Related InterviewSolutions

• Show that there are infinitely many triples (x, y, z) of integers such that x3 + y4 = z31.
• If a, b, c are three real numbers such that | a – b | ≥ | c |, | b – c | ≥ | a |, |c-a|≥|b|, then prove that one of a ,b, c is the sum of the other two.
• The equation 166 × 56 = 8590 is valid in some base b ≥ 10 (that is 1, 6, 5, 8, 9, 0 are digits in base b in the above equation). Find the sum of all possible values of b ≥ 10 satisfying the equation.
• In a book with page numbers from 1 to 100 ,some pages are torn off. The sum of the numbers on the remaining pages is 4949. How many pages are torn off?
• Let p, q be prime numbers such that n3pq – n is a multiple of 3pq for all positive integers n. Find the least possible value of p + q.
• Find the number of all integer-sided isosceles obtuse-angled triangle with perimeter 2008.
• Show that there is no integer a such that a2 – 3a – 19 is divisible by 289.
• Find all integers a, b, c such that a2 = bc + 1, b2 = ca + 1.
• Let a, b, c be positive integers such that a divides b4 , b divides c4 and c divides a4. Prove that abc divides (a + b + c)21 .
• Five distinct 2-digit numbers are in a geometric progression. Find the middle term.