/sci/ - Science & Math

Previous Thread >>10589427

[math]
\text{Three distinct points with integer coordinates lie in the}
\\
\text{plane on a circle of radius }r>0\text{. Show that two of these}
\\
\text{points are separated by a distance of at least }r^{1/3}\text{.}
[/math]

Is there anything you'd like me to change about these daily putnam threads? I'm wide open to suggestions.

Is there anything you'd like me to change about the daily putnam threads? I'm wide open to suggestions.

Daily Putnam Problem >>10589427

Previous Thread >>10583271

[math]
\text{Show that the improper integral}
\\
\quad \qquad \qquad \boxed{\lim_{B\to\infty}\int_{0}^B \sin(x) \sin(x^2)\,dx}
\\
\text{converges.}
[/math]

>>10583431
>wtf this is very hard to read
This problem is pathetically easy, so I added the extra visuals to balance things out.

[math]
{{\text{}^{\displaystyle\color{#781b86}\heartsuit}~~}}
{_{{\displaystyle\color{#781b86}\heartsuit}}}
{{\text{󠛡󠛡}^\text{󠛡󠛡}}_\text{󠛡 }}^{{\displaystyle\color{#65319e}\heartsuit}~~~}
\color{#5147b7}{\mathbf{P\,}}\color{#4852c3}{\mathbf{u \, }}\color{#3e5dcf}{\mathbf{t \, }}\color{#476ac3}{\mathbf{n \, }}\color{#4f77b7}{\mathbf{a \, }}\color{#5884ab}{\mathbf{m \, }}
\color{#60909e}{\mathbf{ ~~ }}
\color{#699d92}{\mathbf{P \, }}\color{#71aa86}{\mathbf{r \, }}\color{#7ab77a}{\mathbf{o \, }}\color{#83b873}{\mathbf{b \, }}\color{#8cb96b}{\mathbf{l \, }}\color{#95ba64}{\mathbf{e \, }}\color{#9fbb5c}{\mathbf{m \, }}
\color{#a8bc55}{\mathbf{ ~~ }}
\color{#b1bd4d}{\mathbf{o \, }}\color{#b7ba4a}{\mathbf{f \, }}
\color{#beb648}{\mathbf{ ~~ }}
\color{#c4b345}{\mathbf{t \, }}\color{#cbb043}{\mathbf{h \, }}\color{#d1ad40}{\mathbf{e \, }}
\color{#d8a93e}{\mathbf{ ~~ }}
\color{#dea63b}{\mathbf{D \, }}\color{#dd9037}{\mathbf{a \, }}\color{#dd7a32}{\mathbf{y}}
~~{{\text{󠛡󠛡}^\text{󠛡󠛡}}\text{󠛡}}^{{\displaystyle\color{#db4d2a}\heartsuit}~}
{_{{\displaystyle\color{#db3725}\heartsuit}~~~}}
{{\text{}^{\displaystyle\color{#da2121}\heartsuit}}}
[/math]
[math]
\qquad\qquad\qquad\quad\qquad\quad \color{#699d92}{\text{>}}\color{#6da48c}{\text{>}}\color{#71aa86}{\text{1}}\color{#76b180}{\text{0}}\color{#7ab77a}{\text{5}}\color{#8cb96b}{\text{8}}\color{#91ba68}{\text{3}}\color{#95ba64}{\text{2}}\color{#9abb60}{\text{7}}\color{#9fbb5c}{\text{1}}
[/math]

[math]\color{#781b86}{\text{P}}\color{#6c2895}{\text{r}}\color{#6135a3}{\text{e}}\color{#5543b2}{\text{v}}\color{#4a50c0}{\text{i}}\color{#3e5dcf}{\text{o}}\color{#4a6fbe}{\text{u}}\color{#5681ad}{\text{s}}\color{#62939c}{\text{ }}\color{#6ea58b}{\text{T}}\color{#7ab77a}{\text{h}}\color{#85b871}{\text{r}}\color{#90b968}{\text{e}}\color{#9bbb5f}{\text{a}}\color{#a6bc56}{\text{d}}\color{#b1bd4d}{\text{ }}\color{#bab849}{\text{>}}\color{#c3b446}{\text{>}}\color{#ccaf42}{\text{1}}\color{#d5ab3f}{\text{0}}\color{#dea63b}{\text{5}}\color{#dd8b36}{\text{7}}\color{#dc7131}{\text{9}}\color{#dc562b}{\text{8}}\color{#db3c26}{\text{9}}\color{#da2121}{\text{4}}[/math]

[math] \color{#781b86}{\tt{The~}}\color{#6a2c98}{\tt{octa}}\color{#5b3cab}{\tt{gon~P_1}}
\color{#4d4dbd}{\tt{P_2 P_3}}\color{#3e5dcf}{\tt{P_4 P_5}}\color{#4d74ba}{\tt{P_6 P_7}}\color{#5c8aa5}{\tt{P_8}}\color{#6ba18f}{\tt{~is~}}\color{#7ab77a}{\tt{ins}}\color{#88b96f}{\tt{crib}}
\color{#96ba64}{\tt{ed}}\color{#a3bc58}{\tt{~in}}\color{#b1bd4d}{\tt{~a~cir}}\color{#bcb749}{\tt{cle,}}
\color{#c8b244}{\tt{~with}}\color{#d3ac40}{\tt{~the}}\color{#dea63b}{\tt{~vert}}\color{#dd8535}{\tt{ices}} [/math]
[math] \color{#6234a1}{\tt{arou}}\color{#5444b4}{\tt{nd~th}}
\color{#4555c6}{\tt{e~cir}}\color{#4668c4}{\tt{cumf}}\color{#557faf}{\tt{eren}}
\color{#64959a}{\tt{ce}}\color{#73ac85}{\tt{~in}}\color{#81b874}{\tt{~the~gi}}
\color{#8fb969}{\tt{ven~o}}\color{#9cbb5e}{\tt{rder.~Gi}}\color{#aabc53}{\tt{ven}}\color{#b7ba4b}{\tt{~tha}}\color{#c2b446}{\tt{t~the}}
\color{#cdaf42}{\tt{~poly}}\color{#d8a93d}{\tt{gon~P_1}}\color{#de9538}{\tt{P_3}}\color{#dd7431}{\tt{P_5}}\color{#dc532b}{\tt{P_7}} [/math]
[math] \color{#4d4dbd}{\tt{is~a}}\color{#3e5dcf}{\tt{~squa}}\color{#4d74ba}{\tt{re~of~a}}\color{#5c8aa5}{\tt{rea~5,~}}\color{#6ba18f}{\tt{and~t}}\color{#7ab77a}{\tt{he~pol}}\color{#88b96f}{\tt{ygon~}}\color{#96ba64}{\tt{P_2}}\color{#a3bc58}{\tt{P_4}}\color{#b1bd4d}{\tt{P_6}}\color{#bcb749}{\tt{P_8}}\color{#c8b244}{\tt{~is~a~}}\color{#d3ac40}{\tt{recta}}\color{#dea63b}{\tt{ngle~}}\color{#dd8535}{\tt{of~}}\color{#dc642e}{\tt{area~4,~}}\color{#db4228}{\tt{find}} [/math]
[math] \color{#4668c4}{\tt{th}}\color{#557faf}{\tt{e~m}}\color{#64959a}{\tt{axi}}\color{#73ac85}{\tt{mum}}\color{#81b874}{\tt{~pos}}\color{#8fb969}{\tt{sib}}\color{#9cbb5e}{\tt{le~ar}}\color{#aabc53}{\tt{ea~of~}}\color{#b7ba4b}{\tt{the~}}\color{#c2b446}{\tt{octa}}\color{#cdaf42}{\tt{gon.}} [/math]

Daily Putnam Problem >>10579894

Previous Thread >>10578703

[math]
\text{Prove that there exist infinitely many integers }n\text{ such that }n,n+1,n+2\text{ are}
\\
\text{each the sum of the squares of two integers. [Example: }0=0^2+0^2\text{, }1=0^2+1^2\text{, }
\\
2=1^2+1^2\text{.]}
[/math]

>>10578722
[math]
\color{red} \heartsuit
[/math]

>>10578727
[math]\mathrm{\LaTeX} \text{ not working? :(}[/math]

[math]\color{RED}{\heartsuit}[/math] Daily Putnam Problem >>10578703

Daily Putnam Problem >>10578703

Previous Thread >>10576533

[math]
\text{Let }A\text{ be a positive real number. What are the possible values of }\sum_{j=0}^\infty x_j^2,
\\
\text{given that }x_0,x_1,\ldots\text{ are positive numbers for which }\sum_{j=0}^\infty x_j=A\text{?}
[/math]

Daily Putnam Problem >>10576533

Previous Thread >>10571410

[math]
\text{Let }S\text{ be a finite set of integers, each greater than 1. Suppose that for each}
\\
\text{integer }n\text{ there is some }s\in S\text{ such that }\gcd(s,n)=1\text{ or }\gcd(s,n)=s\text{. Show}
\\
\text{that there exist }s,t\in S\text{ such that }\gcd(s,t)\text{ is prime.}
[/math]

Daily Putnam Problem >>10571410