inclusionAI/Ring-lite-rl-data huggingface dataset:
Around a circle, an isosceles trapezoid \(ABCD\) is described. The side \(AB\) touches the circle at point \(M\), and the base \(AD\) touches the circle at point \(N\). The segments \(MN\) and \(AC\) intersect at point \(P\), such that \(NP: PM=2\). Find the ratio \(AD: BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:1}
Determine the minimum number of different table arrangements needed for a dinner party with 20 guests, seated in groups of 4, so that every guest meets every other guest at least once.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $PA$, $PB$, and $PC$ be three non-coplanar rays originating from point $P$, with each pair of rays forming a $60^\circ$ angle. A sphere with a radius of 1 is tangent to each of these three rays. Find the distance from the center of the sphere $O$ to point $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Calculate the flux \( \iint \vec{F} \cdot \hat{n} \, ds \) of the vector field \( \vec{F} = 4x \hat{i} - 2y^2 \hat{j} + z^2 \hat{k} \) through the surface \( S = \{(x,y,z) : x^2 + y^2 = 4, z = 0, z = 3\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84\pi}
Determine the minimum number of questions Player B needs to ask to guess the secret numbers selected by Player A, given that Player A chooses a group of single-digit integers \( x_{1}, x_{2}, \cdots, x_{n} \) which can be positive or negative, and Player B can ask about the sum \( a_{1} x_{1} + a_{2} x_{2} + \cdots + a_{n} x_{n} \) with any set of numbers \( a_{1}, a_{2}, \cdots, a_{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit: $$\lim_{n\rightarrow \infty} \dfrac{P_{n-2}\cdot P_{n+2}}{P_{n-1}\cdot P_{n+1}}$$ where $P_{n}$ is the product of the numbers in the nth row of Pascal's Triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^3}
Evaluate the limit: $$\lim_{n \to \infty} \int_0^{\frac{\pi}{2}} \frac{(\sin(x))^{n}}{1-\sin{(x)}}\,\d x.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Calculate the probability of having exactly 40 Type A arrivals before either a single Type B arrival occurs or the time reaches \( t = 3 \). Type A customers arrive at a rate given by \( \lambda_a = 0.7\frac{-t^2+8t+84}{10} \), and Type B customers arrive at a constant rate of \( \lambda_b = 6 \). Use the provided information and the concept of super-positioning to determine this probability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a game involving two players, A and B, two standard six-sided dice are rolled repeatedly. Player A wins if a total of 2 is rolled first, while player B wins if two consecutive totals of 7 are rolled first. What is the probability that player A wins the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{13}}
Find an asymptotic bound for the sum \( \sum_{k = 0}^{\lg n} 2^k \lg(\lg(n) - k) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n)}
Determine the number of solutions \(x(t)\) to the differential equation \(x^{\prime}(t) = x(t) - e^{-t^2}\) that satisfy \(\lim_{t \to \pm \infty}x(t) = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The first floor where the seventh-grade boys live has $x$ rooms. If each room accommodates 6 people, exactly one room remains empty; if each room houses 5 people, 4 people will have nowhere to stay. How many rooms are there in total on the first floor? The equation related to $x$ can be stated as _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Max attended 40 college courses with varying credit hours in 2 years. On average, he attended 5 courses per semester, with each course worth 3 credit hours, but in the last semester, he took two additional courses, each worth 4 credit hours. Sid attended four times as many college courses as Max in the same period, and he took twice as many 4-credit-hour courses as Max in the last semester. What's the total number of credit hours they both completed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{606}
Let $u=\{a_1, \ldots, a_5\}$, where all $a_i \in \{0,1\}$. Let $D(u)$ be the set of all subsequences of $u$ with length 4. Let $A$ be a set of sequences $u_1, \ldots, u_{|A|}$ with the property that $|D(u_i) \cap D(u_j)| = 0$ for all $i \neq j$. What is the maximum value of $|A|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Paul has a collection of 137 baseball cards that he wants to place in his binder. The binder has a variety of pages, with some pages accommodating 4 cards, some accommodating 6 cards, and others accommodating 8 cards. He distributes the cards equally among the page types, starting with the 4-card pages, followed by the 6-card pages, and finally the 8-card pages. How many cards will he have on the last partially-filled page?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the sum of the ten terms in the arithmetic sequence $-3, 4, \dots, 40$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{285}
Find the area of the region that lies inside both curves $r = 1 - \cos \theta$ and $r = \frac{1}{2}$. Express the area in the form $\frac{a \pi}{b} - \frac{c \sqrt{d}}{e}$, where $a$, $b$, $c$, $d$, and $e$ are positive integers. Calculate $a + b + c + d + e$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
The sequence $\{a_{n}\}$ satisfies $a_{{n+1}}+(-1)^{n} a_{n} =2n-1$. Find the sum of the first 80 terms of the sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3240}
Wrap a line segment of 20 cm in length into a hexagon. What is the range of the longest side of the hexagon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{3} \leq x < 10.}
Let $V$ be an infinite-dimensional vector space over the field $\mathbb{F}$, where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$. If the dimension of $V$ is $\alpha$, a cardinal number, determine the dimension of $V^*$, the algebraic dual of $V$. It is known that $\dim V < \dim V^*$ when $V$ is infinite-dimensional.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^\alpha}
Find the integer $n$ such that $8001 < n < 8200$ and $2^n - 1$ divides $2^{k(n-1)! + k^n} - 1$ for all integers $k > n$. What is the value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8191}
Find the dimension of the quotient ring \( \mathbb{C}[x, y] / \langle xy^3, x^2y^2, x^3y, \text{all monomials of degree 5} \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given the sets
$$
A=\{x \mid |x-a|<2\}, \quad B=\left\{x \mid \frac{2x-1}{x+2}<1\right\},
$$
if $A \cap B = A$, find the range of values for the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 1]}
Given the sequence \(\{a_{n}\}\) defined by
\[ a_{n} = \left\lfloor (2+\sqrt{5})^{n} + \frac{1}{2^{n}} \right\rfloor \quad \text{for} \quad n \in \mathbb{Z}_{+}, \]
where \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\). Let \(C\) be a real number such that for any positive integer \(n\),
\[ \sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leq C. \]
Find the minimum value of \(C\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{288}}
Let points \( C \) and \( D \) be the trisection points of \( AB \). At 8:00, person \( A \) starts walking from \( A \) to \( B \) at a constant speed. At 8:12, person \( B \) starts walking from \( B \) to \( A \) at a constant speed. After a few more minutes, person \( C \) starts walking from \( B \) to \( A \) at a constant speed. When \( A \) and \( B \) meet at point \( C \), person \( C \) reaches point \( D \). When \( A \) and \( C \) meet at 8:30, person \( B \) reaches \( A \) exactly at that time. At what time did person \( C \) start walking?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8:16}
Calculate the arc length of the curve given by the equation \(e^{2x+2y}=x-y\) from the point \((0,0)\) to the point \((1,1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
设 $\boldsymbol{u}=\left(u_{1}, u_{2}, u_{3}\right)$ 和 $\boldsymbol{v}=$ $\left(v_{1}, v_{2}, v_{3}\right)$ 是空间向量，满足 $u_{i} 、 v_{i}(i=1,2,3 )$ 均为整数，且
$0.9999<\cos \langle\boldsymbol{u}, \boldsymbol{v}\rangle<1$.
记 $S=\left|u_{1}\right|+\left|u_{2}\right|+\left|u_{3}\right|+\left|v_{1}\right|+\left|v_{2}\right|+\left|v_{3}\right|$.求 $[\sqrt{S}]$ 的最小可能值，其中， $[x]$ 表示不超过实数 $x$ 的最大整数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n$; that is to say, $p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0$ for some real coefficients $a_0, \dots, a_{2n-1}$. Suppose that $p(1/k) = k^2$ for all integers $k$ such that $1 \leq |k| \leq n$. Find all other real numbers $x$ for which $p(1/x) = x^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 1/n!}
On the radius \(AO\) of a circle with center \(O\), a point \(M\) is chosen. On one side of \(AO\) on the circle, points \(B\) and \(C\) are chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \(BC\), given that the radius of the circle is 16 and \(\sin \alpha = \frac{\sqrt{39}}{8}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given that $\sin \alpha \cos \alpha = \frac{1}{8}$, and $\alpha$ is an angle in the third quadrant. Find $\frac{1 - \cos^2 \alpha}{\cos(\frac{3\pi}{2} - \alpha) + \cos \alpha} + \frac{\sin(\alpha - \frac{7\pi}{2}) + \sin(2017\pi - \alpha)}{\tan^2 \alpha - 1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}}{2}}
Evaluate the integral \( \int_R z\,dA \) where \( R \) is the region enclosed by the intersection of the cone \( z^2=x^2+y^2 \) and the cylinder \( x^2+y^2=4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
If the price of a bag of cherries is $5, the price of a bag of olives is $7, and a box of grapes is priced at $11, how much would Jordyn pay for buying 50 bags of cherries with a 12% discount, 75 bags of olives with an 8% discount, and 25 boxes of grapes with a 15% discount, given a 5% sales tax and a 2% service charge on the final amount?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1002.32}
Determine the number of nine-digit odd numbers divisible by 375, where the digits from the second to the second-last are in non-decreasing order. If the number is represented as \( n = \overline{a_1a_2a_3a_4\ldots a_9} \), then \( a_2 \le a_3 \le a_4 \le a_5 \le \ldots \le a_8 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{186}
Determine for which values of \( n \) the integral \( \int_0^1 \frac{x^n}{\sqrt{1-x^4}} \, dx \) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n > -1}
Calculate the surface integral \( \oint_A F \cdot v \, dS(x) \) for the vector field \( F(x,y,z) = (-y+e^{x+z}, 0, e^{x+z}) \) over the set \( A := \delta B_1 \cap(\{x>0, z=0\}\cup\{x=0, z>0\}) \), where \( B_1 \) is the unit ball in \( \mathbb{R}^3 \) and \( v \) is a tangent vector on \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Find \( \epsilon > 0 \) such that for all partitions \( P \) of \([1,2]\) with mesh \( \lambda(P) < \epsilon \), the inequality \(|U_{f,P} - L_{f,P}| < 0.01\) holds, where \( U_{f,P} \) and \( L_{f,P} \) are the upper and lower Darboux sums of the function \( f(x) = \frac{1}{x} \) on \([1,2]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.01}
From point $C$ a tangent is drawn to a circle with radius $2 \sqrt{5}$ and center at point $O$. Point $A$ is the point of tangency. The segment $CO$ intersects the circle at point $B$. From point $B$, a perpendicular is drawn to the line $BC$ and intersects the line $AC$ at point $F$. Find the radius of the circumcircle of triangle $ABC$ if $BF = 2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{30}}{2}}
Find the following limit: 

$$\lim_{x\rightarrow -\infty}{e^{\frac {1}{2-x}}\cdot\frac{x^2+2x-1}{x-2}}-x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the smallest natural number $n$ such that the group $\mathbb{Z}_{6} \times \mathbb{Z}_{20} \times \mathbb{Z}_{45}$ is isomorphic to the direct product of $n$ cyclic groups.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the function expression y = x^2 + 1 and the range {1,3}, how many homologous functions can be defined? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the definite integral: \[ \int_0^4\sqrt{x^2+4}\,dx + \int_2^{2\sqrt{5}}\sqrt{x^2-4}\,dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\sqrt{5}}
Two employees M and N are paid a total of $616 per week by their employer. M is paid a certain percentage more than N, and N is paid $280 per week. What is the percentage of M's salary compared to N's salary?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
For which values of the natural number \( n > 2 \) is the graph \( G_n = (V, E) \), defined by \( V = \{A \subset \{1, 2, \ldots, n\} : |A| = 2\} \) and \( \{A, B\} \in E \iff A \cap B = \emptyset \), connected?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 5}
Find the shortest distance between the lines $\frac{x+2}{2}=\frac{y-1}{3}=\frac{z}{1}$ and $\frac{x-3}{-1}=\frac{y}{1}=\frac{z+1}{2}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5 \sqrt{3}}{3}}
Let \( A = \left( a_{ij} \right) \) be the \( n \times n \) matrix, where \( a_{ij} \) is the remainder of the division of \( i^j + j^i \) by 3 for \( i, j = 1, 2, \ldots, n \). Find the greatest \( n \) for which \(\operatorname{det} A \ne 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the isomorphic form of the quotient group \(\mathbb{C}^\times / S_1\), where \(S_1\) is the unit circle subgroup of the multiplicative group of non-zero complex numbers, defined as \(\{e^{i\theta} \mid \theta \in [0,2\pi)\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}^+}
Find the least positive integer \( k \) such that \( \cos^k \left(\frac{\pi}{2k}\right) \geq \frac{99}{100} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{123}
There are 20 rooms, some with lights on and some with lights off. The occupants of these rooms prefer to match the majority of the rooms. Starting from room one, if the majority of the remaining 19 rooms have their lights on, the occupant will turn the light on; otherwise, they will turn the light off. Initially, there are 10 rooms with lights on and 10 rooms with lights off, and the light in the first room is on. After everyone in these 20 rooms has had a turn, how many rooms will have their lights off?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
7.5. At each vertex of a cube lives a number, not necessarily positive. All eight numbers are distinct. If a number is equal to the sum of the three numbers living in the adjacent vertices, then it is happy. What is the maximum number of happy numbers that can live at the vertices of the cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In an octagon, let the number of obtuse angles be $x$. What is the minimum value of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
a placed 3 sheets with two carbons to get two extra copies of the original . then he decided to get more carbon copies and folded the paper in such a way that the upper half of the sheets were on top of the lower half . then he typed . how many carbon copies did he get ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
We define a positive integer as "special" if it is divisible by all of its non-zero digits. Determine the maximum number of consecutive special numbers that can exist.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
What is the greatest positive integer $m$ such that $ n^2(1+n^2-n^4)\equiv 1\pmod{2^m} $ for all odd integers $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ m = 7 }
Given the functions \( f(x) = e^{x+1} - 1 \) and \( g(x) = 100|f(x)| - \sum_{k=1}^n|f(x^k)| \), where \( n \in \mathbb{N} \), determine the sum of all possible integer values of \( n \) such that \( g(x) \) is differentiable over the entire set of real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Let the set \( M = \{1, 2, \cdots, 2020\} \). The subset \( A \subseteq M \) is such that for any element \( x \) in \( A \), \( 4x \) is not in \( A \). What is the maximum number of elements in the set \( A \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1616}
In the sequence $\{a_n\}$, $a_1= \sqrt{2}$, $a_n= \sqrt{a_{n-1}^2 + 2}\ (n\geqslant 2,\ n\in\mathbb{N}^*)$, let $b_n= \frac{n+1}{a_n^4(n+2)^2}$, and let $S_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. The value of $16S_n+ \frac{1}{(n+1)^2}+ \frac{1}{(n+2)^2}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{4}}
From the digits 0, 1, 2, 3, 4, 5, select 3 digits to form a three-digit number without repeating digits. How many of such three-digit numbers are divisible by 5? (Answer with a number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
10.228. One end of the diameter of a semicircle coincides with the vertex of the angle at the base of an isosceles triangle, and the other lies on this base. Find the radius of the semicircle if it touches one lateral side and divides the other into segments of lengths 5 and 4 cm, measured from the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{\sqrt{11}}}
A bicycle wheel has a certain diameter. It makes 393.1744908390343 complete revolutions in 1 km. What is the diameter of the wheel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.809}
On every kilometer of the highway between the villages Yolkino and Palkino, there is a post with a sign. On one side of the sign, the distance to Yolkino is written, and on the other side, the distance to Palkino is written. Borya noticed that on each post, the sum of all the digits is equal to 13. What is the distance from Yolkino to Palkino?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Calculate the expected hitting time \( t_3 \) for a simple random walk on a 3-dimensional hypercube \( C_3 \), where \( t_3 \) is the expected time to hit a vertex \( B \) starting from a vertex \( A \) with Hamming distance 3. Assume the random walk has a \( 1/n \) chance of moving to each of the \( n \) adjacent vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given the equations \( x^2 - y^2 - u^3 + v^2 + 4 = 0 \) and \( 2xy + y^2 - 2u^2 + 3v^4 + 8 = 0 \), find \( \frac{\partial u}{\partial x} \) at \((x,y) = (2,-1)\). Assume that these equations determine functions \( u(x,y) \) and \( v(x,y) \) near the point \((x,y,u,v) = (2, -1, 2, 1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{32}}
Determine the order of the element \((1,2)+\langle(0,4)\rangle\) in the quotient group \(\Bbb Z_{4}\oplus \Bbb Z_{16}/\langle(0,4)\rangle\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
How many triples $(p,q,n)$ are there such that $1/p+2013/q = n/5$ where $p$, $q$ are prime numbers and $n$ is a positive integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
7}
What is the average case complexity of the stable marriage problem?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n \log n)}
Determine the number of continuous functions \( f(x) \) defined on \([0, 1]\) that satisfy the following conditions: \( \int_0^1 f(x)\,dx = 2 \), \( \int_0^1 (x-0.5)f(x)\,dx = 0 \), and \( \int_0^1 (x-0.5)^2f(x)\,dx = 0.5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In a plane, determine the minimum number of non-parallel straight lines required such that at least one angle formed between any two lines is less than 26 degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
6. How many ways are there to choose integers $a, b$ and $c$ with $a<b<c$ from the list $1,5,8,21,22,27,30,33,37,39,46,50$ so that the product abc is a multiple of 12 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76}
In a guessing game, if the number of possible options, $N$, is even and the target number of options, $M$, is $\frac{N}{2}$, what is the minimum number of guesses required for Bob to ensure that Alice confirms with a YES?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{N}{2} + 1}
On an infinite chessboard, a bishop and a knight are placed on squares in the same row. A meteor storm places a meteor on each square independently with probability \( p \). Neither the bishop nor the knight is hit, but their movements may be obstructed. Find the value of \( p \) such that the expected number of valid squares the bishop can move to equals the expected number of valid squares the knight can move to. Express \( p \) as \( \frac{a}{b} \) for relatively prime positive integers \( a \) and \( b \), and compute \( 100a + b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{102}
Let $X$ and $Y$ be positive integers such that the first few digits of the fractions \( \frac{29}{70}, \frac{70}{169}, \frac{169}{408}, \frac{408}{X}, \frac{X}{Y} \) are all equal to 0.4142. What is \( X + Y \)? Note: The fractions are not equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3363}
Suppose you have 100 distinct video games and 50 distinct friends. You must offer each friend 3 video games from your collection of 100, and each friend will select one of the three games offered to them. Assuming all offers are made simultaneously, you may offer the same game to multiple friends, and each set of 3 games and each game chosen has an equal probability of selection. How many distinct games will be chosen more than once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}