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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:
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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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Brilliant Problem
M11100111001Y1R   8
N a minute ago by The5
Source: Iran TST 2025 Test 3 Problem 3
Find all sequences \( (a_n) \) of natural numbers such that for every pair of natural numbers \( r \) and \( s \), the following inequality holds:
\[ \frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2 \]
8 replies
M11100111001Y1R
May 27, 2025
The5
a minute ago
J
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Nut equation
giangtruong13   2
N 16 minutes ago by Mathzeus1024
Source: Mie black fiends
Solve the quadratic equation: $$[4(\sqrt{(1+x)^3})^2-3\sqrt{1+x^2}](4x^3+3x)=2$$
2 replies
giangtruong13
Apr 1, 2025
Mathzeus1024
16 minutes ago
J
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Euler line problem
m4thbl3nd3r   2
N 17 minutes ago by m4thbl3nd3r
Let $O,H$ be the circumcenter and orthocenter of triangle $ABC$ and $E,F$ be intersections of $OH$ with $AB,AC$. Let $H',O'$ be orthocenter and circumcenter of triangle $AEF$. Prove that $O'H'\parallel BC.$
2 replies
m4thbl3nd3r
an hour ago
m4thbl3nd3r
17 minutes ago
J
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f(x)+f(1-x)=0
ChildFlower   2
N 35 minutes ago by mashumaro
Find all functions $f:\mathbb (0;1] \to\mathbb R$ such that
$$f(x)+f(1-x)=0\; \forall x \in (0;1] $$
2 replies
ChildFlower
Today at 4:10 AM
mashumaro
35 minutes ago
J
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Determine the number $N$ of such distinct necklaces (up to rotation and reflecti
Arytva   0
36 minutes ago
Let $n\ge 3$ be a positive integer. Consider necklaces of length n whose beads are colored in one of three colors, say red, green, or blue, with exactly two beads of each color (so $n=6$). A rotation of the necklace or a reflection (flipping) is considered the same necklace. But now impose the extra condition that no two beads of the same color are adjacent around the circle. Determine the number $N$ of such distinct necklaces (up to rotation and reflection).
0 replies
Arytva
36 minutes ago
0 replies
J
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Geometry
Arytva   0
42 minutes ago
Source: Source?
Let two circles \(\omega_1\) and \(\omega_2\) meet at two distinct points \(X\) and \(Y\). Choose any line \(\ell\) through \(X\), and let \(\ell\) meet \(\omega_1\) again at \(A\) (other than \(X\)) and meet \(\omega_2\) again at \(B\). On \(\omega_1\), let \(M\) be the midpoint of the minor arc \(AY\) (i.e., the point on \(\omega_1\) such that \(\angle AMY\) subtends the arc \(AY\)), and on \(\omega_2\) let \(N\) be the midpoint of the minor arc \(BY\). Prove that
\[ MN \parallel \text{(radical axis of } \omega_1, \omega_2). \]
0 replies
Arytva
42 minutes ago
0 replies
J
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Intersections and concyclic points
Lukaluce   3
N an hour ago by Schintalpati
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.
3 replies
Lukaluce
May 18, 2025
Schintalpati
an hour ago
J
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inequality
SunnyEvan   1
N an hour ago by SunnyEvan
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: $$ \frac{a^2+b^2+2}{a^6b^2+a^2b^6} \geq 2 $$
1 reply
SunnyEvan
3 hours ago
SunnyEvan
an hour ago
J
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Geo problem
lgx57   0
an hour ago
Source: an exercise
Let $M$ is on $AB$ and $N$ is on $AC$.$BM=NC$.
$O_1$ is the circumcenter of $\displaystyle\triangle ABN$, and $O_2$ is the circumcenter of $\triangle AMC$
Line $O_1O_2$ intersects with $AB,AC$ at $P,Q$
Prove that :$AP=AQ$
Graph:https://www.geogebra.org/m/bncrj5mm
0 replies
lgx57
an hour ago
0 replies
J
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Very easy geometry
mihaig   3
N an hour ago by User21837561
Source: Own
Let $\Delta ABC$ with no obtuse angles.
Prove
$$\frac1{\sqrt3}\cdot\left(\cot A+\cot B+\cot C\right)+\left(2-\sqrt 3\right)\sqrt[3]{\cot A\cot B\cot C}\geq\frac2{\sqrt3}.$$
3 replies
mihaig
3 hours ago
User21837561
an hour ago
J
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Weirdly stated but cool collinearity
Rijul saini   4
N 2 hours ago by MathsII-enjoy
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?
4 replies
Rijul saini
Wednesday at 7:09 PM
MathsII-enjoy
2 hours ago
J
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Might be slightly generalizable
Rijul saini   8
N 2 hours ago by SimplisticFormulas
Source: India IMOTC Day 3 Problem 1
Let $ABC$ be an acute angled triangle with orthocenter $H$ and $AB<AC$. Let $T(\ne B,C, H)$ be any other point on the arc $\stackrel{\LARGE\frown}{BHC}$ of the circumcircle of $BHC$ and let line $BT$ intersect line $AC$ at $E(\ne A)$ and let line $CT$ intersect line $AB$ at $F(\ne A)$. Let the circumcircles of $AEF$ and $ABC$ intersect again at $X$ ($\ne A$). Let the lines $XE,XF,XT$ intersect the circumcircle of $(ABC)$ again at $P,Q,R$ ($\ne X$). Prove that the lines $AR,BC,PQ$ concur.
8 replies
1 viewing
Rijul saini
Wednesday at 6:39 PM
SimplisticFormulas
2 hours ago
J
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Beware the degeneracies!
Rijul saini   9
N 2 hours 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
1 viewing
Rijul saini
Wednesday at 6:30 PM
mudok
2 hours ago
J
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A function on a 2D grid
Rijul saini   2
N 2 hours 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
2 hours ago
J
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J
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Easy Quadric equation
VicKmath7   2
N Apr 17, 2025 by RagvaloD
Source: Archimedes Junior 2015
Find all values of the real parameter $a$, so that the equation $x^2+(a-2)x-(a-1)(2a-3)=0$ has two real roots, so that the one is the square of the other.
2 replies
VicKmath7
Mar 16, 2020
RagvaloD
Apr 17, 2025
J
Easy Quadric equation
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Reveal topic tags
algebra
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Source: Archimedes Junior 2015
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VicKmath7
1391 posts
VicKmath7
#1 h  Mar 16, 2020, 5:54 PM
Y by
Find all values of the real parameter $a$, so that the equation $x^2+(a-2)x-(a-1)(2a-3)=0$ has two real roots, so that the one is the square of the other.
Z K Y
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Rainbow1971
35 posts
Rainbow1971
#2 h  Apr 17, 2025, 1:27 PM
Y by
withdrawn
This post has been edited 1 time. Last edited by Rainbow1971, Apr 17, 2025, 8:07 PM
Z K Y
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RagvaloD
4922 posts
RagvaloD
#3 h  Apr 17, 2025, 6:02 PM
Y by
Roots are $a-1,3-2a$

So enough to prove $a-1=(3-2a)^2$ or $3-2a=(a-1)^2$ and so $a=2,\frac{5}{4}, \sqrt{2}, -\sqrt{2}$
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