The real root of the equation $2^{6x} + 2^{3x+2} - 21 = 0$ is

The average of 30 integers is 5. Among these 30 integers, there are exactly 20 which do not exceed 5. What is the highest possible value of the average of these 20 integers?

Let $a, b, x, y$ be real numbers such that $a^2 + b^2 = 25$, $x^2 + y^2 = 169$ and $ax + by = 65$. If $k = ay - bx$, then

In a triangle $ABC$, medians $AD$ and $BE$ are perpendicular to each other, and have lengths 12 cm and 9 cm, respectively. Then, the area of triangle $ABC$, in sq cm, is

Let $a_1, a_2$ be integers such that $a_1 - a_2 + a_3 - a_4 + \dots + (-1)^{n-1} a_n = n$, for $n \ge 1$. Then $a_{51} + a_{52} + \dots + a_{1023}$ equals

How many factors of $2^4 \times 3^5 \times 10^4$ are perfect squares which are greater than 1?

Two circles, each of radius 4 cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is

What is the largest positive integer such that $\dfrac{n^2+7n+12}{n^2-n-12}$ is also positive integer?

In 2010, a library contained a total of 11500 books in two categories - fiction and non-fiction. In 2015, the library contained a total of 12760 books in these two categories. During this period, there was 10% increase in the fiction category while there was 12% increase in the non-fiction category. How many fiction books were in the library in 2015?

Let $f$ be a function such that $f(mn) = f(m)f(n)$ for every positive integers $m$ and $n$. If $f(1), f(2)$ and $f(3)$ are positive integers, $f(1) < f(2)$, and $f(24) = 54$, then $f(18)$ equals