meetforever/GRPO_CL huggingface dataset:
user: Vertex \(E\) of equilateral \(\triangle{ABE}\) is in the interior of unit square \(ABCD\). Let \(R\) be the region consisting of all points inside \(ABCD\) and outside \(\triangle{ABE}\) whose distance from \(\overline{AD}\) is between \(\frac{1}{3}\) and \(\frac{2}{3}\). Find the area of \(R\). The original answer is in the form \(\frac{k-m\sqrt{3}}{n}\), please provide the value of k + m + n.
Answer: 53
user: In the adjoining figure, \( \triangle CDE \) is an equilateral triangle and \( \square ABCD \) and \( \square DEFG \) are squares. If the measure of \( \angle GDA \) is \( x^\circ \), what is the value of \( x \)?
Answer: 120
user: If the angles $a, b, c, d,$ and $e$ are known, what is the expression for the angle $u$ in terms of $a$, $b$, $c$, $d$, and $e$? The original answer is in the form of m*a + n*c + p*d. Please provide the value of m + n + p.
Answer: 3
user: The Tigers beat the Sharks $2$ out of the first $3$ times they played. They then played $N$ more times, and the Sharks ended up winning at least $95\%$ of all the games played. What is the minimum possible value for $N$?
Answer: 37
user: How many ordered pairs $(a, b)$ such that $a$ is a real positive number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\log_b a)^{2017} = \log_b (a^{2017})$?
Answer: 597
user: Three balls are randomly and independently tossed into bins numbered with the positive integers such that for each ball, the probability it is tossed into bin $i$ is $2^{-i}$ for $i = 1, 2, 3, \ldots$. 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. (For example, the balls are evenly spaced if they are tossed into bins $3$, $17$, and $10$.) What is $p+q$?
Answer: 55
user: What is the greatest integer $n$ such that $$n \leq 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{2014}}?$$
Answer: 88
user: Pablo buys popsicles for his friends. The store sells single popsicles for $1$ each, 3-popsicle boxes for $2$, and 5-popsicle boxes for $3$. What is the greatest number of popsicles that Pablo can buy with $8$?
Answer: 13
user: How many $5$-digit positive numbers which contain only odd numbers are there such that there is at least one pair of consecutive digits whose sum is $10$?
Answer: 1845
user: The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111$, $R_5=11111$, etc. When $R_{24}$ is divided by $R_4$, the quotient $Q=\frac{R_{24}}{R_4}$ is an integer whose base-ten representation is a sequence containing only ones and zeros. Find the number of zeros in $Q$.
Answer: 15
user: Juan rolls a fair regular octahedral die marked with the numbers $1$ through $8$. Then Amal rolls a fair six-sided die. Find the probability that the product of the two rolls is a multiple of $3$. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 3
user: Kayla rolls four fair 6-sided dice. Find the probability that at least one of the numbers Kayla rolls is greater than 4 and at least two of the numbers she rolls are greater than 2. Express your answer as a fraction in simplest form, \(\frac{k}{m}\). Please provide the value of k + m.
Answer: 142
user: Three cubes with volumes $1$, $8$, and $27$ are glued together at their faces. Find the smallest possible surface area of the resulting configuration.
Answer: 72
user: Let $f(x) = x^{2}(1-x)^{2}$. What is the value of the sum
\begin{align*}
f\left(\frac{1}{2019}\right)-f\left(\frac{2}{2019}\right)+f\left(\frac{3}{2019}\right)-&f\left(\frac{4}{2019}\right)+\cdots\\
 &\,+f\left(\frac{2017}{2019}\right) - f\left(\frac{2018}{2019}\right)?
\end{align*}
Answer: 0
user: The polynomial $x^3 - 2004x^2 + mx + n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $n$ are possible? The original answer is in the form of k*n, please find the value of k.
Answer: 250500
user: Four rectangular paper strips of length $10$ and width $1$ are placed flat on a table, overlapping perpendicularly as shown. How much area of the table is covered?
Answer: 36
user: From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. Also from $P$, a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$. It is found that $t_1$, the length of the tangent, is the mean proportional between $m$ and $n$. If $m$ and $t$ are integers, find the number of possible integer values for $t$.
Answer: 2
user: Beth adds the consecutive positive integers $a$, $b$, $c$, $d$, and $e$, and finds that the sum is a perfect square. She then adds $b$, $c$, and $d$ and finds that this sum is a perfect cube. What is the smallest possible value of $e$?
Answer: 1127
user: Letters $A, B, C,$ and $D$ represent four different digits from 0, 1, 2, 3, ..., 9. If $\frac{A+B}{C+D}$ is an integer that is as large as possible, what is the value of $A+B$?
Answer: 17
user: A child is sliding out of control with velocity \(v_{c}\) across a frozen lake. He runs head-on into another child, initially at rest, with 3 times the mass of the first child, who holds on so that the two now slide together. The original answer is in the form \(\frac{k \cdot v_{c}}{m}\). Please find the value of \(k + m\).
Answer: 5
user: How many integers $x$ satisfy the equation:
\[
(x^2 - x - 1)^{x + 2} = 1
\]
Answer: 4
user: What is the largest possible value of $8x^2 + 9xy + 18y^2 + 2x + 3y$ such that $4x^2 + 9y^2 = 8$, where $x, y$ are real numbers?
Answer: 26
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 $DE$ is in the form of k\sqrt{m}. Please find the value of k + m.
Answer: 14
user: Represent the hypotenuse of a right triangle by $c$ and the area by $A$. What is the altitude on the hypotenuse? The original answer is in the form of $\frac{mA}{nc}$, where $m$ and $n$ are integers. Please give the value of $m + n$.
Answer: 3
user: What magnitude force does Jonathan need to exert on the physics book to keep the rope from slipping? The answer should be expressed in the form \(\frac{kM g}{m\mu_s}\), where \(k\) and \(m\) are integers. Please find the value of \(k + m\).
Answer: 2
user: Find a rational number between $\sqrt{2}$ and $\sqrt{3}$. If the answer is $x$, what is the value of $10x$?
Answer: 15
user: A square with an area of \(2\) is inscribed within a larger square with an area of \(3\), forming four congruent triangles. Determine the ratio of the shorter leg to the longer leg in the shaded right triangle. The original answer is in the form \(k - \sqrt{m}\). Please provide the value of \(k + m\).
Answer: 5
user: How many positive integers less than $1000$ are there such that they cannot be written as the sum of $2$ or more successive positive integers? Fill in the blank with the correct number.
Answer: 10
user: If the points $(1, y_1)$ and $(-1, y_2)$ lie on the graph of $y = ax^2 + bx + c$, and $y_1 - y_2 = -6$, the original answer is in the form of $k$. Please find the value of $k$.
Answer: -3
user: The fraction \(\frac{2(\sqrt{2} + \sqrt{6})}{3\sqrt{2 + \sqrt{3}}}\) is equal to what value? The answer is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please calculate the value of \(k + m\).
Answer: 7
user: Bertha has $6$ daughters and no sons. Some of her daughters have $6$ daughters, and the rest have none. Bertha has a total of $30$ daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no children? Fill in the blank with the correct number.
Answer: 26
user: Let \(ABCD\) be an isosceles trapezoid with parallel bases \(\overline{AB}\) and \(\overline{CD}\), where \(AB > CD\). Line segments from a point inside \(ABCD\) to the vertices divide the trapezoid into four triangles with areas \(2\), \(3\), \(4\), and \(5\), starting with the triangle with base \(\overline{CD}\) and moving clockwise. The original answer for the ratio \(\frac{AB}{CD}\) is in the form \(k + \sqrt{m}\). Please calculate the value of \(k + m\).
Answer: 4
user: The amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $0.357$, or into $\sqrt{3}$ and $2.5-\sqrt{3}$. Then each number is rounded to its nearest integer, for instance, $2$ and $0$ in the first case above, $2$ and $1$ in the second. What is the probability that the two integers sum to $3$? The original answer is in the form \(\frac{k}{m}\), please give the value of \(k + m\).
Answer: 7
user: Let $K$ be the point of intersection of $AB$ and the line touching the circumcircle of $\triangle ABC$ at $C$ where $m(\widehat {A}) > m(\widehat {B})$. Let $L$ be a point on $[BC]$ such that $m(\widehat{ALB})=m(\widehat{CAK})$, $5|LC|=4|BL|$, and $|KC|=12$. Find the length of $|AK|$ in centimeters, denoted as x. Please provide the value of x.
Answer: 8
user: For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?
Answer: 8
user: A man with mass \(m\) jumps off a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance \(H\), at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant \(k\), and stretches from an original length of \(L_0\) to a final length \(L = L_0 + h\). The maximum tension in the bungee cord is four times the weight of the man.

Determine the spring constant \(k\). The original answer is in the form \(\frac{kmg}{nH}\). Please provide the value of \(k+n\).
Answer: 9
user: The triangle $ABC$ has vertices such that $AB = 3$, $BC = 4$, and $AC = 5$. The inscribed circle is tangent to $AB$ at $C'$, $BC$ at $A'$, and $AC$ at $B'$. The ratio between the area of the triangles $A'B'C'$ and $ABC$ is given in the form \frac{k}{m}. Please find the value of k + m.
Answer: 6
user: For how many integers $x$ does a triangle with side lengths $10$, $24$, and $x$ have all its angles acute?
Answer: 4
user: A very large number of small particles forms a spherical cloud. Initially, they are at rest, have uniform mass density per unit volume $\rho_0$, and occupy a region of radius $r_0$. The cloud collapses due to gravitation; the particles do not interact with each other in any other way.

Find the time that passes until the cloud collapses fully, expressed in the form $\frac{k}{\sqrt{G \rho_0}}$. Please give the value of $10000k$.
Answer: 5427
user: If $m$ and $n$ are the roots of the quadratic equation $x^2 + mx + n = 0$, where $m \neq 0$ and $n \neq 0$, what is the value of the sum of the roots?
Answer: -1
user: How many primes \( p \) are there such that \( 5p(2^{p+1}-1) \) is a perfect square? Provide your answer as a single integer.
Answer: 1
user: In the complex plane, let $A$ be the set of solutions to $z^3 - 8 = 0$ and let $B$ be the set of solutions to $z^3 - 8z^2 - 8z + 64 = 0$. Find the greatest distance between a point of $A$ and a point of $B$. The original answer is in the form $a\sqrt{b}$, please give the value of $ab$.
Answer: 42
user: An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is \$20 or more? Express your answer in the form \(\frac{k}{m}\), where k and m are integers, and give the value of k + m.
Answer: 3
user: 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 $ABCD$? Express your answer in the form $k\sqrt{m}$, and find the value of $k + m$.
Answer: 38
user: Liquid X does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$ cm thick. A rectangular box measuring $6$ cm by $3$ cm by $12$ cm is filled with liquid X. Its contents are poured onto a large body of water. The radius of the resulting circular film is in the form \(\sqrt{\frac{a}{\pi}}\). Please find the value of a + b + c, where a, b, and c are constants such that the expression under the square root is in the form \(\frac{a}{\pi}\).
Answer: 2160
user: $K$ takes 30 minutes less time than $M$ to travel a distance of 30 miles. $K$ travels $\frac{1}{3}$ mile per hour faster than $M$. If $x$ is $K$'s rate of speed in miles per hour, find $K$'s time for the distance in hours. The original answer is in the form \(\frac{k}{x}\), please give the value of \(k\).
Answer: 30
user: Triangle $\triangle ABC$ in the figure has area $10$. Points $D$, $E$, and $F$, all distinct from $A$, $B$, and $C$, are on sides $AB$, $BC$, and $CA$ respectively, and $AD = 2$, $DB = 3$. If triangle $\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then what is that area?
Answer: 6
user: A pyramid has a square base $ABCD$ and vertex $E$. The area of square $ABCD$ is $196$, and the areas of $\triangle ABE$ and $\triangle CDE$ are $105$ and $91$, respectively. Find the volume of the pyramid.
Answer: 784
user: For every positive integer $n$, the function $f$ is defined as follows:
- $f(2n+1) = 2f(2n)$
- $f(2n) = f(2n-1) + 1$
- $f(1) = 0$

Determine the remainder when $f(2005)$ is divided by $5$.
Answer: 1
user: Let $U = 2 \times 2004^{2005}$, $V = 2004^{2005}$, $W = 2003 \times 2004^{2004}$, $X = 2 \times 2004^{2004}$, $Y = 2004^{2004}$, and $Z = 2004^{2003}$. Find the largest value among the expressions $U - V$, $V - W$, $W - X$, $X - Y$, and $Y - Z$. The answer should be given in the form of k - m, where k and m are placeholders for the original expressions. Provide the sum of the alphabetical positions of k and m as the final answer.
Answer: 43
user: A line passes through $A(1,1)$ and $B(100,1000)$. Find the number of other points with integer coordinates that are on the line and strictly between $A$ and $B$.
Answer: 8
user: In our number system, the base is ten. If the base were changed to four, you would count as follows: $1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, \ldots$ What would be the twentieth number in this system?
Answer: 110
user: Sarah is leading a class of $35$ students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands $3$ times, what is the greatest possible number of students that could have been standing before her third wave?
Answer: 31
user: Given the equation \(3x^2 - 4x + k = 0\) with real roots, find the value of \(k\) for which the product of the roots of the equation is a maximum. The original answer is in \(\frac{m}{n}\) format, where \(\frac{m}{n}\) is in its simplest form. Please give the value of \(m + n\).
Answer: 7
user: Find the number of positive integers less than $1000$ that are divisible by neither $5$ nor $7$.
Answer: 686
user: Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walter entered?
Answer: 80
user: $AL$, $BM$, and $CN$ are the medians of $\triangle ABC$. $K$ is the intersection of the medians. If $C, K, L, M$ are concyclic and $AB = \sqrt{3}$, find the length of the median $CN$. The original answer is in the form \(\frac{k\sqrt{3}}{n}\), where k and n are integers. Please find the value of k + n.
Answer: 5
user: The traffic on a certain east-west highway moves at a constant speed of 60 miles per hour in both directions. An eastbound driver passes 20 west-bound vehicles in a five-minute interval. Assume vehicles in the westbound lane are equally spaced. What is the number of westbound vehicles present in a 100-mile section of highway?
Answer: 200
user: Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?
Answer: 103
user: What is the maximum value of the difference between the largest real root and the smallest real root of the equation system \[\begin{array}{rcl} ax^2 + bx+ c &=& 0 \\ bx^2 + cx+ a &=& 0 \\ cx^2 + ax+ b &=& 0 \end{array}\], where at least one of the reals $a,b,c$ is non-zero?
Answer: 0
user: For a rhombus with a side length of 5, the length of one of its diagonals is not larger than $6$, and the length of the other diagonal is not smaller than $6$. Determine the maximum value of the sum of the two diagonals.
Answer: 14
user: In a right triangle with sides \(a\) and \(b\), and hypotenuse \(c\), the altitude drawn on the hypotenuse is \(x\). Find the relationship between \(a\), \(b\), and \(x\) that satisfies \(\frac{k}{x^2} = \frac{m}{a^2} + \frac{n}{b^2}\). Please provide the sum of the coefficients \(k + m + n\).
Answer: 3
user: In triangle $\triangle ABC$ in the adjoining figure, $AD$ and $AE$ trisect $\angle BAC$. The lengths of $BD$, $DE$, and $EC$ are $2$, $3$, and $6$, respectively. Find the length of the shortest side of $\triangle ABC$. The original answer is in the form $x\sqrt{y}$, please give the value of $x + y$.
Answer: 12
user: If $N=\lfloor \frac{2}{5} \rfloor + \lfloor \frac{2^2}{5} \rfloor + \dots + \lfloor \frac{2^{2009}}{5} \rfloor$, find the remainder when $2^{2010}$ is divided by $N$.
Answer: 5024
user: Let $ABC$ be a triangle with $|AB| = 14$, $|BC| = 12$, $|AC| = 10$. Let $D$ be a point on $[AC]$ and $E$ be a point on $[BC]$ such that $|AD| = 4$ and $\text{Area}(ABC) = 2 \times \text{Area}(CDE)$. Find $\text{Area}(ABE)$ in the form of $m\sqrt{n}$, where $m$ and $n$ are integers. Please give the value of $m + n$.
Answer: 10
user: How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation:
\[(1+2a)(2+2b)(2a+b) = 32ab\]?
Answer: 1
user: Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, what number could be obtained?
Answer: 119
user: A plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse is $50\%$ longer than the minor axis, what is the length of the major axis?
Answer: 3
user: For the integer numbers $i, j, k$ satisfying the condition $i^2 + j^2 + k^2 = 2011$, find the largest value of $i + j + k$.
Answer: 77
user: Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. What is the sum of the squares of their ages?
Answer: 42
user: A sector with an acute central angle \(\theta\) is cut from a circle of radius 6. What is the radius of the circle circumscribed about the sector in terms of \(\theta\)? The original answer is in the form \(k \sec \frac{m}{n} \theta\). Please find the value of k + m + n.
Answer: 6
user: Let $p$ be the greatest prime factor of 9991. What is the sum of the digits of $p$?
Answer: 4
user: How many ordered triples of integers \((a, b, c)\), with \(a \ge 2\), \(b \ge 1\), and \(c \ge 0\), satisfy both \(\log_a b = c^{2005}\) and \(a + b + c = 2005\)?
Answer: 2
user: A store prices an item in dollars and cents so that when 4\% sales tax is added, no rounding is necessary because the result is exactly $n$ dollars where $n$ is a positive integer. Find the smallest value of $n$.
Answer: 13
user: A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$. The area of the sheet of wrapping paper can be expressed in the form $k(w+h)^m$. Please find the value of $k + m$.
Answer: 4
user: Assuming \(a\neq3\), \(b\neq4\), and \(c\neq5\), find the integer value of the following expression:
\[\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}\]
Please provide your answer as an integer.
Answer: -1
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. Calculate the integer value of \((t-s) \times 10\).
Answer: -135
user: On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. Find the difference between the mean and median score of the students' scores on this quiz.
Answer: 3
user: Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window?
Answer: 26
user: Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is $50\%$ more than the regular. After both consume $\frac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and $2$ additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?
Answer: 40
user: Suppose $A>B>0$ and A is $x$% greater than $B$. If the answer is in the form \frac{m(A-B)}{nB}, please find the value of m + n.
Answer: 101
user: For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer?
Answer: 7
user: There are four more girls than boys in Ms. Raub's class of $28$ students. What is the ratio of the number of girls to the number of boys in her class? Provide your answer in the form of $a : b$, and then find the value of $a + b$.
Answer: 7
user: Among the following expressions, which one is a perfect square? \(\dfrac{14!15!}{2}\), \(\dfrac{15!16!}{2}\), \(\dfrac{16!17!}{2}\), \(\dfrac{17!18!}{2}\), \(\dfrac{18!19!}{2}\). Provide the expression that represents a perfect square in the form \(\dfrac{k!m!}{2}\), and give the value of k + m.
Answer: 35
user: In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?  
$$  
\begin{array}{cccccc}&A&B&B&C&B\\+&B&C&A&D&A\\\hline&D&B&D&D&D\end{array}
$$  
Provide the number of different possible values for $D$.
Answer: 7
user: Five positive consecutive integers starting with $a$ have an average of $b$. What is the average of 5 consecutive integers that start with $b$? Provide your answer in the form of an expression involving $a$ as $a + k$. Find the value of $k$.
Answer: 4
user: Five positive consecutive integers starting with $a$ have an average of $b$. What is the average of 5 consecutive integers that start with $b$? Provide your answer in the form of $a + k$, where $k$ is an integer. Please find the value of $k$.
Answer: 4
user: In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. Find how many different values are possible for $D$.

\[\begin{array}[t]{r}     ABBCB \\ + \ BCADA \\ \hline     DBDDD \end{array}\]
Answer: 7