AIME I 2019

Consider the integer \( N = 9 + 99 + 999 + 9999 + \cdots.c + 99 \cdots 99 . \) Find the sum of the digits of \(N\) .

Jenn randomly chooses a number \(J\) from \(1 , 2 , 3 , \cdots , 19 , 20\) . Bela then randomly chooses a number \(B\) from \(1 , 2 , 3 , \cdots , 19 , 20\) distinct from \(J\) . The value of \(B - J\) is at least \(2\) with a probability that can be expressed in the form \(\dfrac{m}{n}\) , where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\) .

In \(\triangle P Q R\) , \(P R = 15\) , \(Q R = 20\) , and \(P Q = 25\) . Points \(A\) and \(B\) lie on \(\overline{P Q}\) , points \(C\) and \(D\) lie on \(\overline{Q R}\) , and points \(E\) and \(F\) lie on \(\overline{P R}\) , with \(P A = Q B = Q C = R D = R E = P F = 5\) . Find the area of hexagon \(A B C D E F\) .

A soccer team has \(22\) available players. A fixed set of \(11\) players starts the game, while the other \(11\) are available as substitutes. During the game, the coach may make as many as \(3\) substitutions, where any one of the \(11\) players in the game is replaced by one of the substitutes. No player removed from the game may reenter the game, although a substitute entering the game may be replaced later. No two substitutions can happen at the same time. The players involved and the order of the substitutions matter. Let \(n\) be the number of ways the coach can make substitutions during the game (including the possibility of making no substitutions). Find the remainder when \(n\) is divided by \(1000\) .

A moving particle starts at the point \(( 4 , 4 )\) and moves until it hits one of the coordinate axes for the first time. When the particle is at the point \(( a , b )\) , it moves at random to one of the points \(( a - 1 , b )\) , \(( a , b - 1 )\) , or \(( a - 1 , b - 1 )\) , each with probability \(\dfrac{1}{3}\) , independently of its previous moves. The probability that it will hit the coordinate axes at \(( 0 , 0 )\) is \(m / 3^n\) , where \(m\) and \(n\) are positive integers, and \(m\) is not divisible by \(3\) . Find \(m + n\) .

In convex quadrilateral \(K L M N\) , side \(\overline{M N}\) is perpendicular to diagonal \(\overline{K M}\) , side \(\overline{K L}\) is perpendicular to diagonal \(\overline{L N}\) , \(M N = 65\) , and \(K L = 28\) . The line through \(L\) perpendicular to side \(\overline{K N}\) intersects diagonal \(\overline{K M}\) at \(O\) with \(K O = 8\) . Find \(M O\) .

There are positive integers \(x\) and \(y\) that satisfy the system of equations \( \log_10 x + 2 \log_10 ( \gcd ( x , y ) ) = 60 \) \( \log_10 y + 2 \log_10 ( \text{\lcm} ( x , y ) ) = 570 . \) Let \(m\) be the number of (not necessarily distinct) prime factors in the prime factorization of \(x\) , and let \(n\) be the number of (not necessarily distinct) prime factors in the prime factorization of \(y\) . Find \(3 m + 2 n\) .

Let \(x\) be a real number such that \(\sin^10 x + \cos^10 x = \dfrac{11}{36}\) . Then \(\sin^12 x + \cos^12 x = \dfrac{m}{n}\) where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\) .

Let \(\tau ( n )\) denote the number of positive integer divisors of \(n\) . Find the sum of the six least positive integers \(n\) that are solutions to \(\tau ( n ) + \tau ( n + 1 ) = 7\) .

For distinct complex numbers \(z_1 , z_2 , \cdots , z_673\) , the polynomial \( ( x - z_1 )^3 ( x - z_2 )^3 \cdots.c ( x - z_673 )^3 \) can be expressed as \(x^2019 + 20 x^2018 + 19 x^2017 + g ( x )\) , where \(g ( x )\) is a polynomial with complex coefficients and with degree at most \(2016\) . The value of \( |\displaystyle\sum_{1 \leq j < k \leq 673} z_j z_k| \) can be expressed in the form \(\dfrac{m}{n}\) , where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\) .

In \(\triangle A B C\) , the sides have integer lengths and \(A B = A C\) . Circle \(\omega\) has its center at the incenter of \(\triangle A B C\) . An excircle of \(\triangle A B C\) is a circle in the exterior of \(\triangle A B C\) that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to \(\overline{B C}\) is internally tangent to \(\omega\) , and the other two excircles are both externally tangent to \(\omega\) . Find the minimum possible value of the perimeter of \(\triangle A B C\) .

Given \(f ( z ) = z^2 - 19 z\) , there are complex numbers \(z\) with the property that \(z\) , \(f ( z )\) , and \(f ( f ( z ) )\) are the vertices of a right triangle in the complex plane with a right angle at \(f ( z )\) . There are positive integers \(m\) and \(n\) such that one such value of \(z\) is \(m + \sqrt{n} + 11 i\) . Find \(m + n\) .

Triangle \(A B C\) has side lengths \(A B = 4\) , \(B C = 5\) , and \(C A = 6\) . Points \(D\) and \(E\) are on ray \(A B\) with \(A B < A D < A E\) . The point \(F \neq C\) is a point of intersection of the circumcircles of \(\triangle A C D\) and \(\triangle E B C\) satisfying \(D F = 2\) and \(E F = 7\) . Then \(B E\) can be expressed as \(\dfrac{a + b \sqrt{c}}{d}\) , where \(a\) , \(b\) , \(c\) , and \(d\) are positive integers such that \(a\) and \(d\) are relatively prime, and \(c\) is not divisible by the square of any prime. Find \(a + b + c + d\) .

Find the least odd prime factor of \(2019^8 + 1\) .

Let \(\overline{A B}\) be a chord of a circle \(\omega\) , and let \(P\) be a point on the chord \(\overline{A B}\) . 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 \(P Q\) intersects \(\omega\) at \(X\) and \(Y\) . Assume that \(A P = 5\) , \(P B = 3\) , \(X Y = 11\) , and \(P Q^2 = \dfrac{m}{n}\) , where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\) .