The number of arrangements of 6 people at a round table is (6-1)! = 5! = 120.

The number of ways to choose 2 fruits from 5 is given by 5C2 = 10.

Using the identity a^2 + b^2 = (a + b)^2 - 2ab, we have 70 = 12^2 - 2ab. Thus, 70 = 144 - 2ab, leading to ab = 37.

a^2 = 4 and b^2 = 9, so 4 + 9 = 13.

log_10(0.01) = log_10(10^-2) = -2, so x = -2.

log_10(10^x) = x. Therefore, x = 2.

log_10(20) = log_10(2) + log_10(10) = 0.301 + 1 = 1.301.

log_2(8) = 3 and log_2(4) = 2, thus x = 3 + 2 = 5.

log_2(x) + 2 = 6 implies log_2(x) = 4, so x = 2^4 = 16.

log_3(81) = log_3(3^4) = 4.

log_3(x) = 4 implies x = 3^4 = 81.

log_4(16) = log_4(4^2) = 2.

log_4(256) = log_4(4^4) = 4.

log_a(1) = 0 is true for any base a > 0, except when a = 1.

log_a(16) = 4 implies a^4 = 16. Since 16 = 2^4, we have a^4 = 2^4, thus a = 2.

log_a(3) = 0.5 implies a^0.5 = 3, thus a = 3^2 = 9.

log_a(64) = 6 implies a^6 = 64. Since 64 = 2^6, we have a = 2.

log_b(25) = 2 implies b^2 = 25. Therefore, b = 5.

log_x(64) = 6 implies x^6 = 64. Since 64 = 2^6, we have x = 2.

The sum of the roots is 3 + (-1) = 2, hence p = -2.

Substituting x = 2 into the equation gives 2^2 - 4*2 + k = 0, which simplifies to k = 4.

Using the root 3 in the equation: 3^2 - 7*3 + k = 0, we get k = 6.

The product of the roots is given by c/a. Here, c = 6 and a = 1, so αβ = 6.

The discriminant must be negative for no real roots: 5^2 - 4*1*k < 0, which simplifies to 25 - 4k < 0, or k > 25.

For both roots to be negative, the sum of the roots (which is -5) must be negative, and the product (k) must be positive. Thus, k > 0.

The number of terms in the expansion of (x + y)^n is n + 1. Therefore, n + 1 = 15, which gives n = 14.

By Vieta's formulas, the sum of the roots (a + b + c) of the polynomial x^3 - 6x^2 + 11x - 6 is equal to the coefficient of x^2 with the opposite sign, which is 6.

By Vieta's formulas, the sum of the roots p + q = -b/a.

The equation can be factored as (x + 1)(x + 1) = 0, giving the root -1 with multiplicity 2.

The discriminant is 2^2 - 4*1*1 = 0, indicating that the roots are real and equal.