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2017-07-03 Advanced

         

True or False?

Given any six irrational numbers, there always exist three such that the sum of any two of them is still irrational.

Richard P Feynman starts his journey from O=(0,0)O=(0,0) towards A=(10,6).A=(10,6). He can only go in the +x+x and +y+y directions along the lattice, and he must turn exactly 7 times. An example path is shown above, with the yellow dots indicating turns.

How many ways can Feynman travel from OO to AA in this way?


Bonus: Consider the general case, traveling from (0,0)(0,0) to (p,q)(p,q) with exactly rr turns.

A heavy cylinder of radius RR lies in equilibrium on a smooth surface, separating two liquids of densities ρl\rho_l on the left and ρr\rho_r on the right, with ρr>ρl.\rho_r > \rho_l.

If the height of the liquid on the right is R,R, what is the height hh of the liquid on the left in terms of R?R?

Neglect the surface tensions of the liquids.

Find the smallest positive real xx such that x2xx=6.\big\lfloor x^2 \big\rfloor-x\lfloor x \rfloor=6. If your answer is in the form ab\frac{a}{b}, where aa and bb are coprime positive integers, submit your answer as a+b.a+b.


Notation: \lfloor \cdot \rfloor denotes the floor function.

A uniform solid in the shape of a quarter segment of a sphere of radius RR is released from rest with its diameter vertical, and the center a height of 2R2R above a smooth horizontal floor. The solid strikes the floor in a perfectly elastic collision. The angular velocity of the segment, immediately after its collision with the floor, is of the form ω  =  αgR \omega \; = \; \alpha\sqrt{\frac{g}{R}} for some α>0\alpha > 0, where gg is the constant of gravitational acceleration.

What is the value of α\alpha to 22 decimal places?


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