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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
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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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Functional Equations
burntpizza001   1
N 42 minutes ago by alexheinis
Let $f:R\rightarrow{}R$ is a function satisfying $f(2-x)=f(2+x)$ and $f(20-x)=f(x)$, $\forall x\in R$.

(i) If $f(0) = 5$ then minimum possible number of values of $x$ satisfying $f(x)=5$, for $x\in[0,170]$ is?

(ii) Graph of $y=f(x)$$ is symmetrical about which line?
1 reply
burntpizza001
May 25, 2025
alexheinis
42 minutes ago
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find question
mathematical-forest   0
an hour ago
Are there any contest questions that seem simple but are actually difficult? :-D
0 replies
mathematical-forest
an hour ago
0 replies
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Binomial Sum
P162008   1
N an hour ago by alexheinis
If $f(x,n) = (x + n)^n - \binom{n}{1} (x + n - 1)^n + \binom{n}{2} (x + n - 2)^n - \cdots (-1)^n x^n$ then

A) $\sum_{r=1}^{4} f(4,r) = 33$

B) $\sum_{r=1}^{4} f(r,4) = 96$

C) $\sum_{r=1}^{5} f(5,r) = 129$

D) $\sum_{r=1}^{5} f(r,5) = 258$
1 reply
P162008
May 26, 2025
alexheinis
an hour ago
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Inspired by qrxz17
sqing   2
N an hour ago by MathsII-enjoy
Source: Own
Let $a, b,c>0 ,(a^2+b^2+c^2)^2 - 2(a^4+b^4+c^4) = 27 $. Prove that $$a+b+c\geq 3\sqrt {3}$$
2 replies
sqing
2 hours ago
MathsII-enjoy
an hour ago
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D,E,F are collinear.
TUAN2k8   1
N an hour ago by Beelzebub
Source: Own
Help me with this:
1 reply
TUAN2k8
Yesterday at 1:07 AM
Beelzebub
an hour ago
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Inspired by qrxz17
sqing   2
N an hour ago by MathsII-enjoy
Source: Own
Let $ a,b,c $ be reals such that $ (a^2+b^2)^2 + (b^2+c^2)^2 +(c^2+a^2)^2 = 28 $ and $ (a^2+b^2+c^2)^2 =16. $ Find the value of $ a^2(a^2-1) + b^2(b^2-1)+c^2(c^2-1).$
2 replies
sqing
3 hours ago
MathsII-enjoy
an hour ago
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Prove DK and BC are perpendicular.
yunxiu   63
N an hour ago by sknsdkvnkdvf
Source: 2012 European Girls’ Mathematical Olympiad P1
Let $ABC$ be a triangle with circumcentre $O$. The points $D,E,F$ lie in the interiors of the sides $BC,CA,AB$ respectively, such that $DE$ is perpendicular to $CO$ and $DF$ is perpendicular to $BO$. (By interior we mean, for example, that the point $D$ lies on the line $BC$ and $D$ is between $B$ and $C$ on that line.)
Let $K$ be the circumcentre of triangle $AFE$. Prove that the lines $DK$ and $BC$ are perpendicular.

Netherlands (Merlijn Staps)
63 replies
yunxiu
Apr 13, 2012
sknsdkvnkdvf
an hour ago
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Geometry with fix circle
falantrng   34
N an hour ago by Aiden-1089
Source: RMM 2018 Problem 6
Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.
34 replies
falantrng
Feb 25, 2018
Aiden-1089
an hour ago
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Set of perfect powers is irreducible
Assassino9931   2
N 2 hours ago by navid
Source: Al-Khwarizmi International Junior Olympiad 2025 P4
For two sets of integers $X$ and $Y$ we define $X\cdot Y$ as the set of all products of an element of $X$ and an element of $Y$. For example, if $X=\{1, 2, 4\}$ and $Y=\{3, 4, 6\}$ then $X\cdot Y=\{3, 4, 6, 8, 12, 16, 24\}.$ We call a set $S$ of positive integers good if there do not exist sets $A,B$ of positive integers, each with at least two elements and such that the sets $A\cdot B$ and $S$ are the same. Prove that the set of perfect powers greater than or equal to $2025$ is good.

(In any of the sets $A$, $B$, $A\cdot B$ no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form $n^k$, where $n>1$ and $k > 1$ are integers.)

Lajos Hajdu and Andras Sarkozy, Hungary
2 replies
Assassino9931
May 9, 2025
navid
2 hours ago
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Stop Projecting your insecurities
naman12   53
N 2 hours ago by EeEeRUT
Source: 2022 USA TST #2
Let $ABC$ be an acute triangle. Let $M$ be the midpoint of side $BC$, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Suppose that the common external tangents to the circumcircles of triangles $BME$ and $CMF$ intersect at a point $K$, and that $K$ lies on the circumcircle of $ABC$. Prove that line $AK$ is perpendicular to line $BC$.

Kevin Cong
53 replies
naman12
Dec 12, 2022
EeEeRUT
2 hours ago
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Roots of unity
Henryfamz   1
N 2 hours ago by Mathzeus1024
Compute $$\sec^4\frac\pi7+\sec^4\frac{2\pi}7+\sec^4\frac{3\pi}7$$
1 reply
Henryfamz
May 13, 2025
Mathzeus1024
2 hours ago
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Shortest number theory you might've seen in your life
AlperenINAN   11
N 2 hours ago by Assassino9931
Source: Turkey JBMO TST 2025 P4
Let $p$ and $q$ be prime numbers. Prove that if $pq(p+1)(q+1)+1$ is a perfect square, then $pq + 1$ is also a perfect square.
11 replies
AlperenINAN
May 11, 2025
Assassino9931
2 hours ago
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AP calc?
Thayaden   31
N 2 hours ago by Mathzeus1024
How are we all feeling on AP calc guys?
31 replies
Thayaden
May 20, 2025
Mathzeus1024
2 hours ago
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Inequalities
sqing   3
N 3 hours ago by sqing
Let $a,b,c,d>0,a^2 + d^2-ad = (b + c)^2 $ aand $ a^2 + b^2 = c^2 + d^2.$ Prove that$$ \frac{ab+cd}{ad+bc} \geq \frac{ 4}{5}$$
3 replies
sqing
Yesterday at 2:47 AM
sqing
3 hours ago
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Theory of Equations
P162008   4
N Apr 25, 2025 by vanstraelen
Let $a,b,c,d$ and $e\in [-2,2]$ such that $\sum_{cyc} a = 0, \sum_{cyc} a^3 = 0, \sum_{cyc} a^5 = 10.$ Find the value of $\sum_{cyc} a^2.$
4 replies
P162008
Apr 23, 2025
vanstraelen
Apr 25, 2025
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Theory of Equations
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P162008
226 posts
P162008
#1 h  Apr 23, 2025, 11:27 AM
Y by
Let $a,b,c,d$ and $e\in [-2,2]$ such that $\sum_{cyc} a = 0, \sum_{cyc} a^3 = 0, \sum_{cyc} a^5 = 10.$ Find the value of $\sum_{cyc} a^2.$
This post has been edited 2 times. Last edited by P162008, Apr 23, 2025, 11:28 AM
Reason: Typo
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vanstraelen
9063 posts
vanstraelen
#2 h  Apr 23, 2025, 7:48 PM • 1 Y
Y by P162008
Polynomial $P(x)=x^{5}-5x^{3}+5x-2$.
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P162008
226 posts
P162008
#3 h  Apr 23, 2025, 10:18 PM
Y by
vanstraelen wrote:
Polynomial $P(x)=x^{5}-5x^{3}+5x-2$.

Yeah correct
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P162008
226 posts
P162008
#4 h  Apr 25, 2025, 7:46 AM
Y by
vanstraelen wrote:
Polynomial $P(x)=x^{5}-5x^{3}+5x-2$.

What was your approach?
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vanstraelen
9063 posts
vanstraelen
#5 h  Apr 25, 2025, 3:11 PM
Y by
$P(x)=x^{5}+p_{4}x^{4}+p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0}$.

$a$ is a root $\Rightarrow a^{5}+p_{4}a^{4}+p_{3}a^{3}+p_{2}a^{2}+p_{1}a+p_{0}=0$, etc.
$\sum_{cyc} a^5+p_{4} \cdot \sum_{cyc} a^4+p_{3} \cdot \sum_{cyc} a^3+p_{2} \cdot \sum_{cyc} a^2+p_{1} \sum_{cyc} a+5p_{0}=0$
or $10+p_{4} \cdot \sum_{cyc} a^4+p_{2} \cdot \sum_{cyc} a^2+5p_{0}=0$.
Let $p_{4}=p_{2}=0$, then $p_{0}=-2$.

New equation: $P(x)=x^{5}+px^{3}+qx-2$.
We want $x=2$ is a root, then $q=-15-4p$ and $P(x)=x^{5}+px^{3}-(15+4p)x-2$.
$P(x)=(x-2)[x^{4}+2x^{3}+(p+4)x^{2}+(2p+8)x+1]$.
We try $P(x)=(x-2)[x^{4}+2x^{3}+(p+4)x^{2}+(2p+8)x+1]=(x-2)(x^{2}+tx-1)^{2}$
and we find $t=1,p=-5$.
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