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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:
[list][*]June 5th, Thursday, 7:30pm ET: Open Discussion with Ben Kornell and Andrew Sutherland, Art of Problem Solving's incoming CEO Ben Kornell and CPO Andrew Sutherland host an Ask Me Anything-style chat. Come ask your questions and get to know our incoming CEO & CPO!
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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
J
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Very easy geometry
mihaig   0
a few seconds ago
Source: Own
Let $\Delta ABC$ with no obtuse angles.
Prove
$$\frac1{\sqrt3}\cdot\left(\cot A+\cot B+\cot C\right)+\left(2-\sqrt 3\right)\sqrt[3]{\cot A\cot B\cot C}\geq\frac2{\sqrt3}.$$
0 replies
mihaig
a few seconds ago
0 replies
J
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A scalene triangle and nine point circle
ariopro1387   3
N 4 minutes ago by Giabach298
Source: Iran Team selection test 2025 - P12
In a scalene triangle $ABC$, points $Y$ and $X$ lie on $AC$ and $BC$ respectively such that $BC \perp XY$. Points $Z$ and $T$ are the reflections of $X$ and $Y$ with respect to the midpoints of sides $BC$ and $AC$, respectively. Point $P$ lies on segment $ZT$ such that the circumcenter of triangle $XZP$ coincides with the circumcenter of triangle $ABC$.
Prove that the nine-point circle of triangle $ABC$ passes through the midpoint of segment $XP$.
3 replies
ariopro1387
May 27, 2025
Giabach298
4 minutes ago
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inequality
SunnyEvan   0
11 minutes ago
Let $ a,b > 0 ,$ such that : $ a+b \geq \frac{3(a^4+b^4)}{a^2+b^2+1}\sqrt{\frac{\frac{1}{a}+\frac{1}{b}}{a+b}}.$
Prove that: $$ \frac{a^2+b^2+2}{a^6b^2+a^2b^6} \geq 2 $$
0 replies
SunnyEvan
11 minutes ago
0 replies
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inequality
SunnyEvan   6
N 31 minutes ago by SunnyEvan
Let $ x,y \geq 0 ,$ such that : $ \frac{x^2}{x^3+y}+\frac{y^2}{x+y^3} \geq 1 .$
Prove that : $$ x^2+y^2-xy \leq x+y $$$$ (x+\frac{1}{2})^2+(x+\frac{1}{2})^2 \leq \frac{5}{2} $$$$ (x+1)^2+(y+1)^2 \leq 5 $$$$ (x+2)^2+(y+2)^2 \leq 13 $$
6 replies
SunnyEvan
Yesterday at 1:51 PM
SunnyEvan
31 minutes ago
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Bushy and Jumpy and the unhappy walnut reordering
popcorn1   54
N 38 minutes ago by monval
Source: IMO 2021 P5
Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$.

Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.
54 replies
popcorn1
Jul 20, 2021
monval
38 minutes ago
J
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Write down sum or product of two numbers
Rijul saini   3
N an hour ago by math_comb01
Source: India IMOTC Practice Test 2 Problem 3
Suppose Alice's grimoire has the number $1$ written on the first page and $n$ empty pages. Suppose in each of the next $n$ seconds, Alice can flip to the next page, and write down the sum or product of two numbers (possibly the same) which are already written in her grimoire.

Let $F(n)$ be the largest possible number such that for any $k < F(n)$, Alice can write down the number $k$ on the last page of her grimoire. Prove that there exists a positive integer $N$ such that for all $n>N$, we have that \[n^{0.99n}\leqslant F(n)\leqslant n^{1.01n}.\]
Proposed by Rohan Goyal and Pranjal Srivastava
3 replies
Rijul saini
Wednesday at 6:56 PM
math_comb01
an hour ago
J
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gcd (a^n+b,b^n+a) is constant
EthanWYX2009   83
N an hour ago by Adywastaken
Source: 2024 IMO P2
Determine all pairs $(a,b)$ of positive integers for which there exist positive integers $g$ and $N$ such that
$$\gcd (a^n+b,b^n+a)=g$$holds for all integers $n\geqslant N.$ (Note that $\gcd(x, y)$ denotes the greatest common divisor of integers $x$ and $y.$)

Proposed by Valentio Iverson, Indonesia
83 replies
EthanWYX2009
Jul 16, 2024
Adywastaken
an hour ago
J
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Bugs Bunny at it again
Rijul saini   8
N an hour ago by quantam13
Source: LMAO 2025 Day 2 Problem 1
Bugs Bunny wants to choose a number $k$ such that every collection of $k$ consecutive positive integers contains an integer whose sum of digits is divisible by $2025$.

Find the smallest positive integer $k$ for which he can do this, or prove that none exist.

Proposed by Saikat Debnath and MV Adhitya
8 replies
Rijul saini
Wednesday at 7:01 PM
quantam13
an hour ago
J
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The Bank of Bath
TelMarin   101
N 2 hours ago by monval
Source: IMO 2019, problem 5
The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$, then he turns over the $k$th coin from the left; otherwise, all coins show $T$ and he stops. For example, if $n=3$ the process starting with the configuration $THT$ would be $THT \to HHT \to HTT \to TTT$, which stops after three operations.

(a) Show that, for each initial configuration, Harry stops after a finite number of operations.

(b) For each initial configuration $C$, let $L(C)$ be the number of operations before Harry stops. For example, $L(THT) = 3$ and $L(TTT) = 0$. Determine the average value of $L(C)$ over all $2^n$ possible initial configurations $C$.

Proposed by David Altizio, USA
101 replies
TelMarin
Jul 17, 2019
monval
2 hours ago
J
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Intersections and concyclic points
Lukaluce   2
N 2 hours ago by AylyGayypow009
Source: 2025 Junior Macedonian Mathematical Olympiad P2
Let $B_1$ be the foot of the altitude from the vertex $B$ in the acute-angled $\triangle ABC$. Let $D$ be the midpoint of side $AB$, and $O$ be the circumcentre of $\triangle ABC$. Line $B_1D$ meets line $CO$ at $E$. Prove that the points $B, C, B_1$, and $E$ lie on a circle.
2 replies
Lukaluce
May 18, 2025
AylyGayypow009
2 hours ago
J
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Calvin needs to cover all squares
Rijul saini   5
N 2 hours ago by quantam13
Source: India IMOTC 2025 Day 2 Problem 1
Consider a $2025\times 2025$ board where we identify the squares with pairs $(i,j)$ where $i$ and $j$ denote the row and column number of that square, respectively.

Calvin picks two positive integers $a,b<2025$ and places a pawn at the bottom left corner (i.e. on $(1,1)$) and makes the following moves. In his $k$-th move, he moves the pawn from $(i,j)$ to either $(i+a,j)$ or $(i,j+a)$ if $k$ is odd and to either $(i+b,j)$ and $(i,j+b)$ if $k$ is even. Here all the numbers are taken modulo $2025$. Find the number of pairs $(a,b)$ that Calvin could have picked such that he can make moves so that the pawn covers all the squares on the board without being on any square twice.

Proposed by Tejaswi Navilarekallu
5 replies
Rijul saini
Wednesday at 6:35 PM
quantam13
2 hours ago
J
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Inspired by SunnyEvan
sqing   0
2 hours ago
Source: Own
Let $ x,y \geq 0 , \frac{x^2}{x^3+y}+\frac{y^2}{x+y^3} \geq 1 .$ Prove that
$$ (x-\frac{1}{2})^2+(y+\frac{1}{2})^2 \leq \frac{5}{2} $$$$ (x-1)^2+(y+1)^2 \leq 5 $$$$ (x-2)^2+(y+2)^2 \leq 13$$$$ (x-\frac{1}{2})^2+(y+1)^2 \leq \frac{17}{4} $$$$ (x-1)^2+(y+2)^2 \leq 10 $$$$ (x-\frac{1}{2})^2+(y+2)^2 \leq \frac{37}{4} $$
0 replies
sqing
2 hours ago
0 replies
J
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Beware the degeneracies!
Rijul saini   8
N 2 hours ago by AR17296174
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
8 replies
Rijul saini
Wednesday at 6:30 PM
AR17296174
2 hours ago
J
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Griphook the globin plays a game
mathscrazy   19
N 2 hours ago by heheman
Source: INMO 2025/5
Greedy goblin Griphook has a regular $2000$-gon, whose every vertex has a single coin. In a move, he chooses a vertex, removes one coin each from the two adjacent vertices, and adds one coin to the chosen vertex, keeping the remaining coin for himself. He can only make such a move if both adjacent vertices have at least one coin. Griphook stops only when he cannot make any more moves. What is the maximum and minimum number of coins he could have collected?

Proposed by Pranjal Srivastava and Rohan Goyal
19 replies
mathscrazy
Jan 19, 2025
heheman
2 hours ago
J
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J
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Geometry Handout is finally done!
SimplisticFormulas   2
N Apr 24, 2025 by parmenides51
If there’s any typo or problem you think will be a nice addition, do send here!
handout, geometry
2 replies
SimplisticFormulas
Apr 24, 2025
parmenides51
Apr 24, 2025
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Geometry Handout is finally done!
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Reveal topic tags
geometry
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SimplisticFormulas
132 posts
SimplisticFormulas
#1 h  Apr 24, 2025, 4:58 PM • 5 Y
Y by GeoKing, parmenides51, Funcshun840, NicoN9, L13832
If there’s any typo or problem you think will be a nice addition, do send here!
handout, geometry
Attachments:
A Guide To Be Good At Geometry.pdf (265kb)
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AshAuktober
1016 posts
AshAuktober
#2 h  Apr 24, 2025, 5:02 PM • 1 Y
Y by GeoKing
ORZ ORZ ORZ XIOOOOOOIX
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parmenides51
30653 posts
parmenides51
#3 h  Apr 24, 2025, 9:38 PM • 2 Y
Y by Rounak_iitr, AshAuktober
posts like this remind me that a geo handout collection should be made, in order those worthy posts to be collected at one place

someday, I will create one
This post has been edited 1 time. Last edited by parmenides51, Apr 25, 2025, 12:30 AM
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