2

Part of 2001 Putnam

Problems(2)

Source:

k

\mathcal{C}_k

is biased so that, when tossed, it has probability

\tfrac{1}{(2k+1)}

of falling heads. If the

n

coins are tossed, what is the probability that the number of heads is odd? Express the answer as a rational function

n

Putnamprobabilityfunctioninductionrational functioncollege contests

Source:

Find all pairs of real numbers

(x,y)

satisfying the system of equations: \begin{align*}\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2)\\ \frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4)\end{align*}

PutnamUSAMTSalgebrasystem of equationscollege contests