inclusionAI/Ring-lite-rl-data huggingface dataset:
2. Let $n$ be a three-digit positive integer that does not contain the digit 0. If any permutation of the units, tens, and hundreds digits of $n$ does not form a three-digit number that is a multiple of 4, find the number of such $n$.
(54th Ukrainian Mathematical Olympiad)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{283}
将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+d e f$ 是奇数的概率为 $\qquad$.
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}
Let $f$ be a one-to-one function from the set of natural numbers to itself such that $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. What is the least possible value of $f(999)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Find the maximal possible number of integers you can choose from the set \( \{1, 2, \ldots, 100\} \) such that no product of any non-empty subset of these numbers is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Given two points on the surface of a sphere, and a circle $k$ which passes through exactly one of these points, how many circles are there on the sphere that pass through both points and are tangent to $k$?
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Answer: \boxed{1}
For each positive integer $n$, let $s(n)$ be the sum of the digits of $n$. Find the smallest positive integer $k$ such that \[ s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9999}
The *rank* of a rational number  $q$  is the unique  $k$  for which  $q=\frac{1}{a_1}+\cdots+\frac{1}{a_k}$ , where each  $a_i$  is the smallest positive integer  $q$  such that  $q\geq \frac{1}{a_1}+\cdots+\frac{1}{a_i}$ .  Let  $q$  be the largest rational number less than  $\frac{1}{4}$  with rank  $3$ , and suppose the expression for  $q$  is  $\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}$ .  Find the ordered triple  $(a_1,a_2,a_3)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(5, 21, 421)}
A certain number, when increased by 20%, equals twice the result of 20 less than the number decreased by 20%. What is that number? Express your answer as a whole number.
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Answer: \boxed{80}
Determine the mass percentage of Ca in a mixture containing 20 g of Calcium carbonate (CaCO3) and 10 g of Magnesium chloride (MgCl2), assuming that the mixture is anhydrous and there are no impurities present.
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Answer: \boxed{26.69\%}
Given a function $f(x) = \frac{ax+b}{cx+d}$ ($c \neq 0$), the center of symmetry for the graph of $f(x)$ is $(-\frac{d}{c}, \frac{a}{c})$. It is known that $f(x) = \frac{2-2x}{2x-1}$ and the general term of the sequence $\{a_n\}$ is $a_n = f(\frac{n}{2017})$ for $n \in \mathbb{N}$. Calculate the sum of the first 2017 terms of this sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2016}
There is a peculiar computer with a button. If the current number on the screen is a multiple of 3, pressing the button will divide it by 3. If the current number is not a multiple of 3, pressing the button will multiply it by 6. Xiaoming pressed the button 6 times without looking at the screen, and the final number displayed on the computer was 12. What is the smallest possible initial number on the computer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Two numbers a and b with 0 ≤ a ≤ 1 and 0 ≤ b ≤ 1 are chosen at random. The number c is defined by c = 2a + 2b. The numbers a, b, and c are each rounded to the nearest integer to give A, B, and C, respectively. Find the probability that 2A + 2B = C. Express your answer as a fraction in lowest terms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{16}}
Given a regular 200-sided polygon \( A_{1} A_{2} \cdots A_{200} \), connect the diagonals \( A_{i} A_{i+9} \) for \( i=1, 2, \cdots, 200 \), where \( A_{i+200}=A_{i} \) for \( i=1,2, \cdots, 9 \). How many different intersection points are there inside the regular 200-sided polygon formed by these 200 diagonals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{800}
There are line segments of lengths 1, 2, 3, ..., 9, each of a kind. How many different ways can you select several of them to form a square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Five numbers are interconnected. The first number is less than the second number by 40%, the third number is 25% more than the first number. The fourth number is 20% less than the second number but 30% more than the third number. The fifth number is 50% more than the fourth number. What is the percentage difference between the second and fifth numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46.25\%}
Kevin starts with the vectors \((1,0)\) and \((0,1)\) and at each time step, he replaces one of the vectors with their sum. Find the cotangent of the minimum possible angle between the vectors after 8 time steps.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{987}
At the end-of-term math exam of a certain school, the math scores follow a normal distribution $N\left(76,16\right)$. The exam scores are divided into four levels $A$, $B$, $C$, and $D$ in the proportions of $15\%$, $35\%$, $35\%$, and $15\%$ respectively. Students with scores greater than or equal to $83$ points are classified as level $A$. What should be the score range for level $B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[76,83)}
We consider a chessboard of size $8 \times 8$ where the squares are alternately colored white and black. An infernal rook is a piece that can attack the squares of its color located on its row, as well as the squares of the other color located in its column. What is the maximum number of infernal rooks that can be placed on the chessboard such that no two infernal rooks can ever attack each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
The following list shows every number for which more than half of its digits are the digit $2$, in increasing order:
\[
2, 22, 122, 202, 212, 220, 221, 222, 223, 224, \dots
\]
If the $n$th term in the list is $2022$, what is $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Two water droplets fall freely one after another from a $300 \mathrm{~m}$ high cliff. The first droplet has already fallen $\frac{1}{1000} \mathrm{~mm}$ when the second one starts falling.

How many millimeters apart will the two droplets be at the moment the first one reaches the base of the cliff? (The result should be calculated to an accuracy of $\frac{1}{10} \mathrm{~mm}$. Air resistance, etc., should not be taken into account.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["34.6"]}
a certain debt will be paid in 52 installments from january 1 to december 31 of a certain year . each of the first 25 payments is to be $ 500 ; each of the remaining payments is to be $ 100 more than each of the first 25 payments . what is the average ( arithmetic mean ) payment that will be made on the debt for the year ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$551.92}
The ancient Chinese theory of the "Five Elements" posits that there are five kinds of properties in materials: metal, wood, earth, water, and fire. According to the theory, metal overcomes wood, wood overcomes earth, earth overcomes water, water overcomes fire, and fire overcomes metal. Considering these properties, how many different arrangements of the five elements in a row are possible such that no two consecutive materials have a conquering relationship? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
James buys a weight vest for some amount. He then buys 200 pounds of weight plates for $1.2 per pound. A 200-pound weight vest would cost $700 but there is a $100 discount. He saves $110 with his vest. How much did the weight vest cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$250}
Find the limit of the sequence $\left\{\frac{I_{2n}}{I_{2n+1}}\right\}_{n=1}^{\infty}$, where $I_n=\int_0^{\frac{\pi}{2}}\sin^n x \, dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the complex numbers \( z_1 \) and \( z_2 \) that satisfy \(\left|z_1\right| = \left|z_1 - 2 z_2\right|\) and \( z_1 \overline{z_2} = \sqrt{3}(1 - \mathrm{i}) \), where \(\mathrm{i}\) is the imaginary unit, find \(\frac{z_1}{z_2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \mathrm{i}}
Player A and player B are playing Go. If player A plays as black first, the probability of player A winning is $\frac{4}{5}$; if player B plays as black first, the probability of player B winning is $\frac{2}{3}$. Assuming each game is independent and there are no draws, in a two-game match where the loser of the first game plays first in the second game, and the choice of who plays as black in the first game is equally likely between player A and player B, the probability of player A winning the first game and player B winning the second game is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{45}}
How to evaluate the integral $$\int_\Omega xye^{x^2-y^2}dxdy$$ over the domain $\Omega = \{(x,y)\mid 1\leq x^2-y^2\leq 9, 0\leq x \leq 4, y\geq 0\}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2e(e^8-2)}
Each side of a triangle is divided into three equal parts. The division points serve as the vertices of two triangles, whose intersection is a hexagon. Find the area of this hexagon if the area of the given triangle is $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{9}S}
Four pieces of lumber are placed in parallel positions, perpendicular to line M:

- Piece W is 5m long
- Piece X is 3m long and its left end is 3m from line M
- Piece Y is 5m long and is 2m from line M
- Piece Z is 4m long and is 1.5m 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. Calculate the length, in meters, of the part of piece W to the left of the cut. Express your answer as a decimal number with two places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.75}
All edges of a tetrahedron have a length of $\sqrt{2}$, and the four vertices are on the same sphere. Calculate the surface area of this sphere. Express your answer in terms of $\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \pi}
Real numbers  $x,y,z$  satisfy   $$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+x+y+z=0 $$   and none of them lies in the open interval  $(-1,1)$ . Find the maximum value of  $x+y+z$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
已知实系数多项式 f(x)=x^3+bx^2+cx+d 有复根 z_1, z_2, z_3，且满足 f(2)\geq f(-3)+45. 则 \displaystyle\prod_{k=1}^3(z_k^2+z_k+1) 的最小值为__________.（树懒）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A $3 \times 3$ grid is to be painted with three colors: red, green, and blue, such that:
1. No two squares that share an edge are the same color.
2. No two corner squares on the same edge of the grid have the same color.

For example, the upper-left and bottom-left squares cannot both be red, as that would violate condition (ii). In how many ways can this be done? Note that rotations and reflections are considered distinct colorings.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
10 white and 20 black chips are arranged in a circle. It is allowed to swap any two chips that have exactly three chips between them. Two arrangements of chips (in these 30 positions) are called equivalent if one can be transformed into the other by several such swaps. How many non-equivalent arrangements are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A man spends 1/4 of his salary on food, 1/8 of his salary on house rent, 3/10 of his salary on clothes, and 1/6 of his salary on transportation expenses. The man still has $35,000 left with him. Find his total salary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$221,052.63}
Aerith timed herself solving a contest and noted the time both as days:hours:minutes:seconds and in seconds. For example, if she spent 1,000,000 seconds, she recorded it as 11:13:46:40 and 1,000,000 seconds. Bob subtracts these numbers, ignoring punctuation. In this case, he computes:

\[ 11134640 - 1000000 = 10134640 \]

What is the largest number that always must divide his result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
How many ring homomorphisms are there from $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}$ into $\mathbb{Z}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Lolita can drink some boxes of milk during weekdays. During Saturdays, she can drink twice the number of boxes of milk than on the weekdays, and during Sundays, she can drink thrice the number of boxes of milk than on the weekdays. She drinks 30 boxes of milk per week. How many boxes of milk does she drink on weekdays?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Brielle has a wall clock in his home that rings every 3 hours. If it will first ring at a certain time in the morning, it will ring 8 times in a day. At what time does the clock first ring?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12:00}
Problem 2. Pasha wrote the numbers $4,5,6, \ldots, 14$ on the board, and then erased one or several of them. It turned out that the remaining numbers on the board cannot be divided into several groups such that the sums of the numbers in the groups are equal. What is the greatest possible value of the sum of the remaining numbers on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Let $p=101$. The sum \[\sum_{k=1}^{10}\frac{1}{\binom{p}{k}}\] can be written as a fraction of the form $\dfrac{a}{p!}$, where $a$ is a positive integer. Compute $a \pmod{p}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Two equal-armed balances are shown. The first balance has 3 circles balancing 5 squares, and the second balance has 2 circles balancing 3 squares plus 1 circle. How many squares would it take to balance one circle? Express your answer as a single number of squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Trains $A$ and $B$ are on the same track a distance $100$ miles apart heading towards one another, each at a speed of $50$ miles per hour. A fly starting out at the front of train $A$ flies towards train $B$ at a speed of $75$ miles per hour. Upon reaching train $B$, the fly turns around and flies towards train $A$, again at $75$ miles per hour. The fly continues flying back and forth between the two trains at $75$ miles per hour until the two trains hit each other. How many minutes does the fly spend closer to train $A$ than to train $B$ before getting squashed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{130}{3}}
The lengths of the sides of a trapezoid are 2, 10, 10, and 20. Find the length of the segment connecting the midpoints of the diagonals.

(It is known that these points are distinct). If there are multiple possible answers, list them in ascending order separated by commas or semicolons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Every second, Andrea writes down a random digit uniformly chosen from the set $\{1,2,3,4\}$. She stops when the last two numbers she has written sum to a prime number. What is the probability that the last number she writes down is 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15/44}
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? Provide your answer as a simplified expression in the form of \(\frac{a+b\sqrt{c}}{d}\). Please find the value of a + b + c + d.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Each integer $1$ through $9$ is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. What digit is most likely to be the units digit of the **sum** of Jack's integer and Jill's integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a small pond, there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, where $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$, it will be eaten by a patiently waiting snake. If the frog reaches pad $10$, it will exit the pond, never to return. Find the probability that the frog will escape being eaten by the snake. The original answer is in \frac{m}{n} format, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{209}
Calculate: $-1^{2}\times \left(-2024\right)^{0}+(\frac{1}{3})^{-1}-|-5|+\left(-4\right)^{2023}\times \left(-0.25\right)^{2022}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
Given a board with size $25 \times 25$. Some $1 \times 1$ squares are marked, so that for each $13 \times 13$ and $4 \times 4$ sub-boards, there are at least $\frac{1}{2}$ of the squares in the sub-board that are marked. Find the least possible number of marked squares in the entire board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{313}
已知$A = \{ x \vert x ^ { 2 } - 4 x + 3 < 0 , x \in R \} , B = \{ x \vert 2 ^ { 1 - x } + a \leq 0 , x ^ { 2 } - 2 ( a + 7 ) x + 5$≤0,x∈R}，若A⊆B，则实数a的取值范围是 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4≤a≤-1}
现有长为150cm的铁丝，要截成n(n&gt;2)小段，每段的长为不小于1（cm）的整数．如果其中任意3小段都不能拼成三角形，则n的最大值为
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The equation  $$ (x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016) $$  is written on the board, with  $2016$  linear factors on each side. What is the least possible value of  $k$  for which it is possible to erase exactly  $k$  of these  $4032$  linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k = 2016}
Joe likes to play video games and he normally spends an average of $50 a month on video games. However, the cost of the games sometimes varies from $40 to $60 per game. Joe also sells his games for an average of $30 each once he is done playing them, but the resale price varies between $20 and $40. In addition to his video game expenses, Joe has a monthly subscription to an online gaming platform, which costs him $15 per month. If he starts with $240, how many months can he continue purchasing and reselling games with variable pricing and pay for his gaming subscription expenses before he is out of money?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given $p:(x-m)^{2} \lt 9$, $q:\log _{4}(x+3) \lt 1$, if $\neg q$ is a necessary but not sufficient condition for $\neg p$, then the range of $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2,0]}
Susan made a block with small cubes of 5 cubic cm volume to make a block 7 small cubes long, 7 small cubes wide, and 6 small cubes deep. She realized that she has used more small cubes than she really needed. How many fewer cubes could she have glued together to look like a block with the same dimensions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
A, B, C, and D obtained the top four positions (without ties) in the school, and they made the following statements:
A: "I am neither first nor second."
B: "My position is adjacent to C's position."
C: "I am neither second nor third."
D: "My position is adjacent to B's position."

It is known that A, B, C, and D respectively obtained the places $A, B, C, D$. Determine the four-digit number $\overrightarrow{\mathrm{ABCD}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4213}
在一个 10$\times10$ 的方格表中有一个由4$n$个 1 x 1 的小方格组成的图形，它既可被$n$
个“田”型的图形覆盖，也可被$n$个“田”或
“毌”型(可以旋转)的图形覆盖.求正整数$n$ 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a point \( P \) inside \( \triangle ABC \), perpendiculars are drawn from \( P \) to \( BC, CA, \) and \( AB \) with feet \( D, E, \) and \( F \) respectively. Semicircles are constructed externally on diameters \( AF, BF, BD, CD, CE, \) and \( AE \). These six semicircles have areas denoted \( S_1, S_2, S_3, S_4, S_5, \) and \( S_6 \). Given that \( S_5 - S_6 = 2 \) and \( S_1 - S_2 = 1 \), find \( S_4 - S_3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the function $f(x)=x^{3}+ax^{2}+bx+a$ has an extremum at $x=1$ with the value $10$, find $f(2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Each edge of a triangular pyramid (tetrahedron) is equal to \( a \). Find the maximum area that the orthogonal projection of this pyramid on a horizontal plane can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^2}{2}}
函数$f(x)={{\log }_{a}}({{x}^{2}}-4x+3)$（$a>0$，$a\ne 1$）在区间$[m,+\infty )$上存在反函数，则实数$m$的取值范围是____
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(3,+\infty )$}
Find the mass percentage of I in Aluminum iodide.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{93.38\%}
Given that the perimeter of $\triangle ABC$ is 12, and its sides are $a$, $b$, and $c$, with $b$ being the longest side, find the range of possible values for $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 < b < 6}
A river boat travels at a constant speed from point A to point B. Along the riverbank, there is a road. The boat captain observes that every 30 minutes, a bus overtakes the boat from behind, and every 10 minutes, a bus approaches from the opposite direction. Assuming that the buses depart from points A and B uniformly and travel at a constant speed, calculate the interval time (in minutes) between each bus departure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}