inclusionAI/Ring-lite-rl-data huggingface dataset:
if the function q is defined by the formula q = 7w / (6mx (z^3)), by what factor will q be multiplied if w is quadrupled, m is doubled, x is reduced to half, and z is raised to the power of 4?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4/z}
Let $\angle XOY = \frac{\pi}{2}$; $P$ is a point inside $\angle XOY$ and we have $OP = 1; \angle XOP = \frac{\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of integers $a$ with $1 \leq a \leq 1007$ such that both $a$ and $a+1$ are quadratic residues modulo $1009$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{251}
How many of the first $1000$ positive integers can be written as the sum of finitely many distinct numbers from the sequence $3^0$, $3^1$, $3^2$, $\ldots$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Compute the largest prime factor of $357! + 358! + 359! + 360!$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{379}
Alice and Carol each have a rectangular sheet of paper. Alice has a sheet of paper measuring 10 inches by 12 inches and rolls it into a tube by taping the two 10-inch sides together. Carol rolls her sheet, which measures 8 inches by 15 inches, by taping the two 15-inch sides together. Calculate $\pi$ times the positive difference of the volumes of the two tubes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Given a sequence {a_n} that satisfies a_n = 3^n - λ * 2^n (where λ is a constant, n ∈ ℕ+), find the range of the real number λ such that the sequence {a_n} is monotonically increasing. Express your answer as an inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{λ < 3}
Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest number possible of black edges is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3$}
Positive integers \( x_{1}, \ldots, x_{m} \) (not necessarily distinct) are written on a blackboard. It is known that each of the numbers \( F_{1}, \ldots, F_{2018} \) can be represented as a sum of one or more of the numbers on the blackboard. What is the smallest possible value of \( m \)?

(Here \( F_{1}, \ldots, F_{2018} \) are the first 2018 Fibonacci numbers: \( F_{1} = F_{2} = 1, F_{k+1} = F_{k} + F_{k-1} \) for \( k > 1 \)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1009}
The diagram shows the ellipse whose equation is \(x^{2}+y^{2}-xy+x-4y=12\). The curve cuts the \(y\)-axis at points \(A\) and \(C\) and cuts the \(x\)-axis at points \(B\) and \(D\). Calculate the area of the inscribed quadrilateral \(ABCD\). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Vasya has 100 bank cards. Vasya knows that one of the cards has 1 ruble, another card has 2 rubles, and so on, up to 100 rubles on the last card, but he doesn't know which card has which amount. Vasya can insert a card into an ATM and request a certain amount. The ATM gives the requested amount if it is available on the card, gives nothing if the amount is not available, and the card is eaten in any case. Furthermore, the ATM does not show how much money was on the card. What is the maximum amount Vasya can guarantee to get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2550 rubles}
Kamal's salary went through the following adjustments: decreased by 50%, increased by 30%, decreased by 20%, decreased by 10%, and finally increased by 25%. What is the overall percentage change in his salary?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41.5\%}
Brian has the same percentage of white stones and black stones in his collection of 100 stones as he does of grey stones and green stones in a separate collection. He has 40 grey stones and a certain number of green stones. If he has more white stones than black ones and has 60 white stones, how many green stones does he have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Given $$x^{ \frac {1}{2}} + x^{- \frac {1}{2}} = 3$$, find the value of $$\frac {x^{ \frac {3}{2}} + x^{- \frac {3}{2}} + 2}{x^{-1} + x + 3}$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that $a_n a_{n-2} - a_{n-1}^2 + a_n - n a_{n-2} = -n^2 + 3n - 1$ and initial conditions $a_0 = 1$, $a_1 = 3$, find $a_{20}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1048596}
Given that \( p \) is a prime number, the decimal part of \( \sqrt{p} \) is \( x \). The decimal part of \( \frac{1}{x} \) is \( \frac{\sqrt{p} - 31}{75} \). Find all prime numbers \( p \) that satisfy these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2011}
The total length of the highway from Lishan Town to the provincial capital is 189 kilometers, passing through the county town. The county town is 54 kilometers away from Lishan Town. In the morning at 8:30, a bus departs from Lishan Town to the county town and arrives at 9:15. After a 15-minute stop, it heads to the provincial capital, arriving by 11:00 AM. Another bus departs from the provincial capital directly to Lishan Town at 9:00 AM on the same day, traveling at 60 kilometers per hour. When the two buses meet, the one traveling from the provincial capital to Lishan Town has been traveling for how many minutes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Let the natural number $N$ be a perfect square, which has at least three digits, its last two digits are not $00$, and after removing these two digits, the remaining number is still a perfect square. Then, the maximum value of $N$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1681}
Lesha's summer cottage has the shape of a nonagon with three pairs of equal and parallel sides. Lesha knows that the area of the triangle with vertices at the midpoints of the remaining sides of the nonagon is 12 sotkas. Help him find the area of the entire summer cottage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
On a rectangular table 5 units long and 2 units wide, a ball is rolled from point P at an angle of 45° to PQ and bounces off SR. The ball continues to bounce off the sides at 45° until it reaches S. Calculate the number of bounces of the ball required to reach point S.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Chongqing hotpot, also known as tripe hotpot or spicy hotpot, is one of the traditional ways of eating in China. It originated on the banks of the Jialing River in Chongqing at the end of the Ming Dynasty and the beginning of the Qing Dynasty. This dish was also a rough dining style for dock workers and boatmen at Chaotianmen and other docks. The main ingredients include beef tripe, pig throat, duck intestines, and beef blood. Today, Chongqing hotpot has become a shining business card for Chongqing. During the May Day holiday, Xiao Mei, Xiao Hong, and Xiao Li traveled to Chongqing. When they returned home, they planned to buy hotpot base materials as souvenirs. A certain store has a total of $a$ different brands of hotpot base materials, including Qiaotou brand hotpot, De Zhuang brand hotpot, Qiu Xia brand hotpot, etc. Each brand has only three flavors: spicy, clear soup, and mushroom soup. The price of the spicy flavor for each brand of hotpot base material is $25$ yuan per pack, the price of the clear soup flavor is $m$ yuan per pack, and the price of the mushroom soup flavor is $n$ yuan per pack. It is known that $33\leqslant m \lt n\leqslant 37$, where $m$ and $n$ are both positive integers. Xiao Mei and Xiao Hong each spent $600$ yuan at checkout. They bought different quantities of mushroom soup flavor hotpot base materials. Xiao Li spent $700$ yuan. They each bought one pack of each flavor for each brand, and for the same brand, they chose different flavors. If all three flavors of hotpot base materials were purchased, what is the maximum amount Xiao Li could have spent on buying clear soup and mushroom soup flavor hotpot base materials?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{675}
22. [12] Find the number of ordered triples $(a, b, c)$ of pairwise distinct integers such that $-31 \leq a, b, c \leq 31$ and $a+b+c>0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{117690}
Five guys each have a positive integer (the integers are not necessarily distinct). The greatest common divisor of any two guys' numbers is always more than 1, but the greatest common divisor of all the numbers is 1. What is the minimum possible value of the product of the numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32400}
Amar takes as much time in running 15 meters as a car takes in covering 40 meters. What will be the distance covered by Amar during the time the car covers a certain distance in kilometers, if Amar covers 712.5 meters when the car covers that distance?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{712.5}
Determine the number of pairs of integers, $(a, b)$, with $1 \leq a \leq 100$ so that the line with equation $b=a x-4 y$ passes through point $(r, 0)$, where $r$ is a real number with $0 \leq r \leq 3$, and passes through point $(s, 4)$, where $s$ is a real number with $2 \leq s \leq 4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6595}
Find the least $\lambda$ for which the sequence $\{b_n\}$ defined by $b_1=1$, $b_2=\lambda-1$ and $b_{n+2}=\lambda(b_{n+1}-b_n)$ is always positive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the notebooks of Peter and Nick, two numbers are written. Initially, these two numbers are 1 and 2 for Peter and 3 and 4 for Nick. Once a minute, Peter writes a quadratic trinomial $f(x)$, the roots of which are the two numbers in his notebook, while Nick writes a quadratic trinomial $g(x)$ the roots of which are the numbers in [i]his[/i] notebook. If the equation $f(x)=g(x)$ has two distinct roots, one of the two boys replaces the numbers in his notebook by those two roots. Otherwise, nothing happens. If Peter once made one of his numbers 5, what did the other one of his numbers become?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{14}{5}}
In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled triangles with $\angle AEB=\angle BEC = \angle CED = 45^\circ$ and $AE=32$. Find the length of $CE.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are $57$, $60$, and $91$. What is the fourth term of this sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$206$}
Elbert and Yaiza each draw $10$ cards from a $20$-card deck with cards numbered $1, 2, 3, \ldots, 20$. Then, starting with the player with the card numbered $1$, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, they lose and the game ends.

Given that Yaiza lost and $5$ cards were placed in total, compute the number of ways the cards could have been initially distributed. (The order of cards in a player’s hand does not matter.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{324}
\( n \) is a positive integer that is not greater than 100 and not less than 10, and \( n \) is a multiple of the sum of its digits. How many such \( n \) are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Write down an irrational number that is greater than $-3$ ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Given the quadratic equation \( ax^2 + bx + c \) and the table of values \( 6300, 6481, 6664, 6851, 7040, 7231, 7424, 7619, 7816 \) for a sequence of equally spaced increasing values of \( x \), determine the function value that does not belong to the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6851}
Inside a truncated cone with a height of 8, there is a sphere $O_{1}$ with a radius of 2. The center of sphere $O_{1}$ lies on the axis of the truncated cone, and sphere $O_{1}$ is tangent to the upper base and the side surface of the truncated cone. Another sphere $O_{2}$ with a radius of 3 can also be placed inside the truncated cone, such that sphere $O_{2}$ is tangent to sphere $O_{1}$, the lower base, and the side surface of the truncated cone, each at exactly one point. Determine the maximum number of additional spheres with a radius of 3 that can be placed inside the truncated cone, excluding sphere $O_{2}$. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Inside a truncated cone with a height of 8, there is a sphere O1 with a radius of 2. The center of sphere O1 lies on the axis of the truncated cone, and sphere O1 is tangent to the upper base and the side surface of the truncated cone. Another sphere O2 with a radius of 3 can also be placed inside the truncated cone, such that sphere O2 is tangent to sphere O1, the lower base, and the side surface of the truncated cone, each at exactly one point. What is the maximum number of additional spheres with a radius of 3 that can be placed inside the truncated cone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A TV show costs $100,000 per episode for the first season and twice that much for every other season.  The first season had 12 episodes, and every season after that had 50% more episodes except the last season, which had 24 episodes.  The show had 5 seasons.  How much did it cost to produce all the episodes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$16,800,000}
Given that for any real number $b$, the line $y=kx+b$ always intersects the hyperbola $x^2-2y^2=1$, find the range of values for the real number $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{\sqrt{2}}{2} < k < \frac{\sqrt{2}}{2}}
Ostap Bender organized an elephant distribution for the residents in the city of Fuks. 28 members of a union and 37 non-union members came for the distribution. Ostap distributed the elephants equally among all union members and also equally among all non-union members.

It turned out that there was only one possible way to distribute the elephants (such that all elephants were distributed). What is the largest number of elephants that Ostap Bender could have? (It is assumed that each attendee received at least one elephant.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1036}
Let \( f(x) = ax^2 + bx + c \), where \( a, b, c \in \mathbf{R} \) and \( a > 100 \). Determine the maximum number of integer values \( x \) can take such that \( |f(x)| \leqslant 50 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A \cup B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Point \( M \) lies on the edge \( AB \) of cube \( ABCD A_1 B_1 C_1 D_1 \). A rectangle \( MNLK \) is inscribed in square \( ABCD \) such that one of its vertices is point \( M \) and the other three vertices are located on different sides of the square base. Rectangle \( M_1 N_1 L_1 K_1 \) is the orthogonal projection of rectangle \( MNLK \) onto the plane \( A_1 B_1 C_1 D_1 \) of the top face. The plane of quadrilateral \( MK_1 L_1 N \) forms an angle \( \alpha \) with the plane of the cube's base, where \( \cos \alpha=\sqrt{\frac{2}{11}} \). Find the ratio \( AM : MB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2}
Find the sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$. The original answer is in the form an(b + c), where a, b, and c are constants. Please give the value of a + b + c.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Marvin had a birthday on Tuesday, May 27 in the leap year 2008. Calculate the next year when his birthday will fall on a Saturday.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
The polynomial of seven variables  $$ Q(x_1,x_2,\ldots,x_7)=(x_1+x_2+\ldots+x_7)^2+2(x_1^2+x_2^2+\ldots+x_7^2) $$  is represented as the sum of seven squares of the polynomials with nonnegative integer coefficients:  $$ Q(x_1,\ldots,x_7)=P_1(x_1,\ldots,x_7)^2+P_2(x_1,\ldots,x_7)^2+\ldots+P_7(x_1,\ldots,x_7)^2. $$  Find all possible values of  $P_1(1,1,\ldots,1)$ .

*(A. Yuran)*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $T:\mathbb R^4\to \mathbb R^4$ be a linear map such that the null space of $T$ is $$\{(x,y,w,z)\in \mathbb R^4 :x+y+w+z=0\}$$ and the rank of $(T-4I_4)$ is 3. If the minimal polynomial of $T$ is $x(x-4)^{\alpha}$, then what is the value of $\alpha$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given a right prism $ABC-A_{1}B_{1}C_{1}$ with height $3$, whose base is an equilateral triangle with side length $1$, find the volume of the conical frustum $B-AB_{1}C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{4}}
For some integer values of n, the value of the expression 4n + 7 is an integer greater than 1 and less than a certain number. There are 30 such values of n. What is the upper bound for the expression to satisfy the condition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
If the line $l: y=x+a$ divides the circumference of the circle $C: x^{2}+y^{2}=1$ into two arc segments with a length ratio of $1:3$, then all possible values of the real number $a$ are ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 1}
The horizontal and vertical distances between adjacent points equal 1 unit.  What is the area of triangle $ABC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Gopi initially gives his servant a salary of Rs. 90 plus one turban for one year. However, after every 3 months, the servant receives a 5% raise in his cash salary based on his initial salary. The servant then leaves after completing a total of 9 months of service and receives Rs. 80 and the turban as his payment. Find the price of the turban, considering the raise in salary after every 3 months.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9.125}
In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D$, $E$, and $F$ lie on sides $BC$, $AC$, and $AB$, respectively, and $CE=CD$ and $BF=BD$, then if $\measuredangle EDF$ equals $x^\circ$, what is the value of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Clever Cat sleeps under the oak tree from noon till midnight and tells stories from midnight till noon. A poster on the tree above him says "Two hours ago, Clever Cat was doing the same thing as he will be doing in one hour's time". Calculate the number of hours a day the poster tells the truth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
The radii of two circles are $3$ and $5$ respectively. When these two circles intersect, the range of values for the distance $d$ between their centers is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 < d < 8}
A train that is 200 meters long takes 21.04615384615385 seconds to pass a bridge of a certain length, if the speed of the train is 65 km/hour. What is the length of the bridge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Paul went to a shop to buy some groceries. He bought some bread for $2. The butter had a 10% discount, so he bought it for 10% less than its original price of $3, and juice for two times the price of the bread. Additionally, there is a 5% sales tax on the total cost of items. He had $15 for his shopping. How much money did Paul have left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$5.86}
Simplify the expression: 

\[
\sqrt[3]{\frac{2 x^{2}}{9+18 x+9 x^{2}}} \cdot \sqrt{\frac{(x+1) \sqrt[3]{1-x}}{x}} \cdot \sqrt[3]{\frac{3 \sqrt{1-x^{2}}}{2 x \sqrt{x}}}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[3]{\frac{1-x}{3 x}}}
Let \( x > 1, y > 1 \) and \( z > 1 \) be positive integers for which the following equation
\[ 1! + 2! + 3! + \ldots + x! = y^2 \]
is satisfied. Find the largest possible value of \( x + y + z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A certain number of ladies can do a piece of work in 12 days. A different number of ladies will do half of that work in 3 days. What is the ratio of the second group of ladies to the first group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
在平面直角坐标系 $x O y$ 中，点集 $K=\{(x, y) \mid x, y=-1,0,1\}$ .在 $K$ 中随机取出三个点，则这三点中存在两点之间距离为 $\sqrt{5}$ 的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
4. Natural numbers $a$, $b$, and $c$, greater than 2022, are such that $a+b$ is divisible by $c-2022$, $a+c$ is divisible by $b-2022$, and $b+c$ is divisible by $a-2022$. What is the greatest value that the number $a+b+c$ can take? (S. Berlov)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2022\cdot85}