MathDB

4

Part of 2011 Postal Coaching

Problems(6)

Show existence of method to buy 9 tickets.

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12/31/2011
In a lottery, a person must select six distinct numbers from 1,2,3,…,361, 2, 3,\dots, 36 to put on a ticket. The lottery commitee will then draw six distinct numbers randomly from 1,2,3,…,361, 2, 3, \ldots, 36. Any ticket with numbers not containing any of these 66 numbers is a winning ticket. Show that there is a scheme of buying 99 tickets guaranteeing at least one winning ticket, but 88 tickets are not enough to guarantee a winning ticket in general.
combinatorics unsolvedcombinatorics
Inequality with three positive reals of product 1

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12/31/2011
For all a,b,c>0a, b, c > 0 and abc=1abc = 1, prove that 1a(a+1)+ab(ab+1)+1b(b+1)+bc(bc+1)+1c(c+1)+ca(ca+1)≥34\frac{1}{a(a+1)+ab(ab+1)}+\frac{1}{b(b+1)+bc(bc+1)}+\frac{1}{c(c+1)+ca(ca+1)}\ge\frac{3}{4}
inequalitiesinequalities unsolved
No four chosen vertices form trapezium or rectangle

Source:

12/31/2011
Consider 201122011^2 points arranged in the form of a 2011×20112011 \times 2011 grid. What is the maximum number of points that can be chosen among them so that no four of them form the vertices of either an isosceles trapezium or a rectangle whose parallel sides are parallel to the grid lines?
geometrytrapezoidrectangleparallelogramcombinatorics unsolvedcombinatorics
Equation has integer solution

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12/31/2011
Let a,b,ca, b, c be positive integers for which ac=b2+b+1ac = b^2 + b + 1 Prove that the equation ax2−(2b+1)xy+cy2=1ax^2 - (2b + 1)xy + cy^2 = 1 has an integer solution.
number theory unsolvednumber theory
Find all n tuples of positive integers

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12/31/2011
Let n>1n > 1 be a positive integer. Find all nn-tuples (a1,a2,…,an)(a_1 , a_2 ,\ldots, a_n ) of positive integers which are pairwise distinct, pairwise coprime, and such that for each ii in the range 1≤i≤n1 \le i \le n, (a1+a2+…+an)∣(a1i+a2i+…+ani)(a_1 + a_2 + \ldots + a_n )|(a_1^i + a_2^i + \ldots + a_n^i ).
number theory unsolvednumber theory
Number of ways of placing balls subject to condition

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12/31/2011
Suppose there are nn boxes in a row and place nn balls in them one in each. The balls are colored red, blue or green. In how many ways can we place the balls subject to the condition that any box BB has at least one adjacent box having a ball of the same color as the ball in BB? [Assume that balls in each color are available abundantly.]
combinatorics unsolvedcombinatorics