inclusionAI/Ring-lite-rl-data huggingface dataset:
A school offers 10 elective courses, among which courses A, B, and C have the same class time. According to the school's rules, each student must choose three courses. Find the number of different elective schemes available for each student. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
Find the number of triples of sets $(A, B, C)$ such that:

(a) $A, B, C \subseteq \{1, 2, 3, \dots , 8 \}$.
(b) $|A \cap B| = |B \cap C| = |C \cap A| = 2$.
(c) $|A| = |B| = |C| = 4$.

Here, $|S|$ denotes the number of elements in the set $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45360}
Let \(ABC\) be a triangle with \(AB = 13\), \(BC = 14\), \(CA = 15\). Let \(H\) be the orthocenter of \(ABC\). Find the distance between the circumcenters of triangles \(AHB\) and \(AHC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Compute: $\int_{\lvert z \rvert = \frac{1}{2}} \frac{dz}{z \sin\left( \frac{1}{z} \right)}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Two sides of a regular polygon with $n$ sides, when extended, meet at an angle of $28$ degrees. What is the smallest possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
7 For non-negative real numbers $x_{i}(i=1,2, \cdots, n)$ satisfying $x_{1}+x_{2}+\cdots+x_{n}=1$, find the maximum value of $\sum_{j=1}^{n}\left(x_{j}^{4}-x_{j}^{5}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{12}}
Quadrilateral $ABCD$ has $AB = BC = CD$, $m\angle ABC = 70^\circ$ and $m\angle BCD = 170^\circ$. If the answer is $x^\circ$, what is the value of $x$ for the degree measure of $\angle BAD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
In how many ways can 2 black, 3 white, and 4 red balls be arranged in a row such that no black ball is next to a white ball?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Sixty points are marked on a circle, divided as follows: thirty are red, twenty are blue, and ten are green. These points divide the circle into sixty arcs. Each arc is assigned a number based on the colors of its endpoints:

- An arc between a red and a green point is assigned the number $1$.
- An arc between a red and a blue point is assigned the number $2$.
- An arc between a blue and a green point is assigned the number $3$.
- An arc between two points of the same color is assigned the number $0$.

What is the greatest possible sum of all the numbers assigned to the arcs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given that the graph of the inverse function of \( y = -\frac{x-a}{x-a-1} \) is centrally symmetric with respect to the point \((-1,3)\), determine the real number \( a \). Express your answer as a single real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
174. \(n^{2}\) integers from 1 to \(n^{2}\) are written in a square table of size \(n \times n\): number 1 is in any position in the table; number 2 belongs to the row whose number is equal to the column number containing 1; number 3 belongs to the row whose number matches the column number containing 2, and so on. By how much does the sum of the numbers in the row containing number 1 differ from the sum of the numbers in the column containing the number \(n^{2}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 - n}
Given that point \( F \) is the right focus of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (\(a > b > 0\)), and the eccentricity of the ellipse is \(\frac{\sqrt{3}}{2}\), a line \( l \) passing through point \( F \) intersects the ellipse at points \( A \) and \( B \) (point \( A \) is above the \( x \)-axis), and \(\overrightarrow{A F} = 3 \overrightarrow{F B}\). Find the slope of the line \( l \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\sqrt{2}}
Call an integer $k$ debatable if the number of odd factors of $k$ is a power of two. What is the largest positive integer $n$ such that there exists $n$ consecutive debatable numbers? (Here, a power of two is defined to be any number of the form $2^m$, where $m$ is a nonnegative integer.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
You are given a positive integer  $k$  and not necessarily distinct positive integers  $a_1, a_2 , a_3 , \ldots,
a_k$ . It turned out that for any coloring of all positive integers from  $1$  to  $2021$  in one of the  $k$  colors so that there are exactly  $a_1$  numbers of the first color,  $a_2$  numbers of the second color,  $\ldots$ , and  $a_k$  numbers of the  $k$ -th color, there is always a number  $x \in \{1, 2, \ldots, 2021\}$ , such that the total number of numbers colored in the same color as  $x$  is exactly  $x$ . What are the possible values of  $k$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2021}
Let $ABCD$ be a unit square and let $E$ and $F$ be points inside $ABCD$ such that the line containing $\overline{EF}$ is parallel to $\overline{AB}$. Point $E$ is closer to $\overline{AD}$ than point $F$ is to $\overline{AD}$. The line containing $\overline{EF}$ also bisects the square into two rectangles of equal area. Suppose $[AEFB] = [DEFC] = 2[AED] = 2[BFC]$. The length of segment $\overline{EF}$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In a box, there are 100 balls of different colors: 28 red balls, 20 green balls, 12 yellow balls, 20 blue balls, 10 white balls, and 10 black balls. How many balls must be drawn randomly from the box to ensure that at least 15 of them are of the same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
设A（2,0）为平面上一定点，$P ( \sin ( 2 t - 6 0 ^ { \circ } ) , \cos ( 2 t - 6 0 ^ { \circ } )$为动点，则当t由15°变到45°时，线段AP所扫过的图形的面积是
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac { \pi } { 6 }$}
Given the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 (a > 0, b > 0)$ with its right focus at $F$. If a line passing through point $F$ with a slope angle of $60^\circ$ intersects the right branch of the hyperbola at exactly one point, then the range of the eccentricity of this hyperbola is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2,+\infty)}
The graph of the function $f(x) = \log_a x + 1$ must pass through a fixed point. Determine the coordinates of this fixed point. Express your answer as an ordered pair $(x, y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,1)}
The complex number $z=(a-2)+(a+1)i$, where $a\in\mathbb{R}$, corresponds to a point located in the second quadrant. The range of $|z|$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[ \frac{3\sqrt{2}}{2},3\right]}
Let \( x_{1}, x_{2} \) be the roots of the equation \( x^{2} - x - 3 = 0 \). Find \(\left(x_{1}^{5} - 20\right) \cdot \left(3 x_{2}^{4} - 2 x_{2} - 35\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1063}
Given a constant $a \gt 0$, the graph of the function $f(x)=\frac{2^x}{{{2^x}+ax}}$ passes through the points $P(p,\frac{6}{5})$ and $Q(q,-\frac{1}{5})$. If $2^{p+q}=16pq$, then $a=$____
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
pipe a that can fill a tank in an hour and pipe b that can fill the tank in 15 are opened simultaneously when the tank is empty . pipe b is shut 15 minutes before the tank overflows . when will the tank overflow ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
How many subsets with $3$ elements of the set $S = \{1, 2, 3, \ldots, 19, 20\}$ exist, such that their product is divisible by $4$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{795}
Two trains 130 m and another train of unknown length run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time which they take to cross each other is 10.439164866810657 seconds. What is the length of the second train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Suppose a society has the following rule: every couple has exactly two children, with each child independently having a 50% chance of being male or female. If a couple has one male and one female child, what is the probability that the male line will eventually become extinct?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
An event is a hit or a miss. The first event is a hit, the second is a miss. Thereafter, the probability of a hit equals the proportion of hits in the previous trials. For example, the probability of a hit in the third trial is 1/2. What is the probability of exactly 50 hits in the first 100 trials?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{99}}
若在 1,2,3,\cdots,18 的任意一个排列中，总能找到连续 6 个数之和不小于 m，则实数 m 的最大值为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{57}
Given that a positive number is singular if its representation as a product of powers of distinct prime numbers contains no even powers other than $0$, determine the maximum number of consecutive singular numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given that the positive integers \( a, b, c \) satisfy \( 2017 \geqslant 10a \geqslant 100b \geqslant 1000c \), find the number of possible triples \( (a, b, c) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{574}
Determine the pressure force of water on a vertical parabolic segment, whose base is $4 \mathrm{m}$ and is located at the water surface, and its vertex is at a depth of $4 \mathrm{м}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{167424 \, \text{N}}
Find $\sum_{k=0}^{\infty}\left\lfloor\frac{1+\sqrt{\frac{2000000}{4^{k}}}}{2}\right\rfloor$ where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1414}
Find the angle of inclination of the tangent line to the curve $y=\frac{1}{3}x^3-5$ at the point $(1,-\frac{3}{2})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
The poetry lecture lasted 2 hours and $m$ minutes. The positions of the hour and minute hands on the clock at the end of the lecture are exactly swapped from their positions at the beginning of the lecture. If $[x]$ denotes the integer part of the decimal number $x$, find $[m]=$ $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
We have a group of $n$ kids. For each pair of kids, at least one has sent a message to the other one. For each kid $A$, among the kids to whom $A$ has sent a message, exactly $25\%$ have sent a message to $A$. How many possible two-digit values of $n$ are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
In rectangle $LMNO$, points $P$ and $Q$ quadruple $\overline{LN}$, and points $R$ and $S$ quadruple $\overline{MO}$. Point $P$ is at $\frac{1}{4}$ the length of $\overline{LN}$ from $L$, and point $Q$ is at $\frac{1}{4}$ length from $P$. Similarly, $R$ is $\frac{1}{4}$ the length of $\overline{MO}$ from $M$, and $S$ is $\frac{1}{4}$ length from $R$. Given $LM = 4$, and $LO = MO = 3$. Find the area of quadrilateral $PRSQ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.75}
Three rays emanate from a single point and form pairs of angles of $60^{\circ}$. A sphere with a radius of one unit touches all three rays. Calculate the distance from the center of the sphere to the initial point of the rays.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Determine the number of integers \( D \) such that whenever \( a \) and \( b \) are both real numbers with \( -\frac{1}{4} < a, b < \frac{1}{4} \), then \( \left|a^{2} - D b^{2}\right| < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Let $\triangle XYZ$ be a triangle in the plane, and let $W$ be a point outside the plane of $\triangle XYZ$, so that $WXYZ$ is a pyramid whose faces are all triangles.

Suppose that the edges of $WXYZ$ have lengths of either $24$ or $49$, and no face of $WXYZ$ is equilateral. Determine the surface area of the pyramid $WXYZ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48 \sqrt{2257}}
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 $m+n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{194}
In the polar coordinate system, randomly select two curves from the four curves $C\_1$: $ρ=1$, $C\_2$: $θ= \frac {π}{3}(ρ>=0)$, $C\_3$: $ρ=\cos θ$, $C\_4$: $ρ\sin θ=1$. Let the number of intersection points of these two curves be the random variable $ξ$. Find the mathematical expectation $Eξ=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{14}}
Let \(ABCDA'B'C'D'\) be a rectangular prism with \(|AB|=2|BC|\). \(E\) is a point on the edge \([BB']\) satisfying \(|EB'|=6|EB|\). Let \(F\) and \(F'\) be the feet of the perpendiculars from \(E\) at \(\triangle AEC\) and \(\triangle A'EC'\), respectively. If \(m(\widehat{FEF'})=60^{\circ}\), find the value of \(|BC|/|BE|\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\). Please give the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
A circle centers at the origin and has a certain radius. There are 12 points on the circle that have integer coordinates. What is the radius of the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
已知函数$f(x)=\cos 2x+a(\sin x-\cos x)+3x+2019$在$[0,\pi ]$上单调递增，则实数$\textit{a}$值范围为_________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$[-\frac{\sqrt{2}}{2},3]$}
Point $(1,1)$ is inside the circle $(x-a)^{2}+(y+a)^{2}=4$. Then, the range of the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1 < a < 1}
Given the proposition $p$: $\exists x_{0} > 0$, $2^{x_{0}} \geqslant 3$, what is the negation of $p$? Express your answer as a logical statement using proper notation for quantifiers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\forall x > 0, 2^{x} < 3}
Evaluate the definite integral 3 $$\int_{ -1 }^{ 1 }$$(sinx + x<sup>2</sup>)dx = \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In $\triangle ABC$, let $D$, $E$, and $F$ be the midpoints of $BC$, $CA$, and $AB$, respectively. $G$ is the centroid of the triangle. How many non-similar triangles $ABC$ are there for each value of $\angle BAC$ so that $AEGF$ is a cyclic quadrilateral?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In the Cartesian coordinate system, the point (2, t) is located in the bottom-right region of the line x - 2y + 4 = 0, including the boundary. Determine the range of t. Express your answer as an inequality in terms of t.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t\geq3}
Let $a < b < c$ be the solutions of the equation $2016 x^{3} - 4 x + \frac{3}{\sqrt{2016}} = 0$. Determine the value of $-1 / (a b^{2} c)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1354752}
在平面直角坐标系内, $M \in\left\{(x, y) \left\lvert\, \frac{x^{2}}{200}+\frac{y^{2}}{8} \leq 1\right.\right\}, A(2,1)$, 若 $\Delta O M A$ 的面积不超过 $3$, 则满足条件的整点 $M$ 个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Arrange the numbers 1, 2, ..., 7 in a row such that the sum of the first item, the sum of the first two items, the sum of the first three items, and so on, up to the sum of all seven items are not divisible by 3. How many different arrangements are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
On side \( AC \) of \( \triangle ABC \), take a point \( D \) such that \( \angle BDC = \angle ABC \). If \( BC = 1 \), what is the minimum possible distance between the circumcenter of \( \triangle ABC \) and the circumcenter of \( \triangle ABD \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
200. Let $t$ be an arbitrary positive number; the number of irreducible fractions $\frac{p}{q}$, where the numerator $p$ and the denominator $q$ do not exceed $t$, is denoted by $d(t)$,

## What is the sum

$-S=d\left(\frac{100}{1}\right)+d\left(\frac{100}{2}\right)+d\left(\frac{100}{3}\right)+\ldots+d\left(\frac{100}{99}\right)+d\left(\frac{100}{100}\right) ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}
实数 a,b,c 满足 ab+bc+ca=44，求 (a^2+4)(b^2+4)(c^2+4) 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6400}
What is the smallest tower of 100s that exceeds a tower of 100 threes? In other words, let \( a_1 = 3 \), \( a_2 = 3^3 \), and \( a_{n+1} = 3^{a_n} \). Similarly, let \( b_1 = 100 \), \( b_2 = 100^{100} \), and so on. What is the smallest \( n \) for which \( b_n > a_{100} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
A company gathered at a meeting. Let's call a person sociable if, in this company, they have at least 20 acquaintances, with at least two of those acquaintances knowing each other. Let's call a person shy if, in this company, they have at least 20 non-acquaintances, with at least two of those non-acquaintances not knowing each other. It turned out that in the gathered company, there are neither sociable nor shy people. What is the maximum number of people that can be in this company?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
A group of 40 boys and 28 girls stand hand in hand in a circle facing inwards. Exactly 18 of the boys give their right hand to a girl. How many boys give their left hand to a girl?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let $a_1=5$ and let $$a_{n+1}=\frac{a_n^2}{a_n^2-4a_n+6}$$ Find the biggest integer $m$ not bigger than $a_{2018}$, that is $m\leq a_{2018}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A company is lining a 2300-foot path with a fence. Every 8 feet, a fence pole is placed. The entire path is lined with a fence except for three bridges, with lengths of 48 feet, 58 feet, and 62 feet. How many fence poles will be placed along the whole path on both sides, including the poles at each corner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{534}
Determine the number of connected components of the set $T=\{(a,b,c)\in\Bbb{R}^3: a^2b^2c^2=4\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The sides of triangle $PQR$ are in the ratio of $3:4:5$. Segment $QS$ is the angle bisector drawn to the shortest side, dividing it into segments $PS$ and $SR$. What is the length, in inches, of the longer subsegment of side $PR$ if the length of side $PR$ is $15$ inches? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{60}{7}}
Among all proper fractions where both the numerator and the denominator are two-digit numbers, find the smallest fraction that is greater than $\frac{5}{6}$. Provide the numerator of this fraction in your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
Arrange the numbers 1, 2, 3, 4, 5, 6 in a row, and denote the $i$-th number as $a_i$ ($i=1,2,\ldots,6$). If $a_1 \neq 1$, $a_3 \neq 3$, $a_5 \neq 5$, and $a_1 < a_3 < a_5$, then the number of different permutation methods is ______ (numeric answer).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given $y= \sqrt {\sin x}+ \sqrt {\cos x- \frac {1}{2}}$, find its domain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2k\pi, \frac {\pi}{3}+2k\pi]}
Private Petrov took a bucket of unpeeled potatoes and peeled them in 1 hour. During this process, 25% of the potatoes went to peels. How long did it take him to collect half a bucket of peeled potatoes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40 \text{ minutes}}
$2023$  players participated in a tennis tournament, and any two players played exactly one match. There was no draw in any match, and no player won all the other players. If a player  $A$  satisfies the following condition, let  $A$  be "skilled player".**(Condition)** For each player  $B$  who won  $A$ , there is a player  $C$  who won  $B$  and lost to  $A$ . 

It turned out there are exactly  $N(\geq 0)$  skilled player. Find the minimum value of  $N$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the line $l: mx+y-2m-1=0$ and the circle $C: x^{2}+y^{2}-2x-4y=0$, when the chord cut by line $l$ from circle $C$ is the shortest, the value of the real number $m$ is \_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let $P(x)=x^{4}+2 x^{3}-13 x^{2}-14 x+24$ be a polynomial with roots $r_{1}, r_{2}, r_{3}, r_{4}$. Let $Q$ be the quartic polynomial with roots $r_{1}^{2}, r_{2}^{2}, r_{3}^{2}, r_{4}^{2}$, such that the coefficient of the $x^{4}$ term of $Q$ is 1. Simplify the quotient $Q\left(x^{2}\right) / P(x)$, leaving your answer in terms of $x$. (You may assume that $x$ is not equal to any of $\left.r_{1}, r_{2}, r_{3}, r_{4}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$x^{4}-2 x^{3}-13 x^{2}+14 x+24$}
Melanie is preparing her special meatballs for her guests tonight. The secret to their delicious flavor is the homemade breadcrumbs. For these breadcrumbs, Melanie broke down 2 slices of bread into pieces of differing proportions. She first divides the first slice into half, then tore that half into a third. She divided the other half into quarters. For the second slice, she did it a bit differently. Initially, she divided it into thirds and then tore one of the thirds into another half, while she divided the leftover two thirds into fifths. How many pieces of bread will Melanie put into the blender to make her breadcrumbs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}