnjoyer
Source: Balkan MO 2025 P3
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[f(x+yf(x))+y = xy + f(x+y).\]
Proposed by Giannis Galamatis, Greece
34 replies
falantrng
Apr 27, 2025
LenaEnjoyer
2 hours ago
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Miklos Schweitzer 1971_7
ehsan2004   1
N 2 hours ago by pi_quadrat_sechstel
Let $ n \geq 2$ be an integer, let $ S$ be a set of $ n$ elements, and let $ A_i , \; 1\leq i \leq m$, be distinct subsets of $ S$ of size at least $ 2$ such that \[ A_i \cap A_j \not= \emptyset, A_i \cap A_k \not= \emptyset, A_j \cap A_k \not= \emptyset, \;\textrm{imply}\ \;A_i \cap A_j \cap A_k \not= \emptyset \ .\] Show that $ m \leq 2^{n-1}-1$.

P. Erdos
1 reply
ehsan2004
Oct 29, 2008
pi_quadrat_sechstel
2 hours ago
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Functional equation with a twist (it's number theory)
Davdav1232   0
2 hours ago
Source: Israel TST 8 2025 p2
Prove that for all primes \( p \) such that \( p \equiv 3 \pmod{4} \) or \( p \equiv 5 \pmod{8} \), there exist integers
\[ 1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p \]such that
\[ \prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}. \]
0 replies
Davdav1232
2 hours ago
0 replies
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Grid combi with T-tetrominos
Davdav1232   0
2 hours ago
Source: Israel TST 8 2025 p1
Let \( f(N) \) denote the maximum number of \( T \)-tetrominoes that can be placed on an \( N \times N \) board such that each \( T \)-tetromino covers at least one cell that is not covered by any other \( T \)-tetromino.

Find the smallest real number \( c \) such that
\[ f(N) \leq cN^2 \]for all positive integers \( N \).
0 replies
Davdav1232
2 hours ago
0 replies
J
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forced vertices in graphs
Davdav1232   0
2 hours ago
Source: Israel TST 7 2025 p2
Let \( G \) be a graph colored using \( k \) colors. We say that a vertex is forced if it has neighbors in all the other \( k - 1 \) colors.

Prove that for any \( 2024 \)-regular graph \( G \), there exists a coloring using \( 2025 \) colors such that at least \( 1013 \) of the colors have a forced vertex of that color.

Note: The graph coloring must be valid, this means no \( 2 \) vertices of the same color may be adjacent.
0 replies
Davdav1232
2 hours ago
0 replies
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Can this sequence be bounded?
darij grinberg   70
N 2 hours ago by ezpotd
Source: German pre-TST 2005, problem 4, ISL 2004, algebra problem 2
Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals.

Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded?

Proposed by Mihai Bălună, Romania
70 replies
darij grinberg
Jan 19, 2005
ezpotd
2 hours ago
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find angle
TBazar   4
N 2 hours ago by vanstraelen
Given $ABC$ triangle with $AC>BC$. We take $M$, $N$ point on AC, AB respectively such that $AM=BC$, $CM=BN$. $BM$, $AN$ lines intersect at point $K$. If $2\angle AKM=\angle ACB$, find $\angle ACB$
4 replies
TBazar
Today at 6:57 AM
vanstraelen
2 hours ago
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Polys with int coefficients
adihaya   4
N 3 hours ago by sangsidhya
Source: 2012 INMO (India National Olympiad), Problem #3
Define a sequence $<f_0 (x), f_1 (x), f_2 (x), \dots>$ of functions by $$f_0 (x) = 1$$$$f_1(x)=x$$$$(f_n(x))^2 - 1 = f_{n+1}(x) f_{n-1}(x)$$for $n \ge 1$. Prove that each $f_n (x)$ is a polynomial with integer coefficients.
4 replies
adihaya
Mar 30, 2016
sangsidhya
3 hours ago
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Miklos Schweitzer 1968_9
ehsan2004   1
N 3 hours ago by pi_quadrat_sechstel
Let $ f(x)$ be a real function such that
\[ \lim_{x \rightarrow +\infty} \frac{f(x)}{e^x}=1\]
and $ |f''(x)|\leq c|f'(x)|$ for all sufficiently large $ x$. Prove that \[ \lim_{x \rightarrow +\infty} \frac{f'(x)}{e^x}=1.\]

P. Erdos
1 reply
ehsan2004
Oct 8, 2008
pi_quadrat_sechstel
3 hours ago
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Putnam 1956 B7
sqrtX   7
N 4 hours ago by bjump
Source: Putnam 1956
The polynomials $P(z)$ and $Q(z)$ with complex coefficients have the same set of numbers for their zeros but possibly different multiplicities. The same is true for the polynomials
$$P(z)+1 \;\; \text{and} \;\; Q(z)+1.$$Prove that $P(z)=Q(z).$
7 replies
sqrtX
Jul 5, 2022
bjump
4 hours ago
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Linear Space Decomposition
Suan_16   1
N 4 hours ago by loup blanc
Let $A$ be a linear transformation on linear space $V$ satisfying:$$A^l=0$$but $$A^{l-1} \neq 0$$, and $V_0$ is the eigensubspace of eigenvalue $0$. Prove that $V$ can be decomposed to $dim V_0$ $A$-cyclic subspace's direct sum.

Click to reveal hidden text
1 reply
Suan_16
Apr 18, 2025
loup blanc
4 hours ago
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invertible matrix
jokerjoestar   2
N Mar 31, 2025 by SatisfiedMagma
Let matrix $A \in M_n(R)$ such that:$A^{2025}=2025×A$. Prove that $A-E$ is invertible
2 replies
jokerjoestar
Mar 27, 2025
SatisfiedMagma
Mar 31, 2025
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invertible matrix
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jokerjoestar
150 posts
jokerjoestar
#1 h  Mar 27, 2025, 6:50 AM
Y by
Let matrix $A \in M_n(R)$ such that:$A^{2025}=2025×A$. Prove that $A-E$ is invertible
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alexheinis
10583 posts
alexheinis
#2 h  Mar 27, 2025, 9:31 AM • 3 Y
Y by jokerjoestar, SatisfiedMagma, MihaiT
The minimal polynomial divides $x^{2025}-2025x=x(x^{2024}-2025)$ hence it is coprime to $x-1$.
Or let's do it explicitly here: if $Ax=x$ then $x=A^{2025}(x)=2025 Ax=2025 x$. Hence $x=0$.
This post has been edited 1 time. Last edited by alexheinis, Mar 27, 2025, 9:46 AM
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SatisfiedMagma
458 posts
SatisfiedMagma
#3 h  Mar 31, 2025, 10:29 AM
Y by
Here's another solution found by my batchmate Amit Prakash Jena.

Solution: Let $A - I = X$, then observe that
\begin{align*} A^{2025} &= (X+I)^{2025} \\ 2025A &= \sum_{i = 0}^{2025}X^i \binom{2025}{i} \\ 2025(X+I) &= X^{2025} + \ldots + 2025X + I \\ \cancel{2025X} + 2024I &= X\underbrace{(X^{2024} + \ldots + \binom{2025}{2}X)}_{\coloneqq B} + \cancel{2025X} \end{align*}This clearly means that $X\cdot \frac{B}{2024} = I$, proving the invertibility of $X$ since everything is a square matrix. $\blacksquare$
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