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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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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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geometry party
pnf   0
an hour ago
https://artofproblemsolving.com/community/c4260013_geometry_party
0 replies
pnf
an hour ago
0 replies
J
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All heads to tails?
smartvong   0
an hour ago
Source: CEMC Euclid Contest 2025
An equilateral triangle is formed using $n$ rows of coins. There is 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on, up to $n$ coins in the $n$th row. Initially, all of the coins show heads (H). Carley plays a game in which, on each turn, she chooses three mutually adjacent coins and flips these three coins over. To win the game, all of the coins must be showing tails (T) after a sequence of turns. An example game with 4 rows of coins after a sequence of two turns is shown.

IMAGE

Below (a), (b) and (c), you will find instructions about how to refer to these turns in your solutions.

(a) If there are 3 rows of coins, give a sequence of 4 turns that results in a win.

(b) Suppose that there are 4 rows of coins. Determine whether or not there is a sequence of turns that results in a win.

(c) Determine all values of $n$ for which it is possible to win the game starting with $n$ rows of coins.

Note: For a triangle with 4 rows of coins, there are 9 possibilities for the set of three coins that Carley can flip on a given turn. These 9 possibilities are shown as shaded triangles below:

IMAGE

IMAGE

[You should use the names for these moves shown inside the 9 shaded triangles when answering (b). You should adapt this naming convention in a suitable way when answering parts (a) and (c).]
0 replies
smartvong
an hour ago
0 replies
J
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Inequality with three variables
crazyfehmy   14
N 2 hours ago by TopGbulliedU
Source: Turkey JBMO Team Selection Test 2013, P4
For all positive real numbers $a, b, c$ satisfying $a+b+c=1$, prove that

\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]
14 replies
crazyfehmy
May 31, 2013
TopGbulliedU
2 hours ago
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Factorial: n!|a^n+1
Nima Ahmadi Pour   66
N 2 hours ago by cursed_tangent1434
Source: IMO Shortlist 2005 N4, Iran preparation exam
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property:
\[ n!\mid a^n + 1 \]

Proposed by Carlos Caicedo, Colombia
66 replies
Nima Ahmadi Pour
Apr 24, 2006
cursed_tangent1434
2 hours ago
J
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Eazy equation clap
giangtruong13   0
2 hours ago
Find all $x,y,z$ satisfy that: $$\frac{x}{y+z}=2x-1; \frac{y}{x+z}=3y-1;\frac{z}{x+y}=5x-1$$
0 replies
giangtruong13
2 hours ago
0 replies
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Problems for v_p(n)
xytunghoanh   15
N 2 hours ago by AshAuktober
Hello everyone. I need some easy problems for $v_p(n)$ (use in a problem for junior) to practice. Can anyone share to me?
Thanks :>
15 replies
xytunghoanh
5 hours ago
AshAuktober
2 hours ago
J
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BMO2 1997 Q4
rstather   1
N 2 hours ago by aidan0626
Source: BMO2 1997 Q4
Problem Statement:
The set S= {1/r : r = 1, 2, 3,...} of reciprocals of the positive integers contains arithmetic progressions of various lengths. For instance, 1/20, 1/8, 1/5 is such a progression, of length 3 (and common difference 3/40). Moreover, this is a maximal progression in S of length 3 since it cannot be extended to the left or right within S (−1/40 and 11/40 not
being members of S).
(i) Find a maximal progression in
S of length 1996.
(ii) Is there a maximal progression in
S of length 1997?


For part (i) I constructed the sequence: 1/1996!, 2/1996!, 3/1996!, ..., 1996/1996!. Which has common difference of 1/1996! and because of the nature of the factorial function, each of them are elements in S, and continuing to the left we get 0, which is not in S, and continuing to the right we get 1997/1996! which is not in S because 1997 is prime. Therefore we have found a maximal progression in S of length 1996 as desired.

For part (ii), let A = (4000!)/(2003!). Now consider the sequence: 2004/A, 2005/A, 2006/A, ... , 4000/A. Each is an element of S, by the nature of the factorial function and the common difference is 1/A. Continuing to the left gives 2003/A, 2003 is prime which is not a factor of A, so 2003/A is not an element of S. Continuing to the right gives 4001/A, 4001 is prime which is not a factor of A, so 4001/A is not an element of S. Therefore we have found a maximal progression in S of length 1997, and so the answer is Yes.


Is this proof sound?
1 reply
rstather
3 hours ago
aidan0626
2 hours ago
J
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Geo to make por people happy
AlephG_64   1
N 3 hours ago by ND_
Source: 2025 Finals Portuguese Mathematical Olympiad P4
Let $ABCD$ be a square with $2cm$ side length and with center $T$. A rhombus $ARTE$ is drawn where point $E$ lies on line $DC$. What is the area of $ARTE$?
1 reply
AlephG_64
5 hours ago
ND_
3 hours ago
J
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Modular Matching Pairs
steven_zhang123   3
N 3 hours ago by flower417477
Source: China TST 2025 P20
Let \( n \) be an odd integer, \( m = \frac{n+1}{2} \). Consider \( 2m \) integers \( a_1, a_2, \ldots, a_m, b_1, b_2, \ldots, b_m \) such that for any \( 1 \leq i < j \leq m \), \( a_i \not\equiv a_j \pmod{n} \) and \( b_i \not\equiv b_j \pmod{n} \). Prove that the number of \( k \in \{0, 1, \ldots, n-1\} \) for which satisfy \( a_i + b_j \equiv k \pmod{n} \) for some \( i \neq j \), $i, j \in \left \{ 1,2,\cdots,m \right \} $ is greater than \( n - \sqrt{n} - \frac{1}{2} \).
3 replies
steven_zhang123
Mar 29, 2025
flower417477
3 hours ago
J
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Find the function!
Johann Peter Dirichlet   13
N 3 hours ago by bin_sherlo
Source: Problem 3, Brazilian Olympic Revenge 2005
Find all functions $f: R \rightarrow R$ such that
\[f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy\]
for all $x,y \in R$
13 replies
Johann Peter Dirichlet
Jun 1, 2005
bin_sherlo
3 hours ago
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inequality
revol_ufiaw   3
N 4 hours ago by MS_asdfgzxcvb
Prove that that for any real $x \ge 0$ and natural number $n$,
$$x^n (n+1)^{n+1} \le n^n (x+1)^{n+1}.$$
3 replies
revol_ufiaw
4 hours ago
MS_asdfgzxcvb
4 hours ago
J
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inequality ( 4 var
SunnyEvan   3
N 4 hours ago by arqady
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3) $$$$ a^4+b^4+c^4+d^4+ \frac{252}{25} \geq \frac{88}{25}(a^3+b^3+c^3+d^3) $$equality cases : ?
3 replies
SunnyEvan
Yesterday at 5:19 AM
arqady
4 hours ago
J
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How to judge a number is prime or not?
mingzhehu   0
4 hours ago
A=（10X1+1）(10X+1），X1，X∈N+
B=（10 X1+3）（10X+7），X∈N，X1∈N
C=(10 X1+9)(10X+9), X∈N，X1∈N
D=(10 X1+1)(10X+3), X1∈N+，X∈N
E=（10 X1+7）(10X+9），X∈N，X1∈N
F=（10 X1+1）（10X+7），X1∈N+，X∈N
G=（10 X1+3）（10X+9），X∈N，X1∈N
H=（10 X1+1）10X+9），X1∈N+，X∈N
I=（10 X1+3）（10X+3），X1∈N，X∈N
J=( 10X1+7)(10X+7),X∈N，X1∈N

For any natural number P∈{P=10N+1，n∈N}，make P=A or B or C
If P can make the roots of function group(ABC) without any root group completely made up of integer, P will be a prime
For any natural number P∈{P=10N+3，n∈N}，make P=D or E
If P can make the roots of function group(DE) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+7，n∈N}，make P=F or G
If P can make the roots of function group(FG) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+9，n∈N}，make P=H or I or J
If P can make the roots of function group(GIJ) without any root group completely made up
of integer, P will be a prime
0 replies
mingzhehu
4 hours ago
0 replies
J
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What is an isogonal conjugate and why is it useful?
EaZ_Shadow   6
N 4 hours ago by maxamc
What is an isogonal conjugate and why is it useful? People use them in Olympiad geometry proofs but I don’t understand why and what is the purpose, as it complicates me because of me not understanding it.
6 replies
EaZ_Shadow
Dec 28, 2024
maxamc
4 hours ago
J
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J
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3 triangles inscribed one another, areas (1998 Argentina OMA L2 p3)
parmenides51   1
N Jun 8, 2020 by vanstraelen
Triangles $A_1B_1C_1$ and $A_2B_2C_2$ have parallel sides: $A_1B_1 \parallel A_2B_2, B_1C_1 \parallel B_2C_2, C_1A_1 \parallel C_2A_2$, and triangle $A_2B_2C_2$ is contained inside triangle $A_1B_1C_1$. A third triangle $ABC$ is inscribed in $A_1B_1C_1$ and circumscribed to $A_2B_2C_2$, that is, points $A, B, C$ belong inside segments $B_1C_1, C_1A_1, A_1B_1$ and points $A_2, B_2, C_2$ belong inside segments $BC, CA, AB$, respectively. If the area of $A_1B_1C_1$ is $63$ and the area of $A_2B_2C_2$ is $7$, find the area of $ABC$.
1 reply
parmenides51
Jun 6, 2020
vanstraelen
Jun 8, 2020
J
3 triangles inscribed one another, areas (1998 Argentina OMA L2 p3)
G H J
Reveal topic tags
geometry
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parmenides51
30629 posts
parmenides51
#1 h  Jun 6, 2020, 6:05 AM • 2 Y
Y by Mango247, Mango247
Triangles $A_1B_1C_1$ and $A_2B_2C_2$ have parallel sides: $A_1B_1 \parallel A_2B_2, B_1C_1 \parallel B_2C_2, C_1A_1 \parallel C_2A_2$, and triangle $A_2B_2C_2$ is contained inside triangle $A_1B_1C_1$. A third triangle $ABC$ is inscribed in $A_1B_1C_1$ and circumscribed to $A_2B_2C_2$, that is, points $A, B, C$ belong inside segments $B_1C_1, C_1A_1, A_1B_1$ and points $A_2, B_2, C_2$ belong inside segments $BC, CA, AB$, respectively. If the area of $A_1B_1C_1$ is $63$ and the area of $A_2B_2C_2$ is $7$, find the area of $ABC$.
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vanstraelen
8949 posts
vanstraelen
#2 h  Jun 8, 2020, 4:00 PM • 2 Y
Y by Mango247, Mango247
Area $\triangle ABC\ =\ 21$.
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