Inequalities
Algebra
Level
3
If p,q,r,s and t are 5 numbers such that :-
1.p+q<r+s
2.q+r<s+t
3.r+s<t+p
4.s+t<p+q
then the largest and the smallest numbers among the given numbers are
q and p respectively
r and q respectively
p and q respectively
r and t respectively
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1 solution
Put p+q < r+s (1), q+r < s+t (2), r+s < t+p (3) and s+t < p+q (4)
(1)(3) => q < t (a)
(2)(4) => r < p
(1) => q - s < r - p < 0 => q < s (b)
(1) => q - r < s - p < 0 => q < r < p (c)
(a)(b)(c) => q is the smallest
(4) => s - p < q - t < 0 => s < p (d)
(4)(b) => 0 < s - q < p - t => t < p (e)
(c)(d)(e) => p is the largest