AIME I 2022

Quadratic polynomials \(P ( x )\) and \(Q ( x )\) have leading coefficients \(2\) and \(- 2 ,\) respectively. The graphs of both polynomials pass through the two points \(( 16 , 54 )\) and \(( 20 , 53 ) .\) Find \(P ( 0 ) + Q ( 0 ) .\)

Find the three-digit positive integer \(\underline{a} \underline{b} \underline{c}\) whose representation in base nine is \(\underline{b} \underline{c} \underline{a}_{\text{nine}} ,\) where \(a ,\) \(b ,\) and \(c\) are (not necessarily distinct) digits.

In isosceles trapezoid \(A B C D ,\) parallel bases \(\overline{A B}\) and \(\overline{C D}\) have lengths \(500\) and \(650 ,\) respectively, and \(A D = B C = 333 .\) The angle bisectors of \(\angle A\) and \(\angle D\) meet at \(P ,\) and the angle bisectors of \(\angle B\) and \(\angle C\) meet at \(Q .\) Find \(P Q .\)

Let \(w = \dfrac{\sqrt{3} + i}{2}\) and \(z = \dfrac{- 1 + i \sqrt{3}}{2} ,\) where \(i = \sqrt{- 1} .\) Find the number of ordered pairs \(( r , s )\) of positive integers not exceeding \(100\) that satisfy the equation \(i \cdot w^r = z^s .\)

A straight river that is \(264\) meters wide flows from west to east at a rate of \(14\) meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of \(D\) meters downstream from Sherry. Relative to the water, Melanie swims at \(80\) meters per minute, and Sherry swims at \(60\) meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find \(D .\)

Find the number of ordered pairs of integers \(( a , b )\) such that the sequence \( 3 , 4 , 5 , a , b , 30 , 40 , 50 \) is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.

Let \(a , b , c , d , e , f , g , h , i\) be distinct integers from \(1\) to \(9 .\) The minimum possible positive value of \( \dfrac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i} \) can be written as \(\dfrac{m}{n} ,\) where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n .\)

Equilateral triangle \(\triangle A B C\) is inscribed in circle \(\omega\) with radius \(18 .\) Circle \(\omega_A\) is tangent to sides \(\overline{A B}\) and \(\overline{A C}\) and is internally tangent to \(\omega .\) Circles \(\omega_B\) and \(\omega_C\) are defined analogously. Circles \(\omega_A ,\) \(\omega_B ,\) and \(\omega_C\) meet in six points---two points for each pair of circles. The three intersection points closest to the vertices of \(\triangle A B C\) are the vertices of a large equilateral triangle in the interior of \(\triangle A B C ,\) and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of \(\triangle A B C .\) The side length of the smaller equilateral triangle can be written as \(\sqrt{a} - \sqrt{b} ,\) where \(a\) and \(b\) are positive integers. Find \(a + b .\)

Ellina has twelve blocks, two each of red ( \(\mathbf{\text{R}}\) ), blue ( \(\mathbf{\text{B}}\) ), yellow ( \(\mathbf{\text{Y}}\) ), green ( \(\mathbf{\text{G}}\) ), orange ( \(\mathbf{\text{O}}\) ), and purple ( \(\mathbf{\text{P}}\) ). Call an arrangement of blocks \(\mathit{\text{even}}\) if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement \( \mathbf{\text{R B B Y G G Y R O P P O}} \) is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is \(\dfrac{m}{n} ,\) where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n .\)

Three spheres with radii \(11 ,\) \(13 ,\) and \(19\) are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at \(A ,\) \(B ,\) and \(C ,\) respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that \(A B^2 = 560 .\) Find \(A C^2 .\)

Let \(A B C D\) be a parallelogram with \(\angle B A D < 90^\circ .\) A circle tangent to sides \(\overline{D A} ,\) \(\overline{A B} ,\) and \(\overline{B C}\) intersects diagonal \(\overline{A C}\) at points \(P\) and \(Q\) with \(A P < A Q ,\) as shown. Suppose that \(A P = 3 ,\) \(P Q = 9 ,\) and \(Q C = 16 .\) Then the area of \(A B C D\) can be expressed 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 \(m + n .\)

For any finite set \(X ,\) let \(\| X \|\) denote the number of elements in \(X .\) Define \( S_n = \sum \| A \cap B \| , \) where the sum is taken over all ordered pairs \(( A , B )\) such that \(A\) and \(B\) are subsets of \({ 1 , 2 , 3 , \cdots , n }\) with \(\| A \| = \| B \| .\) For example, \(S_2 = 4\) because the sum is taken over the pairs of subsets \( ( A , B ) \in {( nothing , nothing ) , ( { 1 } , { 1 } ) , ( { 1 } , { 2 } ) , ( { 2 } , { 1 } ) , ( { 2 } , { 2 } ) , ( { 1 , 2 } , { 1 , 2 } )} , \) giving \(S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4 .\) Let \(\dfrac{S_2022}{S_2021} = \dfrac{p}{q} ,\) where \(p\) and \(q\) are relatively prime positive integers. Find the remainder when \(p + q\) is divided by \(1000 .\)

Let \(S\) be the set of all rational numbers that can be expressed as a repeating decimal in the form \(0 . \overline{a b c d} ,\) where at least one of the digits \(a ,\) \(b ,\) \(c ,\) or \(d\) is nonzero. Let \(N\) be the number of distinct numerators obtained when numbers in \(S\) are written as fractions in lowest terms. For example, both \(4\) and \(410\) are counted among the distinct numerators for numbers in \(S\) because \(0 . \overline{3636} = \dfrac{4}{11}\) and \(0 . \overline{1230} = \dfrac{410}{3333} .\) Find the remainder when \(N\) is divided by \(1000 .\)

Given \(\triangle A B C\) and a point \(P\) on one of its sides, call line \(\ell\) the \(\mathit{\text{splitting line}}\) of \(\triangle A B C\) through \(P\) if \(\ell\) passes through \(P\) and divides \(\triangle A B C\) into two polygons of equal perimeter. Let \(\triangle A B C\) be a triangle where \(B C = 219\) and \(A B\) and \(A C\) are positive integers. Let \(M\) and \(N\) be the midpoints of \(\overline{A B}\) and \(\overline{A C} ,\) respectively, and suppose that the splitting lines of \(\triangle A B C\) through \(M\) and \(N\) intersect at \(30^\circ .\) Find the perimeter of \(\triangle A B C .\)

Let \(x ,\) \(y ,\) and \(z\) be positive real numbers satisfying the system of equations: Then \([( 1 - x ) ( 1 - y ) ( 1 - z )]^2\) can be written as \(\dfrac{m}{n} ,\) where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n .\)