meetforever/GRPO_CL huggingface dataset:
user: The isosceles right triangle \(ABC\) has a right angle at \(C\) and an area of \(12.5\). The rays trisecting \(\angle ACB\) intersect \(AB\) at \(D\) and \(E\). Find the area of \(\bigtriangleup CDE\). The original answer is in the form \(\frac{k - m\sqrt{n}}{p}\). Please provide the value of \(k + m + n + p\).
Answer: 80
user: In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. The original answer for the perimeter of the triangle is in the form $k + m\sqrt{2}$. Please provide the value of $k + m$.
Answer: 24
user: In $\triangle ABC$, $\angle C = 90^\circ$ and $AB = 12$. Squares $ABXY$ and $CBWZ$ are constructed outside of the triangle. The points $X$, $Y$, $Z$, and $W$ lie on a circle. Find the perimeter of the triangle. The original answer is in the form $k + m\sqrt{n}$, please give the value of $k + m + n$.
Answer: 26
user: Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths $5$, $5$, and $8$, while those of $T'$ have lengths $a$, $a$, and $b$. Find the integer closest to the value of $b$.
Answer: 3
user: For every positive integer $n$, let ${mod}_5 (n)$ be the remainder obtained when $n$ is divided by $5$. Define a function $f: \{0,1,2,3,\cdots\} \times \{0,1,2,3,4\} \to \{0,1,2,3,4\}$ recursively as follows:    
$$f(i,j)=\begin{cases}\operatorname{mod}_5(j+1)&\text{if}i=0\text{and}0\le j\le4,\\f(i-1,1)&\text{if}i\ge1\text{and}j=0\text{,and}\\f(i-1,f(i,j-1))&\text{if}i\ge1\text{and}1\le j\le4.\end{cases}$$  
Find the value of $f(2015,2)$.
Answer: 1
user: Tetrahedron \(ABCD\) has \(AB=5\), \(AC=3\), \(BC=4\), \(BD=4\), \(AD=3\), and \(CD=\tfrac{12}{5}\sqrt{2}\). Find the volume of the tetrahedron. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 29
user: Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than $5$ steps left). Suppose Dash takes $19$ fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$?
Answer: 13
user: Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than $5$ steps left). Suppose that Dash takes $19$ fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$?
Answer: 13
user: In the given figure, hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$?
Answer: 12
user: In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. Find the value of $FG + JH + CD$. The original answer is in the form k + \sqrt{m}, please give the value of k + m.
Answer: 6
user: The zeroes of the function \( f(x) = x^2 - ax + 2a \) are integers. What is the sum of the possible values of \( a \)?
Answer: 16
user: Let $n$ be a positive integer greater than $4$ such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. Find the sum of the digits of $s$.
Answer: 8
user: Let $S$ be a square of side length 1. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\frac{1}{2}$ is $\frac{a-b\pi}{c}$, where $a,b,$ and $c$ are positive integers and ${gcd}(a,b,c) = 1$. What is the value of $a+b+c$?
Answer: 59
user: A rectangular box measures $a \times b \times c$, where $a$, $b$, and $c$ are integers and $1\leq a \leq b \leq c$. The volume and the surface area of the box are numerically equal. Find how many ordered triples $(a,b,c)$ are possible.
Answer: 10
user: Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. Find the coordinates of $p_{2015}$ in the form $(x, y)$, and provide the sum $x + y$.
Answer: -9
user: Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. Find the value of $\overline{AR}+\overline{BR}+\overline{CR}+\overline{DR}$.
Answer: 192
user: A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$. Each team plays a $76$ game schedule. How many games does a team play within its own division?
Answer: 48
user: A rectangular box measures $a \times b \times c$, where $a$, $b$, and $c$ are integers and $1\leq a \leq b \leq c$. The volume and the surface area of the box are numerically equal. Find the number of ordered triples $(a,b,c)$ that are possible.
Answer: 10
user: A collection of circles in the upper half-plane, all tangent to the \(x\)-axis, is constructed in layers as follows. Layer \(L_0\) consists of two circles of radii \(70^2\) and \(73^2\) that are externally tangent. For \(k \ge 1\), the circles in \(\bigcup\limits_{j=0}^{k-1}L_j\) are ordered according to their points of tangency with the \(x\)-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer \(L_k\) consists of the \(2^{k-1}\) circles constructed in this way. Let \(S=\bigcup\limits_{j=0}^{6}L_j\), and for every circle \(C\) denote by \(r(C)\) its radius. Find \(\sum\limits_{C\in S} \frac{1}{\sqrt{r(C)}}\). The original answer is in the form of \(\frac{k}{m}\), where the fraction is in its simplest form. Please give the value of k + m.
Answer: 157
user: A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a \sqrt{b} + c \sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$, and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. Find the value of $a+b+c+d$.
Answer: 2024
user: A rectangular box has integer side lengths in the ratio $1: 3: 4$. What could be the volume of the box?
Answer: 96
user: Star lists the whole numbers $1$ through $30$ once. Emilio copies Star's numbers, replacing each occurrence of the digit $2$ by the digit $1$. Star adds her numbers and Emilio adds his numbers. How much larger is Star's sum than Emilio's?
Answer: 103
user: Josh writes the numbers $1,2,3,\dots,99,100$. He marks out $1$, skips the next number $(2)$, marks out $3$, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number $(2)$, skips the next number $(4)$, marks out $6$, skips $8$, marks out $10$, and so on to the end. Josh continues in this manner until only one number remains. What is that number?
Answer: 64
user: Which of the following numbers is not a perfect square: $1^{2016}$, $2^{2017}$, $3^{2018}$, $4^{2019}$, or $5^{2020}$? The original answer is in the form $x^{y}$, please give the value of $x + y$.
Answer: 2019
user: What is the tens digit of $2015^{2016}-2017?$
Answer: 0
user: Find the value of the expression
$100-98+96-94+92-90+\cdots+8-6+4-2.$
Answer: 50
user: Let \( z \) be a complex number satisfying \( 12|z|^2 = 2|z+2|^2 + |z^2+1|^2 + 31 \). The original answer for \( z + \frac{6}{z} \) is in the form of k + m, where k and m are integers. Please find the value of k + m.
Answer: -2
user: The interior of a quadrilateral is bounded by the graphs of \((x+ay)^2 = 4a^2\) and \((ax-y)^2 = a^2\), where \(a\) is a positive real number. Find the area of this region in terms of \(a\), valid for all \(a > 0\). The original answer is in the form \(\frac{ka^2}{a^2+m}\). Please provide the value of \(k + m\).
Answer: 9
user: 已知 $M=\left\{(x, y) \mid x^{2}+2 y^{2}=3\right\}, N=\{(x, y) \mid y=m x+b\}$ 。若对于所有的 $m \in \mathbf{R}$, 均有 $M \cap N \neq \varnothing$, 求 $b$ 的取值范围。答案的形式为\left[-\frac{\sqrt{c}}{d}, \frac{\sqrt{c}}{d}\right]，请给出c和d的绝对值之和。
Answer: 8
user: 已知锐角 $\triangle A B C$ 的外接圆的圆心为 $O$, 点 $O$ 到直线 $B C, C A, A B$ 的距离分别为 $k, m, n$. 若 $B C=a, C A=b, A B=c$, 求 $k: m: n$ 的比率。假设答案为 $p:q:r$ 的形式，请给出 $p+q+r$ 的值。
Answer: 3
user: Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area of Carl's garden in square yards?
Answer: 336
user: A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner rectangle?
Answer: 2
user: A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has a side length of $5$ inches. Find the weight, in ounces, of the second piece that is closest to an integer value. If the weight is w ounces, please provide the value of 10w.
Answer: 333
user: A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is the value of $a+b$?
Answer: 4
user: Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area of Carl’s garden in square yards?
Answer: 336
user: Each of the $100$ students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are $42$ students who cannot sing, $65$ students who cannot dance, and $29$ students who cannot act. How many students have two of these talents?
Answer: 64
user: How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$, and the line $x=5.1$?
Answer: 50
user: At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there were three times as many sets of twins as sets of triplets. How many of these $1000$ babies were in sets of quadruplets? Provide your answer as an integer.
Answer: 100
user: Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. Find the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$.
Answer: 12
user: Two different numbers are randomly selected from the set \(\{ -2, -1, 0, 3, 4, 5\}\) and multiplied together. What is the probability that the product is \(0\)? Express your answer as a fraction in simplest form, \(\frac{k}{m}\), and give the value of \(k + m\).
Answer: 4
user: Karl's car uses a gallon of gas every $35$ miles, and his gas tank holds $14$ gallons when it is full. One day, Karl started with a full tank of gas, drove $350$ miles, bought $8$ gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day in total?
Answer: 525
user: Seven cookies of radius \(1\) inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. The original answer is in the form \(\sqrt{x}\). Please find the value of \(x\).
Answer: 2
user: Circles with centers \(P, Q\) and \(R\), having radii \(1, 2\) and \(3\), respectively, lie on the same side of line \(l\) and are tangent to \(l\) at \(P', Q'\) and \(R'\), respectively, with \(Q'\) between \(P'\) and \(R'\). The circle with center \(Q\) is externally tangent to each of the other two circles. Find the area of triangle \(PQR\). The original answer is in the form \(\sqrt{a} - \sqrt{b}\), please give the value of a + b.
Answer: 8
user: All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?
Answer: 729
user: What is the largest power of $2$ that is a divisor of $13^4 - 11^4$?
Answer: 32
user: In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers? Provide your answer as an integer.
Answer: 7
user: Let \(N\) be a positive multiple of \(5\). One red ball and \(N\) green balls are arranged in a line in random order. Let \(P(N)\) be the probability that at least \(\tfrac{3}{5}\) of the green balls are on the same side of the red ball. Observe that \(P(5)=1\) and that \(P(N)\) approaches \(\tfrac{4}{5}\) as \(N\) grows large. Find the sum of the digits of the least value of \(N\) such that \(P(N) < \tfrac{321}{400}\).
Answer: 12
user: In \(\triangle ABC\) shown in the figure, \(AB=7\), \(BC=8\), \(CA=9\), and \(\overline{AH}\) is an altitude. Points \(D\) and \(E\) lie on sides \(\overline{AC}\) and \(\overline{AB}\), respectively, so that \(\overline{BD}\) and \(\overline{CE}\) are angle bisectors, intersecting \(\overline{AH}\) at \(Q\) and \(P\), respectively. What is the length of \(PQ\) in the form \(\frac{a}{b}\sqrt{c}\), where \(a\), \(b\), and \(c\) are integers? Please find the value of \(a + b + c\).
Answer: 28
user: For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. Find the number of positive integer divisors of the number $81n^4$.
Answer: 325
user: What is the area of the region enclosed by the graph of the equation \(x^2+y^2=|x|+|y|?\) Express your answer in the form \(a\pi + b\), where \(a\) and \(b\) are constants. Please provide the value of \(a + b\).
Answer: 3
user: In rectangle $ABCD,$ $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$ where the greatest common factor of $r,s,$ and $t$ is $1.$ What is the value of $r+s+t$?
Answer: 20
user: Rectangle \(ABCD\) has \(AB=5\) and \(BC=4\). Point \(E\) lies on \(\overline{AB}\) so that \(EB=1\), point \(G\) lies on \(\overline{BC}\) so that \(CG=1\), and point \(F\) lies on \(\overline{CD}\) so that \(DF=2\). Segments \(\overline{AG}\) and \(\overline{AC}\) intersect \(\overline{EF}\) at \(Q\) and \(P\), respectively. The value of \(\frac{PQ}{EF}\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 101
user: Jerry starts at $0$ on the real number line. He tosses a fair coin $8$ times. When he gets heads, he moves $1$ unit in the positive direction; when he gets tails, he moves $1$ unit in the negative direction. The probability that he reaches $4$ at some time during this process is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is the value of $a + b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH.$)
Answer: 151
user: For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b, c,$ and $d$, each to some positive power. What is the value of $N$?
Answer: 14
user: A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the value of $p+q.$
Answer: 109
user: A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$?
Answer: 385
user: The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$?
Answer: 20
user: Circles with centers \(P, Q\) and \(R\), having radii \(1, 2\) and \(3\), respectively, lie on the same side of line \(l\) and are tangent to \(l\) at \(P', Q'\) and \(R'\), respectively, with \(Q'\) between \(P'\) and \(R'\). The circle with center \(Q\) is externally tangent to each of the other two circles. Find the area of triangle \(PQR\). The original answer is in the form \(\sqrt{k} - \sqrt{m}\), please give the value of \(k + m\).
Answer: 8
user: What is the value of \(x\) such that the area of the region enclosed by the graph of the equation \(x^2+y^2=|x|+|y|\) is \(\pi + x\)?
Answer: 2
user: A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side?
Answer: 500
user: Let \(ABCD\) be a unit square. Let \(Q_1\) be the midpoint of \(\overline{CD}\). For \(i=1,2,\dots,\) let \(P_i\) be the intersection of \(\overline{AQ_i}\) and \(\overline{BD}\), and let \(Q_{i+1}\) be the foot of the perpendicular from \(P_i\) to \(\overline{CD}\). Find the value of 
\[\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_i P_i \,.\]
The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 5
user: A top hat contains $3$ red chips and $2$ green chips. Chips are drawn randomly, one at a time without replacement, until all $3$ of the reds are drawn or until both green chips are drawn. What is the probability that the $3$ reds are drawn? Express your answer in the form of \(\frac{k}{m}\) as a common fraction, and give the value of \(k + m\).
Answer: 7
user: For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?
Answer: 325
user: A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. Find the number of sets of three teams $\{A, B, C\}$ in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$.
Answer: 385
user: How many ordered triples $(x,y,z)$ of positive integers satisfy ${lcm}(x,y) = 72, {lcm}(x,z) = 600$ and $lcm(y,z)=900$?
Answer: 15
user: What is the volume of the region in three-dimensional space defined by the inequalities \( |x|+|y|+|z|\le1 \) and \( |x|+|y|+|z-1|\le1 \)? Express your answer in the form \(\frac{k}{m}\) where k and m are integers. Please provide the value of \(k + m\).
Answer: 7
user: A binary operation $\diamondsuit$ has the properties that $a\diamondsuit (b\diamondsuit c) = (a\diamondsuit b)\cdot c$ and that $a\diamondsuit a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \diamondsuit (6\diamondsuit x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is the value of $p+q$?
Answer: 109
user: In regular hexagon \(ABCDEF\), points \(W\), \(X\), \(Y\), and \(Z\) are chosen on sides \(\overline{BC}\), \(\overline{CD}\), \(\overline{EF}\), and \(\overline{FA}\) respectively, so lines \(AB\), \(ZW\), \(YX\), and \(ED\) are parallel and equally spaced. The original answer for the ratio of the area of hexagon \(WCXYFZ\) to the area of hexagon \(ABCDEF\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 38
user: Three numbers in the interval \([0,1]\) are chosen independently and at random. Find the probability that the chosen numbers are the side lengths of a triangle with positive area. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 3
user: Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measure of $\angle CED$? If the answer is $x^\circ$, what is the value of $x$?
Answer: 120
user: A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. Find the length of the fourth side.
Answer: 500
user: How many four-digit integers $abcd$, with $a \neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$.
Answer: 17
user: There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is this value of $b$?
Answer: 9
user: There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\gcd(a, b, c, d) = 77$ and $\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$? Please provide your answer as an integer without any commas.
Answer: 27720
user: How many ordered triples $(x,y,z)$ of positive integers satisfy ${lcm}(x,y) = 72, {lcm}(x,z) = 600 { and ~lcm}(y,z)=900$? Provide your answer as an integer.
Answer: 15
user: The sequence \((a_n)\) is defined recursively by \(a_0=1\), \(a_1=\sqrt[19]{2}\), and \(a_n=a_{n-1}a_{n-2}^2\) for \(n\geq 2\). Find the smallest positive integer \(k\) such that the product \(a_1a_2\cdots a_k\) is an integer.
Answer: 17