MATH MODELLING -Level
Problem 1Determine 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<b<c.Find y.
Problem 5
If a, b, c, d are real numbers such that a^2 +b^2 +c^2 +d^2<= 1, find the maximum of the expression:
(a+b)^4 +(a+c) ^4+(b+c)^4 +(b+d) ^4+(c+d) ^4. Problem 6
Find the least natural number k such that for any a ? 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 ofsquares 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. problem 1 ”：
7960Table[{n, Sum, {ii, 1, Ceiling@Log}]}, {n, 7950, 8000}]
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