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In an 8×8 grid of points, what is the maximum number of points that we can select such that no four selected points are the corners of a rectangle whose sides are parallel to the edges of the grid?
If the three blue points are selected, then the red point cannot be selected.
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Consider a uniform rod of mass M and length L, free to rotate around a frictionless axis passing through its center and going into the page. Initially, the rod is stationary in the horizontal position, as shown in the diagram below.
Now, a small bullet of mass m moving with velocity v hits the rod at its extreme end and sticks to it. The system rotates vertically through some angle θ before it momentarily comes to rest. If this angle can be expressed (in degrees) as θ=α+arcsin((M+γm)gLβmv2), where g denotes the gravitational acceleration and α, β, and γ are positive integer constants with α in degrees, then find the value of α+β+γ.
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Does there exist a function f:R→R which satisfies
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Let {an} be a sequence of real numbers satisfying {a0=1an+1=4+3an+an2−2for n≥0. Let S=n=0∑∞an.
This problem is based on a recent Putnam contest problem.
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A ball with mass m is thrown from the origin at speed V0 toward the right on an exotic planet where the strength of gravity is g′=10g=1 m/s2.
Let α be the largest possible angle such that, for all θ<α, the distance between the ball and its launch point will be strictly increasing for t>0.
What is tan2α, to two decimal places?
Details and Assumptions:
Bonus: Generalize this angle for arbitrary values of V0, m, and g′.
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