/sci/ - Science & Math

>>10702199
Sometimes people post solutions, just check the previous threads for proof. Regarding my motivation, I'm just mostly trying to contribute actual quality/wholesome oc to my favorite board on 4chan

>>10701314
this

this

that kid made a heartfelt effort to keep the putnam threads alive while I was gone... LMC has my vote... and he has my admiration...

[math] \mathbf{ \color{ red }{ \heartsuit } } [/math]

Daily Putnam Problem >>10702148

Previous Thread >>10695011

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.

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.

Part B >>10695011

Part B >>10695011

Part A >>10694987

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.

Daily Putnam Thread - Part A >>10694987

Previous Thread >>10623607

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.

Daily Putnam Problem >>10623607

Previous Thread >>10621806

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

Daily Putnam Problem >>10621806

Previous Thread >>10615013

[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}
\\
\text{variable } z \text{, with real coefficients } c_k \text{. Suppose that } |P(i)|<1 \text{. Prove that there}
\\
\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]

Daily Putnam/Olympiad Problem >>10617778

Previous Thread >>10615013