Question

3. A shop contains twelve samples of read shirts, seven samples of white shirts, and N samples of blue shirts. Suppose that the smallest K such that choosing K samples from the collection guarantees that you have six samples of the same color of shirt is K-15. What is N? 4. Show that among any n1 positive integers not exceeding 2nthere must be integer that divides one of the other integers 1 an 5. Given three non-zero real numbers x, y and z. Prove that xty vtz z+x > 1 + + 6. Show that whenever 25 girls and 25 boys are seated around a circular table, there is always a person both of whose neighbors are boys. 7. A lattice point in 2D plane is a point having both coordinates as integers. Given 3n lattice points in a 2D Cartesian plane, Prove that one can always select 2n points out of these 3n points such that P is also a lattice point, where P is defined as follows: 2n 2n (3Σx, Σ) i-1 Where x, and y, are the x and y coordinates of the given points

Please send me solutions for the above five questions.
The questions are based on Pigeonhole Principle.

Answer 1

3)
N = 4
as
12 red , 7 white and N blue
smallest K = 15 such that you have 6 sample of same color

so we can choose 5 maximum red shirts , 5 white shirts to not have 6 sample of same color
and now if N is 5 or more then we can choose 5 blue shirts
then we will have total 5 +5 + 5 = 15 shirts but no 6 shirts of same color

but it is given K = 15 ,
hence N = 4
then
we can choose 5 + 5 + 4 = 14 shirts which are of different color
now any shirt of red or white will make it six shirts of same color

Note that N can't be less than 4
as in that case
minimum value of K won't be 15 , it will be less than that

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