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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
Double integration
Tricky123   2
N 6 minutes ago by Mathzeus1024
Q)
\[\iint_{R} \sin(xy) \,dx\,dy, \quad R = \left[0, \frac{\pi}{2}\right] \times \left[0, \frac{\pi}{2}\right]\]
How to solve the problem like this I am using the substitution method but its seems like very complicated in the last
Please help me
2 replies
Tricky123
May 18, 2025
Mathzeus1024
6 minutes ago
How can I know the sequences's convergence value?
Madunglecha   2
N 17 minutes ago by Madunglecha
What is the convergence value of the sequence??
(n^2)*ln(n+1/n)-n
2 replies
Madunglecha
3 hours ago
Madunglecha
17 minutes ago
Unsolving differential equation
Madunglecha   3
N 2 hours ago by solyaris
For parameter t
I made a differential equation :
y"=y*(x')^2
for here, '&" is derivate and second order derivate for t
could anyone tell me what is equation between y&x?
3 replies
Madunglecha
May 18, 2025
solyaris
2 hours ago
Prove the statement
Butterfly   11
N 3 hours ago by solyaris
Given an infinite sequence $\{x_n\} \subseteq  [0,1]$, there exists some constant $C$, for any $r>0$, among the sequence $x_n$ and $x_m$ could be chosen to satisfy $|n-m|\ge r $ and $|x_n-x_m|<\frac{C}{|n-m|}$.
11 replies
Butterfly
May 7, 2025
solyaris
3 hours ago
Max and min of ab+bc+ca-abc
Tiira   6
N 5 hours ago by MathsII-enjoy
a, b and c are three non-negative reel numbers such that a+b+c=1.
What are the extremums of
ab+bc+ca-abc
?
6 replies
Tiira
Jan 29, 2021
MathsII-enjoy
5 hours ago
Polynomial Minimization
ReticulatedPython   4
N Today at 1:27 AM by jasperE3
Consider the polynomial $$p(x)=x^{n+1}-x^{n}$$, where $x, n \in \mathbb{R+}.$

(a) Prove that the minimum value of $p(x)$ always occurs at $x=\frac{n}{n+1}.$
4 replies
ReticulatedPython
May 6, 2025
jasperE3
Today at 1:27 AM
Trig Identity
gauss202   2
N Today at 1:24 AM by MathIQ.
Simplify $\dfrac{1-\cos \theta + \sin \theta}{\sqrt{1 - \cos \theta + \sin \theta - \sin \theta \cos \theta}}$
2 replies
gauss202
May 14, 2025
MathIQ.
Today at 1:24 AM
2018 Mock ARML I --7 2^n | \prod^{2048}_{k=0} C(2k , k)
parmenides51   3
N Today at 12:57 AM by MathIQ.
Find the largest integer $n$ such that $2^n$ divides $\prod^{2048}_{k=0} {2k \choose k}$.
3 replies
parmenides51
Jan 17, 2024
MathIQ.
Today at 12:57 AM
f(2x+y)=f(x+2y)
jasperE3   3
N Today at 12:46 AM by MathIQ.
Find all functions $f:\mathbb R^+\to\mathbb R^+$ such that:
$$f(2x+y)=f(x+2y)$$for all $x,y>0$.
3 replies
jasperE3
Yesterday at 8:58 PM
MathIQ.
Today at 12:46 AM
System of Equations
P162008   1
N Yesterday at 8:31 PM by alexheinis
If $a,b$ and $c$ are real numbers such that

$(a + b)(b + c) = -1$

$(a - b)^2 + (a^2 - b^2)^2 = 85$

$(b - c)^2 + (b^2 - c^2)^2 = 75$

Compute $(a - c)^2 + (a^2 - c^2)^2.$
1 reply
P162008
Monday at 10:48 AM
alexheinis
Yesterday at 8:31 PM
Might be the first equation marathon
steven_zhang123   35
N Yesterday at 7:09 PM by lightsbug
As far as I know, it seems that no one on HSM has organized an equation marathon before. Click to reveal hidden textSo why not give it a try? Click to reveal hidden text Let's start one!
Some basic rules need to be clarified:
$\cdot$ If a problem has not been solved within $5$ days, then others are eligible to post a new probkem.
$\cdot$ Not only simple one-variable equations, but also systems of equations are allowed.
$\cdot$ The difficulty of these equations should be no less than that of typical quadratic one-variable equations. If the problem involves higher degrees or more variables, please ensure that the problem is solvable (i.e., has a definite solution, rather than an approximate one).
$\cdot$ Please indicate the domain of the solution to the equation (e.g., solve in $\mathbb{R}$, solve in $\mathbb{C}$).
Here's an simple yet fun problem, hope you enjoy it :P :
P1
35 replies
steven_zhang123
Jan 20, 2025
lightsbug
Yesterday at 7:09 PM
THREE People Meet at the SAME. TIME.
LilKirb   7
N Yesterday at 5:33 PM by hellohi321
Three people arrive at the same place independently, at a random between $8:00$ and $9:00.$ If each person remains there for $20$ minutes, what's the probability that all three people meet each other?

I'm already familiar with the variant where there are only two people, where you Click to reveal hidden text It was an AIME problem from the 90s I believe. However, I don't know how one could visualize this in a Click to reveal hidden text Help on what to do?
7 replies
LilKirb
Monday at 1:06 PM
hellohi321
Yesterday at 5:33 PM
Quite straightforward
steven_zhang123   1
N Yesterday at 3:16 PM by Mathzeus1024
Given that the sequence $\left \{ a_{n} \right \} $ is an arithmetic sequence, $a_{1}=1$, $a_{2}+a_{3}+\dots+a_{10}=144$. Let the general term $b_{n}$ of the sequence $\left \{ b_{n} \right \}$ be $\log_{a}{(1+\frac{1}{a_{n}} )} ( a > 0  \text{and}  a \ne  1)$, and let $S_{n}$ be the sum of the $n$ terms of the sequence $\left \{ b_{n} \right \}$. Compare the size of $S_{n}$ with $\frac{1}{3} \log_{a}{(1+\frac{1}{a_{n}} )} $.
1 reply
steven_zhang123
Jan 11, 2025
Mathzeus1024
Yesterday at 3:16 PM
Function and Quadratic equations help help help
Ocean_MathGod   1
N Yesterday at 11:26 AM by Mathzeus1024
Consider this parabola: y = x^2 + (2m + 1)x + m(m - 3) where m is constant and -1 ≤ m ≤ 4. A(-m-1, y1), B(m/2, y2), C(-m, y3) are three different points on the parabola. Now rotate the axis of symmetry of the parabola 90 degrees counterclockwise around the origin O to obtain line a. Draw a line from the vertex P of the parabola perpendicular to line a, meeting at point H.

1) express the vertex of the quadratic equation using an expression with m.
2) If, regardless of the value of m, the parabola and the line y=x−km (where k is a constant) have exactly one point of intersection, find the value of k.

3) (where I'm struggling the most) When 1 < PH ≤ 6, compare the values of y1, y2, and y3.
1 reply
Ocean_MathGod
Aug 26, 2024
Mathzeus1024
Yesterday at 11:26 AM
Permutation polynomial
alexheinis   2
N Apr 11, 2025 by djmathman
Let $n$ be a positive integer. Prove that $f=x^3+x$ is a permutation polynomial mod $n$ iff $n$ is a power of 3.
Note: we call $f$ a permutation polynomial mod $n$ iff the induced map $f:Z_n\rightarrow Z_n$ is bijective.
2 replies
alexheinis
Apr 11, 2025
djmathman
Apr 11, 2025
Permutation polynomial
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alexheinis
10618 posts
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Let $n$ be a positive integer. Prove that $f=x^3+x$ is a permutation polynomial mod $n$ iff $n$ is a power of 3.
Note: we call $f$ a permutation polynomial mod $n$ iff the induced map $f:Z_n\rightarrow Z_n$ is bijective.
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alexheinis
10618 posts
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I found this problem a few days ago on AOPS, unfortunately it was deleted while I was typing the second half of the solution. Since I liked the problem I reposted it. Here is my solution.
For $n=1$ it is clear, now suppose that $n=3^k,k\ge 1$ and that $x^3+x\equiv y^3+y(n)$ where $x\not\equiv y$. Then $(x-y)(x^2+xy+y^2+1)\equiv 0(n)$. If $x\equiv y(3)$ then the second factor is coprime to 3, hence $x\equiv y(n)$, contradiction. Otherwise we have $x^2+xy+y^2\equiv -1(3)\iff (x+2y)^2\equiv -1(3)$, impossible. Hence $f$ is injective and bijective.
Now suppose that $n$ has a prime factor $p\not=3$ and write $n=p^ka$ where $k\ge 1,(a,p)=1$. First suppose that $p>3$.

Lemma. The equation $x^2+xy+y^2\equiv -1(p^k)$ has a solution $x,y\in Z$.
Proof. First we show that $x^2+xy+y^2\equiv -1(p)$ has a solution. We can rewrite as $(2x+y)^2+3y^2\equiv -4(p)$, hence $s^2+3t^2\equiv -4(p)$. The sets $\{s^2\},\{-4-3t^2\}\subset F_p$ have ${{p+1}\over 2}$ elts, hence they cannot be disjoint. We find a solution $s,t\in Z$ and we can solve $2x+y\equiv s, y\equiv t(n)$.
Now suppose that $x^2+xy+y^2=tp^k-1$ and we want to modify this solution to find a solution mod $p^{k+1}$.
This is similar to Hensel's lemma. Try $Y=y+\lambda p^k$ then $x^2+xY+Y^2\equiv -1(p^{k+1})\iff \lambda (x+2y)\equiv -t(p)$. We can solve for $\lambda$ unless when $x+2y\equiv 0(p)$. Symmetrically we can find a solution mod $p^{k+1}$ unless when $x+2y\equiv 0(p)$. If both cases apply then $x\equiv y\equiv 0(p)$. This is impossible and we have proved the lemma.

I leave it to the reader to prove the lemma when $p=2$, actually the proof is a bit easier.
To apply the lemma we will need a solution with $x\not\equiv y(p^k)$. Note that $(x,-x-y)$ is also a solution. If $x\equiv y, -x-y (p^k)$ then $y\equiv -2x\implies x\equiv -2x\implies x\equiv 0\implies x\equiv y\equiv 0$, which again is impossible.

Finally, find $x,y\in Z$ with $x^2+xy+y^2\equiv -1(p^k)$ and $x\not\equiv y(p^k)$. Pick $X\in Z$ with $X\equiv y(a), X\equiv x(p^k)$. Then $X^3+X\equiv y^3+y(n)$ and $X\not\equiv y(n)$.

Thanks djmathman below for the link. Like I said it was posted a few days ago (by Chandru I think), and I didn't know it had appeared before.
This post has been edited 2 times. Last edited by alexheinis, Apr 11, 2025, 9:53 PM
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djmathman
7938 posts
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This is APMO 2014 Problem 3.
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