(;) Semicolon Math: Semicolon Math Round 1

Due to a mistake in the original wording, everyone will receive free points for this problem. There will be no penalty for this problem. Sorry about the inconvenience.

A square is inscribed in a ellipse with equation \(\frac{x^2}{16}+\frac{y^2}{9}=1\). Let the area of intersection of the square inscribed in the ellipse and the rhombus formed when the endpoints of the minor axis and major axis is connected be \(a\). If \(a\) can be expressed as \(\frac{m}{n}\) such that \(m\) and \(n\) are relatively prime, find \(m+n\).

For all positive integers \(n\), let \(\sigma(n)\) be the number of divisors of \(n\). Let \[ N = \sum_{d | 20160} (\sigma(d))^{2001}. \]Here, \(d\) is a positive integer and a divisor of \(20160\). Find the remainder when \(N\) is divided by \(1000\).

Define a "cyclic swap" as taking a certain string of numbers, \(a_1, a_2, a_3, \dots a_n\) and then moving the last number to the front, thus becoming \(a_n, a_1, a_2, \dots a_{n-1}\). Daniel takes a list of \(n\) integers from \(1,2,3, \dots n\) where \(n>3\) and performs a cyclic swap for each pair of integers in form \(2i - 1\) and \(2i\) where \(i\) is an integer and \(1 \le 2i - 1 \le n\). Then, he performs a cyclic swap for each triple of integers in form \(3i - 2, 3i - 1,\) and \(3i\) where \(i\) is an integer and \(1\le 3i - 2, 3i - 1, 3i \le n\). He also performs a cyclic swap on the remaining \(n \pmod 3\) integers that he hasn't applied the cyclic swap onto. In general, Daniel performs cyclic swaps for each \(k\)-tuple of integers \((ki - k + 1, ki - k + 2, ki - k + 3, \dots, ki)\) where \(k\) and \(i\) are integers with \(2 \le k \le n\) and \(1 \le ki - k + 1, ki \le n\). He then performs a cyclic swap on the remaining \(n \pmod k\) integers that he hasn't applied the cyclic swap onto.

For example, on \(1,2,3,4,5\) the list would go:

• \(1, 2, 3, 4, 5\)
• \(2, 1, 4, 3, 5\)
• \(4, 2, 1, 5, 3\)
• \(5, 4, 2, 1, 3\)
• \(3, 5, 4, 2, 1\)

The smallest \(n\) in which a number in it's original position on the list is at the same position by the end of the process is \(n=6\). What is the third smallest \(n\)?

The circumradius of acute triangle \(ABC\) is \(1\). Let \(O\) be the circumcenter of \(ABC\) and \(H\) be the orthocenter of \(ABC\). Also, let \(A'\) be the circumcenter of triangle \(BHC\) and \(OA' = \frac{8}{5}\). The length of \(BC\) can be written in form \(\frac{a}{b},\) where \(a\) and \(b\) are positive relatively prime integers. Find \(a+b\).

Define an insphere of a tetrahedron to be the sphere that is internally tangent to all four faces of the tetrahedron, lying entirely in the interior and boundary of the tetrahedron. Define an exsphere of a tetrahedron to be a sphere that is externally tangent to one face of the tetrahedron and also tangent to the planes containing the other three faces. Suppose a tetrahedron has the four vertices \(A=(0,0,0), B =(6,0,0), C=(0,6,0), D=(0,0,6)\). Let \(r\) denote the radius of its insphere. Let \(R_a\) be the radius of the exsphere that is tangent externally to face \(BCD\) and also tangent to the planes containing \(ABC, ABD,\) and \(ACD\). Find \(R_a^2 + r^2 + R_a + r\).

Let \(ABC\) be an acute triangle inscribed in circle \(\omega\). Let \(X\), \(Y\) and \(Z\) be the midpoints of arc \(AB\), arc \(BC\) and arc \(CA\), respectively. Suppose that the circumradius of \(ABC\) has length \(5\) and that the inradius of \(ABC\) has length \(2\). Find \((AX)(BY)(CZ)\).