AIME II 2019

Two different points, \(C\) and \(D\) , lie on the same side of line \(A B\) so that \(\triangle A B C\) and \(\triangle B A D\) are congruent with \(A B = 9 , B C = A D = 10\) , and \(C A = D B = 17\) . The intersection of these two triangular regions has area \(\dfrac{m}{n}\) , where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\) .

Lily pads \(1 , 2 , 3 , \cdots\) lie in a row on a pond. A frog makes a sequence of jumps starting on pad \(1\) . From any pad \(k\) the frog jumps to either pad \(k + 1\) or pad \(k + 2\) chosen randomly with probability \(\dfrac{1}{2}\) and independently of other jumps. The probability that the frog visits pad \(7\) is \(\dfrac{p}{q}\) , where \(p\) and \(q\) are relatively prime positive integers. Find \(p + q\) .

Find the number of \(7\) -tuples of positive integers \(( a , b , c , d , e , f , g )\) that satisfy the following system of equations: \( a b c = 70 \) \( c d e = 71 \) \( e f g = 72 . \)

A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is \(\dfrac{m}{n}\) , where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\) .

Four ambassadors and one advisor for each of them are to be seated at a round table with \(12\) chairs numbered in order \(1\) to \(12\) . Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are \(N\) ways for the \(8\) people to be seated at the table under these conditions. Find the remainder when \(N\) is divided by \(1000\) .

In a Martian civilization, all logarithms whose bases are not specified are assumed to be base \(b\) , for some fixed \(b \geq 2\) . A Martian student writes down \( 3 \log ( \sqrt{x} \log x ) = 56 \) \( \log_{\log x} ( x ) = 54 \) and finds that this system of equations has a single real number solution \(x > 1\) . Find \(b\) .

Triangle \(A B C\) has side lengths \(A B = 120 , B C = 220\) , and \(A C = 180\) . Lines \(\ell_A , \ell_B\) , and \(\ell_C\) are drawn parallel to \(\overline{B C} , \overline{A C}\) , and \(\overline{A B}\) , respectively, such that the intersections of \(\ell_A , \ell_B\) , and \(\ell_C\) with the interior of \(\triangle A B C\) are segments of lengths \(55 , 45\) , and \(15\) , respectively. Find the perimeter of the triangle whose sides lie on lines \(\ell_A , \ell_B\) , and \(\ell_C\) .

The polynomial \(f ( z ) = a z^2018 + b z^2017 + c z^2016\) has real coefficients not exceeding \(2019 ,\) and \(f \left(\dfrac{1 + \sqrt{3} i}{2}\right) = 2015 + 2019 \sqrt{3} i\) . Find the remainder when \(f ( 1 )\) is divided by \(1000\) .

Call a positive integer \(n\) \(k\) - pretty if \(n\) has exactly \(k\) positive divisors and \(n\) is divisible by \(k\) . For example, \(18\) is \(6\) -pretty. Let \(S\) be the sum of the positive integers less than \(2019\) that are \(20\) -pretty. Find \(\dfrac{S}{20}\) .

There is a unique angle \(\theta\) between \(0^\circ\) and \(90^\circ\) such that for nonnegative integers \(n ,\) the value of \(\tan ( 2^n \theta )\) is positive when \(n\) is a multiple of \(3\) , and negative otherwise. The degree measure of \(\theta\) is \(\dfrac{p}{q}\) , where \(p\) and \(q\) are relatively prime positive integers. Find \(p + q\) .

Triangle \(A B C\) has side lengths \(A B = 7 , B C = 8 ,\) and \(C A = 9 .\) Circle \(\omega_1\) passes through \(B\) and is tangent to line \(A C\) at \(A .\) Circle \(\omega_2\) passes through \(C\) and is tangent to line \(A B\) at \(A .\) Let \(K\) be the intersection of circles \(\omega_1\) and \(\omega_2\) not equal to \(A .\) Then \(A K = \dfrac{m}{n} ,\) where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n .\)

For \(n \geq 1\) call a finite sequence \(( a_1 , a_2 , \cdots , a_n )\) of positive integers progressive if \(a_i < a_{i + 1}\) and \(a_i\) divides \(a_{i + 1}\) for \(1 \leq i \leq n - 1\) . Find the number of progressive sequences such that the sum of the terms in the sequence is equal to \(360 .\)

Regular octagon \(A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8\) is inscribed in a circle of area \(1 .\) Point \(P\) lies inside the circle so that the region bounded by \(\overline{P A_1} , \overline{P A_2} ,\) and the minor arc \(A_1 A_2\) of the circle has area \(\dfrac{1}{7} ,\) while the region bounded by \(\overline{P A_3} , \overline{P A_4} ,\) and the minor arc \(A_3 A_4\) of the circle has area \(\dfrac{1}{9} .\) There is a positive integer \(n\) such that the area of the region bounded by \(\overline{P A_6} , \overline{P A_7} ,\) and the minor arc \(A_6 A_7\) of the circle is equal to \(\dfrac{1}{8} - \dfrac{\sqrt{2}}{n} .\) Find \(n .\)

Find the sum of all positive integers \(n\) such that, given an unlimited supply of stamps of denominations \(5 , n ,\) and \(n + 1\) cents, \(91\) cents is the greatest postage that cannot be formed.

In acute triangle \(A B C ,\) points \(P\) and \(Q\) are the feet of the perpendiculars from \(C\) to \(\overline{A B}\) and from \(B\) to \(\overline{A C}\) , respectively. Line \(P Q\) intersects the circumcircle of \(\triangle A B C\) in two distinct points, \(X\) and \(Y\) . Suppose \(X P = 10\) , \(P Q = 25\) , and \(Q Y = 15\) . The value of \(A B \cdot A C\) can be written in the form \(m \sqrt{n}\) where \(m\) and \(n\) are positive integers, and \(n\) is not divisible by the square of any prime. Find \(m + n\) .