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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!
[*]June 9th, Monday, 7:30pm ET, Game Jam: Operation Shuffle!, Come join us to play our second round of Operation Shuffle! If you enjoy number sense, logic, and a healthy dose of luck, this is the game for you. No specific math background is required; all are welcome.[/list]
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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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Beware the degeneracies!
Rijul saini   9
N 3 minutes ago by mudok
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
9 replies
+2 w
Rijul saini
Wednesday at 6:30 PM
mudok
3 minutes ago
J
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Very easy geometry
mihaig   2
N 3 minutes ago by User21837561
Source: Own
Let $\Delta ABC$ with no obtuse angles.
Prove
$$\frac1{\sqrt3}\cdot\left(\cot A+\cot B+\cot C\right)+\left(2-\sqrt 3\right)\sqrt[3]{\cot A\cot B\cot C}\geq\frac2{\sqrt3}.$$
2 replies
+1 w
mihaig
an hour ago
User21837561
3 minutes ago
J
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A function on a 2D grid
Rijul saini   2
N 10 minutes ago by Supercali
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
2 replies
Rijul saini
Wednesday at 6:46 PM
Supercali
10 minutes ago
J
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Weirdly stated but cool collinearity
Rijul saini   3
N 25 minutes ago by YaoAOPS
Source: LMAO Revenge 2025 Day 1 Problem 2
Let Mary choose any non-degenerate $\triangle ABC$. Let $I$ be its incenter, $I_A$ be its $A$-excenter, $N_A$ be midpoint of arc $BAC$, $M$ is the midpoint of $BC$.

Let $H \neq I$ be the intersection of the line $N_AI$ with $(BIC)$, $F$ be the intersection of the angle bisector of $\angle BAC$ with the line $BC$.

Ana now draws the points $P \neq H$ ,the intersection of line $I_AH$ with $(HIN)$ and $Q$ ,the intersection of $(HIM)$ and $(AN_AI_A)$ such that $I_AH < I_AQ$. Ana wins if the points $A, P, Q$ are collinear. Who has a winning strategy?
3 replies
Rijul saini
Wednesday at 7:09 PM
YaoAOPS
25 minutes ago
J
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ISI UGB 2025 P8
SomeonecoolLovesMaths   7
N 29 minutes ago by Ilovesumona
Source: ISI UGB 2025 P8
Let $n \geq 2$ and let $a_1 \leq a_2 \leq \cdots \leq a_n$ be positive integers such that $\sum_{i=1}^{n} a_i = \prod_{i=1}^{n} a_i$. Prove that $\sum_{i=1}^{n} a_i \leq 2n$ and determine when equality holds.
7 replies
SomeonecoolLovesMaths
May 11, 2025
Ilovesumona
29 minutes ago
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k colorings and triangles
Rijul saini   1
N 35 minutes ago by YaoAOPS
Source: LMAO Revenge 2025 Day 1 Problem 3
In the city of Timbuktu, there is an orphanage. It shelters children from the new mysterious disease that causes children to explode. There are m children in the orphanage. To try to cure this disease, a mad scientist named Myla has come up with an innovative cure. She ties every child to every other child using medicinal ropes. Every child is connected to every other child using one of $k$ different ropes. She then performs a experiment that causes $3$ children, each connected to each other with the same type of rope, to be cured. Two experiments are said to be of the same type, if each of the ropes connecting the children has the same medicine imbued in it. She then unties them and lets them go back home.

We let $f(n, k)$ be the minimum m such that Myla can perform at least $n$ experiments of the same type. Prove that:

$i.$ For every $k \in \mathbb N$ there exists a $N_k \in N$ and $a_k, b_k \in \mathbb Z$ such that for all $n > N_k$, \[f(n, k) = a_kn + b_k.\]
$ii.$ Find the value of $a_k$ for every $k \in \mathbb N$.
1 reply
Rijul saini
Wednesday at 7:11 PM
YaoAOPS
35 minutes ago
J
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A beautiful Lemoine point problem
phonghatemath   2
N 44 minutes ago by HyperDunteR
Source: my teacher
Given triangle $ABC$ inscribed in a circle with center $O$. $P$ is any point not on (O). $AP, BP, CP$ intersect $(O)$ at $A', B', C'$. Let $L, L'$ be the Lemoine points of triangle $ABC, A'B'C'$ respectively. Prove that $P, L, L'$ are collinear.
2 replies
phonghatemath
3 hours ago
HyperDunteR
44 minutes ago
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Iran TST Starter
M11100111001Y1R   9
N an hour ago by Giabach298
Source: Iran TST 2025 Test 1 Problem 1
Let \( a_n \) be a sequence of positive real numbers such that for every \( n > 2025 \), we have:
\[ a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i} \]Prove that there exists a natural number \( M \) such that for all \( n > M \), the following holds:
\[ a_n < \frac{1}{1404} \]
9 replies
M11100111001Y1R
May 27, 2025
Giabach298
an hour ago
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A scalene triangle and nine point circle
ariopro1387   3
N an hour 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
an hour ago
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inequality
SunnyEvan   0
an hour 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
an hour ago
0 replies
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inequality
SunnyEvan   6
N 2 hours 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
2 hours ago
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Bushy and Jumpy and the unhappy walnut reordering
popcorn1   54
N 2 hours 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
2 hours ago
J
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Write down sum or product of two numbers
Rijul saini   3
N 2 hours 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
2 hours ago
J
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gcd (a^n+b,b^n+a) is constant
EthanWYX2009   83
N 2 hours 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
2 hours ago
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J
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maximal cube in tetrahedron, one vertex on feet of altitude
jasperE3   0
Jun 15, 2021
Source: Bulgaria 1986 P3
A regular tetrahedron of unit edge is given. Find the volume of the maximal cube contained in the tetrahedron, whose one vertex lies in the feet of an altitude of the tetrahedron.
0 replies
jasperE3
Jun 15, 2021
0 replies
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maximal cube in tetrahedron, one vertex on feet of altitude
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Reveal topic tags
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Source: Bulgaria 1986 P3
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jasperE3
11395 posts
jasperE3
#1 h  Jun 15, 2021, 4:56 PM
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A regular tetrahedron of unit edge is given. Find the volume of the maximal cube contained in the tetrahedron, whose one vertex lies in the feet of an altitude of the tetrahedron.
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