Affirmative

A smart cafe conducted a survey by asking every fourth person entering their store if they already a smart

tablet. On a given day out of 100 total respondents, 80 answered affirmative. Next day the same survey was

conducted by choosing every fifth person entering the store and the number of respondents was again 100. 

Which of the following is the most likely number of respondents answering in the affirmative? 

(a) 64                  (b) 78           (c) 100                   (d) 92

Pairwise Different

101 distinct numbers are chosen among the integers between 0 and 1000. Prove that, among the absolute values of their pairwise di fferences, there are ten di fferent numbers not exceeding 100.

2013

On each of the cards written in 2013 by number, all of these 2013 numbers are diff erent. The cards are turned down by numbers. In a single move is allowed to point out the ten cards and in return will report one of the numbers written on them (do not know what). For what most W guaranteed to be able to fi nd W cards for which we know what numbers are written on each of them?

Monic poly

Let P(x) and Q(x) be (monic) polynomials with real coe fficients (the fi rst coeffi cient being equal to 1), and
deg P(x) = deg Q(x) = 10. Prove that if the equation P(x) = Q(x) has no real solutions, 
then P(x + 1) = Q(x - 1) has a real solution.

Real Solutions

Given three distinct real numbers a, b, and c, show that at least two of the three following
equations: (x - a)(x - b) = x - c;  (x - c)(x - b) = x - a; (x - c)(x - a) = x - b
have real solutions.

Least Perimeter

In triangle ABC, AB = AC and the lengths of the sides are integers. Let E be the mid point of BC. If the length of BE, AE & AB are in A.P., then find the least possible perimeter of triangle ABC. 

Unconventional

An unconventional die having face value of 1,0,-1,2,3 & -2 thrown thrice. find the probability that the sum of the outcome will (i) Zero & (ii) Six