MathDB

8

Part of 1950 Miklós Schweitzer

Problems(2)

Miklos Schweitzer 1950_8

Source: first round of 1950

10/2/2008
Let A \equal{} (a_{ik}) be an n×n n\times n matrix with nonnegative elements such that \sum_{k \equal{} 1}^n a_{ik} \equal{} 1 for i \equal{} 1,...,n. Show that, for every eigenvalue λ \lambda of A A, either ∣λ∣<1 |\lambda| < 1 or there exists a positive integer k k such that \lambda^k \equal{} 1
linear algebramatrixsearchlinear algebra unsolved
Miklos Schweitzer 1950_8

Source: second part of 1950

10/3/2008
A coastal battery sights an enemy cruiser lying one kilometer off the coast and opens fire on it at the rate of one round per minute. After the first shot, the cruiser begins to move away at a speed of 60 60 kilometers an hour. Let the probability of a hit be 0.75x^{ \minus{} 2}, where x x denotes the distance (in kilometers) between the cruiser and the coast (x≥1 x\geq 1), and suppose that the battery goes on firing till the cruiser either sinks or disappears. Further, let the probability of the cruiser sinking after n n hits be 1 \minus{} \frac {1}{4^n} ( n \equal{} 0,1,...). Show that the probability of the cruiser escaping is 223π \frac {2\sqrt {2}}{3\pi}
probabilityprobability and stats