ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls? Express your answer in the form \(\frac{k}{m}\) and calculate the value of \(k + m\).
Answer: 21
Problem: A square with side length \(3\) is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length \(2\) has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle in the form of a fraction \(\frac{k}{m}\)? Please provide the value of \(k + m\).
Answer: 85
Problem: Let $c = \frac{2\pi}{11}.$ What is the value of
$\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?$
Answer: 1
Problem: All the roots of the polynomial \(z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16\) are positive integers, possibly repeated. The original answer for the value of \(B\) can be expressed as k + m, where k and m are integers. Please find the value of k + m.
Answer: -88
Problem: Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA},\overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$ Express your answer in the form $a\sqrt{b}$, where $a$ and $b$ are integers. Please provide the value of a + b.
Answer: 29
Problem: How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations? 
\begin{align*} x^2+3y&=9 \\ (|x|+|y|-4)^2 &= 1 \end{align*}
Find the number of solutions.
Answer: 5
Problem: Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the probability that the product is divisible by $4$? Provide your answer as a simplified fraction in the form \frac{k}{m}. Please give the value of k + m.
Answer: 123
Problem: In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. What is the perimeter of the hexagon? Express your answer in the form $k\sqrt{m}$, and find the value of $k + m$.
Answer: 15
Problem: Values for $A,B,C,$ and $D$ are to be selected from $\{1, 2, 3, 4, 5, 6\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter; for example, the choices $A=3, B=2, C=4, D=1$ is considered the same as the choices $A=4, B=1, C=3, D=2.$) Find the number of ways.
Answer: 90
Problem: The real number $x$ satisfies the equation $x+\frac{1}{x} = \sqrt{5}$. What is the value of $x^{11}-7x^{7}+x^3$?
Answer: 0
Problem: A choir director must select a group of singers from among his $6$ tenors and $8$ basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of $4$, and the group must have at least one singer. Let $N$ be the number of different groups that could be selected. What is the remainder when $N$ is divided by $100$?
Answer: 95
Problem: Isosceles triangle $ABC$ has $AB = AC = 3\sqrt6$, and a circle with radius $5\sqrt2$ is tangent to line $AB$ at $B$ and to line $AC$ at $C$. What is the area of the circle that passes through vertices $A$, $B$, and $C?$ If the area is $x\pi$, what is the value of $x$?
Answer: 26
Problem: Recall that the conjugate of the complex number $w = a + bi$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$, is the complex number $\overline{w} = a - bi$. For any complex number $z$, let $f(z) = 4i\overline{z}$. The polynomial  
$$P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1$$  
has four complex roots: $z_1$, $z_2$, $z_3$, and $z_4$. Let  
$$Q(z) = z^4 + Az^3 + Bz^2 + Cz + D$$ 
be the polynomial whose roots are $f(z_1)$, $f(z_2)$, $f(z_3)$, and $f(z_4)$, where the coefficients $A,$ $B,$ $C,$ and $D$ are complex numbers. Find the value of $B + D$.
Answer: 208
Problem: In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200$.
$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots, 200, 200, \ldots , 200$
Find the median of the numbers in this list.
Answer: 142
Problem: Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357, 89,$ and $5$ are all uphill integers, but $32, 1240,$ and $466$ are not. Find how many uphill integers are divisible by $15$.
Answer: 6
Problem: In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200$.  
$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \cdots, 200, 200, \cdots , 200$  
What is the median of the numbers in this list?
Answer: 142
Problem: Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. If the expected number of balls that occupy their original positions after these two successive transpositions is \( x \), find the value of \( 10x \).
Answer: 22
Problem: An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used?
Answer: 191
Problem: Suppose $a$, $b$, $c$ are positive integers such that \[a+b+c=23\] and \[\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.\] What is the sum of all possible distinct values of $a^2+b^2+c^2$?
Answer: 438
Problem: Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD},BC=CD=43$, and $\overline{AD}\perp\overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$?
Answer: 194
Problem: Let \(ABCD\) be an isosceles trapezoid having parallel bases \(\overline{AB}\) and \(\overline{CD}\) with \(AB>CD.\) Line segments from a point inside \(ABCD\) to the vertices divide the trapezoid into four triangles whose areas are \(2, 3, 4,\) and \(5\) starting with the triangle with base \(\overline{CD}\) and moving clockwise as shown in the diagram below. The original answer for the ratio \(\frac{AB}{CD}\) is in the form \(k+\sqrt{m}\). Please find the value of \(k+m\).
Answer: 4
Problem: An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meters, respectively. What is the height, in meters, of the pillar at $E$?
Answer: 17
Problem: Distinct lines $\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\ell$ to point $P'$, and then $P'$ is reflected about line $m$ to point $P''$. The equation of line $\ell$ is $5x - y = 0$, and the coordinates of $P''$ are $(4,1)$. The equation of line $m$ is in the form $ax + by = 0$. Please find the value of a + b.
Answer: -1
Problem: For how many ordered pairs \((b,c)\) of positive integers does neither \(x^2+bx+c=0\) nor \(x^2+cx+b=0\) have two distinct real solutions?
Answer: 6
Problem: A bug starts at a vertex of a grid made of equilateral triangles of side length \(1\). At each step, the bug moves in one of the \(6\) possible directions along the grid lines randomly and independently with equal probability. What is the probability that after \(5\) moves the bug never will have been more than \(1\) unit away from the starting position? Provide your answer as a fraction in simplest form \(\frac{k}{m}\), and calculate the value of \(k + m\).
Answer: 121
Problem: A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number? Express your answer in the form \(\frac{k}{m}\), where k and m are integers with no common factors other than 1. Find the value of k + m.
Answer: 21
Problem: A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section, the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?
Answer: 84
Problem: Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find the value of $a+b+c$.
Answer: 147
Problem: Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. Find the value of $\frac{p}{q}$.
Answer: 16
Problem: Set $u_0 = \frac{1}{4}$, and for $k \ge 0$ let $u_{k+1}$ be determined by the recurrence 
$$u_{k+1} = 2u_k - 2u_k^2.$$  
  
This sequence tends to a limit; call it $L$. Find the least value of $k$ such that 
$$|u_k-L| \le \frac{1}{2^{1000}}.$$
Answer: 10
Problem: 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 the value of $m + n$?
Answer: 54
Problem: Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac{3}{4}$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\frac{1}{12}$. What is the least possible number of faces on the two dice combined?
Answer: 17
Problem: Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. Find the value of \[f(2) + f(3) + f(4) + f(5)+ f(6).\]
Answer: 8
Problem: Let $x$ be the least real number greater than $1$ such that $\sin(x)= \sin(x^2)$, where the arguments are in degrees. What is $x$ rounded up to the closest integer?
Answer: 13
Problem: Suppose that on a parabola with vertex \(V\) and a focus \(F\) there exists a point \(A\) such that \(AF=20\) and \(AV=21\). Find the sum of all possible values of the length \(FV\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 43
Problem: Let \(Q(z)\) and \(R(z)\) be the unique polynomials such that\[z^{2021}+1=(z^2+z+1)Q(z)+R(z)\]and the degree of \(R\) is less than \(2.\) The original answer is in the form az^1 + b, please find the value of a + b.
Answer: -1
Problem: For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. Find the number of values of $n \le 50$ for which $f_{50}(n) = 12$.
Answer: 10
Problem: Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$, and the lines $BC, DE,$ and $FA$ determine a triangle with area $324\sqrt{3}$. The perimeter of hexagon $ABCDEF$ can be expressed as $m +n\sqrt{p}$, where $m, n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find the value of $m + n + p$.
Answer: 55
Problem: A square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C'$, and edge $\overline{BC}$ intersects edge $\overline{AB}$ at point $E$. Suppose that $C'D = \frac{1}{3}$. What is the perimeter of triangle $\bigtriangleup AEC'$? Please provide your answer as an integer.
Answer: 2
Problem: The five solutions to the equation\[(z-1)(z^2+2z+4)(z^2+4z+6)=0\] may be written in the form $x_k+y_ki$ for $1\le k\le 5,$ where $x_k$ and $y_k$ are real. Let $\mathcal E$ be the unique ellipse that passes through the points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),$ and $(x_5,y_5)$. The eccentricity of $\mathcal E$ can be written in the form $\sqrt{\frac mn}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$. (Recall that the eccentricity of an ellipse $\mathcal E$ is the ratio $\frac ca$, where $2a$ is the length of the major axis of $\mathcal E$ and $2c$ is the distance between its two foci.)
Answer: 7
Problem: Let $ABCD$ be an isosceles trapezoid with $\overline{BC}\parallel \overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\overline{AC}$ with $X$ between $A$ and $Y$, as shown in the figure. Suppose $\angle AXD = \angle BYC = 90^\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. What is the area of $ABCD$ in the form $m\sqrt{n}$, where $m$ and $n$ are integers and $n$ is not divisible by the square of any prime? Please provide the value of $m + n$.
Answer: 38
Problem: For real numbers $x$, let  
$$P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)$$
where $i = \sqrt{-1}$. Find the number of values of $x$ with $0\leq x<2\pi$ for which  
$$P(x)=0.$$
Answer: 0
Problem: Inside a right circular cone with base radius \(5\) and height \(12\) are three congruent spheres with radius \(r\). Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. Find the radius \(r\) of each sphere. The original answer is in the form \(\frac{k-n\sqrt{3}}{m}\). Please give the value of \(k+n+m\).
Answer: 141
Problem: Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly $19$. How many sheets were borrowed?
Answer: 13
Problem: Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Please calculate the value of \(m + n\).
Answer: 471
Problem: For each integer $n\geq 2$, let $S_n$ be the sum of all products $jk$, where $j$ and $k$ are integers and $1\leq j<k\leq n$. Find the sum of the $10$ least values of $n$ such that $S_n$ is divisible by $3$.
Answer: 197
Problem: Azar and Carl play a game of tic-tac-toe. Azar places an $X$ in one of the boxes in a $3$-by-$3$ array of boxes, then Carl places an $O$ in one of the remaining boxes. After that, Azar places an $X$ in one of the remaining boxes, and so on until all boxes are filled or one of the players has three of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third $O$. How many ways can the board look after the game is over?
Answer: 148
Problem: Right triangle \(ABC\) has side lengths \(BC=6\), \(AC=8\), and \(AB=10\). A circle centered at \(O\) is tangent to line \(BC\) at \(B\) and passes through \(A\). A circle centered at \(P\) is tangent to line \(AC\) at \(A\) and passes through \(B\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 47
Problem: Frieda the frog begins a sequence of hops on a \(3 \times 3\) grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge. For example, if Frieda begins in the center square and makes two hops "up", the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 57
Problem: Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,....$ More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find the value of $p+q.$
Answer: 55
Problem: Each of the $5$ sides and the $5$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color? Provide your answer as a fraction in simplest form, \(\frac{k}{m}\), and find the value of \(k + m\).
Answer: 509
Problem: A quadratic polynomial with real coefficients and leading coefficient \(1\) is called \(\emph{disrespectful}\) if the equation \(p(p(x))=0\) is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial \(\tilde{p}(x)\) for which the sum of the roots is maximized. The original answer for \(\tilde{p}(1)\) is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 21
Problem: What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{ 1, 2, 3, \ldots, 30\}$? (For example the set $\{1, 17, 18, 19, 30\}$ has $2$ pairs of consecutive integers.)

Find the average number of pairs of consecutive integers in such a subset. The original answer is in the form \(\frac{k}{m}\). Please provide the value of \(k + m\).
Answer: 5
Problem: Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.

[asy] size(350); defaultpen(linewidth(0.8)+fontsize(11)); real theta = aTan(1.25/2); pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R; draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6)); draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4")); dot("$A$",A,dir(270)); dot("$B$",B,E); dot("$C$",C,N); dot("$D$",D,W); dot("$P$",P,SE); dot("$Q$",Q,NE); dot("$R$",R,N); dot("$S$",S,dir(270)); [/asy]

Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find the value of $m+n+p$.
Answer: 81
Problem: Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$, and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$. The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $B$ and $F \ne B$. What is the length of $CF$?
Answer: 30
Problem: Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Find the value of $a+b.$
Answer: 85
Problem: Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the divisor function.) Let\[f(n)=\frac{d(n)}{\sqrt [3]n}.\]There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$. What is the sum of the digits of $N$?
Answer: 9
Problem: Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1\le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial
\[q(x) = c_3x^3+c_2x^2+c_1x+c_0\]such that $D(m) = q(m)$ for all odd integers $m\ge 5$. Find the value of $c_1$.
Answer: 11
Problem: For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$
Answer: 2
Problem: What is the value of  
$$3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}?$$
The answer is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 142
Problem: What is the value of \[3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}?\] The original answer is in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\).
Answer: 142
Problem: In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ so that $\overline{BP}$ $\perp$ $\overline{AD}$, $AP = 3$, and $PD = 2$. Find the area of $ABCD$. (Note: The figure is not drawn to scale.)
Answer: 20
Problem: How many three-digit positive integers have an odd number of even digits? Provide your answer as an integer.
Answer: 450
Problem: When three positive integers $a$, $b$, and $c$ are multiplied together, their product is $100$. Suppose $a < b < c$. In how many ways can the numbers be chosen?
Answer: 4
Problem: A donkey suffers an attack of hiccups and the first hiccup happens at $4:00$ one afternoon. Suppose that the donkey hiccups regularly every $5$ seconds. Find the number of seconds after $4:58$ when the donkey’s $700$th hiccup occurs.
Answer: 15
Problem: The least common multiple of a positive integer $n$ and $18$ is $180$, and the greatest common divisor of $n$ and $45$ is $15$. What is the sum of the digits of $n$?
Answer: 6
Problem: For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots? Provide the number of such values.
Answer: 8