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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 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 nonnegative integers
Darealzolt   4
N an hour ago by mathprodigy2011
How many nonnegative integers can be written in the form
\[ a_7 \cdot 3^7 + a_6 \cdot 3^6 + a_5 \cdot 3^5 + a_4 \cdot 3^4 + a_3 \cdot 3^3 + a_2 \cdot 3^2 + a_1 \cdot 3^1 + a_0 \cdot 3^0 \]where \( a_i \in \{-1, 0, 1\} \) for \( 0 \le i \le 7 \)?
4 replies
Darealzolt
Today at 4:58 AM
mathprodigy2011
an hour ago
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Weird locus problem
Sedro   4
N an hour ago by ReticulatedPython
Points $A$ and $B$ are in the coordinate plane such that $AB=2$. Let $\mathcal{H}$ denote the locus of all points $P$ in the coordinate plane satisfying $PA\cdot PB=2$, and let $M$ be the midpoint of $AB$. Points $X$ and $Y$ are on $\mathcal{H}$ such that $\angle XMY = 45^\circ$ and $MX\cdot MY=\sqrt{2}$. The value of $MX^4 + MY^4$ can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
4 replies
Sedro
Yesterday at 3:12 AM
ReticulatedPython
an hour ago
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Two lines meeting on circumcircle
Zhero   54
N an hour ago by Ilikeminecraft
Source: ELMO Shortlist 2010, G4; also ELMO #6
Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$.

Amol Aggarwal.
54 replies
Zhero
Jul 5, 2012
Ilikeminecraft
an hour ago
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Angle Formed by Points on the Sides of a Triangle
xeroxia   6
N an hour ago by xeroxia

In triangle $ABC$, points $D$, $E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, such that
$BD = 20$, $DC = 15$, $CE = 13$, $EA = 8$, $AF = 6$, $FB = 22$.

What is the measure of $\angle EDF$?


6 replies
xeroxia
May 10, 2025
xeroxia
an hour ago
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line JK of intersection points of 2 lines passes through the midpoint of BC
parmenides51   4
N 2 hours ago by reni_wee
Source: Rioplatense Olympiad 2018 level 3 p4
Let $ABC$ be an acute triangle with $AC> AB$. be $\Gamma$ the circumcircle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smallest arc $BC$ of this circle. Let $E$ and $F$ points of the segments $AB$ and $AC$ respectively such that $AE = AF$. Let $P \neq A$ be the second intersection point of the circumcircle circumscribed to $AEF$ with $\Gamma$. Let $G$ and $H$ be the intersections of lines $PE$ and $PF$ with $\Gamma$ other than $P$, respectively. Let $J$ and $K$ be the intersection points of lines $DG$ and $DH$ with lines $AB$ and $AC$ respectively. Show that the $JK$ line passes through the midpoint of $BC$
4 replies
parmenides51
Dec 11, 2018
reni_wee
2 hours ago
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non-homogeneous inequality
BrocSoc   0
2 hours ago
$a+b+c+d=4$. Prove that $$ \frac{4}{abcd} \ge \frac{a}{b} + \frac{b}{c} +\frac{c}{d} +\frac{d}{a} $$
Source: Exercise 1.3.2 in https://web.williams.edu/Mathematics/sjmiller/public_html/161/articles/Riasat_BasicsOlympiadInequalities.pdf
0 replies
BrocSoc
2 hours ago
0 replies
J
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An interesting geometry
k.vasilev   19
N 3 hours ago by Ilikeminecraft
Source: All-Russian Olympiad 2019 grade 10 problem 4
Let $ABC$ be an acute-angled triangle with $AC<BC.$ A circle passes through $A$ and $B$ and crosses the segments $AC$ and $BC$ again at $A_1$ and $B_1$ respectively. The circumcircles of $A_1B_1C$ and $ABC$ meet each other at points $P$ and $C.$ The segments $AB_1$ and $A_1B$ intersect at $S.$ Let $Q$ and $R$ be the reflections of $S$ in the lines $CA$ and $CB$ respectively. Prove that the points $P,$ $Q,$ $R,$ and $C$ are concyclic.
19 replies
k.vasilev
Apr 23, 2019
Ilikeminecraft
3 hours ago
J
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Bosnia and Herzegovina JBMO TST 2016 Problem 3
gobathegreat   3
N 4 hours ago by Sh309had
Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2016
Let $O$ be a center of circle which passes through vertices of quadrilateral $ABCD$, which has perpendicular diagonals. Prove that sum of distances of point $O$ to sides of quadrilateral $ABCD$ is equal to half of perimeter of $ABCD$.
3 replies
gobathegreat
Sep 16, 2018
Sh309had
4 hours ago
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ISI UGB 2025 P2
SomeonecoolLovesMaths   8
N Today at 3:26 PM by lakshya2009
Source: ISI UGB 2025 P2
If the interior angles of a triangle $ABC$ satisfy the equality, $$\sin ^2 A + \sin ^2 B + \sin^2 C = 2 \left( \cos ^2 A + \cos ^2 B + \cos ^2 C \right),$$prove that the triangle must have a right angle.
8 replies
SomeonecoolLovesMaths
Yesterday at 11:16 AM
lakshya2009
Today at 3:26 PM
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Points on the sides of cyclic quadrilateral satisfy the angle conditions
AlperenINAN   3
N Today at 3:13 PM by Primeniyazidayi
Source: Turkey JBMO TST 2025 P1
Let $ABCD$ be a cyclic quadrilateral and let the intersection point of lines $AB$ and $CD$ be $E$. Let the points $K$ and $L$ be arbitrary points on sides $CD$ and $AB$ respectively, which satisfy the conditions
$$\angle KAD = \angle KBC \quad \text{and} \quad \angle LDA = \angle LCB.$$Prove that $EK = EL$.
3 replies
AlperenINAN
Yesterday at 7:15 PM
Primeniyazidayi
Today at 3:13 PM
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Help me solve this problem please. Thank you so much!
illybest   1
N Today at 2:25 PM by GreekIdiot
Give two fixed points B and C, and point A moving on the circle (O). Let D be a point on (O) such that AD is perpendicular to BC. Let O' be the point symmetric to O with respect to BC, M be the midpoint of BC, and N ( dinstinct from D) be the intersection of MD with the circumcircle of triangle AOD. Suppose DO' intersects the circle (O) again at S.
a) Prove that the circle (OMN) is tangent to the circle (DNS)
b) Let d be the line tangent to (DNS) at N. Prove that d always passes through a fixed point when A moves along the arc BC of (O)
1 reply
illybest
Sep 8, 2024
GreekIdiot
Today at 2:25 PM
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ISI UGB 2025 P7
SomeonecoolLovesMaths   10
N Today at 11:44 AM by Mathworld314
Source: ISI UGB 2025 P7
Consider a ball that moves inside an acute-angled triangle along a straight line, unit it hits the boundary, which is when it changes direction according to the mirror law, just like a ray of light (angle of incidence = angle of reflection). Prove that there exists a triangular periodic path for the ball, as pictured below.

IMAGE
10 replies
SomeonecoolLovesMaths
Yesterday at 11:28 AM
Mathworld314
Today at 11:44 AM
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Trigo relation in a right angled. ISIBS2011P10
Sayan   8
N Today at 11:35 AM by SomeonecoolLovesMaths
Show that the triangle whose angles satisfy the equality
\[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\]
is right angled.
8 replies
Sayan
Mar 31, 2013
SomeonecoolLovesMaths
Today at 11:35 AM
J
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Pentagon with given diameter, ratio desired
bin_sherlo   2
N Today at 10:55 AM by Umudlu
Source: Türkiye 2025 JBMO TST P7
$ABCDE$ is a pentagon whose vertices lie on circle $\omega$ where $\angle DAB=90^{\circ}$. Let $EB$ and $AC$ intersect at $F$, $EC$ meet $BD$ at $G$. $M$ is the midpoint of arc $AB$ on $\omega$, not containing $C$. If $FG\parallel DE\parallel CM$ holds, then what is the value of $\frac{|GE|}{|GD|}$?
2 replies
bin_sherlo
Yesterday at 7:21 PM
Umudlu
Today at 10:55 AM
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J
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Algebraic Manipulation
Darealzolt   1
N Apr 30, 2025 by Soupboy0
Find the number of pairs of real numbers $a, b, c$ that satisfy the equation $a^4 + b^4 + c^4 + 1 = 4abc$.
1 reply
Darealzolt
Apr 30, 2025
Soupboy0
Apr 30, 2025
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Algebraic Manipulation
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Darealzolt
7 posts
Darealzolt
#1 h  Apr 30, 2025, 1:25 PM
Y by
Find the number of pairs of real numbers $a, b, c$ that satisfy the equation $a^4 + b^4 + c^4 + 1 = 4abc$.
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Soupboy0
393 posts
Soupboy0
#2 h  Apr 30, 2025, 2:36 PM
Y by
$(a, b, c) = (1, 1, 1), (-1, -1, 1)$ and permutations. Note that if $a, b, c > 0$, then $a^4+b^4+c^4+1 \ge 4abc$ by AM-GM, with equality at $a=b=c=1$, which is the only solution for $a, b, c > 0$. Also note that if $1$ or all of $a, b, c$ are negative, then there are no solutions by the Trivial inequality. Also note that when $2$ of $a, b, c$ are negative, then this case is symmetric to $a, b, c > 0$. Therefore, there are $4$ solutions
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