ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: If $x$ and $y$ are positive real numbers such that $(x + \sqrt{x^2 + 1})(y + \sqrt{y^2 + 1}) = 2018$, find the minimum possible value of $x + y$. The original answer is in the form \frac{k}{\sqrt{m}}, please give the value of k + m.
Answer: 4035
Problem: 在集合$\{0, 1, 2, \ldots, 2014\}$中，有多少个整数满足条件：$C^x_{2014} \ge C^{999}_{2014}$？请计算满足该条件的整数个数。
Answer: 17
Problem: The first term of an arithmetic series of consecutive integers is \( k^2 + 1 \). Find the sum of \( 2k + 1 \) terms of this series. The original answer is in the form \( m \cdot k^3 + n \cdot (k + 1)^3 \). Please calculate the value of \( m + n \).
Answer: 2
Problem: A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression. Let $S_n$ represent the sum of the first $n$ terms of the harmonic progression; for example, $S_3$ represents the sum of the first three terms. If the first three terms of a harmonic progression are $3, 4, 6$, find the value of $S_4$.
Answer: 25
Problem: The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the 2005th term of the sequence?
Answer: 250
Problem: A line initially \(1\) inch long grows according to the following law, where the first term is the initial length:
\[
1 + \frac{1}{4}\sqrt{2} + \frac{1}{4} + \frac{1}{16}\sqrt{2} + \frac{1}{16} + \frac{1}{64}\sqrt{2} + \frac{1}{64} + \cdots.
\]
If the growth process continues forever, what is the limit of the length of the line? The original answer is in the format \(\frac{k}{m}(n + \sqrt{p})\), please give the value of k + m + n + p.
Answer: 10
Problem: Points \(A\), \(B\), \(C\), and \(D\) are on a circle with diameter \(1\), and \(X\) is on diameter \(\overline{AD}\). If \(BX = CX\) and \(3 \angle BAC = \angle BXC = 36^{\circ}\), then find \(AX\) in the form of a trigonometric expression \(\cos x^\circ \sin y^\circ \csc z^\circ\). Please give the value of x + y + z.
Answer: 36
Problem: Find the number of positive integral divisors of 2006.
Answer: 8
Problem: Label one disk "1", two disks "2", three disks "3", ..., fifty disks "50". Put these $1+2+3+ \cdots+50=1275$ labeled disks in a box. Disks are then drawn from the box at random without replacement. What is the minimum number of disks that must be drawn to guarantee drawing at least ten disks with the same label?
Answer: 415
Problem: $ABC$ is a triangle with vertices $A=(0,0)$, $B=(36,15)$, and $C=(x,y)$ where both $x$ and $y$ are integers. What is the minimum area of $\triangle ABC$? Provide your answer in the form of \(\frac{k}{m}\) where \(\frac{k}{m}\) is in simplest form. Calculate the value of \(k + m\).
Answer: 5
Problem: What is the largest integer $n$ such that $2^n + 65$ is equal to the square of an integer? Find the value of $n$.
Answer: 10
Problem: For complex numbers \(u=a+bi\) and \(v=c+di\), define the binary operation \(\otimes\) by \[ u \otimes v = ac + bdi. \] Suppose \(z\) is a complex number such that \(z \otimes z = z^{2} + 40\). The original answer is in the form \(k\sqrt{m}\). Please find the value of \(k + m\).
Answer: 7
Problem: In $\triangle ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. $M$ is the midpoint of side $AB$, and $H$ is the foot of the altitude from $A$ to $BC$. If the length of $HM$ is $x$, find the value of $10x$.
Answer: 65
Problem: In the adjoining figure, the triangle \(ABC\) is a right triangle with \(\angle BCA=90^\circ\). Median \(CM\) is perpendicular to median \(BN\), and side \(BC=s\). The length of \(BN\) is in the form \(\frac{k}{m}\sqrt{n}\). Please calculate the value of \(k + m + n\).
Answer: 9
Problem: Let the product $(12)(15)(16)$, with each factor written in base $b$, be equal to $3146$ in base $b$. Let $s = 12 + 15 + 16$, with each term expressed in base $b$. Find the value of $s$ in base $b$.
Answer: 44
Problem: Let $ABC$ be a triangle such that the midpoints of the three altitudes are collinear. If the largest side of the triangle is $10$, what is the largest possible area of the triangle?
Answer: 25
Problem: When two stars are very far apart, their gravitational potential energy is zero. When they are separated by a distance \(d\), the gravitational potential energy of the system is \(U\). If the stars are separated by a distance \(2d\), what is the gravitational potential energy of the system in terms of \(U\)? The original answer is in \(\frac{kU}{m}\) format, please give the value of \(k + m\).
Answer: 3
Problem: A cyclist travels at a constant speed of $22.0\,\text{km/hr}$ except for a $20$ minute stop. The cyclist’s average speed was $17.5\,\text{km/hr}$. How far did the cyclist travel? If the answer is $x\,\text{km}$, what is the value of $x \times 10$?
Answer: 285
Problem: The dimensions of a rectangle $R$ are $a$ and $b$, with $a < b$. It is required to obtain a rectangle with dimensions $x$ and $y$, where $x < a$ and $y < a$, such that its perimeter is one-third that of $R$, and its area is one-third that of $R$. Find the number of such (different) rectangles.
Answer: 0
Problem: Oscar buys 13 pencils and 3 erasers for $\$1.00$. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost of one pencil and one eraser in cents?
Answer: 10
Problem: What is the smallest positive odd integer $n$ such that the product 
\[2^{1/7} \cdot 2^{3/7} \cdot \ldots \cdot 2^{(2n+1)/7}\] 
is greater than $1000$? 
(In the product, the denominators of the exponents are all sevens, and the numerators are the successive odd integers from $1$ to $2n+1$.)
Answer: 9
Problem: Jack and Jill run \(10\) kilometers. They start at the same point, run \(5\) kilometers up a hill, and return to the starting point by the same route. Jack has a \(10\) minute head start and runs at the rate of \(15\) km/hr uphill and \(20\) km/hr downhill. Jill runs \(16\) km/hr uphill and \(22\) km/hr downhill. How far from the top of the hill are they when they pass going in opposite directions? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 62
Problem: How many non-congruent scalene triangles with perimeter $21$ have integer side lengths that form an arithmetic sequence? (In an arithmetic sequence, successive terms differ by the same amount.) Provide your answer as an integer.
Answer: 3
Problem: How many positive integers are there such that $\left \lfloor \frac{m}{11} \right \rfloor = \left \lfloor \frac{m}{10} \right \rfloor$? ($\left \lfloor x \right \rfloor$ denotes the greatest integer not exceeding $x$.)
Answer: 54
Problem: In a regular pyramid with top point $T$ and equilateral base $ABC$, let $P$, $Q$, $R$, $S$ be the midpoints of $[AB]$, $[BC]$, $[CT]$ and $[TA]$, respectively. If $|AB| = 6$ and the altitude of the pyramid is equal to $2\sqrt{15}$, find the area of $PQRS$. The original answer is in the form of k\sqrt{m}, please give the value of k + m.
Answer: 11
Problem: Let $ABCD$ be a quadrilateral such that $m(\widehat{A}) = m(\widehat{D}) = 90^\circ$. Let $M$ be the midpoint of $[DC]$. If $AC\perp BM$, $|DC|=12$, and $|AB|=9$, then what is the length of $|AD|$?
Answer: 6
Problem: Nonempty set $A=\{x \mid 2a+1 \leq x \leq 3a-5\}$, $B=\{x \mid 3 \leq x \leq 22\}$. Find the range of values for $a$ such that $A \subseteq A \cap B$. The answer is in the form $\{a \mid b \leq a \leq c\}$, please find the value of $b + c$.
Answer: 15
Problem: A light flashes in one of three different colors: red, green, and blue. Every $3$ seconds, it flashes green. Every $5$ seconds, it flashes red. Every $7$ seconds, it flashes blue. If it is supposed to flash in two colors at once, it flashes the more infrequent color. How many times has the light flashed green after $671$ seconds?
Answer: 154
Problem: In the expression $c \cdot a^b - d$, the values of $a$, $b$, $c$, and $d$ are $0$, $1$, $2$, and $3$, although not necessarily in that order. What is the maximum possible value of the result?
Answer: 9
Problem: For how many pairs \((a,b)\) from \((1,2)\), \((3,5)\), \((5,7)\), \((7,11)\), does the polynomial \(P(x)=x^5+ax^4+bx^3+bx^2+ax+1\) have exactly one real root?
Answer: 2
Problem: Given that the equation $(m^2-12)x^4-8x^2-4=0$ has no real roots, find the largest value of $m$ in the form $p\sqrt{q}$, where $p$ and $q$ are natural numbers, and $q$ is square-free. Determine the value of $p+q$ as an integer.
Answer: 5
Problem: Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the numbers between \(\frac{1}{2}\) and \(\frac{2}{3}\). Armed with this information, what number should Carol choose to maximize her chance of winning? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 37
Problem: Horses X, Y, and Z are entered in a three-horse race in which ties are not possible. If the odds against X winning are $3\text{-to-}1$ and the odds against Y winning are $2\text{-to-}3$, what are the odds against Z winning? Express your answer in the form $k\text{-to-}m$. Please provide the value of k + m.
Answer: 20
Problem: For how many positive integers $m$ is \[ \frac{2002}{m^2-2} \] a positive integer? Write your answer as a number.
Answer: 3
Problem: For how many primes \( p \), there exist unique integers \( r \) and \( s \) such that for every integer \( x \), the congruence \( x^{3} - x + 2 \equiv (x - r)^{2} (x - s) \pmod{p} \) holds? Find the number of such primes.
Answer: 2
Problem: Let \(ABCD\) be a circumscribed quadrilateral such that \(m(\widehat{A})=m(\widehat{B})=120^\circ\), \(m(\widehat{C})=30^\circ\), and \(|BC|=2\). Find the length of \(|AD|\). The original answer is in the form \(\sqrt{k} - m\). Please give the value of k + m.
Answer: 4
Problem: If \(n\) is a multiple of \(4\), find the sum \(s=1+2i+3i^2+ \ldots +(n+1)i^{n}\), where \(i=\sqrt{-1}\). The original answer is in the form \(\frac{k}{m}(n+a-b\cdot i)\). Please calculate the value of \(k + m + a + b\).
Answer: 6
Problem: In $\triangle PAT$, $\angle P=36^{\circ}$, $\angle A=56^{\circ}$, and $PA=10$. Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA}$, respectively, so that $PU=AG=1$. Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG}$, respectively. Find the degree measure of the acute angle formed by lines $MN$ and $PA$.
Answer: 80
Problem: Ant Amelia starts on the number line at \(0\) and crawls in the following manner. For \(n=1,2,3,\) Amelia chooses a time duration \(t_n\) and an increment \(x_n\) independently and uniformly at random from the interval \((0,1).\) During the \(n\)th step of the process, Amelia moves \(x_n\) units in the positive direction, using up \(t_n\) minutes. If the total elapsed time has exceeded \(1\) minute during the \(n\)th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most \(3\) steps in all. What is the probability that Amelia’s position when she stops will be greater than \(1\)? The answer is in the form \(\frac{k}{m}\), please provide the value of \(k + m\).
Answer: 5
Problem: Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3? Express your answer as a fraction in simplest form, in the format \(\frac{k}{m}\). Please give the value of k + m.
Answer: 265
Problem: If \(x, y \in (-2, 2)\) and \(xy = -1\), find the minimum value of \(u = \frac{4}{4-x^2} + \frac{9}{9-y^2}\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 17
Problem: What are the last two digits of the decimal representation of \(9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}\)?
Answer: 21
Problem: In a certain cross-country meet between two teams of five runners each, a runner who finishes in the $n^{th}$ position contributes $n$ to his team's score. The team with the lower score wins. If there are no ties among the runners, find how many different winning scores are possible.
Answer: 13
Problem: An "n-pointed star" is formed as follows: the sides of a convex polygon are numbered consecutively $1, 2, \ldots, k, \ldots, n$, where $n \geq 5$. For all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, with sides $n+1$ and $n+2$ being respectively identical with sides $1$ and $2$. Prolong the $n$ pairs of sides numbered $k$ and $k+2$ until they meet. (A figure is shown for the case $n=5$)

Let $S$ be the degree-sum of the interior angles at the $n$ points of the star; then find the expression for $S$ in terms of $n$. The original answer is in the form $a(n-b)$, where $a$ and $b$ are integers. Please find the value of $a + b$.
Answer: 184
Problem: If the equation system \[\begin{array}{rcl} f(x) + g(x) &=& 0 \\ f(x)-(g(x))^3 &=& 0 \end{array}\] has more than one real root, where $a,b,c,d$ are reals and $f(x)=x^2 + ax+b$, $g(x)=x^2 + cx + d$, find at most how many distinct real roots the equation $f(x)g(x) = 0$ can have.
Answer: 2
Problem: If \(a+b=b-c=c-a=3\), find the value of \(a+b+c\) in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of \(k + m\).
Answer: 11
Problem: A regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two sides of length $8$. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid? Express your answer in the form $a\sqrt{b}$, where $a$ and $b$ are integers. Please provide the value of $a + b$.
Answer: 57
Problem: Any five points are taken inside or on a square with side length \(1\). Let \(a\) be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than \(a\). The original answer is in the format \(\frac{\sqrt{n}}{m}\). Please calculate the value of \(k + m + n\).
Answer: 5
Problem: A man travels $m$ feet due north at $2$ minutes per mile. He returns due south to his starting point at $2$ miles per minute. Find the average rate in miles per hour for the entire trip.
Answer: 48
Problem: The polynomial \(P(x) = x^3 + ax^2 + bx + c\) has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the \(y\)-intercept of the graph of \(y = P(x)\) is 2, what is the value of \(b\)?
Answer: -11
Problem: The base of an isosceles triangle is $\sqrt{2}$. The medians to the legs intersect each other at right angles. If the area of the triangle is $x$, what is the value of $10x$?
Answer: 15
Problem: How many positive integers less than $1000$ are $6$ times the sum of their digits?
Answer: 1
Problem: The number \(21! = 51,090,942,171,709,440,000\) has over \(60,000\) positive integer divisors. One of them is chosen at random. What is the probability that it is odd? Provide your answer as a fraction in simplest form. The answer should be in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\).
Answer: 20
Problem: In their base $10$ representation, the integer $a$ consists of a sequence of $1985$ eights and the integer $b$ consists of a sequence of $1985$ fives. Find the sum of the digits of the base 10 representation of $9ab$.
Answer: 17865
Problem: In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. The original answer for the perimeter of the hexagon is in the form $k\sqrt{3}$. Please find the value of $k$.
Answer: 12
Problem: The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$. If the first term is an integer, and $n > 1$, find the number of possible values for $n$.
Answer: 5
Problem: An *iterative average* of the numbers \(1\), \(2\), \(3\), \(4\), and \(5\) is computed in the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. Find the difference between the largest and smallest possible values that can be obtained using this procedure. The answer is in the form \(\frac{k}{m}\). Please provide the value of \(k + m\).
Answer: 25
Problem: If the area of \(\triangle ABC\) is \(64\) square units and the geometric mean (mean proportional) between sides \(AB\) and \(AC\) is \(12\) inches, find the value of \(\sin A\). The answer should be in the format \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of \(k + m\).
Answer: 17
Problem: The integers $-1$, $2$, $-3$, $4$, $-5$, $6$ are written on a blackboard. At each move, we erase two numbers $a$ and $b$, then we re-write $2a+b$ and $2b+a$. Find how many of the sextuples $(0,0,0,3,-9,9)$, $(0,1,1,3,6,-6)$, $(0,0,0,3,-6,9)$, $(0,1,1,-3,6,-9)$, $(0,0,2,5,5,6)$ can be gotten.
Answer: 1
Problem: Each morning of her five-day workweek, Jane bought either a $50$-cent muffin or a $75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?
Answer: 2
Problem: A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ as shown, where $AB = 1$. If the radius of the circle is $r$, find the value of $100r$ to the nearest integer.
Answer: 237
Problem: Doug can paint a room in $5$ hours. Dave can paint the same room in $7$ hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $t$ be the total time, in hours, required for them to complete the job working together, including lunch. The equation satisfied by $t$ is of the form \(\left(\frac{k}{m}+\frac{1}{n}\right)(t-p)=q\). Please find the value of \(k + m + n + p + q\).
Answer: 15
Problem: A rhombus is inscribed in triangle $ABC$ such that one of its vertices is $A$ and two of its sides lie along $AB$ and $AC$. If $\overline{AC} = 6$ inches, $\overline{AB} = 12$ inches, and $\overline{BC} = 8$ inches, find the side of the rhombus in inches.
Answer: 4
Problem: Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of 6 meters, and it takes her 36 seconds longer to walk around the outside edge of the track than around the inside edge. If Keiko's speed is \(x\) meters per second, the original answer is in the form \(\frac{k\pi}{m}\). Please find the value of \(k + m\).
Answer: 4
Problem: The distance light travels in one year is approximately \( 5,870,000,000,000 \) miles. If the distance light travels in \( 100 \) years is expressed as \( x \times 10^{12} \text{ miles} \), what is the value of \( x \)?
Answer: 587
Problem: In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200.$

$$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots, 200, 200, \ldots, 200$$

Find the median of the numbers in this list.
Answer: 142
Problem: A park is in the shape of a regular hexagon with sides of \(2\) km each. Starting at a corner, Alice walks along the perimeter of the park for a distance of \(5\) km. How many kilometers is she from her starting point? If the answer is \( \sqrt{x} \), what is the value of \( x \)?
Answer: 13
Problem: There are $2012$ backgammon checkers (stones, pieces) with one side black and the other side white. These checkers are arranged in a line such that no two consecutive checkers are the same color. At each move, we choose two checkers and flip them along with all the checkers between them. What is the minimum number of moves required to make all checkers the same color?
Answer: 1006
Problem: What is the smallest integer $m$ such that $\frac{10!}{m}$ is a perfect square? Provide your answer as an integer.
Answer: 7
Problem: What is the value of $o-w$, if $gun^2 = wowgun$ where $g,n,o,u,w \in \{0,1,2,\dots, 9\}$?
Answer: 3
Problem: Let line $AC$ be perpendicular to line $CE$. Connect $A$ to $D$, the midpoint of $CE$, and connect $E$ to $B$, the midpoint of $AC$. If $AD$ and $EB$ intersect at point $F$, and $\overline{BC} = \overline{CD} = 15$ inches, then find the area of triangle $DFE$ in square inches.
Answer: 75
Problem: Find the number of roots satisfying the equation $\sqrt{5 - x} = x\sqrt{5 - x}$.
Answer: 2
Problem: How many ordered positive integer triples $(x, y, z)$ are there such that the following system of equations holds?
\[
\begin{align*}
3x^2 - 2y^2 - 4z^2 + 54 &= 0, \\
5x^2 - 3y^2 - 7z^2 + 74 &= 0.
\end{align*}
\]
Provide your answer as an integer.
Answer: 2
Problem: Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. If the answer is the total number of minutes from 1:00 PM to when she finishes the third task, what is the value of this number?
Answer: 150
Problem: Two distinct numbers \(a\) and \(b\) are chosen randomly from the set \(\{2, 2^2, 2^3, \ldots, 2^{25}\}\). What is the probability that \(\log_{a} b\) is an integer? Express your answer as a fraction in simplest form, \(\frac{k}{m}\), and find the value of \(k + m\).
Answer: 331
Problem: Ajay is standing at point $A$ near Pontianak, Indonesia, at $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, at $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C$. If the answer is $x^\circ$, what is the value of $x$?
Answer: 120
Problem: Let $ABCD$ be a parallelogram with $\angle ABC=120^\circ$, $AB=16$, and $BC=10$. Extend $\overline{CD}$ through $D$ to $E$ so that $DE=4$. If $\overline{BE}$ intersects $\overline{AD}$ at $F$, then find the length of $FD$.
Answer: 2
Problem: If $y = \sqrt{x^2 - 2x + 1} + \sqrt{x^2 + 2x + 1}$, find the expression for $y$ in terms of $x$. The original answer is in the form |ax + b| + |cx + d|. Please give the value of a + b + c + d.
Answer: 2
Problem: The measures (in degrees) of the interior angles of a convex hexagon form an arithmetic sequence of positive integers. Let $m^{\circ}$ be the measure of the largest interior angle of the hexagon. If the answer is $x^{\circ}$, what is the largest possible value of $x$?
Answer: 175
Problem: A $16$-quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts of the mixture are removed and replaced with pure antifreeze. This process is repeated a third and a fourth time. Find the fractional part of the final mixture that is water. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 337
Problem: Consider the rectangular strip of length $12$, divided into three rectangles. The distance between the centers of two of the rectangles is $4$. What is the length of the other rectangle?
Answer: 4
Problem: Let \(a, b, c, d\) be rational numbers with \(a > 0\). If for every integer \(n \ge 0\), the number \(an^{3} + bn^{2} + cn + d\) is also an integer, find the minimal value of \(a\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 7
Problem: The sum of the distances from one vertex of a square with sides of length two to the midpoints of each of the sides of the square is in the form k + m\sqrt{n}. Please find the value of k + m + n.
Answer: 9
Problem: A circular disc with diameter $D$ is placed on an $8 \times 8$ checkerboard with width $D$ so that the centers coincide. Find the number of checkerboard squares which are completely covered by the disc.
Answer: 32
Problem: Vertex \(E\) of equilateral triangle \(ABE\) is in the interior of square \(ABCD\), and \(F\) is the point of intersection of diagonal \(BD\) and line segment \(AE\). If length \(AB\) is \(\sqrt{1+\sqrt{3}}\), then find the area of \(\triangle ABF\). The original answer is in the form \(\frac{\sqrt{k}}{m}\), please give the value of k + m.
Answer: 5