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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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MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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0 replies
jwelsh
Jul 1, 2025
0 replies
IMC 2021 P8: Maximum number of vectors such that for any 3, 2 are orthogonal
Sumgato   18
N 26 minutes ago by Assassino9931
Source: IMC 2021 P8
Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\mathbb{R}^n$ such that from any three of them, at least two are orthogonal?
18 replies
Sumgato
Aug 5, 2021
Assassino9931
26 minutes ago
Matrix problem
K3012   11
N 27 minutes ago by K3012
\[ A, B \in M_n(\mathbb{C}) \quad n \geq 2 \]
\[ A^2 = B^2 \]
\[ A^3 + BAB = 2I_n \]
1. \( A^3 = I \)

2. \( \lambda_A \neq \lambda_B \Rightarrow A + B = 0 \)
11 replies
K3012
4 hours ago
K3012
27 minutes ago
2023 Putnam B3
giginori   10
N 2 hours ago by os31415
A sequence $y_1, y_2, \ldots, y_k$ of real numbers is called $\textit{zigzag}$ if $k=1$, or if $y_2-y_1, y_3-y_2, \ldots, y_k-y_{k-1}$ are nonzero and alternate in sign. Let $X_1, X_2, \ldots, X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a\left(X_1, X_2, \ldots, X_n\right)$ be the largest value of $k$ for which there exists an increasing sequence of integers $i_1, i_2, \ldots, i_k$ such that $X_{i_1}, X_{i_2}, \ldots X_{i_k}$ is zigzag. Find the expected value of $a\left(X_1, X_2, \ldots, X_n\right)$ for $n \geq 2$.
10 replies
giginori
Dec 3, 2023
os31415
2 hours ago
Putnam 2019 A3
djmathman   15
N 6 hours ago by lpieleanu
Given real numbers $b_0,b_1,\ldots, b_{2019}$ with $b_{2019}\neq 0$, let $z_1,z_2,\ldots, z_{2019}$ be the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019}b_kz^k.
\]Let $\mu = (|z_1|+ \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\ldots, z_{2019}$ to the origin.  Determine the largest constant $M$ such that $\mu\geq M$ for all choices of $b_0,b_1,\ldots, b_{2019}$ that satisfy
\[
1\leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
\]
15 replies
1 viewing
djmathman
Dec 10, 2019
lpieleanu
6 hours ago
Automorphic Characteristic
KHOMNYO2   3
N Today at 4:51 PM by GreenKeeper
Given a group G where $|G| = p + 1$ for some odd prime $p$. It is known that $p \mid |Aut(G)|$. Prove that $p$ must be in the form of $4k + 3$, where $k$ is an integer. Give a group $|G|$ as an example that satisfies the property.
3 replies
KHOMNYO2
Feb 5, 2025
GreenKeeper
Today at 4:51 PM
expected value question
straight   2
N Today at 4:12 PM by grupyorum
Given positive reals $x_1< x_2, \dots< x_n$ randomly and independently picked in $[0,1]$ (you order them after picking). If you define $x_0 = 0$ and $x_{n+1} = 1$, what is
\[\mathbb{E}(\sum_{i=0}^{n}(x_{i+1} - x_i)^2))\]
2 replies
straight
Sep 28, 2024
grupyorum
Today at 4:12 PM
probability
nguyenalex   6
N Today at 4:07 PM by grupyorum
Let $(W_n )_{n\geq 1}$ be a sequence of independent random variables with standard normal distribution. Estimate the probability that $W_1^2 +W_2^2 +...+ W_{n} ^ 2 < n + 2\sqrt{n}$ .
6 replies
nguyenalex
Oct 26, 2024
grupyorum
Today at 4:07 PM
probability
ILOVEMYFAMILY   2
N Today at 3:39 PM by grupyorum
Let $(X_k)$ be a sequence of independent random variables with distribution $X_k \sim \text{Exp}(k)$. Define $M_n = \min_{1 \leq k \leq n} X_k$. Prove that $$\frac{n(n+1)M_n}{\ln n} \overset{P}{\rightarrow} 0$$and $$nM_n \overset{\text{a.s}}{\rightarrow} 0$$as $n \to \infty$.
2 replies
ILOVEMYFAMILY
Oct 29, 2024
grupyorum
Today at 3:39 PM
Convergence in distribution
Ernest532   4
N Today at 3:10 PM by grupyorum
Let $\{X_i\}$ be i.i.d with pdf $\frac{1}{\lvert x\rvert^3}\mathbb{I}_{\{\lvert x\rvert>1\}}$. Prove that $$\frac{S_n}{\sqrt{n\log(n)}}\xrightarrow[n\to\infty]{\text{d}}\mathcal{N}(0,1).$$
4 replies
Ernest532
Jun 30, 2025
grupyorum
Today at 3:10 PM
Putnam 2017 A2
Kent Merryfield   29
N Today at 11:42 AM by Assassino9931
Let $Q_0(x)=1$, $Q_1(x)=x,$ and
\[Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}\]for all $n\ge 2.$ Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients.
29 replies
Kent Merryfield
Dec 3, 2017
Assassino9931
Today at 11:42 AM
Analysis
ratatuy   7
N Today at 9:28 AM by Mathzeus1024
Source: Own
$i)$Prove that
$\sum_{k=0}^{+\infty}{\frac{(-1)^k}{2k+1}}=\frac{\pi}{4}$
$ii)$Find
$\sum_{k=0}^{+\infty}{\frac{(-1)^k}{2k+2}}=?$
$iii)$Find
$\sum_{n=1}^{+\infty}{\frac{(-1)^n}{n}\left(\frac{1}{n+1}-\frac{1}{n+3}+\frac{1}{n+5}+...+\frac{(-1)^k}{n+1+2k}+...\right)}=?$
7 replies
ratatuy
Jul 3, 2020
Mathzeus1024
Today at 9:28 AM
Integer part
P0tat0b0y   1
N Today at 8:06 AM by P0tat0b0y
Source: own
Calculate the $\left[ {{\left( \prod\limits_{k=1}^{n}{\frac{2k}{2k-1}} \right)}^{2}}-\pi n \right]$, for positive $n$ inteher, where $[a]$ it is the integer part of the number $a$!
1 reply
P0tat0b0y
Jul 27, 2025
P0tat0b0y
Today at 8:06 AM
Differentiate the following function with respect to $x:$ $arccos\bigg(\frac{1-x
Vulch   1
N Today at 8:04 AM by alexheinis
Differentiate the following function with respect to $x:$
$\text{arccos}\bigg(\frac{1-x^2}{1+x^2}\bigg),~0<x<1.$

My attempts are in the following attachment.I got the answer $\frac{2x^2}{1+x^2},$ but the answer in book is given as $\frac{2}{1+x^2}.$ Where is my faults?
1 reply
Vulch
Today at 4:33 AM
alexheinis
Today at 8:04 AM
Hard sequence problem
P0tat0b0y   1
N Today at 8:04 AM by P0tat0b0y
Please help! This is not homework! But it would be very important to get an answer!
Is there a sequence ${{a}_{n}}$ for which the following 4 conditions are met:
(1) ${{a}_{1}}\le 4-\pi $
(2) $3,(5)-\frac{{{a}_{2}}}{2}\ge \pi$
(3) $\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n}}}{n}=0$
(4) ${{(2n+2)}^{2}}{{a}_{n}}\ge {{(2n+1)}^{2}}{{a}_{n+1}}+(n+1)\pi ,\forall n\ge {{n}_{0}}$
Thanks in advance!
1 reply
P0tat0b0y
Yesterday at 9:08 AM
P0tat0b0y
Today at 8:04 AM
real analysis intermediate concepts
am_11235...   4
N Jul 2, 2025 by Moubinool
Source: AMM, proposed by Cezar Lupu and Tudorel Lupu
Let $f$ be a positive-valued, concave function on $[0,1]$. Prove that $$\frac34\left(\int_0^1f(x)\,dx\right)^2\leq\frac18+\int_0^1f^3(x)\,dx$$
4 replies
am_11235...
Jul 1, 2025
Moubinool
Jul 2, 2025
real analysis intermediate concepts
G H J
G H BBookmark kLocked kLocked NReply
Source: AMM, proposed by Cezar Lupu and Tudorel Lupu
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am_11235...
4400 posts
#1 • 1 Y
Y by ehuseyinyigit
Let $f$ be a positive-valued, concave function on $[0,1]$. Prove that $$\frac34\left(\int_0^1f(x)\,dx\right)^2\leq\frac18+\int_0^1f^3(x)\,dx$$
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Moubinool
5586 posts
#2
Y by
don’t need concavity
for $x\geq 0$ , $x^3 +1/8-3x^2/4\geq 0$ ==>$ 3x^2/4 \leq 1/8 + x^3$ (*)

with cauchy Schwartz inequality and (*) we have done
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am_11235...
4400 posts
#3
Y by
Moubinool wrote:
don’t need concavity
for $x\geq 0$ , $x^3 +1/8-3x^2/4\geq 0$ ==>$ 3x^2/4 \leq 1/8 + x^3$ (*)

with cauchy Schwartz inequality and (*) we have done

How come? $\left(\int_0^1f(x)\,dx\right)^3\neq\int_0^1f^3(x)\,dx$.
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Moubinool
5586 posts
#4
Y by
$3/4(\int_{0}^{1} f(x)dx)^2 \leq 3/4 \int_{0}^{1} f^2(x)dx \leq 1/8 + \int_{0}^{1} f^3(x) dx $
This post has been edited 1 time. Last edited by Moubinool, Jul 1, 2025, 7:04 PM
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Moubinool
5586 posts
#5
Y by
am_11235... wrote:
Let $f$ be a positive-valued, concave function on $[0,1]$. Prove that $$\frac34\left(\int_0^1f(x)\,dx\right)^2\leq\frac18+\int_0^1f^3(x)\,dx$$

am 11235 what is the correct statement? i did not use concavity
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