inclusionAI/Ring-lite-rl-data huggingface dataset:
Let \(p\) be a prime number and \(\mathbb{F}_{p}\) be the field of residues modulo \(p\). Let \(W\) be the smallest set of polynomials with coefficients in \(\mathbb{F}_{p}\) such that the polynomials \(x+1\) and \(x^{p-2}+x^{p-3}+\cdots+x^{2}+2x+1\) are in \(W\), and for any polynomials \(h_{1}(x)\) and \(h_{2}(x)\) in \(W\) the polynomial \(r(x)\), which is the remainder of \(h_{1}\left(h_{2}(x)\right)\) modulo \(x^{p}-x\), is also in \(W\). How many polynomials are there in \(W\) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p!}
Let $x_{1}, x_{2}, \cdots, x_{1993}$ satisfy $\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+\left|x_{1992}-x_{1993}\right|=1993$ and let $y_{k}=\frac{x_{1}+x_{2}+\cdots+x_{k}}{k}$ for $k=1,2, \cdots, 1993$. What is the maximum possible value of $\left|y_{1}-y_{2}\right|+\left|y_{2}-y_{3}\right|+\cdots+\left|y_{1992}-y_{1993}\right|$?
(1993 Macau Math Olympiad problem)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1992}
If exactly three of the balls match the numbers of their boxes, calculate the number of different ways to place the balls.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
If the solution set of the inequality $-1 < ax^2 + bx + c < 1$ is $(-1, 3)$, then the range of the real number $a$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2} < a < \frac{1}{2}}
A class has 60 students. After an exam, the math scores $\xi$ are normally distributed with mean 110 and variance 102. If $P(100 \leq \xi \leq 110) = 0.35$, then estimate the number of students scoring above 120 in math. Provide your answer as a whole number, noting that an approximate answer is acceptable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the number of real solutions of the equation
\[\frac{x}{100} = \sin x.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Let \( n > m \geq 1 \) be natural numbers such that the groups of the last three digits in the decimal representation of \( 1978^m \) and \( 1978^n \) coincide. Find the ordered pair \( (m, n) \) of such \( m \) and \( n \) for which \( m+n \) is minimal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3, 103)}
A large collection of congruent right triangles is given, each with side lengths $3$, $4$, and $5$. Find the maximal number of such triangles you can place inside a $20 \times 20$ square, with no two triangles intersecting in their interiors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
What is the probability, expressed as a decimal, of drawing one marble which is either green or white from a bag containing 4 green, 3 white, and 8 black marbles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.4667}
From a solid cylinder, whose height is $$ 8\mathrm{cm} $$ and radius is $$ 6\mathrm{cm}, $$ a conical cavity of height $$ 8\mathrm{cm} $$ and of base radius 6cm is hollowed out. Find the total surface area of the remaining solid. [Use $$ \pi=3.14 $$]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{602.88\mathrm{cm}^2}
What is the greatest number of positive integers less than or equal to 2016 that can be chosen such that no two of them differ by 1, 2, or 6?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{576}
What is $x-y$ if a town has 2017 houses, 1820 have a dog, 1651 have a cat, 1182 have a turtle, $x$ is the largest possible number of houses that have a dog, a cat, and a turtle, and $y$ is the smallest possible number of houses that have a dog, a cat, and a turtle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{563}
在一个圆周上给定 $8$ 个点 $A_{1}, A_{2}, \cdots$, $A_{8}$. 求最小的正整数 $n$, 使得以这 $8$ 个点为顶点的任意 $n$ 个三角形中，必存在两个有公共边的三角形。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In a plane, there are 5 points, no three of which are collinear. Connecting these points with line segments, there are at least $\qquad$ intersection points other than the 5 points themselves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Samara and six of her friends heard alligators spotted on the local river and decided to join a search organized by the wildlife service to capture the animals. They split into three groups: Samara, two friends in Group 2, and three friends in Group 3. After searching the whole day, Samara had seen 30 alligators, while her friends in Group 2 had seen a total of 54 alligators with one friend spotting 8 more alligators than the other. The friends in Group 3 had spotted an average of 12 alligators each when in the afternoon, but due to changing conditions, they increased their spotting rate by 25% for the rest of the day, resulting in an additional number of sightings for each friend. Given that each friend in Group 3 ended up spotting the same total number of alligators, calculate the total number of alligators Samara and her friends saw.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{129}
IMO 2000 Problem B1 100 cards are numbered 1 to 100 (each card different) and placed in 3 boxes (at least one card in each box). How many ways can this be done so that if two boxes are selected and a card is taken from each, then the knowledge of their sum alone is always sufficient to identify the third box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
## Task A-3.5.

In a scalene triangle $ABC$, the lengths of two altitudes are equal to the lengths of two medians. What is the ratio of the lengths of the remaining altitude and the remaining median?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{7}}
设f:R→R满足：对任意实数x、y，有f(2x)+f(2y)=f(x+y)f(x-y)，又f(π)=0，且f(x)不恒等于零，则f(x)的周期是 ·
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4π}
Patty is picking peppermints off a tree. They come in two colors: red and white. She picks fewer than $100$ total peppermints but at least one of each color. The white flavor is stronger, so she prefers red to white. Thus, she always picks fewer white peppermints than ten times the number of reds. How many different combinations of peppermints can she go home with?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4446}
Find the sum of all fractions of the form $\frac{1}{xy}$ where $\operatorname{gcd}(x, y) = 1$, $x \leq n$, $y \leq n$, and $x + y > n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Shapovalov A.V.

Thieves Hapok and Glazok are dividing a pile of 100 coins. Hapok grabs a handful of coins from the pile, and Glazok, looking at the handful, decides who of the two will get it. This continues until one of them receives nine handfuls, after which the other takes all the remaining coins (the division may also end if the coins are divided before anyone gets nine handfuls). Hapok can grab any number of coins in a handful. What is the maximum number of coins he can guarantee himself regardless of Glazok's actions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Suppose that $P$ is a polynomial with integer coefficients such that $P(1) = 2$, $P(2) = 3$, and $P(3) = 2016$. If $N$ is the smallest possible positive value of $P(2016)$, find the remainder when $N$ is divided by $2016$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2013}
Usain runs one lap around the school stadium at a constant speed, and photographers Arina and Marina are positioned near the track. After the start, for 4 seconds, Usain was closer to Arina, then for 21 seconds he was closer to Marina, and then until the finish, he was again closer to Arina. How long does it take for Usain to run a whole lap?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42 \, \text{seconds}}
Given the equation $ax^{2}+x+3a+1=0$ with respect to $x$, if there is a root within the interval $(0,3]$, find the range of values for the real number $a$. Express your answer as an interval in standard notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[- \frac{1}{2},- \frac{1}{3}\right]}
Calculate the total surface area of two hemispheres of radius 8 cm each, joined at their bases to form a complete sphere. Assume that one hemisphere is made of a reflective material that doubles the effective surface area for purposes of calculation. Express your answer in terms of $\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{384\pi}
A car traveled a certain percentage of the way from town X to town Y at an average speed of 35 mph. The car traveled at an average speed of 65 mph for the remaining part of the trip. The average speed for the entire trip was 50 mph. What percentage of the trip was traveled at 35 mph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35\%}
I2.2 The surface area of a cube is $b \mathrm{~cm}^{2}$. If the length of each side is increased by $3 \mathrm{~cm}$, its volume is increased by $(2 b-a) \mathrm{cm}^{3}$, find the value of $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}
10. Teacher Yangyang's city has 4 subway lines, each of which intersects with every other line at exactly one point, serving as the transfer station for those two lines. One day, Teacher Yangyang had a sudden idea to start from the subway station near her home (which is not a transfer station), make at least one transfer at each transfer station, and finally return home. How many transfers does Teacher Yangyang need to make to achieve her goal? $\qquad$ times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
How many positive integers $n < 10^6$ are there such that $n$ is equal to twice the square of an integer and is equal to three times the cube of an integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
What is the highest power of 3 that divides the number whose decimal representation consists of $3^n$ ones?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^n}
Jerry is bored one day, so he makes an array of Cocoa pebbles. He makes 8 equal rows with the pebbles remaining in a box. When Kramer drops by and eats one, Jerry yells at him until Kramer realizes he can make 9 equal rows with the remaining pebbles. After Kramer eats another, he finds he can make 10 equal rows with the remaining pebbles. Find the smallest number of pebbles that were in the box in the beginning.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{352}
Given that the equation $\frac{x-3}{x-2}=\frac{m}{2-x}$ has no solution, calculate the exact value of $m$ that makes the equation undefined. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $s(n)$ denote the sum of digits of a positive integer $n$ in base $10$. If $s(m) = 20$ and $s(33m) = 120$, what is the value of $s(3m)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
The lateral face of a regular truncated triangular pyramid forms an acute angle $\alpha$ with the plane of the base. Find the angle between the height and the lateral edge of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arctan(2 \cot(\alpha))}
The hypotenuse of a right triangle measures $9$ inches, and one angle is $30^{\circ}$. What is the number of square inches in the area of the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10.125\sqrt{3}}
To pull a block up an inclined plane, a force three times greater than that needed to drag the block downwards is required, and the sum of these forces is equal to the weight of the block. Determine the coefficient of friction between the block and the plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{6}}
Two trucks each 250 m long are running in opposite directions on parallel paths. One truck has a certain speed, and the other truck has a speed of 30 km/hr. The time taken by the slower truck to pass the driver of the faster one is 35.997120230381576 seconds. What is the speed of the slower truck?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given a sorted list $L$ with $n$ numbers, where indices from $\frac{n}{4}$ to $\frac{3n}{4}$ contain "good" numbers, find the probability that the median of three randomly selected numbers from $L$ (with repetitions allowed) is a "good" number. The probability of selecting a "good" number is $\frac{1}{2}$, a "low bad" number is $\frac{1}{4}$, and a "high bad" number is also $\frac{1}{4}$. Calculate this probability considering all possible cases.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11/16}
Let \( p(x) = c_1 + c_2 \cdot 2^x + c_3 \cdot 3^x + c_4 \cdot 5^x + c_5 \cdot 8^x \). Given that \( p(k) = k \) for \( k = 1, 2, 3, 4, 5 \), compute \( p(6) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-50}
In $\triangle ABC$, point $D$ is on side $AB$, and $\overrightarrow{BD}=\dfrac{1}{2}\overrightarrow{DA}$. Let $\overrightarrow{CB}=a$ and $\overrightarrow{CA}=b$. Calculate $\overrightarrow{CD}$ in terms of $a$ and $b$, expressing your answer as a linear combination of these vectors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}a+ \dfrac{1}{3}b}
Let $f_n$ be the Fibonacci numbers, defined by $f_0 = 1$, $f_1 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n \geq 2$. For each $i$, where $1 \leq i \leq 200$, calculate the greatest common divisor $g_i$ of $f_i$ and $f_{2007}$. Determine the sum of the distinct values of $g_i$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
A positive integer \( n \) is said to be good if \( 3n \) is a re-ordering of the digits of \( n \) when they are expressed in decimal notation. Find a four-digit good integer which is divisible by 11.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2475}
Given the function $f(x) = 2\sin(\omega x - \frac{\pi}{3})$ ($\omega > 0$) and $g(x) = \cos(2x + \varphi)$ ($0 < \varphi < \pi$) have the same axis of symmetry, then the value of $g(\frac{\pi}{3})$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{\sqrt{3}}{2}}
8,9

In a triangular pyramid $P A B C$, the lateral edge $P B$ is perpendicular to the plane of the base $A B C$, $P B=6, A B=B C$ $=\sqrt{15}, A C=2 \sqrt{3}$. A sphere with center $O$ lying on the face $A B P$ touches the planes of the other faces of the pyramid. Find the distance from the center $O$ of the sphere to the edge $A C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{24}{6+\sqrt{15}}}
2. Given $A, B, C$ are any three points on a plane, and $BC=a, CA=b, AB=c, y=$ $\frac{c}{a+b}+\frac{b}{c}$, the minimum value of $y$ is . $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}-\frac{1}{2}}
In a $N \times N$ square table, all integers are arranged according to the following rule: 1 is placed in any cell, 2 is placed in the row whose number is equal to the column number containing 1, 3 is placed in the row whose number is equal to the column number containing 2, and so on. By how much does the sum of the numbers in the column containing $N^{2}$ differ from the sum of the numbers in the row containing 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N^2 - N}
If in the quadratic equation \( x^{2} - 2(a - 3)x - b^{2} + 9 = 0 \), \( a \) and \( b \) are the numbers obtained from rolling a die, what is the probability \( P \) that this quadratic equation has two positive roots?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{18}\)}
A 250 m long train is running at a speed of 55 km/hr. It crossed a platform of a certain length in 50.395968322534195 seconds. What is the length of the platform?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{520}
12.224. In a regular triangular pyramid, the dihedral angle at the base is equal to $\alpha$, the lateral surface area is $S$. Find the distance from the center of the base to the lateral face.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sin\alpha}{3}\sqrt{S\sqrt{3}\cos\alpha}}
在矩形 $A B C D$ 中, $A B=2, A D=1$, 边 $D C$ 上（包含点 $D 、 C$ ）的动点 $P$ 与 $C B$ 延长线上 (包含点 $B$ ) 的动点 $Q$ 满足 $|\overrightarrow{D P}|=|\overrightarrow{B Q}|$, 则向量 $\overrightarrow{P A}$ 与向量 $\overrightarrow{P Q}$ 的数量积 $\overrightarrow{P A} \cdot \overrightarrow{P Q}$ 的最小值为 $\qquad$.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{7}
For how many values of $n$ with $3 \leq n \leq 12$ can a Fano table be created?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
If \( x \) is positive, find the minimum value of \(\frac{\sqrt{x^{4}+x^{2}+2 x+1}+\sqrt{x^{4}-2 x^{3}+5 x^{2}-4 x+1}}{x}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{10}}
A few (at least $5$) integers are placed on a circle such that each integer is divisible by the sum of its neighboring integers. If the sum of all the integers is positive, what is the minimal value of this sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let  
  
$$p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3.$$
  
Suppose that  
  
$$p(0,0) = p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1) = p(1,1) = p(1, - 1) = p(2,2) = 0.$$  
  
There is a point $(\frac {a}{c},\frac {b}{c})$ for which $p\left(\frac {a}{c},\frac {b}{c}\right) = 0$ for all such polynomials, where $a, b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, and $c > 1$. Find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
A certain type of plant is suitable for growth on hillsides with temperatures between 16℃ and 20℃. It is known that for every 100m increase in altitude, the temperature drops by 0.5℃. If the average temperature at the foot of the mountain is 22℃, calculate the suitable altitude range for planting this type of plant. Express your answer as an inequality in meters (m).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400 \leq x \leq 1200}
56. Probability of divisibility. Find the probability that if the digits $0,1,2, \ldots, 9$ are placed in random order in the empty spaces in the sequence of digits

$$
5-383-8-2-936-5-8-203-9-3-76
$$

then the resulting number will be divisible by 396.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The watermelon seller weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg) one by one while waiting for customers, balancing a watermelon on one pan of the scale with one or two weights on the other pan (possibly identical). At the same time, the seller noted down which weights he used. What is the minimum number of different numbers that could have appeared in his records if the weight of each weight is an integer number of kilograms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Tommy's mother lets him ride his bike 2 blocks north of his block, 3 blocks east, 2 blocks west, and 2 blocks south. Each block is rectangular in shape, with a length of 200 meters and a width of 150 meters. Due to a special rule enforced for Tommy, he needs to alternate between going north and east or west for every block he rides.

His friend's father lets his friend ride his bike in an area of 80 square blocks, with square-shaped blocks that have a side length of 180 meters. Tommy's friend has no restrictions on his riding pattern. How many times greater is the area his friend can ride in compared to Tommy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.4}
If $x$ and $y$ are two negative numbers, and $x < y$, then $|x|$ \_\_\_ $|y|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{>}
Let $f(x,y)$ be a polynomial of degree $3$. If $f(0,0)=f(\pm 1,0)=f(0, \pm 1)=0$ and $f(1,-1)=1$, then what is the value of $f(-1,1)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
We say that a number of 20 digits is [i]special[/i] if its impossible to represent it as an product of a number of 10 digits by a number of 11 digits. Find the maximum quantity of consecutive numbers that are specials.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10^9 - 1}
Jan's three-eyed lizard has 3 times more wrinkles than eyes, and seven times more spots than wrinkles. The lizard's cousin has 2 times more wrinkles than eyes, and 5 times more spots than wrinkles. How many fewer eyes do both lizards have combined, compared to the combined total of spots and wrinkles on both lizards?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{102}
Determine all real constants $t$ such that whenever $a, b, c$ are the lengths of the sides of a triangle, then so are $a^{2}+b c t, b^{2}+c a t, c^{2}+a b t$.If the length of the interval in which \(t\) lies is \(\frac{a}{b}\), find \(a + b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Someone forgot the last digit of a phone number and dialed randomly. The digits can be $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$. Calculate the probability of connecting the call in no more than two attempts. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac {1}{5}}
What is the maximum number of principals that can serve at Lincoln High School during an 8-year period, given that each principal serves exactly one 3-year term? Provide your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. That is, for any real $0 \leq a < b \leq 1000$, the probability that the amount of money in a given envelope is between $a$ and $b$ is $\frac{b-a}{1000}$. At any step, you take an envelope and look at its contents. You may choose either to keep the envelope, at which point you finish, or discard it and repeat the process with one less envelope. If you play to optimize your expected winnings, your expected winnings will be $E$. What is $\lfloor E \rfloor$, the greatest integer less than or equal to $E$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{695}
Baker made 61 pastries and 167 cakes. He sold 108 cakes and some pastries. Baker still has 59 cakes. How many pastries did Baker sell?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
From the digits 1 to 9, three single-digit and three double-digit numbers are formed, with no repeated digits. Find the smallest possible average of the resulting set of numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16.5}