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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 11:16 PM
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
Yesterday at 11:16 PM
0 replies
J
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Holomorphic function
Sifan.C.Maths   4
N an hour ago by Sifan.C.Maths
Source: m exercise
Is there a complex function $f$ such that $f$ satisfies two following statements?
(i) f is holomorphic on a domain $\Omega$ which contains $z=0$.
(ii) $f(\dfrac{1}{n})=0$ if $n$ is an odd natural number, $f(\dfrac{1}{n})=2$ if $n$ is an even natural number ($n$ is different from 0).
4 replies
Sifan.C.Maths
2 hours ago
Sifan.C.Maths
an hour ago
J
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Number of roots of boundary preserving unit disk maps
Assassino9931   1
N an hour ago by Alphaamss
Source: Vojtech Jarnik IMC 2025, Category II, P4
Let $D = \{z\in \mathbb{C}: |z| < 1\}$ be the open unit disk in the complex plane and let $f : D \to D$ be a holomorphic function such that $\lim_{|z|\to 1}|f(z)| = 1$. Let the Taylor series of $f$ be $f(z) = \sum_{n=0}^{\infty} a_nz^n$. Prove that the number of zeroes of $f$ (counted with multiplicities) equals $\sum_{n=0}^{\infty} n|a_n|^2$.
1 reply
Assassino9931
5 hours ago
Alphaamss
an hour ago
J
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Holomorphic problem
Sifan.C.Maths   2
N 2 hours ago by Sifan.C.Maths
Source: my exercise
Give a fuction $f(z)=\dfrac{1}{z^{2021}}$. Prove that for all polynomial $P(z)$ on $\mathbb{C}$, we have
$$\max_{|z|=1}|f(z) -P(z)| \ge 1.$$Can I change 1 by a constant $c > 1$?
2 replies
Sifan.C.Maths
Yesterday at 4:20 PM
Sifan.C.Maths
2 hours ago
J
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maximum on the circle
Sifan.C.Maths   0
2 hours ago
Source: exercise
Let $f$ be holomorphic on $|z|>1$, continuous on $|z|\ge1$ and such that the limit $\lim_{z\to\infty} f(z) = a$ exists. Prove that $|f(z)|$ attains its maximum on the circle $|z|=1$.
0 replies
Sifan.C.Maths
2 hours ago
0 replies
J
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System of two matrices of the same rank
Assassino9931   1
N 2 hours ago by RobertRogo
Source: Vojtech Jarnik IMC 2025, Category II, P2
Let $A,B$ be two $n\times n$ complex matrices of the same rank, and let $k$ be a positive integer. Prove that $A^{k+1}B^k = A$ if and only if $B^{k+1}A^k = B$.
1 reply
Assassino9931
5 hours ago
RobertRogo
2 hours ago
J
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Putnam 1952 B1
centslordm   4
N 5 hours ago by Gauler
A mathematical moron is given two sides and the included angle of a triangle and attempts to use the Law of Cosines: $a^2 = b^2 + c^2 - 2bc \cos A,$ to find the third side $a.$ He uses logarithms as follows. He finds $\log b$ and doubles it; adds to that the double of $\log c;$ subtracts the sum of the logarithms of $2, b, c,$ and $\cos A;$ divides the result by $2;$ and takes the anti-logarithm. Although his method may be open to suspicion his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?
4 replies
centslordm
May 30, 2022
Gauler
5 hours ago
J
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Miklós Schweitzer 1956- Problem 8
Coulbert   2
N 5 hours ago by izzystar
8. Let $(a_n)_{n=1}^{\infty}$ be a sequence of positive numbers and suppose that $\sum_{n=1}^{\infty} a_n^2$ is divergent. Let further $0<\epsilon<\frac{1}{2}$. Show that there exists a sequence $(b_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}b_n^2$ is convergent and

$\sum_{n=1}^{N}a_n b_n >(\sum_{n=1}^{N}a_n^2)^{\frac{1}{2}-\epsilon}$

for every positive integer $N$. (S. 8)
2 replies
Coulbert
Oct 11, 2015
izzystar
5 hours ago
J
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Very straightforward linear recurrence
Assassino9931   1
N 5 hours ago by Etkan
Source: Vojtech Jarnik IMC 2025, Category II, P1
Let $x_0=a, x_1= b, x_2 = c$ be given real numbers and let $x_{n+2} = \frac{x_n + x_{n-1}}{2}$ for all $n\geq 1$. Show that the sequence $(x_n)_{n\geq 0}$ converges and find its limit.
1 reply
Assassino9931
5 hours ago
Etkan
5 hours ago
J
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Integration Bee in Czechia
Assassino9931   0
5 hours ago
Source: Vojtech Jarnik IMC 2025, Category II, P3
Evaluate the integral $\int_0^{\infty} \frac{\log(x+2)}{x^2+3x+2}\mathrm{d}x$.
0 replies
Assassino9931
5 hours ago
0 replies
J
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Trace with minimal polynomial x^n + x - 1
Assassino9931   0
5 hours ago
Source: Vojtech Jarnik IMC 2025, Category I, P4
Let $A$ be an $n\times n$ real matrix with minimal polynomial $x^n + x - 1$. Prove that the trace of $(nA^{n-1} + I)^{-1}A^{n-2}$ is zero.
0 replies
Assassino9931
5 hours ago
0 replies
J
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Fast-growing sequences
Assassino9931   0
5 hours ago
Source: Vojtech Jarnik IMC 2025, Category I, P3
Let us call a sequence $(b_1, b_2, \ldots)$ of positive integers fast-growing if $b_{n+1} \geq b_n + 2$ for all $n \geq 1$. Also, for a sequence $a = (a(1), a(2), \ldots)$ of real numbers and a sequence $b = (b_1, b_2, \ldots)$ of positive integers, let us denote
\[ S(a, b) = \sum_{n=1}^{\infty} \left| a(b_n) + a(b_n + 1) + \cdots + a(b_{n+1} - 1) \right|. \]
a) Do there exist two fast-growing sequences $b = (b_1, b_2, \ldots)$, $c = (c_1, c_2, \ldots)$ such that for every sequence $a = (a(1), a(2), \ldots)$, if all the series
\[ \sum_{n=1}^{\infty} a(n), \quad S(a, b) \quad \text{and} \quad S(a, c) \]are convergent, then the series $\sum_{n=1}^{\infty} |a(n)|$ is also convergent?

b) Do there exist three fast-growing sequences $b = (b_1, b_2, \ldots)$, $c = (c_1, c_2, \ldots)$, $d = (d_1, d_2, \ldots)$ such that for every sequence $a = (a(1), a(2), \ldots)$, if all the series
\[ S(a, b), \quad S(a, c) \quad \text{and} \quad S(a, d) \]are convergent, then the series $\sum_{n=1}^{\infty} |a(n)|$ is also convergent?
0 replies
Assassino9931
5 hours ago
0 replies
J
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Strange ring property
sapience   2
N Yesterday at 11:48 PM by RobertRogo
Let \( (A, +, \cdot) \) be a ring with \( Z(A) \) its centre (\( Z = \{ x \in A \mid xy = yx \text{ for any } x, y \in A \} \)), \( U(A) \) the set of invertible elements and \( A^* = A \setminus \{0\} \).
We will say \(A\) has the property \( \mathcal{P} \) if there exists a subgroup \(H \) of group \( (U(A), \cdot) \) such that \( H \subset Z(A) \), \( H \neq A^* \) and \( x^3 = y^3 \) for any \( x, y \in A^* \setminus H \).
Prove the following:
a) any ring with property \( \mathcal{P} \) is commutative;
b) if \(A \) has the property \( \mathcal{P} \), then \( x^3 = 0 \), for any \( x \in A \setminus U(A) \).

Note: \(0 \) and \(1 \) are the identity elements for \(+ \) and \(\cdot \)
2 replies
sapience
Mar 5, 2025
RobertRogo
Yesterday at 11:48 PM
J
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real analysis
ILOVEMYFAMILY   1
N Yesterday at 5:04 PM by Alphaamss
Source: real analysis
For which value of $p\in\mathbb{R}$ does the series $$\sum_{n=1}^{\infty}\ln \left(1+\frac{(-1)^n}{n^p}\right)$$converge (and absolutely converge)?
1 reply
ILOVEMYFAMILY
Dec 2, 2023
Alphaamss
Yesterday at 5:04 PM
J
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Putnam 1962 B5
sqrtX   4
N Yesterday at 4:56 PM by centslordm
Source: Putnam 1962
Prove that for every integer $n$ greater than $1:$
$$\frac{3n+1}{2n+2} < \left( \frac{1}{n} \right)^{n} + \left( \frac{2}{n} \right)^{n}+ \ldots+\left( \frac{n}{n} \right)^{n} <2.$$
4 replies
sqrtX
May 21, 2022
centslordm
Yesterday at 4:56 PM
J
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J
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Prove this recursion!
Entrepreneur   3
N Apr 20, 2025 by quasar_lord
Source: Amit Agarwal
Let $$I_n=\int z^n e^{\frac 1z}dz.$$Prove that $$\color{blue}{I_n=(n+1)!I_0+e^{\frac 1z}\sum_{n=1}^n n! z^{n+1}.}$$
3 replies
Entrepreneur
Jul 31, 2024
quasar_lord
Apr 20, 2025
J
Prove this recursion!
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Entrepreneur
1170 posts
Entrepreneur
#1 h  Jul 31, 2024, 12:27 PM
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Let $$I_n=\int z^n e^{\frac 1z}dz.$$Prove that $$\color{blue}{I_n=(n+1)!I_0+e^{\frac 1z}\sum_{n=1}^n n! z^{n+1}.}$$
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Evan484Ashoori
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Evan484Ashoori
#2 h  Aug 3, 2024, 3:23 PM
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sangsidhya
#3 h  Apr 20, 2025, 5:59 AM
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is he the author of arihant series?
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quasar_lord
241 posts
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#4 h  Apr 20, 2025, 11:06 AM
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