messages listlengths 2 2 | extra_info dict |
|---|---|
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "For every subset $T$ of $U = \\{ 1,2,3,\\ldots,18 \\}$, let $s(T)$ be the sum of the elements of $T$, with $s(\\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets ... | {
"answer": "683"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "An ordered pair $( b , c )$ of integers, each of which has absolute value less than or equal to five, is chosen at random, with each\nsuch ordered pair having an equal likelihood of being... | {
"answer": "\\frac{111}{121}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions:\n(a) If $16$, $15$, or $14$ students... | {
"answer": "148"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$. However, if $68$ is remove... | {
"answer": "649"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A $9 \\times 9 \\times 9$ [cube](https://artofproblemsolving.com/wiki/index.php/Cube) is composed of twenty-seven $3 \\times 3 \\times 3$ cubes. The big cube is ‘tunneled’ as follows: Fir... | {
"answer": "1056"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisf... | {
"answer": "540"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $L$ be the line with slope $\\frac{5}{12}$ that contains the point $A=(24,-1)$, and let $M$ be the line perpendicular to line $L$ that contains the point $B=(5,6)$. The original coor... | {
"answer": "31"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A point $P$ is outside a circle and is $13$ inches from the center. A secant from $P$ cuts the circle at $Q$ and $R$ \nso that the external segment of the secant $PQ$ is $9$ inches and $Q... | {
"answer": "5"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Pegs are put in a board $1$ unit apart both horizontally and vertically. A rubber band is stretched over $4$ pegs as shown in the figure, forming a [quadrilateral](https://artofproblemsol... | {
"answer": "6"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A box contains chips, each of which is red, white, or blue. The number of blue chips is at least half the number of white chips, and at most one third the number of red chips. The number ... | {
"answer": "57"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes th... | {
"answer": "7"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest prime that is the fifth term of an increasing [arithmetic sequence](https://artofproblemsolving.com/wiki/index.php/Arithmetic_sequence), all four preceding terms also be... | {
"answer": "29"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A computer screen shows a $98 \\times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse but... | {
"answer": "98"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "At each of the sixteen circles in the network below stands a student. A total of $3360$ coins are distributed among the sixteen students. All at once, all students give away all their coi... | {
"answer": "280"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and the denominator have a sum of 1000.",
"role": "user"
... | {
"answer": "200"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "What is the largest even integer that cannot be written as the sum of two odd composite numbers?",
"role": "user"
}
] | {
"answer": "38"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be the set of positive integers $N$ with the property that the last four digits of $N$ are $2020,$ and when the last four digits are removed, the result is a divisor of $N.$ For e... | {
"answer": "93"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $(a_1,b_1),$ $(a_2,b_2),$ $\\dots,$ $(a_{100},b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i,j)$ satisfying ... | {
"answer": "197"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "If $\\{a_1,a_2,a_3,\\ldots,a_n\\}$ is a [set](https://artofproblemsolving.com/wiki/index.php/Set) of [real numbers](https://artofproblemsolving.com/wiki/index.php/Real_numbers), indexed s... | {
"answer": "368"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Point $P$ is inside $\\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure below). Given that $AP=6$, $BP=9$, $PD... | {
"answer": "108"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ be the least positive integer for which $149^n-2^n$ is divisible by $3^3\\cdot5^5\\cdot7^7.$ Find the number of positive integer divisors of $n.$",
"role": "user"
}
] | {
"answer": "270"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be the set of [ordered pairs](https://artofproblemsolving.com/wiki/index.php/Ordered_pair) $(x, y)$ such that $0 < x \\le 1, 0<y\\le 1,$ and $\\left[\\log_2{\\left(\\frac 1x\\righ... | {
"answer": "14"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $S\\,$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S\\,$ so that the union of the two subsets is $S\\,$? The order ... | {
"answer": "365"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The [equation](https://artofproblemsolving.com/wiki/index.php/Equation) $2^{333x-2} + 2^{111x+2} = 2^{222x+1} + 1$ has three [real](https://artofproblemsolving.com/wiki/index.php/Real) [r... | {
"answer": "113"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "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 $OP = 200$. Let $Q$ and $R$ be o... | {
"answer": "41"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) = \\max_{\\substack{1\\le k\\le \\frac{n}{2}}} f(n, k)$. Find the r... | {
"answer": "512"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Circles $\\omega_1$ and $\\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\\omega$ is externally tangent to both $\\omega_1$ ... | {
"answer": "672"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $ABCD$ be an isosceles trapezoid with $AD=BC$ and $AB<CD.$ Suppose that the distances from $A$ to the lines $BC,CD,$ and $BD$ are $15,18,$ and $10,$ respectively. Let $K$ be the area ... | {
"answer": "567"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "As shown in the figure below, a regular dodecahedron (the polyhedron consisting of $12$ congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a... | {
"answer": "810"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came bac... | {
"answer": "90"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "What is the least common multiple of ${\\frac{1}{x}}$, $\\frac{1}{2x}$, and $\\frac{1}{3x}$ is $\\frac{1}{6x}$?",
"role": "user"
}
] | {
"answer": "\\frac{1}{6x}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "(Gregory Galperin) A [square](https://artofproblemsolving.com/wiki/index.php/Square) grid on the [Euclidean plane](https://artofproblemsolving.com/wiki/index.php/Cartesian_plane) consists... | {
"answer": "\\text{No}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8$, and the second term of both series can be... | {
"answer": "518"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "For each integer $n\\geq3$, let $f(n)$ be the number of $3$-element subsets of the vertices of the regular $n$-gon that are the vertices of an isosceles triangle (including equilateral tr... | {
"answer": "245"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits 1 through 9 in such a way that the sum of the numbers on every three consecutive vertices is a multip... | {
"answer": "144"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "In rectangle $ABCD$, $AB = 12$ and $BC = 10$. Points $E$ and $F$ lie inside rectangle $ABCD$ so that $BE = 9$, $DF = 8$, $\\overline{BE} \\parallel \\overline{DF}$, $\\overline{EF} \\par... | {
"answer": "36"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Consider arrangements of the $9$ numbers $1, 2, 3, \\dots, 9$ 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 $1$, $... | {
"answer": "360"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Sides $AB,~ BC, ~CD$ and $DA$, respectively, of convex quadrilateral $ABCD$ are extended past $B,~ C ,~ D$ and $A$ to points $B',~C',~ D'$ and $A'$. Also, $AB = BB' = 6,~ BC = CC' = 7, ~C... | {
"answer": "50"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C... | {
"answer": "409"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let the sum of a set of numbers be the sum of its elements. Let $S$ be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of $S$ have the same sum. What is ... | {
"answer": "61"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. T... | {
"answer": "47"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "For a permutation $p = (a_1,a_2,\\ldots,a_9)$ of the digits $1,2,\\ldots,9$, let $s(p)$ denote the sum of the three $3$-digit numbers $a_1a_2a_3$, $a_4a_5a_6$, and $a_7a_8a_9$. Let $m$ be... | {
"answer": "162"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Define a T-grid to be a $3\\times3$ matrix which satisfies the following two properties:\n\n\nExactly five of the entries are $1$'s, and the remaining four entries are $0$'s.\nAmong the e... | {
"answer": "68"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "An $m\\times n\\times p$ rectangular box has half the volume of an $(m + 2)\\times(n + 2)\\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\\le n\\le p.$ What is... | {
"answer": "130"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may be any of the points $(x_n + ... | {
"answer": "373"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The increasing [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) $3, 15, 24, 48, \\ldots\\,$ consists of those [positive](https://artofproblemsolving.com/wiki/index.php/... | {
"answer": "063"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Segments $\\overline{AB}, \\overline{AC},$ and $\\overline{AD}$ are edges of a cube and $\\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\\sqrt{10}... | {
"answer": "192"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubi... | {
"answer": "6"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$ is formed by removing 6-8-10 triangles from the corners of a $23$ $\\times$ $27$ rectangle with ... | {
"answer": "184"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it r... | {
"answer": "351"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1... | {
"answer": "110"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds late... | {
"answer": "52"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A [convex](https://artofproblemsolving.com/wiki/index.php/Convex) [polyhedron](https://artofproblemsolving.com/wiki/index.php/Polyhedron) has for its [faces](https://artofproblemsolving.c... | {
"answer": "840"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "An integer is called snakelike if its decimal representation $a_1a_2a_3\\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is [ odd](https://artofproblemsolving.com/wiki/index.php/Odd_integer) an... | {
"answer": "882"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ be the number of integer values of $x$ such that $P = x^4 + 6x^3 + 11x^2 + 3x + 31$ is the square of an integer. Then $n$ is: \n$\\textbf{(A)}\\ 4 \\qquad \\textbf{(B) }\\ 3 \\qq... | {
"answer": "(D) \\ 1"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The area of the region bounded by the graph of\\[x^2+y^2 = 3|x-y| + 3|x+y|\\]is $m+n\\pi$, where $m$ and $n$ are integers. What is $m + n$?\n$\\textbf{(A)} ~18\\qquad\\textbf{(B)} ~27\\qq... | {
"answer": "54"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the cente... | {
"answer": "\\frac{n+1}{4n-2}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A circular disc with diameter $D$ is placed on an $8\\times 8$ checkerboard with width $D$ so that the centers coincide. The number of checkerboard squares which are completely covered by... | {
"answer": "32"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "At the Rice Mathematics Tournament, 80% of contestants wear blue jeans, 70% wear tennis shoes, and 80% of those who wear blue jeans also wear tennis shoes. What fraction of people wearing... | {
"answer": "\\frac{32}{35}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A number of runners competed in a race. When Ammar finished, there were half as many runners who had finished before him compared to the number who finished behind him. Julia was the 10th... | {
"answer": "31"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A cube with side length $100cm$ is filled with water and has a hole through which the water drains into a cylinder of radius $100cm.$ If the water level in the cube is falling at a ra... | {
"answer": "\\frac{1}{\\pi}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "An equilateral triangle is inscribed inside of a circle of radius $R$ . Find the side length of the triangle",
"role": "user"
}
] | {
"answer": " R\\sqrt{3} "
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "If $0 < x < \\frac{\\pi}{2}$ and $\\frac{\\sin x}{1 + \\cos x} = \\frac{1}{3}$ , what is $\\frac{\\sin 2x}{1 + \\cos 2x}$ ?",
"role": "user"
}
] | {
"answer": "\\frac{3}{4}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of all four-digit integers whose digits are a rearrangement of the digits $1$ , $2$ , $3$ , $4$ , such as $1234$ , $1432$ , or $3124$ .",
"role": "user"
}
] | {
"answer": "66660"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$ . Compute $\\sum_{k=1}^{84} \\frac{x_k}{x_k-1}$ .",
"role": "user"
}
] | {
"answer": "\\frac{77}{2}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Juan wrote a natural number and Maria added a digit $ 1$ to the left and a digit $ 1$ to the right. Maria's number exceeds to the number of Juan in $14789$ . Find the number of Juan.... | {
"answer": "532"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The bisectors of the angles $A$ and $B$ of the triangle $ABC$ intersect the sides $BC$ and $AC$ at points $D$ and $E$ . It is known that $AE + BD = AB$ . Find the angle $\\... | {
"answer": "60^\\circ"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A point $(x,y)$ in the first quadrant lies on a line with intercepts $(a,0)$ and $(0,b)$ , with $a,b > 0$ . Rectangle $M$ has vertices $(0,0)$ , $(x,0)$ , $(x,y)$ , and $(0,y)... | {
"answer": "1"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Floyd looked at a standard $12$ hour analogue clock at $2\\!:\\!36$ . When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved... | {
"answer": "14"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Real numbers are chosen at random from the interval $[0,1].$ If after choosing the $n$ -th number the sum of the numbers so chosen first exceeds $1$ , show that the expected value for... | {
"answer": " e "
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": " Let $z_1$ and $z_2$ be the zeros of the polynomial $f(x) = x^2 + 6x + 11$ . Compute $(1 + z^2_1z_2)(1 + z_1z_2^2)$ .",
"role": "user"
}
] | {
"answer": "1266"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Jerry's favorite number is $97$ . He knows all kinds of interesting facts about $97$ :\n\n\n- $97$ is the largest two-digit prime.\n- Reversing the order of its digits results in anot... | {
"answer": "96"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Two circles $\\mathcal{C}_1$ and $\\mathcal{C}_2$ with centers $(1, 1)$ and $(4, 5)$ and radii $r_1 < r_2$ , respectively, are drawn on the coordinate plane. The product of the s... | {
"answer": "13"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Show that there exists a composite number $n$ such that $a^n \\equiv a \\; \\pmod{n}$ for all $a \\in \\mathbb{Z}$ .",
"role": "user"
}
] | {
"answer": " n = 561 "
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Tina writes four letters to her friends Silas, Jessica, Katie, and Lekan. She prepares an envelope for Silas, an envelope for Jessica, an envelope for Katie, and an envelope for Lekan. Ho... | {
"answer": "\\frac{3}{8}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "For every positive integer $k$ let $a(k)$ be the largest integer such that $2^{a(k)}$ divides $k$ . For every positive integer $n$ determine $a(1)+a(2)+\\cdots+a(2^n)$ .",
"... | {
"answer": " 2^n - 1 "
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "In the accompanying figure $ \\overline{CE}$ and $ \\overline{DE}$ are equal chords of a circle with center $ O$ . Arc $ AB$ is a quarter-circle. Then the ratio of the area of tria... | {
"answer": "2:1"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "What is the smallest possible natural number $n$ for which the equation $x^2 -nx + 2014 = 0$ has integer roots?",
"role": "user"
}
] | {
"answer": "91"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$ .",
"role": "user"
}
] | {
"answer": "4648"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find the remainder when the number of positive divisors of the value $$ (3^{2020}+3^{2021})(3^{2021}+3^{2022})(3^{2022}+3^{2023})(3^{2023}+3^{2024}) $$ is divided by $1000$ .\n\n*Pro... | {
"answer": "783"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Suppose $xy-5x+2y=30$ , where $x$ and $y$ are positive integers. Find the sum of all possible values of $x$ ",
"role": "user"
}
] | {
"answer": "31"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ have $\\angle ABC=67^{\\circ}$ . Point $X$ is chosen such that $AB = XC$ , $\\angle{XAC}=32^\\circ$ , and $\\angle{XCA}=35^\\circ$ . Compute $\\angle{BAC}$ ... | {
"answer": "81^\\circ"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Paul fi\flls in a $7\\times7$ grid with the numbers $1$ through $49$ in a random arrangement. He then erases his work and does the same thing again, to obtain two diff\u000berent ra... | {
"answer": "73.5"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": " $\\textbf{Problem 5.}$ Miguel has two clocks, one clock advances $1$ minute per day and the other one goes $15/10$ minutes per day. \nIf you put them at the same correct time, What ... | {
"answer": "1440"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the... | {
"answer": "79"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $a_1 = 1$ and $a_{n+1} = a_n \\cdot p_n$ for $n \\geq 1$ where $p_n$ is the $n$ th prime number, starting with $p_1 = 2$ . Let $\\tau(x)$ be equal to the number of divisor... | {
"answer": "40"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A group of women working together at the same rate can build a wall in $45$ hours. When the work started, all the women did not start working together. They joined the worked over a per... | {
"answer": "75"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after $t$ minutes he is at position $(t,\\si... | {
"answer": "3\\sqrt{3}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "In an office, at various times during the day the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time,... | {
"answer": "(D)"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $ a$ , $ b$ , $ c$ , $ x$ , $ y$ , and $ z$ be real numbers that satisfy the three equations\r\n\\begin{align*}\r\n 13x + by + cz &= 0 \r\n ax + 23y + cz &= 0 \r\n ax +... | {
"answer": "-2"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $T = TNFTPP$ . When properly sorted, $T - 35$ math books on a shelf are arranged in alphabetical order from left to right. An eager student checked out a... | {
"answer": "2161"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches. \nWhat is the largest possible n... | {
"answer": "16"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "At the Mountain School, Micchell is assigned a *submissiveness rating* of $3.0$ or $4.0$ for each class he takes. His *college potential* is then defined as the average of his submiss... | {
"answer": "160"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $m,n$ be natural numbers such that $\\hspace{2cm} m+3n-5=2LCM(m,n)-11GCD(m,n).$ Find the maximum possible value of $m+n$ .",
"role": "user"
}
] | {
"answer": "70"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find value of $$ \\frac{1}{1+x+xy}+\\frac{1}{1+y+yz}+\\frac{1}{1+z+zx} $$ if $x$ , $y$ and $z$ are real numbers usch that $xyz=1$ ",
"role": "user"
}
] | {
"answer": "1"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "**p1.** What is the maximum possible value of $m$ such that there exist $m$ integers $a_1, a_2, ..., a_m$ where all the decimal representations of $a_1!, a_2!, ..., a_m!$ end with... | {
"answer": "5"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $N$ is a three digit number divisible by $7$ such that upon removing its middle digit, the remaining two digit number is also divisible by $7$ . What is the minimum poss... | {
"answer": "154"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Garfield and Odie are situated at $(0,0)$ and $(25,0)$ , respectively. Suddenly, Garfield and Odie dash in the direction of the point $(9, 12)$ at speeds of $7$ and $10$ units pe... | {
"answer": " 159 "
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "ABC is a triangle with coordinates A =(2, 6), B =(0, 0), and C =(14, 0).\n(a) Let P be the midpoint of AB. Determine the equation of the line perpendicular to AB passing through P.\n(b)... | {
"answer": "5\\sqrt{2}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "In a bag are all natural numbers less than or equal to $999$ whose digits sum to $6$ . What is the probability of drawing a number from the bag that is divisible by $11$ ?",
"role... | {
"answer": "\\frac{1}{7}"
} |
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