Duke Math Meet 2009 : First Relay Round

1A. Find the lowest positive angle $\theta$ that satisfies the equation $\sqrt {1+\cos \theta} = \sin \theta + \cos\theta$ expressed in degrees.

Discussion:

$\sqrt {1 +\cos\theta} = \cos\theta + \sin \theta \Rightarrow \sqrt{2\cos^2 \frac{\theta}{2} } = \sqrt2{\frac{1}{\sqrt2} \cos\theta + \frac{1}{\sqrt2} \sin\theta }$

Now this gives

$\sqrt2 \cos\frac{\theta}{2} = \sqrt2\cos(\theta - \frac{\pi}{4}) \Rightarrow \frac{\theta}{2} = \theta - \frac{\pi}{4}$ or $\frac{\theta}{2} = -\theta + \frac{\pi}{4}$

Thus the possible values of $\theta$ are $90^o$ or $30^o$ .

Since we require the smallest positive angle hence the answer is $30^o$ .

1B Let n be two times the tens digit of TNYWR. Find the coefficient of the $x^{n-1}y^{n+1}$ term in the expansion of $(2x + \frac{y}{2} + 3)^{2n}$

Discussion:

TNYWR is 3. Hence n = 6 Thus we are required to find coefficient of $x^5 y^7$ term in the expansion of $(2x + \frac{y}{2} + 3 )^{12}$

This can be easily found from trinomial expansion. The required term is ${{12}\choose {5}}(2x)^5 {{7}\choose{7}} (\frac{y}{2})^7 = 792 \times 32 \times \frac{1}{128} = 198$

1C Let k be TNYWR, and let n = k/2. Find the smallest integer m greater than n such that 15
divides m and 12 divides the number of positive integer factors of m.

Discussion:

k = 198, hence n = 99.

So we have to look at multiples of 15 greater than 99. We want 12 to divide the number of positive divisors of m.

Suppose $m = p_1^{\alpha_1} p_2^{\alpha_2} ... p_k^{\alpha_k}$ . The number positive divisors of k is $(\alpha_1 +1 )... (\alpha_k + 1)$