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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Pre- Calc
AimlessNonsense   1
N 2 hours ago by Mathzeus1024
A rock got stuck in the tread of my tire and when I was driving 45 miles per hour, the rock came loose
and hit the inside of the wheel well of the car. How fast, in miles per hour, was the rock traveling
when it came out of the tread? (The tire has a diameter of 26 inches.)

I have been trying to figure this out for about 3 hours now and I know I am making some small mistake, but cannot seem to figure out what it is.
1 reply
AimlessNonsense
Aug 30, 2015
Mathzeus1024
2 hours ago
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How to solve this problem
xiangovo   1
N 2 hours ago by loup blanc
Source: website
How many nonzero points are there on x^3y + y^3z + z^3x = 0 over the finite field \mathbb{F}_{5^{18}} up to scaling?
1 reply
xiangovo
Mar 19, 2025
loup blanc
2 hours ago
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Finite solution for x
Rohit-2006   1
N 2 hours ago by Filipjack
$P(t)$ be a non constant polynomial with real coefficients. Prove that the system of simultaneous equations —
$$\int_{0}^{x} P(t)sin t dt =0$$$$\int_{0}^{x}P(t) cos t dt=0$$has finitely many solutions $x$.
1 reply
1 viewing
Rohit-2006
Today at 4:19 AM
Filipjack
2 hours ago
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We know that $\frac{d}{dx}\bigg(\frac{dy}{dx}\bigg)=\frac{d^2 y}{dx^2}.$ Why we
Vulch   1
N 3 hours ago by Aiden-1089
We know that $\frac{d}{dx}\bigg(\frac{dy}{dx}\bigg)=\frac{d^2 y}{dx^2}.$ Why we can't write $\frac{d^2 y}{dx^2}$ as $\frac{d^2 y}{d^2 x^2}?$
1 reply
Vulch
4 hours ago
Aiden-1089
3 hours ago
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complex analysis
functiono   1
N 3 hours ago by Mathzeus1024
Source: exam
find the real number $a$ such that

$\oint_{|z-i|=1} \frac{dz}{z^2-z+a} =\pi$
1 reply
functiono
Jan 15, 2024
Mathzeus1024
3 hours ago
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Computational Calculus
Munmun5   0
3 hours ago
1. Consider the set of all continuous and infinitely differentiable functions $f$ with domain $[0,2025]$ satisfying $f(0)=0,f'(0)=0,f'(2025)=1$ and $f''$ is strictly increasing on $[0,2025]$ Compute smallest real M such that all functions in this set ,$f(2025)<M$ .
2. Polynomials $A(x)=ax^3+abx^2-4x-c,B(x)=bx^3+bcx^2-6x-a,C(x)=cx^3+cax^2-9x-b$ have local extrema at $b,c,a$ respectively. find $abc$ . Here $a,b,c$ are constants .
3. Let $R$ be the region in the complex plane enclosed by curve $$f(x)=e^{ix}+e^{2ix}+\frac{e^{3ix}}{3}$$for $0\leq x\leq 2\pi$. Compute perimeter of $R$ .
0 replies
Munmun5
3 hours ago
0 replies
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Why is this series not the Fourier series of some Riemann integrable function
tohill   0
5 hours ago
$\sum_{n=1}^{\infty}{\frac{\sin nx}{\sqrt{n}}}$ (0<x<2π)
0 replies
tohill
5 hours ago
0 replies
J
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Converging product
mathkiddus   10
N Today at 4:30 AM by HacheB2031
Source: mathkiddus
Evaluate the infinite product, $$\prod_{n=1}^{\infty} \frac{7^n - n}{7^n + n}.$$
10 replies
mathkiddus
Apr 18, 2025
HacheB2031
Today at 4:30 AM
J
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Find the formula
JetFire008   4
N Today at 12:36 AM by HacheB2031
Find a formula in compact form for the general term of the sequence defined recursively by $x_1=1, x_n=x_{n-1}+n-1$ if $n$ is even.
4 replies
JetFire008
Yesterday at 12:23 PM
HacheB2031
Today at 12:36 AM
J
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$f\circ g +g\circ f=0\implies n$ even
al3abijo   4
N Yesterday at 10:37 PM by alexheinis
Let $n$ a positive integer . suppose that there exist two automorphisms $f,g$ of $\mathbb{R}^n$ such that $f\circ g +g\circ f=0$ .
Prove that $n$ is even.
4 replies
al3abijo
Yesterday at 9:05 PM
alexheinis
Yesterday at 10:37 PM
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2025 OMOUS Problem 6
enter16180   2
N Yesterday at 9:06 PM by loup blanc
Source: Open Mathematical Olympiad for University Students (OMOUS-2025)
Let $A=\left(a_{i j}\right)_{i, j=1}^{n} \in M_{n}(\mathbb{R})$ be a positive semi-definite matrix. Prove that the matrix $B=\left(b_{i j}\right)_{i, j=1}^{n} \text {, where }$ $b_{i j}=\arcsin \left(x^{i+j}\right) \cdot a_{i j}$, is also positive semi-definite for all $x \in(0,1)$.
2 replies
enter16180
Apr 18, 2025
loup blanc
Yesterday at 9:06 PM
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Sum of multinomial in sublinear time
programjames1   0
Yesterday at 7:45 PM
Source: Own
A frog begins at the origin, and makes a sequence of hops either two to the right, two up, or one to the right and one up, all with equal probability.

1. What is the probability the frog eventually lands on $(a, b)$?

2. Find an algorithm to compute this in sublinear time.
0 replies
programjames1
Yesterday at 7:45 PM
0 replies
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Find the answer
JetFire008   1
N Yesterday at 6:42 PM by Filipjack
Source: Putnam and Beyond
Find all pairs of real numbers $(a,b)$ such that $ a\lfloor bn \rfloor = b\lfloor an \rfloor$ for all positive integers $n$.
1 reply
JetFire008
Yesterday at 12:31 PM
Filipjack
Yesterday at 6:42 PM
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Pyramid packing in sphere
smartvong   2
N Yesterday at 4:23 PM by smartvong
Source: own
Let $A_1$ and $B$ be two points that are diametrically opposite to each other on a unit sphere. $n$ right square pyramids are fitted along the line segment $\overline{A_1B}$, such that the apex and altitude of each pyramid $i$, where $1\le i\le n$, are $A_i$ and $\overline{A_iA_{i+1}}$ respectively, and the points $A_1, A_2, \dots, A_n, A_{n+1}, B$ are collinear.

(a) Find the maximum total volume of $n$ pyramids, with altitudes of equal length, that can be fitted in the sphere, in terms of $n$.

(b) Find the maximum total volume of $n$ pyramids that can be fitted in the sphere, in terms of $n$.

(c) Find the maximum total volume of the pyramids that can be fitted in the sphere as $n$ tends to infinity.

Note: The altitudes of the pyramids are not necessarily equal in length for (b) and (c).
2 replies
smartvong
Apr 13, 2025
smartvong
Yesterday at 4:23 PM
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J
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Cauchy Integral Problem
Did2   3
N Jan 22, 2024 by Etkan
Evaluate
$$ \int_C \frac{z^2-2 z}{(z+1)^2\left(z^2+4\right)} d z $$where $C$ is the circle $|z|=10$.
3 replies
Did2
Jan 22, 2024
Etkan
Jan 22, 2024
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Cauchy Integral Problem
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Reveal topic tags
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Did2
106 posts
Did2
#1 h  Jan 22, 2024, 3:48 PM
Y by
Evaluate
$$ \int_C \frac{z^2-2 z}{(z+1)^2\left(z^2+4\right)} d z $$where $C$ is the circle $|z|=10$.
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Etkan
1554 posts
Etkan
#2 h  Jan 22, 2024, 4:35 PM
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Did2 wrote:
Evaluate
$$ \int_C \frac{z^2-2 z}{(z+1)^2\left(z^2+4\right)} d z $$where $C$ is the circle $|z|=10$.

The residues at $0$, $2i$ and $-2i$ are $0$, $\frac{7+i}{25}$ and $\frac{7-i}{25}$, respectively, so by the Residue Theorem we get that the answer is $\frac{28\pi i}{25}$.
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vakifjoe
121 posts
vakifjoe
#3 h  Jan 22, 2024, 7:03 PM
Y by
Why do you calculate the residue at $z=0$? $f$ has a pole of order two at $z=-1$. The integral value is zero, which can be easily seen with calculating the residue at infinity.
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Etkan
1554 posts
Etkan
#4 h  Jan 22, 2024, 9:03 PM
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vakifjoe wrote:
Why do you calculate the residue at $z=0$?

Because I messed up with the $z$ term in the numerator and computed the limit as $z\to 0$. The residue at $z=-1$ is $-\frac{14}{25}$, giving the final answer to be $0$.
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