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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 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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Unique NT Function
IndoMathXdZ   33
N 21 minutes ago by N3bula
Source: IMO SL 2018 N6
Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.
33 replies
IndoMathXdZ
Jul 17, 2019
N3bula
21 minutes ago
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angle wanted, right ABC, AM=CB , CN=MB
parmenides51   1
N 33 minutes ago by Mathzeus1024
Source: 2022 European Math Tournament - Senior First + Grand League - Math Battle 1.3
In a right-angled triangle $ABC$, points $M$ and $N$ are taken on the legs $AB$ and $BC$, respectively, so that $AM=CB$ and $CN=MB$. Find the acute angle between line segments $AN$ and $CM$.
1 reply
parmenides51
Dec 19, 2022
Mathzeus1024
33 minutes ago
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Some interestingly hard inequality
ItzsleepyXD   1
N 36 minutes ago by ItzsleepyXD
Source: Own , modified
Let $ a,b,c \in \mathbb{R^+}$. Find the max $t \in \mathbb{R^+}$ that $$ \frac{1}{a^3(b+c)} + \frac{1}{b^3(c+a} + \frac{1}{c^3(a+b)} \geqslant \frac{4}{9}( \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b})^3 + t ( \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} -\frac{3}{2})$$for all $a^2+b^2+c^2 = 3$.

more
1 reply
ItzsleepyXD
Mar 17, 2025
ItzsleepyXD
36 minutes ago
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Inspired by Czech-Polish-Slovak 2017
sqing   2
N an hour ago by lbh_qys
Let $x, y$ be real numbers. Prove that
$$\frac{(xy+1)(x + 2y)}{(x^2 + 1)(2y^2 + 1)} \leq \frac{3}{2\sqrt 2}$$$$\frac{(xy+1)(x +3y)}{(x^2 + 1)(3y^2 + 1)} \leq \frac{2}{ \sqrt 3}$$$$\frac{(xy - 1)(x + 2y)}{(x^2 + 1)(2y^2 + 1)} \leq \frac{1}{\sqrt 2}$$$$\frac{(xy - 1)(x + 3y)}{(x^2 + 1)(3y^2 + 1)} \leq \frac{\sqrt 3}{2}$$
2 replies
sqing
an hour ago
lbh_qys
an hour ago
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Inequality by Po-Ru Loh
v_Enhance   55
N an hour ago by ethan2011
Source: ELMO 2003 Problem 4
Let $x,y,z \ge 1$ be real numbers such that \[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \] Prove that \[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]
55 replies
v_Enhance
Dec 29, 2012
ethan2011
an hour ago
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Serbian selection contest for the BMO 2025 - P4
OgnjenTesic   1
N an hour ago by WallyWalrus
Let $a_1, a_2, \ldots, a_8$ be real numbers. Prove that
$$\sum_{i=1}^{8} \left( a_i^2 + a_i a_{i+2} \right) \geq \sum_{i=1}^{8} \left( a_i a_{i+1} + a_i a_{i+3} \right),$$where the indices are taken modulo 8, i.e., $a_9 = a_1$, $a_{10} = a_2$, and $a_{11} = a_3$. In which cases does equality hold?

Proposed by Vukašin Pantelić and Andrija Živadinović
1 reply
OgnjenTesic
Apr 7, 2025
WallyWalrus
an hour ago
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Projective training on circumscribds
Assassino9931   1
N an hour ago by VicKmath7
Source: Bulgaria Balkan MO TST 2025
Let $ABCD$ be a circumscribed quadrilateral with incircle $k$ and no two opposite angles equal. Let $P$ be an arbitrary point on the diagonal $BD$, which is inside $k$. The segments $AP$ and $CP$ intersect $k$ at $K$ and $L$. The tangents to $k$ at $K$ and $L$ intersect at $S$. Prove that $S$ lies on the line $BD$.
1 reply
Assassino9931
Yesterday at 10:17 PM
VicKmath7
an hour ago
J
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Orthocenter config once again
Assassino9931   7
N an hour ago by VicKmath7
Source: Bulgaria National Olympiad 2025, Day 2, Problem 4
Let \( ABC \) be an acute triangle with \( AB < AC \), midpoint $M$ of side $BC$, altitude \( AD \) (\( D \in BC \)), and orthocenter \( H \). A circle passes through points \( B \) and \( D \), is tangent to line \( AB \), and intersects the circumcircle of triangle \( ABC \) at a second point \( Q \). The circumcircle of triangle \( QDH \) intersects line \( BC \) at a second point \( P \). Prove that the lines \( MH \) and \( AP \) are perpendicular.
7 replies
Assassino9931
Tuesday at 1:53 PM
VicKmath7
an hour ago
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Angle EBA is equal to Angle DCB
WakeUp   6
N an hour ago by Nari_Tom
Source: Baltic Way 2011
Let $ABCD$ be a convex quadrilateral such that $\angle ADB=\angle BDC$. Suppose that a point $E$ on the side $AD$ satisfies the equality
\[AE\cdot ED + BE^2=CD\cdot AE.\]
Show that $\angle EBA=\angle DCB$.
6 replies
WakeUp
Nov 6, 2011
Nari_Tom
an hour ago
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if xy+xz+yz+2xyz+1 prove that...
behdad.math.math   5
N an hour ago by Sadigly
if xy+xz+yz+2xyz+1 prove that x+y+z>=3/2
5 replies
behdad.math.math
Sep 25, 2008
Sadigly
an hour ago
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Inequalities
sqing   7
N 2 hours ago by sqing
Let $ a,b,c $ be real numbers so that $ a+2b+3c=2 $ and $ 2ab+6bc+3ca =1. $ Show that
$$-\frac{1}{6} \leq ab-bc+ ca\leq \frac{1}{2}$$$$\frac{5-\sqrt{61}}{9} \leq a-b+c\leq \frac{5+\sqrt{61}}{9} $$
7 replies
sqing
Yesterday at 2:40 PM
sqing
2 hours ago
J
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Circle and square
Marrelia   1
N 5 hours ago by sunken rock
Given a circle with center $O$, and square $ABCD$. Point $A$ and $B$ are on the circle, and $CD$ is tangent to the circle at point $E$. Let $M$ represent the midpoint of $AD$ and $F$ represent the intersection between $AD$ and circle. Prove that $MF = FD$.
1 reply
Marrelia
Today at 3:00 AM
sunken rock
5 hours ago
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Challenging Trigonometric Sums - AoPS Volume 2 Problem 277
Shiyul   2
N 5 hours ago by sp0rtman00000
Problem #277 (Source: Mu Alpha Theta 1992)

Find $\color[rgb]{0.35,0.35,0.35}\displaystyle\sum_{n=0}^\infty\frac{\sin (nx)}{3^n}$ if $\color[rgb]{0.35,0.35,0.35}\sin x=1/3$ and $\color[rgb]{0.35,0.35,0.35} 0\le x\le \pi/2$.

I know what cosine of x is also positive because of the value of x. I've also tried to see if the value of sin(nx) ever repeats, but it doesn't. Can anyone give me a hint (not the full solution) on how to start on solving this problem? Thank you.
2 replies
Shiyul
Today at 4:44 AM
sp0rtman00000
5 hours ago
J
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law of log
Miranda2829   14
N 5 hours ago by sp0rtman00000
5log (5²) + 8 ˡºᵍ₈4 =

is this answer 6?
14 replies
Miranda2829
Today at 2:12 AM
sp0rtman00000
5 hours ago
J
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J
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Determine the angles of triangle PQR
Amir Hossein   0
Sep 15, 2010
Inside an equilateral triangle $ABC$ one constructs points $P, Q$ and $R$ such that
\[\angle QAB = \angle PBA = 15^\circ,\\ \angle RBC = \angle QCB = 20^\circ,\\ \angle PCA = \angle RAC = 25^\circ.\]
Determine the angles of triangle $PQR.$
0 replies
Amir Hossein
Sep 15, 2010
0 replies
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Determine the angles of triangle PQR
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Amir Hossein
5452 posts
Amir Hossein
#1 h  Sep 15, 2010, 9:58 PM • 2 Y
Y by Adventure10, Mango247
Inside an equilateral triangle $ABC$ one constructs points $P, Q$ and $R$ such that
\[\angle QAB = \angle PBA = 15^\circ,\\ \angle RBC = \angle QCB = 20^\circ,\\ \angle PCA = \angle RAC = 25^\circ.\]
Determine the angles of triangle $PQR.$
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