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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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Equation of integers
jgnr   3
N an hour ago by KTYC
Source: Indonesia Mathematics Olympiad 2005 Day 1 Problem 2
For an arbitrary positive integer $ n$, define $ p(n)$ as the product of the digits of $ n$ (in decimal). Find all positive integers $ n$ such that $ 11p(n)=n^2-2005$.
3 replies
jgnr
Jun 2, 2008
KTYC
an hour ago
J
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Divisibility..
Sadigly   4
N an hour ago by Solar Plexsus
Source: Azerbaijan Junior MO 2025 P2
Find all $4$ consecutive even numbers, such that the square of their product is divisible by the sum of their squares.
4 replies
Sadigly
Yesterday at 7:37 AM
Solar Plexsus
an hour ago
J
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Surjective number theoretic functional equation
snap7822   3
N an hour ago by internationalnick123456
Source: 2025 Taiwan TST Round 3 Independent Study 2-N
Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions:
[list=i]
[*] For all $m, n \in \mathbb{N}$, if $m > n$ and $f(m) > f(n)$, then $f(m-n) = f(n)$;
[*] $f$ is surjective.
[/list]
Find the maximum possible value of $f(2025)$.

Proposed by snap7822
3 replies
snap7822
May 1, 2025
internationalnick123456
an hour ago
J
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FE with devisibility
fadhool   0
an hour ago
if when i solve an fe that is defined in the set of positive integer i found m|f(m) can i set f(m) =km such that k is not constant and of course it depends on m but after some work i find k=c st c is constant is this correct
0 replies
fadhool
an hour ago
0 replies
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Many Equal Sides
mathisreal   3
N an hour ago by QueenArwen
Source: Brazil EGMO TST 2023 #1
Let $ABC$ be a triangle with $BA=BC$ and $\angle ABC=90^{\circ}$. Let $D$ and $E$ be the midpoints of $CA$ and $BA$ respectively. The point $F$ is inside of $\triangle ABC$ such that $\triangle DEF$ is equilateral. Let $X=BF\cap AC$ and $Y=AF\cap DB$. Prove that $DX=YD$.
3 replies
mathisreal
Nov 10, 2022
QueenArwen
an hour ago
J
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LOTS of recurrence!
SatisfiedMagma   4
N an hour ago by Reacheddreams
Source: Indian Statistical Institute Entrance UGB 2023/5
There is a rectangular plot of size $1 \times n$. This has to be covered by three types of tiles - red, blue and black. The red tiles are of size $1 \times 1$, the blue tiles are of size $1 \times 1$ and the black tiles are of size $1 \times 2$. Let $t_n$ denote the number of ways this can be done. For example, clearly $t_1 = 2$ because we can have either a red or a blue tile. Also $t_2 = 5$ since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.

[list=a]
[*]Prove that $t_{2n+1} = t_n(t_{n-1} + t_{n+1})$ for all $n > 1$.

[*]Prove that $t_n = \sum_{d \ge 0} \binom{n-d}{d}2^{n-2d}$ for all $n >0$.
[/list]
Here,
\[ \binom{m}{r} = \begin{cases} \dfrac{m!}{r!(m-r)!}, &\text{ if $0 \le r \le m$,} \\ 0, &\text{ otherwise} \end{cases}\]for integers $m,r$.
4 replies
SatisfiedMagma
May 14, 2023
Reacheddreams
an hour ago
J
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combi/nt
blug   1
N an hour ago by blug
Prove that every positive integer $n$ can be written in the form
$$n=a_1+a_2+...+a_k,$$where $a_m=2^i3^j$ for some non-negative $i, j$ such that
$$a_x\nmid a_y$$for every $x, y\leq k$.
1 reply
blug
Yesterday at 3:37 PM
blug
an hour ago
J
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Inequality, inequality, inequality...
Assassino9931   9
N an hour ago by ZeroHero
Source: Al-Khwarizmi Junior International Olympiad 2025 P6
Let $a,b,c$ be real numbers such that \[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]Find the smallest possible value of $a^2 + b^2 + c^2$.

Binh Luan and Nhan Xet, Vietnam
9 replies
Assassino9931
Today at 9:38 AM
ZeroHero
an hour ago
J
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Vectors in a tilted square
mathwizard888   20
N an hour ago by HamstPan38825
Source: 2016 IMO Shortlist A3
Find all positive integers $n$ such that the following statement holds: Suppose real numbers $a_1$, $a_2$, $\dots$, $a_n$, $b_1$, $b_2$, $\dots$, $b_n$ satisfy $|a_k|+|b_k|=1$ for all $k=1,\dots,n$. Then there exists $\varepsilon_1$, $\varepsilon_2$, $\dots$, $\varepsilon_n$, each of which is either $-1$ or $1$, such that
\[ \left| \sum_{i=1}^n \varepsilon_i a_i \right| + \left| \sum_{i=1}^n \varepsilon_i b_i \right| \le 1. \]
20 replies
mathwizard888
Jul 19, 2017
HamstPan38825
an hour ago
J
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Calculate the distance AD
MTA_2024   3
N an hour ago by MTA_2024
A semi-circle is inscribed in a quadrilateral $ABCD$. The center $O$ of the semi-circle is the midpoint of segment $AD$. We have $AB=9$ and $CD=16$.
Calculate the distance $AD$.
3 replies
MTA_2024
Yesterday at 3:50 PM
MTA_2024
an hour ago
J
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Combi Geo
Adywastaken   0
2 hours ago
Source: NMTC 2024/8
A regular polygon with $100$ vertices is given. To each vertex, a natural number from the set $\{1,2,\dots,49\}$ is assigned. Prove that there are $4$ vertices $A, B, C, D$ such that if the numbers $a, b, c, d$ are assigned to them respectively, then $a+b=c+d$ and $ABCD$ is a parallelogram.
0 replies
Adywastaken
2 hours ago
0 replies
J
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2012 preRMO p17, roots of equation x^3 + 3x + 5 = 0
parmenides51   11
N 2 hours ago by Pengu14
Let $x_1,x_2,x_3$ be the roots of the equation $x^3 + 3x + 5 = 0$. What is the value of the expression
$\left( x_1+\frac{1}{x_1} \right)\left( x_2+\frac{1}{x_2} \right)\left( x_3+\frac{1}{x_3} \right)$ ?
11 replies
parmenides51
Jun 17, 2019
Pengu14
2 hours ago
J
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Interesting question from Al-Khwarezmi olympiad 2024 P3, day1
Adventure1000   3
N 3 hours ago by sqing
Find all $x, y, z \in \left (0, \frac{1}{2}\right )$ such that
$$ \begin{cases} (3 x^{2}+y^{2}) \sqrt{1-4 z^{2}} \geq z; \\ (3 y^{2}+z^{2}) \sqrt{1-4 x^{2}} \geq x; \\ (3 z^{2}+x^{2}) \sqrt{1-4 y^{2}} \geq y. \end{cases} $$Proposed by Ngo Van Trang, Vietnam
3 replies
Adventure1000
May 7, 2025
sqing
3 hours ago
J
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Malaysia MO IDM UiTM 2025
smartvong   1
N 3 hours ago by jasperE3
MO IDM UiTM 2025 (Category C)

Contest Description

Preliminary Round
Section A
1. Given that $2^a + 2^b = 2016$ such that $a, b \in \mathbb{N}$. Find the value of $a$ and $b$.

2. Find the value of $a, b$ and $c$ such that $$\frac{ab}{a + b} = 1, \frac{bc}{b + c} = 2, \frac{ca}{c + a} = 3.$$
3. If the value of $x + \dfrac{1}{x}$ is $\sqrt{3}$, then find the value of
$$x^{1000} + \frac{1}{x^{1000}}$$.

Section B
1. Let $\mathbb{Z}$ be the set of integers. Determine all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that for all integer $a, b$:
$$f(2a) + 2f(b) = f(f(a + b))$$
2. The side lengths $a, b, c$ of a triangle $\triangle ABC$ are positive integers. Let
$$T_n = (a + b + c)^{2n} - (a - b + c)^{2n} - (a + b - c)^{2n} - (a - b - c)^{2n}$$for any positive integer $n$.
If $\dfrac{T_2}{2T_1} = 2023$ and $a > b > c$, determine all possible perimeters of the triangle $\triangle ABC$.

Final Round
Section A
1. Given that the equation $x^2 + (b - 3)x - 2b^2 + 6b - 4 = 0$ has two roots, where one is twice of the other, find all possible values of $b$.

2. Let $$f(y) = \dfrac{y^2}{y^2 + 1}.$$Find the value of $$f\left(\frac{1}{2001}\right) + f\left(\frac{2}{2001}\right) + \cdots + f\left(\frac{2000}{2001}\right) + f\left(\frac{2001}{2001}\right) + f\left(\frac{2001}{2000}\right) + \cdots + f\left(\frac{2001}{2}\right) + f\left(\frac{2001}{1}\right).$$
3. Find the smallest four-digit positive integer $L$ such that $\sqrt{3\sqrt{L}}$ is an integer.

Section B
1. Given that $\tan A : \tan B : \tan C$ is $1 : 2 : 3$ in triangle $\triangle ABC$, find the ratio of the side length $AC$ to the side length $AB$.

2. Prove that $\cos{\frac{2\pi}{5}} + \cos{\frac{4\pi}{5}} = -\dfrac{1}{2}.$
1 reply
smartvong
5 hours ago
jasperE3
3 hours ago
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BMT 2018 Algebra Round Problem 7
IsabeltheCat   5
N Apr 22, 2025 by P162008
Let $$h_n := \sum_{k=0}^n \binom{n}{k} \frac{2^{k+1}}{(k+1)}.$$Find $$\sum_{n=0}^\infty \frac{h_n}{n!}.$$
5 replies
IsabeltheCat
Dec 3, 2018
P162008
Apr 22, 2025
J
BMT 2018 Algebra Round Problem 7
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Reveal topic tags
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IsabeltheCat
4242 posts
IsabeltheCat
#1 h  Dec 3, 2018, 3:36 AM • 2 Y
Y by Adventure10, Mango247
Let $$h_n := \sum_{k=0}^n \binom{n}{k} \frac{2^{k+1}}{(k+1)}.$$Find $$\sum_{n=0}^\infty \frac{h_n}{n!}.$$
This post has been edited 1 time. Last edited by IsabeltheCat, Dec 3, 2018, 11:11 PM
Reason: wrong dummy variable was used previously
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rzlng
99 posts
rzlng
#2 h  Dec 3, 2018, 4:51 AM • 3 Y
Y by Srofller, Adventure10, Mango247
sketch
@below, thanks. fixed
This post has been edited 1 time. Last edited by rzlng, Dec 3, 2018, 5:11 AM
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TomCalc
1635 posts
TomCalc
#3 h  Dec 3, 2018, 5:05 AM • 1 Y
Y by Adventure10
IsabeltheCat wrote:
Find $$\sum_{k=0}^\infty \frac{h_n}{n!}.$$
The second sum has a wrong dummy variable and @above the answer is correct.
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NikoIsLife
9657 posts
NikoIsLife
#4 h  Dec 3, 2018, 10:59 AM • 3 Y
Y by Adventure10, Mango247, soryn
Solution
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IsabeltheCat
4242 posts
IsabeltheCat
#5 h  Dec 3, 2018, 11:11 PM • 2 Y
Y by Adventure10, Mango247
TomCalc wrote:
IsabeltheCat wrote:
Find $$\sum_{k=0}^\infty \frac{h_n}{n!}.$$
The second sum has a wrong dummy variable and @above the answer is correct.

Thanks for pointing that out. I fixed the error in the original post.
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P162008
187 posts
P162008
#6 h  Apr 22, 2025, 6:56 AM
Y by
Storage
This post has been edited 3 times. Last edited by P162008, Apr 28, 2025, 9:40 PM
Reason: Typo
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