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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.

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 <em>does not</em> completely cover the unit square.

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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