Initially, the U-tube shown above is filled with water, and the level on each side is \(\ell_0 = \SI{5}{\centi\meter}.\) After some oil is poured on side \(A,\) the level on that side becomes \(L_A = \SI{7.3}{\centi\meter}.\)

What is \(L_B\) \((\)in \(\si{\centi\meter})?\)

**Assume** that the water and oil do not mix, and that their densities are related by \(\rho_\textrm{oil} = 0.85\rho_\textrm{water}.\)

**True or False?**

For all reals \(k,\) there exist reals \(x,y,\) and \(z\) satisfying \[\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=k.\]

Find the smallest positive integer which ends in 17, is divisible by 17, and whose digits sum to 17.

I will walk from A to B moving only North and East along the grid lines. Then, I will walk back to A along the gridlines, visiting only locations that I have not visited before.

How many paths are there from A to B that allow me to walk back to A in this way?

Select a random point from the interior of a unit square (uniformly across the area). Let \(p\) be the probability that a unit circle centered at that point will completely cover the square.

To three decimal places, what is \(p?\)

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