inclusionAI/Ring-lite-rl-data huggingface dataset:
$\left[\begin{array}{l}{[\text { Theory of algorithms (other). }} \\ {[\quad \text { Estimation + example }}\end{array}\right]$

There are 2004 boxes on the table, each containing one ball. It is known that some of the balls are white, and their number is even. You are allowed to point to any two boxes and ask if there is at least one white ball in them. What is the minimum number of questions needed to guarantee identifying a box that contains a white ball?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2003}
Find the number of positive integers $n$ between $1$ and $1000$, inclusive, satisfying
\[ \lfloor \sqrt{n - 1} \rfloor + 1 = \left\lfloor \sqrt{n + \sqrt{n} + 1}\right\rfloor\]
where $\lfloor n \rfloor$ denotes the greatest integer not exceeding $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{535}
What is the largest power of 1991 which divides \( 1990m + 1992n \), where \( m = 1991^{1992} \) and \( n = 1991^{1990} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1991^{1990}}
After finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the average of these $36$ numbers. The original answer is in the format k:m, where k and m are integers. Please find the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Xiaoqiang and Xiaohua, two classmates, agreed to meet beside the fountain at Daliang Clock Tower Park in the afternoon. They agreed that whoever arrives first must wait for 10 minutes, and if the other person has not arrived by then, they can leave. If Xiaoqiang arrives between 1:20 and 2:00, and assuming Xiaohua arrives sometime between 1:00 and 2:00, with the arrival time within this hour being equally likely, then the probability that they will meet is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac {5}{16}}
You have infinitely many boxes, and you randomly put 3 balls into them. The boxes are labeled $1,2, \ldots$. Each ball has probability $1 / 2^{n}$ of being put into box $n$. The balls are placed independently of each other. What is the probability that some box will contain at least 2 balls?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 / 7}
设关于$\textit{x}$的不等式$a{{x}^{2}}+8(a+1)x+7a+16\ge 0,(a\in Z)$，只有有限个整数解，且0是其中一个解，则全部不等式的整数解的和为____________
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$-10$}
The formula for converting a Fahrenheit temperature  $F$  to the corresponding Celsius temperature  $C$  is  $C=\frac{5}{9}(F-32)$ .  An integer Fahrenheit temperature is converted to Celsius and rounded to the nearest integer; the resulting integer Celsius temperature is converted back to Fahrenheit and rounded to the nearest integer.  For how many integer Fahrenheit temperatures  $T$  with  $32 \leq T \leq 1000$  does the original temperature equal the final temperature?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{539}
A book library charges fifty cents per day on any book borrowed by its members to read outside the library's premises. At the start of May, Celine borrowed three books from the library to read at home for her upcoming examinations. She returned one book 20 days after borrowing it, and the other two stayed at her house for a certain period. She paid a total of $41 for lending the three books. When did she return the other two books?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Given that $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ are unit vectors with an angle of $\frac {2π}{3}$ between them, and $\overrightarrow {a}$ = 3 $\overrightarrow {e_{1}}$ + 2 $\overrightarrow {e_{2}}$, $\overrightarrow {b}$ = 3 $\overrightarrow {e_{2}}$, find the projection of $\overrightarrow {a}$ onto $\overrightarrow {b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {1}{2}}
A bag contains 8 yellow marbles, 7 red marbles, and 5 black marbles. Without looking in the bag, Igor removes N marbles all at once. What is the maximum possible value of N such that, no matter which choice of N marbles he removes, there are at least 4 marbles of one color and at least 3 marbles of another color left in the bag? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the largest $A$ for which there exists a polynomial
$$
P(x)=A x^{4}+B x^{3}+C x^{2}+D x+E
$$
with real coefficients, which satisfies
$$
0<P(x)<1 \text { for }-1<x<1
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Dana Point beach has four times the number of sharks as Newport Beach, and Huntington Beach has half the number of sharks as Dana Point beach. If Newport Beach has 22 sharks and the total shark population across the three beaches was initially 65, what is the percentage increase or decrease in shark populations across the three beaches in comparison to the initial population?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{136.92\%}
Find the mass percentage of H in Dihydrogen monoxide
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11.19\%}
On the board, there are two-digit numbers. Each number is composite, but any two numbers are coprime. What is the maximum number of such numbers that can be written?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
How many unique pairs of parallel edges does a regular octahedron have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
We start with 5000 forints in our pocket to buy gifts, visiting three stores. In each store, we find a gift that we like and purchase it if we have enough money. The prices in each store are independently 1000, 1500, or 2000 forints, each with a probability of $\frac{1}{3}$. What is the probability that we are able to purchase gifts from all three stores and still have money left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{27}}
Yvon has 4 different notebooks and 5 different pens. Determine the number of different possible combinations of notebooks and pens he could bring.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
When a cone is divided by two planes parallel to its base such that its height is cut into three equal parts, what is the ratio of the volumes of the three parts of the cone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:7:19}
Michel starts with the string $H M M T$. An operation consists of either replacing an occurrence of $H$ with $H M$, replacing an occurrence of $M M$ with $M O M$, or replacing an occurrence of $T$ with $M T$. For example, the two strings that can be reached after one operation are $H M M M T$ and $H M O M T$. Compute the number of distinct strings Michel can obtain after exactly 10 operations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
对任意正实数 $a$ ，记函数 $f(x)=|\lg x|$ 在 $[a,+\infty)$ 上的最小值为 $m_{a}$, 函数 $g(x)=\sin \frac{\pi x}{2}$ 在 $[0, a]$ 上的最大值为 $M_{a}$. 若 $M_{a}-m_{a}=\frac{1}{2}$,  $a$ 的值可能为$\frac{a}{b}$ 或 $\sqrt{c}$，请给出 $a + b + c$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given two sets $A$ and $B$ that satisfy the conditions $A\cap B \neq \emptyset$ and $A\cup B = \{1, 2, 3, 4, 5\}$. When $A \neq B$, the pair $(A, B)$ and $(B, A)$ are considered as two different pairs of sets. The total number of such set pairs $(A, B)$ that meet the conditions is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{211}
In $\triangle ABC, D$ and $E$ are the midpoints of $BC$ and $CA$, respectively. $AD$ and $BE$ intersect at $G$. Given that $GEC$D is cyclic, $AB=41$, and $AC=31$, compute $BC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Let \( S = \{1, 2, \ldots, n\} \). Define \( A \) as an arithmetic sequence with at least two terms and a positive common difference, where all terms are in \( S \). Also, adding any other element of \( S \) to \( A \) does not result in an arithmetic sequence with the same common difference as \( A \). Find the number of such sequences \( A \) (a sequence of only two terms is also considered an arithmetic sequence).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\lfloor \frac{n^2}{4} \right\rfloor}
As shown in Figure 1, in the triangular prism $P-ABC$, $PA \perp$ plane $ABC$, $\angle ABC = 120^\circ$, and $PA = 4$. If the radius of the circumscribed sphere of the triangular prism $P-ABC$ is $2\sqrt{2}$, find the tangent value of the angle between the line $PC$ and the plane $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 \sqrt{3}}{3}}
Four pieces of lumber are placed in parallel positions, perpendicular to line M:

- Piece W is 5 m long
- Piece X is 3 m long and its left end is 3 m from line M
- Piece Y is 5 m long and is 2 m from line M
- Piece Z is 4 m long and is 1.5 m from line M

A single cut, perpendicular to the pieces of lumber, is made along the dotted line L. The total length of lumber on each side of L is the same. What is the length, in metres, of the part of piece W to the left of the cut?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.75}
After counting all her money in her local currency, Michelle decided to travel and exchange her savings into whole $100 US Dollar bills. She went to a currency exchange and noticed that the exchange rate was 1 US Dollar equals 0.85 units of her local currency. Additionally, there was a 3% exchange fee on the total amount that she would need to pay to the currency exchange. After completing the exchange, Michelle acquired 8 pieces of $100 US Dollar bills. How much was the total amount of Michelle's savings in her local currency?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$701.03}
Given points $A(4, y_{1})$, $B(\sqrt{2}, y_{2})$, $C(-2, y_{3})$ are all on the graph of the quadratic function $y=(x-2)^{2}-1$, then the relationship between $y_{1}$, $y_{2}$, $y_{3}$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y_{3} \gt y_{1} \gt y_{2}}
A chord \( AB \) of fixed length slides its ends along a circle of radius \( R \). A point \( C \) on this chord, which is at distances \( a \) and \( b \) from the ends \( A \) and \( B \) of the chord respectively, traces another circle as the chord makes a full rotation. Compute the area of the annulus enclosed between the given circle and the circle traced by point \( C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi ab}
The function $g(x)$ satisfies
\[g(x + g(x)) = 5g(x)\] for all \(x\), and \(g(1) = 5\). Find \(g(26)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125}
Rthea, a distant planet, is home to creatures whose DNA consists of two distinguishable strands of bases with a fixed orientation. Each base is one of the letters \(H, M, N, T\), and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Additionally, the bases \(H\) and \(M\) cannot appear next to each other on the same strand; the same restriction applies to \(N\) and \(T\). Determine the number of possible DNA sequences on Rthea.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28812}
The diameter of a circle with radius $r$ is the base of an equilateral triangle. Find the part of the area of the triangle that lies outside the circle.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{r^{2}(3\\sqrt{3}-\\pi)}{6}"]}
Compute the number of ways to completely fill a $3 \times 15$ rectangle with non-overlapping $1 \times 3$ rectangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{189}
The clock hands show exactly one o'clock. A mosquito and a fly are sitting at the same distance from the center on the hour hand and the minute hand respectively. When the hands overlap, the insects swap places. By how many times is the distance traveled by the mosquito in half a day greater than the distance covered by the fly over the same period?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{83}{73}}
Given the function f(x) = e^x + ax, find the range of values for a such that the function has a local maximum point greater than zero. Express your answer as an inequality using standard mathematical notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a < -1}
Given the equation $\log_{\frac{1}{3}}(a-3^x) = x-2$, find the minimum value of the real number $a$ for the equation to have a solution. Express your answer as a single real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In Kaya's teacher's desk, there are 18 pink highlighters, 14 yellow highlighters, 11 blue highlighters, 8 green highlighters, and 7 purple highlighters. However, 3 pink highlighters, 2 yellow highlighters, and 1 blue highlighter are broken, and Kaya has borrowed 1 green and 2 purple highlighters from her teacher as well. After one week, the teacher plans to increase the total number of remaining highlighters by 25%. How many highlighters will be left after the teacher's increase next week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
A seven-digit number has the following properties: the hundreds digit is twice the ten millions digit, the tens digit is twice the hundred thousands digit, the units digit is twice the ten thousands digit, the thousands digit is 0, and it must be divisible by a five-digit number \( a \). What is \( a \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10002}
Given positive numbers $a$ and $b$ satisfying $2a+b=ab$, find the minimum value of $a+2b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given that \( m \) is a constant, and the function
\[ 
f(x) = \sin^{4} x + m \sin x \cdot \cos x + \cos^{4} x 
\]
has the range \(\left[0, \frac{9}{8}\right]\). Find the value of \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 1}
用 $[x]$ 表示不超过实数 x 的最大整数, 若数列 $\left\{a_{n}\right\}$ 满足: $a_{n}=\left[(2+\sqrt{3})^{2^{n}}\right](n \in$
$\left.\mathbf{N}^{*}\right)$ ，则 $a_{2020}$ 的末尾两位数字是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
In a room, there are 2019 aligned switches, each connected to a light bulb. Initially, all the light bulbs are switched on. Then, 2019 people enter the room one by one and perform the following operation: 

- The first person toggles every switch.
- The second person toggles every second switch.
- The third person toggles every third switch.
- This continues until the 2019th person toggles every 2019th switch.

Determine how many light bulbs remain switched on after all 2019 people have entered the room and completed their operations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1975}
Let $S$ be the set of ordered pairs $(x, y)$ such that $0 < x \le 1, 0<y\le 1,$ and $\left[\log_2{\left(\frac 1x\right)}\right]$ and $\left[\log_5{\left(\frac 1y\right)}\right]$ are both even. Given that the area of the graph of $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ The notation $[z]$ denotes the greatest integer that is less than or equal to $z.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$14$}
In an acute triangle \( ABC \), \(\angle A = 30^\circ\). Taking \( BC \) as the diameter, a circle is drawn which intersects \( AB \) and \( AC \) at points \( D \) and \( E \) respectively. Connecting \( DE \) divides triangle \( ABC \) into triangle \( ADE \) and quadrilateral \( BDEC \). Suppose their areas are \( S_1 \) and \( S_2 \) respectively. Find the ratio \( S_1 : S_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
35. In triangle $ABC$, points $M, N$, and $P$ are taken: $M$ and $N$ are on sides $AC$ and $BC$, and $P$ is on segment $MN$, such that

$$
\frac{|AM|}{|MC|}=\frac{|CN|}{|NB|}=\frac{|MP|}{|PN|}
$$

Find the area of triangle $ABC$, if the areas of triangles $AMP$ and $BNP$ are $T$ and $Q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(T^{1/3}+Q^{1/3})^{3}}
Problem 9.5. On an island, there live knights who always tell the truth, and liars who always lie.

One day, 80 residents of the island gathered, each wearing a T-shirt with a number from 1 to 80 (different residents had different numbers). Each of them said one of two phrases:

- "Among those gathered, at least 5 liars have a T-shirt number greater than mine."
- "Among those gathered, at least 5 liars have a T-shirt number less than mine."

What is the smallest number of knights that could have been among these 80 residents?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Given differentiable functions $f(x)$ and $g(x)$ on the interval $[0,2]$, where $f''(x) - g''(x) = 0$, $f'(A) = 2$, $g'(A) = 4$, and $f(B) = 3$, $g(B) = 9$, find the value of $f\left(\frac{3}{2}\right) - g\left(\frac{3}{2}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
A function \( f \) satisfies the following conditions for all nonnegative integers \( x \) and \( y \):

- \( f(0, x) = f(x, 0) = x \)
- If \( x \geq y \geq 0 \), \( f(x, y) = f(x - y, y) + 1 \)
- If \( y \geq x \geq 0 \), \( f(x, y) = f(x, y - x) + 1 \)

Find the maximum value of \( f \) over \( 0 \leq x, y \leq 100 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
Consider a sequence {a_n} satisfying: a_1 = 2, a_{n+1} = 1 - 1/a_n. Let T denote the product of the first n terms of the sequence {a_n}. Calculate T_{2017}.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
设函数$f ( x ) = 4 \sin x \cdot \sin ^ { 2 } ( \frac { \pi } { 4 } + \frac { x } { 2 } ) + \cos 2 x .$.若|f(x)-m|&lt;2成立的充分条件是$\frac { \pi } { 6 } \leq x \leq \frac { 2 \pi } { 3 }$，则实数m的取值范围是 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,4).}
Let $f$ be the unique function defined on the positive integers such that \[\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1\] for all positive integers $n$. What is the value of $f(2023)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Let $A=\{V, W, X, Y, Z, v, w, x, y, z\}$. Find the number of subsets of the 2-configuration \( \{\{V, W\}, \{W, X\}, \{X, Y\}, \{Y, Z\}, \{Z, V\}, \{v, x\}, \{v, y\}, \{w, y\}, \{w, z\}, \{x, z\}, \{V, v\}, \{W, w\}, \{X, x\}, \{Y, y\}, \{Z, z\}\} \) that are consistent of order 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the sum of the coefficients of all even degree terms in x in the expansion of $(x + \sqrt{x^3 - 1})^6 +( x - \sqrt{x^3 -1})^6 $ , ($x>1$)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A train 120 m long crosses a platform with a certain speed in 15 sec. The length of the platform is 130.02 m. What is the speed of the train in kmph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60.0048}
Given the second-order linear differential equation $$y''+2y'+2y=0$$ with initial conditions $y(0)=0$ and $y'(0)=5$, find the maximum value of the function $y(x)$ over the interval $0 \leq x < \infty$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5\sqrt{2}}{2}e^{-\dfrac{\pi}{4}}}
Compute the following expression:
\[ 2(1+2(1+2(1+2(1+2(1+2(1+2(1+2))))))) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{510}
For any positive integer $n$, let $r_n$ denote the greatest odd divisor of $n$. Compute the sum:
\[ T = r_{100} + r_{101} + r_{102} + \cdots + r_{200} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10025}
Count the matches. A friend writes that he bought a small box of short matches, each one inch long. He found that he could arrange them in a triangle whose area in square inches was equal to the number of matches. Then he used 6 matches, and it turned out that from the remaining matches, he could form a new triangle with an area that contained as many square inches as there were matches left. After using another 6 matches, he was able to do the same thing again.

How many matches did he originally have in the box? This number is less than 40.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
A sports team's members have unique numbers taken from the integers 1 to 100. If no member's number is the sum of the numbers of any two other members, nor is it twice the number of any other member, what is the maximum number of people this team can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
$已知f(x)=|x-a|x+|x-2|(x-a).$
$当a=1时，求不等式f(x)<0的解集；$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(-\infty ,1)$}
Vasya invented a new chess piece called the "super-bishop." One "super-bishop" (denote it as $A$) attacks another (denote it as $B$) if they are on the same diagonal, there are no pieces between them, and the next cell on the diagonal beyond "super-bishop" $B$ is empty. For example, in the figure, piece $a$ attacks piece $b$, but does not attack any of the pieces $c, d, e, f,$ or $g$.

What is the maximum number of "super-bishops" that can be placed on a chessboard so that each one is attacking at least one other super-bishop?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
The shortest path from Atown to Cetown runs through Betown. Two signposts on this path are shown, but one of them is broken and a number is missing. Given that the distance from Atown to Betown is 4 km and the distance from Atown to the second signpost is 6 km, calculate the distance that was written on the broken sign. Express your answer in kilometers (km) as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Triangles $\triangle ABC$ and $\triangle PQR$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,10)$, $C(20,0)$, $P(30,0)$, $Q(30,20)$, $R(50,0)$. A rotation of $n$ degrees counterclockwise around the point $(x,y)$ where $0<n<180$, will transform $\triangle ABC$ to $\triangle PQR$. Find $n+x+y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Find the minimum difference between two positive integer factors of 1998. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Given that out of 12 bottles of beverages, 2 have expired, calculate the probability of selecting exactly 1 expired bottle from these 12 bottles. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6}}