meetforever/GRPO_CL huggingface dataset:
user: In square \(ABCD\), points \(E\) and \(H\) lie on \(\overline{AB}\) and \(\overline{DA}\), respectively, so that \(AE=AH.\) Points \(F\) and \(G\) lie on \(\overline{BC}\) and \(\overline{CD}\), respectively, and points \(I\) and \(J\) lie on \(\overline{EH}\) so that \(\overline{FI} \perp \overline{EH}\) and \(\overline{GJ} \perp \overline{EH}\). See the figure below. Triangle \(AEH\), quadrilateral \(BFIE\), quadrilateral \(DHJG\), and pentagon \(FCGJI\) each has area \(1.\) The original answer is in the form \(k - m\sqrt{n}\), please give the value of \(k + m + n\).
Answer: 14
user: How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n$?
Answer: 48
user: How many positive integers $n$ satisfy 
$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$
(Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
Answer: 6
user: For how many positive integers $n \le 1000$ is
$\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$
not divisible by $3$?
Answer: 22
user: What is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1$?
Answer: 201
user: Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that
\[ (2 + i)^n = a_n + b_ni \] for all integers $n\geq 0$, where $i = \sqrt{-1}$. Find the value of \[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,.\] The original answer is in the form \(\frac{k}{m}\). Please give the value of \(k + m\).
Answer: 23
user: What is the maximum value of \(\frac{(2^t-3t)t}{4^t}\) for real values of \(t?\) The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of k + m.
Answer: 13
user: When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.

[asy] size(300); defaultpen(linewidth(0.8)+fontsize(13)); real r = 0.05; draw((0.9,0)--(3.5,0),EndArrow(size=7)); filldraw((4,2.5)--(7,2.5)--(7,-2.5)--(4,-2.5)--cycle,gray(0.65)); fill(circle((5.5,1.25),0.8),white); fill(circle((5.5,1.25),0.5),gray(0.65)); fill((4.3,-r)--(6.7,-r)--(6.7,-1-r)--(4.3,-1-r)--cycle,white); fill((4.3,-1.25+r)--(6.7,-1.25+r)--(6.7,-2.25+r)--(4.3,-2.25+r)--cycle,white); fill((4.6,-0.25-r)--(6.4,-0.25-r)--(6.4,-0.75-r)--(4.6,-0.75-r)--cycle,gray(0.65)); fill((4.6,-1.5+r)--(6.4,-1.5+r)--(6.4,-2+r)--(4.6,-2+r)--cycle,gray(0.65)); label("$N$",(0.45,0)); draw((7.5,1.25)--(11.25,1.25),EndArrow(size=7)); draw((7.5,-1.25)--(11.25,-1.25),EndArrow(size=7)); label("if $N$ is even",(9.25,1.25),N); label("if $N$ is odd",(9.25,-1.25),N); label("$\frac N2$",(12,1.25)); label("$3N+1$",(12.6,-1.25)); [/asy]
For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$\[7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26\]When the same $6$-step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$\[N \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to 1\]
Answer: 83
user: Square \(ABCD\) in the coordinate plane has vertices at the points \(A(1,1), B(-1,1), C(-1,-1),\) and \(D(1,-1).\) Consider the following four transformations:

\(\quad\bullet\qquad\) \(L,\) a rotation of \(90^{\circ}\) counterclockwise around the origin;

\(\quad\bullet\qquad\) \(R,\) a rotation of \(90^{\circ}\) clockwise around the origin;

\(\quad\bullet\qquad\) \(H,\) a reflection across the \(x\)-axis; and

\(\quad\bullet\qquad\) \(V,\) a reflection across the \(y\)-axis.

Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying \(R\) and then \(V\) would send the vertex \(A\) at \((1,1)\) to \((-1,-1)\) and would send the vertex \(B\) at \((-1,1)\) to itself. Find how many sequences of \(20\) transformations chosen from \(\{L, R, H, V\}\) will send all of the labeled vertices back to their original positions. The original answer is in the form \(x * 2^y\), please give the value of \(x + y\).
Answer: 39
user: Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly \(7\). Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice? Express your answer in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please give the value of \(k + m\).
Answer: 43
user: Let \( n \) be the least positive integer greater than 1000 for which
\(\gcd(63, n+120) = 21\) and \(\gcd(n+63, 120) = 60.\)
Find the sum of the digits of \( n \).
Answer: 18
user: How many positive integers $n$ satisfy 
$$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$$
Answer: 6
user: Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\[n = f_1\cdot f_2\cdots f_k,\]where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$?
Answer: 112
user: Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly \(7\). Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice? Express your answer in the form \(\frac{k}{m}\), and find the value of k + m.
Answer: 43
user: The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying
\[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\]
is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p+q$?
Answer: 929
user: For each real number \(a\) with \(0 \leq a \leq 1\), let numbers \(x\) and \(y\) be chosen independently at random from the intervals \([0, a]\) and \([0, 1]\), respectively, and let \(P(a)\) be the probability that
\(\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1\).
Find the maximum value of \(P(a)\). The original answer is in the form \(k - \sqrt{m}\), please give the value of \(k + m\).
Answer: 4
user: At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?
Answer: 32
user: Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch?
Answer: 20
user: What is the area of the shaded figure shown below?
[asy] size(200); defaultpen(linewidth(0.4)+fontsize(12)); pen s = linewidth(0.8)+fontsize(8);  pair O,X,Y; O = origin; X = (6,0); Y = (0,5); fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2)); for (int i=1; i<7; ++i) { draw((i,0)--(i,5), gray+dashed); label("${"+string(i)+"}$", (i,0), 2*S); if (i<6) { draw((0,i)--(6,i), gray+dashed); label("${"+string(i)+"}$", (0,i), 2*W); } } label("$0$", O, 2*SW); draw(O--X+(0.35,0), black+1.5, EndArrow(10)); draw(O--Y+(0,0.35), black+1.5, EndArrow(10)); draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5); [/asy]

Find the area of the shaded figure.
Answer: 6
user: Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have an area of $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch?
Answer: 20
user: What is the maximum number of balls of clay of radius $2$ that can completely fit inside a cube of side length $6$ assuming the balls can be reshaped but not compressed before they are packed in the cube?
Answer: 6
user: The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is the value of $A$?
Answer: 9
user: When a student multiplied the number $66$ by the repeating decimal,  
$\underline{1}.\underline{a} \ \underline{b} \ \underline{a}  \underline{b}\cdots=\underline{1}.\overline{\underline{a} \ \underline{b}},$  
where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66~ times$ $\underline{1}.\underline{a} \ \underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. Find the $2$-digit number $\underline{a} \ \underline{b}.$
Answer: 75
user: The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^\circ$ around the point $(1,5)$ and then reflected about the line $y = -x$. The image of $P$ after these two transformations is at $(-6,3)$. What is the value of $b - a$?
Answer: 7
user: The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is the value of $A?$
Answer: 9
user: Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. Find the number of distinct integers that can be written as the sum of two, not necessarily different, special fractions.
Answer: 11
user: Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at \(4\) miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to \(2\) miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at \(3\) miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet? The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 25
user: The least positive integer with exactly $2021$ distinct positive divisors can be written in the form $m \cdot 6^k$, where $m$ and $k$ are integers and $6$ is not a divisor of $m$. What is the value of $m+k$?
Answer: 58
user: The greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$. Find the sum of the digits of the greatest prime number that is a divisor of $16{,}383$.
Answer: 10
user: As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. If the answer is $x^\circ$, what is the degree measure of $\angle AFE$?
Answer: 170
user: Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. Find how many distinct integers can be written as the sum of two, not necessarily different, special fractions.
Answer: 11
user: When a student multiplied the number $66$ by the repeating decimal,  
$$\underline{1}.\underline{a} \ \underline{b} \ \underline{a} \ \underline{b}\ldots=\underline{1}.\overline{\underline{a} \ \underline{b}},$$  
where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $\underline{1}.\underline{a} \ \underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. Find the $2$-digit number $\underline{a} \ \underline{b}.$
Answer: 75
user: Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. Find the sum of the greatest number and the least number that appear in the second row from the top.
Answer: 367
user: Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines? Provide your answer as an integer.
Answer: 6
user: A two-digit positive integer is said to be cuddly if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?
Answer: 1
user: The greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16{,}383$? Provide your answer as a single integer.
Answer: 10
user: Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M, 32, 33, 34, 35, 36, 37, 38, 39,$ and $40.$ Find the value of $\frac{N}{M}.$
Answer: 74
user: The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure of the vertex angle of this triangle, in degrees?
Answer: 150
user: Which expression is equivalent to

$(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$

Provide your answer in the form $a^{b} - c^{b}$, and then find the value of $a + b + c$.
Answer: 133
user: When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. Find the probability that the sum of the numbers rolled is even. The original answer is in \frac{k}{m} format, please give the value of k + m.
Answer: 13
user: A right rectangular prism whose surface area and volume are numerically equal has edge lengths $\log_{2}x, \log_{3}x,$ and $\log_{4}x.$ What is the integer value of $x$?
Answer: 576
user: Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. If the area of the circle passing through $A$, $O$, and $C$ is $x\pi$, what is the value of $x$?
Answer: 12
user: Two distinct numbers are selected from the set $\{1,2,3,4,\dots,36,37\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers?
Answer: 10
user: $A$ school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $10(t-s)$?
Answer: -135
user: Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by $100$ and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat?
Answer: 27
user: The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\text{nine}}.$ Find the remainder when $N$ is divided by $5$.
Answer: 3
user: Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce?
Answer: 60
user: A laser is placed at the point $(3,5)$. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, then hit the point $(7,5)$. What is the total distance $d$ the beam will travel along this path? If the answer is $a\sqrt{b}$, where $a$ and $b$ are integers and $b$ is not divisible by the square of any prime, what is the value of $a$?
Answer: 10
user: Triangle \(ABC\) has \(AB=13, BC=14\) and \(AC=15\). Let \(P\) be the point on \(\overline{AC}\) such that \(PC=10\). There are exactly two points \(D\) and \(E\) on line \(BP\) such that quadrilaterals \(ABCD\) and \(ABCE\) are trapezoids. The distance \(DE\) can be expressed in the form \(k\sqrt{m}\). Please find the value of \(k + m\).
Answer: 14
user: Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts $42$ steps of the same size from the front of the ship to the back. In terms of Emily's equal steps, what is the length of the ship?
Answer: 70
user: A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. Find the area of the region bounded by these $6$ reflected arcs. The original answer is in the form of k\sqrt{m} - n\pi, please provide the value of k + m + n.
Answer: 7
user: Consider two concentric circles of radius $17$ and $19.$ The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle? The original answer is in the form of k\sqrt{m}, please give the value of k + m.
Answer: 14
user: 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 in the form of \(\frac{k}{m}\) where the fraction is in simplest form. Please give the value of \(k + m\).
Answer: 123
user: Let $N = 34 \cdot 34 \cdot 63 \cdot 270$. What is the value of $y$ if the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$ is $1 : y$?
Answer: 14
user: Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. If the average value of all the integers in the set $S$ is $x$, what is the value of $10x$?
Answer: 368
user: For \(n\) a positive integer, let \(f(n)\) be the quotient obtained when the sum of all positive divisors of \(n\) is divided by \(n.\) For example, \[f(14)=(1+2+7+14)\div 14=\frac{12}{7}\]
Find the value of \(f(768)-f(384).\) The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 193
user: The angle bisector of the acute angle formed at the origin by the graphs of the lines \(y = x\) and \(y=3x\) has equation \(y=kx.\) The original answer is in the form \(\frac{k+\sqrt{m}}{n}\). Please find the value of \(k + m + n\).
Answer: 8
user: How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy 
$1-3\sin\theta+5\cos3\theta = 0?$
Answer: 6
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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