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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 3:18 PM
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0 replies
jlacosta
Yesterday 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
1 viewing
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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Balkan Mathematical Olympiad 2018 P1
microsoft_office_word   46
N 28 minutes ago by EmersonSoriano
Source: BMO 2018
A quadrilateral $ABCD$ is inscribed in a circle $k$ where $AB$ $>$ $CD$,and $AB$ is not paralel to $CD$.Point $M$ is the intersection of diagonals $AC$ and $BD$, and the perpendicular from $M$ to $AB$ intersects the segment $AB$ at a point $E$.If $EM$ bisects the angle $CED$ prove that $AB$ is diameter of $k$.
Proposed by Emil Stoyanov,Bulgaria
46 replies
+1 w
microsoft_office_word
May 9, 2018
EmersonSoriano
28 minutes ago
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f this \8char
v4913   29
N 40 minutes ago by iliya8788
Source: EGMO 2022/2
Let $\mathbb{N}=\{1, 2, 3, \dots\}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold:
(1) $f(ab) = f(a)f(b)$, and
(2) at least two of the numbers $f(a)$, $f(b)$, and $f(a+b)$ are equal.
29 replies
v4913
Apr 9, 2022
iliya8788
40 minutes ago
J
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Chords and tangent circles
math154   27
N an hour ago by Learning11
Source: ELMO Shortlist 2012, G4
Circles $\Omega$ and $\omega$ are internally tangent at point $C$. Chord $AB$ of $\Omega$ is tangent to $\omega$ at $E$, where $E$ is the midpoint of $AB$. Another circle, $\omega_1$ is tangent to $\Omega, \omega,$ and $AB$ at $D,Z,$ and $F$ respectively. Rays $CD$ and $AB$ meet at $P$. If $M$ is the midpoint of major arc $AB$, show that $\tan \angle ZEP = \tfrac{PE}{CM}$.

Ray Li.
27 replies
math154
Jul 2, 2012
Learning11
an hour ago
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D1019 : Dominoes 2*1
Dattier   5
N an hour ago by polishedhardwoodtable
I have a 9*9 grid like this one:

IMAGE

We choose 5 white squares on the lower triangle, 5 black squares on the upper triangle and one on the diagonal, which we remove from the grid.
Like for example here:

IMAGE

Can we completely cover the grid remove from these 11 squares with 2*1 dominoes like this one:

IMAGE
5 replies
Dattier
Mar 26, 2025
polishedhardwoodtable
an hour ago
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Goes through fixed points
CheshireOrb   5
N 2 hours ago by HoRI_DA_GRe8
Source: Vietnam TST 2021 P5
Given a fixed circle $(O)$ and two fixed points $B, C$ on that circle, let $A$ be a moving point on $(O)$ such that $\triangle ABC$ is acute and scalene. Let $I$ be the midpoint of $BC$ and let $AD, BE, CF$ be the three heights of $\triangle ABC$. In two rays $\overrightarrow{FA}, \overrightarrow{EA}$, we pick respectively $M,N$ such that $FM = CE, EN = BF$. Let $L$ be the intersection of $MN$ and $EF$, and let $G \neq L$ be the second intersection of $(LEN)$ and $(LFM)$.

a) Show that the circle $(MNG)$ always goes through a fixed point.

b) Let $AD$ intersects $(O)$ at $K \neq A$. In the tangent line through $D$ of $(DKI)$, we pick $P,Q$ such that $GP \parallel AB, GQ \parallel AC$. Let $T$ be the center of $(GPQ)$. Show that $GT$ always goes through a fixed point.
5 replies
CheshireOrb
Apr 2, 2021
HoRI_DA_GRe8
2 hours ago
J
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Unsolved NT, 3rd time posting
GreekIdiot   8
N 3 hours ago by ektorasmiliotis
Source: own
Solve $5^x-2^y=z^3$ where $x,y,z \in \mathbb Z$
Hint
8 replies
GreekIdiot
Mar 26, 2025
ektorasmiliotis
3 hours ago
J
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n=y^2+108
Havu   6
N 3 hours ago by ektorasmiliotis
Given the positive integer $n = y^2 + 108$ where $y \in \mathbb{N}$.
Prove that $n$ cannot be a perfect cube of a positive integer.
6 replies
Havu
Today at 4:30 PM
ektorasmiliotis
3 hours ago
J
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Valuable subsets of segments in [1;n]
NO_SQUARES   0
3 hours ago
Source: Russian May TST to IMO 2023; group of candidates P6; group of non-candidates P8
The integer $n \geqslant 2$ is given. Let $A$ be set of all $n(n-1)/2$ segments of real line of type $[i, j]$, where $i$ and $j$ are integers, $1\leqslant i<j\leqslant n$. A subset $B \subset A$ is said to be valuable if the intersection of any two segments from $B$ is either empty, or is a segment of nonzero length belonging to $B$. Find the number of valuable subsets of set $A$.
0 replies
NO_SQUARES
3 hours ago
0 replies
J
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Fneqn or Realpoly?
Mathandski   2
N 4 hours ago by jasperE3
Source: India, not sure which year. Found in OTIS pset
Find all polynomials $P$ with real coefficients obeying
\[P(x) P(x+1) = P(x^2 + x + 1)\]for all real numbers $x$.
2 replies
Mathandski
6 hours ago
jasperE3
4 hours ago
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thanks u!
Ruji2018252   2
N 4 hours ago by CHESSR1DER
find all $f: \mathbb{R}\to \mathbb{R}$ and
\[(x-y)[f(x)+f(y)]\leqslant f(x^2-y^2), \forall x,y \in \mathbb{R}\]
2 replies
Ruji2018252
6 hours ago
CHESSR1DER
4 hours ago
J
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Functional equations
hanzo.ei   11
N 6 hours ago by GreekIdiot
Source: Greekldiot
Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$
11 replies
hanzo.ei
Mar 29, 2025
GreekIdiot
6 hours ago
J
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D1018 : Can you do that ?
Dattier   1
N 6 hours ago by Dattier
Source: les dattes à Dattier
We can find $A,B,C$, such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$.

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$?
1 reply
Dattier
Mar 24, 2025
Dattier
6 hours ago
J
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D1010 : How it is possible ?
Dattier   14
N Today at 5:30 PM by ehuseyinyigit
Source: les dattes à Dattier
Is it true that$$\forall n \in \mathbb N^*, (24^n \times B \mod A) \mod 2 = 0 $$?

A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975

B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902
14 replies
Dattier
Mar 10, 2025
ehuseyinyigit
Today at 5:30 PM
J
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iran tst 2018 geometry
Etemadi   10
N Today at 5:05 PM by amirhsz
Source: Iranian TST 2018, second exam day 2, problem 5
Let $\omega$ be the circumcircle of isosceles triangle $ABC$ ($AB=AC$). Points $P$ and $Q$ lie on $\omega$ and $BC$ respectively such that $AP=AQ$ .$AP$ and $BC$ intersect at $R$. Prove that the tangents from $B$ and $C$ to the incircle of $\triangle AQR$ (different from $BC$) are concurrent on $\omega$.

Proposed by Ali Zamani, Hooman Fattahi
10 replies
Etemadi
Apr 17, 2018
amirhsz
Today at 5:05 PM
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J
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Inspired by old results
sqing   3
N Oct 17, 2024 by sqing
Let $a,b,c>0 $ .Prove that
$$\frac{k^2}{a(1+b)}+\frac{1}{b(1+c)}+\frac{k^2}{c(1+a)} \geq \frac{4k-1}{1+abc}$$Where $2\leq k \in N^+.$
$$\frac{4}{a(1+b)}+\frac{1}{b(1+c)}+\frac{4}{c(1+a)} \geq \frac{7}{1+abc}$$$$\frac{1}{a(1+b)}+\frac{2}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{4\sqrt 2-2}{1+abc}$$
3 replies
sqing
Oct 16, 2024
sqing
Oct 17, 2024
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Inspired by old results
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Reveal topic tags
algebra
inequalities proposed
inequalities
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sqing
41401 posts
sqing
#1 h  Oct 16, 2024, 2:42 PM
Y by
Let $a,b,c>0 $ .Prove that
$$\frac{k^2}{a(1+b)}+\frac{1}{b(1+c)}+\frac{k^2}{c(1+a)} \geq \frac{4k-1}{1+abc}$$Where $2\leq k \in N^+.$
$$\frac{4}{a(1+b)}+\frac{1}{b(1+c)}+\frac{4}{c(1+a)} \geq \frac{7}{1+abc}$$$$\frac{1}{a(1+b)}+\frac{2}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{4\sqrt 2-2}{1+abc}$$
This post has been edited 1 time. Last edited by sqing, Oct 16, 2024, 2:58 PM
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Entrepreneur
1167 posts
Entrepreneur
#3 h  Oct 16, 2024, 3:05 PM
Y by
Bump......
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anhduy98
1080 posts
anhduy98
#4 h  Oct 17, 2024, 8:17 AM
Y by
sqing wrote:
Let $a,b,c>0 $ .Prove that
$$\frac{k^2}{a(1+b)}+\frac{1}{b(1+c)}+\frac{k^2}{c(1+a)} \geq \frac{4k-1}{1+abc}$$Where $2\leq k \in N^+.$

$$\bullet LHS=\frac{1}{1+abc}\left(k^2\left (\frac{a(1+b)}{1+a}+\frac{1+a}{a(1+b)}\right)+\left(\frac{k^2b(1+c)}{1+b}+\frac{1+b}{b(1+c)}\right)+\left(\frac{k^2(1+c)}{c(1+a)}+\frac{c(1+a)}{1+c}\right)-2k^2-1 \right)$$
.
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sqing
41401 posts
sqing
#5 h  Oct 17, 2024, 9:17 AM
Y by
Very nice.Thank anhduy98.
k=1: Balkan MO 2006, Problem 1; Crux 1998, proposed by Mohammed Aassila; Kömal; ...
This post has been edited 2 times. Last edited by sqing, Oct 17, 2024, 9:54 AM
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