MaxWong

The axis is just another word for the diameter. Therefore the diameter is 14 units, meaning the radius is 7 units.

Now you can use the formula \(\text{surface area} = 4\pi (\text{radius})^2\) to finish it off.

So, let \(h \times w \times l\) be the dimensions of that prism.

We have \(\begin{cases}2(hw + wl + hl) = 56\\h + w + l = 56\end{cases}\). We are to find \(\sqrt{h^2 + w^2 + l^2}\).

We can use the identity \((h + w + l)^2 = (h^2 + w^2 + l^2) + 2(hw + wl + hl)\) to calculate \(h^2 + w^2 + l^2\), and then take square root. I will leave that as an exercise for you.

If you don't believe the identity is true, try to expand the left hand side and the right hand side. \((h + w + l)^2 = ((h + w) + l)^2 = (h + w)^2 + 2l(h + w) + l^2\) and etc.

Triangle inequality implies \(\begin{cases} 3 + 6 > AC\\ 3 + AC > 6\\ 6 + AC > 3\\ 9 + AC > 12\\ 9 + 12 > AC\\ 12 + AC > 9 \end{cases}\). If you list the integers satisfying all 6 inequalities, the possible values would be 4, 5, 6, 7, 8.

Let w be the length of the top side of the top-left rectangle. Then the left side of the top-left rectangle is \(\dfrac 3w\) and the left side of the bottom-left rectangle is \(\dfrac 7w\). This suggests the heights of the rectangle are in the ratio 3 : 7.

Since the top side of the top-right rectangle and that of the bottom-right rectangle are the same, the area ratio of the top-right rectangle and that of the bottom-right rectangle is also 3 : 7, which means \(x = 12 \times \dfrac 73 = 28\).

We use Euclidean algorithm to simplify \(\gcd(3n + 4, 7n + 15)\).

\(\quad \gcd(3n + 4, 7n + 15)\\ = \gcd(3n+4, n + 7)\\ = \gcd(-17, n + 7)\\ = \gcd(n + 7, 17)\)

If x is an integer, \(\gcd(x, 17)\) can only be either 1 or 17. That means the sum of all possible values required is 1 + 17 = 18.

Let a, b, c, d, e, x be the 6 numbers, with x being the number that is removed.

\(\begin{cases}\dfrac{a + b + c + d + e + x}6 = 14\\\dfrac{a + b +c + d + e}5 = 10\end{cases}\)

That means a + b + c + d + e = 50. Substitute that into the first equation to get \(\dfrac{50 + x}6 = 14\).

Can you work out the value of x from here? That is the number that was removed.

Note that the exterior angles of any convex polygon always add up to 360 degrees.

Since an interior angle and its corresponding exterior angle always add up to 180 degrees, a larger exterior angle means a smaller interior angle. So we are actually looking for the largest exterior angle.

Using the ratio, we calculate that the largest exterior angle is \(360^\circ \times \dfrac{7}{3 + 5 + 7} = 168^\circ\). You can subtract that from 180 degrees to find the smallest interior angle. 

This is one step away from the answer. Please continue on your own, but I hope you have an idea of what I explained, because that would be useful if you are to solve a similar problem entirely on your own.

Let a and b be the two numbers.

\(\begin{cases}a^2 + b^2 = 327\\ab = 85\end{cases}\)

We want to find a + b, but we only have quadratic terms in our equations, so we consider (a + b)^2 instead.

\((a + b)^2 = a^2 +b^2 + 2ab = 327 + 2(85) = 497\)

Suppose x = a + b. Then x^2 = 497. Can you take it from here?

Hint: The value of x is NOT negative since a and b are nonnegative numbers.

Let x be the original number.

Then 5x/(7 + x) = 8. Multiplying by (7 + x) on both sides gives 5x = 8x + 56. You can move the terms around to find the value of x.

This is equivalent to finding the number of solutions to the nine linear Diophantine equations \(a_1 + a_2 + a_3 + \cdots + a_{10} = 10, 20, 30, \cdots, 90\), You can do that using stars-and-bars method, but it will be very tough.