MaxWong

Suppose that \(n^2 + 72 = m^2\), where m is a positive integer.

Moving terms around gives \(m^2 - n^2 = 72\), which in turn means \((m - n)(m+n) = 72\).

Note that m - n and m + n have the same sign (since they differ by an even number).

They cannot be both odd, because their product is an even number. Hence they must be both even.

The positive factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.

So, we get the possible values \(\begin{cases} m + n = 36\\ m - n = 2 \end{cases}\), \(\begin{cases} m + n = 18\\ m - n = 4 \end{cases}\), \(\begin{cases} m + n = 12\\ m - n = 6 \end{cases}\).

The rest should be simple enough to figure out.

The path in question:

\((1,1) \\\to (13, 19)\\(\text{2 steps in between})\\\to (1, 19)\\\text{(3 steps in between)}\\\to(1, 3)\\\to(15,23)\\ \text{(2 steps in between)}\\\to(3, 23)\\\text{(4 steps in between)}\\\to(3, 3)\\\to (17,25)\\\text{(3 steps in between)}\\\to (1, 25)\\\text{(5 steps in between)}\\\to(1, 1)\)

Counting directly, there are 28 different points.

\(\newcommand{\dint}{\displaystyle\int} \begin{array}{rcl} \dint (\ln x)^2 \,dx &=& x(\ln x)^2 - \dint x\,d\left((\ln x)^2\right)\\ &=& x(\ln x)^2 - \dint x\cdot \dfrac{2\ln x}{x}\,dx\\ &=& x(\ln x)^2 - 2\dint \ln x\,dx\\ &=& x(\ln x)^2 - 2\left(x\ln x - \dint x\,d(\ln x)\right)\\ &=& \cdots \end{array}\)

I will leave the rest to you.

As you said, a smoot is 67 inches. 

168 acres is equivalent to 168 * 43560 square feet, which is equivalent to 168 * 43560 * 144 square inches.

Dividing by 67^2 gives around 235000 square smoots.

2y = 2f(x) is just y = f(x), so the point is (9, 7).

By Vieta's formula, \(\text{product of roots} = \dfrac{-k}3\).

But since one zero is the reciprocal of the other, product of roots is 1.

Therefore, \(\dfrac{-k}3 = 1\). That means k = -3.

Note that if three numbers x, y, z form an arithmetic progression, then \(y = \dfrac{x + z}2\).

Also, if three numbers a, b, c form a geometric progression, then \(b^2 = ac\).

Therefore, \(4^2 = 2\cdot x\). Solving gives x = 8.

Can you continue from here?

Use the formula area = (base)(height)/2.

\(\text{Area} = \dfrac{6\text{ in.} \times 7 \text{ in.}}2 = 21 \text{ sq. in.}\)

2y = 2f(x) + 2 is just y = f(x) + 1. That's the image when the graph of y = f(x) is translated 1 unit upwards.

So, translate (9, 7) one unit upwards, and you will get (9, 8). That point must be on the graph of 2y = 2f(x) + 2.

Therefore, the answer is 17.