ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Write $2017$ numbers on the blackboard: $-\frac{1008}{1008}, -\frac{1007}{1008}, \ldots, -\frac{1}{1008}, 0, \frac{1}{1008}, \frac{2}{1008}, \ldots, \frac{1007}{1008}, \frac{1008}{1008}$.

One processes some steps as follows: erase two arbitrary numbers $x, y$ on the blackboard and then write the number $x + 7xy + y$. After $2016$ steps, there is only one number left. Find the last number on the blackboard in the form of a simplified fraction \(-\frac{k}{m}\). Please provide the value of \(k + m\).
Answer: 8
Problem: How many valence electrons are in the pyrophosphate ion, $\ce{P2O7}^{4-}$? Provide your answer as a whole number.
Answer: 56
Problem: The sequence $a_1$, $a_2$, $a_3$, $\dots$ satisfies $a_1 = 19$, $a_9 = 99$, and, for all $n \ge 3$, $a_n$ is the arithmetic mean of the first $n - 1$ terms. Find the value of $a_2$.
Answer: 179
Problem: An engineer is given a fixed volume \(V_m\) of metal with which to construct a spherical pressure vessel. Assuming the vessel has thin walls and is always pressurized to near its bursting point, the amount of gas the vessel can contain, \(n\) (measured in moles), does not depend on the radius \(r\) of the vessel. Instead, it depends only on \(V_m\) (measured in \(\text{m}^3\)), the temperature \(T\) (measured in \(\text{K}\)), the ideal gas constant \(R\) (measured in \(\text{J/(K} \cdot \text{mol})\)), and the tensile strength of the metal \(\sigma\) (measured in \(\text{N/m}^2\)). Express \(n\) in terms of these parameters. The original answer is in the form \(\frac{k}{m} \times \frac{V_m\sigma}{RTn}\). Please give the value of \(k + m + n\).
Answer: 6
Problem: Let $E$ be a point inside the rhombus $ABCD$ such that $|AE|=|EB|$, $m(\widehat{EAB}) = 11^{\circ}$, and $m(\widehat{EBC}) = 71^{\circ}$. If the answer is $x^{\circ}$, find the value of $x$ for $m(\widehat{DCE})$.
Answer: 68
Problem: In a circle with center at \(O\) and radius \(r\), chord \(AB\) is drawn with length equal to \(r\) (units). From \(O\), a perpendicular to \(AB\) meets \(AB\) at \(M\). From \(M\), a perpendicular to \(OA\) meets \(OA\) at \(D\). In terms of \(r\), find the area of triangle \(MDA\) in appropriate square units. The original answer is in the form \(\frac{k r^2 \sqrt{m}}{n}\), where \(k\), \(m\), and \(n\) are integers. Please provide the value of \(k + m + n\).
Answer: 36
Problem: If the sum of digits in the decimal representation of a positive integer $n$ is $111$ and the sum of digits in the decimal representation of $7002n$ is $990$, what is the sum of digits in the decimal representation of $2003n$?
Answer: 555
Problem: A bug starts at a vertex of a grid made of equilateral triangles with a side length of $1$. At each step, the bug moves in one of the $6$ possible directions along the grid lines randomly and independently with equal probability. Find the probability that after $5$ moves, the bug will never have been more than $1$ unit away from the starting position. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 121
Problem: For how many primes $p$ less than $15$, there exist integer triples $(m,n,k)$ such that:
\[
\begin{array}{rcl}
m+n+k &\equiv& 0 \pmod{p} \\
mn+mk+nk &\equiv& 1 \pmod{p} \\
mnk &\equiv& 2 \pmod{p}.
\end{array}
\]
Find the number of such primes $p$.
Answer: 3
Problem: Let $ABCD$ be a convex quadrilateral such that $m (\widehat{ADB}) = 15^{\circ}$ and $m (\widehat{BCD}) = 90^{\circ}$. The diagonals of the quadrilateral are perpendicular at $E$. Let $P$ be a point on $|AE|$ such that $|EC| = 4$, $|EA| = 8$, and $|EP| = 2$. If the answer is $x^\circ$, what is the value of $x$?
Answer: 75
Problem: The acronym AMC is shown in the rectangular grid below with grid lines spaced \(1\) unit apart. In units, find the sum of the lengths of the line segments that form the acronym AMC. The original answer is in the form \(a + b\sqrt{c}\), please provide the value of \(a + b + c\).
Answer: 19
Problem: How many ordered triples \((x, y, z)\) of integers satisfy the following system of equations?

\[
\begin{align*}
x^2 - 3xy + 2yz - z^2 &= 31 \\
-x^2 + 6yz + 2z^2 &= 44 \\
x^2 + xy + 8z^2 &= 100
\end{align*}
\]

Find the number of ordered triples.
Answer: 0
Problem: Let $C$ and $D$ be points on the circle with center $O$ and diameter $[AB]$, where $C$ and $D$ are on different semicircles with diameter $[AB]$. Let $H$ be the foot of the perpendicular from $B$ to $[CD]$. If $|AO| = 13$, $|AC| = 24$, and $|HD| = 12$, what is $\widehat{DCB}$ in degrees? If the answer is $x^\circ$, what is the value of $x$?
Answer: 30
Problem: Find the number of distinct ordered pairs $(x,y)$, where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4-10x^2y^2+9=0$.
Answer: 3
Problem: Points $A = (3,9), B = (1,1), C = (5,3),$ and $D = (a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD$. The quadrilateral formed by joining the midpoints of $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA}$ is a square. Find the sum of the coordinates of point $D$.
Answer: 10
Problem: A function $f$ from the integers to the integers is defined as follows:
\[ f(n) = \begin{cases} n + 3 & \text{if } n \text{ is odd} \\
\frac{n}{2} & \text{if } n \text{ is even} \end{cases} \]
Suppose $k$ is odd and $f(f(f(k))) = 27$. Find the sum of the digits of $k$.
Answer: 6
Problem: Let $(a_n)_{n=1}^{\infty}$ be a real sequence such that $a_1=1$, $a_3=4$, and for every $n \geq 2$, $a_{n+1} + a_{n-1} = 2a_n + 1$. Find the value of $a_{2011}$.
Answer: 2021056
Problem: Larry always orders pizza with exactly two of his three favorite toppings: pepperoni, bacon, and sausage. If he has ordered a total of $600$ pizzas and has had each topping equally often, how many pizzas has he ordered with pepperoni? Provide your answer as an integer.
Answer: 400
Problem: Find the value of \( \frac {1}{2 - \frac {1}{2 - \frac {1}{2 - \frac12}}} \). The original answer is in the form \(\frac{k}{m}\). Please provide the value of \(k + m\).
Answer: 9
Problem: What is the hundreds digit of the sum below?

$$1 + 12 + 123 + 1234 + 12345 + 123456 + 1234567 + 12345678 + 123456789$$
Answer: 2
Problem: Four coplanar regular polygons share a common vertex but have no interior points in common. Each polygon is adjacent to two of the other polygons, and each pair of adjacent polygons has a common side length of $1$. How many possible perimeters are there for all such configurations? Provide your answer as an integer.
Answer: 4
Problem: A train, an hour after starting, meets with an accident which detains it a half hour, after which it proceeds at $\frac{3}{4}$ of its former rate and arrives $3\frac{1}{2}$ hours late. Had the accident happened $90$ miles farther along the line, it would have arrived only $3$ hours late. Find the length of the trip in miles.
Answer: 600
Problem: A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile, the speed is constant. If the second mile is traversed in $2$ hours, find the time, in hours, needed to traverse the $n$th mile. The original answer is in the form $k(n-m)$, please give the value of $k + m$.
Answer: 3
Problem: Two circles with radii 1 and 2 touch each other and a line as shown in the figure. In the region between the circles and the line, there is a circle with radius $r$ which touches the two circles and the line. The original answer is in the form $m - n\sqrt{2}$, where $m$ and $n$ are integers. Please find the value of $m + n$.
Answer: 10
Problem: In tetrahedron $ABCD$, where $AB=1$, $CD=3$, the distance between $AB$ and $CD$ is $2$, and the intersection angle between $AB$ and $CD$ is $\frac{\pi}{3}$, find the volume of tetrahedron $ABCD$. The original answer is in \frac{\sqrt{k}}{m} format, please give the value of k + m.
Answer: 5
Problem: A point object of mass \(m\) is connected to a cylinder of radius \(R\) via a massless rope. At time \(t = 0\), the object is moving with an initial velocity \(v_0\) perpendicular to the rope. The rope has a length \(L_0\) and a non-zero tension. All motion occurs on a horizontal frictionless surface. The cylinder remains stationary and does not rotate. The object moves in such a way that the rope slowly winds up around the cylinder. The rope will break when the tension exceeds \(T_{max}\). Express your answers in terms of \(T_{max}\), \(m\), \(L_0\), \(R\), and \(v_0\). The original answer is in the form \(\frac{k m^2 v_0^3}{n T_{max}}\). Please calculate the value of \(k + m + n\).
Answer: 4
Problem: Let \(S_1\) be the area of the regular pentagon \(ABCDE\). And let \(S_2\) be the area of the regular pentagon whose sides lie on the lines \(AC, CE, EB, BD, DA\). The original answer is in the form \(\frac{k+m\sqrt{5}}{n}\). Please find the value of \(k+m+n\).
Answer: 12
Problem: A circle passes through the vertices of a triangle with side-lengths of \(7\frac{1}{2}, 10, 12\frac{1}{2}\). The radius of the circle can be expressed in the form \(\frac{k}{m}\). Find the value of \(k + m\).
Answer: 29
Problem: If \(V = gt + V_0\) and \(S = \frac{1}{2}gt^2 + V_0t\), find the expression for \(t\) in terms of \(S\), \(V\), and \(V_0\). The original answer is in the form \(\frac{kS}{m(V + V_0)}\), please give the value of k + m.
Answer: 3
Problem: If the decimal representation of $2^n$ starts with $7$, what is the first digit in the decimal representation of $5^n$? Provide your answer as a single digit.
Answer: 1
Problem: We draw a circle with radius $5$ on a gridded paper where the grid consists of squares with sides of length $1$. The center of the circle is placed in the middle of one of the squares. Find the number of squares through which the circle passes. Please provide your answer as an integer.
Answer: 40
Problem: If $a$, $b$, and $d$ are the lengths of a side, the shortest diagonal, and the longest diagonal, respectively, of a regular nonagon (see adjoining figure), determine the relationship between these variables in the form of an equation. The original answer is in the form $d = m \cdot a + n \cdot b + p$. Please give the value of $m + n + p$.
Answer: 2
Problem: How many pairs of integers \((x, y)\) are there such that \(2x + 5y = xy - 1\)?
Answer: 4
Problem: Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$. Find the probability that Flora will eventually land at $10$. The original answer is in \frac{k}{m} format, please give the value of k + m.
Answer: 3
Problem: A right circular cone and a right cylinder with the same height $20$ do not have the same circular base, but the circles are coplanar, and their centers are the same. If the cone and the cylinder are on the same side of the plane and their base radii are $20$ and $10$, respectively, find the ratio of the volume of the part of the cone inside the cylinder over the volume of the part of the cone outside the cylinder.
Answer: 1
Problem: Triangle \(ABC\) is a right triangle with \(\angle ACB\) as its right angle, \(m\angle ABC = 60^\circ\), and \(AB = 10\). Let \(P\) be randomly chosen inside \(\triangle ABC\), and extend \(\overline{BP}\) to meet \(\overline{AC}\) at \(D\). Find the probability that \(BD > 5\sqrt{2}\). The original answer is in the form \(\frac{k - \sqrt{m}}{n}\), please provide the value of k + m + n.
Answer: 9
Problem: The integer $k$ is a \textit{good number}, if we can divide a square into $k$ squares. Find how many good numbers not greater than $2006$ there are.
Answer: 2003
Problem: For how many positive integer values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities \[ \begin{cases} 2x > 3x - 3 \\ 3x - a > -6 \end{cases} \]?
Answer: 3
Problem: Side \(AB\) of triangle \(ABC\) has length \(8\) inches. Line \(DEF\) is drawn parallel to \(AB\) so that \(D\) is on segment \(AC\), and \(E\) is on segment \(BC\). Line \(AE\) extended bisects angle \(FEC\). If \(DE\) has length \(5\) inches, then find the length of \(CE\) in inches. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 43
Problem: Acute-angled triangle \(ABC\) is inscribed in a circle with center at \(O\). The arcs are given as follows: \(\overarc{AB} = 120^\circ\) and \(\overarc{BC} = 72^\circ\). A point \(E\) is located on the minor arc \(AC\) such that \(OE\) is perpendicular to \(AC\). Determine the ratio of the magnitudes of angles \(\angle OBE\) and \(\angle BAC\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 4
Problem: If $78$ is divided into three parts which are proportional to $1, \frac{1}{3}, \frac{1}{6}$, find the value of the middle part in its simplest fractional form. The original answer is in \frac{k}{m} format, please give the value of k + m.
Answer: 55
Problem: The point $(-3, 2)$ is rotated $90^\circ$ clockwise around the origin to point $B$. Point $B$ is then reflected over the line $y = x$ to point $C$. What are the coordinates of $C$? Provide your answer as an ordered pair $(k, m)$, and then find the sum of k and m.
Answer: 5
Problem: What is the least possible value of \((xy-1)^2+(x+y)^2\) for real numbers \(x\) and \(y\)?
Answer: 1
Problem: A permutation $ (a_1, a_2, a_3, a_4, a_5)$ of $ (1, 2, 3, 4, 5)$ is heavy-tailed if $ a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations?
Answer: 48
Problem: The solutions to the equations $z^2=4+4\sqrt{15}i$ and $z^2=2+2\sqrt{3}i$, where $i=\sqrt{-1}$, form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\sqrt{q}-r\sqrt{s}$, where $p$, $q$, $r$, and $s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number. Find the value of $p+q+r+s$.
Answer: 20
Problem: The graph, $G$ of $y = \log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Find an equation for $G'$. The original answer is in the form $y = a^{-bx}$, please give the value of a + b.
Answer: 11
Problem: In how many ways can $17$ identical red and $10$ identical white balls be distributed into $4$ distinct boxes such that the number of red balls is greater than the number of white balls in each box? Provide your answer as an integer.
Answer: 5720
Problem: How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $3$ times their perimeters?
Answer: 6
Problem: For all positive integers $n$, the function $f$ satisfies the following conditions:
- $f(1) = 1$
- $f(2n + 1) = 2f(n)$
- $f(2n) = 3f(n) + 2$

Determine how many positive integers $x \leq 100$ make the value of $f(x)$ odd.
Answer: 7
Problem: The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \, \text{cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d \, \text{cm}$. The square has a perimeter greater than 0. How many positive integers are NOT possible values for $d$?
Answer: 663
Problem: A quadrilateral has consecutive sides of lengths $70$, $90$, $130$, and $110$. It is both inscribed in a circle and has a circle inscribed within it. The point of tangency of the inscribed circle to the side of length $130$ divides that side into segments of lengths $x$ and $y$. Find the value of $|x-y|$.
Answer: 13
Problem: In the diagram below, for each row except the bottom row, the number in each cell is determined by the sum of the two numbers beneath it. Find the sum of all cells denoted with a question mark.
```latex
\begin{asy}
unitsize(2cm);
path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle;
draw(box); label("$2$",(0,0));
draw(shift(1,0)*box); label("$?$",(1,0));
draw(shift(2,0)*box); label("$?$",(2,0));
draw(shift(3,0)*box); label("$?$",(3,0));
draw(shift(0.5,0.4)*box); label("$4$",(0.5,0.4));
draw(shift(1.5,0.4)*box); label("$?$",(1.5,0.4));
draw(shift(2.5,0.4)*box); label("$?$",(2.5,0.4));
draw(shift(1,0.8)*box); label("$5$",(1,0.8));
draw(shift(2,0.8)*box); label("$?$",(2,0.8));
draw(shift(1.5,1.2)*box); label("$9$",(1.5,1.2));
\end{asy}
```
Answer: 13
Problem: What is the perimeter of the right triangle whose exradius of the hypotenuse is $30$?
Answer: 60
Problem: How many pairs of positive integers $(x, y)$ satisfy the equation $2^x - y^2 = 4$?
Answer: 1
Problem: $X$, $Y$, and $Z$ are pairwise disjoint sets of people. The average ages of people in the sets $X$, $Y$, $Z$, $X \cup Y$, $X \cup Z$, and $Y \cup Z$ are given in the table below.

\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Set} & X & Y & Z & X \cup Y & X \cup Z & Y \cup Z \\
\hline
\text{Average age of people in the set} & 37 & 23 & 41 & 29 & 39.5 & 33 \\
\hline
\end{array}
\]

Find the average age of the people in set $X \cup Y \cup Z$.
Answer: 34
Problem: Let $ABCD$ be a cyclic quadrilateral with $|AB|=26$, $|BC|=10$, $m(\widehat{ABD})=45^\circ$, $m(\widehat{ACB})=90^\circ$. Find the area of $\triangle DAC$.
Answer: 84
Problem: If \( y = x + \frac{1}{x} \), then transform the equation \( x^4 + x^3 - 4x^2 + x + 1 = 0 \) into the form \( x^2(y^2 + y - k) = 0 \). What is the value of \( k \)?
Answer: 6
Problem: $ABCD$ is a square and $M$ and $N$ are the midpoints of $BC$ and $CD$ respectively. Find the value of $\sin \theta$. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

```asy
\draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
\draw((0,0)--(2,1));
\draw((0,0)--(1,2));
\label("A", (0,0), SW);
\label("B", (0,2), NW);
\label("C", (2,2), NE);
\label("D", (2,0), SE);
\label("M", (1,2), N);
\label("N", (2,1), E);
\label("$\theta$", (.5,.5), SW);
```
Answer: 8
Problem: Let $p(x) = x^{2} + bx + c$, where $b$ and $c$ are integers. If $p(x)$ is a factor of both $x^{4} + 6x^{2} + 25$ and $3x^{4} + 4x^{2} + 28x + 5$, find the value of $p(1)$.
Answer: 4
Problem: Let \(ABCD\) be a rectangle and let \(\overline{DM}\) be a segment perpendicular to the plane of \(ABCD\). Suppose that \(\overline{DM}\) has integer length, and the lengths of \(\overline{MA}, \overline{MC},\) and \(\overline{MB}\) are consecutive odd positive integers (in this order). Find the volume of pyramid \(MABCD\). The original answer is in the form \(k\sqrt{m}\), please provide the value of \(k + m\).
Answer: 29
Problem: Let \(ABCD\) be a unit square. Let \(Q_1\) be the midpoint of \(\overline{CD}\). For \(i=1,2,\dots,\) let \(P_i\) be the intersection of \(\overline{AQ_i}\) and \(\overline{BD}\), and let \(Q_{i+1}\) be the foot of the perpendicular from \(P_i\) to \(\overline{CD}\). The sum of the areas of \(\triangle DQ_i P_i\) for \(i=1\) to \(\infty\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 5
Problem: A large sphere is on a horizontal field on a sunny day. At a certain time, the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant, a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.) The original answer is in the form \( k\sqrt{m} - n \). Please give the value of \( k + m + n \).
Answer: 35
Problem: Given the sets of consecutive integers $\{1\}$, $\{2, 3\}$, $\{4, 5, 6\}$, $\{7, 8, 9, 10\}$, $\cdots$, where each set contains one more element than the preceding one, and where the first element of each set is one more than the last element of the preceding set. Let $S_n$ be the sum of the elements in the $N$th set. Find the value of $S_{21}$.
Answer: 4641
Problem: Several sets of prime numbers, such as $\{7, 83, 421, 659\}$, use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
Answer: 207
Problem: What is the product of all the solutions to the equation $$\log_{7x}2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?$$ Provide your answer as an integer.
Answer: 1
Problem: In $\triangle ABC$, we have $AC = BC = 7$ and $AB = 2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 8$. What is the length of $BD$?
Answer: 3
Problem: Let \(E\), \(F\), \(G\) be points on sides \([AB]\), \([BC]\), \([CD]\) of the rectangle \(ABCD\), respectively, such that \(|BF|=|FQ|\), \(m(\widehat{FGE})=90^\circ\), \(|BC|=\frac{4\sqrt{3}}{5}\), and \(|EF|=\sqrt{5}\). The value of \(|BF|\) is in the form \(\frac{\sqrt{m} - \sqrt{n}}{p}\). Please find the value of \(m + n + p\).
Answer: 16
Problem: Diagonal $DB$ of rectangle $ABCD$ is divided into $3$ segments of length $1$ by parallel lines $L$ and $L'$ that pass through $A$ and $C$ and are perpendicular to $DB$. If the area of $ABCD$, rounded to the nearest tenth, is $x$, what is the value of $10x$?
Answer: 42
Problem: Let \[f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.\] Express $f(n+1)-f(n-1)$ in terms of $f(n)$. The original answer is in the form of a single variable k, please provide the value of k.
Answer: 1
Problem: If the five-digit number $3AB76$ is divisible by $9$ and $A < B < 6$, find the value of $B - A$.
Answer: 2
Problem: Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. The probability of obtaining a six on any toss is \(\frac{1}{6}\), independent of the outcome of any other toss. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 116
Problem: Let $p(x)$ be a polynomial of degree $4$ with leading coefficient $1$. Suppose $p(1)=1$, $p(2)=2$, $p(3)=3$, and $p(4)=4$. Find the value of $p(5)$.
Answer: 29
Problem: At Rachelle's school, an A counts as 4 points, a B as 3 points, a C as 2 points, and a D as 1 point. Her GPA for the four classes she is taking is calculated as the total sum of points divided by 4. She is certain to get As in both Mathematics and Science, and at least a C in each of English and History. She thinks she has a \frac{1}{6} chance of getting an A in English, and a \frac{1}{4} chance of getting a B. In History, she has a \frac{1}{4} chance of getting an A, and a \frac{1}{3} chance of getting a B, independently of her grades in English. What is the probability that Rachelle will achieve a GPA of at least 3.5? Provide your answer in the form of \frac{k}{m} and give the value of k + m.
Answer: 35
Problem: In right triangle \(ABC\), \(BC = 5\), \(AC = 12\), and \(AM = x\); \(\overline{MN} \perp \overline{AC}\), \(\overline{NP} \perp \overline{BC}\); \(N\) is on \(AB\). If \(y = MN + NP\), one-half the perimeter of rectangle \(MCPN\), find the value of \(y\). The original answer is in the form \(\frac{k - mx}{n}\). Please provide the value of \(k + m + n\).
Answer: 163
Problem: A fair die is rolled six times. Find the probability of rolling at least a five at least five times. Express your answer as a fraction in simplest form. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 742
Problem: If the function \( f \) satisfies the equation \( f(x) + f\left( \frac{1}{\sqrt[3]{1-x^3}}\right) = x^3 \) for every real \( x \neq 1 \), find the value of \( f(-1) \). The original answer is in the form \(\frac{k}{m}\), where the fraction is in its simplest form. Please provide the value of \( k + m \).
Answer: 5
Problem: A square has sides of length $10$, and a circle centered at one of its vertices has radius $10$. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in the form \(a + b\pi\), where \(a\) and \(b\) are integers. Please give the value of \(a + b\).
Answer: 175
Problem: For what value of $n$ is the equation $i + 2i^2 + 3i^3 + \cdots + ni^n = 48 + 49i$ satisfied? Note: here $i = \sqrt{-1}$.
Answer: 97