/sci/ - Science & Math

Any putnam fags take the exam today?

Discuss putnam

Let [math]N[/math] be the positive integer with [math]\mathrm{1998}[/math] decimal digits, all of them 1; that is, [eqn]N=1111...11.[/eqn]Find the thousandth digit after the decimal point of [math]\sqrt{N}[/math]

You have 10 idenitcal ropes, each rope with a red and a blue end. You randomly connect all the rope's ends, but red ends can only be connected to blue ends and vice versa. What's the expected number of loops of rope at the end?

Previous thread: >>12539169

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

Previous thread: >>12535217

Whatever happened to the Putnam general? Anyway, here it is.

Has there been anyone that did unexpectedly well on the Putnam?
Has there been any very eccentric fellows?

I want to study for the Putnam exam because I'm a bad enough dude who thinks he can get a high score. What maths do I need to study? Is it doable when taking a math undergrad?

http://www.math.harvard.edu/putnam/index.html

seems the daily putnam page is discontinued

Does anyone hear wanna post this year's putnam questions, if they have em

HOW THE FUCK IS THIS ALLOWED. UUUUUUUUUUUUUUUUUUUUUUUUUUUJJJJJJJJJJJJJJJJJJJJJBBBBBBBBBBBBBBBLLLLLLLLLRRRRRRRRRRRRRR I AM NOT STUPID I CAN DO IT EVEN THOUGH I GO TO A GARBAGE SCHOOL REEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE

What happened to the Putnam poster? I liked making futile attempts at solving them. Anyway, here's today's:

http://www.math.harvard.edu/putnam/index.html

Suppose that in a certain society, each pair of persons can be classified as either [math] amicable [/math] or [math] hostile [/math]. We shall say that each member of an amicable pair is a [math] friend [/math] of the other, and each member of a hostile pair is a [math] foe [/math] of the other. Suppose that the society has [math] \, n \, [/math] persons and [math] \, q \, [/math] amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include [math] \, q(1 - 4q/n^2) \, [/math] or fewer amicable pairs.

Given a nonisosceles, nonright triangle [math] \, ABC, \, [/math] let [math] \, O \, [/math] denote the center of its circumscribed circle, and let [math] \, A_1, \, B_1, \, [/math] and [math] \, C_1 \, [/math] be the midpoints of sides [math] \, BC, \, CA, \, [/math] and [math] \, AB, \, [/math] respectively. Point [math] \, A_2 \, [/math] is located on the ray [math] \, OA_1 \, [/math] so that [math] \, \Delta OAA_1 \, [/math] is similar to [math] \, \Delta OA_2A [/math] . Points [math] \, B_2 \, [/math] and [math] \, C_2 \, [/math] on rays [math] \, OB_1 \, [/math] and [math] \, OC_1, \, [/math] respectively, are defined similarly. Prove that lines [math] \, AA_2, \, BB_2, \, [/math] and [math] \, CC_2 \, [/math] are concurrent, i.e. these three lines intersect at a point.

A calculator is broken so that the only keys that still work are the [math] \, \sin, \; \cos, [/math] [math] \tan, \; \sin^{-1}, \; \cos^{-1}, \, [/math] and [math] \, \tan^{-1} \, [/math] buttons. The display initially shows 0. Given any positive rational number [math] \, q, \, [/math] show that pressing some finite sequence of buttons will yield [math] \, q [/math]. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

You already know who it is. Back at it again.
Previous /lmc/ Thread : >>10651693
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WHADDUP BOIS YOU ALREADY KNOW WHO IT IS IT'S LIL'MINECART HERE

BACK AT IT AGAIN

Previous Thread >>10618126

[math]
\text{Let } ABC \text{ be an acute-angled triangle whose side lengths satisfy the inequalities}
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AB < AC < BC \text{. If point } I \text{ is the center of the inscribed circle of triangle}
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ABC \text{ and point } O \text{ is the center of the circumscribed circle, prove that line } IO
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\text{intersects segments } AB \text{ and } BC \text{.}
[/math]

[math]
\text{Let }P(z)=z^{n} + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n \text{ be a polynomial in the complex}
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\text{variable } z \text{, with real coefficients } c_k \text{. Suppose that } |P(i)|<1 \text{. Prove that there}
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\text{exist real numbers } a \text{ and } b \text{ such that } P(a+bi)=0 \text{ and } (a^2+b^2+1)^2 < 4b^2 + 1 \text{.}
[/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]

[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]

[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]

[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]