inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $P$ be the transition matrix of a random walk on the $n$-cycle, where $n$ is odd. Find the smallest value of $t$ such that $Pt(x, y) > 0$ for all states $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n-1}
Suppose $a_{1}, a_{2}, \ldots, a_{100}$ are positive real numbers such that $$a_{k}=\frac{k a_{k-1}}{a_{k-1}-(k-1)}$$ for $k=2,3, \ldots, 100$. Given that $a_{20}=a_{23}$, compute $a_{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{215}
Pat Peano has plenty of 0\u0027s, 1\u0027s, 3\u0027s, 4\u0027s, 5\u0027s, 6\u0027s, 7\u0027s, 8\u0027s and 9\u0027s, but he has only twenty-two 2\u0027s.  How far can he number the pages of his scrapbook with these digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
112
}
In a certain area, there are 100,000 households, among which there are 99,000 ordinary households and 1,000 high-income households. A simple random sampling method is used to select 990 households from the ordinary households and 100 households from the high-income households for a survey. It was found that a total of 120 households own 3 or more sets of housing, among which there are 40 ordinary households and 80 high-income households. Based on these data and combining your statistical knowledge, what do you think is a reasonable estimate of the proportion of households in the area that own 3 or more sets of housing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.8\%}
Evaluate the limit: $$ \lim _{x\to \infty }\left(\frac{\arctan \left(1-\cos \left(\frac{1}{x}\right)\right)-\sin \:\left(\frac{1}{2x^2}\right)-e^{-x}}{\ln\left(1+\frac{1}{x^2}\right)-\arctan \left(\frac{1}{x^2}\right)}\right) $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Find a closed form expression for the triple sum:

$$\sum_{n_1=0}^\infty\sum_{n_2=0}^\infty\sum_{n=0}^\infty \binom{n_1+n}{n}\binom{n_2+n}{n}x_1^{n_1}x_2^{n_2}x^n,$$

where $|x_1|, |x_2|, |x|, |x_1 + x_2 + x| < 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{x_1x_2 - x_1 - x_2 - x + 1}}
Given the differential equation \( x \cdot \ln(x) \frac{dy}{dx} + y = 2x \cdot \ln(x) \) for \( x \ge 1 \), find the value of \( y(e) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the value of $\int^{\pi/2}_0 \frac{\log(1+x\sin^2\theta)}{\sin^2\theta}d\theta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\pi(\sqrt{1+x}-1)\)}
Rosa received a bottle of perfume in the shape of a cylinder with a base radius of $7 \mathrm{~cm}$ and a height of $10 \mathrm{~cm}$. After using the perfume for two weeks, 0.45 liters remain in the bottle. What fraction represents the volume of perfume Rosa has already used?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{49 \pi - 45}{49 \pi}}
Starting with the display "1," calculate the fewest number of keystrokes needed to reach "240" using only the keys [+1] and [x2].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
When a plane intersects a cube, what is the shape of the intersection that cannot be formed? Express your answer as the name of the impossible shape.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Heptagon}
Given the complex numbers \( z_{1} = -\sqrt{3} - i \), \( z_{2} = 3 + \sqrt{3} i \), and \( z = (2 + \cos \theta) + i \sin \theta \), find the minimum value of \( \left|z - z_{1}\right| + \left|z - z_{2}\right| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + 2\sqrt{3}}
Given \( n \) line segments in space, no three of which are parallel to the same plane, and for which the midpoint connection between any two segments is the common perpendicular of those segments, find the maximum possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
从正 15 边形的顶点中选出 3 个连成钝角三角形，不同的选法有多少种？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{315}
On the party, every boy gave $1$ candy to every girl, and every girl gave $1$ candy to every boy. Then, every boy ate $2$ candies, and every girl ate $3$ candies. It is known that $\frac{1}{4}$ of all candies were eaten. Find the greatest possible number of children at the party.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Given a point P $(x, y)$ on the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$, find the minimum distance from point P to the line $2x + y - 10 = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
2015 people are sitting around a round table, each of whom is either a knight or a liar. Knights always tell the truth, liars always lie. Each person is given a card with a unique number written on it. After looking at the cards of their neighbors, everyone at the table says, "My number is greater than both of my neighbors' numbers." Following this, $k$ of the people said, "My number is less than both of my neighbors' numbers." What is the maximum possible value of $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2013}
Let $p<q<r$ be three integers such that $p,q,r$ is a geometric progression and $p,r,q$ is an arithmetic progression. Find the smallest possible value of $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Arithmetic sequences $(a_n)$ and $(b_n)$ have integer terms with $a_1 = b_1 = 1 < a_2 \le b_2$ and $a_nb_n = 2010$ for some $n$. What is the largest possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Every city in a certain state is directly connected by air with at most three other cities in the state, but one can get from any city to any other city with at most one change of plane. What is the maximum possible number of cities?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a village, 60% of the population are adults and 40% are children. Out of the adults, 70% are employed, while 30% are unemployed. The employed adult population is 18,000. If the adult population grows by 5% over the next year, what will be the new total population of the village after one year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45,000}
Given an isosceles triangle \(ABC\) with \(AB = AC\) and \(\angle ABC = 53^\circ\), find the measure of \(\angle BAM\). Point \(K\) is such that \(C\) is the midpoint of segment \(AK\). Point \(M\) is chosen such that:

- \(B\) and \(M\) are on the same side of line \(AC\);
- \(KM = AB\);
- \(\angle MAK\) is the maximum possible.

How many degrees is \(\angle BAM\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
There are $12$ students in a classroom; $6$ of them are Democrats and $6$ of them are Republicans. Every hour, the students are randomly separated into four groups of three for political debates. If a group contains students from both parties, the minority in the group will change their political alignment to that of the majority at the end of the debate. What is the expected amount of time needed for all $12$ students to have the same political alignment, in hours?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A palindrome is a string that does not change when its characters are written in reverse order. Let S be a 40-digit string consisting only of 0's and 1's, chosen uniformly at random out of all such strings. Let $E$ be the expected number of nonempty contiguous substrings of $S$ which are palindromes. Compute the value of $\lfloor E\rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{113}
Determine all real numbers \(x\) such that the series \( \sum_{k=1}^{\infty}(k-\sqrt{k^2+1})(x+2)^k \) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-3, -1)}
Determine the possible values of $n$ for which a graph $G$ with $n$ vertices can have more components and more edges than another graph $G'$ with the same number of vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 5}
Consider a Markov chain $Z_k$ with 6 states, structured as a pyramid where each node has a probability of $1/2$ to move to a neighboring node. Starting from state $1$ with $P(Z_0=1)=1$, calculate the expected time $E[T_B]$ to first reach one of the bottom states, which are $3$, $4$, or $5$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the area enclosed by the curve formed by the intersection of the plane $z=2$ with the ellipsoid $\frac{x^2}{25} + \frac{y^2}{1} + \frac{z^2}{5} = 1$ in the $xOy$ plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
In a queuing theory scenario, two facilities, A and B, offer identical services with single servers. The service times follow negative exponential distributions with mean service times of 1 minute for facility A and 4 minutes for facility B. A total of 60 customers per hour seek service at both facilities, with arrivals following a Poisson distribution. Customers, uninformed of queue lengths, independently choose a facility aiming to minimize their average waiting time plus service time. Eventually, a steady state distribution of customers across both facilities is reached. Determine the probability \( p \) that a customer selects facility A.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{8}}
Evaluate the triple integral \( \iiint x\, \text{d}x \) over the region bounded by the surfaces \( z = x^2 + y^2 \) and \( z = -4x - 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4\pi}
Calculate the work done by the vector field \( \vec{F}(x,y,z) = (2xe^y + 2x)\vec{i} + (x^2e^y)\vec{j} + (3z^2)\vec{k} \) along the curve \( C: \vec{r}(t) = \frac{2t}{1+t^2}\vec{i} + (t-t^3)\vec{j} + \frac{2}{1+t^2}\vec{k} \) from \( t = 0 \) to \( t = 1 \). Use the formula for work done by a vector field along a curve: 
\[ W = \int_0^1 \vec{F}(f(t), g(t), h(t)) \cdot \vec{r}\;'(t) \; \operatorname d\!t \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
Find the radius of convergence of the series \( \sum_{n=0}^{\infty} \left(\frac{z}{1+z}\right)^n. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the maximum value of \( t \) such that the following integral equals zero:
\[
\int_{-1}^{t}\left[\sin\left(x^{3} - 9x^{2} + 18x\right) + x^{2} -6x + \frac{11}{3}\right] \, \mathrm{d}x = 0
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find a number $n$ such that $n$ is a product of three prime numbers and $n$ is a square modulo 389.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Evaluate the function \( g(x) \) defined as \( g(x) = \tan \left(\frac{1}{2}\arcsin\left(\frac{2f(x)}{1+(f(x))^2}\right)\right) \), where \( f(x) \) is given by
\[ f(x) = \lim_{n\rightarrow \infty}\frac{x^2+2(x+1)^{2n}}{(x+1)^{2n+1}+x^2+1} \]
with \( n\in \mathbb{N} \). Determine the number of points where \( g(x) \) is not differentiable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the value of $p_i$ that maximizes the probability of obtaining a sum of $k$ when summing $n$ independent Bernoulli random variables $X_1, X_2, \ldots, X_n$ with success probabilities $p_1, p_2, \ldots, p_n$. Provide your answer for $p_i$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{k}{n}}
Determine the number of ring homomorphisms from \(\mathbb{Z}[i]\) to \(\mathbb{Z}/85\mathbb{Z}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
For a positive real number $r$, consider the regular surface $M_r$ defined by \( M_r = \{(x,y,z) \mid x^2+y^2=z, 0<x, 0<y, z<r^2\} \). Calculate the integral of the Gaussian curvature \( K \) over \( M_r \), i.e., \( \int_{M_r} K \, dA \). Determine the limit of this integral as \( r \) approaches infinity: \( \lim_{r \rightarrow \infty} \int_{M_r} K \, dA \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Evaluate the integral of the function \( f(z) = z^2 + 2z - 3 \) over the ellipse defined by \( |z-2i|+|z+2i|=12 \), using the parametrization \( z(t) = \sqrt{35} \sin t + 6i\cos t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the conditional expectation $E[X|X>Y]$ where $X$ and $Y$ are independent standard normal variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{\sqrt{\pi}}}
Determine the number of cosets of the principal congruence subgroup of level 2 in \( \mathbb{P}SL(2,\mathbb{Z}) \) without using the reduction mod 2 map.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the degree of the image of a hypersurface $H \subset \mathbb{P}^n$ of degree $b$ under the Veronese map $\nu_{n,d}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b d^{n-1}}
In a cube $ABCD-A'B'C'D'$ with an edge length of $6$, point $E$ is the midpoint of edge $A'B'$, and point $F$ lies on edge $A'D'$ such that $|A'F|=2|D'F|$. Calculate the volume of the solid $A'EF-ABD$. \( \boxed{\text{Volume}} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{69}
Determine the number of regions formed by the intersection of $n$ unit spheres in $\mathbb{R}^3$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n^3 - 3n^2 + 8n}{3}}
Find the minimum value of the polynomial \(x^5 + 4x^4 + 3x^3 + 2x^2 + x + 1\) within the interval \((-2, 2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 - 5\sqrt{5}}
Evaluate the integral \( \int_{C}\frac{y^3dx-xy^2dy}{(x^2+y^2)^2} \) over the curve \( C = \frac{x^2}{16} + \frac{y^2}{9} = 1 \), oriented clockwise, using Green's Theorem. Note that the vector field is not defined at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Calculate the norm of the integral operator $B$ defined by $Bf(x) = \int_{\mathbb{R}} k(x,y)f(y) \, dy$, where $k(x,y) = \frac{1}{\sqrt{2\pi}} e^{-\frac{(x-y)^2}{2}}$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the second degree Taylor polynomial of the function \( g(x,y) = e^{f(x,y) - 5} \sin(y) \) at the point \((1,0)\), where \( f: \mathbb{R}^2 \to \mathbb{R} \) is a \( C^2 \) function with the second degree Taylor polynomial centered at \((1,0)\) given by \( P(x,y) = 4 + x + xy + \frac{y^2}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{xy + y^2}
Annie takes a 6 question test, with each question having two parts each worth 1 point. On each part, she receives one of nine letter grades {A, B, C, D, E, F, G, H, I} that correspond to a unique numerical score. For each question, she receives the sum of her numerical scores on both parts. She knows that A corresponds to 1, E corresponds to 0.5, and I corresponds to 0.

When she receives her test, she realizes that she got two of each of A, E, and I, and she is able to determine the numerical score corresponding to all 9 markings. If $ n $ is the number of ways she can receive letter grades, what is the exponent of 2 in the prime factorization of $ n $?

2020 CCA Math Bonanza Individual Round #10
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
11
}
Unit circles $O_1$ and $O_2$ are tangent to each other and line $L$. Circle $O_n$ is tangent to $O_1$, $O_{n-1}$, and $L$. Find the radius of $O_{20}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{361}}
12 Let $A$ be an $m \times m$ matrix. Let $X_{i}$ be the set of entries in the $i$th row and $Y_{j}$ the set of entries in the $j$th column, $1 \leq i, j \leq m$. $A$ is called cool if the sets $X_{1}, \ldots, X_{m}, Y_{1}, \ldots, Y_{m}$ are all different. Determine the smallest value for $n$ so that a cool $2005 \times 2005$ matrix with entries from the set $\{1,2, \ldots, n\}$ exists.

## Solution
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
6. Find the smallest natural number $n$ such that in the decimal representation of $\sqrt{n}$, two nines immediately follow the decimal point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2600}
已知平面直角坐标系中的点集  
$$  
L=\{(x, y) \mid x, y \in\{1,2, \ldots, 100\}\}.  
$$  
设 $A$ 是由平面上若干个凸多边形组成的集合，满足： $A$ 中每个凸多边形的所有顶点均属于 $L$, 并且 $L$ 中的每个点都恰好是 $A$ 中一个凸多边形的顶点.  
求 $A$ 中所有凸多边形的面积之和的最小可能值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1667}
Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = \frac{x^2}{16}$. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola $P_1$ is tangent to $P_2$, which is tangent to $P_3$, which is tangent to $P_4$, which is tangent to $P_5$, which is tangent to $P_6$, which is tangent to $P_1$. What is the diameter of the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
In class 8 'G', there are several failing students, but Vovochka is performing the worst. The teachers' council decided that either Vovochka must improve his grades by the end of the term, or he will be expelled. If Vovochka improves his grades, the percentage of failing students in the class will be $24\%$, and if he is expelled, the percentage of failing students will be $25\%$. What is the current percentage of failing students in class 8 'G'?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28\%}
Zkov G.

A bank serves a million customers, the list of whom is known to Ostap Bender. Each has a six-digit PIN code, and different customers have different codes. In one move, Ostap Bender can choose any customer he has not yet chosen and peek at the digits of the code at any $N$ positions (he can choose different positions for different customers). Ostap wants to find out the code of the millionaire Koreiko. For what smallest $N$ can he guarantee to do this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Dr. Alex, a computer scientist, is exploring the technical feasibility of teleportation devices described in a popular sci-fi author's books. In the novel, each teleportation device requires 5 energy crystals to function, and each crystal can only power the device for 3 hours. 

Dr. Alex plans to test the devices for a full day (24 hours) to analyze their feasibility. If she has a supply of 50 energy crystals, how many devices can she test for the entire 24-hour period without running out of crystals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Lele's family raises some chickens and ducks. Grabbing any 6 of them, there are at least 2 that are not ducks; Grabbing any 9 of them, there is at least 1 that is a duck. What is the maximum number of chickens and ducks in Lele's family?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
若使不等式$x ^ { 2 } - 4 x + 3 < 0$和$x ^ { 2 } - 6 x + 8 < 0$同时成立的x值使关于x的不等式$2 x ^ { 2 } - 9 x + a < 0$也成立，求实数a的取值范围.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a≤9}
Given a triangle \( \triangle ABC \) with internal angles \( A, B, \) and \( C \), what is the sum of the numerator and denominator when the maximum value of \( (\sin A \cos B + \sin B \cos C + \sin C \cos A)^{2} \) is expressed in its simplest fractional form?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
We draw a triangle inside a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures $75^\circ$. 

Next, we draw a second triangle congruent to the first, also with one vertex at the center of the circle and the other vertices on the circumference, rotated $75^\circ$ clockwise from the first triangle so that it shares a side with the first triangle. 

We continue this process to draw a third, fourth, and fifth triangle, each rotated $75^\circ$ clockwise from the previous triangle. 

The base of the fifth triangle will intersect the base of the first triangle. What is the degree measure of the obtuse angle formed by this intersection?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
A teacher grades students' tests by using a certain ratio to subtract the number of incorrect responses from the number of correct responses. Student A answers each of the 100 questions on her test and receives a score of 73. She answered 91 questions correctly. What is the ratio of the number of incorrect responses subtracted to the number of correct responses?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 30 inhabitants of this island sat around a round table. Each of them said one of two phrases: "My neighbor on the left is a liar" or "My neighbor on the right is a liar." What is the minimum number of knights that can be at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let  $AXYZB$  be a regular pentagon with area  $5$  inscribed in a circle with center  $O$ .  Let  $Y'$  denote the reflection of  $Y$  over  $\overline{AB}$  and suppose  $C$  is the center of a circle passing through  $A$ ,  $Y'$  and  $B$ . Compute the area of triangle  $ABC$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that the graph of the function $y=e^{-2x+1}$ is symmetric with respect to a certain line $l$ to the graph of the function $y=\frac{{\ln(-x-1)-3}}{2}$, if $P$ and $Q$ are two moving points on their graphs, then the minimum distance between these two points is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}(4+\ln2)}{2}}
Task 2. Find the number of ways to color all natural numbers from 1 to 20 in blue and red such that both colors are used and the product of all red numbers is coprime with the product of all blue numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
实数x、y满足$x ^ { 2 } + y ^ { 2 } = 20 $，则xy+8x+y的最大值是
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
What would the 25th number be in a numeric system where the base is five?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}