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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
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
May 1, 2025
0 replies
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Orthocenter
jayme   4
N 6 minutes ago by Sadigly
Dear Mathlinkers,

1. ABC an acuatangle triangle
2. H the orthcenter of ABC
3. DEF the orthic triangle of ABC
4. A* the midpoint of AH
5. X the point of intersection of AH and EF.

Prove : X is the orthocenter of A*BC.

Sincerely
Jean-Louis
4 replies
jayme
Mar 25, 2015
Sadigly
6 minutes ago
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Concurrency
Dadgarnia   29
N 8 minutes ago by blueprimes
Source: Iranian TST 2020, second exam day 2, problem 4
Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent.

Proposed by Alireza Dadgarnia
29 replies
Dadgarnia
Mar 12, 2020
blueprimes
8 minutes ago
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Good Numbers
ilovemath04   30
N 22 minutes ago by ihategeo_1969
Source: ISL 2019 N5
Let $a$ be a positive integer. We say that a positive integer $b$ is $a$-good if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.
30 replies
ilovemath04
Sep 22, 2020
ihategeo_1969
22 minutes ago
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Sequences problem
BBNoDollar   1
N 42 minutes ago by BBNoDollar
Source: Mathematical Gazette Contest
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
1 reply
BBNoDollar
6 hours ago
BBNoDollar
42 minutes ago
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s(I)=2019
math90   8
N 3 hours 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
3 hours ago
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Cauchy's functional equation with f({max{x,y})=max{f(x),f(y)}
tom-nowy   1
N 5 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
5 hours ago
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A problem in point set topology
tobylong   1
N Today at 4:59 PM 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
1 viewing
tobylong
Today at 3:14 AM
alexheinis
Today at 4:59 PM
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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
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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
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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
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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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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
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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
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Combinatorial Sum
P162008   1
N Today at 12:56 PM by Mathzeus1024
Evaluate $\sum_{n=0}^{\infty} \frac{2^n + 1}{(2n + 1) \binom{2n}{n}}$
1 reply
P162008
Apr 24, 2025
Mathzeus1024
Today at 12:56 PM
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Combinatorial Sum
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P162008
175 posts
P162008
#1 h  Apr 24, 2025, 1:02 AM
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Evaluate $\sum_{n=0}^{\infty} \frac{2^n + 1}{(2n + 1) \binom{2n}{n}}$
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Mathzeus1024
862 posts
Mathzeus1024
#2 h  Today at 12:56 PM
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$\sum_{n=0}^{\infty} \frac{2^n + 1}{(2n + 1) \binom{2n}{n}} = \sum_{n=0}^{\infty} \frac{2^n + 1}{\binom{2n}{n}}\cdot \int_{0}^{1} x^{2n} dx = \int_{0}^{1} \sum_{n=0}^{\infty} \frac{(\sqrt{2}x)^{2n} + x^{2n}}{\binom{2n}{n}} dx$;

or $\int_{0}^{1} \frac{4x \cdot \arcsin(x/2) +4\sqrt{4-x^2}}{(4-x^2)^{3/2}} + \frac{2x \cdot \arcsin(x/\sqrt{2})+2\sqrt{2-x^2}}{(2-x^2)^{3/2}} dx$;

or $\frac{4\arcsin(x/2)}{\sqrt{4-x^2}} + \frac{2\arcsin(x/\sqrt{2})}{\sqrt{2-x^2}}|_{0}^{1} = \textcolor{red}{\frac{(9+4\sqrt{3})\pi}{18}}$.
This post has been edited 5 times. Last edited by Mathzeus1024, Today at 1:10 PM
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