# Recent questions in Math

Which city has the median population value?
City Population in Millions Melbourne 3.2 Bangkok 7.2 Nairobi Paris 9.6 São Paulo 17.7 Tokyo 28.0 Seattle 2.1
For each positive integer n , defind H(n) as the following. Prove that if n>=2, then H(n) is not an integer

$H(n)=1+ \frac{1}{2}+\frac{1}{3}+ ...+\frac{1}{n}$

Prove the following statement:

$V \subseteq W \Leftrightarrow P(V) \subseteq P(W)$

Where P(X) is the power set of X.

Prove the following using mathematical induction

$\sum_{i=1}^{n}i^3 = (\sum_{i=1}^{n}i)^2$

The area of an equilateral triangle is increasing at the rate of 5 m^2/hr.

Find the rate at which the height is changing when the area is

$\frac{64}{\sqrt3} m^2$

Suppose Aaron is pumping water into a tank (in the shape of an inverted right circular cone) at a rate of 1600 ft^3/min.
If the altitude is 10 ft and the radius of the base is 5 ft, find the rate at which the radius is changing when the height of the water is 7 ft.
Solve equation

$e^x - e^{-x} = e$

Find y(1) if

$\frac{dy}{dx} = x y^2 \text { and } y(0) =1.$

Suppose that f is a function that has a continuous second derivative and that satisfies

$f(0) = 4, f(1) =3, f'(0) = 5, f'(1) = 7, f''(0) =8 \text{ and } f''(1) = 11.$

Show that

$\int_0^1f(x) f''(x)dx \leq1$

Show that for any positive continuous function f on [0, a]

$\int_{0}^{a} \frac{f(x)}{f(x) + f(a-x)} dx = a/2$

Find the value of abc if a+b+c=0 and

$a^3+b^3+c^3 =216$

What is c if b and c are constants and

$(x+2)(x+b) = x^2 + cx + 6$