inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the parabola C: y²=4x, its focus is F, and its directrix is l. Point A is on l, and line segment AF intersects parabola C at point B. If $$\overrightarrow {FA}=3 \overrightarrow {FB}$$, find the length of $$\overrightarrow {AF}$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A function \( f \) is defined on the complex numbers by \( f(z) = (a + bi)z^2 \), where \( a \) and \( b \) are real numbers. The function has the property that for each complex number \( z \), \( f(z) \) is equidistant from both \( z \) and the origin. Given that \( |a+bi| = 5 \), find \( b^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{99}{4}}
In the right triangle \(ABC\) with \(\angle B = 90^\circ\), \(P\) is a point on the angle bisector of \(\angle A\) inside \(\triangle ABC\). Point \(M\) (distinct from \(A\) and \(B\)) lies on the side \(AB\). The lines \(AP\), \(CP\), and \(MP\) intersect sides \(BC\), \(AB\), and \(AC\) at points \(D\), \(E\), and \(N\) respectively. Given that \(\angle MPB = \angle PCN\) and \(\angle NPC = \angle MBP\), find \(\frac{S_{\triangle APC}}{S_{ACDE}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{544}
Given $a^{2}+b^{2}=6ab$, find $\frac{a+b}{a-b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \sqrt{2}}
Suppose \( n \) is a natural number. For any real numbers \( x, y, z \), the inequality \(\left(x^{2} + y^{2} + z^{2}\right) \leqslant n \left(x^{4} + y^{4} + z^{4}\right)\) always holds. Then the minimum value of \( n \) is \(\qquad\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the largest constant  $K\geq 0$  such that   $$ \frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2 $$   holds for all positive real numbers  $a,b,c$  such that  $ab+bc+ca=abc$ .

*Proposed by Orif Ibrogimov (Czech Technical University of Prague).*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{K = 18}
Given the inequality ax^2 + bx + c < 0 with solution set (-∞, α) ∪ (β, +∞) and α < β < 0, find the solution set for the inequality cx^2 - bx + a > 0. Express your answer in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(-\frac{1}{\alpha}, -\frac{1}{\beta}\right)}
19 The sequence of positive integers $\left\{a_{n}\right\}$ satisfies: for any positive integers $m, n$, if $m \mid n, m<n$, then $a_{m} \mid a_{n}$, and $a_{m}<a_{n}$. Find the minimum possible value of $a_{2000}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
Find the sum of the reciprocals of the distances from the centroid of the triangle formed by the intersection of the plane $lx+my+nz=p$ (where $l^2+m^2+n^2=1$) with the coordinate axes to the points where the perpendicular from the centroid to the plane meets the coordinate planes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{p}}
There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that
 ( $1$ ) for each $k,$ the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i + n$ is seat $i$ ); 
 ( $2$ ) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break. Find the number of possible values of $n$ with $1 < n < 1000.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{332}
[Pairing and Grouping; Bijections] $[$ Evenness and Oddness $]$

Given a 29-digit number $X=\overline{a_{1}} \overline{-} \overline{-} \overline{29}\left(0 \leq a_{k} \leq 9, a_{1} \neq 0\right)$. It is known that for every $k$ the digit $a_{k}$ appears in the representation of this number $a_{30-k}$ times (for example, if $a_{10}=7$, then the digit $a_{20}$ appears seven times). Find the sum of the digits of the number $X$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{201}
已知正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的棱长为 1, 以顶点 $A$ 为球心， $\frac{2 \sqrt{3}}{3}$ 为半径作一个球，则球面与正方体的表面相交所得到的曲线的长等于原始答案的形式为 $\frac{m \sqrt{k}}{n} \pi$，请给出 m + n +k的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In triangle \( \triangle ABC \), given that \( 2 \sin A + 3 \cos B = 4 \) and \( 3 \sin B + 2 \cos A = \sqrt{3} \), find the measure of angle \( \angle C \) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
It is known that in a right triangle \(c^{2}=a^{2}+b^{2}\), where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse. Find the acute angles of the triangle if \(\left(\frac{c}{a+b}\right)^{2}=0.5\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Among all fractions of the form \(\frac{m}{n}\), where \(m\) and \(n\) are four-digit numbers with the same sum of digits, the largest one was chosen. Find it and write your answer as an improper fraction (do not round or simplify it!).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9900}{1089}}
Person A and Person B are standing 30 meters apart, facing each other. They play "Rock, Paper, Scissors." The winner moves forward 3 meters, the loser moves back 2 meters, and in the case of a tie, both move forward 1 meter. After 15 rounds, Person A is 17 meters from the starting point, and Person B is 2 meters from the starting point. How many times did Person A win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
If \( a, b, c \) are integers (where \( 0 < c < 90 \)), and \( \sqrt{9 - 8 \sin 50^{\circ}} = a + b \sin c^{\circ} \), find the value of \( \frac{a+b}{c} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
7. Let the set $T=\left\{(a, b, c) \mid a, b, c \in \mathbf{N}^{*}\right.$, and $a, b, c$ can form the side lengths of some triangle $\}$. Then the value of $\sum_{(a, b, c) \in T} \frac{2^{a}}{3^{b} \cdot 5^{c}}$ is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{21}}
Janet played a song on her clarinet in 3 minutes and 20 seconds. Janet’s goal is to play the song at a 25%
faster rate. Find the number of seconds it will take Janet to play the song when she reaches her goal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Given column vectors $v_1, v_2, v_3 \in \mathbb{R}^2$, if for all distinct $i, j \in \{1, 2, 3\}$, the minimum eigenvalue of the matrix $v_iv_i^T + v_jv_j^T$ satisfies $\lambda_{\min}(v_iv_i^T+v_jv_j^T) \leq 1$, what is an upper bound for $\lambda_{\min}(v_1v_1^T+v_2v_2^T+v_3v_3^T)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the following premises:
1. The diagonals of a parallelogram are equal.
2. A square is a parallelogram.

Using syllogistic reasoning, deduce a logical conclusion about the diagonals of a square. Express your answer as a clear and concise statement.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{The diagonals of a square are equal.}
If a three-digit number $m$ satisfies that the digits in each place are all different and not zero, then this three-digit number is called a "mutually exclusive number." Remove the unit digit of the "mutually exclusive number" $m$ to get a two-digit number ${m'}$, and let the difference between ${m'}$ and the unit digit of $m$ be denoted as $F(m)$, and the difference between the tens digit and the unit digit of $m$ be denoted as $G(m)$. It is known that a three-digit positive integer $m=20(5x+1)+2y$ (where $x$ and $y$ are integers, and $1\leqslant x\leqslant 9, 1\leqslant y\leqslant 9$) is a "mutually exclusive number," $\frac{{F(m)}}{{G(m)}}$ is an integer and can be divided by $13$. Then the maximum value of the "mutually exclusive number" $m$ that satisfies the conditions is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{932}
4.94. Solve the Cauchy problem: $y^{\prime \prime}-8 y^{3}=0 ; y(0)=-1 ; y^{\prime}(0)=2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{1+2x}}
Dorothy is 15 years old and wants to go to a museum with her family. Her family consists of her, her younger brother, her parents, and her grandfather. The regular ticket cost is $10. People of a certain age or younger have a discount. Dorothy had $70 before the trip and had $26 left after the trip. What is the discount percentage for people 18 years old or younger?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30\%}
Find the value of the ratio $$\frac{\int_0^1 x^n \sqrt{1-x^2} dx}{\int_0^1 x^{n-2} \sqrt{1-x^2} dx}$$ using appropriate techniques.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n-1}{n+2}}
Given that the last initial of Mr. and Mrs. Alpha's baby's monogram is 'A', determine the number of possible monograms in alphabetical order with no letter repeated.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$. To the nearest integer the area of triangle $ABC$ is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$79$}
Two circles intersect at points \( A \) and \( B \). Chords \( A C \) and \( A D \) are drawn in each of these circles such that each chord in one circle is tangent to the other circle. Find \( A B \) if \( C B = a \) and \( D B = b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{ab}}
In year $N$, the $300^{\text{th}}$ day of the year is a Tuesday. In year $N+1$, the $200^{\text{th}}$ day is also a Tuesday. On what day of the week did the $100$<sup>th</sup> day of year $N-1$ occur?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\text{Thursday}$}
A rhombus has a side length \( a \) and an acute angle of \( 60^\circ \). A circle is inscribed in the rhombus. Determine the area of the quadrilateral whose vertices are the points of tangency of the circle with the sides of the rhombus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3a^2 \sqrt{3}}{16}}
For any positive integer \( n \), let \( D_n \) be the set of all positive divisors of \( n \), and define \( f_i(n) \) as the number of elements in the set
\[ F_i(n) = \{ a \in D_n \mid a \equiv i \pmod{4} \}. \]
Find the smallest positive integer \( m \) such that
\[ 2 f_1(m) - f_2(m) = 2017. \]
(The 14th China Southeastern Mathematical Olympiad)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \times 5^{2016}}
If 5 + 3 + 2 = 151022, 9 + 2 + 4 = 183652, then 7 + 2 + some number = 143547. What is the third number in the third equation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Circle $O$ has chord $A B$. A circle is tangent to $O$ at $T$ and tangent to $A B$ at $X$ such that $A X=2 X B$. What is \frac{A T}{B T}?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Define a sequence of integers $a_1, a_2, \dots, a_k$ where every term $a_i \in \{1,2\}$, and let $S$ denote their sum. Another sequence of integers $t_1, t_2, \ldots, t_k$ is defined by \[t_i=\sqrt{a_i(S-a_i)},\] for all $t_i$. Suppose that \(\sum_{1 \leq i \leq k} t_i=4000.\) Find the value of \(\sum_{1 \leq i \leq k} a^2_i.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{476}
Teams A and B each have 7 members who appear in a predetermined order to compete in a Go game relay. The first player of each team competes first; the loser is eliminated, and the winner advances to compete against the next player of the losing team. This process continues until all the players of one team are eliminated. The other team wins, forming a competition sequence. What is the total number of possible competition sequences? $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3432}
In triangle $ABC$, $AB = 7$, $BC = 24$, and the area of triangle $ABC$ is 84 square units. Given that the length of median $AM$ from $A$ to $BC$ is 12.5, find $AC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Grandma Olga has 6 daughters and 5 sons. If all her daughters each have 10 sons, 9 daughters, and 4 step-children (2 sons and 2 daughters), and each of her sons has 8 daughters, 7 sons, and 3 in-laws (who have an additional 2 children each, 1 son and 1 daughter), how many grandchildren and extended family members does she have in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
Suppose that
\[
\prod_{n=1}^{\infty}\left(\frac{1+i\cot\left(\frac{n\pi}{2n+1}\right)}{1-i\cot\left(\frac{n\pi}{2n+1}\right)}\right)^{\frac{1}{n}} = \left(\frac{p}{q}\right)^{i \pi},
\]
where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Xiaoming is riding a bicycle, while Xiaoming's father is walking. They start from locations $A$ and $B$ respectively, moving towards each other. After meeting, Xiaoming continues for another 18 minutes to reach $B$. It is known that Xiaoming's cycling speed is 4 times that of his father's walking speed, and it takes Xiaoming's father a certain number of minutes to walk from the meeting point to $A$. How long does Xiaoming's father need to walk from the meeting point to $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288 \text{ minutes}}
5 men are equal to as many women as are equal to some boys. All of them earn Rs. 150 only. Men's wages are Rs. 10. How many boys are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
We color an equilateral triangle by coloring each edge with one of $k \geq 1$ colors. Find a formula for the number of orbits under the action of $D_6$, the dihedral group of $6$ elements, on the resulting set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{6} (2k+k^3+3k^2)\)}
$\left.\frac{\text { Auxiliary similar triangles }}{[\quad \text { Law of Cosines }}\right]$

In triangle $ABC$, a point $D$ is taken on side $AC$, such that $AD=3$, $\cos \angle BDC=13/20$, and $\angle B+\angle ADB=180^{\circ}$. Find the perimeter of triangle $ABC$ if $BC=2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
The distance between towns A and B is 999 km. At every kilometer of the road that connects A and B a sign shows the distances to A and B as follows:
$$\boxed{0|999}\quad\boxed{1|998}\quad\boxed{2|997}\quad\ldots\quad\boxed{998|1}\quad\boxed{999|0}$$
How many signs are there, with both distances written with the help of only two distinct digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
已知 \( \sin \alpha+2 \cos \beta=2 \). 则\( y=\sin \left(\alpha+\frac{\pi}{4}\right)+2 \sin \left(\beta+\frac{\pi}{4}\right) \)的值域为 $\left[\sqrt{a}-\frac{\sqrt{b}}{c}, \sqrt{d}+\frac{\sqrt{e}}{f}\right]$.请提供a+b+c+d+e+f的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
In triangle \( KLM \), a bisector \( MN \) is drawn. A circle passes through vertex \( M \), touches side \( KL \) at point \( N \), intersects side \( KM \) at point \( P \), and side \( LM \) at point \( Q \). Segments \( KP \), \( QM \), and \( LQ \) are equal to \( k \), \( m \), and \( q \) respectively. Find \( MN \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{km(m+q)}{q}}}
Point \( M \) is the midpoint of the hypotenuse \( AC \) of right triangle \( ABC \). Points \( P \) and \( Q \) lie on lines \( AB \) and \( BC \), respectively, such that \( AP = PM \) and \( CQ = QM \). Find the measure of angle \( \angle PQM \) given that \( \angle BAC = 17^\circ \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17^\circ}
In triangle \(ABC\), \(\angle ABC=50^\circ\) and \(\angle ACB=70^\circ\). Let \(D\) be the midpoint of side \(BC\). A circle is tangent to \(BC\) at \(B\) and is also tangent to segment \(AD\); this circle intersects \(AB\) again at \(P\). Another circle is tangent to \(BC\) at \(C\) and is also tangent to segment \(AD\); this circle intersects \(AC\) again at \(Q\). Find \(\angle APQ\) (in degrees).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Color the vertices of a quadrilateral pyramid so that the endpoints of each edge are different colors. If there are only 5 colors available, what is the total number of distinct coloring methods?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{420}
Find the smallest natural number that can be expressed as the sum of 2002 natural addends with the same sum of digits and as the sum of 2003 natural addends with the same sum of digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10010}
John needs to get a new seeing-eye dog. The adoption fee cost $150 for an untrained dog, but there is a 10% discount available. It then takes 12 weeks of training which costs $250 a week.  However, a local non-profit organization offers a 15% subsidy for half of the training costs. After the training, she needs certification which costs $3000 but her insurance covers 90% of that. Meanwhile, she also needs to purchase additional supplies for the dog, such as a harness, leash, and food, costing $200. What is her out-of-pocket cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$3410}
Simplify the expression and then evaluate: $(a-2b)(a^2+2ab+4b^2)-a(a-5b)(a+3b)$, where $a=-1$ and $b=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-21}
Find the sum, 'P', in terms of the unknown interest rate 'r', when the difference between compound interest and simple interest on a sum for 2 years is Rs. 65, with interest rates of 10% p.a. for the first year and 'r' % p.a. for the second year, without knowing the actual value of 'r'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65000/r}
A rectangle measuring 19 cm by 65 cm is divided by lines parallel to its sides into small squares with a side length of 1 cm. Into how many parts will this rectangle be divided if a diagonal is also drawn within it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1318}
If the internal angles of $\Delta ABC$ satisfy $\sin A+ \sqrt{2}\sin B=2\sin C$, then the minimum value of $\cos C$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{ \sqrt{6}- \sqrt{2}}{4}}
Given a parallelogram \(A B C D\) with \(\angle B = 111^\circ\) and \(B C = B D\). On the segment \(B C\), there is a point \(H\) such that \(\angle B H D = 90^\circ\). Point \(M\) is the midpoint of side \(A B\). Find the angle \(A M H\). Provide the answer in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
Sang-jun has 10 more books on economics than a quarter of his total books, 5 fewer books on social studies than 3/5 of the rest, and 12 books on science. If Sang-jun has 13 books that are not economics, social studies, or science, how many books does Sang-jun have in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Calculate the smallest root \(x_0\) of the equation

$$
x^{2}-\sqrt{\lg x +100}=0
$$

(with a relative error of no more than \(10^{-390} \%\)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10^{-100}}
Let $a$ be a complex number, and set $\alpha$, $\beta$, and $\gamma$ to be the roots of the polynomial $x^3 - x^2 + ax - 1$. Suppose $$(\alpha^3+1)(\beta^3+1)(\gamma^3+1) = 2018.$$ Compute the product of all possible values of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2009}
There is a circle with a circumference of 1 meter and a point marked on it. Two cockroaches start running in the same direction from the marked point at different speeds. 

- Whenever the faster cockroach catches up with the slower one, the slower one instantly turns around and starts running in the opposite direction with the same speed.
- Whenever they meet face-to-face, the faster one instantly turns around and starts running in the opposite direction with the same speed.

Determine how far from the marked point their 100th meeting could occur.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
How many natural numbers exist such that their square results in a thirty-digit number with the following property: When divided from left to right into three groups of ten digits, the numbers formed by the middle group and the right group are both four times the number formed by the left group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18377}
Calculate the minimum distance of the cyclic code of length 9 generated by the polynomial \( g(x) = x^7 + x^6 + x^4 + x^3 + x + 1 \) over GF(2).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $N$ be the number of ways of distributing $52$ identical balls into $4$ distinguishable boxes such that no box is empty and the difference between the number of balls in any two of the boxes is not a multiple of $6$. If $N = 100a + b$, where $a, b$ are positive integers less than $100$, find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
The sum of the integer parts of all positive real solutions \( x \) that satisfy \( x^{4}-x^{3}-2 \sqrt{5} x^{2}-7 x^{2}+\sqrt{5} x+3 x+7 \sqrt{5}+17=0 \) is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In a right-angled triangle $ABC$ with the right angle at $C$, the angle bisector $BD$ and the altitude $CH$ are drawn. A perpendicular $CK$ is dropped from vertex $C$ to the angle bisector $BD$. Find the angle $HCK$ if $BK: KD = 3:1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{30^\circ}.}
A duty mechanic descended on a downward-moving metro escalator. His entire journey from the upper platform to the lower one lasted $24 \mathrm{s}$. Then he ascended and descended again at the same pace, but this time on a stationary escalator. It is known that the descent took $42 \mathrm{s}$. How many seconds would it take for a person to descend on the downward-moving escalator while standing on a step?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}