inclusionAI/Ring-lite-rl-data huggingface dataset:
Given that the composition of linear transformations $T_1: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $T_2: \mathbb{R}^m \rightarrow \mathbb{R}^m$ is bijective, find the ranks of $T_1$ and $T_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
A stock's price follows a model where, if the current price is \( s \), after one period, it will be either \( us \) with probability \( p \) or \( ds \) with probability \( 1 - p \). Assuming successive movements are independent, approximate the probability that the stock's price will be up at least 30% after the next 1000 periods, given \( u = 1.012 \), \( d = 0.990 \), and \( p = 0.52 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.999}
What is the maximum number of real zeros that a polyquaral with a degree of 5 and 3 nested square roots can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let \( f \) be a function analytic on the annulus \( \{ z \in \mathbb{C} : 1/2 < |z| < 2 \} \) except for a simple pole at \( z = 1 \). The residue of \( f \) at \( z = 1 \) is 1. Let \( \sum_n a_n z^n \) and \( \sum_n b_n z^n \) be the Laurent series expansions of \( f \) on the annuli \( \{ z : 1/2 < |z| < 1 \} \) and \( \{ z : 1 < |z| < 2 \} \), respectively. Compute \( b_n - a_n \) for every integer \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Carol stands at a flag pole that is $21$ feet tall. She begins to walk in the direction of the flag's shadow to say hi to her friends. When she has walked $10$ feet, her shadow passes the flag's shadow. Given that Carol is exactly $5$ feet tall, how long in feet is her shadow?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{25}{8}$}
Determine the distribution of the random variable \( W = \sqrt{\sum_{i=1}^{n} X_i^2} \), where \( X_1, X_2, \ldots, X_n \) are independent and identically distributed as \( \mathcal{N}(0,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\chi(n)}
Calculate \( E(B_{1}^2B_{2}B_{3}) \), where \( B_{i} \) represents Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the value of the following expression: \[ \frac{\int_0^\pi x^3\ln(\sin x)\,dx}{\int_0^\pi x^2\ln(\sqrt{2}(\sin x))\,dx} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{2}}
On a circular board, five numbers $2, 0, 1, 9, 0$ are written in the order specified clockwise (the last zero is written next to the first two). At each step, the sum of each pair of neighboring numbers is inserted between them. After 5 steps, calculate the difference between the sum of all numbers from the first zero (initially between 2 and 1) to the second zero (initially between 9 and 2) when traversing the circle clockwise, and the sum of all the other numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1944}
Find the determinant of the weighted orthogonal projection matrix $J = A \left( A^\intercal W A \right)^{-1} A^\intercal W$, where $A$ is a non-square matrix, and $W$, $J$, and $A^\intercal W A$ are square matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
For the series \( e^{\tan x} \), estimate the series by finding the best integer value \( k \) such that:
\[ e^{\tan x} = 1 + x + \frac{x^2}{2!} + o(x^k) \]
as \( x \to 0 \). Determine the largest integer \( k \) for which this statement holds true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the characteristic polynomial of a matrix \( A \) as \( p(\lambda) = (\lambda+1)(\lambda-2)^2(\lambda+3)^2 \), find \( \det(A^{-1}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{36}}
Suppose \( x,y,z \) , and \( w \) are positive reals such that
 \( {x}^{2} + {y}^{2} - \frac{xy}{2} = {w}^{2} + {z}^{2} + \frac{wz}{2} = {36} \)
 \( {xz} + {yw} = {30}. \)
 Find the largest possible value of \( {\left( xy + wz\right) }^{2} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
960
}
Ana and Banana play a game on a $1 \times 2024$ board, with squares initially painted white. Ana starts the game. Each move by Ana consists of choosing any $k \leq m$ white squares on the board and painting them all green. Each move by Banana consists of choosing any sequence of consecutive green squares and painting them all white. Determine the smallest positive integer $m \leq 2024$ for which Ana can guarantee that, after one of her moves, the entire board will be painted green.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1013}
A flagpole is originally5meters tall. A hurricane snaps the flagpole at a point \( x \) meters above the ground so that the upper part,
 still attached to the stump,touches the ground 1 meter away from the base. What is \( x \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2.4
}
Evaluate \( \alpha^2(\alpha^2+\alpha+1) \) in the Galois field \( GF(27) \), where \( \alpha \) is a root of the irreducible polynomial \( x^3+2x+1 \) over \( \mathbb{Z}_3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\alpha^2 + 2}
Determine the fundamental period of the continuous time signal given by:
\[ x(t) = \sum_{n=-\infty}^{\infty}e^{-(2t-n)}u(2t-n) \] where \( u(t) \) is the unit step function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Compute the limit \( \lim_{x\rightarrow\infty}\frac{x}{\overline{F}_{X}(x)}\int_{x}^{\infty}\overline{F}_{X}(s)ds \), where \( X\sim\text{Norm}(0,1) \) is a standard normally distributed random variable and \( \overline{F}_{X}(x) = 1 - F_{X}(x) \) is the survival function of \( X \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $(B_t)_{t \ge 0}$ be a standard Brownian motion. Define the stopping time \( T := \inf\{t \ge 0 : B_t = 5t - 2\} \). Calculate \( E[T] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{5}}
A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example, $28$ is a perfect number because $1 + 2 + 4 + 7 + 14 = 28$. Let $n_i$ denote the ith smallest perfect number. Define$$f(x) =\sum_{i|n_x}\sum_{j|n_i}\frac{1}{j}$$(where $\sum_{i|n_x}$ means we sum over all positive integers $i$ that are divisors of $n_x$). Compute $f(2)$, given there are at least $50 $perfect numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A regular triangular prism $ABC A_{1} B_{1} C_{1}$ with base $ABC$ and lateral edges $A A_{1}, B B_{1}, C C_{1}$ is inscribed in a sphere of radius 3. Given that segment $CD$ is a diameter of this sphere and $A D = 2 \sqrt{6}$, find the volume of the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\sqrt{15}}
Let \( \{z_1, z_2, z_3, z_4, z_5\} \) be the solutions of the equation \( z^5 = 2i \). Calculate the value of \( x^2 = \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} + \frac{1}{z_4} + \frac{1}{z_5} + i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i}
Given the joint moment generating function \(M_{X,Y}(t,u) = \exp[2t+3u+t^2+atu+2u^2]\), determine the value of \(a\) that makes the random variables \(X + 2Y\) and \(2X - Y\) independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Given the sequence defined by \(x_{n+1} = 3x_n + \sqrt{8x^2_n + 2}\), if it is known that \(x_{2017} + x_{2023} = 990\), find the value of \(x_{2020}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the fraction of permutations of the letters A through G that, when inserted into an initially empty binary search tree, will produce the same structure as the binary search tree derived from the sequence A, E, F, G, B, D, C.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{504}}
Let \( f: \Bbb{N} \to \Bbb{N} \) be a strictly increasing function such that \( f(f(n)) = 3n \) for all \( n \in \Bbb{N} \). Find \( f(2001) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3816}
Determine an upper bound for the tree width of an \(n \times n\) grid graph using the elimination method, where vertices are removed and their neighbors are connected.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Determine the number of distinct Jordan forms for a $6 \times 6$ matrix with the characteristic polynomial $(x+2)^4 (x-1)^2$, considering the ordering of both eigenvalues and blocks.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Let \( x = e^{2ir\pi/7} \) where \( r \equiv 0, \pm1, \pm2, \pm3 \pmod{7} \). Evaluate the sum: 
\[ \sum_{r=1}^6 \frac{x^r}{1+x^{2r}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
Seven thieves steal diamonds and hide in the jungle. During the night, two thieves try to divide the diamonds between themselves but find one extra diamond. They wake a third thief, but still have one extra diamond. This pattern continues until they wake the seventh thief, at which point the diamonds are evenly distributed. How many diamonds did they steal in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{301}
Evaluate \(x + 2y + 3z\) given that the following limit is a finite non-zero number:

\[\lim_{h \to 0}\frac { \int _{ 2 }^{ h+2 }{ g(t)dt-h(xg(2)+yg(2+h/3)+zg(2+h)) } }{ { h }^{ 4 } }\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{4}}
Evaluate the line integral \( \int_{C} f \, dl \) along the curve \( C \), where \( C \) is the intersection of the paraboloid \( z = x^2 + y^2 \) and the plane \( 2x + 2y - z + 2 = 0 \). The curve starts at point \((3,1,10)\) and ends at \((1,3,10)\). The function is given by \( f(x,y,z) = z - y^2 - 2x - 1 \). The projection of \( C \) in the \( xy \)-plane is the circle \((x-1)^2 + (y-1)^2 = 4\), and the parametric representation of \( C \) is:
\[
\begin{split}
x &= 1 + 2\cos t \\
y &= 1 + 2\sin t \\
z &= 6 + 4\cos t + 4\sin t
\end{split}
\]
with \( t \) ranging from the angle corresponding to point \((3,1,10)\) to the angle corresponding to point \((1,3,10)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the distance from the vector \((1,0,0,1)\) to the subspace \(L = (\text{Sp}\{(1,0,2,1),(3,1,4,4),(1,1,0,2)\})^{\perp}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{14}}{3}}
Determine the maximum distance between any two norms on \(\mathbb{R}^n\) and a Euclidean norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{n}}
Find the kernel of the homomorphism \( A[Y] \to A_a \) defined by sending \( Y \) to \( 1/a \), where \( a \) is an element of \( A \). Determine the ideal \( I \) in \( A[Y] \) that forms the kernel of this homomorphism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(aY - 1)}
Evaluate the integral: $$\int_1^{+\infty}\frac{e^{2^u}-e^{-2^u}}{2u\ln(2)}\,\d u$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Compute the side length of the largest cube contained in the region\n\n\\[ \\{(x, y, z) : x^2+y^2+z^2 \\le 25 \\text{ and } x, y \\ge 0 \\} \\]\n\nof three-dimensional space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
10/3
}
Let $a,b,c,d,e$ be nonnegative integers such that $625a+250b+100c+40d+16e=15^3$.  What is the maximum possible value of $a+b+c+d+e$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{153}
Carol was sending out birthday invitations to her friends. Each package of invitations she bought had 5 invitations in it and she bought 3 packs. If she wants to invite 23 friends, how many extra invitations will she need to buy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A necklace consists of 100 blue and several red beads. It is known that every segment of the necklace containing 8 blue beads contain also at least 5 red beads. What minimum number of red beads can be in the necklace?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72
}
Find the maximum error bound for the Taylor polynomial approximation of the function \( f(x) = x\sin(3x) \) at \( a = 0 \) with \( n = 4 \) over the interval \(-0.7 \le x \le 0.7\). The fourth-degree Taylor polynomial is given by \( T_4(x) = 3x^2 - \frac{9x^4}{2} \). Use the fifth derivative of \( f(x) \), which is \( 81(3x\cos(3x) + 5\sin(3x)) \), to find the error bound \( |R_4(x)| \le \) some error value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.587}
Identify a nonabelian group that contains an element which is its own inverse and is not the identity element. Provide an example without using the dihedral group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_3}
Find the sum of all positive integers $B$ such that $(111)_B = (aabbcc)_6$, where $a, b, c$ represent distinct base $6$ digits, with $a \neq 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{237}
In an election with five candidates using a ranked-choice voting system, Candidate A secured 38% of the first-choice votes, Candidate B had 28%, and the remaining candidates (C, D, and E) each secured 11%, 15%, and 8% respectively. If after the first round, the candidate with the fewest votes (Candidate E) was eliminated and their votes were redistributed according to voters' second choices, Candidates A and B received an additional 5% each, while candidate D secured another 2% of the redistributed votes. In the end, the winning candidate had a total of 7,320 votes. Assuming no votes were spoiled, calculate the total number of valid votes cast in the election.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17,023}
A group of students decided to collect as many paise from each member of the group as is the number of members. If the total collection amounts to Rs. 48.75 and it is known that 20% of the group members contributed an additional amount equivalent to half the number of members, calculate the number of members in the group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Example 4 Given that $E$ is the midpoint of side $AB$ of quadrilateral $ABCD$, $BC=CD=CE$, $\angle B=75^{\circ}$, $\angle D=90^{\circ}$. Find the degree measure of $\angle DAB$. ${ }^{[4]}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Let $S_n(R)$ and $V_n(R)$ denote the surface area and volume of an $n$-sphere with radius $R$ respectively. It is well known that $$S_n(R) = \frac{n \pi^{\frac{n}{2}}R^{n-1}}{\Gamma\left(\frac{n}{2}+1\right)}\quad\text{and}\quad V_n(R) = \frac{\pi^{\frac{n}{2}}R^{n}}{\Gamma\left(\frac{n}{2}+1\right)}$$ where $\Gamma(z)$ is the gamma function: $$\Gamma(z) = \int_0^\infty e^{-x} x^{z-1}\, dx.$$ Let $v$ and $s$ be positive integers such that $S_s(1)$ and $V_v(1)$ are the maximum. What is $v-s?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Find the maximum value of $p$ for natural numbers $m$, $n$, and $p$ that satisfy the equation \(2mnp=(m+2)(n+2)(p+2)\) and the inequality \(m \leq n \leq p\), given that \(m\) cannot be equal to 1 or 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130}
In projective space $\mathbb{P}^n$, determine the minimum possible dimension of the intersection of a subspace of dimension $p$ and a subspace of dimension $q$, given that $p + q < n$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
In an isosceles triangle $ABC$, where $A$ is the top vertex, $B$ is the right vertex, and $C$ is the left vertex, it is given that $AC = AB$. The triangle is inscribed in a circle with center $O$. The diameter $AD$ passes through $O$ and intersects $CB$ at $E$. If $AC = \sqrt{\frac{15}{2}}$ and $OE = 1$, find the radius of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Find the area of the portion of the conical surface defined by the equation \(x^2 + y^2 = z^2\) that is bounded between the planes \(z = 0\) and \(x + 2z = 3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{6}\pi}
Calculate the fraction of a flux within a $20$ degree angle range ($0.111\pi$ radians) for a flux distribution given by $I(\phi) = I_0 \cos^2(\phi)$, centered at $\phi = 0$. The flux fraction is the ratio of the integrals $\int_{-\theta}^{\theta} I(\phi) \, d\phi$ and $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} I(\phi) \, d\phi$, where $\theta = 20$ degrees ($0.111\pi$ radians). Use the integral of $\cos^2(\phi)$ as $\frac{\cos(\phi)\sin(\phi)+\phi}{2}$ to compute this ratio.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.427}
Numbers $a$, $b$, and $c$ form an arithmetic sequence if $b - a = c - b$. Let $a$, $b$, and $c$ be positive integers forming an arithmetic sequence with $a < b < c$. Let $f(x) = ax^2 + bx + c$. Two distinct real numbers $r$ and $s$ satisfy $f(r) = s$ and $f(s) = r$. If $rs = 2017$, determine the smallest possible value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine the maximum number of points that can be placed in \(\mathbb{R}^n\) such that no \(n+1\) points lie in the same \(\mathbb{R}^{n-1}\)-plane and the Euclidean distance between every pair of points is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 1}
In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$, say $C_1$, and then the midpoint of $A_1D$, which is $A_2$. He continues in this fashion indefinitely. The total length of his path, if $AB=5$ and $BC=12$, is of the form $a + b\sqrt{c}$. Find $\frac{abc}{4}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{793}
A shopping mall installed 5 colored lights, each shining in a different color: red, orange, yellow, green, or blue. The sequence in which they shine is not fixed. Let's define the ordered shining of these 5 lights once as a "flash". In each flash, only one light shines per second, and the interval between two consecutive flashes is 5 seconds. Calculate the minimum time required to achieve all different flashes. Express your answer in seconds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1195}
A merchant bought some goods at a discount of $20\%$ of the list price. He wants to mark them at such a price that he can give a discount of $20\%$ of the marked price and still make a profit of $20\%$ of the selling price. The per cent of the list price at which he should mark them is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$125$}
5 / 8 of the population of the country of Venezia lives in Montague province, while the rest lives in Capulet province. In the upcoming election, 80% of Montague residents support Romeo, while 70% of Capulet residents support Juliet; each resident of Venezia supports exactly one of these two candidates. What is the probability that a Juliet supporter chosen at random resides in Capulet, rounded if necessary to the nearest percent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68\%}
Inside the circle \(\omega\), there is a circle \(\omega_{1}\) that touches it at point \(K\). Circle \(\omega_{2}\) touches circle \(\omega_{1}\) at point \(L\) and intersects circle \(\omega\) at points \(M\) and \(N\). It turns out that points \(K\), \(L\), and \(M\) are collinear. Find the radius of circle \(\omega\) if the radii of circles \(\omega_{1}\) and \(\omega_{2}\) are 4 and 7, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
If 5 students are made to sit in a row, then 6 do not have seats. If some students are made to sit in a row, then 3 rows are empty. Each row has to have 6 students for all students to be seated. How many students were made to sit in a row in the second scenario?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$, $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$. Compute the area of the pentagon.

[i]Greece[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Yvan and Zoé are playing the following game. Let $n \in \mathbb{N}$. The integers from 1 to $n$ are written on $n$ cards arranged in order. Yvan removes one. Zoé then removes 2 consecutive ones. Next, Yvan removes 3 consecutive ones, and Zoé finishes by removing 4 consecutive ones.

What is the smallest value of $n$ for which Zoé can ensure she can play both of her turns?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let $X$ be a projective variety over $\mathbf{C}$ of dimension $n$. Consider the blow-up $\pi: Y \to X$ at a smooth point $x \in X$. Find a formula for the intersection number $E^n$, where $E$ is the exceptional divisor.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^{n-1}}
Evaluate the integral \( \int_{-L}^L e^{i\frac{(m-n)\pi}{L}x} \, dx \), where \( 0 < L \in \mathbb{R} \) and \( m, n \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2L \delta_{mn}}
Evaluate the limit: $$\lim_{n\to \infty}\frac{1}{\sqrt[3]{{n^3}+2n-1}}+\frac{1}{\sqrt[3]{{n^3}+2n}}+\cdots+\frac{1}{\sqrt[3]{{n^3}+4n}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $\oplus$ denote the xor binary operation. Define $x \star y = (x+y) - (x \oplus y).$ Compute 
\[ \sum_{k=1}^{63} (k \star 45). \]

**Remark:** The xor operation works as follows: when considered in binary, the $k$th binary digit of $a \oplus b$ is $1$ exactly when the $k$th binary digits of $a$ and $b$ are different. For example, $5 \oplus 12 = 0101_2 \oplus 1100_2 = 1001_2 = 9.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2880}
Compute the result of the tensor product $M/IM \otimes_R R/I$, where $R$ is a ring, $I$ is an ideal in $R$, and $M$ is an $R$-module.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M/IM}
Calculate the Hamming distance of a linear block code with parameters (7,2), where the code is 7 bits long, has 2 data bits, and 5 error correction bits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
X deposits $1 at time $t=0$ with a force of interest given by \( \delta_t = \frac{t^2}{k} \). Y deposits the same amount at the same time with a nominal interest rate of 8% per annum, convertible semiannually. At \( t=5 \), their accumulated values are equal. Determine the value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106.24}