sungyub/dapo-math-17k-verl huggingface dataset:
user: 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. What is $m+n$?

Answer: 7
user: In this diagram, $AB$ and $AC$ are the equal sides of an isosceles triangle $ABC$, in which an equilateral triangle $DEF$ is inscribed. Designate angle $BFD$ by $a$, angle $ADE$ by $b$, and angle $FEC$ by $c$. The relationship between $a$, $b$, and $c$ is such that $k(b + c) = 2a$. Please find the value of $k + m + n$, where $m$ and $n$ are the coefficients of $b$ and $c$, respectively.

Answer: 3
user: On a standard die, one of the dots is removed at random, with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots? Express your answer as a fraction in simplest form, \frac{k}{m}, and provide the value of k + m.

Answer: 32
user: The ratio of the interior angles of two regular polygons with sides of unit length is $3:2$. How many such pairs of polygons are there?

Answer: 3
user: In checking the petty cash, a clerk counts $q$ quarters, $d$ dimes, $n$ nickels, and $c$ cents. Later, he discovers that $x$ of the nickels were counted as quarters and $x$ of the dimes were counted as cents. To correct the total obtained, by how many cents must the clerk adjust the total? The original answer is in the form $kx$, please give the value of $k$.

Answer: 11
user: Given that $\angle BAC = 80^\circ$, $|AB| = |AC|$, $K \in [AB]$, $L \in [AB]$, $|AB|^2 = |AK| \cdot |AL|$, and $|BL| = |BC|$, find $\angle KCB$. If the answer is $x^\circ$, what is the value of $x$?

Answer: 25
user: Given the polynomial $a_0x^n + a_1x^{n-1} + \cdots + a_{n-1}x + a_n$, where $n$ is a non-negative integer, and $a_0$ is a positive integer. The remaining coefficients $a_1, a_2, \ldots, a_n$ are integers or zero. Define $h = n + a_0 + |a_1| + |a_2| + \cdots + |a_n|$. Find the number of polynomials for which $h = 3$.

Answer: 5
user: Let $f(x) = |x-2| + |x-4| - |2x-6|$ for $2 \leq x \leq 8$. Find the sum of the largest and smallest values of $f(x)$.

Answer: 2
user: A planet orbits around a star $S$, as shown in the figure. The semi-major axis of the orbit is $a$. The perigee, namely the shortest distance between the planet and the star, is $0.5a$. When the planet passes point $P$ (on the line through the star and perpendicular to the major axis), its speed is $v_1$. What is the value of $v_2$ in terms of $v_1$ when it passes the perigee? The original answer is in the form \(\frac{k}{\sqrt{m}}v_1\). Please calculate the value of \(k + m\).

Answer: 8
user: Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$ respectively. What is the area of the hexagon? The answer should be in the form $k\sqrt{m}$, where k and m are integers. Please find the value of k + m.

Answer: 28
user: In \(\triangle ABC\), \(\angle C = 90^\circ\), \(AC = 6\), and \(BC = 8\). Points \(D\) and \(E\) are on \(\overline{AB}\) and \(\overline{BC}\), respectively, and \(\angle BED = 90^\circ\). If \(DE = 4\), find the value of \(BD\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 23
user: A right triangle $ABC$ with hypotenuse $AB$ has side $AC = 15$. Altitude $CH$ divides $AB$ into segments $AH$ and $HB$, with $HB = 16$. Find the area of $\triangle ABC$.

Answer: 150
user: In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0)$, $(0,3)$, $(3,3)$, $(3,1)$, $(5,1)$, and $(5,0)$. Find the slope of the line through the origin that divides the area of this region exactly in half. The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is a simplified fraction. Please give the value of \(k + m\).

Answer: 16
user: How many ordered integer pairs $(x,y)$ are there such that $xy = 4(y^2 + x)$?

Answer: 14
user: Let \(ABC\) be a triangle such that \(|AB| = 8\) and \(|AC| = 2|BC|\). What is the largest value of the altitude from side \([AB]\)? The original answer is in the form \(\frac{k}{m}\). Please provide the value of \(k + m\).

Answer: 19
user: A positive integer $n$ has 60 divisors and $7n$ has 80 divisors. What is the greatest integer $k$ such that $7^k$ divides $n$?

Answer: 2
user: How many ordered integer pairs $(x, y)$, where $0 \leq x, y < 31$, satisfy the equation $(x^2 - 18)^2 \equiv y^2 \pmod{31}$? Provide your answer as an integer.

Answer: 60
user: Lily has an unfair coin that has a \(\frac{2}{3}\) probability of showing heads and a \(\frac{1}{3}\) probability of showing tails. She flips the coin twice. What is the probability that the first flip is heads while the second is tails? Provide your answer in the form of \(\frac{k}{m}\), where the fraction is in simplest form, and find the value of k + m.

Answer: 11
user: Find the number of counterexamples to the statement: 
\[ \text{If } N \text{ is an odd positive integer the sum of whose digits is 4 and none of whose digits is 0, then } N \text{ is prime.} \]
Provide your answer as an integer.

Answer: 2
user: Jamal wants to store $30$ computer files on floppy disks, each of which has a capacity of $1.44$ megabytes (MB). Three of his files require $0.8$ MB of memory each, $12$ more require $0.7$ MB each, and the remaining $15$ require $0.4$ MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?

Answer: 13
user: Distinct points \(A\) and \(B\) are on a semicircle with diameter \(MN\) and center \(C\). The point \(P\) is on \(CN\) and \(\angle CAP = \angle CBP = 10^{\circ}\). If \(\stackrel{\frown}{MA} = 40^{\circ}\), then \(\stackrel{\frown}{BN}\) equals how many degrees? Please provide your answer as an integer representing the number of degrees.

Answer: 20
user: The base of a triangle is $15$ inches. Two lines are drawn parallel to the base, terminating in the other two sides, and dividing the triangle into three equal areas. Find the length of the parallel closer to the base in inches. The answer should be in the form $k\sqrt{m}$, where k and m are integers. Please calculate the value of k + m.

Answer: 11
user: Find the set of values of $m$ for which $x^2 + 3xy + x + my - m$ has two factors, with integer coefficients, which are linear in $x$ and $y$. The original answer is in the form of $\{p, q\}$, where $p$ and $q$ are integers. Please provide the value of $p + q$.

Answer: 12
user: Figure \(ABCD\) is a trapezoid with \(AB \parallel DC\), \(AB = 5\), \(BC = 3 \sqrt{2}\), \(\angle BCD = 45^\circ\), and \(\angle CDA = 60^\circ\). Find the length of \(DC\). The original answer is in the form \(m + \sqrt{n}\). Please give the value of \(m + n\).

Answer: 11
user: 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 \(-a \cdot z + b\). Please find the value of \(a + b\).

Answer: 1
user: In a square $ABCD$ with side length $2$ units, $E$ is the midpoint of $AD$. A point $F$ is located on $BE$ such that $CF \perp BE$. If the area of the quadrilateral $CDEF$ is $x$ and $x$ is multiplied by 10, what is the value of $10x$?

Answer: 22
user: The buttons \( \{\times, +, \div\} \) on a calculator have their functions swapped. A button instead performs one of the other two functions; no two buttons have the same function. The calculator claims that \( 2 + 3 \div 4 = 10 \) and \( 4 \times 2 \div 3 = 5 \). What does \( 4 + 3 \times 2 \div 1 \) equal on this calculator? Find the value.

Answer: 7
user: On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was $15$ inches. If the answer is $x$ inches, what is the value of $100x$?

Answer: 34
user: Line $\frac{x}{4}+\frac{y}{3}=1$ and ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ intersect at $A$ and $B$. A point on the ellipse $P$ satisfies that the area of $\triangle PAB$ is $3$. Find the number of such points.

Answer: 2
user: $P$ is the intersection point of diagonals of cyclic quadrilateral $ABCD$. The circumcenters of $\triangle APB$ and $\triangle CPD$ lie on the circumcircle of $ABCD$. If $AC + BD = 18$, find the area of $ABCD$. The original answer is in the form \(\frac{k\sqrt{m}}{n}\), please give the value of k + m + n.

Answer: 88
user: For all integers $n$ greater than $1$, define $a_n = \frac{1}{\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$. Find the value of $b - c$.

Answer: -1
user: Let $(a_n)_{n=1}^\infty$ be a real sequence such that \[a_n = (n-1)a_1 + (n-2)a_2 + \dots + 2a_{n-2} + a_{n-1}\] for every $n \geq 3$. If $a_{2011} = 2011$ and $a_{2012} = 2012$, find the value of $a_{2013}$.

Answer: 4025
user: The sides of $\triangle ABC$ form an arithmetic sequence of integers. The incircle $I$ is tangent to $AB$, $BC$, and $CA$ at points $D$, $E$, and $F$, respectively. Given that $DB = \frac{3}{2}$ and $FA = \frac{1}{2}$, find the radius of the incircle $I$. Express your answer in the form \frac{\sqrt{m}}{n}, where the fraction is in its simplest form. Calculate the value of m + n.

Answer: 21
user: How many ordered pairs of positive integers $(x,y)$ are there such that $y^2 - x^2 = 2y + 7x + 4$?

Answer: 1
user: A ball of radius \(R\) and mass \(m\) is magically put inside a thin shell of the same mass and radius \(2R\). The system is at rest on a horizontal frictionless surface initially. When the ball is, again magically, released inside the shell, it sloshes around in the shell and eventually stops at the bottom of the shell. How far does the shell move from its initial contact point with the surface? The original answer is in \(\frac{a}{b}\) format, please give the value of a + b.

Answer: 3
user: For how many integer values of $m$ do the lines $13x + 11y = 700$ and $y = mx - 1$ intersect at a point with integer-valued coordinates? Provide your answer as an integer.

Answer: 2
user: Find the value of the product \[ \frac{8}{4} \cdot \frac{12}{8} \cdot \frac{16}{12} \cdots \frac{4n + 4}{4n} \cdots \frac{2008}{2004}. \]

Answer: 502
user: A subset of the integers $1, 2, \ldots, 100$ has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?

Answer: 76
user: An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $16$, and one of the base angles is $\arcsin(0.8)$. Find the area of the trapezoid.

Answer: 80
user: $N$ is a $50$-digit number (in the decimal scale). All digits except the $26^{\text{th}}$ digit (from the left) are $1$. If $N$ is divisible by $13$, what is the value of the $26^{\text{th}}$ digit?

Answer: 3
user: If the sum $1 + 2 + 3 + \cdots + K$ is a perfect square $N^2$ and if $N$ is less than $100$, find all possible values for $K$. The original answer is in the form of $k_1, k_2, k_3$. Please give the sum of $k_1, k_2, k_3$.

Answer: 58
user: A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $(a, b)$?

Answer: 2
user: Two circles with radii $15$ and $95$ have their centers $d$ units apart. The external tangents to the circles intersect at an angle of $60$ degrees. What is the distance $d$ between the centers of the circles?

Answer: 160
user: Using a table of a certain height, two identical blocks of wood are placed as shown in Figure 1. The length $r$ is found to be $32$ inches. After rearranging the blocks as in Figure 2, the length $s$ is found to be $28$ inches. How high is the table in inches?

Answer: 30
user: Bernardo randomly picks $3$ distinct numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a $3$-digit number. Silvia randomly picks $3$ distinct numbers from the set $\{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a $3$-digit number. Find the probability that Bernardo's number is larger than Silvia's number. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 93
user: Let \(H\) be the intersection of altitudes in triangle \(ABC\). If \(\angle B = \angle C = \alpha\) and \(O\) is the center of the circle passing through \(A\), \(H\), and \(C\), then find \(\angle HOC\) in terms of \(\alpha\) and express your answer in degrees. The original answer is in the form \(k^\circ - m\alpha\), please give the value of \(k + m\).

Answer: 182
user: What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$? Provide your answer as an integer.

Answer: 8
user: The graphs of $2x + 3y - 6 = 0$, $4x - 3y - 6 = 0$, $x = 2$, and $y = \frac{2}{3}$ intersect in how many points?

Answer: 1
user: Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres? The original answer is in the form of $k + \sqrt{m}$, where $k$ and $m$ are integers. Please find the value of $k + m$.

Answer: 4
user: Find the integer $n$ that satisfies $n^{29} \equiv 7 \pmod{65}$.

Answer: 37
user: How many pairs of integers $(m,n)$ are there such that $mn + n + 14 = (m-1)^2$? Provide the number of such pairs.

Answer: 8
user: If circular arcs $AC$ and $BC$ have centers at $B$ and $A$, respectively, then there exists a circle tangent to both $\stackrel{\frown}{AC}$ and $\stackrel{\frown}{BC}$, and to $\overline{AB}$. If the length of $\stackrel{\frown}{BC}$ is $12$, find the circumference of the circle.

Answer: 27
user: Let $p$ and $q$ be two consecutive terms of the sequence of odd primes. Find the minimum number of positive divisors of $p + q$.

Answer: 4
user: $\triangle ABC$ is a triangle and $D,E,F$ are points on $BC$, $CA$, $AB$ respectively. It is given that $BF=BD$, $CD=CE$ and $\angle BAC=48^{\circ}$. If the answer is $x^\circ$, find the value of $x$ for the angle $\angle EDF$.

Answer: 66
user: There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer $n$ between 1 and 25. Betül draws $n$ balls from the bag one by one such that no ball is put back after it is drawn. If exactly 2 of the $n$ balls are blue and the second blue ball is drawn at the $n^{th}$ order, Ahmet wins; otherwise, Betül wins. To increase the possibility of winning, what integer should Ahmet choose?

Answer: 13
user: What are the last three digits of $49^{303} \cdot 3993^{202} \cdot 39^{606}$?

Answer: 561
user: A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?

Answer: 26
user: All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined?

\[
\begin{array}{c c c c} 
{} & \textbf{Adams} & \textbf{Baker} & \textbf{Adams \text{ and } Baker} \\
\textbf{Boys:} & 71 & 81 & 79 \\
\textbf{Girls:} & 76 & 90 & ? \\
\textbf{Boys \text{ and } Girls:} & 74 & 84 & \\
\end{array}
\]

Find the average score for the girls at the two schools combined.

Answer: 84
user: In the following base-10 equation, each letter represents a unique digit: $AM \cdot PM = ZZZ$. Find the sum of $A + M + P + Z$.

Answer: 21
user: In how many ways can $10$ distinct books be placed onto a $3$-shelf bookcase such that no shelf is empty? Express your answer in the form \( n \cdot 10! \), where \( n \) is an integer.

Answer: 36
user: What is the product of all odd positive integers less than 10000? Express your answer in the form of a factorial expression divided by powers of 2 and factorials. The original answer is in the format \(\frac{x!}{2^y \cdot z!}\). Please find the value of \(x + y + z\).

Answer: 20000
user: What is the area of the region enclosed by the graph of the equations $x^2 - 14x + 3y + 70 = 21 + 11y - y^2$ that lies below the line $y = x - 3$? If the area is $A\pi$, what is the value of $A$?

Answer: 8
user: Given the line $y = \frac{3}{4}x + 6$ and a line $L$ parallel to the given line and $4$ units from it. Find a possible equation for line $L$. The original answer is in the form $y = \frac{3}{4}x + c$, please give the value of c.

Answer: 1
user: What is the smallest possible area, in square centimeters, of a rectangle that can completely contain the shape formed by joining six squares of side length $8$ cm as shown below?

\[ \begin{asy} 
size(5cm,0);
draw((0,2)--(0,3));
draw((1,1)--(1,3));
draw((2,0)--(2,3));
draw((3,0)--(3,2));
draw((4,0)--(4,1));
draw((2,0)--(4,0));
draw((1,1)--(4,1));
draw((0,2)--(3,2));
draw((0,3)--(2,3));
\end{asy} \]

Answer: 768
user: In unit square \(ABCD\), the inscribed circle \(\omega\) intersects \(\overline{CD}\) at \(M\), and \(\overline{AM}\) intersects \(\omega\) at a point \(P\) different from \(M\). Find the length of \(AP\). The original answer is in the form \(\frac{k\sqrt{m}}{n}\), where the fraction is in its simplest form. Please calculate the value of \(k + m + n\).

Answer: 16
user: If $w$ is one of the imaginary roots of the equation $x^3=1$, find the value of the product $(1-w+w^2)(1+w-w^2)$.

Answer: 4
user: A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds \(T_{max}\). One end of the cord is attached to a fixed point, the other is attached to an object of mass \(3m\). If a second, smaller object of mass \(m\) moving at an initial speed \(v_0\) strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of \(v_f\). All motion occurs on a horizontal, frictionless surface.

Find the ratio of the total kinetic energy of the system of two masses after the perfectly elastic collision and the cord has broken to the initial kinetic energy of the smaller mass prior to the collision. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 7
user: In a restaurant, a meal consists of one sandwich and one optional drink. In other words, a sandwich is necessary for a meal but a drink is not necessary. There are two types of sandwiches and two types of drinks. How many possible meals can be purchased?

Answer: 6
user: Yan is somewhere between his home and the stadium. To get to the stadium, he can either walk directly to the stadium or walk home first and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium? Express your answer in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest form, and provide the value of k + m.

Answer: 7
user: The point $(-1, -2)$ is rotated $270^{\circ}$ counterclockwise about the point $(3, 1)$. What are the coordinates of its new position? If the new position is $(x, y)$, please give the coordinates in the form of $(a, b)$ and find the value of $a + b$.

Answer: 5
user: What is the sum of distinct remainders when $(2n-1)^{502}+(2n+1)^{502}+(2n+3)^{502}$ is divided by $2012$ where $n$ is a positive integer?

Answer: 1514
user: Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. Find the probability that the product is divisible by $4$. The original answer is in the form \(\frac{k}{m}\), where k and m are integers. Please calculate the value of k + m.

Answer: 123
user: My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, find the expected score of the losing team. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 5
user: Real numbers \(a\) and \(b\) are chosen with \(1 < a < b\) such that no triangle with positive area has side lengths \(1, a, b\) or \(\frac{1}{b}, \frac{1}{a}, 1\). The original answer is in the form \(\frac{k+\sqrt{n}}{m}\). Find the value of \(k + m + n\).

Answer: 10
user: Two circles are externally tangent. Lines \(\overline{PAB}\) and \(\overline{PA'B'}\) are common tangents with \(A\) and \(A'\) on the smaller circle and \(B\) and \(B'\) on the larger circle. If \(PA = AB = 4\), then what is the value of \(x\) if the area of the smaller circle is \(x\pi\)?

Answer: 2
user: $AB$ is the diameter of a circle centered at $O$. $C$ is a point on the circle such that angle $\angle BOC$ is $60^\circ$. If the diameter of the circle is $5$ inches, find the length of chord $AC$, expressed in inches. The answer should be in the form \(\frac{k\sqrt{m}}{n}\), where \(k\), \(m\), and \(n\) are integers. Please calculate the value of \(k + m + n\).

Answer: 10
user: Find the sum of the roots of $\tan^2 x - 9\tan x + 1 = 0$ that are between $x = 0$ and $x = 2\pi$ radians. If the answer is $k\pi$, what is the value of $k$?

Answer: 3
user: For how many integers $n$, with $2 \leq n \leq 80$, is $\frac{(n-1)n(n+1)}{8}$ equal to an integer? Provide the number of such integers.

Answer: 49
user: Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. Find the number of possible values of $k$.

Answer: 1
user: $A(-1,1)$, $B$, and $C$ are points on the hyperbola $x^2 - y^2 = 1$. If $\triangle ABC$ is a regular triangle, find the area of $\triangle ABC$. The original answer is in the form of k\sqrt{m}, please give the value of k + m.

Answer: 6
user: On a large piece of paper, Dana creates a "rectangular spiral" by drawing line segments of lengths, in cm, of 1, 1, 2, 2, 3, 3, 4, 4, ... as shown. Dana's pen runs out of ink after the total of all the lengths he has drawn is 3000 cm. What is the length of the longest line segment that Dana draws, in cm?

Answer: 54
user: Older television screens have an aspect ratio of $4:3$. That is, the ratio of the width to the height is $4:3$. The aspect ratio of many movies is not $4:3$, so they are sometimes shown on a television screen by 'letterboxing' - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of $2:1$ and is shown on an older television screen with a $27$-inch diagonal. If the height of each darkened strip is $x$ inches, what is the value of $10x$?

Answer: 27
user: How many pairs of positive integers $(x,y)$ are there such that $\sqrt{xy} - 71\sqrt{x} + 30 = 0$?

Answer: 8
user: Some of $A,B,C,D,$ and $E$ are truth tellers, and the others are liars. Truth tellers always tell the truth. Liars always lie. We know $A$ is a truth teller. According to the conversation below:

$B$: I'm a truth teller.

$C$: $D$ is a truth teller.

$D$: $B$ and $E$ are not both truth tellers.

$E$: $A$ and $B$ are truth tellers.

How many truth tellers are there?

Answer: 3
user: Arithmetic sequences $(a_n)$ and $(b_n)$ have integer terms with $a_1 = b_1 = 1 < a_2 \le b_2$ and $a_nb_n = 2010$ for some $n$. What is the largest possible value of $n$?

Answer: 8