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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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0 replies
jwelsh
Jul 1, 2025
0 replies
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
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
Divsibility (combinatorics)
cmtappu96   8
N 27 minutes ago by MODBreathing_FirstForm
Find the number of $4$-digit numbers (in base $10$) having non-zero digits and which are divisible by $4$ but not by $8$.
8 replies
cmtappu96
Dec 5, 2010
MODBreathing_FirstForm
27 minutes ago
combinatorial clusters
Cats_on_a_computer   0
34 minutes ago
Source: THM, Richard Earl
Let $m,n$ be positive integers. For each $k=1,2,\dots,m$, define
\[
\Delta_k \;=\;
\Bigl\lfloor \tfrac{k\,n}{m}\Bigr\rfloor
\;-\;
\Bigl\lfloor \tfrac{(k-1)\,n}{m}\Bigr\rfloor.
\]Observe that $\Delta_k\in\{0,1\}$ and $\sum_{k=1}^m\Delta_k=n$. We view the indices $1,2,\dots,m$ cyclically, so that $\Delta_{m+1}=\Delta_1$. A *cluster* is a maximal cyclic block of consecutive indices all of whose $\Delta_k=1$. If there are $d$ such clusters, write their lengths (in cyclic order) as
\[
\ell_1,\ell_2,\dots,\ell_d.
\]Prove the following:


(i) $d = \gcd(m,n)$.
(ii) Each cluster‐length $\ell_i$ equals either $\lfloor n/d\rfloor$ or $\lceil n/d\rceil$, and exactly
\[
    n \;-\; d\,\bigl\lfloor\tfrac{n}{d}\bigr\rfloor
  \]of the $\ell_i$s are equal to $\lceil n/d\rceil$ (the remaining $d - \bigl(n - d\lfloor n/d\rfloor\bigr)$ clusters have length $\lfloor n/d\rfloor$).
0 replies
Cats_on_a_computer
34 minutes ago
0 replies
Geometry finale: radical axis bisects D-altitude
v_Enhance   52
N an hour ago by ravengsd
Source: USA TSTST 2016 Problem 6, by Danielle Wang
Let $ABC$ be a triangle with incenter $I$, and whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$, respectively. Let $K$ be the foot of the altitude from $D$ to $\overline{EF}$. Suppose that the circumcircle of $\triangle AIB$ meets the incircle at two distinct points $C_1$ and $C_2$, while the circumcircle of $\triangle AIC$ meets the incircle at two distinct points $B_1$ and $B_2$. Prove that the radical axis of the circumcircles of $\triangle BB_1B_2$ and $\triangle CC_1C_2$ passes through the midpoint $M$ of $\overline{DK}$.

Proposed by Danielle Wang
52 replies
v_Enhance
Jun 29, 2016
ravengsd
an hour ago
Know me where i m wrong
Not__Infinity   7
N an hour ago by wangyanliluke
So, the problem is


Show that there is no integer solution to this expression
(a+b)((a^2)+(b^2)) = 2001.

My solution:-

Now, factoring 2001 gives 3×23×29= 2001

Case 1: (a+b)((a^2)+(b^2)) = 3 ×667.

Solving for b gives and irrational. Let me give the proof.

Take a+b=3.....(¡) and ((a^2)+(b^2)) = 667...........(¡¡)

Rearrange (¡) for a, which is a = 3 - b.

substitute value of a in (¡¡) and you'll get an quadratic equation. Solving it will give and irrational value of b.

Doing this repeatedly, making cases from these factors above mentioned and comparing, we get that no integer value will be obtained. Hence proved.

Correct me if i am wrong.

Source is some thread which idk. But i find this interesting thing.
7 replies
Not__Infinity
Jul 12, 2025
wangyanliluke
an hour ago
Hard version of 2024 CSMC-3
EthanWYX2009   0
an hour ago
Source: 2023 December 谜之竞赛-6, also from an essay by Szemeredi
Determine the smallest positive real number \( \alpha \) such that there exists a constant \( c \) satisfying: For any prime \( p \) and any \( k \geq cp^\alpha \) distinct integers \( a_1, \cdots, a_k \) modulo \( p \), there exists a non-empty subset \( I \subseteq \{1, 2, \cdots, k\} \) such that
\[\sum_{i \in I} a_i \equiv 0 \pmod{p}.\]Proposed by Hanqing Huang, Peking University
0 replies
EthanWYX2009
an hour ago
0 replies
IOQM Problem
JetFire008   14
N an hour ago by mahyar_ais
Source: IOQM 2020
Find the sum of all positive integers $n$ for which $|2^n+5^n-65|$ is a perfect square.
14 replies
JetFire008
Yesterday at 11:06 AM
mahyar_ais
an hour ago
Nice and Hard Inequality
EthanWYX2009   0
an hour ago
Source: 2023 December 谜之竞赛-2
Determine the minimum real number \( c \) such that for any positive real numbers \( a_1, a_2, \cdots, a_{2023} \) satisfying $\sum_{i=1}^{2023} a_i = 2023$, the following inequality holds:
\[\sum_{1 \leq i < j \leq 2023} \{a_i a_j\} \leq c.\]Proposed by Tianqin Li, High School Affiliated to Renmin University of China
0 replies
EthanWYX2009
an hour ago
0 replies
beam of light passes through every cell without a barrier
Scilyse   0
an hour ago
Source: 2025 Apr 谜之竞赛-5
Fix an integer $n \geq 3$. An $n \times n$ chessboard is comprised of $n^2$ cells. A laser is placed at the midpoint of a side of a cell. This laser emits a beam of light that initially forms a $45^\circ$ angle with this side. When the beam hits a side of the table or a side of a cell containing a barrier, it reflects back according to the law of reflection. We say that the beam passes through a cell if it passes through the strict interior of the cell.

Let $M$ be a positive integer. Suppose that we can position the laser and place barriers in $M$ of the cells so that the beam passes through every cell not containing a barrier. Find the minimum possible value of $M$.
0 replies
Scilyse
an hour ago
0 replies
Inspired by the slimshadyyy.3.60's one.
arqady   7
N an hour ago by Cats_on_a_computer
Let $a\geq b\geq c\geq0$ and $a^2+b^2+c^2+abc=4$. Prove that:
$$a+b+c+\frac{1}{\sqrt2}\left(\sqrt{a}-\sqrt{c}\right)^2\geq3.$$
7 replies
arqady
Apr 1, 2025
Cats_on_a_computer
an hour ago
Concurrency property in an earlier configuration
RANDOM__USER   2
N 2 hours ago by keglesnit
Source: Own
Let \(D\) be an arbitrary point on the side \(BC\) in a triangle \(\triangle{ABC}\). Let \(E\) and \(F\) be the intersection of the lines parallel to \(AC\) and \(AB\) through \(D\) with \(AB\) and \(AC\). Let \(G\) be the intersection of \((AFE)\) with \((ABC)\). Let \(M\) be the midpoint of \(BC\) and \(X\) the intersection of \(AM\) with \((ABC)\). Let \(J\) be the intersection of \((XFG)\) with \(AC\). Prove that \(XB\), \(AD\) and \(JM\) are concurrent at \(P\).

IMAGE

Note: This is another property of a configuration I posted before where one needed to prove that \(X, D\) and \(G\) are collinear. There are surprisingly many properties in the configuration posted earlier :P
2 replies
RANDOM__USER
Yesterday at 9:03 PM
keglesnit
2 hours ago
Concyclic points
RANDOM__USER   2
N 2 hours ago by X.Luser
Source: Own
Let \(M\) be the midpoint of \(BC\) in \(\triangle{ABC}\). Let \(X\) be the intersection of \(AM\) with \((ABC)\). Let \(D\) and \(E\) be the intersection of lines through \(X\) parallel to \(AB\) and \(AC\) with \(AC\) and \(AB\). Let \(Y\) be the intersection of \((ABC)\) with \((XDE)\). Let \(P\) and \(Q\) be the intersection of \(DX\) and \(EX\) with the tangent to \((ABC)\) at \(A\). Prove that \(XYPQ\) is cyclic.

IMAGE
2 replies
RANDOM__USER
Today at 5:30 AM
X.Luser
2 hours ago
2017 N4 type Problem
zqy648   0
2 hours ago
Source: 2023 New Year 谜之竞赛-3
Let prime number $p>3$ and $t$ be a positive integer. Show that
\[\binom{p^{t+1}}{p^t}\equiv\binom{p^t}{p^{t-1}}\pmod {p^{3t+2}}.\]
0 replies
zqy648
2 hours ago
0 replies
old and easy imo inequality
Valentin Vornicu   220
N 2 hours ago by Abhi9624
Source: IMO 2000, Problem 2, IMO Shortlist 2000, A1
Let $ a, b, c$ be positive real numbers so that $ abc = 1$. Prove that
\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]
220 replies
Valentin Vornicu
Oct 24, 2005
Abhi9624
2 hours ago
G5 with upgrade?
AlephG_64   2
N 2 hours ago by bin_sherlo
Source: 2nd AGOsl G9
Let $ABC$ be a triangle with incenter $I$ and contact triangle $DEF$. Bisector of $\angle B$ meets $AC$ at $B_1$ and bisector of $\angle C$ meets $AB$ at $C_1$. Tangent at $A$ meets $EF$ at $P$. $Q$ is the projection of $D$ into $EF$. $M$ is the midpoint of $AP$. Circle $(AQI)$ meets $FE$ again at $N$. Prove that $MN \parallel C_1B_1$

Proposed by AlephG_64
2 replies
AlephG_64
Dec 22, 2024
bin_sherlo
2 hours ago
A second final attempt to make a combinatorics problem
JARP091   2
N May 29, 2025 by JARP091
Source: At the time of writing this problem I do not know the source if any
Arthur Morgan is playing a game.

He has $n$ eggs, each with a hardness value $k_1, k_2, \dots, k_n$, where $\{k_1, k_2, \dots, k_n\}$ is a permutation of the set $\{1, 2, \dots, n\}$. He is throwing the eggs from an $m$-floor building.

When the $i$-th egg is dropped from the $j$-th floor, its new hardness becomes
\[
\left\lfloor \frac{k_i}{j+1} \right\rfloor.
\]If $\left\lfloor \frac{k_i}{j+1} \right\rfloor = 0$, then the egg breaks and cannot be used again.

Arthur can drop each egg from a particular floor at most once.
For which values of $n$ and $m$ can Arthur always determine the correct ordering of the eggs according to their initial hardness values?
Note: The problem might be wrong or too easy
2 replies
JARP091
May 25, 2025
JARP091
May 29, 2025
A second final attempt to make a combinatorics problem
G H J
Source: At the time of writing this problem I do not know the source if any
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
JARP091
124 posts
#1
Y by
Arthur Morgan is playing a game.

He has $n$ eggs, each with a hardness value $k_1, k_2, \dots, k_n$, where $\{k_1, k_2, \dots, k_n\}$ is a permutation of the set $\{1, 2, \dots, n\}$. He is throwing the eggs from an $m$-floor building.

When the $i$-th egg is dropped from the $j$-th floor, its new hardness becomes
\[
\left\lfloor \frac{k_i}{j+1} \right\rfloor.
\]If $\left\lfloor \frac{k_i}{j+1} \right\rfloor = 0$, then the egg breaks and cannot be used again.

Arthur can drop each egg from a particular floor at most once.
For which values of $n$ and $m$ can Arthur always determine the correct ordering of the eggs according to their initial hardness values?
Note: The problem might be wrong or too easy
This post has been edited 1 time. Last edited by JARP091, May 25, 2025, 2:46 PM
Reason: Wrongly LaTeXted
Z K Y
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JARP091
124 posts
#2
Y by
Bump for this problem
Z K Y
The post below has been deleted. Click to close.
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JARP091
124 posts
#3
Y by
Drop something from 1, if it breaks then its 0

if it doesnt break, it is currently 1, we have one more one, and we have a number we are trying to find out

we cant drop the number we are interested in from floor 1, otherwise we lose information

hence drop another egg from floor 2, if it doesnt break, it was 3, we started with a 2, and ther last egg is 1, if it breaks, then we either dropped a 1, or we dropped a 2, and so the possible outputs are (1,0,2) (1,0,3) (1,0,1), now we cant figure out what state the last egg is in, so it is impossible.

For n $>$ 3, n = 3 is a subproblem that cannot be solved.

Hence only possible solutions are:

i) n = 1, m $\geq$ 1

ii) n = 2, m $\geq$ 2
Z K Y
N Quick Reply
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