inclusionAI/Ring-lite-rl-data huggingface dataset:
Evaluate the integral \( \int_0^1 t^{\frac{1}{z}}(1-\ln t^{\frac{1}{z}})\,dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t}
Let $\triangle ABC$ be a right triangle at $A$ with circumcircle $\omega$. The tangents to $\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT = CT = 25$, $BC = 34$, and $TX^2 + TY^2 + XY^2 = 1975$. Find $XY^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{987.5}
For discrete probability distributions \( P \) and \( Q \) defined on the same probability space \( \mathcal{X} = \{0, 1, \ldots, n\} \), where \( P = [p, 0, \ldots, 0, 1-p] \) and \( Q \) follows a binomial distribution \( Q = \binom{n}{k} q^k (1-q)^{n-k} \) for all \( k \in \{0, 1, 2, \ldots, n\} \), determine the parameter \( q \) that minimizes the Kullback-Leibler divergence \( D_{\text{KL}}(P \| Q) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - p}
Given that $\tan (α+80^{\circ})=4\sin 420^{\circ}$, find the value of $\tan (α+20^{\circ})$. Express your answer as a fraction or decimal, including the square root symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{3}}{7}}
Let $f(x)=cx(x-1)$, where $c$ is a positive real number.  We use $f^n(x)$ to denote the polynomial obtained by composing $f$ with itself $n$ times.  For every positive integer $n$, all the roots of $f^n(x)$ are real.  What is the smallest possible value of $c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the values of $a$ for which the fixed point iteration converges to the root $\alpha_2 > 0$ of the equation $f_a(x) = 0$, where $f_a(x) = (1−a)x+ax^3$ and $\varphi(x) = ax(1 − x^2)$. Provide your answer as a set of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 2)}
Find an upper bound for the integral \( \int_{0}^{1} \frac{\cos x}{\sqrt{x}}\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{977}{540}}
A regular dodecagon \( P_1, P_2, \ldots, P_{12} \) is inscribed in a unit circle with center \( O \). Let \( X \) be the intersection of \( P_1 P_5 \) and \( OP_2 \), and let \( Y \) be the intersection of \( P_1 P_5 \) and \( OP_4 \). Let \( A \) be the area of the region bounded by \( XY \), \( XP_2 \), \( YP_4 \), and the minor arc \( \widehat{P_2 P_4} \). Compute \( \lfloor 120A \rfloor \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Compute $\frac{x^{10} - 32x^5 + 1024}{x^5 - 32}$ when $x=8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32768}
Determine the number of distinct colorings of the vertices of an 8-cycle graph using 10 colors, where each color can be used an unlimited number of times. Two colorings are considered identical if one can be transformed into the other by a symmetry of the cycle. Use Burnside's lemma to find the solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6278140}
The average height of 35 boys in a class was calculated as 180 cm. It was later found that the height of one of the boys in the class was wrongly written as 166 cm whereas his actual height was a certain height. The actual average height of the boys in the class is 178 cm (rounded to two decimal places). What is the actual height of the boy whose height was wrongly written?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{236}
There are 3 teachers who have all assigned homework. Determine the number of possible situations where 4 students are doing homework at the same time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{4}}
Given the joint probability density function \(f_{X,Y}(x,y)=\begin{cases}12(y-x^2),&0<x<y<1\\0,&\text{elsewhere.}\end{cases}\), compute the probability \(P(Y > 1/2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{41}{48}}
Let \( T': C_0(\mathbb{R}) \rightarrow C_0(\mathbb{R}) \) be defined by \( T'(\psi)(x) = \phi(x)\psi(x) \), where \( \phi \in C([0,1]) \) is fixed. Find \( ||T'|| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\|\phi\|_\infty}
In triangle \(ABC\), point \(N\) lies on side \(AB\) such that \(AN = 3NB\); the median \(AM\) intersects \(CN\) at point \(O\). Find \(AB\) if \(AM = CN = 7\) cm and \(\angle NOM = 60^\circ\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{7}}
Determine the number of solutions to the equation 10 sin(x + π/6) = x within the bounds -10 ≤ x ≤ 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate the limit \( \lim_{x \to 0} \frac{[1-F_X(x)]^2}{f_X(x)} \) using L'Hopital's Rule, where \( F_X(x) \) is the cumulative distribution function and \( f_X(x) \) is the probability density function of a random variable \( X \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
If $A$ is a permutation matrix, what is the rank of $A+A^2+A^3+...+A^n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Four friends — Alex, Betty, Clara, and Dave — participated in a relay race by running in pairs, with one pair sitting out each race. Dave ran in 8 races, which was more than any other friend, and Betty ran in 3 races, which was fewer than any other friend. Determine the total number of races those pairs completed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given that $α∈[\dfrac{π}{2}, \dfrac{3π}{2}]$, $β∈[-\dfrac{π}{2}, 0]$, and the equations $(α-\dfrac{π}{2})^{3}-\sin α-2=0$ and $8β^{3}+2\cos^{2}β+1=0$ hold, find the value of $\sin(\dfrac{α}{2}+β)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{2}}{2}}
How many irreducible fractions with a numerator of 2015 are there that are less than \( \frac{1}{2015} \) and greater than \( \frac{1}{2016} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1440}
Let $n$ be a positive integer. Find the smallest positive integer $k$ such that for any set $S$ of $n$ points in the interior of the unit square, there exists a set of $k$ rectangles such that the following hold:
The sides of each rectangle are parallel to the sides of the unit square.
Each point in $S$ is not in the interior of any rectangle.
Each point in the interior of the unit square but not in $S$ is in the interior of at least one of the $k$ rectangles
(The interior of a polygon does not contain its boundary.)

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n+2
}
Derive an upper bound for \(|P(\min_i X_i>c)-P(\min_i Y_i>c)|\) in terms of \(\alpha\), given that \(|P(X_i>c)-P(Y_i>c)| \leq \alpha\) for fixed \(c\) and random variables \(X_i\) and \(Y_i\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n\alpha}
Let $F$ be a finite field of order $q$. Consider $U(3,F)$, the multiplicative group of $3 \times 3$ upper-triangular matrices with entries from $F$ and all diagonal entries equal to 1. Determine whether the number of distinct conjugacy classes of $U(3,F)$ is a polynomial in $q$ with integer coefficients.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q^2 + q - 1}
For a positive integer $K$, define a sequence, $\{a_n\}$, as following: $a_1 = K$ and
$a_{n+1} =a_n -1$ if $a_n$ is even
$a_{n+1} =\frac{a_n - 1}{2}$ if $a_n$ is odd ,  for all $n \ge 1$. 
Find the smallest value of $K$, which makes $a_{2005}$ the first term equal to $0$.




Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{1003} - 2}
A garrison of 2000 men has provisions for 120 days. At the end of 25 days, a reinforcement arrives, and it is now found that the provisions will last only for 35 days more. What is the reinforcement?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3429}
There were 3 candidates in an election and they received 3136, 7636, and some votes respectively. The winning candidate got 51.910714285714285% of the total votes in that election. How many votes did the winning candidate receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7636}
No matter how $n$ real numbers on the interval $[1,2013]$ are selected, if it is possible to find a scalene polygon such that its sides are equal to some of the numbers selected, what is the least possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
13
}
等差数列$\left\{ {{a}_{n}} \right\}$的公差$\textit{d}$≠0，$\textit{a3}$是$\textit{a}_{2}$，$\textit{a5}$的等比中项，已知数列$\textit{a}_{2}$，$\textit{a4}$，${{a}_{{{k1}}}}$，${{a}_{{{k2}}}}$，……，${{a}_{{{kn}}}}$，……为等比数列，数列$\left\{ {{k}_{n}} \right\}$的前$\textit{n}$项和记为$\textit{T$_{n}$}$，则2$\textit{T$_{n}$}$＋9=_______
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{${{3}^{n+2}}+2n$}
A group of some adults and 70 children go for trekking. There is a meal for either 70 adults or 90 children, and 42 adults have their meal. The total number of children that can be catered with the remaining food is 36. How many adults went for trekking?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
A segment \( AB \) of unit length, which is a chord of a sphere with radius 1, is positioned at an angle of \( \pi / 3 \) to the diameter \( CD \) of this sphere. The distance from the end \( C \) of the diameter to the nearest end \( A \) of the chord \( AB \) is \( \sqrt{2} \). Determine the length of segment \( BD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Let \( h(x) = \int_{-x}^x f(t) \, dt \) where \( f \) is an even function. Given that \( h(x-3) = f(x) \), find the value of \( \int_{-x}^{x-3} h(t) \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a Hilbert space, consider open sets $A_1, B_1, A_2,$ and $B_2$ such that $A_1 \cap B_1 = A_2 \cap B_2 = \emptyset$ and $A_1 \cup B_1 = A_2 \cup B_2$. Furthermore, both $A_1$ and $A_2$ contain the origin (0) and are connected. Determine if $A_1 = A_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A_1 = A_2}
Determine the greatest common divisor (gcd) of all numbers of the form $p^8 - 1$, where $p$ is a prime number greater than 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{480}
Two solid bodies rotate about stationary, mutually perpendicular, and intersecting axes with constant angular velocities \(X\) and \(Y\). Determine the angular acceleration of one body relative to the other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{XY}
Consider the linear operator \( T: V \to V \) defined by differentiation, where \( V = \{p(x) \in F[x] \ | \ \deg(p(x)) \le n\} \) is the set of polynomials over a field \( F \) with degree at most \( n \). Determine if this operator has any eigenvalues other than \( \lambda = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given \(a, b, c \in \mathbf{R}^{+}\) (positive real numbers) such that \(a b c + a + c = b\), determine the maximum value of \(p = \frac{2}{a^{2}+1} - \frac{2}{b^{2}+1} + \frac{3}{c^{2}+1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{3}}
A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2, 3, 4}
Determine whether the dimension of any open subset \( U \) of \( \mathbb{R}^n \) is equal to \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
A satellite of mass 'm' revolves around Earth in a circular orbit of radius $r_0$ with velocity $v_0$. At a certain point, its direction changes by an angle $	heta = \cos^{-1}\left(\frac{3}{5}\right)$, while maintaining its speed. This change causes the satellite to enter an elliptical orbit. Calculate the ratio of the speed at the perigee to the speed at the apogee of this elliptical orbit. Use the following formulas for elliptical orbit velocities:

$$V_{\text{max}} = \sqrt{\frac{GM}{a} \frac{1+e}{1-e}}$$
$$V_{\text{min}} = \sqrt{\frac{GM}{a} \frac{1-e}{1+e}}$$

Determine how the eccentricity $e$ can be calculated using the angle $\theta$. Provide the ratio of the speed at the perigee to the speed at the apogee.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine whether the inequality \( S = \sum_{i,j=1}^n \frac{a_i a_j}{i+j} \ge 0 \) holds for any sequence \((a_1, a_2, \ldots, a_n) \in \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S \geq 0}
A die is initially placed with the number 6 on top. It is then turned randomly to one of the four adjacent faces. Let $P_n$ be the probability that after $n$ such turns, the die will again show the number 6 on top. Determine the limit $\lim_{n \to \infty} P_n$. \( \text{Express your answer as a single value.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Consider the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$. Its right focus and one endpoint of its conjugate axis are $F$ and $A$, respectively. Let $P$ be a point on the left branch of the hyperbola $C$. If the minimum value of the perimeter of $\triangle APF$ is $6b$, then find the eccentricity of the hyperbola $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{85}}{7}}
Nicky draws a regular 15-gon, 21-gon, and 35-gon such that all vertices of all three polygons lie on the same circle. Let $n$ be the number of distinct vertices on the circle. Find the sum of all possible values of $n$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{57}
Find the number of real solutions to $\sin 6 \pi x = x.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A horizontal cylindrical tank has a diameter of 3 feet. The tank is filled with gas, and a dipstick is used to measure the gas level. Determine the height on the dipstick that corresponds to the tank being 3/8 full.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.20}
What statement directly negates the proposition "There are at most two solutions"? Provide your answer as a complete sentence that logically contradicts the original statement.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{There are at least three solutions.}
In a bicycle race between two competitors, let $Y(t)$ denote the amount of time (in seconds) by which the racer that started in the inside position is ahead when $100t$ percent of the race has been completed, where $0 \leq t \leq 1$. The process $Y(t, 0 \leq t \leq 1)$ can be modeled as a Brownian motion with variance parameter $\sigma^2$. If the inside racer is leading by $\sigma$ seconds at the midpoint of the race, what is the probability that she is the winner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Phi(\sqrt{2})}
We defined an operation denoted by $*$ on the integers, which satisfies the following conditions: 

1) $x * 0 = x$ for every integer $x$;
2) $0 * y = -y$ for every integer $y$;
3) $((x+1) * y) + (x * (y+1)) = 3(x * y) - x y + 2 y$ for every integer $x$ and $y$.

Determine the result of the operation $19 * 90$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1639}
Peter and Andrew like to run in the morning. Peter runs some miles more than Andrew's 2 miles. After 5 days, they have both run 35 miles. How many more miles does Peter run than Andrew each day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine if the infinite sum \( \frac{1}{20}\sum_{n=1}^{\infty}\left(n^2+n\right)\left(\frac{4}{5}\right)^{n-1} \) can be simplified further.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{25}{2}}
Given that $F_1$ and $F_2$ are the left and right foci of a hyperbola $E$, and point $P$ is on $E$, with $\angle F_1 P F_2 = \frac{\pi}{6}$ and $(\overrightarrow{F_2 F_1} + \overrightarrow{F_2 P}) \cdot \overrightarrow{F_1 P} = 0$, determine the eccentricity $e$ of hyperbola $E$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3} + 1}{2}}
Evaluate the improper integral $$\int_{-\infty}^{\infty}\frac{\cos(x)}{4x^2-\pi^2}dx$$ This integral is from a Kyoto University exam and can potentially be solved using techniques like the Residue Theorem or by decomposition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}}
A reaction in which a carboxylic acid reacts with an alcohol to form an organic compound and water is called
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
esterification
}
In an equilateral triangle divided into smaller equilateral triangles with side length 1 by lines parallel to its sides, each side is divided into $n$ parts. Determine the maximum number of segments of length 1 that can be marked such that no triangle has all its sides marked. Assume $n=5$ in the given figure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Consider a constrained random walk starting at 0, where each minute the value either increases or decreases by 1, but cannot go below 0. Determine if the expected value of this walk is infinite if it continues indefinitely.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
The area of a triangle ABC is 40 cm^2. Points D, E, and F are on sides AB, BC, and CA respectively. If AD = 3 cm, DB = 5 cm, and the area of triangle ABE is equal to the area of quadrilateral DBEF, find the area of triangle AEC in cm^2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $f: \Bbb R \rightarrow \Bbb R$ be a differentiable function with $f'$ uniformly continuous. Suppose $\lim\limits_{x\rightarrow \infty}f(x)=L$ for some $L$. Does $\lim\limits_{x\rightarrow \infty}f'(x)$ exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( T, U, V, W \) be vector spaces over a field \( \mathbb{F} \), and let \( \alpha: T \to U \) and \( \beta: V \to W \) be fixed linear maps with ranks \( r \) and \( s \) respectively. If these spaces are finite-dimensional, determine the rank of the linear map \( \Phi: \mathcal{L}(U,V) \to \mathcal{L}(T,W) \) defined by \( \Phi(\theta) = \beta \circ \theta \circ \alpha \). Express your answer in terms of \( r \) and \( s \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{rs}
Given the function $g$ defined by the table and given that $v_0=3$ and $v_{n+1} = g(v_n + 1)$ for $n \ge 0$, find $v_{2004}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of nonisomorphic magmas of order 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3330}
设 $m$ 是正整数, $n=2^{m}-1$, 数轴上 $n$个点所成的集合为 $P_{n}=\{1,2, \cdots, n\}$ 。
一只蚱蜢在这些点上跳跃，每步从一个点跳到与之相邻的点. 求 $m$ 的最大值, 使得对任意 $x 、 y \in P_{n}$, 从点 $x$ 跳 2012 步到点 $y$ 的跳法种数为偶数（允许中途经过点 $x 、 y$ ）。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
5. Let the set $I=\{1,2, \cdots, 2020\}$. We define
$$
\begin{array}{l}
W=\{w(a, b)=(a+b)+a b \mid a, b \in I\} \cap I, \\
Y=\{y(a, b)=(a+b) \cdot a b \mid a, b \in I\} \cap I, \\
X=W \cap Y,
\end{array}
$$

as the "Wu" set, "Yue" set, and "Xizi" set, respectively. Find the sum of the largest and smallest numbers in the Xizi set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020}
Given that $\angle A , \angle B , \angle C$ are the internal angles of the triangle $ABC$, and the vector $\boldsymbol{\alpha}=\left(\cos \frac{A-B}{2}, \sqrt{3} \sin \frac{A+B}{2}\right)$ with $|\boldsymbol{\alpha}|= \sqrt{2}$. If when $\angle C$ is maximized, there exists a moving point $M$ such that $|\overrightarrow{M A}|$, $|\overrightarrow{A B}|$, $|\overrightarrow{M B}|$ form an arithmetic sequence, what is the maximum value of $\frac{|\overrightarrow{M C}|}{|\overrightarrow{A B}|}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{3} + \sqrt{2}}{4}}
Let $X$ be a random variable with an absolutely continuous cumulative distribution function $F(x)$. Determine the distribution of the random variable $Y = \log(1 - F(X))$. Provide the probability density function of $Y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^y}
A clock has an hour hand of length 3 and a minute hand of length 4. From 1:00 am to 1:00 pm of the same day, find the number of occurrences when the distance between the tips of the two hands is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
Given the ideal $I$ in $\mathbb{C}[x,y]$ generated by $y^2-x^2$ and $y^2+x^2$, determine the dimension of the vector space $\mathbb{C}[x,y]/I$ over $\mathbb{C}$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $a$, $b$, $c$, and $d$ be positive real numbers such that $abcd=17$. Let $m$ be the minimum possible value of \[a^2 + b^2 + c^2 + a(b+c+d) + b(c+d) + cd.\] Compute $\lfloor 17m \rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{594}
Evaluate the double integral \( \iint_D 2x \cos(2(x-y)x) \, dx \, dy \), where \( D \) is the rectangle with corners at \((\pi/6, 2+\pi/6), (\pi/6, 4+\pi/6), (\pi/4, 4+\pi/4), \text{ and } (\pi/4, 2+\pi/4)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{5}{16}}
Solve the following equation:

$$
1.2065^{x}+1.2065^{x+1}+1.2065^{x+2}+1.2065^{x+3}=1.2065^{10}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x\approx1}
Determine whether the limit \( \lim_{y\rightarrow\infty}\left(\zeta\left(\frac{1}{2}+iy\right) + \frac{1}{2}\left(\frac{1}{y/\pi}\right)^{1/2+iy} - \sum_{n=1}^{[y/\pi]}\frac{1}{n^{1/2+iy}}\right) \) converges to 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The first and second terms of the Fibonacci sequence are both 1. Starting from the third term, each term is the sum of the two preceding terms. Numbers appearing in the Fibonacci sequence are called Fibonacci numbers. In how many different ways can 121 be expressed as the sum of distinct Fibonacci numbers? (The same addends in a different order are considered the same way.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Consider the set of all four-digit rising numbers using the digits 1 through 7. Find the digit that the 35th number in the list from smallest to largest does not contain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}