AIME II 2024

Among the \(900\) residents of Aimeville, there are \(195\) who own a diamond ring, \(367\) who own a set of golf clubs, and \(562\) who own a garden spade. In addition, each of the \(900\) residents owns a bag of candy hearts. There are \(437\) residents who own exactly two of these things, and \(234\) residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.

A list of positive integers has the following properties: \(bullet\) The sum of the items in the list is \(30\) . \(bullet\) The unique mode of the list is \(9\) . \(bullet\) The median of the list is a positive integer that does not appear in the list itself. Find the sum of the squares of all the items in the list.

Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is \(999\) , and the sum of the three numbers formed by reading top to bottom is \(99\) . The grid below is an example of such an arrangement because \(8 + 991 = 999\) and \(9 + 9 + 81 = 99\) . \( 0 & 0 & 8 9 & 9 & 1 \)

Let \(x, y\) and \(z\) be positive real numbers that satisfy the following system of equations: \( \log_2\left(\dfrac{x}{y z}\right) = \dfrac{1}{2} \) \( \log_2\left(\dfrac{y}{x z}\right) = \dfrac{1}{3} \) \( \log_2\left(\dfrac{z}{x y}\right) = \dfrac{1}{4} \) Then the value of \(|\log_2(x^4 y^3 z^2)|\) is \(\dfrac{m}{n}\) where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\).

Let \(A B C D E F\) be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments \(\overline{A B}\) , \(\overline{C D}\) , and \(\overline{E F}\) has side lengths \(200 , 240 ,\) and \(300\) . Find the side length of the hexagon.

Alice chooses a set \(A\) of positive integers. Then Bob lists all finite nonempty sets \(B\) of positive integers with the property that the maximum element of \(B\) belongs to \(A\) . Bob's list has \(2024\) sets. Find the sum of the elements of \(A\) .

Let \(N\) be the greatest four-digit integer with the property that whenever one of its digits is changed to \(1\) , the resulting number is divisible by \(7\) . Let \(Q\) and \(R\) be the quotient and remainder, respectively, when \(N\) is divided by \(1000\) . Find \(Q + R\) .

Torus \(T\) is the surface produced by revolving a circle with radius 3 around an axis in the plane of the circle that is a distance 6 from the center of the circle (so like a donut). Let \(S\) be a sphere with a radius 11. When \(T\) rests on the inside of \(S\) , it is internally tangent to \(S\) along a circle with radius \(r_i\) , and when \(T\) rests on the outside of \(S\) , it is externally tangent to \(S\) along a circle with radius \(r_o\) . The difference \(r_i - r_o\) can be written as \(\dfrac{m}{n}\) , where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n\) .

Let \(\triangle\) \(A B C\) have incenter \(I\) and circumcenter \(O\) with \(\overline{I A} \perp \overline{O I}\) , circumradius \(13\) , and inradius \(6\) . Find \(A B \cdot A C\) .

Find the number of triples of nonnegative integers \(( a , b , c )\) satisfying \(a + b + c = 300\) and \( a^2 b + a^2 c + b^2 a + b^2 c + c^2 a + c^2 b = 6 , 000 , 000 . \)

Let \(O ( 0 , 0 ) , A \left( \dfrac{1}{2} , 0 \right) ,\) and \(B \left( 0 , \dfrac{\sqrt{3}}{2} \right)\) be points in the coordinate plane. Let \(\mathcal{F}\) be the family of segments \(\overline{P Q}\) of unit length lying in the first quadrant with \(P\) on the \(x\) -axis and \(Q\) on the \(y\) -axis. There is a unique point \(C\) on \(\overline{A B} ,\) distinct from \(A\) and \(B ,\) that does not belong to any segment from \(\mathcal{F}\) other than \(\overline{A B}\) . Then \(O C^2 = \dfrac{p}{q}\) , where \(p\) and \(q\) are relatively prime positive integers. Find \(p + q\) .

Let \(\omega \neq 1\) be a 13th root of unity. Find the remainder when \( \displaystyle\prod_{k = 0}^12 ( 2 - 2 \omega^k + \omega^{2 k} ) \) is divided by \1000.

Let \(b \geq 2\) be an integer. Call a positive integer \(n\) \(b \mathit{\text{-eautiful}}\) if it has exactly two digits when expressed in base \(b\) , and these two digits sum to \(\sqrt{n}\) . For example, \(81\) is \(13\) -eautiful because \(81 = \underline{6}\) \(\underline{3}_13\) and \(6 + 3 = \sqrt{81}\) . Find the least integer \(b \geq 2\) for which there are more than ten \(b\) -eautiful integers.

Find the number of rectangles that can be formed inside a fixed regular dodecagon ( \(12\) -gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles.