Prove that x⁴ + 3x³ + 6x² + 9x + 12 cannot be expressed as a product of two polynomials of degree 2 with integer coefficients.

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 ________ .

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

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

(A) 52 

(B) 62 

(C) 54 

(D) 68

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?

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

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.

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

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 _______.