answer: 72

tier: AIME


problem: For real $x$ with $-8\le x\le 25$, define $F(x)=|x+4|+2|x-10|+|x-12|+2|x-21|$. Find the minimum value of $F(x)$.

answer: 38


problem: Find the product of all real roots of \[x^2-5x+15=8\sqrt{x^2-5x}\].

answer: 225


problem: A circle has radius $25$. Two parallel chords of lengths $48$ and $30$ lie on opposite sides of the center. Find the distance between the chords.

answer: 27


problem: Complex numbers $x$ and $y$ satisfy $x^2+y^2=91$ and $x^3+y^3=836$. Find the largest real value that $x+y$ can have.

answer: 11


problem: A sequence is defined by $a_1=8$, $a_2=27$, and $a_n=3a_{n-1}+6a_{n-2}$ for $n\ge3$. Find the remainder when $a_{829}$ is divided by $1000$.

answer: 293


problem: Of $31$ people seated around a circular table, three are chosen uniformly at random. The probability that at least two chosen people were adjacent is $\frac{p}{q}$ in lowest terms. Find $p+q$.

answer: 173


problem: Find the largest two-digit prime divisor of $\binom{164}{82}$.

answer: 97


problem: Find the minimum value of $\dfrac{36x^2\sin^2x+25}{x\sin x}$ for $0<x<\pi$.

answer: 60


problem: Consider $5$-digit decimal integers beginning with $8$ that have exactly one pair of equal digits, with every other digit occurring once. Find the remainder when the number of such integers is divided by $1000$.

answer: 40


problem: A rectangular box has side lengths $23$, $25$, and $h$. The centers of the three faces meeting at one vertex form a triangle of area $\frac{579}{8}$. Find $h$.

answer: 2


problem: In a circle, a diameter has the two-digit base-$10$ length $\overline{ab}_{10}$, and a perpendicular chord has length $\overline{ba}_{10}$. The diameter is longer, the two digits sum to $11$, and the distance from the chord to the center is rational. Find the diameter's value in base $10$.

answer: 65


problem: For every nonempty subset of $\{1,2,\ldots,14\}$, list its elements in decreasing order and alternately add and subtract, beginning with addition. Find the remainder when the sum of these alternating sums is divided by $1000$.

answer: 688


problem: In a circle, a diameter has the two-digit base-$9$ length $\overline{ab}_{9}$, and a perpendicular chord has length $\overline{ba}_{9}$. The diameter is longer, the two digits sum to $9$, and the distance from the chord to the center is rational. Find the diameter's value in base $10$.

answer: 65


problem: In a circle, a diameter has the two-digit base-$8$ length $\overline{ab}_{8}$, and a perpendicular chord has length $\overline{ba}_{8}$. The diameter is longer, the two digits sum to $7$, and the distance from the chord to the center is rational. Find the diameter's value in base $10$.

answer: 35


problem: The $8$ positive integer terms of an arithmetic sequence have total $420$, and the greatest term is $2$ times the least term. Find the greatest term.

answer: 70


problem: How many odd integers from $250$ through $999$ have no repeated decimal digit and have digit sum divisible by $5$?

answer: 54


problem: Find the area of a triangle with side lengths $10,13,13$.

answer: 60


problem: Positive integers $a\le b$ satisfy $\gcd(a,b)=3$ and $\operatorname{lcm}(a,b)=228$. Find the sum of all possible values of $a+b$.

answer: 300


problem: Let $X,Y,Z>1$ and $W>1$ satisfy $\log_X W=792$, $\log_Y W=352$, and $\log_{X^{3}YZ^{2}}W=96$. Determine $\log_Z W$.

answer: 528


problem: For real $x$ with $-21\le x\le 26$, define $F(x)=2|x+17|+3|x-8|+|x-18|+3|x-22|$. Find the minimum value of $F(x)$.

answer: 102


problem: A polynomial $P$ has degree at most $4$ and satisfies $P(0)=1$, $P(1)=-1$, $P(2)=-47$, $P(3)=-257$, $P(4)=-823$. Find the remainder when $P(5)$ is divided by $1000$.

answer: 991


problem: The nonreal roots of $z^2-7z+24=0$ are $\alpha$ and $\beta$. Find the remainder when $\alpha^{36}+\beta^{36}$ is divided by $1000$.

answer: 977


problem: A rectangular box has side lengths $3$, $8$, and $h$. The centers of the three faces meeting at one vertex form a triangle of area $\frac{61}{4}$. Find $h$.

answer: 14


problem: How many pairs of positive integers $(x,y)$ with $x\le y$ satisfy $\frac1x+\frac1y=\frac1{147}$?

answer: 8


problem: Three distinct vertices of a regular $10$-gon are chosen uniformly at random. The probability that they form an obtuse triangle is $\frac{m}{q}$ in lowest terms. Find $m+q$.

answer: 3


problem: How many subsets $S$ of $\{1,2,\ldots,10\}$ satisfy $\sum_{s\in S}s\equiv 4\pmod{7}$? The empty set is allowed.

answer: 146


problem: Angles $u$ and $v$ satisfy $\tan u+\tan v=34$ and $\cot u+\cot v=68$. Find $\tan(u+v)$.

answer: 68


problem: Find the smallest positive integer whose cube has last three digits $237$.

answer: 933


problem: A faculty has three departments: department 1 has 2 men and 3 women; department 2 has 3 men and 2 women; department 3 has 3 men and 3 women. A six-person committee must contain exactly two people from each department and exactly three men. How many committees are possible?

answer: 558


problem: A $6\times 5\times 9$ rectangular block is painted on all six faces and cut into unit cubes. How many unit cubes have exactly two painted faces?

answer: 56


problem: A faculty has three departments: department 1 has 2 men and 3 women; department 2 has 3 men and 2 women; department 3 has 4 men and 2 women. A six-person committee must contain exactly two people from each department and exactly three men. How many committees are possible?

answer: 536


problem: How many rectangles have all four vertices among the vertices of a regular $32$-gon?

answer: 120


problem: A polynomial $P$ has degree at most $4$ and satisfies $P(0)=-2$, $P(1)=8$, $P(2)=30$, $P(3)=40$, $P(4)=-10$. Find the remainder when $P(5)$ is divided by $1000$.

answer: 808