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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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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
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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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a_k+a_n/1+a_ka_n is constant when k+n is constant
gghx   2
N 15 minutes ago by lightsynth123
Source: SMO senior 2024 Q5
Let $a_1,a_2,\dots$ be a sequence of positive numbers satisfying, for any positive integers $k,l,m,n$ such that $k+n=m+l$, $$\frac{a_k+a_n}{1+a_ka_n}=\frac{a_m+a_l}{1+a_ma_l}.$$Show that there exist positive numbers $b,c$ so that $b\le a_n\le c$ for any positive integer $n$.
2 replies
gghx
Aug 3, 2024
lightsynth123
15 minutes ago
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Euler line of incircle touching points /Reposted/
Eagle116   5
N 20 minutes ago by Tsikaloudakis
Let $ABC$ be a triangle with incentre $I$ and circumcentre $O$. Let $D,E,F$ be the touchpoints of the incircle with $BC$, $CA$, $AB$ respectively. Prove that $OI$ is the Euler line of $\vartriangle DEF$.
5 replies
Eagle116
Yesterday at 2:48 PM
Tsikaloudakis
20 minutes ago
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Bunch of midpoints
Retemoeg   0
24 minutes ago
Source: Own
Let $ABC$ be a scalene triangle with orthocenter $H$ and medial triangle $MNP$. Let $F$ be a point on $AC$ such that $\angle HMF = 90^{\circ}$. If $L$ is the midpoint of segment $BF$, show that $\triangle NLP$ is isoceles.
0 replies
Retemoeg
24 minutes ago
0 replies
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Floor sequence
va2010   86
N 38 minutes ago by math-olympiad-clown
Source: 2015 ISL N1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \]contains at least one integer term.
86 replies
va2010
Jul 7, 2016
math-olympiad-clown
38 minutes ago
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Inequality from my inequality training.
Orkhan-Ashraf_2002   3
N 41 minutes ago by SunnyEvan
Let $a,b,c$ non-negative real numbers,but $ab+bc+ca\not=$0.Prove that
\[1\leq \frac{a+b}{a+4b+c}+\frac{b+c}{b+4c+a}+\frac{c+a}{c+4a+b}\leq \frac{4}{3}\]
3 replies
Orkhan-Ashraf_2002
Aug 21, 2016
SunnyEvan
41 minutes ago
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SL 2015 G1: Prove that IJ=AH
Problem_Penetrator   135
N an hour ago by math-olympiad-clown
Source: IMO 2015 Shortlist, G1
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.
135 replies
Problem_Penetrator
Jul 7, 2016
math-olympiad-clown
an hour ago
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Easy diophantine
gghx   3
N an hour ago by lightsynth123
Source: SMO senior 2024 Q2 / SMO junior 2024 Q5
Find all integer solutions of the equation $$y^2+2y=x^4+20x^3+104x^2+40x+2003.$$
Note: has appeared many times before, see here
3 replies
gghx
Aug 3, 2024
lightsynth123
an hour ago
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Easy IMO 2023 NT
799786   132
N an hour ago by anudeep
Source: IMO 2023 P1
Determine all composite integers $n>1$ that satisfy the following property: if $d_1$, $d_2$, $\ldots$, $d_k$ are all the positive divisors of $n$ with $1 = d_1 < d_2 < \cdots < d_k = n$, then $d_i$ divides $d_{i+1} + d_{i+2}$ for every $1 \leq i \leq k - 2$.
132 replies
799786
Jul 8, 2023
anudeep
an hour ago
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Equal Distances in an Isosceles Setting
mojyla222   1
N an hour ago by Retemoeg
Source: IDMC 2025 P4
Let $ABC$ be an isosceles triangle with $AB=AC$. The circle $\omega_1$, passing through $B$ and $C$, intersects segment $AB$ at $K\neq B$. The circle $\omega_2$ is tangent to $BC$ at $B$ and passes through $K$. Let $M$ and $N$ be the midpoints of segments $AB$ and $AC$, respectively. The line $MN$ intersects $\omega_1$ and $\omega_2$ at points $P$ and $Q$, respectively, where $P$ and $Q$ are the intersections closer to $M$. Prove that $MP=MQ$.

Proposed by Hooman Fattahi
1 reply
1 viewing
mojyla222
3 hours ago
Retemoeg
an hour ago
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Circles tangent to AD and AB intersect on AC
gghx   4
N 2 hours ago by lightsynth123
Source: SMO senior 2024 Q1
In an acute triangle $ABC$, $AC>AB$, $D$ is the point on $BC$ such that $AD=AB$. Let $\omega_1$ be the circle through $C$ tangent to $AD$ at $D$, and $\omega_2$ the circle through $C$ tangent to $AB$ at $B$. Let $F(\ne C)$ be the second intersection of $\omega_1$ and $\omega_2$. Prove that $F$ lies on $AC$.
4 replies
gghx
Aug 3, 2024
lightsynth123
2 hours ago
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IMO 2023 P2
799786   90
N 2 hours ago by wu2481632
Source: IMO 2023 P2
Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$.
90 replies
799786
Jul 8, 2023
wu2481632
2 hours ago
J
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New Year Calls Between Two Families
mojyla222   1
N 2 hours ago by YaoAOPS
Source: IDMC 2025 P1
Let $m,n$ be natural numbers. As the New Year arrives, it is a cherished tradition among Iranian families to call their relatives and exchange Nowruz greetings. Just moments after the New Year begins, a family with $n$ members calls a related family with $m$ members to celebrate the occasion. Since the two families are close, every member of one family wishes to speak with every member of the other family to exchange greetings. Thus, during this single call, all $mn$ distinct one-on-one conversations must take place exactly once.

Each family has only one telephone in their home, meaning the phone must be passed between family members as needed. For example, if the mothers of both families start the conversation, and then the mother from one family wants to speak with a child from the other family, only one phone needs to be passed. However, if after the mothers' conversation, the fathers from both families wish to speak with each other, then both phones must be handed over — once in each household — resulting in two phone passes.

What are the minimum and maximum possible numbers of times the telephones are passed in total (across both families) during the entire process?

Proposed by Soroush Behroozifar
1 reply
mojyla222
3 hours ago
YaoAOPS
2 hours ago
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Paint and Optimize: A Grid Strategy Problem
mojyla222   1
N 2 hours ago by YaoAOPS
Source: Iran 2025 second round p2
Ali and Shayan are playing a turn-based game on an infinite grid. Initially, all cells are white. Ali starts the game, and in the first turn, he colors one unit square black. In the following turns, each player must color a white square that shares at least one side with a black square. The game continues for exactly 2808 turns, after which each player has made 1404 moves. Let $A$ be the set of black cells at the end of the game. Ali and Shayan respectively aim to minimize and maximise the perimeter of the shape $A$ by playing optimally. (The perimeter of shape $A$ is defined as the total length of the boundary segments between a black and a white cell.)

What are the possible values of the perimeter of $A$, assuming both players play optimally?
1 reply
mojyla222
3 hours ago
YaoAOPS
2 hours ago
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Problem3
samithayohan   114
N 2 hours ago by Retemoeg
Source: IMO 2015 problem 3
Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma $ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^{\circ}$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order.

Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

Proposed by Ukraine
114 replies
samithayohan
Jul 10, 2015
Retemoeg
2 hours ago
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2024 8's
Marius_Avion_De_Vanatoare   3
N Apr 3, 2025 by EVKV
Source: Moldova JTST 2024 P2
Prove that the number $ \underbrace{88\dots8}_\text{2024\; \textrm{times}}$ is divisible by 2024.
3 replies
Marius_Avion_De_Vanatoare
Jun 10, 2024
EVKV
Apr 3, 2025
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2024 8's
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Reveal topic tags
number theory
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Source: Moldova JTST 2024 P2
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Marius_Avion_De_Vanatoare
55 posts
Marius_Avion_De_Vanatoare
#1 h  Jun 10, 2024, 2:28 PM
Y by
Prove that the number $ \underbrace{88\dots8}_\text{2024\; \textrm{times}}$ is divisible by 2024.
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MathLuis
1498 posts
MathLuis
#2 h  Jun 10, 2024, 5:23 PM • 1 Y
Y by megarnie
We basically want to prove $2024 \mid 8 \cdot \frac{10^{2024}-1}{9}$, or just $2277 \mid 10^{2024}-1$.
Now since $2277=3^2 \cdot 11 \cdot 23$ we have $\phi(2277)=\phi(3^2) \cdot \phi(11) \cdot \phi(23)=6 \cdot 10 \cdot 22=1320$, now since $\gcd(2024, 1320)=88$ we must prove $2277 \mid 10^{88}-1$. Now from here we will split the division in prime powers which makes sense due to crt.
$3^2 \mid 10^{22k}-1, 11 \mid 10^{22k}-1, 23 \mid 10^{22k}-1$ all hold trivially, the last one from fermat's theorem, therefore we have $2277 \mid 10^{22k}-1$ for all positive integers $k$ and ofc when $k=4$ we indeed get $2277 \mid 10^{88}-1 \mid 10^{2024}-1$ which implies the original problem thus we are done :cool:.
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aidan0626
1852 posts
aidan0626
#3 h  Jun 10, 2024, 5:29 PM
Y by
We have $2024=8\cdot11\cdot23$. Clearly, the number is divisible by 8, and we can see that it is also divisible by 11 from the divisibility rule. Now, all we have to show is that it is divisible by 23 now, or that $23\mid \frac{8}{9}(10^{2024}-1)$. Since both 8 and 9 are relatively prime to 23, this is the same as showing that $23\mid 10^{2024}-1.$ This is equivalent to $10^{2024}\equiv1\pmod{23}$, which is true by Fermat's.
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EVKV
49 posts
EVKV
#4 h  Apr 3, 2025, 2:04 AM
Y by
The number is equivalent to $8(\frac{10^{2024}-1}{9})$
$2024=11$x$23$x$2^{2}$
clearly 4 divides it
$10\equiv 1$ $mod$ $11$
so 11 divides it too
$8(\frac{10^{2024}-1}{9})$ $\equiv$ $8(\frac{(10^{22})^{92}-1}{9})$ $\equiv$ $8(\frac{(1)^{92}-1}{9})$ $\equiv$ $1$ $mod$ $23$

as $gcd(9,2024) = 1$ it has a valid inverse allowing this
This post has been edited 1 time. Last edited by EVKV, Apr 3, 2025, 2:05 AM
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