| {"id":"aimepp-aime-0001","problem":"Let $X,Y,Z>1$ and $W>1$ satisfy $\\log_X W=54$, $\\log_Y W=24$, and $\\log_{X^{3}Y^{2}Z^{2}}W=6$. Determine $\\log_Z W$.","answer":72,"answer_str":"072","tier":"AIME"} |
| {"id":"aimepp-aime-0002","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,"answer_str":"038","tier":"AIME"} |
| {"id":"aimepp-aime-0003","problem":"Find the product of all real roots of \\[x^2-5x+15=8\\sqrt{x^2-5x}\\].","answer":225,"answer_str":"225","tier":"AIME"} |
| {"id":"aimepp-aime-0004","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,"answer_str":"027","tier":"AIME"} |
| {"id":"aimepp-aime-0005","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,"answer_str":"011","tier":"AIME"} |
| {"id":"aimepp-aime-0006","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,"answer_str":"293","tier":"AIME"} |
| {"id":"aimepp-aime-0007","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,"answer_str":"173","tier":"AIME"} |
| {"id":"aimepp-aime-0008","problem":"Find the largest two-digit prime divisor of $\\binom{164}{82}$.","answer":97,"answer_str":"097","tier":"AIME"} |
| {"id":"aimepp-aime-0009","problem":"Find the minimum value of $\\dfrac{36x^2\\sin^2x+25}{x\\sin x}$ for $0<x<\\pi$.","answer":60,"answer_str":"060","tier":"AIME"} |
| {"id":"aimepp-aime-0010","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,"answer_str":"040","tier":"AIME"} |
| {"id":"aimepp-aime-0011","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,"answer_str":"002","tier":"AIME"} |
| {"id":"aimepp-aime-0012","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,"answer_str":"065","tier":"AIME"} |
| {"id":"aimepp-aime-0013","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,"answer_str":"688","tier":"AIME"} |
| {"id":"aimepp-aime-0014","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,"answer_str":"065","tier":"AIME"} |
| {"id":"aimepp-aime-0015","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,"answer_str":"035","tier":"AIME"} |
| {"id":"aimepp-aime-0016","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,"answer_str":"070","tier":"AIME"} |
| {"id":"aimepp-aime-0017","problem":"How many odd integers from $250$ through $999$ have no repeated decimal digit and have digit sum divisible by $5$?","answer":54,"answer_str":"054","tier":"AIME"} |
| {"id":"aimepp-aime-0018","problem":"Find the area of a triangle with side lengths $10,13,13$.","answer":60,"answer_str":"060","tier":"AIME"} |
| {"id":"aimepp-aime-0019","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,"answer_str":"300","tier":"AIME"} |
| {"id":"aimepp-aime-0020","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,"answer_str":"528","tier":"AIME"} |
| {"id":"aimepp-aime-0021","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,"answer_str":"102","tier":"AIME"} |
| {"id":"aimepp-aime-0022","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,"answer_str":"991","tier":"AIME"} |
| {"id":"aimepp-aime-0023","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,"answer_str":"977","tier":"AIME"} |
| {"id":"aimepp-aime-0024","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,"answer_str":"014","tier":"AIME"} |
| {"id":"aimepp-aime-0025","problem":"How many pairs of positive integers $(x,y)$ with $x\\le y$ satisfy $\\frac1x+\\frac1y=\\frac1{147}$?","answer":8,"answer_str":"008","tier":"AIME"} |
| {"id":"aimepp-aime-0026","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,"answer_str":"003","tier":"AIME"} |
| {"id":"aimepp-aime-0027","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,"answer_str":"146","tier":"AIME"} |
| {"id":"aimepp-aime-0028","problem":"Angles $u$ and $v$ satisfy $\\tan u+\\tan v=34$ and $\\cot u+\\cot v=68$. Find $\\tan(u+v)$.","answer":68,"answer_str":"068","tier":"AIME"} |
| {"id":"aimepp-aime-0029","problem":"Find the smallest positive integer whose cube has last three digits $237$.","answer":933,"answer_str":"933","tier":"AIME"} |
| {"id":"aimepp-aime-0030","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,"answer_str":"558","tier":"AIME"} |
| {"id":"aimepp-aime-0031","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,"answer_str":"056","tier":"AIME"} |
| {"id":"aimepp-aime-0032","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,"answer_str":"536","tier":"AIME"} |
| {"id":"aimepp-aime-0033","problem":"How many rectangles have all four vertices among the vertices of a regular $32$-gon?","answer":120,"answer_str":"120","tier":"AIME"} |
| {"id":"aimepp-aime-0034","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,"answer_str":"808","tier":"AIME"} |
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