inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the probability that a 4 × 4 square grid becomes a single uniform color (all white or all black) after rotation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{32768}}
Given that \(\det(A) = \frac{1}{24}\), find the value of \(\det\left(\left(\frac{1}{3}A\right)^{-1} - 120 A^*\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-192}
Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, 

\[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1998}
a and b are two multiples of a certain number, and q is the set of consecutive integers between a and b, inclusive. If q contains 10 multiples of that number, there are 19 multiples of 7 in q. What is that certain number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Using $n^{3}$ unit cubes, a cube with edge length $n$ is formed $(n \in \mathbf{N}, n \geqslant 2)$. A line is said to pass through a unit cube if it passes through any point inside the unit cube. What is the maximum number of unit cubes that a single line can pass through?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3n-2}
在平面直角坐标系中，甲从原点出发，到点 (4,4)；乙从点 (2,0) 出发，到点 (6,4). 已知两人同时出发，且每次向上或向右走一步. 则甲、乙的行走路线没有公共点的概率为__________.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{19}
Evaluate the limit: \[ I = \lim_{t\to0^{+}}\lim_{x\to+\infty}\dfrac{\displaystyle\int_{0}^{\sqrt{t}}dx\int_{x^2}^{t}\sin{y^2}dy}{\left[\left(\dfrac{2}{\pi}\arctan{\dfrac{x}{t^2}}\right)^x-1\right]\arctan{t^{\frac{3}{2}}}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{7}}
Two three-letter strings, $aaa$ and $bbb$ , are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a $1/3$ chance of being received incorrectly, as an $a$ when it should have been a $b$ , or as a $b$ when it should be an $a$ . However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let $S_a^{}$ be the three-letter string received when $aaa$ is transmitted and let $S_b^{}$ be the three-letter string received when $bbb$ is transmitted. Let $p$ be the probability that $S_a^{}$ comes before $S_b^{}$ in alphabetical order. When $p$ is written as a fraction in lowest terms , what is its numerator ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{532}
Let $n \geq 2$ be a positive integer. Each cell of an $n \times n$ board is colored red or blue. We place dominoes on the board, each covering two cells. We call a domino plain if it lies on two red or two blue cells, and colorful if it lies on one red and one blue cell. Find the largest positive integer $k$ with the following property: no matter how the red/blue coloring of the board is done, it is always possible to place $k$ non-overlapping dominoes on the board that are either all plain or all colorful.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\left\\lfloor\\frac{n^{2}}{4}\\right\\rfloor"]}
Determine the ring to which \( R[X,Y,Z,T]/(X-Y^2,T-Y^4,T^3-Z) \) is isomorphic after substituting \( X=Y^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R[Y]}
Let  $\triangle ABC$  be a triangle with side length  $BC= 4\sqrt{6}$ . Denote  $\omega$  as the circumcircle of  $\triangle{ABC}$ . Point  $D$  lies on  $\omega$  such that  $AD$  is the diameter of  $\omega$ . Let  $N$  be the midpoint of arc  $BC$  that contains  $A$ .  $H$  is the intersection of the altitudes in  $\triangle{ABC}$  and it is given that  $HN = HD= 6$ . If the area of  $\triangle{ABC}$  can be expressed as  $\frac{a\sqrt{b}}{c}$ , where  $a,b,c$  are positive integers with  $a$  and  $c$  relatively prime and  $b$  not divisible by the square of any prime, compute  $a+b+c$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
Evaluate the limit of the sum \( \sum_{n=1}^\infty \frac{\cos(n)}{n^k} \) as \( k \to 0^+ \) using the polylogarithm function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
A piece of iron wire with a length of $80cm$ is randomly cut into three segments. Calculate the probability that each segment has a length of no less than $20cm$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{1}{16}"]}
Consider the set of digits \( \{1, 2, 3, \ldots, 8\} \). How many permutations of these digits have the property that the product of any five consecutive digits is divisible by 5?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10080}
A train 110 m long traveling at 60 kmph takes 16.7986561075114 seconds to cross a bridge of a certain length. How long is the bridge in meters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{170}
Determine the structure of the quotient group $\mathbb{Z}^2/((m,n)\mathbb{Z})$, where $m$ and $n$ are integers greater than one and coprime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Let  $n\ge2$  be an integer and let  $x_1,x_2,\ldots,x_n$  be real numbers. Consider  $N=\binom n2$  sums  $x_i+x_j$ ,  $1\le i<j\le n$ , and denote them by  $y_1,y_2,\ldots,y_N$  (in an arbitrary order). For which  $n$  are the numbers  $x_1,x_2,\ldots,x_n$  uniquely determined by the numbers  $y_1,y_2,\ldots,y_N$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 3}
Determine the dimension of the cut space for an undirected graph, given that the cut space is a vector space over \( \mathbb{Z}_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - c}
Determine the number of homomorphisms from the group \(C_5 \times C_4 \times C_4\) onto \(C_{10}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A family of 5 members has an average age of some years. If the age of the youngest member is 10 years, and the average age of the family at the time of the birth of the youngest member was 12.5 years, what is the current average age of the family?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let $A=\mathbb{C}[x,y]/(x^3-y^2)$, and consider the maximal ideal $m=(\bar{x} - 1, \bar{y} - 1)$ of $A$. Determine the minimal number of generators, $\mu(mA_m)$, for the ideal $mA_m$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given a regular triangular pyramid \( PABC \) (with \( P \) as the apex) with a base side length \( a \) and lateral edge \( b \) ( \( b > a \)). A sphere lies above the plane of the base \( ABC \), touches this plane at point \( A \), and also touches the lateral edge \( PB \). Find the radius of the sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a(2b - a) \sqrt{3}}{2 \sqrt{3b^2 - a^2}}}
Find the value of $x$ that satisfies the equation $\log(x) = 2x$. Express your answer as a decimal rounded to three decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.278}
Given the graph of $y=\sqrt{2}\sin 3x$, calculate the horizontal shift required to obtain the graph of $y=\sin 3x+\cos 3x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{12}}
Calculate the line integral of the vector field \( F = (2x\sin(\pi y)-e^z, \pi x^2\cos(\pi y)-3e^z, -xe^z) \) along the curve \( C \) from \((0,0,0)\) to \((1,1,\ln(2))\), where the curve is defined by the conditions \( z=\ln(1+x) \) and \( y=x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{13}{2}}
Let $S_{k}=1^{k}+2^{k}+3^{k}+…+n^{k}$, when $k=1$, $2$, $3…$, observe the following equations:
$S_{1}= \frac {1}{2}n^{2}+ \frac {1}{2}n$, $S_{2}= \frac {1}{3}n^{3}+ \frac {1}{2}n^{2}+ \frac {1}{6}n$, $S_{3}= \frac {1}{4}n^{4}+ \frac {1}{2}n^{3}+ \frac {1}{4}n^{2}$,
$S\;\_{4}= \frac {1}{5}n^{5}+ \frac {1}{2}n^{4}+ \frac {1}{3}n^{3}- \frac {1}{30}n$, $S\_{5}= \frac {1}{6}n^{6}+An^{5}+Bn^{4}- \frac {1}{12}n^{2}$, $…$,
It can be inferred that $A-B=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {1}{12}}
Find the kernel of the derivative operator $D$ for polynomials in $F[x]$, where $F$ is a ring with characteristic $p > 0$. What is the set of polynomials that are mapped to zero by this operator?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F[x^p]}
Given that point $P$ is the intersection point of the lines $l_{1}$: $mx-ny-5m+n=0$ and $l_{2}$: $nx+my-5m-n=0$ ($m$,$n\in R$, $m^{2}+n^{2}\neq 0$), and point $Q$ is a moving point on the circle $C$: $\left(x+1\right)^{2}+y^{2}=1$, calculate the maximum value of $|PQ|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 + 2\sqrt{2}}
We have an empty equilateral triangle with length of a side $l$. We put the triangle, horizontally, over a sphere of radius $r$. Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of $l$ and $r$)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\sqrt{r^2 + \\frac{l^2}{4}}"]}
What is the computational complexity of finding the $n$-th derivative of $f(x)$, where $f(x)$ is defined as the product of two functions $g(x)$ and $g(1-x)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^2)}
Determine the order of growth for the binomial coefficient \( {}^n C_k \) when approximated by a polynomial. Consider the case where \( n \gg k \) and provide the general order of growth for \( {}^n C_k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n^k)}
What is the official name for an additive abelian group equipped with a multiplication that is distributive over addition but not necessarily associative?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{non-associative ring}
Determine the minimum degree \( n \) for which there exists a polynomial \( P(x) \) of degree not exceeding \( n \) such that the creative potential \( c_i = P(a_i) \) of all 6 selected candidates is strictly greater than that of the remaining 7 candidates, given that the scores \( a_1, a_2, \ldots, a_{13} \) are distinct.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
How many times can the trigonometric series $\sum_{k=1}^{\infty} \frac{\cos k x}{k^{4}}$ be differentiated term by term?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
12. Chocolate Coupons. A chocolate candy factory places a coupon in each box of candies. For every ten coupons collected, a customer receives a free box of candies. What fraction of the cost of a box does one coupon represent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{1}{9}"]}
Let \(f\) be a function defined on the interval \((a,b)\) such that \(f' + f^2 + 1 \ge 0\). Given that \(\lim_{x\to a}f(x)=\infty\) and \(\lim_{x\to b}f(x)=-\infty\), determine the possible interval for \(b-a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\pi, \infty)}
Evaluate the double integral \( \iint_{D} f(x,y) \, dA \) over the region \( D \) bounded by the lines \( x=0 \), \( y=0 \), and \( x+y=1 \), where \( f(x,y) \) is defined as:

\[f(x,y) = \begin{cases} x^2 & x \geq y \\ y^2 & y > x \end{cases}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{48}}
Evaluate the limit, if it exists:
\[\lim_{x\rightarrow\infty}\Delta^x_{k=1}\sin\left(\frac{k\pi}{2x}\right)\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the complexity of the recurrence relation \( T(n) = \sqrt{2n}T(\sqrt{2n} )+ n^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^2)}
Evaluate the limit of the integral: $$\lim_{n\to\infty}\int_0^{2n\pi}\frac{(n+x)\sin(x/n)}{x(1+x)^2}\,dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $(X_i, P_i)$, with $i$ being an integer, be an independent random variable such that $X_i|P_i$ ~ $Bernoulli(P_i)$ and $P_i$ ~ $Beta(\alpha, \beta)$. What is the variance of $Y = \sum_{i=1}^n X_i$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n \alpha \beta}{(\alpha+\beta)^2}}
Evaluate the limit using the Stolz-Cesàro theorem:
\[ \lim_{n\to \infty} \frac{2021(1^{2020}+2^{2020}+3^{2020}+\ldots+n^{2020}) - n^{2021}}{2021(1^{2019}+2^{2019}+3^{2019}+\ldots+n^{2019})} = L \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1010}
Provide an example of an absolutely convergent series \( \sum_{n=0}^\infty |r_n| \) with \( r_n \in \mathbb{Q} \) such that \( \sum_{n=0}^\infty r_n \) converges to a real number that is not rational, i.e., \( \sum_{n=0}^\infty r_n \in \mathbb{R} \setminus \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sum_{n=0}^\infty \frac{1}{n!}}
Calculate the line integral \( \int_{C} \vec{F} \cdot \vec{n} \, ds \) where \( \vec{F}(x, y) = x^{10} \vec{i} + (3x - 10x^9y) \vec{j} \) and \( \vec{n} \) is the unit normal vector to the curve \( C \). The curve \( C \) is defined by \( x^2 + y^2 = 1 \) and is restricted to the first quadrant. Provide the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
a, b, and c started a business with different capitals. At the end of the year, the profit share of b is Rs. 1800, and the difference between the profit shares of a and c is Rs. 720. If a invested Rs. 8000 and b invested Rs. 10000, how much did c invest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4000}
Evaluate the limit: \[ \lim_{x\to 0}\left(\frac{1}{(\sin^{-1}(x))^2} - \frac{1}{x^2}\right) \] without using L'Hôpital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{3}}
Find the value of \( y(\frac{1}{2}) \) for the function \( y(x) \) that minimizes the integral \( \int_{0}^{1} e^{-(y'-x)} + (1+y)y' \, dx \), given that \( y(0) = 0 \) and \( y(1) = \frac{1}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
There is a strip \( 1 \times 99 \) divided into 99 cells of size \( 1 \times 1 \), which are colored alternately in black and white. It is allowed to repaint all the cells of any checkerboard rectangle \( 1 \times k \) at the same time. What is the minimum number of repaintings required to make the entire strip a single color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Let $T: V \rightarrow V$ be a linear operator with the minimal polynomial $m(x) = (x-1)x^8$. Find the minimal polynomial of $T^3$. \( \text{Provide your answer as a polynomial.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x - 1)x^3}
Find the value of $a+b+c$ given the limit:
\[\lim_{x \to \infty}\frac{a(2x^3-x^2)+b(x^3+5x^2-1)-c(3x^3+x^2)}{a(5x^4-x)-bx^4+c(4x^4+1)+2x^2+5x}=1\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{58}{109}}
Let $ABCDEF$ be a regular hexagon and let point $O$ be the center of the hexagon. How many ways can you color these seven points either red or blue such that there doesn’t exist any equilateral triangle with vertices of all the same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6
}
If the one-variable quadratic equation $x^{2}-4x+1-t=0$ (where $t$ is a real number) has solutions in the range $0 \lt x \lt \frac{7}{2}$, then the range of values for $t$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3\leqslant t < 1}
Calculate the surface area generated by revolving the polar curve \( r = \sqrt{\cos 2\theta} \) around the y-axis, with the limits of integration from \( \theta = 0 \) to \( \theta = \frac{\pi}{4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \sqrt{2}}
Find the smallest natural number \( n \) such that the group \( \mathbb{Z}_{6} \times \mathbb{Z}_{20} \times \mathbb{Z}_{45} \) is isomorphic to a direct product of \( n \) cyclic groups.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. Let $D$, $E$, and $F$ be the feet of the altitude, angle bisector, and median from $A$ to $BC$, respectively. If $DE = 3$ and $EF = 5$, compute the length of $BC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Determine the dimension of the affine algebraic set \(X = V(f_1, f_2, f_3) \subseteq \mathbb{A}^6\), where:
\[f_1 = x_1x_5 - x_4x_2, \quad f_2 = x_1x_6 - x_4x_3, \quad f_3 = x_2x_6 - x_5x_3.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the supremum of \(|f(1)|\) for functions \(f\colon \Bbb C\setminus\{0\}\to \Bbb C\) that are analytic and satisfy \(|f(z)|\le 5|z|^{−\sqrt2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There are $n \geq 2$ coins, each with a different positive integer value. Call an integer $m$ sticky if some subset of these $n$ coins have total value $m$. We call the entire set of coins a stick if all the sticky numbers form a consecutive range of integers. Compute the minimum total value of a stick across all sticks containing a coin of value 100.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{199}
已知复数 $z$ 满足 $|z|=3$. 则 $\left|\frac{z^{2}+1}{z^{2}+\mathrm{i} z+z+\mathrm{i}}\right|$ 的最大值为原始答案的形式为\(\dfrac{m\sqrt{n}+\sqrt{k}}{q}\)，请给出m + n + k+q的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
Consider the system of equations

\[
8x - 6y = c,
\]
\[
12y - 18x = d.
\]

If this system has a solution \((x, y)\) where both \(x\) and \(y\) are nonzero, find the value of \(\frac{c}{d}\), assuming \(d\) is nonzero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{4}{9}}
Determine the smallest positive value of $x,$ in degrees, for which
\[\tan (x + 100^{\circ}) = \tan (x + 50^{\circ}) \tan x \tan (x - 50^{\circ}).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
Calculate the value of the Laplace operator \( \Delta\left(\frac{1}{r^2}\right) \) where \( r = |x| = \sqrt{x_1^2 + x_2^2 + x_3^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{r^4}}
A scientist begins an experiment with a cell culture that starts with some integer number of identical cells. After the first second, one of the cells dies, and every two seconds from there another cell will die (so one cell dies every odd-numbered second from the starting time). Furthermore, after exactly 60 seconds, all of the living cells simultaneously split into two identical copies of itself, and this continues to happen every 60 seconds thereafter. After performing the experiment for awhile, the scientist realizes the population of the culture will be unbounded and quickly shuts down the experiment before the cells take over the world. What is the smallest number of cells that the experiment could have started with?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
From 6 students of different heights, 5 are chosen to stand in a row from left to right for a photo. The requirement is that the students standing in even positions must be taller than those in adjacent odd positions. Calculate the total number of different possible photos that can be taken under this arrangement.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
A circle having radius \( r_1 \) centered at point \( N \) is tangent to a circle of radius \( r_2 \) centered at \( M \). Let \( l \) and \( j \) be the two common external tangent lines to the two circles. A circle centered at \( P \) with radius \( r_2 \) is externally tangent to circle \( N \) at the point at which \( l \) coincides with circle \( N \), and line \( k \) is externally tangent to \( P \) and \( N \) such that points \( M, N \), and \( P \) all lie on the same side of \( k \). For what ratio \( r_1 / r_2 \) are \( j \) and \( k \) parallel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the integral \( \oint_{|z| = 4} \frac{e^{1/(1 - z)}}{z^2} \, dz \) using the Residue Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( \{x\} \) denote the fractional part of \( x \). For example, \( \{5.5\} = 0.5 \). Find the smallest prime \( p \) such that the inequality \[ \sum_{n=1}^{p^2}\left\{\frac{n^p}{p^2}\right\} > 2016 \] holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Evaluate the line integral \( \int_C x \, dy - y \, dx \) where \( C \) consists of a straight line segment from \((-2, 0)\) to \((0, 0)\), another straight line segment from \((0, 0)\) to \((0, -2)\), and the counterclockwise portion of a circle with radius 2, centered at the origin, starting from \((0, -2)\) and ending at \((-2, 0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\pi}
Find the number of pairs \((n, C)\) of positive integers such that \(C \leq 100\) and \(n^2 + n + C\) is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
A natural number  $k > 1$  is called *good* if there exist natural numbers  $$ a_1 < a_2 < \cdots < a_k $$  such that  $$ \dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots +  \dfrac{1}{\sqrt{a_k}} = 1 $$  .

Let  $f(n)$  be the sum of the first  $n$  *[good* numbers,  $n \geq$  1. Find the sum of all values of  $n$  for which  $f(n+5)/f(n)$  is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Determine the minimum number of coin tosses needed to test if the probability of landing on heads is at most 0.05 with at least 95% confidence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59}
Given a convex figure in the Cartesian plane that is symmetric with respect to both axes, a rectangle $A$ is constructed inside it with maximum area. The rectangle is then enlarged with its center at the center of the rectangle and a ratio of lambda such that it covers the convex figure. Determine the smallest value of lambda that ensures this condition for all such convex figures.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit of the integral as \( n \to \infty \):

\[ I_n = \int_0^3 \sin\left(\frac{\pi}{3} \sin\left(\frac{\pi}{3} \sin \left(\frac{\pi}{3} \dots \sin\left(\frac{\pi}{3} x\right) \dots \right)\right)\right)dx \] 

where the sine function is nested \( n \) times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}