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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
1 viewing
jlacosta
Yesterday at 11:16 PM
0 replies
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Queue geo
vincentwant   5
N 16 minutes ago by Ilikeminecraft
Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $Y, Z$ be the feet of the altitudes from $B, C$ to $AC, AB$ respectively. Let $D$ be the midpoint of $BC$. Let $\omega_1$ be the circle with diameter $AD$. Let $Q\neq A$ be the intersection of $(ABC)$ and $\omega$. Let $H$ be the orthocenter of $ABC$. Let $K$ be the intersection of $AQ$ and $BC$. Let $l_1,l_2$ be the lines through $Q$ tangent to $\omega,(AYZ)$ respectively. Let $I$ be the intersection of $l_1$ and $KH$. Let $P$ be the intersection of $l_2$ and $YZ$. Let $l$ be the line through $I$ parallel to $HD$ and let $O'$ be the reflection of $O$ across $l$. Prove that $O'P$ is tangent to $(KPQ)$.
5 replies
+1 w
vincentwant
Wednesday at 3:54 PM
Ilikeminecraft
16 minutes ago
J
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A sequence containing every natural number exactly once
Pomegranat   3
N 27 minutes ago by YaoAOPS
Source: Own
Does there exist a sequence \( \{a_n\}_{n=1}^{\infty} \), which is a permutation of the natural numbers (that is, each natural number appears exactly once), such that for every \( n \in \mathbb{N} \), the sum of the first \( n \) terms is divisible by \( n \)?
3 replies
Pomegranat
2 hours ago
YaoAOPS
27 minutes ago
J
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An inequality with n of Maximum with Minimum
Qing-Cloud   2
N 27 minutes ago by Qing-Cloud
Let $ n $ be a positive integer divisible by 3, and let $ x_1, x_2, \dots, x_n $ be non-negative real numbers satisfying $ \sum_{i=1}^n x_i = 1 $. Define
\[ M_n = \min_{1 \le i \le n} \{x_i x_{3i}\}, \]where the indices are taken modulo $ n $. Determine the maximum possible value of $ M_n $.
2 replies
Qing-Cloud
Yesterday at 2:18 PM
Qing-Cloud
27 minutes ago
J
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Iranian Second Round P2
AlirezaOpmc   10
N an hour ago by Lemmas
Source: Iran 2nd-round MO 2018 - P2
Let $n$ be odd natural number and $x_1,x_2,\cdots,x_n$ be pairwise distinct numbers. Prove that someone can divide the difference of these number into two sets with equal sum.
( $X=\{\mid x_i-x_j \mid | i<j\}$ )
10 replies
AlirezaOpmc
Apr 26, 2018
Lemmas
an hour ago
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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 4 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
4 hours ago
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Miklós Schweitzer 1956- Problem 8
Coulbert   2
N 4 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
4 hours ago
J
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Very straightforward linear recurrence
Assassino9931   1
N 4 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
4 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 6 hours ago 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
6 hours ago
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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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Determine all the functions
Martin.s   2
N Apr 28, 2025 by Blackbeam999


Determine all the functions $f: \mathbb{R} \to \mathbb{R}$ such that

\[ f(x^2 \cdot f(x) + f(y)) = f(f(x^3)) + y \]
for all $x, y \in \mathbb{R}$.


2 replies
Martin.s
Aug 14, 2024
Blackbeam999
Apr 28, 2025
J
Determine all the functions
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Martin.s
1536 posts
Martin.s
#1 h  Aug 14, 2024, 7:22 PM
Y by
Determine all the functions $f: \mathbb{R} \to \mathbb{R}$ such that

\[ f(x^2 \cdot f(x) + f(y)) = f(f(x^3)) + y \]
for all $x, y \in \mathbb{R}$.
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RagvaloD
4912 posts
RagvaloD
#2 h  Aug 14, 2024, 7:30 PM • 1 Y
Y by Martin.s
$x=0: f(f(y))=y$
So equation is $f(x^2f(x)+f(y))=x^3+y$ or $f(x^3+y)=x^2f(x)+f(y)$
$y=0:f(x^3)=x^2f(x)+f(0)$ so $f(x+y)=f(x)+f(y)-f(0)$ for every $x,y$
$x^3+y=f(x^2f(x)+f(y))=f(x^2f(x))+f(y)-f(0)$
$x=0: f(y)=y$
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Blackbeam999
16 posts
Blackbeam999
#3 h  Apr 28, 2025, 11:22 AM
Y by
RagvaloD wrote:
$x=0: f(f(y))=y$
So equation is $f(x^2f(x)+f(y))=x^3+y$ or $f(x^3+y)=x^2f(x)+f(y)$
$y=0:f(x^3)=x^2f(x)+f(0)$ so $f(x+y)=f(x)+f(y)-f(0)$ for every $x,y$
$x^3+y=f(x^2f(x)+f(y))=f(x^2f(x))+f(y)-f(0)$
$x=0: f(y)=y$

In last step it should be f(x^2f(x))+f(f(y))-f(0) not f(x^2f(x))+f(y)-f(0)
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