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We will post fixes to any problems AFTER the contest has finished, to make it fair to everyone.
The fixes will be available in the problem statements.
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Problem 1: Integer roots
Problem status: not solved
Value (if solved during a running contest): 70
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 \(A\)?
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Problem 2: Distance between centroids
Problem status: not solved
Value (if solved during a running contest): 150
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 \(a + b\).
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Problem 3: Parabolas, lines, and circles
Problem status: not solved
Value (if solved during a running contest): 200
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 \(m+n.\)
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Problem 4: Overlapping triangles
Problem status: not solved
Value (if solved during a running contest): 225
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 \(PABQ\).
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Problem 5: Order of permutation
Problem status: not solved
Value (if solved during a running contest): 300
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.
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Problem 6: p-adic valuation
Problem status: not solved
Value (if solved during a running contest): 300
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 \(m+n\).
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