**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?

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