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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
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[*]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
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Problem 5
blug   4
N 4 minutes ago by Sir_Cumcircle
Source: Czech-Polish-Slovak Junior Match 2025 Problem 5
For every integer $n\geq 1$ prove that
$$\frac{1}{n+1}-\frac{2}{n+2}+\frac{3}{n+3}-\frac{4}{n+4}+...+\frac{2n-1}{3n-1}>\frac{1}{3}.$$
4 replies
blug
May 19, 2025
Sir_Cumcircle
4 minutes ago
J
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Cool integer FE
Rijul saini   2
N 22 minutes ago by ZVFrozel
Source: LMAO Revenge 2025 Day 1 Problem 1
Alice has a function $f : \mathbb N \rightarrow \mathbb N$ such that for all naturals $a, b$ the function satisfies:
\[a + b \mid a^{f(a)} + b^{f(b)} \]Bob wants to find all possible functions Alice could have. Help Bob and find all functions that Alice could have.
2 replies
Rijul saini
Yesterday at 7:06 PM
ZVFrozel
22 minutes ago
J
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A beautiful collinearity regarding three wonderful points
math_pi_rate   10
N 35 minutes ago by alexanderchew
Source: Own
Let $\triangle DEF$ be the medial triangle of an acute-angle triangle $\triangle ABC$. Suppose the line through $A$ perpendicular to $AB$ meet $EF$ at $A_B$. Define $A_C,B_A,B_C,C_A,C_B$ analogously. Let $B_CC_B \cap BC=X_A$. Similarly define $X_B$ and $X_C$. Suppose the circle with diameter $BC$ meet the $A$-altitude at $A'$, where $A'$ lies inside $\triangle ABC$. Define $B'$ and $C'$ similarly. Let $N$ be the circumcenter of $\triangle DEF$, and let $\omega_A$ be the circle with diameter $X_AN$, which meets $\odot (X_A,A')$ at $A_1,A_2$. Similarly define $\omega_B,B_1,B_2$ and $\omega_C,C_1,C_2$.
1) Show that $X_A,X_B,X_C$ are collinear.
2) Prove that $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a circle centered at $N$.
3) Prove that $\omega_A,\omega_B,\omega_C$ are coaxial.
4) Show that the line joining $X_A,X_B,X_C$ is perpendicular to the radical axis of $\omega_A,\omega_B,\omega_C$.
10 replies
math_pi_rate
Nov 8, 2018
alexanderchew
35 minutes ago
J
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Tricky FE
Rijul saini   4
N 36 minutes ago by YaoAOPS
Source: LMAO 2025 Day 1 Problem 1
Let $\mathbb{R}$ denote the set of all real numbers. Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that
$$f(xy) + f(f(y)) = f((x + 1)f(y))$$for all real numbers $x$, $y$.

Proposed by MV Adhitya and Kanav Talwar
4 replies
Rijul saini
Yesterday at 6:58 PM
YaoAOPS
36 minutes ago
J
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Quotient of Polynomials is Quadratic
tastymath75025   26
N an hour ago by pi271828
Source: USA TSTST 2017 Problem 3, by Linus Hamilton and Calvin Deng
Consider solutions to the equation \[x^2-cx+1 = \dfrac{f(x)}{g(x)},\]where $f$ and $g$ are polynomials with nonnegative real coefficients. For each $c>0$, determine the minimum possible degree of $f$, or show that no such $f,g$ exist.

Proposed by Linus Hamilton and Calvin Deng
26 replies
tastymath75025
Jun 29, 2017
pi271828
an hour ago
J
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Bugs Bunny at it again
Rijul saini   4
N an hour ago by ThatApollo777
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
4 replies
Rijul saini
Yesterday at 7:01 PM
ThatApollo777
an hour ago
J
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Orthocenters equidistant from circumcenter
Rijul saini   5
N an hour ago by YaoAOPS
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
5 replies
1 viewing
Rijul saini
Yesterday at 6:31 PM
YaoAOPS
an hour ago
J
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Six variables (2)
Nguyenhuyen_AG   1
N an hour ago by lbh_qys
Let $a, \, b, \,c, \, x, \, y, \, z$ be six positive real numbers. Prove that
\[a^2+b^2+c^2+\frac{4(ax+by+cz)\sqrt{ab+bc+ca}}{x+y+z} \geqslant 2(ab+bc+ca).\]
1 reply
Nguyenhuyen_AG
2 hours ago
lbh_qys
an hour ago
J
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The line is a common tangent
Rijul saini   3
N an hour ago by pingupignu
Source: India IMOTC 2025 Day 4 Problem 3
Let $ABCD$ be a cyclic quadrilateral with circumcentre $O$ and circumcircle $\Gamma$. Let $T$ be the intersection of tangents at $B$ and $C$ to $\Gamma$. Let $\omega$ be the circumcircle of triangle $TBC$ and let $M(\neq T)$, $N(\neq T)$ denote the second intersections of $TA,TD$ with $\omega$ respectively. Let $AD$ and $BC$ intersect at $E$ and $\Omega$ be the circumcircle of triangle $EMN$. If $AD$ intersects $\Omega$ again at $X \neq E$, prove that the line tangent to $\Omega$ at $X$ is also tangent to $\omega$.

Proposed by Malay Mahajan and Siddharth Choppara
3 replies
1 viewing
Rijul saini
Yesterday at 6:47 PM
pingupignu
an hour ago
J
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One of P or Q lies on circle
Rijul saini   6
N an hour ago by ZVFrozel
Source: LMAO 2025 Day 1 Problem 3
Let $ABC$ be an acute triangle with orthocenter $H$. Let $M$ be the midpoint of $BC$, and $K$ be the intersection of the tangents from $B$ and $C$ to the circumcircle of $ABC$. Denote by $\Omega$ the circle centered at $H$ and tangent to line $AM$.

Suppose $AK$ intersects $\Omega$ at two distinct points $X$, $Y$.
Lines $BX$ and $CY$ meet at $P$, while lines $BY$ and $CX$ meet at $Q$. Prove that either $P$ or $Q$ lies on $\Omega$.

Proposed by MV Adhitya, Archit Manas and Arnav Nanal
6 replies
+1 w
Rijul saini
Yesterday at 6:59 PM
ZVFrozel
an hour ago
J
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Polynomial strategy game
Rijul saini   1
N an hour ago by everythingpi3141592
Source: India IMOTC 2025 Day 1 Problem 3
Let $N \geqslant 2024!$ be a positive integer. Alice and Bob play the following game, with Alice going first after which they alternate turns. They determine the numbers $a_0,a_1, a_2, \ldots, a_{2025}$ in the following way.

On the $k$th turn, the player whose turn it is sets $a_{k-1}$ to be an integer such that:
$\bullet$ $1\leqslant a_{k-1}\leqslant N$
$\bullet$ There exists a polynomial $P$ with integer coefficients and $ P(i) = a_i$ for $0 \leqslant i \leqslant k-1$

Alice wins if and only if Bob is unable to pick a value in one of his moves i.e. $a_{1}, a_3,\ldots$. In particular, she also loses if Bob is able to pick $a_{2025}$ successfully.
Determine all values of $N$ for which Alice can ensure that she wins regardless of Bob's strategy.

Proposed by Atul Shatavart Nadig and Rohan Goyal
1 reply
Rijul saini
Yesterday at 6:31 PM
everythingpi3141592
an hour ago
J
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One of the lines is tangent
Rijul saini   4
N an hour ago by ZVFrozel
Source: LMAO 2025 Day 2 Problem 2
Let $ABC$ be a scalene triangle with incircle $\omega$. Denote by $N$ the midpoint of arc $BAC$ in the circumcircle of $ABC$, and by $D$ the point where the $A$-excircle touches $BC$. Suppose the circumcircle of $AND$ meets $BC$ again at $P \neq D$ and intersects $\omega$ at two points $X$, $Y$.

Prove that either $PX$ or $PY$ is tangent to $\omega$.

Proposed by Sanjana Philo Chacko
4 replies
Rijul saini
Yesterday at 7:02 PM
ZVFrozel
an hour ago
J
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Circumcenter lies on altitude
ABCDE   59
N 2 hours ago by Ilikeminecraft
Source: 2016 ELMO Problem 2
Oscar is drawing diagrams with trash can lids and sticks. He draws a triangle $ABC$ and a point $D$ such that $DB$ and $DC$ are tangent to the circumcircle of $ABC$. Let $B'$ be the reflection of $B$ over $AC$ and $C'$ be the reflection of $C$ over $AB$. If $O$ is the circumcenter of $DB'C'$, help Oscar prove that $AO$ is perpendicular to $BC$.

James Lin
59 replies
ABCDE
Jun 24, 2016
Ilikeminecraft
2 hours ago
J
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OreINMO: My stepfunction cannot be this linear
anantmudgal09   15
N 2 hours ago by shendrew7
Source: INMO 2023 P3
Let $\mathbb N$ denote the set of all positive integers. Find all real numbers $c$ for which there exists a function $f:\mathbb N\to \mathbb N$ satisfying:
[list]
[*] for any $x,a\in\mathbb N$, the quantity $\frac{f(x+a)-f(x)}{a}$ is an integer if and only if $a=1$;
[*] for all $x\in \mathbb N$, we have $|f(x)-cx|<2023$.
[/list]

Proposed by Sutanay Bhattacharya
15 replies
anantmudgal09
Jan 15, 2023
shendrew7
2 hours ago
J
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J
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Easy diophantine
gghx   3
N Apr 20, 2025 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
Apr 20, 2025
J
Easy diophantine
G H J
Reveal topic tags
number theory
L
G H BBookmark kLocked kLocked NReply
Source: SMO senior 2024 Q2 / SMO junior 2024 Q5
The post below has been deleted. Click to close.
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gghx
1072 posts
gghx
#1 h  Aug 3, 2024, 2:34 AM • 1 Y
Y by Khoi_dang-vu_2009
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
This post has been edited 2 times. Last edited by gghx, Oct 12, 2024, 10:45 AM
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tait1k27
490 posts
tait1k27
#2 h  Oct 17, 2024, 3:31 AM • 1 Y
Y by AnthonyDraude
$$(y+1)^2=(x^2+10x+2)^2+2000$$
Z K Y
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COCBSGGCTG3
8 posts
COCBSGGCTG3
#3 h  Feb 24, 2025, 4:18 PM
Y by
Bordering Method
Z K Y
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lightsynth123
19 posts
lightsynth123
#4 h  Apr 20, 2025, 6:17 AM
Y by
By completing the square, it is easy to find out that
$$(y-x^2-10x-1)(y+x^2+10x+3) = 2000$$which is then easy to solve via factoring.
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