# Problems of the Week

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# 2017-11-20 Intermediate

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?

The green path on the left allows me to walk back to A visiting only new locations, but the orange path on the right does not.

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 $p?$

On the left, the unit circle completely covers the unit square. On the right, the unit circle does not completely cover the unit square.

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