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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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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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How many ways can we indistribute n different marbles into 6 identical boxes
Taiharward   6
N an hour ago by Taiharward
How many ways can we distribute n indifferent marbles into 6 identical boxes and one jar?
6 replies
Taiharward
Today at 2:14 AM
Taiharward
an hour ago
J
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Combinatoric
spiderman0   1
N 2 hours ago by MathBot101101
Let $ S = \{1, 2, 3, \ldots, 2024\}.$ Find the maximum positive integer $n \geq 2$ such that for every subset $T \subset S$ with n elements, there always exist two elements a, b in T such that:

$|\sqrt{a} - \sqrt{b}| < \frac{1}{2} \sqrt{a - b}$
1 reply
spiderman0
Yesterday at 7:46 AM
MathBot101101
2 hours ago
J
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Combinatorial proof
MathBot101101   10
N 2 hours ago by MathBot101101
Is there a way to prove
\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}=1-\frac{1}{{n+1)!}
without induction and using only combinatorial arguments?

Induction proof wasn't quite as pleasing for me.
10 replies
MathBot101101
Apr 20, 2025
MathBot101101
2 hours ago
J
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Inequalities
sqing   13
N 2 hours ago by sqing
Let $ a,b,c> 0 $ and $ ab+bc+ca\leq 3abc . $ Prove that
$$ a+ b^2+c\leq a^2+ b^3+c^2 $$$$ a+ b^{11}+c\leq a^2+ b^{12}+c^2 $$
13 replies
sqing
Yesterday at 1:54 PM
sqing
2 hours ago
J
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trolling geometry problem
iStud   4
N Yesterday at 3:11 AM by GreenTea2593
Source: Monthly Contest KTOM April P3 Essay
Given a cyclic quadrilateral $ABCD$ with $BC<AD$ and $CD<AB$. Lines $BC$ and $AD$ intersect at $X$, and lines $CD$ and $AB$ intersect at $Y$. Let $E,F,G,H$ be the midpoints of sides $AB,BC,CD,DA$, respectively. Let $S$ and $T$ be points on segment $EG$ and $FH$, respectively, so that $XS$ is the angle bisector of $\angle{DXA}$ and $YT$ is the angle bisector of $\angle{DYA}$. Prove that $TS$ is parallel to $BD$ if and only if $AC$ divides $ABCD$ into two triangles with equal area.
4 replies
iStud
Monday at 9:28 PM
GreenTea2593
Yesterday at 3:11 AM
J
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Equal distances between pairs of orthocenters in cyclic quad
Shu   2
N Apr 20, 2025 by Nari_Tom
Source: XVII Tuymaada Mathematical Olympiad (2010), Senior Level
In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.
2 replies
Shu
Jul 31, 2011
Nari_Tom
Apr 20, 2025
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NEPAL TST 2025 DAY 2
Tony_stark0094   9
N Apr 17, 2025 by hectorleo123
Consider an acute triangle $\Delta ABC$. Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.

Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$, respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$. Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$, and by $Q$ the intersection of ray $EB$ with $\Gamma$.

If $O$ is the circumcenter of $\Delta ABC$, prove that $O$, $D$, and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.

$\textbf{Proposed by Kritesh Dhakal, Nepal.}$
9 replies
Tony_stark0094
Apr 12, 2025
hectorleo123
Apr 17, 2025
J
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Quad formed by orthocenters has same area (all 7's!)
v_Enhance   35
N Apr 16, 2025 by Wictro
Source: USA January TST for the 55th IMO 2014
Let $ABCD$ be a cyclic quadrilateral, and let $E$, $F$, $G$, and $H$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. Let $W$, $X$, $Y$ and $Z$ be the orthocenters of triangles $AHE$, $BEF$, $CFG$ and $DGH$, respectively. Prove that the quadrilaterals $ABCD$ and $WXYZ$ have the same area.
35 replies
v_Enhance
Apr 28, 2014
Wictro
Apr 16, 2025
J
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Nice Quadrilateral Geo
amuthup   52
N Apr 14, 2025 by Frd_19_Hsnzde
Source: 2021 ISL G4
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
52 replies
amuthup
Jul 12, 2022
Frd_19_Hsnzde
Apr 14, 2025
J
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IMO Problem 4
iandrei   105
N Apr 13, 2025 by cj13609517288
Source: IMO ShortList 2003, geometry problem 1
Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.
105 replies
iandrei
Jul 14, 2003
cj13609517288
Apr 13, 2025
J
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cyclic quadrilateral starting with right triangle
parmenides51   4
N Apr 12, 2025 by Rounak_iitr
Source: Australian MO 2015
Let $ABC$ be a triangle with $ACB = 90^o$. The points $D$ and $Z$ lie on the side $AB$ such that $CD$ is perpendicular to $AB$ and $AC = AZ$. The line that bisects $BAC$ meets $CB$ and $CZ$ at $X$ and $Y$ , respectively. Prove that the quadrilateral $BXYD$ is cyclic.
4 replies
parmenides51
Sep 24, 2018
Rounak_iitr
Apr 12, 2025
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H not needed
dchenmathcounts   46
N Apr 11, 2025 by endless_abyss
Source: USEMO 2019/1
Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar.

Robin Son
46 replies
dchenmathcounts
May 23, 2020
endless_abyss
Apr 11, 2025
J
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Diagonal of a pentagon that divides it into a triangle and a cyclic quadrilatera
EmersonSoriano   0
Apr 5, 2025
Source: 2017 Peru Southern Cone TST P1
We say that a diagonal of a convex pentagon is good if it divides the pentagon into a triangle and a circumscribable quadrilateral. What is the maximum number of good diagonals that a convex pentagon can have?

Clarification: A polygon is circumscribable if there is a circle tangent to each of its sides.
0 replies
EmersonSoriano
Apr 5, 2025
0 replies
J
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The locus of P with supplementary angles condition
WakeUp   3
N Apr 4, 2025 by Nari_Tom
Source: Baltic Way 2001
Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.
3 replies
WakeUp
Nov 17, 2010
Nari_Tom
Apr 4, 2025
J
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J
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Find all tuples of real numbers
nhathhuyyp5c   0
Apr 13, 2025
Let \( n \geq 2 \) be an integer. Find all tuples of real numbers \( (a_1, a_2, \ldots, a_n) \) such that
\[ 2a_1 - 3a_2,\ 2a_2 - 3a_3,\ \ldots,\ 2a_n - 3a_1 \]is a permutation of \( a_1, a_2, \ldots, a_n \).
0 replies
nhathhuyyp5c
Apr 13, 2025
0 replies
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Find all tuples of real numbers
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nhathhuyyp5c
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nhathhuyyp5c
#1 h  Apr 13, 2025, 8:03 AM
Y by
Let \( n \geq 2 \) be an integer. Find all tuples of real numbers \( (a_1, a_2, \ldots, a_n) \) such that
\[ 2a_1 - 3a_2,\ 2a_2 - 3a_3,\ \ldots,\ 2a_n - 3a_1 \]is a permutation of \( a_1, a_2, \ldots, a_n \).
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