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# Problems of the Week

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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 $$O=(0,0)$$ towards $$A=(10,6).$$ He can only go in the $$+x$$ and $$+y$$ 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 $$O$$ to $$A$$ in this way?

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

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

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

Neglect the surface tensions of the liquids.

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


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

A uniform solid in the shape of a quarter segment of a sphere of radius $$R$$ is released from rest with its diameter vertical, and the center a height of $$2R$$ 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 $\omega \; = \; \alpha\sqrt{\frac{g}{R}}$ for some $$\alpha > 0$$, where $$g$$ is the constant of gravitational acceleration.

What is the value of $$\alpha$$ to $$2$$ decimal places?

Inspiration

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