Two circles are tangents in a triangles with angle 60

Let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be two sequences of positive numbers. Show that the following statements are equivalent:
[list=1]
[*]There is a sequence $(c_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}{\frac{a_n}{c_n}}$ and $\sum_{n=1}^{\infty}{\frac{c_n}{b_n}}$ both converge;[/*]
[*] $\sum_{n=1}^{\infty}{\sqrt{\frac{a_n}{b_n}}}$ converges.[/*]
[/list]

Proposed by Tomáš Bárta, Charles University, Prague

3 replies

ThE-dArK-lOrD

Jul 24, 2018

Fibonacci_math

3 hours ago

Infinite series involving tau function

bakkune 1

N 3 hours ago by Safal

For each positive integer $n$ , let $\tau(n)$ be the number of positive divisors of $n$ . Evaluate
$\sum_{n=1}^{+\infty} (-1)^n \frac{\tau(n)}{n}$

1 reply

bakkune

Today at 4:35 AM

Safal

3 hours ago

two solutions

τρικλινο 4

N 5 hours ago by Safal

in a book:CORE MATHS for A-LEVEL ON PAGE 41 i found the following

1st solution

$x^2-5x=0$

$x(x-5)=0$

hence x=0 or x=5

2nd solution

$x^2-5x=0$

$x-5=0$ dividing by x

hence the solution x=0 has been lost

is the above correct?

4 replies

τρικλινο

Yesterday at 6:20 PM

Safal

5 hours ago

high school math

aothatday 5

N Today at 3:32 AM by aothatday

Let $x_n$ be a positive root of the equation $x_n^n=x^2+x+1$ . Prove that the following sequence converges: $n^2(x_n-x_{ n+1})$

5 replies

aothatday

Apr 10, 2025

aothatday

Today at 3:32 AM

Putnam 2003 B1

btilm305 13

N Today at 12:08 AM by clarkculus

Do there exist polynomials $a(x)$ , $b(x)$ , $c(y)$ , $d(y)$ such that $1 + xy + x^2y^2= a(x)c(y) + b(x)d(y)$ holds identically?

13 replies

btilm305

Jun 23, 2011

clarkculus

Today at 12:08 AM

High School Integration Extravaganza Problem Set

Riemann123 12

N Yesterday at 5:20 PM by jkim0656

Source: River Hill High School Spring Integration Bee

Hello AoPS!

Along with user geodash2, I have organized another high-school integration bee (River Hill High School Spring Integration Bee) and wanted to share the problems!

We had enough folks for two concurrent rooms, hence the two sets. (ARML kids from across the county came.)

Keep in mind that these integrals were written for a high-school contest-math audience. I hope you find them enjoyable and insightful; enjoy!

[center]Warm Up Problems[/center]
$\int_{1}^{2} \frac{x^{3}+x^2}{x^5}dx$ $\int_{2025}^{2025^{2025}}\frac{1}{\ln\left(2025\right)\cdot x}dx$ $\int(\sin^2(x)+\cos^2(x)+\sec^2(x)+\csc^2(x))dx$ $\int_{-2025.2025}^{2025.2025}\sin^{2025}(2025x)\cos^{2025}(2025x)dx$ $\int_{\frac \pi 6}^{\frac \pi 3} \tan(\theta)^2d\theta$ $\int \frac{1+\sqrt{t}}{1+t}dt$ -----
[center]Easier Division Set 1[/center]
$\int \frac{x^{2}+2x+1}{x^{3}+3x^{2}+3x+3}dx$ $\int_{0}^{\frac{3\pi}{2}}\left(\frac{\pi}{2}-x\right)\sin\left(x\right)dx$ $\int_{-\pi/2}^{\pi/2}x^3e^{-x^2}\cos(x^2)\sin^2(x)dx$ $\int\frac{1}{\sqrt{12-t^{2}+4t}}dt$ $\int \frac{\sqrt{e^{8x}}}{e^{8x}-1}dx$ -----
[center]Easier Division Set 2[/center]
$\int \frac{e^x}{e^{2x}+1} dx$ $\int_{-5}^5\sqrt{25-u^2}du$ $\int_{-\frac12}^\frac121+x+x^2+x^3\ldots dx$ $\int \cos(\cos(\cos(\ln \theta)))\sin(\cos(\ln \theta))\sin(\ln \theta)\frac{1}{\theta}d\theta$ $\int_{0}^{\frac{1}{6}}\frac{8^{2x}}{64^{2x}-8^{\left(2x+\frac{1}{3}\right)}+2}dx$ -----
[center]Harder Division Set 1[/center]
$\int_{0}^{\frac{\pi}{2}}\frac{\sin\left(x\right)}{\sin\left(x\right)+\cos\left(x\right)}+\frac{\sin\left(\frac{\pi}{2}-x\right)}{\sin\left(\frac{\pi}{2}-x\right)+\cos\left(\frac{\pi}{2}-x\right)}dx$ $\int_0^{\infty}e^{-x}\Bigl(\cos(20x)+\sin(20x)\Bigr) dx$ $\lim_{n\to \infty}\frac{1}{n}\int_{1}^{n}\sin(nt)^2dt$ $\int_{x=0}^{x=1}\left( \int_{y=-x}^{y=x} \frac{y^2}{x^2+y^2}dy\right)dx$ $\int_{0}^{13}\left\lceil\log_{10}\left(2^{\lceil x\rceil }x\right)\right\rceil dx$ -----
[center]Harder Division Set 2[/center]
$\int \frac{6x^2}{x^6+2x^3+2}dx$ $\int -\sin(2\theta)\cos(\theta)d\theta$ $\int_{0}^{5}\sin(\frac{\pi}2 \lfloor{x}\rfloor x) dx$ $\int_{0}^{1} \frac{\sin^{-1}(\sqrt{x})^2}{\sqrt{x-x^2}}dx$ $\int\left(\cot(\theta)+\tan(\theta)\right)^2\cot(2\theta)^{100}d\theta$ -----
[center]Bonanza Round (ie Fun/Hard/Weird Problems) (In No Particular Order)[/center]
$\int \ln\left\{\sqrt[7]{x}^\frac1{\ln\left\{\sqrt[5]{x}^\frac1{\ln\left\{\sqrt[3]{x}^\frac1{\ln\left\{\sqrt{x}\right\}}\right\}}\right\}}\right\}dx$ $\int_{1}^{{e}^{\pi}} \cos(\ln(\sqrt{u}))du$ $\int_e^{\infty}\frac {1-x\ln{x}}{xe^x}dx$ $\int_{0}^{1}\frac{e^{x}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{1}}}}\times\frac{e^{-\frac{x^{2}}{2}}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{2}}}}\times\frac{e^{\frac{x^{3}}{3}}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{3}}}}\times\frac{e^{-\frac{x^{4}}{4}}}{\left(x^{2}+3x+2\right)^{\frac{1}{2^{4}}}} \ldots \,dx$
For $x$ on the domain $-0.2025\leq x\leq 0.2025$ it is known that $\displaystyle f(x)=\sin\left(\int_{0}^x \sqrt[3]{\cos\left(\frac{\pi}{2} t\right)^3+26}\ dt\right)$ is invertible. What is $\displaystyle (f^{-1})'(0)$ ?

12 replies

Riemann123

Friday at 2:11 PM

jkim0656

Yesterday at 5:20 PM

limiting behavior of the generalization of IMO 1968/6 for arbitrary powers

revol_ufiaw 1

N Yesterday at 3:17 PM by alexheinis

Source: inspired by IMO 1968/6

Define $f : \mathbb{N} \rightarrow \mathbb{N}$ by
$f(n) = \sum_{i\ge 0} \bigg\lfloor \frac{n + a^i}{a^{i+1}}\bigg\rfloor=\bigg\lfloor \frac{n + 1}{a} \bigg\rfloor + \bigg\lfloor \frac{n + a}{a^2} \bigg\rfloor + \bigg\lfloor \frac{n + a^2}{a^3} \bigg\rfloor + \cdots$ for some fixed $a \in \mathbb{N}$ . Prove that
$\lim_{n \rightarrow \infty} \frac{f(n)}{n/(a-1)} = 1.$
[P.S.: IMO 1968/6 asks to prove $f(n) = n$ for $a = 2$ .]

1 reply

revol_ufiaw

Yesterday at 1:43 PM

alexheinis

Yesterday at 3:17 PM

Putnam 2015 B4

Kent Merryfield 22

N Yesterday at 2:58 PM by lpieleanu

Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c.$ Express $\sum_{(a,b,c)\in T}\frac{2^a}{3^b5^c}$ as a rational number in lowest terms.

22 replies

Kent Merryfield

Dec 6, 2015

lpieleanu

Yesterday at 2:58 PM

A real analysis Problem from contest

Safal 2

N Yesterday at 11:17 AM by Safal

Source: Random.

Let $f: (0,\infty)\rightarrow \mathbb{R}$ be a function such that $\lim_{x\rightarrow \infty} f(x)=1$ and $f(x+1)=f(x)$ for all $x\in (0,\infty)$

Prove or disprove the following statements.

1. $f$ is continuous.
2. $f$ is bounded.

Is My Idea correct?

Suppose $f(a)\neq f(b)$ for some $a,b$ in domain then also we know second condition given in problem implies $f(a+n)=f(a)$ for all $n\in\mathbb{N}$ . By first condition $\lim_{n\rightarrow\infty} f(a+n)=1=f(a)$ . Similarly, $f(b)=1$ . Contradicting the assumption.
$f(a)\neq f(b)$ . Thus we must have $f(a)=f(b)=1$ for all $a,b$ in domain.Proving both the statements.

2 replies

Safal

Yesterday at 8:34 AM

Safal

Yesterday at 11:17 AM

maximal determinant

EthanWYX2009 4

N Yesterday at 10:59 AM by loup blanc

Source: 2023 Aug taca-9

Let matrix
$A=\begin{bmatrix} 1&1&1&1&1\\1&-1&1&-1&1\\?&?&?&?&?\\?&?&?&?&?\\?&?&?&?&?\end{bmatrix}\in\mathbb R^{5\times 5}$ satisfy $\text{tr} (AA^T)=28.$ Determine the maximum value of $\det A.$

4 replies

EthanWYX2009

Apr 9, 2025

loup blanc

Yesterday at 10:59 AM

Two circles are tangents in a triangles with angle 60

nAalniaOMliO 1

N Mar 29, 2025 by sunken rock

Source: Belarusian National Olympiad 2025

In a triangle $ABC$ angle $\angle BAC = 60^{\circ}$ . Point $M$ is the midpoint of $BC$ , and $D$ is the foot of altitude from point $A$ . Points $T$ and $P$ are marked such that $TBC$ is equilateral, and $\angle BPD=\angle DPC = 30^{\circ}$ and this points lie in the same half-plane with respect to $BC$ , not in the same as $A$ .
Prove that the circumcircles of $ADP$ and $AMT$ are tangent.
Ivan Korshunau

1 reply