inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $P_0P_5Q_5Q_0$ be a rectangular chocolate bar, one half dark chocolate and one half white chocolate, as shown in the diagram below. We randomly select $4$ points on the segment $P_0P_5$, and immediately after selecting those points, we label those $4$ selected points $P_1, P_2, P_3, P_4$ from left to right. Similarly, we randomly select $4$ points on the segment $Q_0Q_5$, and immediately after selecting those points, we label those $4$ points $Q_1, Q_2, Q_3, Q_4$ from left to right. The segments $P_1Q_1, P_2Q_2, P_3Q_3, P_4Q_4$ divide the rectangular chocolate bar into $5$ smaller trapezoidal pieces of chocolate. The probability that exactly $3$ pieces of chocolate contain both dark and white chocolate can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[Diagram in the individuals file for this exam on the Chmmc website]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
If the solution set of the inequality $ax^2+bx+a^2-1\leq0$ with respect to $x$ is $[-1,+\infty)$, then the values of the real numbers $a$ and $b$ are respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0, -1}
Consider the set \( \left\{ 1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{64} \right\} \). Define an operation that replaces two elements \( a \) and \( b \) with \( \frac{2ab}{a+b} \). If, after a number of steps, all elements in the set become equal, what is the value of each element?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{65}}
Determine the condition on the value of 'a' such that the characteristic polynomial \(x^3 - 3x + a\) has three distinct real roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 2)}
Find the remainder when \( \sum_{n=1}^{2024^{2024}} f(n) \) is divided by 1000, where \( f(n) \) denotes the number of divisors of \( 2024^{2024} \) that are either less than \( n \) or share at least one prime factor with \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{224}
A net of a cube is shown with one integer on each face. A larger cube is constructed using 27 copies of this cube. What is the minimum possible sum of all of the integers showing on the six faces of the larger cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Tim is participating in the following three math contests.  On each contest his score is the number of correct answers.
     $\bullet$  The Local Area Inspirational Math Exam consists of 15 problems.
     $\bullet$  The Further Away Regional Math League has 10 problems.
     $\bullet$  The Distance-Optimized Math Open has 50 problems.
For every positive integer  $n$ , Tim knows the answer to the  $n$ th problems on each contest (which are pairwise distinct), if they exist; however, these answers have been randomly permuted so that he does not know which answer corresponds to which contest.  Unaware of the shuffling, he competes with his modified answers.  Compute the expected value of the sum of his scores on all three contests.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
For each permutation $(a_1,a_2,\dots,a_{11})$ of the numbers $1,2,3,4,5,6,7,8,9,10,11$, we can determine at least $k$ of $a_i$s when we get $(a_1+a_3, a_2+a_4,a_3+a_5,\dots,a_8+a_{10},a_9+a_{11})$. Determine the maximum possible value of $k$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the interval of convergence for the power series \( \sum_{n=0}^{\infty}\left(\frac{1}{\sqrt{3n}}-\frac{1}{n!}\right)(x+2)^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-3, -1)}
Approximate the probability that a simple symmetric random walk \( \{X_n\} \) with \( p = 1/2 \) satisfies \( X_n \in [-3\sqrt{n}, 2\sqrt{n}] \) for large \( n \). Use the central limit theorem to express this probability in terms of a standard normal distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Phi(2) - \Phi(-3)}
Points  $A$ ,  $B$ ,  $C$ , and  $D$  lie on a circle such that chords  $\overline{AC}$  and  $\overline{BD}$  intersect at a point  $E$  inside the circle.  Suppose that  $\angle ADE =\angle CBE = 75^\circ$ ,  $BE=4$ , and  $DE=8$ .  The value of  $AB^2$  can be written in the form  $a+b\sqrt{c}$  for positive integers  $a$ ,  $b$ , and  $c$  such that  $c$  is not divisible by the square of any prime. Find  $a+b+c$ .

*Proposed by Tony Kim*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{115}
A rectangular plot $ABCD$ is divided into two rectangles as shown in the diagram and is contracted to two households, Jia and Yi. Jia's vegetable greenhouse has the same area as Yi’s chicken farm, while the remaining part of Jia's area is 96 acres more than Yi's. Given that $BF = 3 CF$, what is the total area of the rectangular plot $ABCD$ in acres?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192}
From a $6 \times 6$ squared grid, gray triangles were cut out. What is the area of the remaining figure? The side length of each cell is 1 cm. Provide the answer in square centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$, respectively. They then compute the polynomial $f(x)=x^{3}+p x^{2}+q x+r$ with roots $a, b, c$. If the expected value of the sum of the squares of the coefficients of $f(x)$ is $\frac{m}{n}$ for relatively prime positive integers $m, n$, find the remainder when $m+n$ is divided by 1000 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{551}
Suppose that the series \(\sum a_n x^n\) and \(\sum b_n x^n\) have radii of convergence of 1 and 2, respectively. Determine the radius of convergence of the series \(\sum a_n b_n x^n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the value of \( \mu \) that minimizes the expected cost of fines and cleanup for a bottle filling process. A bottle is considered underfilled if it contains less than 500 ml and overfilled if it contains more than 560 ml. The purchaser fines the supplier \( \$1 \) per underfilled bottle, and it costs the manufacturer \( \$2 \) to clean up an overfilled bottle. Given that the variance \( \sigma^2 = 20^2 \), the expected cost \( E[X] \) is given by:
\[
E[X] = \phi\left(\frac{500-\mu}{20}\right) + 2\left[1-\phi\left(\frac{560-\mu}{20}\right)\right]
\]
Find \( \mu \) such that the derivative of \( E[X] \) with respect to \( \mu \) is zero:
\[
0 = -\frac{1}{20} \phi\left(\frac{500-\mu}{20}\right) + \frac{2}{20}\phi\left(\frac{560-\mu}{20}\right)
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{525.38}
On a large one-way street, there are two traffic lights one after the other; each traffic light is set up so that the time interval when it shows a green light is two-thirds of the total operating time of the traffic light. A motorist noticed that when he drives through the first traffic light during the green light at his usual speed, in 3 out of 4 cases, the second traffic light does not stop him. Suppose now that the motorist passes the first traffic light during the red light. What is the probability that the second traffic light will also be red?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Quadrilateral  $ABCD$  has both an inscribed and a circumscribed circle and sidelengths  $BC = 4, CD = 5, DA = 6$ . Find the area of  $ABCD$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\sqrt{6}}
Determine the cardinality of the set \( E = \{a: \exists m \text{ such that } a_n = s_n \forall n \ge m \} \), where \( s = (s_1, s_2, \ldots) \) is an infinite binary word with \( s_p = 1 \) if \( p \) is a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
Seven students count from 1 to 1000 as follows: Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000. Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers. Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers. Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers. Finally, George says the only number that no one else says. What number does George say?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$365$}
In the triangle \( \triangle ABC \), \( \angle BAC = 50^\circ \), and \( BE = BD \), \( CF = CD \). Find the measure of \( \angle EDF \) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65^\circ}
The center of the circle that touches the side \(BC\) of triangle \(ABC\) at point \(B\) and passes through point \(A\) lies on segment \(AC\). Find the area of triangle \(ABC\), given that \(BC = 6\) and \(AC = 9\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{135}{13}}
Write either  $1$  or  $-1$  in each of the cells of a  $(2n) \times (2n)$ -table, in such a way that there are exactly  $2n^2$  entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be  $M$ . Determine the largest possible value of  $M$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A rectangular piece of paper $A B C D$ is folded and flattened as shown in the diagram, so that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ precisely landing on side $A B$. Given $\angle 1 = 22^\circ$, find the measure of $\angle 2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Let  $a$  and  $b$  be two positive reals such that the following inequality
    \[ ax^3 + by^2 \geq xy - 1 \] is satisfied for any positive reals  $x, y \geq 1$ . Determine the smallest possible value of  $a^2 + b$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3\sqrt{3}}}
\[
\frac{\sin ^{2}\left(135^{\circ}-\alpha\right)-\sin ^{2}\left(210^{\circ}-\alpha\right)-\sin 195^{\circ} \cos \left(165^{\circ}-2 \alpha\right)}{\cos ^{2}\left(225^{\circ}+\alpha\right)-\cos ^{2}\left(210^{\circ}-\alpha\right)+\sin 15^{\circ} \sin \left(75^{\circ}-2 \alpha\right)}=-1
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Calculate the surface area of the region of the plane given by \( z = 1 + x + 2y \) that lies inside the cone defined by \( z = \sqrt{2x^2 + 10y^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\sqrt{3}\pi}
Four coplanar regular polygons share a common vertex but have no interior points in common. Each polygon is adjacent to two of the other polygons, and each pair of adjacent polygons has a common side length of $1$. How many possible perimeters are there for all such configurations? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
in measuring the sides of a rectangle , one side is taken 6 % in excess , and the other 5 % in deficit . find the error percent in the area calculated from these measurements .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.7\%}
Consider the sequence $\{a_n\}$ defined by $a_1 = 1$, $a_2 = \frac{1}{2}$, and for any positive integer $n$, \(a_{n+2} = a_{n+1} - \frac{a_n a_{n+1}}{2}.\) Determine whether the limit $\lim_{n\to\infty} na_n$ exists, and if it does, find its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $R$ be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find 8 positive integers \( n_1, n_2, \ldots, n_8 \) such that we can express every integer \( n \) with \( |n| < 1986 \) as \( a_1 n_1 + \cdots + a_8 n_8 \) with each \( a_i = 0, \pm 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 3, 3^2, 3^3, 3^4, 3^5, 3^6, 3^7}
Flat Albert and his buddy Mike are watching the game on Sunday afternoon. Albert is drinking lemonade from a two-dimensional cup which is an isosceles triangle whose height and base measure 9 cm and 6 cm; the opening of the cup corresponds to the base, which points upwards. Every minute after the game begins, the following takes place: if $n$ minutes have elapsed, Albert stirs his drink vigorously and takes a sip of height $\frac{1}{n^{2}} \mathrm{~cm}$. Shortly afterwards, while Albert is busy watching the game, Mike adds cranberry juice to the cup until it's once again full in an attempt to create Mike's cranberry lemonade. Albert takes sips precisely every minute, and his first sip is exactly one minute after the game begins. After an infinite amount of time, let $A$ denote the amount of cranberry juice that has been poured (in square centimeters). Find the integer nearest $\frac{27}{\pi^{2}} A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial  $f(x)$  with the following properties:

(a) for any integer  $n$ ,  $f(n)$  is an integer;
(b) the degree of  $f(x)$  is less than  $187$ .

Alice knows that  $f(x)$  satisfies (a) and (b), but she does not know  $f(x)$ . In every turn, Alice picks a number  $k$  from the set  $\{1,2,\ldots,187\}$ , and Bob will tell Alice the value of  $f(k)$ . Find the smallest positive integer  $N$  so that Alice always knows for sure the parity of  $f(0)$  within  $N$  turns.

*Proposed by YaWNeeT*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{187}
Juan rolls a fair regular octahedral die marked with the numbers 1 through 8. Amal now rolls a fair twelve-sided die marked with the numbers 1 through 12. What is the probability that the product of the two rolls is a multiple of 4?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{16}}
Find the values of $p$ for which the limit \( \lim_{(x, y) \to (0, 0)} \frac{|x| \cdot |y|^p}{x^2 + y^2} \) exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > 1}
Given $\displaystyle Q_r(\theta) = \frac{r \sin \theta}{1 - 2r\cos \theta + r^2}$, which is the imaginary part of $\displaystyle\frac{1}{1-re^{i\theta}}$, evaluate the integral $\frac{1}{2\pi}\int_{-\pi}^{\pi}Q_r^2(\theta) \, d\theta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{r^2}{2(1-r^2)}}
2018 little ducklings numbered 1 through 2018 are standing in a line, with each holding a slip of paper with a nonnegative number on it; it is given that ducklings 1 and 2018 have the number zero.  At some point, ducklings 2 through 2017 change their number to equal the average of the numbers of the ducklings to their left and right.  Suppose the new numbers on the ducklings sum to 1000.  What is the maximum possible sum of the original numbers on all 2018 slips?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
In the expansion of the binomial ${(\sqrt{x}-\frac{1}{{2x}}})^n$, only the coefficient of the 4th term is the largest. The constant term in the expansion is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{15}{4}"]}
Consider all acute-angled triangles with a given side \( a \) and angle \( \alpha \).

What is the maximum of the sum of the squares of the lengths of sides \( b \) and \( c \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^2}{2 \sin^2(\alpha/2)}}
Find a multiple of 23, which when divided by 1821 and 24 leaves remainders 710 and 13 respectively. The least such multiple is 3024. What is the divisor we are finding the multiple of?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Evaluate the expression \(\frac{29\int_0^1 (1-x^4)^7\,dx}{4\int_0^1 (1-x^4)^6\,dx}\) without expanding the integrals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors. Suppose $M$ can be written as $\prod_{k=1}^{n}p_k^{a_k}$, where the $p_k$'s are distinct primes and the $a_k$'s are positive integers. Find $\sum_{k=1}^{n}(p_k + a_k)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
Compute the limit \( \lim\limits_{s\rightarrow0^{+}}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{s}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
A square has coordinates at $(0, 0)$, $(4, 0)$, $(0, 4)$, and $(4, 4)$. Rohith is interested in circles of radius $ r$ centered at the point $(1, 2)$. There is a range of radii $a < r < b$ where Rohith’s circle intersects the square at exactly $6$ points, where $a$ and $b$ are positive real numbers. Then $b - a$ can be written in the form $m +\sqrt{n}$, where $m$ and $n$ are integers. Compute $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3
}
What is the smallest value of \( n \) such that the symmetric group \( S_n \) contains an even permutation of order 312? Given that there exists an 8-cycle, a 3-cycle, and a 13-cycle in the group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Given that $θ$ is an angle in the second quadrant, and $\sin θ$ and $\cos θ$ are the two roots of the equation $2x^2 + (\sqrt{3} - 1)x + m = 0$ (where $m \in \mathbb{R}$), find the value of $\sin θ - \cos θ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1 + \sqrt{3}}{2}}
Determine the values of $p$ for which the series \( \sum\limits_{n=1}^\infty\left(\dfrac1{n^p}\sum\limits_{k=1}^nk^a\right) \) converges, given that \( a > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > a + 2}
Determine whether the sequence \(\sqrt[n]{n!}\) converges or diverges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find an integer $n$ such that the decimal representation of the number $5^{n}$ contains at least 1968 consecutive zeros.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1968}
Find a polynomial in \(\Bbb{Q}[x]\) that is irreducible over \(\Bbb{Q}\), has at least one linear factor over \(\Bbb{R}\), and at least one irreducible quadratic factor over \(\Bbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^3 - 2}
Determine the values of \( p \in \mathbb{R}_{\geq 0} \) for which the integral \( \int_0^\infty x\sin(x^p) \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, \infty)}
The trapezoid $ABCD$ satisfies $AB \parallel CD$, $AB = 70$, $AD = 32$ and $BC = 49$. We also know that $\angle ABC = 3 \angle ADC$. How long is the base $CD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
133
}
Let \( r=\sqrt{\frac{\sqrt{53}}{2}+\frac{3}{2}} \). There is a unique triple of positive integers \((a, b, c)\) such that 
\[
r^{100}=2 r^{98}+14 r^{96}+11 r^{94}-r^{50}+a r^{46}+b r^{44}+c r^{40}
\]
What is the value of \( a^{2}+b^{2}+c^{2} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15339}
Last year, Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
The number $2017$ is prime. Let $S=\sum\limits_{k=0}^{62}\binom{2014}{k}$. What is the remainder when $S$ is divided by $2017$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024
}
Determine the cardinality of the set $S$ of all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x) \neq 0$ for only finitely many $x \in \mathbb{R}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
There are 100 countries participating in an olympiad. Each country is willing to communicate in exactly \( n \) languages. If each set of 20 countries can communicate in at least one common language, and no language is common to all 100 countries, what is the minimum possible value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Each time you click a toggle switch, the switch either turns from *off* to *on* or from *on* to *off*. Suppose that you start with three toggle switches with one of them *on* and two of them *off*. On each move, you randomly select one of the three switches and click it. Let $m$ and $n$ be relatively prime positive integers such that $\frac{m}{n}$ is the probability that after four such clicks, one switch will be *on* and two of them will be *off*. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{142}
Find the limit of the sequence \((a_n)_{n\geq1}\), where \(a_n=\frac{S_n}{n^\alpha}\) and \(S_n=\sum_{k=1}^{n^2} [\sqrt{k}]\), with \(\alpha\in\mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
On the side \(CD\) of trapezoid \(ABCD\) (where \(AD \parallel BC\)), point \(M\) is marked. A perpendicular \(AH\) is dropped from vertex \(A\) to segment \(BM\). It turns out that \(AD = HD\). Find the length of segment \(AD\) if it is known that \(BC = 16\), \(CM = 8\), and \(MD = 9\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
1. On a $5 \times 7$ grid, 9 cells are marked. We will call a pair of cells with a common side interesting if exactly one cell in the pair is marked. What is the maximum number of interesting pairs that can be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Given the function f(x) = ln(x - 1/2ax^2 - x), if x = 1 is the extreme point of f(x), find the value of a. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
On my sister's birthday, she was 126 cm in height. She had grown a certain percentage since the year before when she was 119.7 cm tall. What was the growth percentage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.26\%}
The numbers \(a, b, c, d\) belong to the interval \([-10.5, 10.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{462}