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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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Find min
lgx57   4
N 4 minutes ago by sqing
Source: Own
Find min of $\dfrac{a^2}{ab+1}+\dfrac{b^2+2}{a+b}$
4 replies
lgx57
Yesterday at 3:01 PM
sqing
4 minutes ago
J
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At most 1 Nonzero Solution
FireBreathers   0
4 minutes ago
Source: https://artofproblemsolving.com/community/c4h3340223p30944256
Let them be $a_1,a_2,...,a_{2023}$ be real numbers. Not all zero. Prove that $\sqrt{1+a_1x}+\sqrt{1+a_2x}+...\sqrt{1+a_{2023}x} = 2023$ has at most $1$ nonzero real root.
0 replies
FireBreathers
4 minutes ago
0 replies
J
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an exponential inequality with two variables
teresafang   1
N 13 minutes ago by CHESSR1DER
x and y are positive real numbers.prove that [(x^y)/y]^(1/2)+[(y^x)/x]^(1/2)>=2.
sorry.I’m not good at English.Also I don’t know how to use Letax.
1 reply
teresafang
32 minutes ago
CHESSR1DER
13 minutes ago
J
H
Arbitrary point on BC and its relation with orthocenter
falantrng   31
N 40 minutes ago by NZP_IMOCOMP4
Source: Balkan MO 2025 P2
In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus
31 replies
falantrng
Apr 27, 2025
NZP_IMOCOMP4
40 minutes ago
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No more topics!
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problem interesting
Cobedangiu   9
N May 1, 2025 by Cobedangiu
Let $a=3k^2+3k+1 (a,k \in N)$
$i)$ Prove that: $a^2$ is the sum of $3$ square numbers
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers
9 replies
Cobedangiu
Apr 30, 2025
Cobedangiu
May 1, 2025
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problem interesting
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Reveal topic tags
algebra
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Cobedangiu
66 posts
Cobedangiu
#1 h  Apr 30, 2025, 5:06 AM
Y by
Let $a=3k^2+3k+1 (a,k \in N)$
$i)$ Prove that: $a^2$ is the sum of $3$ square numbers
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers
This post has been edited 1 time. Last edited by Cobedangiu, Apr 30, 2025, 5:06 AM
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Cobedangiu
66 posts
Cobedangiu
#2 h  Apr 30, 2025, 6:12 AM
Y by
Cobedangiu wrote:
Let $a=3k^2+3k+1 (a,k \in N)$
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers

.....
This post has been edited 1 time. Last edited by Cobedangiu, Apr 30, 2025, 5:04 PM
Reason: .
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Cobedangiu
66 posts
Cobedangiu
#3 h  Apr 30, 2025, 10:55 AM
Y by
Cobedangiu wrote:
Let $a=3k^2+3k+1 (a,k \in N)$
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers

no one?
This post has been edited 1 time. Last edited by Cobedangiu, Apr 30, 2025, 5:04 PM
Reason: .
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tom-nowy
119 posts
tom-nowy
#4 h  Apr 30, 2025, 12:26 PM • 1 Y
Y by Cobedangiu
$ i)$ here
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Cobedangiu
66 posts
Cobedangiu
#5 h  May 1, 2025, 1:03 AM
Y by
anyone have solution for next part?
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compoly2010
3 posts
compoly2010
#6 h  May 1, 2025, 1:08 AM
Y by
What do the three dots between b and a symbolise?
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Cobedangiu
66 posts
Cobedangiu
#7 h  May 1, 2025, 1:19 AM
Y by
compoly2010 wrote:
What do the three dots between b and a symbolise?

divisible
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vincentwant
1372 posts
vincentwant
#8 h  May 1, 2025, 1:37 AM
Y by
https://en.wikipedia.org/wiki/Legendre%27s_three-square_theorem

Let $f(x)$ be equal to $x$ divided by the largest power of $4$ that divides $x$. Legendre's three-square theorem says that $x$ is expressible as the sum of three squares iff $f(x)$ is not $7$ mod $8$.

Observe that $f(x^2)$ is never $7$ mod $8$, so even $n$ always works. If $n$ is odd, if $f(b^n)$ is $7$ mod $8$, then $\nu_2(b^n)$ is even and thus $\nu_2(b)$ is even. Thus $f(b)$ is odd. Observe that if $f(b)$ is odd then $f(b^n)=f(b)^n$. Thus if $f(b^n)$ is $7$ mod $8$ then $f(b)^n$ is $7$ mod $8$. However, any odd number to an odd power is congruent to itself mod $8$, so this means $f(b)\equiv7\pmod8$, contradiction.
This post has been edited 1 time. Last edited by vincentwant, May 1, 2025, 1:45 AM
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tom-nowy
119 posts
tom-nowy
#9 h  May 1, 2025, 9:36 AM
Y by
Perhaps "three square numbers" means three positive squares. If zero were allowed, Question 1 would just be $a^2 = (3k^2 + 3k + 1)^2 + 0^2 + 0^2$, which would make it trivial.
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Cobedangiu
66 posts
Cobedangiu
#10 h  May 1, 2025, 9:40 AM
Y by
tom-nowy wrote:
Perhaps "three square numbers" means three positive squares. If zero were allowed, Question 1 would just be $a^2 = (3k^2 + 3k + 1)^2 + 0^2 + 0^2$, which would make it trivial.

it is not allowed
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