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.

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 _______.

a, b, c are three natural numbers such that a x b x c = 27846. If a/6 = b + 4 = c – 4, find a + b + c.

ABC is a triangle in which the angles are in the ratio 3 : 4 : 5 . PQR is a triangle in which the angles are in the ratio 5 : 6 : 7. The difference between the least angle of ABC and the least angle of PQR is aº. Then a = _________

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 ________ .

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 sum of the ages of a man and his wife is six times the sum of the ages of their children. Two years ago the sum of their ages was ten times the sum of the ages of their children. Six years hence the sum of their ages will be three times the sum of the ages of their children. How many children do they have?

The difference between the biggest and the smallest three digit number each of which has different digits is

(A) 864 

(B) 875 

(C) 885 

(D) 895

We have four sets S₁, S₂, S₃, S₄ each containing a number of parallel lines. The set S_i contains i + 1 parallel lines i = 1,2,3,4. A line in S_i is not parallel to lines in S_j when i ≠ j. In how many points do these lines intersect ?

(A) 54 

(B) 63 

(C) 71

(D) 95

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.