Summer Problem Solving Marathon Question #42

[Value = 4 points]

A certain biased coin has a fixed probability of coming up heads, where that probability is neither 0 nor 1. When the coin is flipped five times, the probability of getting exactly one head is the same as the probability of getting exactly two heads. What is the probability of getting exactly three heads?

Summer Problem Solving Marathon Solution #41

Adding all three equations, we obtain:

(k+2)x + (k+2)y + (k+2)z = 3k

or:

(k+2)(x+y+z) = 3k

When k = -2, this becomes 0 = -6, which is not true for any values of x, y, and z. So there is no solution to the system of equations when k = -2.

For any other value of k, we have x + y + z = 3k/(k+2). So by setting x = y = z = k/(k+2), we obtain a solution.'

Thus k = -2 is the only value for which there is no solution in x, y, and z to the system of equations.

Summer Problem Solving Marathon Solution #40

Because 3 and 5 are both odd, 3¹¹ and 5¹³ are also both odd. Thus 3¹¹ + 5¹³ is even, and is divisible by 2. So 2 is the smallest prime factor of the sum.

Summer Problem Solving Marathon Solution #39

Consider first the horizontal movement of the bug. The bug moves horizontally on each odd-numbered move, with the horizontal movements alternating positive and negative directions. On the first move, the bug moves 1 unit horizontally, on the third -1/4 horizontally, on the fifth 1/16 horizontally, on the seventh -1/64 horizontally, and so on.

Combining successive positive and negative movements, we have movement of 3/4, and then 3/64, and so on. This is an infinite geometric sequence with initial term 3/4 and common ration 1/16. Its sum is thus (3/4)/(1 - 1/16) = (3/4)/(15/16) = 4/5.

Next consider the vertical movement of the bug. The bug moves vertically on each even-numbered move, with the vertical movement alternating positive and negative directions. On the second move, the bug moves 1/2 unit vertically, on the fourth -1/8, on the sixth 1/32, on the eighth -1/128, and so on.

Combining successive positive and negative movements, we have movements of 3/8, and then 3/128, and so on. This is an infinite geometric sequence with initial terms 3/8 and common ratio 1/16. Its sum is thus (3/8)/(1 - 1/16) = (3/8)/(15/16) = 2/5.

In the limit, then, the bug approaches the point (4/5, 2/5).

Summer Problem Solving Marathon Question #41

[Value = 5 points]

Find the set of all values of k such that the following system of equations has no solutions in x, y, and z:

kx + y + z = k
x + ky + z = k
x + y + kz = k

Summer Problem Solving Marathon Question #40

[Value = 1 point]

What is the smallest prime factor of 3¹¹ + 5¹³?

Summer Problem Solving Marathon Solution #38

Let x + iy be an arbitrary root of the polynomial. Then the reciprocal of that root is 1/(x + iy). We then convert the denominator to a real number by multiplying by the complex conjugate x - iy, to obtain:

(x - iy)/((x + iy)(x - iy)) = (x - iy)/(x² + y²)

But because the root is on the circle centered at 0 + 0i with radius 1, we have, by the Pythagorean theorem, x² + y² = 1. So the reciprocal of the root is x - iy.

But this is the complex conjugate of the original root. Complex roots of polynomials with real-valued coefficients always comes in pairs by complex conjugates, so the reciprocal of the original root is another root of the polynomial.

Thus the reciprocals of the roots of the polynomial are just the roots of the polynomial. We thus want the sum of the roots of the polynomial. But the sum of the roots of any polynomial with a leading coefficient of 1 is the negative of the coefficient of the next-to-largest power of the variable. Hence the sum of the roots, which is also the sum of the reciprocals of the roots, is -a.