inclusionAI/Ring-lite-rl-data huggingface dataset:
The decreasing interval of the function $f(x) = -x^2 - x + 4$ ($x \in \mathbb{R}$) is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(-\frac{1}{2}, +\infty\right)}
Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6, BC=5=DA,$ and $CD=4.$ Draw circles of radius $3$ centered at $A$ and $B,$ and circles of radius $2$ centered at $C$ and $D.$ A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\frac{-k+m\sqrt{n}}{p},$ where $k, m, n,$ and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k+m+n+p.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{134}
\( \triangle ABC \) has side lengths \( 13 \), \( 14 \), and \( 15 \). Let the feet of the altitudes from \( A \), \( B \), and \( C \) be \( D \), \( E \), and \( F \), respectively. The circumcircle of \( \triangle DEF \) intersects \( AD \), \( BE \), and \( CF \) at \( I \), \( J \), and \( K \) respectively. What is the area of \( \triangle IJK \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0)$, $(0,3)$, $(3,3)$, $(3,1)$, $(5,1)$, and $(5,0)$. Find the slope of the line through the origin that divides the area of this region exactly in half. The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is a simplified fraction. Please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In a betting game, you can win or lose a quantity $x$. The probability of winning a single bet is constant, $p$. You start with a wealth of $x$, which you bet in the first bet. What is the probability of losing all the money, i.e. of ruin, in an infinite number of bets, as a function of $p$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1-p}{p}\)}
Calculate the value of $\frac{\tan 22.5^{\circ}}{1 - \tan^2 22.5^{\circ}}$. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
For $k=1, 2, \ldots, 2011$, we denote $S_k = \frac{1}{k} + \frac{1}{k+1} + \cdots + \frac{1}{2011}$. Compute the sum $S_1 + S_1^2 + S_2^2 + \cdots + S_{2011}^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4022}
Given that $\operatorname{tg} \theta$ and $\operatorname{ctg} \theta$ are the real roots of the equation $2x^{2} - 2kx = 3 - k^{2}$, and $\alpha < \theta < \frac{5 \pi}{4}$, find the value of $\cos \theta - \sin \theta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\sqrt{\frac{5 - 2\sqrt{5}}{5}}}
P.I. Find the largest natural number from which it is impossible to obtain a number divisible by 11 by erasing digits.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{987654321}
The math teacher of the ninth grade teaches an average of 6 tutoring classes per month. If the classes are taught by female teachers, each female teacher should teach 15 classes per month. If the classes are taught only by male teachers, how many classes should each male teacher teach per month? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
5. Point $M$ divides the diagonal $A C$ of square $A B C D$ in the ratio $M C: A M=1: 4$. A line passing through point $M$ divides the square into two parts, the areas of which are in the ratio $1: 11$. In what ratio does this line divide the perimeter of the square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{19}}
It is known that unequal numbers $a$, $b$, and $c$ are successive members of an arithmetic progression. All of them are greater than $1000$ and all are squares of natural numbers. Find the smallest possible value of $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2500}
Walter fell from the eighth platform of some scaffolding and fell past David after falling 4 meters. If he fell for an additional three more times that depth before hitting the ground, and the platforms are evenly spaced out in height, what platform was David on?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $m\in R$, the moving straight line passing through the fixed point $A$ with equation $x+my-2=0$ intersects the moving straight line passing through the fixed point $B$ with equation $mx-y+4=0$ at point $P\left(x,y\right)$. Find the maximum value of $|PA|\cdot |PB|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $m$, $n$, $a$, and $k$ be positive integers with $k > 1$ such that the equation \(5^m + 63n + 49 = a^k\) holds. Find the minimum value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given that $x > 2$, $y > 0$ and they satisfy $2^{x \cdot 2^{y}} = 16$, find the minimum value of $\frac {2}{x-2}$ + $\frac {2}{y}$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a regular triangular pyramid \( S A B C \). Point \( S \) is the apex of the pyramid, \( AB = 1 \), \( AS = 2 \), \( BM \) is the median of triangle \( ABC \), and \( AD \) is the angle bisector of triangle \( SAB \). Find the length of segment \( DM \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{31}}{6}}
Given two sets of real numbers \( A = \{a_{1}, a_{2}, \cdots, a_{100} \} \) and \( B = \{b_{1}, b_{2}, \cdots, b_{50} \} \), find the number of mappings \( f \) from \( A \) to \( B \) that make every element in \( B \) have a preimage, and satisfy the condition:
$$
f(a_{1}) \leq f(a_{2}) \leq \cdots \leq f(a_{100})
$$
Express your answer as a combination (e.g., C(n, k)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{99}{49}}
Circle $B$, which has radius 2008, is tangent to horizontal line $A$ at point $P$. Circle $C_1$ has radius 1 and is tangent both to circle $B$ and to line $A$ at a point to the right of point $P$. Circle $C_2$ has radius larger than 1 and is tangent to line $A$ and both circles $B$ and $C_1$. For $n > 1$, circle $C_n$ is tangent to line $A$ and both circles $B$ and $C_{n-1}$. Find the largest value of $n$ such that this sequence of circles can be constructed through circle $C_n$ where the $n$ circles are all tangent to line $A$ at points to the right of $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
The line $l: x = my + n\ (n > 0)$ passes through point $A(4, 4)$. If the diameter of the circumcircle of the feasible region is 8, then the value of the real number $n$ is _________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the infinite series $$\sum^{\infty}_{k=1}\frac{(-1)^{k-1}}{k}\sum^{\infty}_{r=0}\frac{1}{k2^{r}+1}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For integers $0 \le m,n \le 64$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor \frac{m}{2^k} \right\rfloor$ and $\left\lfloor \frac{n}{2^k} \right\rfloor$ are both odd integers. Consider a $65 \times 65$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 65$) is:
\[ (-1)^{\alpha(i-1, j-1)}. \]
Compute the remainder when $\det M$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{208}
Tom has a list of nine integers: 1, 2, 3, 4, 5, 6, 7, 8, and 9. He creates a second list by adding 2 to some of the integers in the first list and by adding 5 to all of the other integers in the first list. What is the smallest number of different integers he can obtain in the second list?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In triangle $ABC$, let sides $a$, $b$, and $c$ be opposite to angles $A$, $B$, and $C$ respectively. Given that $\sin A = 2\sin B$ and $a + b = \sqrt{3}c$, find the magnitude of angle $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^{\circ}}
In △ABC, the sides opposite to angles A, B, C are a, b, c, respectively, and the altitude on side BC is $\frac{a}{2}$. When $\frac{c}{b} + \frac{b}{c}$ reaches its maximum value, the value of angle A is _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
Let $p$, $q$, $r$, $s$, $t$, and $u$ be positive integers with $p+q+r+s+t+u = 2023$. Let $N$ be the largest of the sum $p+q$, $q+r$, $r+s$, $s+t$ and $t+u$. What is the smallest possible value of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{810}
Find the number of real solutions of the equation
\[
\frac{x}{50} = \sin x.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
$14 \cdot 36$ In the sequence $\left[\frac{1^{2}}{1980}\right],\left[\frac{2^{2}}{1980}\right],\left[\frac{3^{2}}{1980}\right], \cdots,\left[\frac{1980^{2}}{1980}\right]$, how many different numbers are there?
(6th All-Russian Mathematical Olympiad, 1980)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1486}
Let $\omega$ be an incircle of triangle $ABC$. Let $D$ be a point on segment $BC$, which is tangent to $\omega$. Let $X$ be an intersection of $AD$ and $\omega$ against $D$. If $AX : XD : BC = 1 : 3 : 10$, a radius of $\omega$ is $1$, find the length of segment $XD$. Note that $YZ$ expresses the length of segment $YZ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$XD= \dfrac{3 \sqrt{10}}{5}$}
On a computer screen is the single character a. The computer has two keys: c (copy) and \( p \) (paste), which may be pressed in any sequence. Pressing \( p \) increases the number of a's on screen by the number that were there the last time \( c \) was pressed. \( c \) doesn't change the number of a's on screen. Determine the fewest number of keystrokes required to attain at least 2018 a's on screen. (Note: pressing \( p \) before the first press of \( c \) does nothing).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
A dealer offers a cash discount of 15% and still makes a certain profit when he further allows 25 articles to be sold at the cost price of 20 articles to a particular sticky bargainer. His articles were listed 100% above the cost price. What is the profit percentage made by the dealer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36\%}
## Task 2.

In a certain archipelago, there are 2017 islands named $1, 2, \ldots, 2017$. Two agencies, Red Dragon and Blue Eye, are negotiating the schedule of ferry routes between the islands. For each pair of islands, exactly one agency will organize a ferry route, and this route will only operate from the island with the smaller number to the island with the larger number.

A ferry route schedule is good if there do not exist two islands with labels $A < B$ such that it is possible to travel from island $A$ to island $B$ using only ferries from Red Dragon, and also using only ferries from Blue Eye.

Determine the total number of good ferry route schedules.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017!}
Let $ABCDEFGHIJ$ be a regular 10-sided polygon with all its vertices on a circle centered at $O$ with a radius of 5. The diagonals $AD$ and $BE$ intersect at $P$, and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the length of the segment $PQ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given the condition p: $|x|>1$ and condition q: $x<m$, if $\lnot p$ is a necessary but not sufficient condition for $\lnot q$, then the range of the real number $m$ is $\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \in (-\infty, -1]}
a 270 m long train running at the speed of 120 km / hr crosses another train running in opposite direction at the speed of 80 km / hr in 9 sec . what is the length of the other train ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{230}
A bus leaves from city $A$ to city $B$, which is 240 km away from $A$, at a speed of 40 km/h. At the same time, a car leaves from $B$ heading towards $A$ at a speed of $v$ km/h. Half an hour after meeting the bus, the car turns back towards $B$ without reaching city $A$ and continues moving towards $B$ at the same speed. Determine all values of $v$ for which the car arrives in city $B$ before the bus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{v \in (56, 120)}
In a triangle \( \triangle ABC \), \(a\), \(b\), and \(c\) are the sides opposite to angles \(A\), \(B\), and \(C\) respectively, with \(B= \dfrac {2\pi}{3}\). If \(a^{2}+c^{2}=4ac\), then find the value of \( \dfrac {\sin (A+C)}{\sin A\sin C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{10\sqrt{3}}{3}}
For odd positive integers $n$, define $f(n)$ to be the smallest odd integer greater than $n$ that is not relatively prime to $n$. Compute the smallest $n$ such that $f(f(n))$ is not divisible by $3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{161}
A rectangular tiled patio is composed of 30 square tiles. The rectangular patio will be rearranged so that there will be 2 fewer columns of tiles and 4 more rows of tiles. After the change in layout, the patio will still have 30 tiles, and it will still be rectangular. How many rows are in the tile patio before the change in layout?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1}  \] is itself an integer.

[i]Laurentiu Panaitopol, Romania[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(5, 3)}
A car traveling with 9/13 of its actual speed covers 120 km in 2 hours, 19 minutes, and 36 seconds. Find the actual speed of the car.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74.5}
.4 + .02 + .006 =
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{.426}
In Italy, they produce clocks in which the hour hand completes one full rotation per day, and the minute hand completes 24 rotations per day. As usual, the minute hand is longer than the hour hand. Consider all positions of the two hands and the zero marking on the Italian clock that coincide with those on a regular clock. How many such positions exist on the Italian clock within a day? (The zero marking on the Italian clock represents 24 hours, while on a regular clock it represents 12 hours.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the  $2019$ th strictly positive integer  $n$  such that  $\binom{2n}{n}$  is not divisible by  $5$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37805}
The difference between the cube and the square of a number has the form $a b c a b c$ (in the decimal system). What is this number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78}
We consider 2021 lines in the plane, no two of which are parallel and no three of which are concurrent. Let E be the set of their intersection points. We want to assign a color to each point in E such that any two points on the same line, whose connecting segment contains no other point of E, have different colors. What is the minimum number of colors needed to achieve such a coloring?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
$记\triangle ABC的内角A，B，C的对边分别为a,b,c,已知\sin C \cdot \sin(A-B) = \sin B \cdot \sin(C-A).$
$若a=5，cos A=\frac{25}{31}，求\triangle ABC的周长.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$14$}
Let  $S$  be the set of 81 points  $(x, y)$  such that  $x$  and  $y$  are integers from  $-4$  through  $4$ .  Let  $A$ ,  $B$ , and  $C$  be random points chosen independently from  $S$ , with each of the 81 points being equally likely.  (The points  $A$ ,  $B$ , and  $C$  do not have to be different.)  Let  $K$  be the area of the (possibly degenerate) triangle  $ABC$ .  What is the expected value (average value) of  $K^2$  ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{200}{3}}
A fenced, rectangular field measures $24~meters$ by $52~ meters$. An agricultural researcher has $1994$ meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partitioned using all or some of the $1994$ meters of fence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{702}
We consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a sequence $(d_i)_{1\le i\le k}$ with $$1=d_1<d_2<\dots <d_k=n\quad (k\ge 6).$$
Find all positive integers $n$ such that $$n=d_5^2+d_6^2.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that $f(f(1995)) = 95$, $f(xy) = f(x)f(y)$, and $f(x) \le x$ for all $x, y$. Find all possible values of $f(1995)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
Given the function $f(x)=2\sin ( \frac {π}{6}x+ \frac {π}{3}) (2 < x < 10)$, suppose its graph intersects with the $x$-axis at point $A$, and the line $l$ passing through point $A$ intersects with the graph of $f(x)$ at points $B$ and $C$. If $O$ is the coordinate origin, find the value of $( \overrightarrow{OB}+ \overrightarrow{OC})⋅ \overrightarrow{OA}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
In the number \(2 * 0 * 1 * 6 * 0 *\), each of the 5 asterisks needs to be replaced by any of the digits \(0,1,2,3,4,5,6,7,8\) (digits can repeat) such that the resulting 10-digit number is divisible by 18. How many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3645}
Points  $ K$ ,  $ L$ ,  $ M$ , and  $ N$  lie in the plane of the square  $ ABCD$  so that  $ AKB$ ,  $ BLC$ ,  $ CMD$ , and  $ DNA$  are equilateral triangles. If  $ ABCD$  has an area of  $ 16$ , find the area of  $ KLMN$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32 + 16\sqrt{3}}
A circle of radius $R$, passing through the vertices $A, B$ and $C$ of a rectangular trapezoid $A B C D$ ( $\angle A=\angle B=90^{\circ}$ ) intersects the segments $A D$ and $C D$ at points $M$ and $N$, respectively, such that $A M: A D=C N: C D=1: 3$. Find the area of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\sqrt{5}R^{2}}{3}}
Four college students are applying for jobs at three companies, with each student being employed by at most one company. How many ways are there for each company to employ at least one student? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
What three-digit positive integer is one more than a multiple of 3, 4, 5, 6, and 7?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{421}
A certain school held a radio gymnastics competition, with scoring criteria including spirit, neatness, and standard of movements. The total score is calculated in a ratio of $2:3:5$ for the three criteria. It is known that Class 8-1 scored $8$ points, $9$ points, and $10$ points for the three criteria respectively. What is the total score of Class 8-1 in this competition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9.3 \text{ points}}
Find the minimum value of the expression \((\sqrt{2(1+\cos 2x)} - \sqrt{36 - 4\sqrt{5}} \sin x + 2) \cdot (3 + 2\sqrt{10 - \sqrt{5}} \cos y - \cos 2y)\). If the answer is not an integer, round it to the nearest whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-27}
Pedro must choose two irreducible fractions, each with a positive numerator and denominator, such that:

1. The sum of the fractions is equal to $2$.
2. The sum of the numerators of the fractions is equal to $1000$.

In how many ways can Pedro do this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Mitya is going to fold a square sheet of paper $A B C D$. Mitya calls a fold beautiful if the side $A B$ intersects the side $C D$ and the four resulting right triangles are equal.

Before that, Vanya selects a random point $F$ on the sheet. Find the probability that Mitya can make a beautiful fold that passes through point $F$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Alex is working on a science project and has taken pictures of all of the plants in a park near her home. She wants to find out how many of each plant she has pictures of, but she has 1200 pictures, and counting them all will take her a long time. Her teacher suggested she use a program on the computer that can automatically tell her what plant is in each picture and count them for her. However, the processing speed of the program varies depending on the type of plant and has a specific rate per hour.

- For tree pictures, it takes 1.5 minutes to process each picture, with a rate of 35 pictures per hour.
- For flower pictures, it takes 2.5 minutes to process each picture, with a rate of 20 pictures per hour.
- For grass pictures, it takes 1 minute to process each picture, with a rate of 50 pictures per hour.
- For shrub pictures, it takes 3 minutes to process each picture, with a rate of 15 pictures per hour.
- For fern pictures, it takes 2 minutes to process each picture, with a rate of 25 pictures per hour.

There are 300 pictures of trees, 400 pictures of flowers, 250 pictures of grass, 150 pictures of shrubs, and 100 pictures of ferns. Calculate the total processing time for all pictures in hours considering the processing rate per hour for each plant category.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47.57}
Sequences $a_n$ and $b_n$ are defined for all positive integers $n$ such that $a_1 = 5$, $b_1 = 7$,  

$$a_{n+1} = \frac{\sqrt{(a_n+b_n-1)^2+(a_n-b_n+1)^2}}{2},$$

and 

$$b_{n+1} = \frac{\sqrt{(a_n+b_n+1)^2+(a_n-b_n-1)^2}}{2}.$$  

How many integers $n$ from 1 to 1000 satisfy the property that $a_n, b_n$ form the legs of a right triangle with a hypotenuse that has integer length?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Given a quadratic function $f(x)=ax^{2}+bx+c$ ($a,b,c \in \mathbb{R}$), when $x \in [0,1]$, $|f(x)| \leqslant 1$, then the maximum value of $(a+b)c$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {1}{4}}