There are n petrol pumps in a circle of circumference L, distributed in an arbitrary manner. Each petrol pump gives $a_i$ quantity of petrol s.t. $\sum_i a_i = L$. You have a car which consumes $x$ quantity for covering distance $x$. You start at an petrol pump with empty tank. Show that there exists an petrol pump starting at which you would be able to go round the circle. (Asked by Rustomji)
Showing posts with label Pigeon hole. Show all posts
Showing posts with label Pigeon hole. Show all posts
Friday, 6 May 2016
Tuesday, 9 September 2014
Maths puzzle #2 - Divisibility by 100
Given any 51 integers, prove that there exist two integers a, b such that a^2 - b^2 is divisible by 100
Solution: Start thinking by why only 51? If we divide any number by 50 it will give 50 possible remainders, which means there are two integers a, b which give the same remainder. So, a = 50q1+r and b = 50q2+r where 0<= r < 50.
a^2 - b^2 = (a-b)*(a+b) = 50(q1-q2) * 2(25(q1+q2)+r), hence divisibility by 100
Source: Manish asked me this problem, who was in turn asked by Rustam, who read it in Mathematical circles: the russian experience
Solution: Start thinking by why only 51? If we divide any number by 50 it will give 50 possible remainders, which means there are two integers a, b which give the same remainder. So, a = 50q1+r and b = 50q2+r where 0<= r < 50.
a^2 - b^2 = (a-b)*(a+b) = 50(q1-q2) * 2(25(q1+q2)+r), hence divisibility by 100
Source: Manish asked me this problem, who was in turn asked by Rustam, who read it in Mathematical circles: the russian experience
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