Let a, b, c be real numbers, not all of them are equal. Prove that if a + b + c = 0, then a² + ab + b² = b² + bc + c² = c² + ca + a².

Prove the converse, if a² + ab + b² = b² + bc + c² = c² = ca + a², then a + b + c = 0.

The numbers 2, 3, 12, 14, 15, 20, 21 may be divided into two sets so that the product of the numbers in each set is the same. What is this product?

(A) 420 

(B) 1260 

(C) 2520 

(D) 6720

The four digit number 8ab9 is a perfect square. The value of a² + b² is

(A) 52 

(B) 62 

(C) 54 

(D) 68

Vishva plays football every 4th day. He played on a Tuesday . He plays football on a Tuesday again in _________ days.

The number of ways in which 26 identical chocolates be distributed between Amy, Bob, Cathy and Daniel so that each receives at least one chocolate and Amy receives more chocolates than Bob is _______.

Samrud had to multiply a number by 35. By mistake he multiplied by 53 and got a result 720 more. The new product is _________.

There are 5 cards. Five positive integers (may be different or equal) are written on these cards, one on each card. Abhiram finds the sum of the numbers on every pair of cards. He obtains only three different totals 57, 70, 83. Find the largest integer written on a card.

A cube has edge length x (an integer). three faces meeting at a corner are painted blue. The cube is then cut into smaller cubes of unit length. If exactly 343 of these cubes have no faces painted blue, then the value of x is ________ .

In an elementary school 26% of the students are girls. If there are 240 less girls than boys, then the strength of the school is _________

If x²(y + z) = a², y²(z + x) = b², z²(x + y) = c², xyz = a b c prove that a² + b² + c² + 2abc = 1