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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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Inequality with 3 variables and a special condition
Nuran2010   3
N 19 minutes ago by sqing
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
For positive real numbers $a,b,c$ we have $3abc \geq ab+bc+ca$.
Prove that:

$\frac{1}{a^3+b^3+c}+\frac{1}{b^3+c^3+a}+\frac{1}{c^3+a^3+b} \leq \frac{3}{a+b+c}$.

Determine the equality case.
3 replies
Nuran2010
Tuesday at 5:06 PM
sqing
19 minutes ago
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Inspired by JK1603JK and arqady
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b,c $ be reals such that $ abc\neq 0$ and $ a+b+c=0. $ Prove that
$$\left|\frac{a-2b}{c}\right|+\left|\frac{b-2c}{a} \right|+\left|\frac{c-2a}{b} \right|\ge \frac{1+3\sqrt{13+16\sqrt{2}}}{2}$$$$\left|\frac{a-3b}{c}\right|+\left|\frac{b-3c}{a}\right|+\left|\frac{c-3a}{b}\right|\ge 1+2\sqrt{13+16\sqrt{2}} $$
1 reply
sqing
an hour ago
sqing
an hour ago
J
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Inequalities
sqing   1
N an hour ago by sqing
Let $ a,b,c>0 . $ Prove that
$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq 4\left(\frac{a+b}{b+c}+ \frac{b+c}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{32}{9}\left(\frac{a+b}{b+c}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{8}{3}\left( \frac{a+b}{b+c}+ \frac{b+c}{c+a}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a^2}{b^2}\right)\left(1+\frac{b^2}{c^2}\right)\left(1+\frac{c^2}{a^2}\right )\geq \frac{8}{3}\left( \frac{a^2+bc}{b^2+ca}+\frac{b^2+ca}{c^2+ab}+\frac{c^2+ab}{a^2+bc}\right)$$
1 reply
sqing
Today at 12:20 AM
sqing
an hour ago
J
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An easiest problem ever
Asilbek777   0
an hour ago
Simplify
0 replies
Asilbek777
an hour ago
0 replies
J
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Many Reflections form Cyclic
FireBreathers   0
an hour ago
Let $ABCD$ be a cyclic quadrilateral. The point $E$ is the reflection of $B$ $w.r.t$ the intersection of $AD$ and $BC$, the point $F$ is the reflection of $B$ $w.r.t$ midpoint of $CD$. Also let $G$ be the reflection of $A$ $w.r.t$ midpoint of $CE$. Show that $C,E,F,G,$ concyclic.
0 replies
FireBreathers
an hour ago
0 replies
J
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Inequalities
sqing   6
N an hour ago by sqing
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$ (1-abc) (1-a)(1-b)(1-c) \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c) \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840 $$
6 replies
sqing
Jul 12, 2024
sqing
an hour ago
J
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A problem with a rectangle
Raul_S_Baz   11
N an hour ago by Raul_S_Baz
On the sides AB and AD of the rectangle ABCD, points M and N are taken such that MB = ND. Let P be the intersection of BN and CD, and Q be the intersection of DM and CB. How can we prove that PQ || MN?
IMAGE
11 replies
Raul_S_Baz
Apr 26, 2025
Raul_S_Baz
an hour ago
J
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6 variable inequality
ChuongTk17   4
N 2 hours ago by arqady
Source: Own
Given real numbers a,b,c,d,e,f in the interval [-1;1] and positive x,y,z,t such that $$2xya+2xzb+2xtc+2yzd+2yte+2ztf=x^2+y^2+z^2+t^2$$. Prove that: $$a+b+c+d+e+f \leq 2$$
4 replies
ChuongTk17
Nov 29, 2024
arqady
2 hours ago
J
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(14n+25)/(2n+1) 'is a perfect square - Portugal OPM 2017 p1
parmenides51   3
N 2 hours ago by justaguy_69
Determine all integer values of n for which the number $\frac{14n+25}{2n+1}$ 'is a perfect square.
3 replies
parmenides51
May 15, 2024
justaguy_69
2 hours ago
J
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APMO 2015 P1
aditya21   62
N 2 hours ago by Tonne
Source: APMO 2015
Let $ABC$ be a triangle, and let $D$ be a point on side $BC$. A line through $D$ intersects side $AB$ at $X$ and ray $AC$ at $Y$ . The circumcircle of triangle $BXD$ intersects the circumcircle $\omega$ of triangle $ABC$ again at point $Z$ distinct from point $B$. The lines $ZD$ and $ZY$ intersect $\omega$ again at $V$ and $W$ respectively.
Prove that $AB = V W$

Proposed by Warut Suksompong, Thailand
62 replies
aditya21
Mar 30, 2015
Tonne
2 hours ago
J
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Or statement function
ItzsleepyXD   2
N 3 hours ago by cursed_tangent1434
Source: Own , Mock Thailand Mathematic Olympiad P2
Find all $f: \mathbb{R} \to \mathbb{Z^+}$ such that $$f(x+f(y))=f(x)+f(y)+1\quad\text{ or }\quad f(x)+f(y)-1$$for all real number $x$ and $y$
2 replies
ItzsleepyXD
Yesterday at 9:07 AM
cursed_tangent1434
3 hours ago
J
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Trivial fun Equilateral
ItzsleepyXD   4
N 3 hours ago by cursed_tangent1434
Source: Own , Mock Thailand Mathematic Olympiad P1
Let $ABC$ be a scalene triangle with point $P$ and $Q$ on the plane such that $\triangle BPC , \triangle CQB$ is an equilateral . Let $AB$ intersect $CP$ and $CQ$ at $X$ and $Z$ respectively and $AC$ intersect $BP$ and $BQ$ at $Y$ and $W$ respectively .
Prove that $XY\parallel ZW$
4 replies
ItzsleepyXD
Yesterday at 9:05 AM
cursed_tangent1434
3 hours ago
J
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Geometry Proof
Jackson0423   2
N 3 hours ago by aidan0626
In triangle \( \triangle ABC \), point \( P \) on \( AB \) satisfies \( DB = BC \) and \( \angle DCA = 30^\circ \).
Let \( X \) be the point where the perpendicular from \( B \) to line \( DC \) meets the angle bisector of \( \angle BCA \).
Then, the relation \( AD \cdot DC = BD \cdot AX \) holds.

Prove that \( \triangle ABC \) is an isosceles triangle.
2 replies
Jackson0423
Yesterday at 4:17 PM
aidan0626
3 hours ago
J
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Do not try to case bash lol
ItzsleepyXD   2
N 3 hours ago by cursed_tangent1434
Source: Own , Mock Thailand Mathematic Olympiad P3
Let $n,d\geqslant 6$ be a positive integer such that $d\mid 6^{n!}+1$ .
Prove that $d>2n+6$ .
2 replies
ItzsleepyXD
Yesterday at 9:08 AM
cursed_tangent1434
3 hours ago
J
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J
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midpoint equidistant from // lines - 2018 Rusanovsky Lyceum Olympiad 8.5
parmenides51   1
N Jul 7, 2021 by sunken rock
Let lines $\ell_1,\ell_2,\ell_3$ be parallel. The straight line $\ell$ intersects $\ell_1,\ell_2,\ell_3$ at points $A,B,C$ respectively. The points $E$ and $D$ lies on the lines $\ell_1,\ell_3$ respectively such that $BE = BC$, $AB = DB$. Prove that the midpoint of the angle bisector $BL$ of the $BDE$ triangle lies at the same distance from the straight lines $\ell_1$ and $\ell_3$.

(M. Plotnikov)
1 reply
parmenides51
Jul 6, 2021
sunken rock
Jul 7, 2021
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midpoint equidistant from // lines - 2018 Rusanovsky Lyceum Olympiad 8.5
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Reveal topic tags
geometry
equal segments
Parallel Lines
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parmenides51
30650 posts
parmenides51
#1 h  Jul 6, 2021, 6:40 PM • 1 Y
Y by HWenslawski
Let lines $\ell_1,\ell_2,\ell_3$ be parallel. The straight line $\ell$ intersects $\ell_1,\ell_2,\ell_3$ at points $A,B,C$ respectively. The points $E$ and $D$ lies on the lines $\ell_1,\ell_3$ respectively such that $BE = BC$, $AB = DB$. Prove that the midpoint of the angle bisector $BL$ of the $BDE$ triangle lies at the same distance from the straight lines $\ell_1$ and $\ell_3$.

(M. Plotnikov)
This post has been edited 1 time. Last edited by parmenides51, Jul 7, 2021, 8:14 AM
Reason: latex
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sunken rock
4390 posts
sunken rock
#2 h  Jul 7, 2021, 8:13 AM
Y by
A solution at https://stanfulger.blogspot.com/2021/07/2018-rusanovsky-lyc-olympiad-m.html

Best regards,
sunken rock
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