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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!

Summer camps are starting this month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have a transformative summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
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0 replies
jlacosta
Jun 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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A function on a 2D grid
Rijul saini   1
N 7 minutes ago by guptaamitu1
Source: India IMOTC 2025 Day 4 Problem 2
Does there exist a function $f:\{1,2,...,2025\}^2 \rightarrow \{1,2,...,2025\}$ such that:

$\bullet$ for any positive integer $i \leqslant 2025$, the numbers $f(i,1),f(i,2),...,f(i,2025)$ are all distinct, and
$\bullet$ for any positive integers $i \leqslant 2025$ and $j \leqslant 2024$, $f(f(i,j),f(i,j+1))=i$?

Proposed by Shantanu Nene
1 reply
Rijul saini
Yesterday at 6:46 PM
guptaamitu1
7 minutes ago
J
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Cute geometry
Rijul saini   5
N 9 minutes ago by everythingpi3141592
Source: India IMOTC Practice Test 1 Problem 3
Let scalene $\triangle ABC$ have altitudes $BE, CF,$ circumcenter $O$ and orthocenter $H$. Let $R$ be a point on line $AO$. The points $P,Q$ are on lines $AB,AC$ respectively such that $RE \perp EP$ and $RF \perp FQ$. Prove that $PQ$ is perpendicular to $RH$.

Proposed by Rijul Saini
5 replies
+1 w
Rijul saini
Yesterday at 6:51 PM
everythingpi3141592
9 minutes ago
J
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Orthocenters equidistant from circumcenter
Rijul saini   6
N 13 minutes ago by guptaamitu1
Source: India IMOTC 2025 Day 1 Problem 2
In triangle $ABC$, consider points $A_1,A_2$ on line $BC$ such that $A_1,B,C,A_2$ are in that order and $A_1B=AC$ and $CA_2=AB$. Similarly consider points $B_1,B_2$ on line $AC$, and $C_1,C_2$ on line $AB$. Prove that orthocenters of triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equidistant from the circumcenter of $ABC$.

Proposed by Shantanu Nene
6 replies
1 viewing
Rijul saini
Yesterday at 6:31 PM
guptaamitu1
13 minutes ago
J
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My Unsolved Problem
ZeltaQN2008   3
N 34 minutes ago by Funcshun840
Source: IDK
Let \( ABC \) be an acute triangle inscribed in its circumcircle \( (O) \), and let \( (I) \) be its incircle. Let \( K \) be the point where the $A-mixtilinear$ incircle of triangle $ABC$ touches \((O)\). Suppose line \( OI \) intersects segment \( AK \) at \( P \), and intersects line \( BC \) at \( Q \). Let the line through \( I \) perpendicular to \( BC \) intersect line \( KQ \) at \( A' \). Prove that: \[AI \parallel PA'.\]
3 replies
ZeltaQN2008
Yesterday at 1:23 PM
Funcshun840
34 minutes ago
J
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24 convex quadrilaterals
popcorn1   23
N an hour ago by ezpotd
Source: IMO Shortlist 2020 C2
In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that
[list]
[*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and
[*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color.
[/list]
23 replies
popcorn1
Jul 20, 2021
ezpotd
an hour ago
J
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Turkey TST 2015 P1
aloski1687   5
N an hour ago by Mathgloggers
Source: Turkey TST 2015
Let $l, m, n$ be positive integers and $p$ be prime. If $p^{2l-1}m(mn+1)^2 + m^2$ is a perfect square, prove that $m$ is also a perfect square.
5 replies
aloski1687
Apr 1, 2015
Mathgloggers
an hour ago
J
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Beware the degeneracies!
Rijul saini   4
N an hour ago by ND_
Source: India IMOTC 2025 Day 1 Problem 1
Let $a,b,c$ be real numbers satisfying $$\max \{a(b^2+c^2),b(c^2+a^2),c(a^2+b^2) \} \leqslant 2abc+1$$Prove that $$a(b^2+c^2)+b(c^2+a^2)+c(a^2+b^2) \leqslant 6abc+2$$and determine all cases of equality.

Proposed by Shantanu Nene
4 replies
Rijul saini
Yesterday at 6:30 PM
ND_
an hour ago
J
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2024 IMO P6
IndoMathXdZ   39
N an hour ago by monval
Source: 2024 IMO P6
Let $\mathbb{Q}$ be the set of rational numbers. A function $f: \mathbb{Q} \to \mathbb{Q}$ is called aquaesulian if the following property holds: for every $x,y \in \mathbb{Q}$,
\[ f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y). \]Show that there exists an integer $c$ such that for any aquaesulian function $f$ there are at most $c$ different rational numbers of the form $f(r) + f(-r)$ for some rational number $r$, and find the smallest possible value of $c$.
39 replies
IndoMathXdZ
Jul 17, 2024
monval
an hour ago
J
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Linetown Mayor Admits Orz
Rijul saini   1
N an hour ago by YaoAOPS
Source: LMAO 2025 Day 1 Problem 2
Having won the elections in Linetown, Turbo the Snail has become mayor, and one of the most pressing issues he needs to work on is the road network. Linetown can be represented as a configuration of $2025$ lines
in the plane, of which no two are parallel and no three are concurrent.

There is one house in Linetown for each pairwise intersection of two lines. The $2025$ lines are used as roads by the townsfolk. In the past, the roads in Linetown used to be two-way, but this often led to residents accidentally cycling back to where they started.

Turbo wants to make each of the $2025$ roads one-way such that it is impossible for any resident to start at a house, follow the roads in the correct directions, and end up back at the original house. In how many ways can Turbo achieve this?

Proposed by Archit Manas
1 reply
Rijul saini
Yesterday at 6:59 PM
YaoAOPS
an hour ago
J
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Functional equation: f(xf(y)+f(x)f(y))=xf(y)+f(xy)
Behappy0918   2
N an hour ago by Behappy0918
Find all function $f: \mathbb{R} \to \mathbb{R}$ such that for all $x, y\in\mathbb{R}$, $$f(xf(y)+f(x)f(y))=xf(y)+f(xy)$$
2 replies
Behappy0918
Tuesday at 12:24 PM
Behappy0918
an hour ago
J
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Painting Beads on Necklace
amuthup   47
N 2 hours ago by ezpotd
Source: 2021 ISL C2
Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible.

Carl Schildkraut, USA
47 replies
amuthup
Jul 12, 2022
ezpotd
2 hours ago
J
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Onto the altitude'
TheUltimate123   4
N 2 hours ago by EpicBird08
Source: Extension of nukelauncher's and my Mock AIME #15 (https://artofproblemsolving.com/community/c875089h1825979p12212193)
In triangle $ABC$, let $D$, $E$, and $F$ denote the feet of the altitudes from $A$, $B$, and $C$, respectively, and let $O$ denote the circumcenter of $\triangle ABC$. Points $X$ and $Y$ denote the projections of $E$ and $F$, respectively, onto $\overline{AD}$, and $Z=\overline{AO}\cap\overline{EF}$. There exists a point $T$ such that $\angle DTZ=90^\circ$ and $AZ=AT$. If $P=\overline{AD}\cap\overline{ZT}$ and $Q$ lies on $\overline{EF}$ such that $\overline{PQ}\parallel\overline{BC}$, prove that line $AQ$ bisects $\overline{BC}$.
4 replies
TheUltimate123
May 19, 2019
EpicBird08
2 hours ago
J
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The Bank of Oslo
mathisreaI   60
N 2 hours ago by ezpotd
Source: IMO 2022 Problem 1
The Bank of Oslo issues two types of coin: aluminum (denoted A) and bronze (denoted B). Marianne has $n$ aluminum coins and $n$ bronze coins arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer $k \leq 2n$, Gilberty repeatedly performs the following operation: he identifies the longest chain containing the $k^{th}$ coin from the left and moves all coins in that chain to the left end of the row. For example, if $n=4$ and $k=4$, the process starting from the ordering $AABBBABA$ would be $AABBBABA \to BBBAAABA \to AAABBBBA \to BBBBAAAA \to ...$

Find all pairs $(n,k)$ with $1 \leq k \leq 2n$ such that for every initial ordering, at some moment during the process, the leftmost $n$ coins will all be of the same type.
60 replies
mathisreaI
Jul 13, 2022
ezpotd
2 hours ago
J
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2-var inequality
sqing   2
N 2 hours ago by Rohit-2006
Source: Own
Let $ a,b\geq 0 $ and $\frac{1}{a^2+3} + \frac{1}{b^2+3} -ab\leq \frac{1}{2}.$ Prove that
$$ a^2+ab+b^2 \geq \frac{3(\sqrt{57}-7)}{4}$$Let $ a,b\geq 0 $ and $\frac{a}{b^2+3} + \frac{b}{a^2+3} +ab\leq \frac{1}{2}.$ Prove that
$$ a^2+ab+b^2 \leq \frac{9}{4}$$Let $ a,b\geq 0 $ and $ \frac{a}{b^3+3}+\frac{b}{a^3+3}-ab\leq \frac{1}{2}.$ Prove that
$$ a^2+ab+b^2 \geq \frac{9}{4}$$
2 replies
sqing
Yesterday at 12:55 PM
Rohit-2006
2 hours ago
J
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J
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Interesting inequalities
sqing   3
N Apr 23, 2025 by sqing
Source: Own
Let $ a,b> 0 $ and $ a+b\leq 2ab . $ Prove that
$$\frac{ 9a^2- ab +9b^2 }{ a^2(1+b^4)}\leq\frac{17 }{2}$$$$\frac{a- ab+b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$$$\frac{2a- 3ab+2b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$
3 replies
sqing
Apr 23, 2025
sqing
Apr 23, 2025
J
Interesting inequalities
G H J
Reveal topic tags
inequalities proposed
inequalities
algebra
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Source: Own
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sqing
42590 posts
sqing
#1 h  Apr 23, 2025, 9:05 AM
Y by
Let $ a,b> 0 $ and $ a+b\leq 2ab . $ Prove that
$$\frac{ 9a^2- ab +9b^2 }{ a^2(1+b^4)}\leq\frac{17 }{2}$$$$\frac{a- ab+b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$$$\frac{2a- 3ab+2b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$
This post has been edited 1 time. Last edited by sqing, Apr 23, 2025, 9:07 AM
Z K Y
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sqing
42590 posts
sqing
#2 h  Apr 23, 2025, 9:22 AM
Y by
Let $ a,b> 0 $ and $ a+b\leq 2ab . $ Prove that
$$\frac{ 9a^2- a^2b^2 +9b^2 }{ a^2(1+b^4)}\leq\frac{17 }{2}$$$$\frac{a- a^2b^2+b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$$$\frac{2a- 3a^2b^2+2b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$
Z K Y
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lbh_qys
616 posts
lbh_qys
#3 h  Apr 23, 2025, 9:24 AM
Y by
sqing wrote:
Let $ a,b> 0 $ and $ a+b\leq 2ab . $ Prove that
$$\frac{ 9a^2- ab +9b^2 }{ a^2(1+b^4)}\leq\frac{17 }{2}$$$$\frac{a- ab+b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$$$\frac{2a- 3ab+2b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$
sqing wrote:
Let $ a,b> 0 $ and $ a+b\leq 2ab . $ Prove that
$$\frac{ 9a^2- a^2b^2 +9b^2 }{ a^2(1+b^4)}\leq\frac{17 }{2}$$$$\frac{a- a^2b^2+b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$$$\frac{2a- 3a^2b^2+2b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$

Just use $1+b^4 \geq 2b^2$.
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sqing
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sqing
#4 h  Apr 23, 2025, 12:49 PM
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Thank lbh_qys.
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