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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Thursday 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
Thursday at 11:16 PM
0 replies
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Does the sequence log(1+sink)/k converge?
tom-nowy   6
N 7 minutes ago by Dattier
Source: Question arising while viewing https://artofproblemsolving.com/community/c7h3556569
Does the sequence $$ \frac{\ln(1+\sin k)}{k} \;\;\;(k=1,2,3,\ldots) $$converge?
6 replies
tom-nowy
Apr 30, 2025
Dattier
7 minutes ago
J
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Rolles theorem
sasu1ke   0
17 minutes ago

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that
\[ f(0) = 2, \quad f'(0) = -2, \quad \text{and} \quad f(1) = 1. \]Prove that there exists a point \( \xi \in (0, 1) \) such that
\[ f(\xi) \cdot f'(\xi) + f''(\xi) = 0. \]

0 replies
sasu1ke
17 minutes ago
0 replies
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s(I)=2019
math90   8
N 37 minutes ago by MathSaiyan
Source: IMC 2019 Day 2 P8
Let $x_1,\ldots,x_n$ be real numbers. For any set $I\subset\{1,2,…,n\}$ let $s(I)=\sum_{i\in I}x_i$. Assume that the function $I\to s(I)$ takes on at least $1.8^n$ values where $I$ runs over all $2^n$ subsets of $\{1,2,…,n\}$. Prove that the number of sets $I\subset \{1,2,…,n\}$ for which $s(I)=2019$ does not exceed $1.7^n$.

Proposed by Fedor Part and Fedor Petrov, St. Petersburg State University
8 replies
math90
Jul 31, 2019
MathSaiyan
37 minutes ago
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Cauchy's functional equation with f({max{x,y})=max{f(x),f(y)}
tom-nowy   1
N 2 hours ago by Filipjack
Source: https://x.com/D_atWork/status/1788496152855560470, Problem 4
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying the following two conditions for all $x,y \in \mathbb{R}$:
\[ f(x+y)=f(x)+f(y), \;\;\; f \left( \max \{x, y \} \right) = \max \left\{ f(x),f(y) \right\}. \]
1 reply
tom-nowy
Today at 2:23 PM
Filipjack
2 hours ago
J
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A problem in point set topology
tobylong   1
N 4 hours ago by alexheinis
Source: Basic Topology, Armstrong
Let $f:X\to Y$ be a closed map with the property that the inverse image of each point in $Y$ is a compact subset of $X$. Prove that $f^{-1}(K)$ is compact whenever $K$ is compact in $Y$.
1 reply
tobylong
Today at 3:14 AM
alexheinis
4 hours ago
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D1026 : An equivalent
Dattier   0
Today at 1:39 PM
Source: les dattes à Dattier
Let $u_0=1$ and $\forall n \in \mathbb N, u_{2n+1}=\ln(1+u_{2n}), u_{2n+2}=\sin(u_{2n+1})$.

Find an equivalent of $u_n$.
0 replies
Dattier
Today at 1:39 PM
0 replies
J
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Summation
Saucepan_man02   2
N Today at 12:57 PM by Etkan
If $P = \sum_{r=1}^{50} \sum_{k=1}^{r} (-1)^{r-1} \frac{\binom{50}{r}}{k}$, then find the value of $P$.

Ans
2 replies
Saucepan_man02
Today at 9:40 AM
Etkan
Today at 12:57 PM
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sequences, n-sum of type 1/2
jasperE3   1
N Today at 11:38 AM by pi_quadrat_sechstel
Source: Putnam 1991 A6
An $n$-sum of type $1$ is a finite sequence of positive integers $a_1,a_2,\ldots,a_r$, such that:
$(1)$ $a_1+a_2+\ldots+a_r=n$;
$(2)$ $a_1>a_2+a_3,a_2>a_3+a_4,\ldots, a_{r-2}>a_{r-1}+a_r$, and $a_{r-1}>a_r$. For example, there are five $7$-sums of type $1$, namely: $7$; $6,1$; $5,2$; $4,3$; $4,2,1$. An $n$-sum of type $2$ is a finite sequence of positive integers $b_1,b_2,\ldots,b_s$ such that:
$(1)$ $b_1+b_2+\ldots+b_s=n$;
$(2)$ $b_1\ge b_2\ge\ldots\ge b_s$;
$(3)$ each $b_i$ is in the sequence $1,2,4,\ldots,g_j,\ldots$ defined by $g_1=1$, $g_2=2$, $g_j=g_{j-1}+g_{j-2}+1$; and
$(4)$ if $b_1=g_k$, then $1,2,4,\ldots,g_k$ is a subsequence. For example, there are five $7$-sums of type $2$, namely: $4,2,1$; $2,2,2,1$; $2,2,1,1,1$; $2,1,1,1,1,1$; $1,1,1,1,1,1,1$. Prove that for $n\ge1$ the number of type $1$ and type $2$ $n$-sums is the same.
1 reply
jasperE3
Aug 20, 2021
pi_quadrat_sechstel
Today at 11:38 AM
J
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Equation of Matrices which have same rank
PureRun89   3
N Today at 8:51 AM by pi_quadrat_sechstel
Source: Gazeta Mathematica

Let $A,B \in \mathbb{C}_{n \times n}$ and $rank(A)=rank(B)$.
Given that there exists positive integer $k$ such that
$$A^{k+1} B^k=A.$$Prove that
$$B^{k+1} A^k=B.$$
(Note: The submition of the problem is end so I post this)
3 replies
PureRun89
May 18, 2023
pi_quadrat_sechstel
Today at 8:51 AM
J
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D1023 : MVT 2.0
Dattier   1
N Today at 7:55 AM by Dattier
Source: les dattes à Dattier
Let $f \in C(\mathbb R)$ derivable on $\mathbb R$ with $$\forall x \in \mathbb R,\forall h \geq 0, f(x)-3f(x+h)+3f(x+2h)-f(x+3h) \geq 0$$
Is it true that $$\forall (a,b) \in\mathbb R^2, |f(a)-f(b)|\leq \max\left(\left|f'\left(\dfrac{a+b} 2\right)\right|,\dfrac {|f'(a)+f'(b)|}{2}\right)\times |a-b|$$
1 reply
Dattier
Apr 29, 2025
Dattier
Today at 7:55 AM
J
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Equivalent condition of the uniformly continuous fo a function
Alphaamss   0
Today at 7:35 AM
Source: Personal
Let $f_{a,b}(x)=x^a\cos(x^b),x\in(0,\infty)$. Get all the $(a,b)\in\mathbb R^2$ such that $f_{a,b}$ is uniformly continuous on $(0,\infty)$.
0 replies
Alphaamss
Today at 7:35 AM
0 replies
J
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Find all continuous functions
bakkune   2
N Today at 7:22 AM by bakkune
Source: Own
Find all continuous function $f, g\colon\mathbb{R}\to\mathbb{R}$ satisfied
$$ (x - k)f(x) = \int_k^x g(y)\mathrm{d}y $$for all $x\in\mathbb{R}$ and all $k\in\mathbb{Z}$.
2 replies
bakkune
Today at 6:02 AM
bakkune
Today at 7:22 AM
J
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ISI 2019 : Problem #2
integrated_JRC   40
N Today at 6:56 AM by Sammy27
Source: I.S.I. 2019
Let $f:(0,\infty)\to\mathbb{R}$ be defined by $$f(x)=\lim_{n\to\infty}\cos^n\bigg(\frac{1}{n^x}\bigg)$$(a) Show that $f$ has exactly one point of discontinuity.
(b) Evaluate $f$ at its point of discontinuity.
40 replies
integrated_JRC
May 5, 2019
Sammy27
Today at 6:56 AM
J
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Cube Colouring Problems
Saucepan_man02   0
Today at 6:44 AM
Could anyone kindly post some problems (and hopefully along the solution thread/final answer) related to combinatorial colouring of cube?
0 replies
Saucepan_man02
Today at 6:44 AM
0 replies
J
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J
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Integrable function: + and - on every subinterval.
SPQ   3
N Apr 23, 2025 by solyaris
Provide a function integrable on [a, b] such that f takes on positive and negative values on every subinterval (c, d) of [a, b]. Prove your function satisfies both conditions.
3 replies
SPQ
Apr 23, 2025
solyaris
Apr 23, 2025
J
Integrable function: + and - on every subinterval.
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SPQ
5 posts
SPQ
#1 h  Apr 23, 2025, 2:40 AM
Y by
Provide a function integrable on [a, b] such that f takes on positive and negative values on every subinterval (c, d) of [a, b]. Prove your function satisfies both conditions.
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greenturtle3141
3557 posts
greenturtle3141
#2 h  Apr 23, 2025, 4:26 AM
Y by
Really depends what you mean by "integrable", but for the lebesgue case the characteristic function on rationals works just fine
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SPQ
5 posts
SPQ
#3 h  Apr 23, 2025, 5:07 AM
Y by
Thanks for the suggestion. You are right that the characteristic function of the rationals in [a, b] is Lebesgue integrable. However, the characteristic function doesn't satisfy the second condition: it doesn't take on negative values anywhere; it only takes on the values 1 and 0.

Also, just to clarify, I was referring to Riemann integrability.
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solyaris
632 posts
solyaris
#4 h  Apr 23, 2025, 7:06 AM
Y by
@above: But of course you can slightly modify the charactersistic function (so that it takes values $1,-1$ instead of $1,0$) to give a counterexample.

For Riemann integrability you only have to introduce a further slight modification: Take $f(x) = 0$ for irrational $x$, and for rational $x = \frac k n$
with integer $k,n$ and $n > 0$ in reduced form, define $f(\frac k n) = -\frac 1 n$ for odd $n$ and $f(\frac k n) = \frac 1 n$ for even $n$. It is not hard to see that $f$ is Riemann-integrable and takes positive and negative values in any open interval.
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