AIME I 2017

Fifteen distinct points are designated on \(\triangle A B C\) : the 3 vertices \(A\) , \(B\) , and \(C\) \(3\) other points on side \(\overline{A B}\) \(4\) other points on side \(\overline{B C}\) and \(5\) other points on side \(\overline{C A}\) . Find the number of triangles with positive area whose vertices are among these \(15\) points.

When each of \(702\) , \(787\) , and \(855\) is divided by the positive integer \(m\) , the remainder is always the positive integer \(r\) . When each of \(412\) , \(722\) , and \(815\) is divided by the positive integer \(n\) , the remainder is always the positive integer \(s \neq r\) . Find \(m + n + r + s\) .

For a positive integer \(n\) , let \(d_n\) be the units digit of \(1 + 2 + \cdots + n\) . Find the remainder when \( \displaystyle\sum_{n = 1}^2017 d_n \) is divided by \(1000\) .

A pyramid has a triangular base with side lengths \(20\) , \(20\) , and \(24\) . The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length \(25\) . The volume of the pyramid is \(m \sqrt{n}\) , where \(m\) and \(n\) are positive integers, and \(n\) is not divisible by the square of any prime. Find \(m + n\) .

A rational number written in base eight is \(\underline{a} \underline{b} . \underline{c} \underline{d}\) , where all digits are nonzero. The same number in base twelve is \(\underline{b} \underline{b} . \underline{b} \underline{a}\) . Find the base-ten number \(\underline{a} \underline{b} \underline{c}\) .

A circle circumscribes an isosceles triangle whose two congruent angles have degree measure \(x\) . Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is \(\dfrac{14}{25}\) . Find the difference between the largest and smallest possible values of \(x\) .

For nonnegative integers \(a\) and \(b\) with \(a + b \leq 6\) , let \(T ( a , b ) = \binom{6}{a} \binom{6}{b} \binom{6}{a + b}\) . Let \(S\) denote the sum of all \(T ( a , b )\) , where \(a\) and \(b\) are nonnegative integers with \(a + b \leq 6\) . Find the remainder when \(S\) is divided by \(1000\) .

Two real numbers \(a\) and \(b\) are chosen independently and uniformly at random from the interval \(( 0 , 75 )\) . Let \(O\) and \(P\) be two points on the plane with \(O P = 200\) . Let \(Q\) and \(R\) be on the same side of line \(O P\) such that the degree measures of \(\angle P O Q\) and \(\angle P O R\) are \(a\) and \(b\) respectively, and \(\angle O Q P\) and \(\angle O R P\) are both right angles. The probability that \(Q R \leq 100\) is equal to \(\dfrac{m}{n}\) , where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\) .

Let \(a_10 = 10\) , and for each positive integer \(n > 10\) let \(a_n = 100 a_{n - 1} + n\) . Find the least positive \(n > 10\) such that \(a_n\) is a multiple of \(99\) .

Let \(z_1 = 18 + 83 i\) , \(z_2 = 18 + 39 i ,\) and \(z_3 = 78 + 99 i ,\) where \(i = \sqrt{- 1}\) . Let \(z\) be the unique complex number with the properties that \(\dfrac{z_3 - z_1}{z_2 - z_1} \cdot \dfrac{z - z_2}{z - z_3}\) is a real number and the imaginary part of \(z\) is the greatest possible. Find the real part of \(z\) .

Consider arrangements of the \(9\) numbers \(1 , 2 , 3 , \cdots , 9\) in a \(3 \times 3\) array. For each such arrangement, let \(a_1\) , \(a_2\) , and \(a_3\) be the medians of the numbers in rows \(1\) , \(2\) , and \(3\) respectively, and let \(m\) be the median of \({ a_1 , a_2 , a_3 }\) . Let \(Q\) be the number of arrangements for which \(m = 5\) . Find the remainder when \(Q\) is divided by \(1000\) .

Call a set \(S\) product-free if there do not exist \(a , b , c \in S\) (not necessarily distinct) such that \(a b = c\) . For example, the empty set and the set \({ 16 , 20 }\) are product-free, whereas the sets \({ 4 , 16 }\) and \({ 2 , 8 , 16 }\) are not product-free. Find the number of product-free subsets of the set \({ 1 , 2 , 3 , 4 , \cdots , 7 , 8 , 9 , 10 }\) .

For every \(m \geq 2\) , let \(Q ( m )\) be the least positive integer with the following property: For every \(n \geq Q ( m )\) , there is always a perfect cube \(k^3\) in the range \(n < k^3 \leq m n\) . Find the remainder when \( \displaystyle\sum_{m = 2}^2017 Q ( m ) \) is divided by \(1000\) .

Let \(a > 1\) and \(x > 1\) satisfy \(\log_a ( \log_a ( \log_a 2 ) + \log_a 24 - 128 ) = 128\) and \(\log_a ( \log_a x ) = 256\) . Find the remainder when \(x\) is divided by \(1000\) .

The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths \(2 \sqrt{3}\) , \(5\) , and \(\sqrt{37}\) , as shown, is \(\dfrac{m \sqrt{p}}{n}\) , where \(m\) , \(n\) , and \(p\) are positive integers, \(m\) and \(n\) are relatively prime, and \(p\) is not divisible by the square of any prime. Find \(m + n + p\) .