inclusionAI/Ring-lite-rl-data huggingface dataset:
Given that the variables $a$, $b$, and $c$ are independent and identically distributed random variables with an exponential distribution with a common parameter $1$, calculate the probability that the polynomial $ax^2 + bx + c$ has real roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Determine the number of alternating subsets of the set \( \{1, 2, 3, \ldots, n\} \) for any natural number \( n \), and calculate \( A(33) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9227465}
Within the real numbers, solve the equation$$x^5 + \lfloor x \rfloor =  20$$where $\lfloor x \rfloor$ denotes the largest whole number less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x=\sqrt[5]{19}}
Find the root of the equation \( 8t^3 - 4t^2 - 4t + 1 = 0 \) that lies in the interval \( \left(0, \frac{\pi}{13}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2225}
Determine the conflict-free chromatic number \(\chi_{\text{cf}}(\mathbb{R}, \tau)\) for the real numbers \(\mathbb{R}\) with the Euclidean topology \(\tau\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given three random variables $X$, $Y$, and $Z$ with $\mathsf{Corr}(X, Y)=0.9$ and $\mathsf{Corr}(Y, Z)=0.8$, determine the minimum possible value of $\mathsf{Corr}(X, Z)$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.458}
Compute the limit \( \lim\limits_{h\to0^+}{h\sum_{n=1}^\infty f(nh)} \) as an improper integral, where \( f(x)=\frac{\ln(x)}{1+x^2} \). Note that \( f \) is increasing on \( (0,\sqrt{e}] \) and decreasing on \( [\sqrt{e},+\infty) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the number of permutations in the symmetric group $S_{12}$ that have the cycle structure $(2,2,0,1,2,0,0,0,0,0)$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the greatest number \( k \) such that for any sequence of \( k \) points in the plane, \( 2023-k \) points can be added so that the sequence of \( 2023 \) points is carioca.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
Evaluate the definite integral: $$\int^{\pi/4}_{0}\frac{8 \cos(2t)}{\sqrt {9-5 \sin t (2t)}}~dt.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{5}}
Given that \(t^3\) and \(t e^t \sin(t)\) are solutions to an nth-order linear homogeneous differential equation with constant coefficients, determine the minimum possible value of \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the quadratic character of 293 modulo 379.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Find the whole number value of $x$ that maximizes the expression \[ \frac{100\left\lfloor\frac{10000}{\left\lfloor \frac{100000}{2x}\right\rfloor+1}\right\rfloor}{x} \] given that $100000$ is divisible by $x$. \(x\) must be a divisor of $100000$. What is the maximum value of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
Evaluate the integral \( \int_C \tan z \, dz \), where \( C \) is the positively oriented circle \( |z|=2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4\pi i}
Provide an example of a polynomial in the vector space of polynomials with degree at most 20 that is not in the image of the second derivative operator.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^{20}}
Find the number of distinct conjugacy classes of subgroups of the symmetric group $S_5$ that are isomorphic to the Klein four-group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral \( \int_{0}^{\infty} \frac{\sin x - x \cos x}{x^2 + \sin^2 x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
$n$ is a integer and $a_1, a_2, \ldots, a_n\in[-1,1]$ are real numbers with $ \sum_{i=1}^{n}a_{i}=0$ ,try to find the maximum value of
$$ \sum_{1\leq i ,  j \leq n , i\ne j}|a_{i}-a^2_j|$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( n(n-1) \).

}
Determine the number of elements of order 6 in the automorphism group \( \text{Aut}(\mathbb{Z}_{720}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the value of $y$ that satisfies the equation: \[ y = \sqrt[4]{10000 - \frac{1}{10000 - \frac{1}{10000 - \cdots}}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine an approximate upper bound for the maximum number of balls in a bin when dropping \( n \log n \) balls into \( n \) bins randomly, ignoring distribution dependencies and integer constraints.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(\log n)}
Evaluate the integral \( \int_{-1}^{1} (4t^4 - 4t^2 - 1) e^{-{(t^2 -1)}^2} \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Calculate the residue of the function \( f(z) = \frac{1}{z^2\sin(\pi z)} \) at the triple pole \( z=0 \) without using the Laurent series expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{6}}
Find the smallest positive $m$ for which there are at least 11 even and 11 odd positive integers $n$ so that $\tfrac{n^3+m}{n+2}$ is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
m=8
}
Place 2005 points on the circumference of a circle. Two points \( P, Q \) are said to form a pair of neighbors if the chord \( P Q \) subtends an angle of at most \( 10^{\circ} \) at the center. Determine the smallest number of pairs of neighbors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110275}
Calculate the expected number of games played in a tennis match between two equally skilled players, where a player must win three consecutive games to win the match. Each game is independent, with each player having a 50% chance of winning any given game.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the number of codewords of weight 4 in a binary $[31,22,5]$ linear code, given that each weight 2 coset can contain at most one weight 4 codeword.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the field over which all 2x2 orthogonal matrices are diagonalizable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{C}}
Calculate the volume of the region in the first octant bounded by the planes \( x + y + z = 9 \), \( 2x + 3y = 18 \), and \( x + 3y = 9 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{81}{2}}
Consider a Markov Chain based random number generator with 5 points, starting at the 1st point. Each point has a 25% chance to move forward, a 25% chance to move backward, and a 50% chance to move to an adding point. The adding points add 1 to the final number and return to their connected point. The 1st point cannot move backward and instead returns to itself. If the generator moves forward from the 5th point, it reaches a final absorbing point, ending the process. Calculate the expected number of times the generator visits the adding points before reaching the final absorbing point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Calculate the limit: \[ \lim _{n\to \infty }\frac{2^{4n+1} \ln(2n^3+1)+n^5\cdot 10^n}{15^n+4^{2n-1} \ln (5n)} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Given the parity check matrix \(H=\begin{pmatrix} 2 & 0 & 1 & 1 & 1 \\ 0 & 2 & 2 & 0 & 0 \\ 0 & 0 & 2 & 2 & 1 \\ 1 & 1 & 2 & 2 & 0 \end{pmatrix}\), determine the minimum distance \(d(C)\) of the ternary linear code \(C\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the value of \(a\) such that the equation \(x^2 = a^x\) has exactly one real root.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\frac{2}{e}}}
Evaluate the integral \( \int_0^{2\pi} \cos(\cos(z))\cosh(\sin(z)) \, dz \) using a corollary of Cauchy's Integral Formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
How many 4-digit numbers exist such that deleting any one of its digits results in a 3-digit number that is a divisor of the original 4-digit number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Calculate the tensor product \( \mathbb{Q} / \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Using Taylor's theorem, determine the largest positive real value \( r \) for which the Maclaurin polynomial \( P_6(x) \) of \( f(x) = (\sin(x))^2 \) has an error of less than 0.2 on the interval \( 0 < x < r \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[8]{63}}
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \( f(x + y) + f(z) = f(x) + f(y + z) \) for all \( x, y, z \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = ax + b}
令 $a_{n}$ 表示距离 $\sqrt{n}$ 最近的正整数，其中 $n \in \mathbf{N}^{*}$ ，若 $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}= 2016$，求正整数 $n$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1017072}
In a right triangle \( ABC \) with \( AC = 16 \) and \( BC = 12 \), a circle with center at \( B \) and radius \( BC \) is drawn. A tangent to this circle is constructed parallel to the hypotenuse \( AB \) (the tangent and the triangle lie on opposite sides of the hypotenuse). The leg \( BC \) is extended to intersect this tangent. Determine by how much the leg is extended.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. The original answer is in the form \(\frac{k\sqrt{m} - n}{p}\), where \(k\), \(m\), \(n\), and \(p\) are integers. What is the value of \(k + m + n + p\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let \( K(x) \) be the number of irreducible fractions \( \frac{a}{b} \) such that \( a < x \) and \( b < x \) (where \( a \) and \( b \) are natural numbers).

For example, \( K\left(\frac{5}{2}\right) = 3 \) (the fractions are \( 1, 2, \frac{1}{2} \)).

Compute the sum \( K(100) + K\left(\frac{100}{2}\right) + K\left(\frac{100}{3}\right) + \ldots + K\left(\frac{100}{99}\right) + K\left(\frac{100}{100}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9801}
Let $a, b, c, d, e, f$ be integers selected from the set $\{1,2, \ldots, 100\}$, uniformly and at random with replacement. Set $M=a+2 b+4 c+8 d+16 e+32 f$. What is the expected value of the remainder when $M$ is divided by 64?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{63}{2}}
In a certain month, Mrs. Bil's earnings were a percentage of the Bil family's total income. In June, Mrs. Bil earned 10 percent more than in May. If the rest of the family's income was the same both months, then in June, Mrs. Bil's earnings were approximately 71.96 percent of the Bil family's total income. What was the percentage of Mrs. Bil's earnings in the first month?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70.00}
四个半径为 1 的球两两相切，则它们外切正四面体的棱长为多少？答案的形式为 $k(1+\sqrt{m})$，请给出 $k+m$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In a 200 x 200 square grid, each cell contains a real number with an absolute value not exceeding 1. The total sum of all numbers in the grid is zero. Determine the smallest value of \( S \) such that the sum of the numbers in at least one row or one column is necessarily not exceeding \( S \) in absolute value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
In an \(m \times 10\) grid, each cell is filled with either a 0 or a 1 such that each column contains exactly three 1s. Let the sum of the numbers in the \(i^{th}\) row be denoted by \(x_i\). Furthermore, for any two columns, there is always at least one row where both columns have a 1. Let \(x = \max_{1 \leq i \leq m} \{x_i\}\). Find the minimum value of \(x\), denoted by \(x_{\text{min}} =\) ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Two projective planes, each with two punctures, are joined along the two punctures. What is the Euler characteristic of the resulting surface?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
The school director, custodian, and parent committee each bought a carpet for the school assembly hall, which has dimensions $10 \times 10$. Without agreeing with each other, they decided to place all three carpets as follows: the first carpet of size $6 \times 8$ in one corner, the second carpet of size $6 \times 6$ in the opposite corner, and the third carpet of size $5 \times 7$ in one of the remaining corners (all measurements are in meters).

Find the area of the hall covered by carpets in three layers (provide the answer in square meters).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
If the function $f(x) = \ln{x} + ax^2 - 2x$ has two distinct extreme value points, find the range of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \in \left(0, \frac{1}{2}\right)}
Given the function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(f(x+y)) = f(x^2) + f(y^2)$, and $f(f(2020)) = 1010$, find the value of $f(2025)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{505}
On a plane, a line \(\ell\) and two circles \(c_1\) and \(c_2\) of different radii are given such that \(\ell\) touches both circles at point \(P\). A point \(M \neq P\) on \(\ell\) is chosen such that the angle \(Q_1MQ_2\) is maximized, where \(Q_1\) and \(Q_2\) are the points of tangency of the tangent lines drawn from \(M\) to \(c_1\) and \(c_2\), respectively, other than \(\ell\). Find the value of \(\angle PMQ_1 + \angle PMQ_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180^\circ}
Foma and Erema were traveling along a straight road to Moscow in a cart at a constant speed.

- At 12:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "82."
- At 13:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "71."
- At 15:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "46."

It is known that Erema rounded the distance to the nearest whole number each time, and if there were two nearest whole numbers, he could choose either.

At 16:00, Foma asked again: "How many versts to Moscow?" This time Erema gave an exact answer, without rounding. What was Erema's answer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34.5\ \text{versts}}
Positive integers \(a_1, a_2, \cdots, a_{2006}\) are such that the ratios \(\frac{a_1}{a_2}, \frac{a_2}{a_3}, \cdots, \frac{a_{2005}}{a_{2006}}\) are pairwise distinct. Determine the minimum number of distinct integers among \(a_1, a_2, \cdots, a_{2006}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Find a possible expression for the function $f(x)$ such that the following equation holds:

$$\int f(x)\sin(2x)dx=-\frac{1}{2}f(x)\cos(2x)+\int \frac{1}{4}\cos(2x)dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}x}
A set $D$ of positive integers is called *indifferent* if there are at least two integers in the set, and for any two distinct elements $x,y \in D$, their positive difference $|x-y|$ is also in $D$. Let $M(x)$ be the smallest size of an indifferent set whose largest element is $x$. Compute the sum $M(2) + M(3) + \dots + M(100)$. \(\text{Proposed by Yannick Yao}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1257}
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. If $1 \leqslant a_5 \leqslant 4$ and $2 \leqslant a_6 \leqslant 3$, then the range of values for $S_6$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-12, 42]}
Evaluate the integral: \[ \int_0^{\frac{\pi}{2}}\left(\frac{x}{\sin x}\right)^2\text{d}x \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \ln 2}
There are 3 cities, and roads must be constructed so that every pair of cities is connected. Assume that all the roads connecting any two cities have independent costs that are exponentially distributed with parameter one. Find the expected value of the minimal cost to connect all cities. That is, find \( E(c_3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{6}}
Yoongi wants to subtract 57 from a four-digit number. Yoongi has mistaken 9 for 6 in the unit place of a four-digit number. When the difference Yoongi gets is 1819, find the correct result.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1822}
Let $F_{1}(-c,0)$ and $F_{2}(c,0)$ be the two foci of the ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a > b > 0)$. Let $P$ be a point where the circle with diameter $F_{1}F_{2}$ intersects the ellipse. If $\angle PF_{1}F_{2}=2\angle PF_{2}F_{1}$, then the eccentricity of the ellipse equals \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt {3}-1}
Calculate: $108^\circ 18' 25'' - 56^\circ 23' 32'' =$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51^\circ 54' 53''}
4. Determine the smallest natural number $n$ with the following property: If we choose $n$ different natural numbers less than 2006, there are two among them such that the ratio of the sum and the difference of their squares is greater than three.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Given xyz = 1, then what is ( 5 ^ (x + y + z) ^ 2 ) / ( 5 ^ (x - y + 2z) )
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given that $x > 0$ and $y > 0$, and $2x + y = 2$, find the minimum value of $\frac{2}{x + 1} + \frac{1}{y}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{4}}
Given the joint density function \(f_{X,Y}(x,y)=\left\{ \begin{array}{ll} 2 & \mbox{when } x > 0 , y > 0 \text{ and } x+y<2 \\ 0 & \mbox{otherwise } \end{array} \right.\), determine the covariance of X and Y, \(\text{Cov}(X,Y)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{9}}
Let \( T : \mathbb{R}^4 \to \mathbb{R}^4 \) be a linear transformation such that there exists a vector \( v \in \mathbb{R}^4 \) for which \( v, T(v), T(T(v)) \) are linearly independent and \( T(T(T(v))) = 0 \). Determine the rank of the transformation \( T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the values of \(\beta\) for which the improper integral \[ \iint_{\Bbb R^2}\frac{dxdy}{(1+x^2-xy+y^2)^\beta} \] converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\beta > 1}
在直三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, $A B=1, B C=C C_{1}=\sqrt{3}, \angle A B C=90^{\circ}$, 点 $P$ 是平面 $A B C$上一动点, 则 $A_{1} P+\frac{1}{2} P C$ 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}
Given positive integers \( a, b, c, \) and \( d \) such that \( a > b > c > d \) and \( a + b + c + d = 2004 \), as well as \( a^2 - b^2 + c^2 - d^2 = 2004 \), what is the minimum value of \( a \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{503}
The quadrilateral \(ABCD\) is a rhombus with side length \(a\), and \(\angle BAD=60^\circ\). After folding along the diagonal \(BD\) to form a dihedral angle of \(120^\circ\) \(A-BD-C\), the distance between \(AC\) and \(BD\) is \(\quad\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{4} a}
Al told Bob that he was thinking of  $2011$  distinct positive integers. He also told Bob the sum of those  $2011$  distinct positive integers. From this information, Bob was able to determine all  $2011$  integers. How many possible sums could Al have told Bob?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find all natural numbers that are increased ninefold if a zero is inserted between the units digit and the tens digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
[Ratio of areas of triangles with a common angle] Auxiliary similar triangles $\quad]$

In triangle $A B C$, angle $C$ is $30^{\circ}$, and angle $A$ is acute. A line perpendicular to side $B C$ is drawn, cutting off triangle $C N M$ from triangle $A B C$ (point $N$ lies between vertices $B$ and $C$). The areas of triangles $C N M$ and $A B C$ are in the ratio $3: 16$. Segment $M N$ is half the height $B H$ of triangle $A B C$. Find the ratio $A H: H C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
A Moskvich car was sent to transport mail from the post office to the airfield. The plane carrying the mail landed earlier than expected, and the delivered mail was sent to the post office by a passing truck. After driving for 30 minutes, the truck met the Moskvich on the road, which received the mail and turned back immediately without delay. The Moskvich arrived at the post office 20 minutes earlier than usual. How many minutes earlier than the scheduled time did the plane land?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
A box contains 9 good products and 3 defective products. One product is taken out at a time, and it is not put back in the box after being taken out. What is the probability that 3 defective products have been taken out before 2 good products are taken out?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{55}}
Given the following equations: observe each equation: $a+b=1$, $a^{2}+b^{2}=3$, $a^{3}+b^{3}=4$, $a^{4}+b^{4}=7$, $a^{5}+b^{5}=11$, $...$, then according to this pattern, we can get $a^{6}+b^{6}=$ $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Positive integers $a$, $b$, $c$, $d$, and $e$ satisfy the equations:

1. $(a + 1)(3bc + 1) = d + 3e + 1$
2. $(b + 1)(3ca + 1) = 3d + e + 13$
3. $(c + 1)(3ab + 1) = 4(26 - d - e) - 1$

Find $d^2 + e^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{146}
Let $ABC$ be a triangle with incenter $I$ such that $AB = 20$ and $AC = 19$. Point $P \neq A$ lies on line $AB$ and point $Q \neq A$ lies on line $AC$. Suppose that $IA = IP = IQ$ and that line $PQ$ passes through the midpoint of side $BC$. Suppose that $BC = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3902}