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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 \((\alpha_1 +1 )… (\alpha_k + 1) \)

The first multiple of 15 greater than 99 is \(105 = 15 \times 7 \) . By inspection we see that m = 150.

October 10, 2014

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