If m and n are integers such that (m + 2n)(2m + n) = 27, then the maximum possible value of 2m − 3n is

If log₆₄x² + log₈√y + 3log₅₁₂(y√z) = 4, where x,y and z are positive real numbers, then the minimum possible value of (x + y + z) is

 The average number of copies of a book sold per day by a shopkeeper is 60 in the initial seven days and 63 in the initial eight days, after the book launch. On the ninth day, she sells 11 copies less than the eighth day, and the average number of copies sold per day from second day to ninth day becomes 66. The number of copies sold on the first day of the book launch is

If 9x² + 2x − 3 − 4(3x² + 2x − 2) + 27 = 0, then the product of all possible values of x is

Let ABCDEF be a regular hexagon and P and Q be the midpoints of AB and CD, respectively. Then, the ratio of the areas of trapezium PBCQ and hexagon ABCDEF is

If a,b,c and d are integers such that their sum is 46 , then the minimum possible value of (a − b)² + (a − c)² + (a − d)² is

Let a_n be the n^th  term of a decreasing infinite geometric progression. If a₁ + a₂ + a₃ = 52 and a₁a₂ + a₂a₃ + a₃a₁ = 624, then the sum of this geometric progression is

Suppose a,b,c are three distinct natural numbers, such that 3ac = 8(a + b). Then, the smallest possible value of 3a + 2b + c is

Two tangents drawn from a point P touch a circle with center O at points Q and R. Points A and B lie on PQ and PR, respectively, such that AB is also a tangent to the same circle. If ∠AOB = 50^∘, then ∠APB, in degrees, equals

The number of divisors of (2⁶ × 3⁵ × 5³ × 7²), which are of the form (3r + 1), where r is a non-negative integer, is