inclusionAI/Ring-lite-rl-data huggingface dataset:
Tom's cat needs an expensive surgery. He has had pet insurance for 24 months that cost $20 per month. The procedure cost $5000 but the insurance covers a certain percentage of this. He saved $3520 by having insurance. What percentage of the procedure cost does the insurance not cover?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
Mary works in a restaurant a maximum of 70 hours a week. For the first 40 hours, she is paid $10 per hour. For the next 20 hours (overtime), she is paid at a rate which is 30% higher than her regular rate. For any additional hours worked beyond 60, her pay rate is increased to 60% higher than her regular rate. Furthermore, there are tiered deductions for taxes and insurance based on her earnings:

- For hourly earnings up to $400, she pays a flat 15% in taxes and $50 for insurance.
- For hourly earnings between $401 and $600, she pays an additional 10% tax on that portion.
- For hourly earnings above $600, she pays an additional 25% tax on that portion.

On weekends, Mary gets a flat bonus of $75 for every 8-hour shift she works. If Mary works a combination of regular shifts and overtime during the week, and a single 8-hour shift on weekends, how much will she earn in a week after accounting for deductions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$691.25}
In a pirate's trunk, there are 5 chests. In each chest, there are 4 boxes, and in each box, there are 10 gold coins. The trunk, the chests, and the boxes are all locked. Blind Pew unlocks 9 locks and takes all the coins in all the boxes he unlocks. What is the smallest number of gold coins he could take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
In a zoo, there were 200 parrots. One day, they each made a statement in turn. Starting from the second parrot, all statements were: "Among the previous statements, more than 70% are false." How many false statements did the parrots make in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140}
Given that a hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ has only one common point with the parabola $y=x^{2}+1$, calculate the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
Find all values that the expression

$$
5 \arcsin x - 2 \arccos y
$$

can take under the condition \(x^{2} + y^{2} = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[-\frac{7\pi}{2}, \frac{3\pi}{2}\right]}
In the cube $ABCD-A_1B_1C_1D_1$, among all the edges, face diagonals, and space diagonals, how many line segments are perpendicular to $AB$? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In the tetrahedron \( ABCD \), it is known that \( \angle ADB = \angle BDC = \angle CDA = 60^\circ \), \( AD = BD = 3 \), and \( CD = 2 \). Find the radius of the circumscribed sphere of the tetrahedron \( ABCD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Call a positive integer $N \ge 2$ "special" if for every $k$ such that $2 \leq k \leq N$, $N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). How many special integers are there less than $100$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Let $ABCD$ be a quadrilateral such that $\angle ABC = \angle  CDA = 90^o$, and $BC = 7$. Let $E$ and $F$ be on $BD$ such that $AE$ and $CF$ are perpendicular to BD. Suppose that $BE = 3$. Determine the product of the smallest and largest possible lengths of $DF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In the Cartesian coordinate system \(xOy\), point \(A\) is on the circle \(x^{2} + y^{2} = 1\), and the coordinates of point \(B\) are \((3,0)\). If point \(C\) makes \(\triangle ABC\) an equilateral triangle, then the maximum value of \(|OC|\) is \(\quad\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A regular tetrahedron has a square shadow of area 16 when projected onto a flat surface (light is shone perpendicular onto the plane). Compute the sidelength of the regular tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \sqrt{2}}
a train is 300 meter long is running at a speed of 35 km / hour . in what time will it pass a bridge of 115 meter length ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42.7}
On March $6$, $2002$, during the celebrations of the $500$th anniversary of the birth of mathematician Pedro Nunes, ten people entered the Viva bookstore for science. Each person bought exactly $3$ different books. Furthermore, any two people bought at least one copy of the same book. "The Adventures of Mathematics" by Pedro Nunes was one of the books that achieved the highest number of sales that morning. What is the smallest value this number could have taken?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, and each head always either tells the truth or lies consistently. Moreover, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each dragon said the following:

- 1st head: "To my left is a green dragon."
- 2nd head: "To my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."

What is the maximum number of red dragons that could have been at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{176}
Given a tetrahedron \(K L M N\) with \(K L = 5\), \(N M = 6\), \(\angle L M N = \angle K N M = 35^\circ\), and \(\angle L N M = \angle K M N = 55^\circ\). A sphere is circumscribed around the tetrahedron. Consider the set of all points on this sphere such that the sum of spherical distances from these points to \(K\), \(L\), \(M\), and \(N\) is no more than \(6 \pi\). What is the area of this set? Note that the spherical distance between two points on a sphere is the length of the shortest arc of a great circle connecting these points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18\pi}
Line $l_1$ has equation $4x - 3y = 2$ and passes through point $D = (-2, -3)$. Line $l_2$ has equation $y = 2$ and intersects line $l_1$ at point $E$. Line $l_3$ has a positive slope, passes through point $D$, and meets $l_2$ at point $F$. The area of $\triangle DEF$ is $6$. What is the slope of $l_3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{25}{32}}
Find the number of positive integers $n \le 2014$ such that there exists an integer $x$ satisfying the condition that $\frac{x+n}{x-n}$ is an odd perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{503}
Evin's calculator is broken and can only perform three operations:

1. Given a number $x$, output $2x$.
2. Given a number $x$, output $4x + 1$.
3. Given a number $x$, output $8x + 3$.

Initially starting with the number $0$, determine how many distinct numbers at most $128$ can be generated using these operations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{82}
Let  $n$  be a positive integer with  $k\ge22$  divisors  $1=d_{1}< d_{2}< \cdots < d_{k}=n$ , all different. Determine all  $n$  such that \[{d_{7}}^{2}+{d_{10}}^{2}= \left( \frac{n}{d_{22}}\right)^{2}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^3 * 3 * 5 * 17}
In trapezoid  $ABCD$ ,  $AD \parallel BC$  and  $\angle ABC + \angle CDA = 270^{\circ}$ .  Compute  $AB^2$  given that  $AB \cdot \tan(\angle BCD) = 20$  and  $CD = 13$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{260}
Find the function $g(x)$ if it satisfies the equation $$(e^x-1)g(4x)=\left(e^{4x}-1\right)g(x)$$ and $g'(0) = 2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(e^x - 1)}
Find the smallest number written using only the digits 0 and 1 that is divisible by the product of the six smallest natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1111111110000}
In the Cartesian coordinate plane, define d(P, Q) = |x₁ - x₂| + |y₁ - y₂| as the "manhattan distance" between two points P(x₁, y₁) and Q(x₂, y₂). Find the minimum "manhattan distance" between a point P on the ellipse $\frac{x^2}{2} + y^2 = 1$ and a point Q on the line 3x + 4y - 12 = 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{12-\sqrt{34}}{4}}
Find \( k \) such that, for all \( n \), the following expression is a perfect square:

$$
4 n^{2} + k n + 9
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
How many ways are there to label the faces of a regular octahedron with the integers 18, using each exactly once, so that any two faces that share an edge have numbers that are relatively prime? Physically realizable rotations are considered indistinguishable, but physically unrealizable reflections are considered different.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In the faculty of reverse-engineering, 240 second-year students study numeric methods, 423 second-year students study automatic control of airborne vehicles, and 134 second-year students study them both. There are 663 students in the faculty. What percentage of the total students are second-year students?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79.79\%}
Among the nine digits 1, 2, 3, …, 9, select four digits to form a four-digit number where the first and last digits are odd. How many such four-digit numbers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{840}
One mole of an ideal monoatomic gas is first heated isobarically. In this process, it does 20 J of work. Then, it is heated isothermally, receiving the same amount of heat as in the first case. How much work (in joules) does the gas do in the second case?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50 \text{ J}}
Let \(ABCDEF\) be a regular hexagon with side length \(1\). Denote by \(X\), \(Y\), and \(Z\) the midpoints of sides \(\overline{AB}\), \(\overline{CD}\), and \(\overline{EF}\), respectively. Find the area of the convex hexagon whose interior is the intersection of the interiors of \(\triangle ACE\) and \(\triangle XYZ\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\), where \(\frac{k}{m}\) is in its simplest form. Please give the value of \(k + m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Define the \textit{bigness} of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer $N$ for which there exists a rectangular prism with integer side lengths and \textit{bigness} $N$ and another one with integer side lengths and \textit{bigness} $N + 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
(8) Let $S(n)$ denote the sum of the digits of the positive integer $n$, then $\sum_{n=1}^{2011} S(n)=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["28072"]}
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from $3$ pears and $8$ ounces of orange juice from $2$ oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What is the percentage of the blend that is pear juice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Find the smallest natural number \( n \) such that every convex 100-gon can be represented as the intersection of \( n \) triangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Given an odd prime number \( p \), there exists a positive integer \( k \) such that \( \sqrt{k - p k} \) is also a positive integer. Find the value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(p+1)^2}{4}}
Find the minimum value of k such that every k-element subset of S = {1, 2,..., 32} contains three different elements a, b, c such that a divides b and b divides c. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
在 $\triangle A B C$ 中, 若 $O$ 为形外一点, 满足 $\angle B O C=2 \angle B A C$, 线段 $O C$ 与线段 $A B$ 交于 $D$, 且 $O B=O C=3$, $O D=2$, 求 $B D \cdot A D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the limit $\lim_{x\to0^+}\frac{x^{x^x}- (\sin x)^{x^x}}{x^4}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6}}
A product was introduced into a shopping mall for sale. In the first year, the mall did not charge a management fee, so the product was priced at 70 yuan per piece, with an annual sales volume of 11.8 thousand pieces. Starting from the second year, the mall charged a management fee of x% of the sales amount (i.e., for every 100 yuan of sales, x yuan was charged as a fee). As a result, the price per piece of the product increased by 70 * x% / (1 - x%) yuan compared to the first year, and the expected annual sales volume decreased by x thousand pieces. Find the maximum value of x such that the management fee collected by the mall from this product in the second year is not less than 140,000 yuan.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
有限项等差数列的公差为 $4$ ，第二项起各项之和+首项的平方 $<100$ ，则该数列可能有几项?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let  $x,y,z$  is positive. Solve: $\begin{cases}{x\left( {6 - y} \right) = 9}
{y\left( {6 - z} \right) = 9}
{z\left( {6 - x} \right) = 9}\end{cases}$ 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ x = 3 }
Let $a, b, c$, and $d$ be real numbers. The six sums of two numbers $x$ and $y$, different from the previous four, are $117$, $510$, $411$, and $252$, in no particular order. Determine the maximum possible value of $x + y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{804}
Let the positive integer \( n \) satisfy \( n \leq 2016 \), and \( \left\{\frac{n}{2}\right\}+\left\{\frac{n}{4}\right\}+\left\{\frac{n}{6}\right\}+\left\{\frac{n}{12}\right\}=3 \). The number of such \( n \) is \(\quad\). Here, \(\{x\} = x - \lfloor x \rfloor\), where \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
Petya and Vasya are playing a game. Initially, each of the 2022 boxes contains one matchstick. In one move, all matchsticks from any non-empty box can be transferred to any other non-empty box. They take turns, with Petya starting. The winner is the one who, after their move, has at least half of all the matchsticks in one box for the first time. Who will win with the correct play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Vasya}
After a typhoon, 13 trees in Haley's backyard died. If she had grown 3 trees initially, how many more trees died in the typhoon than those that survived?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Let the sum of the first $n$ terms of a geometric sequence be $S_n$. If the ratio $S_{12} : S_6 = 1 : 2$, then the ratio $S_{18} : S_6 =$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 : 4}
The symbol \( R_{k} \) represents an integer whose decimal representation consists of \( k \) consecutive 1s. For example, \( R_{3} = 111 \), \( R_{5} = 11111 \), and so on. If \( R_{4} \) divides \( R_{24} \), the quotient \( Q = \frac{R_{24}}{R_{4}} \) is an integer, and its decimal representation contains only the digits 1 and 0. How many zeros are in \( Q \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $D$ be a point on the side $[BC]$ of $\triangle ABC$ such that $|AB| + |BD| = |AC|$ and $m(\widehat{BAD}) = m(\widehat{DAC}) = 30^\circ$. If the answer is $x^\circ$, what is the value of $x$ for $m(\widehat{ACB})$? Please provide the value of $x$ as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Given \( f_{1}(x)=|1-2 x| \) for \( x \in [0,1] \) and \( f_{n}(x)=f_{1}(f_{n-1}(x)) \), determine the number of solutions to the equation \( f_{2005}(x)=\frac{1}{2} x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2005}}
A regular triangular pyramid \(SABC\) is given, with the edge of its base equal to 1. Medians of the lateral faces are drawn from the vertices \(A\) and \(B\) of the base \(ABC\), and these medians do not intersect. It is known that the edges of a certain cube lie on the lines containing these medians. Find the length of the lateral edge of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6}}{2}}
Four points \( B, A, E, L \) are on a straight line. \( G \) is a point off the line such that \(\angle B A G = 120^\circ\) and \(\angle G E L = 80^\circ\). Find the reflex angle at \( G \). Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{340}
Let $\Omega_1$ be a circle with center $O$, and let $AB$ be the diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$, different from $O$. Suppose another circle $\Omega_2$ with center $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$, intersecting $\Omega_1$ again at $A_1$ and $B_1$, respectively, such that $A_1$ and $B_1$ are on opposite sides of $AB$. Given that $A_1B = 5$, $AB_1 = 15$, and $OP = 10$, find the radius of $\Omega_1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Calculate $\lim\limits_{n\to\infty} V_{n}$ ($\forall n\in \mathbb{N}$), where $V_{n}=\left(\sum\limits_{k=0}^{2n}\sqrt{k}\right)-(2n+1)\sqrt{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
In a tournament with 12 chess players, each player plays against every other player exactly once. After the tournament, each player creates a series of lists: the first list contains only their own name, the second list includes their name and the names of all opponents they defeated, and each subsequent list includes all names from the previous list plus the names of players defeated by anyone on the previous list. It is known that for every player, the eleventh and twelfth lists have different numbers of names. Determine the number of games that ended in a draw during the tournament.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Suppose the scores of an exam follow a normal distribution, and the correlation coefficient \( \rho = 0.8 \) for exam1 and exam2. If a student's score percentile rank (PR) is 90% in exam1, estimate his/her PR in exam2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
在三棱锥 $P-A B C$ 中, 三条棱 $P A$ 、 $P B 、 P C$ 两两垂直，且 $P A=1, P B=P C=2$. 若 $Q$ 为三棱锥 $P-A B C$ 外接球的球面上任一点, 则 $Q$ 到面 $A B C$ 距离的最大值减去$\frac{\sqrt{6}}6$的结果的2倍为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \(ABC\) be an acute-angled triangle with \(AB \neq BC\) and circumcircle \(k\). Let \(P\) and \(Q\) be the intersections of \(k\) with the angle bisector and the external angle bisector of \(\angle CBA\) respectively. Let \(D\) be the intersection of \(AC\) and \(PQ\). Determine the ratio \(AD:DC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How many ways can you fill a table of size $n \times n$ with integers such that each cell contains the total number of even numbers in its row and column, excluding itself? Two tables are considered different if they differ in at least one cell.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $N = 12!$ and denote by $X$ the set of positive divisors of $N$ other than $1$. A pseudo-ultrafilter $U$ is a nonempty subset of $X$ such that for any $a,b \in X$:
\begin{itemize}
\item If $a$ divides $b$ and $a \in U$ then $b \in U$.
\item If $a,b \in U$ then $\gcd(a,b) \in U$.
\item If $a,b \notin U$ then $\operatorname{lcm}(a,b) \notin U$.
\end{itemize}
How many such pseudo-ultrafilters are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
A sum fetched a total simple interest of Rs. 6016.75 at the rate of 8% per annum in 5 years. What is the sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15041.875}
Molly has a set of cards numbered 1 to 12. She places eight of them at the vertices of an octagon so that the sum of every pair of numbers at opposite ends of an edge of the octagon is a multiple of 3. Identify the numbers that Molly did not place on the vertices of the octagon. Express your answer as a list of numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3, 6, 9, 12}
Distinct points \(A\) and \(B\) are on a semicircle with diameter \(MN\) and center \(C\). The point \(P\) is on \(CN\) and \(\angle CAP = \angle CBP = 10^{\circ}\). If \(\stackrel{\frown}{MA} = 40^{\circ}\), then \(\stackrel{\frown}{BN}\) equals how many degrees? Please provide your answer as an integer representing the number of degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
已知集合 $A, B, C \subseteq\{1,2,3, \cdots, 2020\}$, 且 $A \subseteq B \subseteq C$, 求有序集合组 $(A, B, C)$ 的个数。答案的形式为x^y，请给出x+y的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2024}
## Task A-1.1.

Determine the smallest natural number $a$ such that the value of the expression

$$
\frac{n^{8}+n^{6}+n^{4}+n^{2}+a}{n^{2}-1}
$$

for $n=2014$ is an integer divisible by 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\cdot2014^{2}-6}
How can we determine the number of possible arrangements of the pieces in a $2 \times 2 \times 2$ Rubik's cube, considering both their orientations and relative positions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3,674,160}
From a group of 5 male teachers and 4 female teachers, select 3 teachers to be class leaders for 3 different classes (1 class leader per class). The selection must include both male and female teachers. How many different appointment plans are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{420}
There were some roses and 84 orchids in the vase. Jessica cut some more roses and orchids from her flower garden. There are now 91 orchids and 14 roses in the vase. How many roses were initially in the vase before Jessica cut more?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}