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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.
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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.
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A heavy cylinder of radius R lies in equilibrium on a smooth surface, separating two liquids of densities ρl on the left and ρr on the right, with ρr>ρl.
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.
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Find the smallest positive real x such that ⌊x2⌋−x⌊x⌋=6. If your answer is in the form ba, where a and b are coprime positive integers, submit your answer as a+b.
Notation: ⌊⋅⌋ denotes the floor function.
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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 ω=αRg for some α>0, where g is the constant of gravitational acceleration.
What is the value of α to 2 decimal places?
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