Let \(S\) be the set of positive integers \(N\) with the property that the last four digits of \(N\) are \(2020\), and when the last four digits are removed, the result is a divisor of \(N\). For example, \(42,020\) is in \(S\) because \(4\) is a divisor of \(42,020\). Find the sum of all the digits of all the numbers in \(S\). For example, the number \(42,020\) contributes \(4+2+0+2+0=8\) to this total.
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Source: AIME 2020 I Q4
Elo Rating: 1500.00
Let \(\overline{AB}\) be a chord of a circle \(\omega\), and let \(P\) be a point on the chord \(\overline{AB}\). Circle \(\omega_1\) passes through \(A\) and \(P\) and is internally tangent to \(\omega\). Circle \(\omega_2\) passes through \(B\) and \(P\) and is internally tangent to \(\omega\). Circles \(\omega_1\) and \(\omega_2\) intersect at points \(P\) and \(Q\). Line \(PQ\) intersects \(\omega\) at \(X\) and \(Y\). Assume that \(AP=5\), \(PB=3\), \(XY=11\), and \(PQ^2 = \tfrac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).
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Source: AIME 2019 I Q15
Elo Rating: 1500.00