/sci/ - Science & Math

Previous Thread >>10543223

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\text{Consider the power series expansion}
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\qquad \qquad \qquad \qquad \qquad \qquad \frac{1}{1-2x-x^2} = \sum_{n=0}^\infty a_n x^n.
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\text{Prove that, for each integer }n\geq 0\text{, there is an integer }m\text{ such that}
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\qquad \quad \qquad \qquad \qquad \qquad \qquad a_n^2 + a_{n+1}^2 = a_m .
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Daily Putnam Problem >>10543223

Previous Thread >>10536135

[math]
\text{Let }p(x)\text{ be a polynomial that is nonnegative for all real }x\text{. Prove that for some}
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k\text{, there are polynomials }f_1(x),\dots,f_k(x)\text{ such that}
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\qquad \qquad \qquad \qquad \qquad \qquad p(x) = \sum_{j=1}^k (f_j(x))^2.
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Daily Putnam Problem >>10536135

Previous Thread >>10532228

[math]
\text{Find polynomials }f(x)\text{,}g(x)\text{, and }h(x)\text{, if they exist, such that for all }x\text{,}
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|f(x)|-|g(x)|+h(x) = \begin{cases} -1 & \mbox{if $x<-1$} \\
3x+2 & \mbox{if $-1 \leq x \leq 0$} \\
-2x+2 & \mbox{if $x>0$.}
\end{cases}

[/math]

Daily Putnam Problem >>10532228

Previous Thread >>10529139

[math]
\text{Prove that, for any integers }a, b, c\text{, there exists a positive integer }n\text{ such that }
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\sqrt{n^3+an^2+bn+c}\text{ is not an integer.}
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Daily Putnam Problem >>10529139

Previous Thread >>10526643

[math]
\text{Let }N\text{ be the positive integer with 1998 decimal digits,}
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\text{all of them 1; that is,}
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\qquad \qquad \qquad \qquad N=1111\cdots 11.
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\text{Find the thousandth digit after the decimal point of }\sqrt N\text{.}
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Daily Putnam Problem >>10526643

Previous Thread >>10523587

[math]
\text{Find necessary and sufficient conditions on positive integers }m\text{ and }n\text{ so that}
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\qquad \qquad \qquad \qquad \qquad \sum_{i=0}^{mn-1} (-1)^{\lfloor i/m \rfloor +\lfloor i/n\rfloor}=0.
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Daily Putnam Problem >>10523587

Previous Thread >>10520885

[math]
\text{Let }H\text{ be the unit hemisphere } \{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}\text{, }C\text{ the}
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\text{unit circle }\{(x,y,0):x^2+y^2=1\}\text{, and }P\text{ the regular pentagon inscribed in}
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C\text{. Determine the surface area of that portion of }H\text{ lying over the planar}
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\text{region inside }P\text{, and write your answer in the form }A \sin\alpha + B \cos\beta\text{, where}
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A,B,\alpha,\beta\text{ are real numbers.}
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Daily Putnam Problem >>10520885

Previous Thread >>10518553

[math]
\text{Given a point }(a,b)\text{ with }0<b<a\text{, determine the minimum perimeter of a}
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\text{triangle with one vertex at }(a,b)\text{, one on the }x\text{-axis, and one on the line }y=x\text{.}
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\text{You may assume that a triangle of minimum perimeter exists.}
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Daily Putnam Problem >>10518553