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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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[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
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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
J
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Inequalities
sqing   5
N 14 minutes ago by sqing
Let $ a,b>0, a^2+ab+b^2 \geq 6 $. Prove that
$$a^4+ab+b^4\geq 10$$Let $ a,b>0, a^2+ab+b^2 \leq \sqrt{10} $. Prove that
$$a^4+ab+b^4 \leq 10$$Let $ a,b>0, a^2+ab+b^2 \geq \frac{15}{2} $. Prove that
$$ a^4-ab+b^4\geq 10$$Let $ a,b>0, a^2+ab+b^2 \leq \sqrt{10} $. Prove that
$$-\frac{1}{8}\leq a^4-ab+b^4\leq 10$$
5 replies
1 viewing
sqing
Yesterday at 2:42 PM
sqing
14 minutes ago
J
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Cube Sphere
vanstraelen   0
an hour ago

Given the cube $\left(\begin{array}{ll} EFGH \\ ABCD \end{array}\right)$ with edge $6$ cm.
Find the volume of the sphere passing through $A,B,C,D$ and tangent to the plane $(EFGH)$.
0 replies
vanstraelen
an hour ago
0 replies
J
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How many pairs
Ecrin_eren   6
N an hour ago by Ecrin_eren


Let n be a natural number and p be a prime number. How many different pairs (n, p) satisfy the equation:

p + 2^p + 3 = n^2 ?



6 replies
Ecrin_eren
May 2, 2025
Ecrin_eren
an hour ago
J
H
Square number
linkxink0603   2
N 2 hours ago by linkxink0603
Find m is positive interger such that m^4+3^m is square number
2 replies
linkxink0603
3 hours ago
linkxink0603
2 hours ago
J
No more topics!
H
J
H
Identity Proof
jjsunpu   3
N Apr 17, 2025 by jjsunpu
Hi this is my identity I name it Excalibur

for example

for k^2

it's 2n - 1

for k^3

it's 3n^2 -3n + 1
3 replies
jjsunpu
Apr 16, 2025
jjsunpu
Apr 17, 2025
J
Identity Proof
G H J
Reveal topic tags
induction
L
G H BBookmark kLocked kLocked NReply
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jjsunpu
130 posts
jjsunpu
#1 h  Apr 16, 2025, 10:35 AM
Y by
Hi this is my identity I name it Excalibur

for example

for k^2

it's 2n - 1

for k^3

it's 3n^2 -3n + 1
Attachments:
This post has been edited 2 times. Last edited by jjsunpu, Apr 18, 2025, 1:44 AM
Reason: changes
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jjsunpu
130 posts
jjsunpu
#2 h  Apr 16, 2025, 10:38 AM
Y by
does anyone know if someone already found this?
Z K Y
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fruitmonster97
2492 posts
fruitmonster97
#3 h  Apr 16, 2025, 12:46 PM
Y by
I doubt this has been solved previously, but it seems a little silly:
\[x^n-(x-1)^n+(x-1)^n-(x-2)^n+\cdots+2^n-1^n+1^n-0^n=x^n-0^n=x^n\]
Z K Y
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jjsunpu
130 posts
jjsunpu
#4 h  Apr 17, 2025, 11:33 AM
Y by
oh that's sad
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