(;) Semicolon Math: Semicolon Math Round 2

Let \(P(x) = x^4 + Ax^3 + Cx^2 + Dx + 24\) have only positive integer roots, possibly with repetition. What is the sum of all possible values of \(A\)?

Let \(ABCD\) be a square with side length \(1\). Let \(E\) be a point on side \(BC\) such that \(BE = EC\). Similarly, let \(F\) and \(G\) be points on side \(CD\) such that \(DF = FG = GC\). Let the intersection of \(AF\) and \(DE\) be \(H\) and the intersection of \(AG\) and \(DE\) be \(I\).

Define \(G_1\) to be the centroid of \(\triangle ADH\) and \(G_2\) to be the centroid of \(\triangle AHI.\) If \(G_1G_2\) can be written in form \(\frac{\sqrt{a}}{b}\) where \(a\) is a positive integer not divisible by the square of any prime and \(b\) is a positive integer, find \(a + b\).

Let \(A\) and \(B\) be the points of intersection between the line \(x+y=-1\) and the circle with radius \(5\) centered at \((0,0).\) Let \(f(x) = ax^2+bx+c,\) where \(a, b, c\) are real numbers with \(a > 0\) pass through \(A\) and \(B.\) Additionally, let the minimum value of \(f(x)\) be located at \(C\) and the \(x\)-coordinate of \(C\) be \(1.\)

Let \(k\) be the sum of roots of \(f(x)\) and let \(D\) and \(E\) be the intersection between the line \(x+y=k\) and the circle with radius \(5\) centered at \((0,0).\) The value \(CD^2 + CE^2\) can be expressed as \(\frac{m}{n}\) where \(\gcd(m,n)=1\). Find \(m+n.\)

Segment \(PQ\) has length \(65\). Let \(A\) and \(B\) be points on the same side of line \(PQ\), and let \(E\) be the intersection of \(AQ\) and \(BP\). Given \(AP = 39\), \(AQ = 52\), \(BP = 60\), \(BQ = 25\), find the area of quadrilateral \(PABQ\).

Let the set \(\{1,2,3, \ldots, n\}\) be denoted by \([n]\). A permutation of \([n]\) is a function \(f\) from \([n]\) to \([n]\) such that distinct elements of the domain map to distinct elements in the codomain, i.e. \(f(x) \neq f(y)\) if \(x \neq y\). Find the number of permutations \(\sigma(x)\) from \([6]\) to \([6]\) with \(\sigma^6 = e\), where \(e\) is the identity permutation defined by \(e(x) = x\) for all \(x\) in \([6]\). That is, applying the permutation six times results in the identity map.

For all primes \(p\) and positive integers \(n\), denote \(v_p(n)\) by the greatest integer \(k\) such that \(p^k|n\). A nonnegative integer \(n\) is chosen at random from \(S=\{0, 1, \ldots, 2^{2021}-1\}\). The expected value of \(v_2\left(\binom{2n}{n}\right)\) can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).