inclusionAI/Ring-lite-rl-data huggingface dataset:
A rectangular ribbon, 2 cm wide, is cut into 10 smaller ribbons that are also 2 cm wide and then arranged into a pattern as shown in the figure. The lengths of the 4 shortest ribbons in the middle are the same. How long was the original rectangular ribbon, in cm?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80 \text{ cm}}
Three distinct lines are drawn in the plane. Suppose there exist exactly \( n \) circles in the plane tangent to all three lines. Find all possible values of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0, 2, 4}
Each integer in  $\{1, 2, 3, . . . , 2020\}$  is coloured in such a way that, for all positive integers  $a$  and  $b$ such that  $a + b \leq 2020$ , the numbers  $a$ ,  $b$  and  $a + b$  are not coloured with three different colours.
Determine the maximum number of colours that can be used.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Determine all sequences $(x_1,x_2,\cdots,x_{2011})$ of positive integers such that for every positive integer $n$ there is an integer $a$ with $$x_1^n+2x_2^n+\cdots+2011x_{2011}^n=a^{n+1}+1.$$ Please provide the value of $x_1 + x_2 + \cdots + x_{2011}$ for the sequence that satisfies the condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2011}
Five identical rectangles are arranged to form a larger rectangle PQRS, as shown. The area of PQRS is 4000. Calculate the length, x, of each of the identical rectangles. Round your answer to the nearest whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
In the nearly past year of 2015, there are 7 consecutive days where the sum of the dates is 100. What are the dates of these 7 days?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29, 30, 31, 1, 2, 3, 4}
Let  $ABCD$  be an isosceles trapezoid with  $AD\parallel BC$ .  Points  $P$  and  $Q$  are placed on segments  $\overline{CD}$  and  $\overline{DA}$  respectively such that  $AP\perp CD$  and  $BQ\perp DA$ , and point  $X$  is the intersection of these two altitudes.  Suppose that  $BX=3$  and  $XQ=1$ .  Compute the largest possible area of  $ABCD$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
In the diagram, the area of square \( QRST \) is 36. Also, the length of \( PQ \) is one-half of the length of \( QR \). Calculate the perimeter of rectangle \( PRSU \). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find all solutions to $x^{4}+2 x^{3}+2 x^{2}+2 x+1=0$ (including non-real solutions).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1, i, -i}
Among equilateral triangles, parallelograms, rectangles, isosceles trapezoids, and circles, those that are both axisymmetric and centrosymmetric are     .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{rectangles, circles}}
Let $a$ and $b$ be real numbers. Let $r,$ $s,$ and $t$ be the roots of \[f(x) = x^3 + ax^2 + bx - 1,\]and then let $g(x) = x^3 + mx^2 + nx + p$ be a polynomial with roots $r^2,$ $s^2,$ and $t^2.$ If $g(-1) = -5,$ find the greatest possible value for $b.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1+\sqrt5}
Let \( y = f(x) \) be a thrice differentiable function defined on \( \mathbb{R} \) such that \( f(x) = 0 \) has at least 5 distinct zeros. Find the minimum number of zeros of the equation \( f(x) + 6f'(x) + 12f''(x) + 8f'''(x) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Professor Ciconete drew the following points on the board:

Next, he asked his students how many squares with vertices on these points can be drawn. What is the correct answer to Professor Ciconete's question?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
求所有的三元正整数组 $(a, b, c)$ 满足 $\frac{a}{2^{a}}=\frac{b}{2^{b}}+\frac{c}{2^{c}}$. 请提供满足条件的三元组元素和的总和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
As shown in the diagram, through point \( D \) draw \( D O \) bisecting \( \angle A D C \). Through point \( B \) draw \( B O \) bisecting \( \angle A B C \), with \( B O \) and \( D O \) intersecting at point \( O \). Given that \( \angle A = 35^\circ \) and \( \angle O = 42^\circ \), find \( \angle C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49^\circ}
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps.  However, instead of walking up the steps one at a time, both Cozy and Dash jump.  Cozy goes two steps up with each jump (though if necessary, he will just jump the last step).  Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than $5$ steps left).  Suppose Dash takes $19$ fewer jumps than Cozy to reach the top of the staircase.  Let $s$ denote the sum of all possible numbers of steps this staircase can have.  What is the sum of the digits of $s$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
$\triangle ABC$  has side lengths  $AB=20$ ,  $BC=15$ , and  $CA=7$ . Let the altitudes of  $\triangle ABC$  be  $AD$ ,  $BE$ , and  $CF$ . What is the distance between the orthocenter (intersection of the altitudes) of  $\triangle ABC$  and the incenter of  $\triangle DEF$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let \( A \) be a \( 6 \times 6 \) diagonal matrix with characteristic polynomial \( x(x+1)^2(x-1)^3 \). Find the dimension of the set \( S = \{ B_{6\times 6} : AB = BA \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given a parabola $y=ax^{2}+bx+c\left(a\neq 0\right)$ with its vertex at $\left(-3,2\right)$ and opening downwards, determine the type of extremum (maximum or minimum) and its value. Express your answer in the format "The parabola has a [maximum/minimum] value of [value]."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{The parabola has a maximum value of 2.}
Let  $S=\{1,2,3,\cdots,100\}$ . Find the maximum value of integer  $k$ , such that there exist  $k$  different nonempty subsets of  $S$  satisfying the condition: for any two of the  $k$  subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2^{99} - 1 }
Determine the order of the smallest nonabelian simple group that is not isomorphic to $A_5$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
A cube is made by soldering twelve $3$-inch lengths of wire properly at the vertices of the cube. If a fly alights at one of the vertices and then walks along the edges, the greatest distance it could travel before coming to any vertex a second time, without retracing any distance, is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$24\text{ in.}$}
How many coins can be placed on a  $10 \times 10$  board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Consider an  $8\times 8$  grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e. it cannot stay in the same square). What is the expected number of minutes until the rook reaches the upper right corner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
The figure shows a quadrilateral \(ABCD\) in which \(AD = DC\) and \(\angle ADC = \angle ABC = 90^\circ\). The point \(E\) is the foot of the perpendicular from \(D\) to \(AB\). The length \(DE\) is 25. What is the area of quadrilateral \(ABCD\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{625}
Determine the radius of convergence of the series \( \sum_{n=2}^{\infty} \frac{n^{2n}}{4^n(2n+1)!} (3-2x)^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{e^2}}
Determine the region of convergence for the series \( \sum_{n=1}^{\infty} \frac{z^n}{n (\log n)^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|z| \leq 1}
The diagrams below show a $2 \times 2$ grid made up of four $1 \times 1$ squares. Two paths are illustrated on the grid, starting from the lower left corner to the upper right corner, with lengths $4$ and $6$ respectively. A path may not intersect itself by moving to a point where it has already been. Find the sum of the lengths of all possible paths from the lower left corner to the upper right corner of the grid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
In the diagram, points Q and R lie on PS and Angle QWR = 38 degrees. If Angle TQP = Angle TQW = x degrees and Angle VRS = Angle VRW = y degrees, and U is the point of intersection of TQ extended and VR extended, find the measure of Angle QUR. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
For positive integers  $m$  and  $n$ , the decimal representation for the fraction  $\tfrac{m}{n}$  begins  $0.711$  followed by other digits. Find the least possible value for  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
As shown in the figure, an isosceles triangle $ABC$ is folded along $EF$, such that vertex $A$ coincides with the midpoint $D$ of the base. If the perimeter of $\triangle ABC$ is 16 cm, and the perimeter of quadrilateral $BCEF$ is 10 cm, find the length of $BC$ in cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!" 

Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I'd know for certain what it is?"

Cat says, "Nope! However, if I now told you the units digit of my favorite number, you'd know which one it is!"

Claire says, "Now I know your favorite number!" What is Cat's favorite number?

*Proposed by Andrew Wu*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
I have some natural numbers whose only common divisor is 1. If I add 50 to each of them, I obtain two-digit numbers. If I subtract 32 from each of them, I also obtain natural two-digit numbers. What are the numbers that I have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42, 43, 44, 45, 46, 47, 48, 49}
Given an acute triangle  $ABC$  with altituties AD and BE. O circumcinter of  $ABC$ .If o lies on the segment DE then find the value of  $sinAsinBcosC$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
What is the perimeter of the boundary of the region consisting of all points which can be expressed as (2u-3w, v+4w) with $0 \leq u \leq 1$, $0 \leq v \leq 1$, and $0 \leq w \leq 1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$16$}
The sides and vertices of a pentagon are labelled with the numbers  $1$  through  $10$  so that the sum of the numbers on every side is the same. What is the smallest possible value of this sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Two spheres with equal mass and charge are initially at a distance of \( l \) from each other when they are released without an initial velocity. After \( t \) seconds, the distance between them doubles. How much time will it take for the distance between the spheres to double if they are initially released from a distance of \( 3l \)? Air resistance can be neglected.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \sqrt{3} t}
将正四棱锥的顶点染色，要求同一条棱的两个端点不同色．如果只有5种颜色可供使用，那么不同的染色方法共有多少种？（假设经过绕对称轴旋转后可以变相同的染色方法是同一种染色方法）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Determine the splitting field and its degree of extension over $\mathbb{Q}$ for the polynomial $x^4+2x^2-4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Triangle $A B C$ is given with $A B=13, B C=14, C A=15$. Let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Let $G$ be the foot of the altitude from $A$ in triangle $A F E$. Find $A G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{396}{65}}
Find the number of permutations  $( p_1, p_2, p_3 , p_4 , p_5 , p_6)$  of  $1, 2 ,3,4,5,6$  such that for any  $k, 1 \leq k \leq 5$ ,  $(p_1, \ldots, p_k)$  does not form a permutation of  $1 , 2, \ldots, k$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{461}
Find the maximum value of real number  $A$  such that   $$ 3x^2 + y^2 + 1 \geq A(x^2 + xy + x) $$   for all positive integers  $x, y.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{3}}
In the following diagram, $AB = 1$. The radius of the circle with center $C$ can be expressed as $\frac{p}{q}$. Determine $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
What is the smallest number of additional squares that must be shaded in the given figure so that it has at least one line of symmetry and rotational symmetry of order 2? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A 600 meter long train crosses a signal post in 40 seconds. How long will it take to cross a bridge of a certain length at the same speed if it takes 8 minutes to cross the bridge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6600}
7. Let the dihedral angle between a lateral face and the base of a regular quadrilateral pyramid $P-ABCD$ be $\alpha$, and the dihedral angle between two adjacent lateral faces be $\beta$. Then $2 \cos \beta + \cos 2 \alpha =$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Suppose convex hexagon  $ \text{HEXAGN}$  has  $ 120^\circ$ -rotational symmetry about a point  $ P$ —that is, if you rotate it  $ 120^\circ$  about  $ P$ , it doesn't change. If  $ PX\equal{}1$ , find the area of triangle  $ \triangle{GHX}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{4}}
Determine all pairs of natural numbers  $a$  and  $b$  such that  $\frac{a+1}{b}$   and   $\frac{b+1}{a}$  they are natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 1), (1, 2), (2, 3), (2, 1), (3, 2)}
The intersecting squares from left to right have sides of lengths 12, 9, 7, and 3, respectively. By how much is the sum of the black areas greater than the sum of the gray areas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103}
In the first pencil case, there is a lilac pen, a green pencil, and a red eraser; in the second, a blue pen, a green pencil, and a yellow eraser; in the third, a lilac pen, an orange pencil, and a yellow eraser. The contents of these pencil cases are characterized by the following pattern: in each pair of pencil cases, exactly one pair of items matches in both color and function. What should be in the fourth pencil case to maintain this pattern? (Each pencil case contains exactly three items: a pen, a pencil, and an eraser.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{blue pen, orange pencil, red eraser}
We say that a polygon  $P$  is inscribed in another polygon  $Q$  when all the vertices of  $P$  belong to the perimeter of  $Q$ . We also say in this case that  $Q$  is circumscribed to  $P$ . Given a triangle  $T$ , let  $\ell$  be the largest side of a square inscribed in  $T$   and  $L$  is the shortest side of a square circumscribed to  $T$  . Find the smallest possible value of the ratio  $L/\ell$  .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{3\sqrt{3}}{\pi}$}
Given that there are  $24$  primes between  $3$  and  $100$ , inclusive, what is the number of ordered pairs  $(p, a)$  with  $p$  prime,  $3 \le p < 100$ , and  $1 \le a < p$  such that  $p \mid (a^{p-2} - a)$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
What is the minimum number of factors that need to be crossed out from the number \( 99! = 1 \cdot 2 \cdot \ldots \cdot 99 \) so that the product of the remaining factors ends in \( 2 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
## Task A-2.4.

Five sides of a wooden cube are painted blue, while one side remains unpainted. The cube is then cut into smaller, congruent cubes, of which 649 have exactly one blue side. How many of the smaller cubes have exactly two blue sides?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{92}
Given $x \in \mathbb{R}$, $n \in \mathbb{Z}$, and $f(\sin x) = \sin(4n+1)x$, find $f(\cos x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cos(4n + 1)x}
The distance from the curve $C_1: y = \sqrt{x} + a$ to the line $l: x - 2y = 0$ is $\sqrt{5}$. Determine the value of the real number $a$. Express your answer as a single real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Consider a regular $n$-gon $A_1A_2\ldots A_n$ with area $S$. Let us draw the lines $l_1,l_2,\ldots,l_n$ perpendicular to the plane of the $n$-gon at $A_1,A_2,\ldots,A_n$ respectively. Points $B_1,B_2,\ldots,B_n$ are selected on lines $l_1,l_2,\ldots,l_n$ respectively so that:

(i) $B_1,B_2,\ldots,B_n$ are all on the same side of the plane of the $n$-gon;
(ii) Points $B_1,B_2,\ldots,B_n$ lie on a single plane;
(iii) $A_1B_1=h_1,A_2B_2=h_2,\ldots,A_nB_n=h_n$.

Express the volume of polyhedron $A_1A_2\ldots A_nB_1B_2\ldots B_n$ as a function in $S,h_1,\ldots,h_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["S \\cdot \\frac{\\sum_{i=1}^n h_i}{n}"]}
Petya and Vasya are playing a game on a $20 \times 20$ grid. On each turn, a player chooses a cell where all four sides are not colored, and colors all four sides either in red and blue in any order (for example, they can color all sides in one color). No segments longer than one cell side should be created in one color. The player who cannot make a move loses the game. Petya goes first. Which player has a winning strategy, regardless of how the opponent plays?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Vasya}
Given that the ratio of AD to AB is 4:3, AB is 36 inches, determine the ratio of the area of the rectangle to the combined area of the semicircles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{16}{3\pi}}
For how many positive integers $n \le 1000$ is$\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$not divisible by $3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Given $\sqrt{20} \approx 4.472, \sqrt{2} \approx 1.414$, find $-\sqrt{0.2} \approx$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-0.4472}
The area of the largest triangle that can be inscribed in a semi-circle whose radius is $r$ is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$r^{2}$}
Square $ABCD$ is constructed along diameter $AB$ of a semicircle, where both the square and semicircle are coplanar. Line segment $AB$ has a length of 8 centimeters. If point $M$ is the midpoint of arc $AB$, what is the length of segment $MD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{10}}
What is the range of the inclination angle of a straight line? Express your answer as an interval, using appropriate mathematical notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, \pi)}
As shown in the figure, a large rectangle is divided into 9 smaller rectangles. The areas of the three small rectangles located at the corners are 9, 15, and 12, respectively. Find the area of the small rectangle located at the fourth corner.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Eliana spent 90 dollars on fuel this week. During the week, the cost of fuel per gallon increased from 3 dollars to 4 dollars. If Eliana refilled three times at ascending costs by using the same quantity of fuel in each refill, how much fuel was Eliana able to purchase during her first and third refills combined?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
A student did not notice the multiplication sign between two three-digit numbers and wrote a six-digit number, which turned out to be seven times greater than their product. Find these numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{143}
Nancy takes 4 antacids per day when she eats Thai food, 3 antacids per day when she eats Indian food, 2 antacids per day when she eats Mexican food, 1 antacid per day when she eats Italian food, and no antacid on days she eats other cuisines. If Nancy eats Thai twice a week, Indian three times a week, Mexican twice a week, and Italian once a week, how many antacids does she take per month, considering a month has 30 days and she follows her eating schedule consistently?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
The distance from Steve's house to work is 28 km. On the way back, Steve drives at a certain speed ratio compared to the way to work. Altogether, Steve is spending 6 hours a day on the roads, and his speed on the way back from work is 14 km/h. What is the ratio of Steve's speed on the way back to his speed on the way to work?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
Point D is from AC of triangle ABC so that 2AD=DC. Let DE be perpendicular to BC and AE
intersects BD at F. It is known that triangle BEF is equilateral. Find <ADB?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
A square with an integer side was cut into 2020 squares. It is known that the areas of 2019 squares are equal to 1, and the area of the 2020th square is not 1. Find all possible values that the area of the 2020th square can take. In the answer, provide the smallest of the obtained values of the area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}