MATH MODELLING -Level

northwolves

Problem 1 Determine the least possible value of the natural number n such that n! =1.2.3…n ends in exactly 1987 zeros. Problem 2 In the set of 20 elements {1,2,3,4,5,6,7,8,9,A,B,C,D,J,K,L,U,X,Y,Z} we have made an aleatory suite of 28 throws..What is the probability that the sequence “Cuba July 1987” turn on in this order in the suite already? Problem 3* For any integer r>=1, determine the smallest integer h( r )>=1 such that for any partition of the set {1,2,…h( r )} in r classes, there are integers a>=0, 1<=x<=y, so that a+x, a+y and a+x+y belong to the same class. Problem 4 It is given that x = -2272, y = 10^3 +10^2 c+10b+a and z = 1 satisfy the equation ax+by+cz = 1, where a, b, c are positive integers with a

northwolves

Problem 6 Find the least natural number k such that for any a ? [0,1] and any natural number n a ^k (1-a) ^n < 1/(n+1) ^3 holds true. Problem 7 Solve the equation 28^x = 19^y +87 ^z where x, y, z are integers. Problem 8* A game consists in pushing a flat stone along a sequence of squares S0, S1, S2, .... , which are arranged in linear order. The stone is initially placed on square S0. When the stone stops on a square Sk it is pushed again in the same direction and so on until it reaches square S1987 or goes beyond it; then the game stops. Each time the stone is pushed, the probability that it will advance exactly n squares is 1/2^n . Determine the probability that the stone will stop exaclty on square S1987. Problem 9 Let S1 and S2 be two spheres with distinct radii, which touch externally. The spheres lie inside a cone C, and each sphere touches the cone in a full circle. Inside the cone, there are n solid spheres arranged in a ring in such a way that each solid sphere touches the cone C, both of the spheres S1 and S2 externally as well as the two neighbouring solid spheres. What are the possible values of n? Problem 10 N points are given arbitrarily in a plane. And no 3 of them are collinear. If we draw segments between these points; which is the minimum number if segments that can be colored red in such a way that, if we choose any 4 points, among the segments corresponding to them it can be found a red triangle.

wayne

problem 1 ”： 7960

1. Table[{n, Sum[Floor[n/5^ii], {ii, 1, Ceiling@Log[5, n]}]}, {n, 7950, 8000}]

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