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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
an hour ago
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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next 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 an enriching 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)!

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[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]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
an hour ago
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
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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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Do not try to bash on beautiful geometry
ItzsleepyXD   8
N 32 minutes ago by Ianis
Source: Own , Mock Thailand Mathematic Olympiad P9
Let $ABC$be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$
8 replies
ItzsleepyXD
Wednesday at 9:30 AM
Ianis
32 minutes ago
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Sum of points' powers
Suntafayato   3
N 42 minutes ago by Ianis
Given 2 circles $\omega_1, \omega_2$, find the locus of all points $P$ such that $\mathcal{P}ow(P, \omega_1) + \mathcal{P}ow(P, \omega_2) = 0$ (i.e: sum of powers of point $P$ with respect to the two circles $\omega_1, \omega_2$ is zero).
3 replies
Suntafayato
Mar 24, 2020
Ianis
42 minutes ago
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PAMO 2017 Shortlst: Sum of maxima of adjacent pairs in permutation
DylanN   1
N an hour ago by MelonGirl
Source: 2017 Pan-African Shortlist - I4
Find the maximum and minimum of the expression
\[ \max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1), \]where $(a_1, a_2, \dots, a_n)$ runs over the set of permutations of $(1, 2, \dots, n)$.
1 reply
DylanN
May 5, 2019
MelonGirl
an hour ago
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Bijective quartic modulo p
DottedCaculator   12
N 2 hours ago by MathLuis
Source: ELMO 2024/6
For a prime $p$, let $\mathbb{F}_p$ denote the integers modulo $p$, and let $\mathbb{F}_p[x]$ be the set of polynomials with coefficients in $\mathbb{F}_p$. Find all $p$ for which there exists a quartic polynomial $P(x) \in \mathbb{F}_p[x]$ such that for all integers $k$, there exists some integer $\ell$ such that $P(\ell) \equiv k \pmod p$. (Note that there are $p^4(p-1)$ quartic polynomials in $\mathbb{F}_p[x]$ in total.)

Aprameya Tripathy
12 replies
DottedCaculator
Jun 21, 2024
MathLuis
2 hours ago
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Show that AB/AC=BF/FC
syk0526   76
N 3 hours ago by reni_wee
Source: APMO 2012 #4
Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold.

(Here we denote $XY$ the length of the line segment $XY$.)
76 replies
syk0526
Apr 2, 2012
reni_wee
3 hours ago
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Geometry with known distances
nAalniaOMliO   3
N 3 hours ago by TheBaiano
Source: Belarusian National Olympiad 2025
In a rectangle $ABCD$ two not intersecting circles $\omega_1$ and $\omega_2$ are drawn such that $\omega_1$ is tangent to $AB$ and $AD$ at points $P$ and $S$ respectively, and $\omega_2$ is tangent to $CB$ and $CD$ at $T$ and $Q$ respectively. It is known that $PQ=11, ST=10, BD=14$.
Find the distance between centers of circles $\omega_1$ and $\omega_2$.
3 replies
nAalniaOMliO
Mar 28, 2025
TheBaiano
3 hours ago
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Reflections of AB, AC with respect to BC and angle bisector of A
falantrng   28
N 3 hours ago by lksb
Source: BMO 2024 Problem 1
Let $ABC$ be an acute-angled triangle with $AC > AB$ and let $D$ be the foot of the
$A$-angle bisector on $BC$. The reflections of lines $AB$ and $AC$ in line $BC$ meet $AC$ and $AB$ at points
$E$ and $F$ respectively. A line through $D$ meets $AC$ and $AB$ at $G$ and $H$ respectively such that $G$
lies strictly between $A$ and $C$ while $H$ lies strictly between $B$ and $F$. Prove that the circumcircles of
$\triangle EDG$ and $\triangle FDH$ are tangent to each other.
28 replies
falantrng
Apr 29, 2024
lksb
3 hours ago
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Coincide
giangtruong13   4
N 3 hours ago by vsarg
Source: Hanoi Specialized School's Math Test (Round 2 - Phase 1)
Let $ABCD$ be a trapezoid inscribed in circle $(O)$, $AD||BC, AD < BC$. Let $P$ is the symmetric point of $A$ across $BC$, $AP$ intersects $BC$ at $K$. Let $M$ is midpoint of $BC$ and $H$ is orthocenter of triangle $ABC$. On $BD$ take a point $F$ so that $AF||HM$. Prove that: $ FK,AC,PD$ coincide
4 replies
giangtruong13
Apr 27, 2025
vsarg
3 hours ago
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Show that CK is parallel to AB
Math-lover123   45
N 4 hours ago by reni_wee
Source: Sharygin First Round 2013, Problem 16
The incircle of triangle $ABC$ touches $BC$, $CA$, $AB$ at points $A_1$, $B_1$, $C_1$, respectively. The perpendicular from the incenter $I$ to the median from vertex $C$ meets the line $A_1B_1$ in point $K$. Prove that $CK$ is parallel to $AB$.
45 replies
Math-lover123
Apr 8, 2013
reni_wee
4 hours ago
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Infinite sum of the angles made by the points A_i,B_i
WakeUp   2
N 4 hours ago by Rohit-2006
Source: Baltic Way 1997
On two parallel lines, the distinct points $A_1,A_2,A_3,\ldots $ respectively $B_1,B_2,B_3,\ldots $ are marked in such a way that $|A_iA_{i+1}|=1$ and $|B_iB_{i+1}|=2$ for $i=1,2,\ldots $. Provided that $A_1A_2B_1=\alpha$, find the infinite sum $\angle A_1B_1A_2+\angle A_2B_2A_3+\angle A_3B_3A_4+\ldots $
2 replies
WakeUp
Jan 28, 2011
Rohit-2006
4 hours ago
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Tilted Students Thoroughly Splash Tiger part 2
DottedCaculator   18
N 6 hours ago by MathLuis
Source: ELMO 2024/5
In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$, let $M$ be the midpoint of $\overline{BC}$. Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$. Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$. Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$, and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$. Prove that $A$, $Y$, and $Z$ are collinear.

Tiger Zhang
18 replies
DottedCaculator
Jun 21, 2024
MathLuis
6 hours ago
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Find area!
ComplexPhi   4
N 6 hours ago by TigerOnion
Let $O_1$ be a point in the exterior of the circle $\omega$ of center $O$ and radius $R$ , and let $O_1N$ , $O_1D$ be the tangent segments from $O_1$ to the circle. On the segment $O_1N$ consider the point $B$ such that $BN=R$ .Let the line from $B$ parallel to $ON$ intersect the segment $O_1D$ at $C$ . If $A$ is a point on the segment $O_1D$ other than $C$ so that $BC=BA=a$ , and if the incircle of the triangle $ABC$ has radius $r$ , then find the area of $\triangle ABC$ in terms of $a ,R ,r$.
4 replies
ComplexPhi
Feb 4, 2015
TigerOnion
6 hours ago
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Isi 2016 geometry
zizou10   22
N Yesterday at 6:26 PM by kamatadu
Source: ISI BSTAT 2016 #5
Prove that there exists a right angle triangle with rational sides and area $d$ if and only if $x^2,y^2$ and $z^2$ are squares of rational numbers and are in Arithmetic Progression

Here $d$ is an integer.
22 replies
zizou10
May 8, 2016
kamatadu
Yesterday at 6:26 PM
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IMO Shortlist 2012, Geometry 2
lyukhson   88
N Yesterday at 6:11 PM by zuat.e
Source: IMO Shortlist 2012, Geometry 2
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.
88 replies
lyukhson
Jul 29, 2013
zuat.e
Yesterday at 6:11 PM
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I need help with this problem
VIATON   1
N Apr 18, 2025 by sqing
Let $x,y$ satisfy:
$\frac{4}{(x-1)^2 + (y-2)^2 +4} + \frac{9}{(x-2)^2 + (y-4)^2 +9} = 1$
$[(x-2)^2+(y-4)^2][ (x-1)^2 + (y-2)^2] = 36$
Find Max of :$x+y$
1 reply
VIATON
Apr 18, 2025
sqing
Apr 18, 2025
J
I need help with this problem
G H J
Reveal topic tags
algebra
Inequality
inequalities proposed
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VIATON
140 posts
VIATON
#1 h  Apr 18, 2025, 7:09 AM
Y by
Let $x,y$ satisfy:
$\frac{4}{(x-1)^2 + (y-2)^2 +4} + \frac{9}{(x-2)^2 + (y-4)^2 +9} = 1$
$[(x-2)^2+(y-4)^2][ (x-1)^2 + (y-2)^2] = 36$
Find Max of :$x+y$
This post has been edited 1 time. Last edited by VIATON, Apr 18, 2025, 7:09 AM
Reason: .
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sqing
41923 posts
sqing
#2 h  Apr 18, 2025, 2:20 PM
Y by
Let $ x,y $ be reals satisfy $\frac{4}{(x-1)^2 + (y-2)^2 +4} + \frac{9}{(x-2)^2 + (y-4)^2 +9} = 1 $ and $ [(x-2)^2+(y-4)^2][ (x-1)^2 + (y-2)^2] = 36. $ Prove that
$$0.749054...\leq x+y \leq 8.25095...$$*
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