inclusionAI/Ring-lite-rl-data huggingface dataset:
Calculate the expected value of \(\frac{\max\{|X|,|Y|\}}{\min\{|X|,|Y|\}}\) where \(X\) and \(Y\) are independent random variables each following a standard normal distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let \( n > 2 \) be an integer. In a country there are \( n \) cities and every two of them are connected by a direct road. Each road is
 assigned an integer from the set \( \{ 1,2,\ldots ,m\} \) (different roads may be assigned the same number). The priority of a city is the
 sum of the numbers assigned to roads which lead to it. Find the smallest \( m \) for which it is possible that all cities have a differer
 priority.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
m = 3
}
A circular birthday cake is divided into eight equal wedges. In how many distinct ways can red and green candles be placed on the cake, with one candle on each wedge, considering the symmetries of the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the sum of the two smallest possible values of \( {x}^{ \circ  } \) (in degrees) that satisfy the
 following equation if \( x \) is greater than \( {2017}^{ \circ  } \) :
 \( {\cos }^{5}{9x} + {\cos }^{5}x = {32}{\cos }^{5}{5x}{\cos }^{5}{4x} + 5{\cos }^{2}{9x}{\cos }^{2}x\left( {\cos {9x} + \cos x}\right) . \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4064
}
Consider a group $G$ of order $105$. Determine the number of Sylow 3-subgroups of $G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find a Gröbner basis for the ideal $I \cap \mathbb{Q}[y]$ in the polynomial ring $\mathbb{Q}[x,y]$, where $I=\langle x^2-y,y^2-x,x^5-x^2\rangle$, using the lexicographic ordering $y > x$. Provide your answer as a set of polynomials.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{y^4 - y\}}
Determine whether the series \( \sum_{n=0}^{\infty} \frac{(-1)^n (2n)!x^n}{(n!)^2 4^n} \) converges or diverges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1]}
Given the differential equation \( y'(x, y_0) = y^2(x, y_0) + y(x, y_0) \sin(x) \), where \( y_0 \) is the value of \( y \) at \( x = 0 \), find \( \frac{\partial y}{\partial y_0} \bigg|_{y_0=0} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{1 - \cos x}}
Determine the maximum number of real roots of the equation $e^x + P(x) = 0$, where $P(x)$ is a polynomial of degree $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 1}
Let \( P \) be a point inside regular pentagon \( ABCDE \) such that \( \angle PAB = 48^\circ \) and \( \angle PDC = 42^\circ \). Find \( \angle BPC \), in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Suppose $f(x)$ is a function such that $f(2x)=f(x+y)f(x-y)$ for all $x,y \in \mathbb{R}$. Given that $f(10)=4, f(0)\neq 0$, and $f'(0)=2$, find the value of $f'(10)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $n$ be a fixed positive integer. For integer $m$ satisfying $|m| \le n$, define $S_m=\displaystyle\sum_{\substack{i-j=m \\0 \le i,j \le n}} \dfrac{1}{2^{i+j}}$. Then\[\lim_{n\to \infty} \left( S_{-n}^2 + S_{-n+1}^2 +... + S_n^2 \right)\]can be expressed in the form $\dfrac{p}{q}$ for relatively prime positive integers $p,q$. Compute $100p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
8027
}
What is the coefficient of c in the equation that has the largest possible value of cc as 47 when the equation is in the form of a constant times c plus the square of the difference between a variable d and 12 equals 235?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A $41 \times 41$ grid contains a hidden tank. A fighter plane fires at one cell at a time. If the shot hits, the tank moves to an adjacent cell (sharing a side); if the shot misses, the tank remains in place. The tank must be hit twice to be destroyed. What is the minimum number of shots needed to guarantee the destruction of the tank?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2522}
设 $M$ 是由有限个正整数构成的集合, 且 $M=A_{1} \cup A_{2} \cup \cdots \cup A_{20}=B_{1} \cup B_{2} \cup \cdots \cup B_{20}$, 这里 $A_{i} \neq \varnothing, B_{i} \neq \varnothing, i=1,2, \cdots, 20$, 并对任意的 $1 \leq i<j \leq 20$, 都有 $A_{i} \cap A_{j}=\varnothing, B_{i} \cap B_{j}=\varnothing$.

已知对任意的 $1 \leq i \leq 20,1 \leq j \leq 20$, 若 $A_{i} \cap B_{j}=\varnothing$, 则 $\left|A_{i} \cup B_{j}\right| \geq 18$, 求集合 $M$ 的元素个数的最小值（这里， $|X|$ 表示集合 $X$ 的元素个数）.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$180$}
$\text{For how many integers}0\leq n<2013\mathrm{,is}n^4+2n^3-20n^2+2n-21\text{divisible by}2013?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6
}
Evaluate the triple integral: \[ \int_0^1 \int_1^2 \int_0^{x+y} 12(4x+y+3z)^2 \, dz \, dy \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1272}
Let $N$ be the number of non-empty subsets $T$ of $S = \{1, 2, 3, 4, . . . , 2020\}$ satisfying $\max (T) >1000$. Compute the largest integer $k$ such that $3^k$ divides $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2
}
Let $a_0, a_1, \dots, a_{19} \in \mathbb{R}$ and \( P(x) = x^{20} + \sum_{i=0}^{19}a_ix^i, x \in \mathbb{R}. \) If \( P(x)=P(-x) \) for all \( x \in \mathbb{R} \), and \( P(k)=k^2, \) for \( k=0, 1, 2, \dots, 9 \), find \( \lim_{x\rightarrow 0} \frac{P(x)}{\sin^2x}. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - (9!)^2}
Let  $k$  be a positive integer with the following property: For every subset  $A$  of  $\{1,2,\ldots, 25\}$  with  $|A|=k$ , we can find distinct elements  $x$  and  $y$  of  $A$  such that  $\tfrac23\leq\tfrac xy\leq\tfrac 32$ .  Find the smallest possible value of  $k$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Centuries ago, the pirate Captain Blackboard buried a vast amount of treasure in a single cell of a $2 \times 4$ grid-structured island. Treasure was buried in a single cell of an $M\times N$ ($2\le M$, $N$) grid. You and your crew have reached the island and have brought special treasure detectors to find the cell with the treasure For each detector, you can set it up to scan a specific subgrid $[a,b]\times[c,d]$ with $1\le a\le b\le 2$ and $1\le c\le d\le 4$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready.
What is the minimum $Q$ required to gaurantee to determine the location of the Blackboard’s legendary treasure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3
}
Determine the Lebesgue measure of the Luzin set \( A \), which consists of numbers in the interval \([0,1]\) whose continued fraction expansion contains an infinite subsequence of integers such that each integer is divisible by the previous one.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In the expansion of \((x^2+x+1)^{54}\), how many coefficients are divisible by 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Elyse had 100 pieces of gum. She gave half to her brother Rick. Rick then gave a third of his to his friend Shane. Later, Shane received 10 more pieces of gum from his cousin. Shane chewed 11 pieces of gum and decided to share half of what he had left with his sister, Sarah. How many pieces of gum does Shane have left now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Compute the prime factorization of  $1007021035035021007001$ .  (You should write your answer in the form  $p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$  where  $p_1,\ldots,p_k$  are distinct prime numbers and  $e_1,\ldots,e_k$  are positive integers.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7^7 \times 11^7 \times 13^7}
Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let \( O \) be a point in three-dimensional space, and let \( l_1, l_2, l_3 \) be mutually perpendicular straight lines passing through \( O \). Let \( S \) denote the sphere with center \( O \) and radius \( R \). For every point \( M \) on \( S \), let \( S_M \) denote the sphere with center \( M \) and radius \( R \). We denote by \( P_1, P_2, P_3 \) the intersection of \( S_M \) with the straight lines \( l_1, l_2, l_3 \), respectively. Define \( P_1 \neq O \) if \( l_1 \) meets \( S_M \) at two distinct points and \( P_1 = O \) otherwise (similarly for \( P_2 \) and \( P_3 \)). What is the set of centers of gravity of the (possibly degenerate) triangles \( P_1 P_2 P_3 \) as \( M \) runs through the points of \( S \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2R}{3}}
[ Work Problems.]

Three workers are digging a pit. They work in turns, and each of them works as long as it takes the other two to dig half of the pit. Working this way, they dug the pit. How many times faster would three workers dig the same pit if they worked simultaneously?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5}
We have a Poisson process of intensity $\lambda = 4$. We have the following event: $A$: "Two marks appear with a separation of $\frac{1}{12}$ or less". We need to calculate the probability of exactly two "marks" appearing between $0$ and $\frac{1}{2}$, and $A$ occurring at the same time. That is, the probability of exactly two marks occurring in $(0, \frac{1}{2})$ with a separation of $\frac{1}{12}$ or less.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{11}{18}e^{-2}\)}
Determine all non-negative integers \( n \) for which there exists a non-periodic function \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfying the functional equation \( f(x+1) + f(x-1) = \sqrt{n} f(x) \) for all \( x \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 4}
Find the smallest integer $n$ such that the inequality $0.\overline{3}_n < \log_{n}{3}$ holds for all integers greater than $n$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the integral \( \int_0^1 \int_0^1 \frac{xy\ln(y)}{xy-1}\ dx \ dy \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\zeta(3) - 1}
The sums of the three columns and the three rows of the table are the same.

| 4 | 9 | 2 |
| - | - | - |
| 8 | 1 | 6 |
| 3 | 5 | 7 |

What is the minimum number of cells in the table that must be changed so that all six new sums are different from each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In a convex quadrilateral \(ABCD\), let \(M\) be the midpoint of segment \(BC\). Given that \(\angle AMD = 90^\circ\), \(\angle ADM = 15^\circ\), and \(AD = AB + CD\), find the measure of \(\angle BAD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150^\circ}
A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$ ?

[asy]
draw((0,2)--(2,2)--(2,0)--(0,0)--cycle);
draw((0,0.3)--(0.3,2)--(2,1.7)--(1.7,0)--cycle);
label("$a$",(-0.1,0.15));
label("$b$",(-0.1,1.15));
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ ab = \frac{1}{2} \]}
A circle of radius \(r\) passes through both foci of, and exactly four points on, the ellipse with equation \(x^2 + 16y^2 = 16\). The set of all possible values of \(r\) is an interval \([a, b)\). The original answer is in the form \(\sqrt{a} + b\). Please find the value of \(a + b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
There are several soldiers forming a rectangular formation with exactly eight columns. If adding 120 people or removing 120 people from the formation can both form a square formation, how many soldiers are there in the original rectangular formation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{136}
A train 110 m long traveling at 72 kmph takes 13.998880089592832 seconds to cross a bridge of certain length. How long is the bridge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{169.98}
Seventy percent of a train's passengers are women, and fifteen percent of those women are in the luxury compartment. What is the number of women in the luxury compartment if the train is carrying 300 passengers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
An ant starts at one vertex of a tetrahedron. Each minute it walks along a random edge to an adjacent vertex. What is the probability that after one hour the ant winds up at the same vertex it started at?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3^{59} + 1}{4 \cdot 3^{59}}}
Dan went to the grocery store to buy ingredients for several batches of cakes. He bought 9 dozen eggs, 6 pounds of flour, and 3 cases of butter with each case containing 24 sticks of butter. In addition, he purchased 12 bottles of vanilla extract with each bottle containing 1 ounce. Calculate the total number of individual items Dan bought.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{198}
A teacher wrote a sequence of consecutive odd numbers starting from 1 on the blackboard: $1, 3, 5, 7, 9, 11, \cdots$ After writing, the teacher erased two numbers, dividing the sequence into three segments. If the sums of the first two segments are 961 and 1001 respectively, what is the sum of the two erased odd numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{154}
A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. The original answer is in the form $\frac{a\sqrt{b}-c}{d}$, please give the value of a + b + c + d.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
For an integer  $k$  let  $T_k$  denote the number of  $k$ -tuples of integers  $(x_1,x_2,...x_k)$  with  $0\le x_i < 73$  for each  $i$ , such that  $73|x_1^2+x_2^2+...+x_k^2-1$ . Compute the remainder when  $T_1+T_2+...+T_{2017}$  is divided by  $2017$ .

[i]Proposed by Vincent Huang
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
For each positive integer $n$ , let $f(n) = \sum\limits_{k = 1}^{100} \lfloor \log_{10} (kn) \rfloor$ . Find the largest value of $n$ for which $f(n) \le 300$ . 
Note: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{109}
A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
In the final round of a giraffe beauty contest, two giraffes named Tall and Spotted have made it to this stage. There are 105 voters divided into 5 districts, each district divided into 7 sections, with each section having 3 voters. Voters select the winner in their section by majority vote; in a district, the giraffe winning the majority of sections wins the district; finally, the giraffe winning the majority of districts is declared the contest winner. The giraffe named Tall won. What is the minimum number of voters who could have voted for him?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
$ABCDEFGH$ is a cube having a side length of 2. $P$ is the midpoint of $EF$, as shown. The area of $\triangle APB$ is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt{8}$}
Problem 8. Given an isosceles triangle $P Q R(P Q=Q R)$ with the angle at the vertex equal to $108^{\circ}$. Point $O$ is located inside triangle $P Q R$ such that $\angle O R P=30^{\circ}$, and $\angle O P R=24^{\circ}$. Find the measure of angle $\angle Q O R$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Given the function f(x) = sin2x + 2√3cos²x - √3, and the function g(x) = mcos(2x - π/6) - 2m + 3 (m > 0), if there exist x₁, x₂ ∈ [0, π/4] such that f(x₁) = g(x₂) holds true, then the range of the real number m is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{2}{3}, 2]}
Calculate the areas of the figures bounded by the lines given in polar coordinates.

$$
r=\cos \phi, \quad r=\sin \phi, \quad \left(0 \leq \phi \leq \frac{\pi}{2}\right)
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Initially, a natural number $n$ is written on the blackboard. Then, at each minute, [i]Neymar[/i] chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that [i]Neymar[/i] will never be able to write on the blackboard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2033 }
For the walls of the house, John would use 12 large planks of wood and 10 small planks. If large planks together need 15 pieces of nails to be secured and small planks together need 5 nails, how many nails does John need for the house wall?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
The light glows for every some seconds. Between 1:57:58 and 3:20:47 AM, it glowed 354.92857142857144 times. What is the interval in seconds between each glow?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
How many non-isomorphic Frobenius groups of order 100 exist with a kernel of $C_{25}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the function $f(x)=ax^{3}+2bx^{2}+3cx+4d$, where $a,b,c,d$ are real numbers, $a < 0$, and $c > 0$, is an odd function, and when $x\in[0,1]$, the range of $f(x)$ is $[0,1]$. Find the maximum value of $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{2}}
Two spheres touch the plane of triangle \(ABC\) at points \(A\) and \(B\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 9, and the distance between their centers is \(\sqrt{305}\). The center of a third sphere with a radius of 7 is at point \(C\), and it externally touches each of the first two spheres. Find the radius of the circumcircle of triangle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{14} \approx 7.48}
Find the number of binary sequences of length $n$ that satisfy the pattern $x_1 \ge x_2 \le x_3 \ge x_4 \le x_5 \ge \ldots$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_{n+2}}
Find the closed-form expression for the series: \[ \sum_{n=1}^\infty \frac{1}{1 + 2^2 + 3^2 + \ldots + n^2} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18 - 24 \ln 2}
Evaluate the limit: $$\lim\limits_{x\to 0}\frac{1}{2x}\int_{0}^{x}\left[1-\cos\left(\frac{2}{u}\right)\right]du$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
For a matrix $M \in \mathbb{R}^{n \times n}_{\geq 0}$ with nonnegative entries, define $m$ as the smallest positive integer such that all the entries of $M^m$ are strictly positive, if such an $m$ exists. Determine, as a function of $n$, the maximum possible finite value of $m$ over all possible choices of $M$. Express your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n-1)^2 + 1}
A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A=(-3, 2)$ and $B=(3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
195

}
Find one integer solution to the Diophantine equation \(18x + 14y + 63z = 5\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3, 1, -1)}
We define \( {\mathbb{F}}_{101}\left\lbrack  x\right\rbrack \) as the set of all polynomials in \( x \) with coefficients in \( {\mathbb{F}}_{101} \) (the integers modulo101with usual addition and
 subtraction),so that two polynomials are equal if and only if the coefficients of \( {x}^{k} \) are equal in \( {\mathbb{F}}_{101} \) for each nonnegative integer
 \( k \) . For example, \( \left( {x + 3}\right) \left( {{100x} + 5}\right)  = {100}{x}^{2} + {2x} + {15} \) in \( {\mathbb{F}}_{101}\left\lbrack  x\right\rbrack \) because the corresponding coefficients are equal modu
 101.
 We say that \( f\left( x\right)  \in  {\mathbb{F}}_{101}\left\lbrack  x\right\rbrack \) is \\emph\{lucky\} if it has degree at most 1000 and there exist \( g\left( x\right) ,h\left( x\right)  \in  {\mathbb{F}}_{101}\left\lbrack  x\right\rbrack \) such that
 \( f\left( x\right)  = g\left( x\right) \left( {{x}^{1001} - 1}\right)  + h{\left( x\right) }^{101} - h\left( x\right) \)
 in \( {\mathbb{F}}_{101}\left\lbrack  x\right\rbrack \) . Find the number of lucky polynomials.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( {101}^{954} \)}
In triangle $ABC$, point $P$ is inside the triangle such that lines $AP$, $BP$, and $CP$ intersect sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Given that $AP = a$, $BP = b$, $CP = c$, $PD = PE = PF = 3$, and $a + b + c = 43$, find the value of $abc$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{441}
Determine the number of zeros of the function \( g(z) = z^4 + iz^3 + 1 \) in the region \( \{ z \in \mathbb{C} : \text{Re}(z) > 0, \text{Im}(z) > 0 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the maximal possible dimension of a subspace \( U \subseteq V \), where \( V \) is the vector space of functions from \([0,1]\) to \( \mathbb{R} \), and \( U \) consists of monotone functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the cardinality of a Hilbert space $H$ that has an orthonormal basis with cardinality $\mathfrak{c}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}