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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)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

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
Periodic sequence
EeEeRUT   8
N 4 minutes ago by PHSH
Source: Isl 2024 A5
Find all periodic sequence $a_1,a_2,\dots$ of real numbers such that the following conditions hold for all $n\geqslant 1$:$$a_{n+2}+a_{n}^2=a_n+a_{n+1}^2\quad\text{and}\quad |a_{n+1}-a_n|\leqslant 1.$$
Proposed by Dorlir Ahmeti, Kosovo
8 replies
EeEeRUT
Jul 16, 2025
PHSH
4 minutes ago
Bonza functions
KevinYang2.71   56
N 4 minutes ago by MR_D33R
Source: 2025 IMO P3
Let $\mathbb{N}$ denote the set of positive integers. A function $f\colon\mathbb{N}\to\mathbb{N}$ is said to be bonza if
\[
f(a)~~\text{divides}~~b^a-f(b)^{f(a)}
\]for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\leqslant cn$ for all bonza functions $f$ and all positive integers $n$.

Proposed by Lorenzo Sarria, Colombia
56 replies
KevinYang2.71
Jul 15, 2025
MR_D33R
4 minutes ago
British Math Olympiad Round 1 2005, Q6
maths_dodo   1
N 18 minutes ago by cubres
Hello, I'm a bit stuck on this little problem...

Let T be a set of 2005 coplanar points with no three collinear. Show that, for any of the 2005 points, the number of triangles it lies strictly within, whose vertices are points in T, is even.

I tried for some small cases like 4 or 5 points, and I'm trying to use these to help me with the larger case.

(British Math Olympiad Round 1 2005, Problem 6)
1 reply
maths_dodo
4 hours ago
cubres
18 minutes ago
A rather famous lemma in geometry
Tintarn   4
N 36 minutes ago by DenizErsan_84
Source: Bundeswettbewerb Mathematik 2022, Round 1 - Problem 3
A circle $k$ touches a larger circle $K$ from inside in a point $P$. Let $Q$ be point on $k$ different from $P$. The line tangent to $k$ at $Q$ intersects $K$ in $A$ and $B$.
Show that the line $PQ$ bisects $\angle APB$.
4 replies
Tintarn
Mar 30, 2022
DenizErsan_84
36 minutes ago
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
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
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
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
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
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
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
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
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
Are all solutions normal ?
loup blanc   0
Thursday at 8:47 PM
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
Thursday at 8:47 PM
0 replies
Inverses and inversion
Vivouaf   2
N Jun 28, 2025 by Vivouaf
Let $\Omega, \omega$ be circles and $A$ a point in the plane. Denote by $A'$, the inverse of $A$ wrt $\omega$. Perform an inversion at $\Omega$ which sends $\omega$ to $\omega'$, $A$ to $B$, $A'$ to $B'$. Show that $B'$ remains the inverse of $B$ wrt $\omega'$.
2 replies
Vivouaf
Jun 26, 2025
Vivouaf
Jun 28, 2025
Inverses and inversion
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Vivouaf
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Let $\Omega, \omega$ be circles and $A$ a point in the plane. Denote by $A'$, the inverse of $A$ wrt $\omega$. Perform an inversion at $\Omega$ which sends $\omega$ to $\omega'$, $A$ to $B$, $A'$ to $B'$. Show that $B'$ remains the inverse of $B$ wrt $\omega'$.
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sami1618
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#2 • 1 Y
Y by Vivouaf
Solution. Place in the complex plane so that $\Omega$ is the unit-circle and $w$ is centered along the $x$ axis. Let $c\in \mathbb{R}$ be the location of the center of $w$ and $r$ be it's radius. Inversion about $\Omega$ has equation $z\mapsto \frac{1}{\overline{z}}$, while inversion about $w$ has equation $$z\mapsto c+\frac{r^2}{\overline{z}-c}.$$The circle $w$ has diameter along the $x$-axis of $c-r$ to $c+r$. Thus $w'$ has diameter along the $x$-axis of $d_1=\frac{1}{c-r}$ to $d_2=\frac{1}{c+r}$. Inversion about $w'$ hence has the equation $$z\mapsto \frac{1}{2}(d_1+d_2)+\frac{\frac{1}{4}(d_1-d_2)^2}{\overline{z}-\frac{1}{2}(d_1+d_2)}=\frac{c}{c^2-r^2}+\frac{r^2}{(c^2-r^2)(\overline{z}(c^2-r^2)-c)}.$$Let $A=a$. Then $a'=c+\frac{r^2}{\overline{a}-c}$ and hence $$b=\frac{1}{\overline{a}}\quad\text{and}\quad b'=\frac{a-c}{ac-c^2+r^2}.$$The inverse of $b$ in $w'$ is given by $$\frac{1}{c^2-r^2}\left(c+\frac{r^2}{\overline{b}(c^2-r^2)-c}\right)=\frac{1}{c^2-r^2}\left(c+\frac{ar^2}{c^2-r^2-ac}\right)=\frac{c-a}{c^2-r^2-ac}=b',$$as desired.
This post has been edited 1 time. Last edited by sami1618, Jun 27, 2025, 10:35 PM
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Vivouaf
70 posts
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Here's a less calculatory solution :
Adjusting the radius of $\Omega$, we can suppose $\omega$ is mapped onto itself (this is possible since adjusting the radius is the same as doing a homothety at $P$ center of $\Omega$). We still have $(ABA'B')$, and we only need to show $O-A'-B'$ where $O$ is te center of $\omega$. Denote by $P'$ the inverse of $P$ wrt $\omega$. Then it's well known that $PP' \cdot PO = \mathbb{P}_\omega(P)$ (sketch of proof : introducing one tangent from $P$ and using Pythagore), and since $O-A-B$, we have $(PP'A'B')$ and $(OP'A'A)$ after inverting at $P$. Now, $\widehat{OA'P'} = \widehat{OAP'} = \widehat{BPP'} = \widehat{B'A'P'}$, thus $O-A'-B'$ and since $ABA'B'$ is cyclic, we indeed have $OA' \cdot OB' = OA \cdot OB$, i. e. $A'$ and $B'$ are inverses wrt $\omega$, as desired.
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