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1950 Miklós Schweitzer

Part of Miklós Schweitzer

Subcontests

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Miklos Schweitzer 1950_8

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

Miklos Schweitzer 1950_8

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}
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Miklos Schweitzer 1950_6

Prove the following identity for determinants: |c_{ik} \plus{} a_i \plus{} b_k \plus{} 1|_{i,k \equal{} 1,...,n} \plus{} |c_{ik}|_{i,k \equal{} 1,...,n} \equal{} |c_{ik} \plus{} a_i \plus{} b_k|_{i,k \equal{} 1,...,n} \plus{} |c_{ik} \plus{} 1|_{i,k \equal{} 1,...,n}

Miklos Schweitzer 1950_6

Consider an arc of a planar curve; let the radius of curvature at any point of the arc be a differentiable function of the arc length and its derivative be everywhere different from zero; moreover, let the total curvature be less than π2 \frac{\pi}{2}. Let P1,P2,P3,P4,P5 P_1,P_2,P_3,P_4,P_5 and P6 P_6 be any points on this arc, subject to the only condition that the radius of curvature at Pk P_k is greater than at Pj P_j if j<k j<k. Prove that the radius of the circle passing through the points P1,P3 P_1,P_3 and P5 P_5 is less than the radius of the circle through P2,P4 P_2,P_4 and P6 P_6
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Miklos Schweitzer 1950_3

Let E E be a system of n^2 \plus{} 1 closed intervals of the real line. Show that E E has either a subsystem consisting of n \plus{} 1 elements which are monotonically ordered with respect to inclusion or a subsystem consisting of n \plus{} 1 elements none of which contains another element of the subsystem.

Miklos Schweitzer 1950_3

For any system x1,x2,...,xn x_1,x_2,...,x_n of positive real numbers, let f_1(x_1,x_2,...,x_n) \equal{} x_1, and f_{\nu} \equal{} \frac {x_1 \plus{} 2x_2 \plus{} \cdots \plus{} \nu x_{\nu}}{\nu \plus{} (\nu \minus{} 1)x_1 \plus{} (\nu \minus{} 2)x_2 \plus{} \cdots \plus{} 1\cdot x_{\nu \minus{} 1}} for \nu \equal{} 2,3,...,n. Show that for any ϵ>0 \epsilon > 0, a positive integer n0<n0(ϵ) n_0 < n_0(\epsilon) can be found such that for every n>n0 n > n_0 there exists a system x1′,x2′,...,xn′ x_1',x_2',...,x_n' of positive real numbers with x_1' \plus{} x_2' \plus{} \cdots \plus{} x_n' \equal{} 1 and fν(x1′,x2′,...,xn′)≤ϵ f_{\nu}(x_1',x_2',...,x_n')\le \epsilon for \nu \equal{} 1,2,...,n .
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