/sci/ - Science & Math

[math]
\text{The 20 members of a local tennis club have scheduled exactly 14 two-person}
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\text{games among themselves, with each member playing in at least one game.}
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\text{Prove that within this schedule there must be a set of 6 games with 12 distinct}
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\text{players.}
[/math]

Daily Putnam Problem >>10615013

Previous Thread >>10613291

[math]
\text{For each positive integer } n \text{, let}
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\qquad \qquad \qquad \qquad \displaystyle S_n = 1 + \frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n},
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\qquad \qquad \qquad \qquad \displaystyle T_n = S_1+S_2+S_3+\cdots+S_n ,
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\qquad \qquad \qquad \qquad \displaystyle U_n = \frac{T_1}{2}+\frac{T_2}{3}+\frac{T_3}{4}+\cdots + \frac{T_n}{n+1}.
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\text{Find, with proof, integers }0<a,b,c,d<1000000\text{ such that }T_{1988}=a S_{1989} - b
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\text{and }U_{1988}=c S_{1989} - d \text{.}
[/math]

Here's a more readable version.

[math]
\text{For each positive integer } n \text{, let}
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\displaystyle S_n = 1 + \frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n},
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\displaystyle T_n = S_1+S_2+S_3+\cdots+S_n ,
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\displaystyle U_n = \frac{T_1}{2}+\frac{T_2}{3}+\frac{T_3}{4}+\cdots + \frac{T_n}{n+1}.
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\text{Find, with proof, integers }
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0<a,b,c,d<1000000\text{ such that }
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T_{1988}=a S_{1989} - b

\text{ and }U_{1988}=c S_{1989} - d \text{.}
[/math]

Daily Putnam Problem >>10613291

Previous Thread >>10610385

[math]
\text{Let }B\text{ be a set of more than }2^{n+1}/n\text{ distinct points with coordinates of the}
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\text{form }(\pm 1,\pm 1,\ldots,\pm 1)\text{ in }n\text{-dimensional space with }n\geq 3\text{. Show that there are}
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\text{three distinct points in }B\text{ which are the vertices of an equilateral triangle.}
[/math]

Daily Putnam Problem >>10610385

Previous Thread >>10607175

[math]
\text{Let }S_0\text{ be a finite set of positive integers. We define finite sets }S_1,S_2,\ldots \text{ of}
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\text{positive integers as follows: the integer }a\text{ is in }S_{n+1}\text{ if and only if exactly one of}
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a-1 \text{ or }a\text{ is in }S_n \text{. Show that there exist infinitely many integers }N \text{ for which}
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S_N=S_0\cup\{N+a: a\in S_0\}.
[/math]

Daily Putnam Problem >>10607175

Previous Thread >>10603810

[math]
\text{Let }f(x)\text{ be a continuous function such that }f(2x^2-
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1)=2xf(x)\text{ for all }x\text{. Show that }f(x)=0\text{ for }-1\leq
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x\leq 1\text{.}
[/math]

Daily Putnam Problem >>10607140

Previous Thread >>10603810

[math]
\text{Let }f(x)\text{ be a continuous function such that }f(2x^2-
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1)=2xf(x)\text{ for all }x\text{. Show that }f(x)=0\text{ for }-1\leq
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x\leq 1\text{.}
[/math]

Daily Putnam Problem >>10603810

Previous Thread >>10602626

[math]
\text{Let }f(t)=\sum_{j=1}^N a_j \sin(2\pi jt)\text{, where each }a_j\text{ is real and }a_N\text{ is not equal to 0.}
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\text{Let }N_k\text{ denote the number of zeroes (including multiplicities) of } \displaystyle \frac{d^k f}{dt^k}\text{. Prove}
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\text{that}
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\qquad \qquad \qquad \qquad \displaystyle N_0\leq N_1\leq N_2\leq \cdots \mbox{ and } \lim_{k\to\infty} N_k = 2N.
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\text{[Editorial clarification: only zeroes in }[0, 1)\text{ should be counted.]}
[/math]

Daily Putnam Problem >>10602626

Previous Thread >>10592377

[math]
\text{Prove that the expression}
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\qquad \qquad \qquad \displaystyle\frac{gcd(m,n)}{n}\binom{n}{m}
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\text{is an integer for all pairs of integers }n\geq m\geq 1 \text{.}
[/math]

Daily Putnam Problem >>10592377