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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
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[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
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0 replies
jlacosta
Mar 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
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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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Nice problem
hanzo.ei   1
N 17 minutes ago by Mathzeus1024
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that
\[ f(xy) = f(x)f(y) \;-\; f(x + y) \;+\; 1, \quad \forall x, y \in \mathbb{R}. \]
1 reply
2 viewing
hanzo.ei
2 hours ago
Mathzeus1024
17 minutes ago
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ortho conf DEF, radius MD, intersect ME,MF, collinear H,K,L
star-1ord   0
32 minutes ago
Source: Estonia Final Round 2025 12-3
Let $ABC$ be an acute-angled triangle with $|AB|<|AC|$. The altitudes $AD,BE$ and $CF$ intersect at $H$. Let $M$ be the midpoint of $BC$. Point $K$ is chosen on the extension of $EM$ beyond $M$ and point $L$ is chosen on the segment $FM$ such that $|MK|=|ML|=|MD|$. Prove that points $K, L$ and $H$ are collinear.

a little harder version
0 replies
star-1ord
32 minutes ago
0 replies
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Funny system of equations in three variables
Tintarn   10
N an hour ago by Marcus_Zhang
Source: Baltic Way 2020, Problem 5
Find all real numbers $x,y,z$ so that
\begin{align*} x^2 y + y^2 z + z^2 &= 0 \\ z^3 + z^2 y + z y^3 + x^2 y &= \frac{1}{4}(x^4 + y^4). \end{align*}
10 replies
Tintarn
Nov 14, 2020
Marcus_Zhang
an hour ago
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a^{2m}+a^{n}+1 is perfect square
kmh1   1
N 2 hours ago by kmh1
Source: own
Find all positive integer triplets $(a,m,n)$ such that $2m>n$ and $a^{2m}+a^{n}+1$ is a perfect square.
1 reply
kmh1
Mar 20, 2025
kmh1
2 hours ago
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Interesting problem
deraxenrovalo   0
2 hours ago
Given $\triangle$$ABC$ with circumcenter $O$$.\;$Let $P$ be an arbitrary point on $(BOC)$ such that $P$ is outside $(ABC)$$.\;$Let $Q$ be an arbitrary point on $(ABC)$$.\;$$AB$ cuts $(ACP)$ again at $E$ and $AC$ cuts $(ABP)$ again at $F$$.\;$The intersection of $BF$ and $CE$ is $R$$.\;$Let $X$ and $Y$ be the intersection of $EF$ with $(PQC)$ and $(PQR)$ respectively such that $X$, $Y$, $P$ are pairwise distinct.
Show that : $(APX)$, $(BPY)$, $(QPE)$ are coaxial circles

hint
0 replies
deraxenrovalo
2 hours ago
0 replies
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Vieta Jumping Unsolved(Reposted)
Eagle116   0
2 hours ago
Source: MONT, Vieta Jumping part
The question is:
Let $x_1$, $x_2$, $\dots$, $x_n$ be $n$ integers. If $k>n$ is an integer, prove that the only solution to
$$x_1^2 + x_2^2 + \dots + x_n^2 = kx_1x_2\dots x_n $$is is $x_1 = x_2 = \dots = x_n = 0$.
0 replies
Eagle116
2 hours ago
0 replies
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Geometry with parallel lines.
falantrng   32
N 2 hours ago by endless_abyss
Source: RMM 2018,D1 P1
Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .
32 replies
+1 w
falantrng
Feb 24, 2018
endless_abyss
2 hours ago
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sum divides n-th moment
navi_09220114   1
N 2 hours ago by ja.
Source: Own. Malaysian IMO TST 2025 P9
Given four distinct positive integers $a<b<c<d$ such that $\gcd(a,b,c,d)=1$, find the maximum possible number of integers $1\le n\le 2025$ such that $$a+b+c+d\mid a^n+b^n+c^n+d^n$$
Proposed by Ivan Chan Kai Chin
1 reply
navi_09220114
Yesterday at 1:07 PM
ja.
2 hours ago
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Find all functions
Jackson0423   0
3 hours ago
Find all functions F:R->R such that
1/(F(F(x))-F(x))=F(x)
I know x+1/x works..
0 replies
Jackson0423
3 hours ago
0 replies
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2x+1 is a perfect square but the following x+1 integers are not.
Sumgato   7
N 3 hours ago by Davut1102
Source: Spain Mathematical Olympiad 2018 P1
Find all positive integers $x$ such that $2x+1$ is a perfect square but none of the integers $2x+2, 2x+3, \ldots, 3x+2$ are perfect squares.
7 replies
Sumgato
Mar 17, 2018
Davut1102
3 hours ago
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Prove that P1(x), P2(x) ,... Pn(x) = k has no root
truongphatt2668   2
N 3 hours ago by truongphatt2668
Let $n \in \mathbb{N}^*$ and $P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$ such that $\mathrm{deg} P_i = 2, \forall i = \overline{1,n}$. Prove that exists many $k \in \mathbb{N}$ such that every equation: $P_i(x) = k, \forall i = \overline{1,n}$ has no real roots
2 replies
truongphatt2668
Today at 2:26 AM
truongphatt2668
3 hours ago
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Geo: incircle, escircle, isotomic conjugate
XAN4   1
N 3 hours ago by deraxenrovalo
Source: Own
For $\triangle{ABC}$, Its incircle $\odot I$ and $A-$escircle $\odot I_A$ are tangent to $BC$ at $D$ and $E$ respectively. $AI$ intersects line $BC$ at $J$. Line $AD$ intersects $\odot I$ at $F$, and line $AE$ intersects $\odot I_A$ at $G$. Line $FG$ intersects $BC$ at $H$. Prove that $BJ=CH$.
1 reply
XAN4
Mar 19, 2025
deraxenrovalo
3 hours ago
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special sets
ChubbyTomato426   0
3 hours ago
Let $n$ be a positive integer. A subset $\{a, b, c, d\} \subseteq \{1, 2, . . . , 4n\}$ with four distinct elements is special if there exists a rearrangement $(x, y, z, w)$ of $(a, b, c, d)$ such that $xy -zw = 1$. Prove that the set $\{1, 2, . . . , 4n \}$ cannot be partitioned into $n$ special disjoint sets.
0 replies
ChubbyTomato426
3 hours ago
0 replies
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2, 4, 5-Nim
cjquines0   2
N 3 hours ago by Mathdreams
Source: Philippines MO 2016/4
Two players, \(A\) (first player) and \(B\), take alternate turns in playing a game using 2016 chips as follows: the player whose turn it is, must remove \(s\) chips from the remaining pile of chips, where \(s \in \{ 2,4,5 \}\). No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?
2 replies
cjquines0
Jan 21, 2017
Mathdreams
3 hours ago
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min of a+b+c+ {3/(ab+bc+ca) if a,b,c>0 with a^2+b^2+c^2=1
parmenides51   1
N Nov 13, 2022 by grupyorum
Source: Mathcenter Contest / Oly - Thai Forum 2012 (R1) p2 sl-9 https://artofproblemsolving.com/community/c3196914_mathcenter_contest
Let $a,b,c \in \mathbb{R}^+$ where $a^2+b^2+c^2=1$. Find the minimum value of .
$$a+b+c+\frac{3}{ab+bc+ca}$$(PP-nine)
1 reply
parmenides51
Nov 13, 2022
grupyorum
Nov 13, 2022
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min of a+b+c+ {3/(ab+bc+ca) if a,b,c>0 with a^2+b^2+c^2=1
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Reveal topic tags
algebra
inequalities
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Source: Mathcenter Contest / Oly - Thai Forum 2012 (R1) p2 sl-9 https://artofproblemsolving.com/community/c3196914_mathcenter_contest
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parmenides51
30628 posts
parmenides51
#1 h  Nov 13, 2022, 5:42 PM • 1 Y
Y by teomihai
Let $a,b,c \in \mathbb{R}^+$ where $a^2+b^2+c^2=1$. Find the minimum value of .
$$a+b+c+\frac{3}{ab+bc+ca}$$(PP-nine)
This post has been edited 1 time. Last edited by parmenides51, Dec 5, 2022, 3:29 PM
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grupyorum
1400 posts
grupyorum
#2 h  Nov 13, 2022, 6:05 PM • 1 Y
Y by Mango247
Solution with calculus. Let $t=a+b+c$. Then $1<t\le\sqrt{3}$, and the given quantity is $\phi(t):=t+6/(t^2-1)$. Taking derivatives, we get $\phi'(t) = 1-3/(t-1)^2+3/(t+1)^2$. It is not hard to verify $\phi'<0$ on $(1,\sqrt{3})$ (in fact it is negative on a much broader interval, see here.). As a result, $\phi$ is minimized when $t=\sqrt{3}$, yielding the value $3+\sqrt{3}$.
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