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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

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[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
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0 replies
jwelsh
Jul 1, 2025
0 replies
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When is it a prime
Ecrin_eren   2
N 28 minutes ago by Kempu33334
For how many integer values of a the expression a^8
+ 2^(2+2^a) is a prime number?
2 replies
Ecrin_eren
2 hours ago
Kempu33334
28 minutes ago
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last digit
Noname23   1
N 30 minutes ago by Kempu33334
Let $a=3^0+3^1+3^2+...+3^{2020}$. Find the last 2 digits of:
$(a+3)^{a+3}$
1 reply
Noname23
Today at 11:41 AM
Kempu33334
30 minutes ago
J
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Basic Inequalities Doubt
JetFire008   4
N an hour ago by mathprodigy2011
If $x+y=1$, find the maximum value of $x^2+y^2=1$.
I saw a solution to this question where Titu's lemma was applied and the answer was $\frac{1}{2}$. But my doubt is can't we apply other inequality to get the maximum result? or did they us titu's lemma because the given information can fit only in this lemma?
4 replies
JetFire008
Yesterday at 7:52 AM
mathprodigy2011
an hour ago
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1\p=1/a^2+1/b^2 diophantine with prime p (Greece Juniors 2017 p3)
parmenides51   4
N 3 hours ago by AylyGayypow009
Find all triplets $(a,b,p)$ where $a,b$ are positive integers and $p$ is a prime number satisfying: $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$
4 replies
parmenides51
Mar 16, 2020
AylyGayypow009
3 hours ago
J
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Spectral radius
ILOVEMYFAMILY   1
N Today at 11:54 AM by alexheinis
Let $A \in \mathbb{R}^{n \times n}$. The spectral radius of $A$, denoted by $\rho(A)$, is defined as
\[ \rho(A) = \max_i |\lambda_i| \]where $\lambda_i$ are all the eigenvalues of the matrix $A$.
Let $A \in \mathbb{R}^{n \times n}$. There exists a norm $\|\cdot\|$ such that $\|A\| < 1$ if and only if the spectral radius of $A$ satisfies the condition $\rho(A) < 1$.
1 reply
ILOVEMYFAMILY
Yesterday at 1:41 PM
alexheinis
Today at 11:54 AM
J
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Putnam 2014 A4
Kent Merryfield   37
N Today at 3:36 AM by numbertheory97
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$) Determine the smallest possible value of the probability of the event $X=0.$
37 replies
Kent Merryfield
Dec 7, 2014
numbertheory97
Today at 3:36 AM
J
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Find the value of $x$ in the following matrix problem: $\begin{bmatrix}x-5-1\en
Vulch   3
N Today at 3:06 AM by Etkan
Find the value of $x$ in the following matrix problem:

$\begin{bmatrix}x&-5&-1\end{bmatrix}\begin{bmatrix}1&0&2\\0&2&1\\2&0&3\end{bmatrix}\begin{bmatrix}x\\4\\1\end{bmatrix}=0$
3 replies
Vulch
Today at 1:58 AM
Etkan
Today at 3:06 AM
J
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Putnam 2003 B3
btilm305   36
N Today at 2:36 AM by SirAppel
Show that for each positive integer n, \[n!=\prod_{i=1}^n \; \text{lcm} \; \{1, 2, \ldots, \left\lfloor\frac{n}{i} \right\rfloor\}\](Here lcm denotes the least common multiple, and $\lfloor x\rfloor$ denotes the greatest integer $\le x$.)
36 replies
btilm305
Jun 23, 2011
SirAppel
Today at 2:36 AM
J
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irritative method
ILOVEMYFAMILY   3
N Today at 12:11 AM by ILOVEMYFAMILY
If $\|T\| < 1$ for any natural matrix norm and $c$ is a given vector, then the sequence $\{x^{(k)}\}_{k=0}^{\infty}$ defined by
\[ x^{(k)} = T x^{(k-1)} + c \]converges, for any $x^{(0)} \in \mathbb{R}^n$, to a vector $x \in \mathbb{R}^n$, with $x = Tx + c$, and the following error bounds hold:
a) $\|x - x^{(k)}\| \leq \|T\|^k \|x^{(0)} - x\|$
b) $\|x - x^{(k)}\| \leq \frac{\|T\|^k}{1 - \|T\|} \|x^{(1)} - x^{(0)}\|$
3 replies
ILOVEMYFAMILY
Yesterday at 9:43 AM
ILOVEMYFAMILY
Today at 12:11 AM
J
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Eccentricity Sleuthing
Mathzeus1024   1
N Yesterday at 7:53 PM by vanstraelen
A nice problem involving conics.
1 reply
Mathzeus1024
Jul 15, 2025
vanstraelen
Yesterday at 7:53 PM
J
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Numerical analysis
Tricky123   0
Yesterday at 4:09 PM
Q) the exponent n of binary digits gives a range of 0 to $2^{n}-1$
But how we get it if any one give me the concept of approach the problem? Help
0 replies
Tricky123
Yesterday at 4:09 PM
0 replies
J
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Irritative methods
ILOVEMYFAMILY   0
Yesterday at 1:43 PM
Let $A \in \mathbb{R}^{n \times n}$. Prove that:
1) If $A$ is a diagonally dominant matrix, then both the Jacobi and Gauss-Seidel methods converge, and the Gauss-Seidel method converges faster in the sense that $\rho(T_{GS}) < \rho(T_J)$, where $T_{GS}$ and $T_J$ are the iteration matrices defined by $x^{(k+1)}=Tx^k+b$ for each method
2) If $A$ is a symmetric positive definite matrix, then both the Jacobi and Gauss-Seidel methods converge.
0 replies
ILOVEMYFAMILY
Yesterday at 1:43 PM
0 replies
J
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analysis
We2592   1
N Yesterday at 12:56 AM by alexheinis
Q) find the value of the integration $I=\int_{a}^{b} \frac{e^{-|x|}}{1+(sinhx)^2}$
1 reply
We2592
Jul 17, 2025
alexheinis
Yesterday at 12:56 AM
J
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Are all solutions normal ?
loup blanc   0
Jul 17, 2025
This post is linked to this one
https://artofproblemsolving.com/community/c7t290f7h3608120_matrix_equation
Let $Z=\{A\in M_n(\mathbb{C}) ; (AA^*)^2=A^4\}$.
If $A\in Z$ is a normal matrix, then $A$ is unitarily similar to $diag(H_p,S_{n-p})$,
where $H$ is hermitian and $S$ is skew-hermitian.
But are there other solutions? In other words, is $A$ necessarily normal?
I don't know the answer.
0 replies
loup blanc
Jul 17, 2025
0 replies
J
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J
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Root of Unity filtering
Kyj9981   5
N Jul 2, 2025 by Andyluo
Evaluate
$\binom{2017}{0}+\binom{2017}{3}+\cdots+\binom{2017}{2016}$

Source: Ray Li Handout on Roots of Unity

Answer
5 replies
Kyj9981
Jul 2, 2025
Andyluo
Jul 2, 2025
J
Root of Unity filtering
G H J
Reveal topic tags
root of unity
Root of unity filter
algebra
binomial coefficient
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Kyj9981
43 posts
Kyj9981
#1 h  Jul 2, 2025, 3:22 PM
Y by
Evaluate
$\binom{2017}{0}+\binom{2017}{3}+\cdots+\binom{2017}{2016}$

Source: Ray Li Handout on Roots of Unity

Answer
This post has been edited 2 times. Last edited by Kyj9981, Jul 2, 2025, 3:56 PM
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StrahdVonZarovich
2390 posts
StrahdVonZarovich
#2 h  Jul 2, 2025, 3:28 PM
Y by
to be clear, is the problem to find $\sum^{2016}_{n=0}\binom{2017}{3n}?$
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pineconee
393 posts
pineconee
#3 h  Jul 2, 2025, 3:32 PM
Y by
Let $\omega=\exp(2\pi i/3)$.
\begin{align*} \sum_{\underset{0\leq x \leq 2017}{3\mid x}} \dbinom{2017}x &= \dfrac13 \sum_{0\leq x \leq 2017}\dbinom{2017}x(1+\omega^x+\omega^{2x}) \\ &= \dfrac13\left[\sum_{0\leq x \leq 2017}\dbinom{2017}x + \sum_{0\leq x \leq 2017}\dbinom{2017}x\omega^x + \sum_{0\leq x \leq 2017}\dbinom{2017}x\omega^{2x}\right] \\ &= \dfrac13(2^{2017} + (1+\omega)^{2017} + (1+\omega^2)^{2017}) \\ &= \dfrac13(2^{2017} + (-\omega)^{4034} + (-\omega)^{2017}) \\ &= \dfrac13(2^{2017} + 1) \end{align*}
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Kyj9981
43 posts
Kyj9981
#4 h  Jul 2, 2025, 3:38 PM
Y by
Solution
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Shan3t
531 posts
Shan3t
#5 h  Jul 2, 2025, 3:55 PM
Y by
Let $S$ be the sum $\binom{2017}{0}+\binom{2017}{3}+\cdots+\binom{2017}{2016}.$

$\sum_{k=0}^n\dbinom{n}{k} = (1+1)^n = 2^n.$
$\sum_{k=0}^n\dbinom{n}{k}\omega = (1+\omega)^n.$
$\sum_{k=0}^n\dbinom{n}{k}\omega^2 = (1+\omega^2)^n.$
Adding gives:
$\sum_{k=0}^n\dbinom{n}k(1+\omega+\omega^2) = 3S.$(I suggest writing out each of the summations to see why.) Now $3S = 2^n+(1+\omega)^n+(1+\omega^2)^n.$ We can simplify this to $3S = 2^n+(-\omega^2)^n+(-\omega)^n,$ because $1+\omega+\omega^2 = 0.$ Replacing $n=2017,$ gives $3S = 2^{2017}-\omega^{4034}-\omega^{2017} = 2^{2017}-\omega^2-\omega-1+1 = 2^{2017}+1.$
Hence:
$S=\frac{2^{2017}+1}3.$
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Andyluo
1067 posts
Andyluo
#6 h  Jul 2, 2025, 4:02 PM
Y by
could we get the handout

nvm https://cs.stanford.edu/~rayyli/static/contest/lectures/Ray%20Li%20rootsofunity.pdf
This post has been edited 1 time. Last edited by Andyluo, Jul 2, 2025, 4:02 PM
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