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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Wednesday at 3:18 PM
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0 replies
jlacosta
Wednesday at 3:18 PM
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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1 area = 2025 points
giangtruong13   0
a minute ago
In a plane give a set $H$ that has 8097 distinct points with area of a triangle that has 3 points belong to $H$ all $ \leq 1$. Prove that there exists a triangle $G$ that has the area $\leq 1 $ contains at least 2025 points that belong to $H$( each of that 2025 points can be inside the triangle or lie on the edge of triangle $G$)X
0 replies
giangtruong13
a minute ago
0 replies
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Burak0609
Burak0609   0
8 minutes ago
So if $2 \nmid n\implies$ $2d_2+d_4+d_5=d_7$ is even it's contradiction. I mean $2 \mid n and d_2=2$.
If $3\mid n \implies d_3=3$ and $(d_6+d_7)^2=n+1,3d_6d_7=n \implies d_6^2-d_6d_7+d_7^2=1$,we can see the only solution is$d_6=d_7=1$ and it is contradiction.
If $4 \mid n d_3=4$ and $(d_6+d_7)^2-4d_6d_7=1 \implies d_7=d_6+1$. So $n=4d_6(d_6+1)$.İt means $8 \mid n$.
If $d_6=8 n=4.8.9=288$ but $3 \nmid n$.İt is contradiction.
If $d_5=8$ we have 2 option. Firstly $d_4=5 \implies 2d_2+d_4+d_5=17=d_7 d_6=16$ but $10 \mid n$ is contradiction. Secondly $d_4=7 \implies d_7=2.2+7+8=19 and d_6=18$ but $3 \nmid n$ is contradiction. I mean $d_4=8 \implies d_7=d_5+12, n=4(d_5+11)(d_5+12) and d_5 \mid n=4(d_5+11)(d_5+12)$. So $d_5 \mid 4.11.12 \implies d_5 \mid 16.11$. If $d_5=16 d_6=27$ but $3 \nmid n$ is contradiction. I mean $d_5=11,d_6=22,d_7=23$. The only solution is $n=2024$.
0 replies
Burak0609
8 minutes ago
0 replies
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Good Partitions
va2010   25
N 31 minutes ago by lelouchvigeo
Source: 2015 ISL C3
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
25 replies
va2010
Jul 7, 2016
lelouchvigeo
31 minutes ago
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An inequality on triangles sides
nAalniaOMliO   7
N an hour ago by navier3072
Source: Belarusian National Olympiad 2025
Numbers $a,b,c$ are lengths of sides of some triangle. Prove the inequality$$\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c} \geq \frac{a+b}{2c}+\frac{b+c}{2a}+\frac{c+a}{2b}$$
7 replies
nAalniaOMliO
Mar 28, 2025
navier3072
an hour ago
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D is incenter
Layaliya   3
N an hour ago by rong2020
Source: From my friend in Indonesia
Given an acute triangle \( ABC \) where \( AB > AC \). Point \( O \) is the circumcenter of triangle \( ABC \), and \( P \) is the projection of point \( A \) onto line \( BC \). The midpoints of \( BC \), \( CA \), and \( AB \) are \( D \), \( E \), and \( F \), respectively. The line \( AO \) intersects \( DE \) and \( DF \) at points \( Q \) and \( R \), respectively. Prove that \( D \) is the incenter of triangle \( PQR \).
3 replies
Layaliya
Yesterday at 11:03 AM
rong2020
an hour ago
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Constructing sequences
SMOJ   6
N 2 hours ago by lightsynth123
Source: 2018 Singapore Mathematical Olympiad Senior Q5
Starting with any $n$-tuple $R_0$, $n\ge 1$, of symbols from $A,B,C$, we define a sequence $R_0, R_1, R_2,\ldots,$ according to the following rule: If $R_j= (x_1,x_2,\ldots,x_n)$, then $R_{j+1}= (y_1,y_2,\ldots,y_n)$, where $y_i=x_i$ if $x_i=x_{i+1}$ (taking $x_{n+1}=x_1$) and $y_i$ is the symbol other than $x_i, x_{i+1}$ if $x_i\neq x_{i+1}$. Find all positive integers $n>1$ for which there exists some integer $m>0$ such that $R_m=R_0$.
6 replies
SMOJ
Mar 31, 2020
lightsynth123
2 hours ago
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Orthocenter is the midpoint of the altitude
plagueis   6
N 2 hours ago by FrancoGiosefAG
Source: Mexican Quarantine Mathematical Olympiad P4
Let $ABC$ be an acute triangle with orthocenter $H$. Let $A_1$, $B_1$ and $C_1$ be the feet of the altitudes of triangle $ABC$ opposite to vertices $A$, $B$, and $C$ respectively. Let $B_2$ and $C_2$ be the midpoints of $BB_1$ and $CC_1$, respectively. Let $O$ be the intersection of lines $BC_2$ and $CB_2$. Prove that $O$ is the circumcenter of triangle $ABC$ if and only if $H$ is the midpoint of $AA_1$.

Proposed by Dorlir Ahmeti
6 replies
plagueis
Apr 26, 2020
FrancoGiosefAG
2 hours ago
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Inspired by JK1603JK
sqing   3
N 2 hours ago by SunnyEvan
Source: Own
Let $ a,b,c\geq 0 $ and $ab+bc+ca=1.$ Prove that$$\frac{abc-2}{abc-1}\ge \frac{4(a^2b+b^2c+c^2a)}{a^3b+b^3c+c^3a+1} $$
3 replies
1 viewing
sqing
5 hours ago
SunnyEvan
2 hours ago
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polynomial
tiendat004   0
2 hours ago
Let $p$ and $q$ be two prime numbers, with $p$ being a divisor of $q-1$. Prove that there exist integers $a,b,c,d$ such that the polynomial $x^p+cx+d$ is divisible by the polynomial $x^2+ax+b$ with $c$ is a multiple of $q$ and $b\neq 0$.
0 replies
tiendat004
2 hours ago
0 replies
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IMO 2018 Problem 1
juckter   168
N 2 hours ago by Trasher_Cheeser12321
Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.

Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece
168 replies
juckter
Jul 9, 2018
Trasher_Cheeser12321
2 hours ago
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An epitome of config geo
AndreiVila   9
N 3 hours ago by ihategeo_1969
Source: The Golden Digits Contest, December 2024, P3
Let $ABC$ be a scalene acute triangle with incenter $I$ and circumcircle $\Omega$. $M$ is the midpoint of small arc $BC$ on$\Omega$ and $N$ is the projection of $I$ onto the line passing through the midpoints of $AB$ and $AC$. A circle $\omega$ with center $Q$ is internally tangent to $\Omega$ at $A$, and touches segment $BC$. If the circle with diameter $IM$ meets $\Omega$ again at $J$, prove that $JI$ bisects $\angle QJN$.

Proposed by David Anghel
9 replies
AndreiVila
Dec 22, 2024
ihategeo_1969
3 hours ago
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Beautiful problem
luutrongphuc   0
3 hours ago
Let triangle $ABC$ be circumscribed about circle $(I)$, and let $H$ be the orthocenter of $\triangle ABC$. The circle $(I)$ touches line $BC$ at $D$. The tangent to the circle $(BHC)$ at $H$ meets $BC$ at $S$. Let $J$ be the midpoint of $HI$, and let the line $DJ$ meet $(I)$ again at $X$. The tangent to $(I)$ parallel to $BC$ meets the line $AX$ at $T$. Prove that $ST$ is tangent to $(I)$.
0 replies
luutrongphuc
3 hours ago
0 replies
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functional equation
tiendat004   0
3 hours ago
Let $a,b$ be two nonzero real numbers such that $a^2\neq b^2-2b-1.$ Consider the functions $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ satisfying the condition $$g(f(x+by))=a[f(x)+2yg(x)]+2xg(y)+bg(y)[y+g(y)],\quad\forall x,y\in\mathbb{R}.$$(a) Prove that there exists a real number $c$ such that $$g(bx)=\dfrac{b}{a}g(x)+cx,\quad\forall x\in\mathbb{R}.$$(b) Find all functions $f$ and $g$ that satisfy the given condition.
0 replies
tiendat004
3 hours ago
0 replies
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inequality ( 4 var
SunnyEvan   0
3 hours ago
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 : ?
0 replies
SunnyEvan
3 hours ago
0 replies
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Polynomials
Pao_de_sal   2
N Apr 1, 2025 by ektorasmiliotis
find all natural numbers n such that the polynomial x²ⁿ + xⁿ + 1 is divisible by x² + x + 1
2 replies
Pao_de_sal
Apr 1, 2025
ektorasmiliotis
Apr 1, 2025
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Polynomials
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Reveal topic tags
Polynomials
algebra
polynomial
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Pao_de_sal
1 post
Pao_de_sal
#1 h  Apr 1, 2025, 8:35 PM • 1 Y
Y by RainbowJessa
find all natural numbers n such that the polynomial x²ⁿ + xⁿ + 1 is divisible by x² + x + 1
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BS2012
970 posts
BS2012
#2 h  Apr 1, 2025, 8:41 PM • 1 Y
Y by RainbowJessa
Let $\omega$ be a primitive third root of unity that satisfies $\omega\neq 1$ and $\omega^3=1.$ The roots of $x^2+x+1$ are $\omega$ and $\omega^2.$ Then, the statement is only true if
$$\omega^{2n}+\omega^n+\omega^0=\omega^{4n}+\omega^{2n}+\omega^0=0.$$(using that $1=\omega^0$ here)
Also, note that $\omega^{4n}=\left(\omega^3\right)^n\omega^n=\omega^n,$ so both sums are equal.
Additionally, note that if $n$ is not a multiple of $3,$ $0,$ $n,$ and $2n$ cover all residues mod $3,$ however if $n$ is a multiple of $3$ then $\omega^{2n}+\omega^n+\omega^0=1.$ Thus, the answer is all $n$ that are not multiple of $3.$
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ektorasmiliotis
101 posts
ektorasmiliotis
#3 h  Apr 1, 2025, 8:52 PM • 1 Y
Y by RainbowJessa
Pao_de_sal wrote:
find all natural numbers n such that the polynomial x²ⁿ + xⁿ + 1 is divisible by x² + x + 1

GERMAN MO - 1984
if 4^n+2^n+1 is a prime, prove that n=3^k
This post has been edited 1 time. Last edited by ektorasmiliotis, Apr 1, 2025, 8:52 PM
Reason: typo
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