sungyub/dapo-math-17k-verl huggingface dataset:
user: Let $p_1 = 2, p_2 = 3, p_3 = 5 ...$ be the sequence of prime numbers. Find the least positive even integer $n$ so that $p_1 + p_2 + p_3 + ... + p_n$ is not prime.

Answer: 8
user: The polynomial $P(x)=a_0+a_1x+a_2x^2+...+a_8x^8+2009x^9$ has the property that $P(\tfrac{1}{k})=\tfrac{1}{k}$ for $k=1,2,3,4,5,6,7,8,9$. There are relatively prime positive integers $m$ and $n$ such that $P(\tfrac{1}{10})=\tfrac{m}{n}$. Find $n-10m$.

Answer: 20090
user: Towers grow at points along a line. All towers start with height 0 and grow at the rate of one meter per second. As soon as any two adjacent towers are each at least 1 meter tall, a new tower begins to grow at a point along the line exactly half way between those two adjacent towers. Before time 0 there are no towers, but at time 0 the first two towers begin to grow at two points along the line. Find the total of all the heights of all towers at time 10 seconds.

Answer: 1033
user: How many distinct four letter arrangements can be formed by rearranging the letters found in the word [b]FLUFFY[/b]? For example, FLYF and ULFY are two possible arrangements.

Answer: 72
user: Bill bought 13 notebooks, 26 pens, and 19 markers for 25 dollars. Paula bought 27 notebooks, 18 pens, and 31 markers for 31 dollars. How many dollars would it cost Greg to buy 24 notebooks, 120 pens, and 52 markers?

Answer: 88
user: Five men and seven women stand in a line in random order. Let m and n be relatively prime positive integers so that $\tfrac{m}{n}$ is the probability that each man stands next to at least one woman. Find $m + n.$

Answer: 287
user: Let $ABCD$ be a trapezoid where $AB$ is parallel to $CD.$ Let $P$ be the intersection of diagonal $AC$ and diagonal $BD.$ If the area of triangle $PAB$ is $16,$ and the area of triangle $PCD$ is $25,$ find the area of the trapezoid.

Answer: 81
user: Find the smallest possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions 

$\star$ each of the pairs of integers $(a, b), (b, c), (c, d),$ and $(d, e)$ are [b]not[/b] relatively prime

$\star$ all other pairs of the five integers [b]are[/b] relatively prime.

Answer: 47
user: There are positive integers $b$ and $c$ such that the polynomial $2x^2 + bx + c$ has two real roots which differ by $30.$ Find the least possible value of $b + c.$

Answer: 126
user: If $a$ and $b$ are positive integers such that $a \cdot b = 2400,$ find the least possible value of $a + b.$

Answer: 98
user: Suppose that $f$ is a function such that $3f(x)- 5xf \left(\frac{1}{x}\right)= x - 7$ for all non-zero real numbers $x.$ Find $f(2010).$

Answer: 4021
user: The prime factorization of $12 = 2 \cdot 2 \cdot 3$ has three prime factors. Find the number of prime factors in the factorization of $12! = 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1.$

Answer: 19
user: Let $S$ be the set of all $10-$term arithmetic progressions that include the numbers $4$ and $10.$ For example, $(-2,1, 4,7,10,13, 16,19,22,25)$ and $(10,8\tfrac{1}{2}, 7,5\tfrac{1}{2}, 4, 2\tfrac{1}{2},1,\tfrac{1}{2},-2, -3\tfrac{1}{2})$ are both members of $S.$ Find the sum of all values of $a_{10}$ for each $(a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, a_{10}) \in S,$ that is, $\sum_{a_1, a_2, a_3, ... , a_{10} \in S} a_{10}.$

Answer: 630
user: In the number arrangement
\[\begin{array}{ccccc}
\texttt{1}&&&&\\
\texttt{2}&\texttt{3}&&&\\
\texttt{4}&\texttt{5}&\texttt{6}&&\\
\texttt{7}&\texttt{8}&\texttt{9}&\texttt{10}&\\
\texttt{11}&\texttt{12}&\texttt{13}&\texttt{14}&\texttt{15}\\
\vdots&&&&
\end{array}\]
what is the number that will appear directly below the number $2010$?

Answer: 2073
user: Find the number of ordered pairs of integers $(m, n)$ that satisfy $20m-10n = mn$.

Answer: 24
user: When $4 \cos \theta - 3 \sin \theta = \tfrac{13}{3},$ it follows that $7 \cos 2\theta - 24 \sin 2\theta = \tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Answer: 122
user: Sally's salary in 2006 was $\$37,500$. For 2007 she got a salary increase of $x$ percent. For 2008 she got another salary increase of $x$ percent. For 2009 she got a salary decrease of $2x$ percent. Her 2009 salary is $\$34,825$. Suppose instead, Sally had gotten a $2x$ percent salary decrease for 2007, an $x$ percent salary increase for 2008, and an $x$ percent salary increase for 2009. What would her 2009 salary be then?

Answer: 34825
user: There are relatively prime positive integers $p$ and $q$ such that $\dfrac{p}{q}=\displaystyle\sum_{n=3}^{\infty} \dfrac{1}{n^5-5n^3+4n}$. Find $p+q$.

Answer: 97
user: Let $a$ and $b$ be real numbers satisfying $2(\sin a + \cos a) \sin b = 3 - \cos b$. Find $3 \tan^2a+4\tan^2 b$.

Answer: 35
user: Alan, Barb, Cory, and Doug are on the golf team, Doug, Emma, Fran, and Greg are on the swim team, and Greg, Hope, Inga, and Alan are on the tennis team. These nine people sit in a circle in random order. The probability that no two people from the same team sit next to each other is $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Answer: 2521
user: In the coordinate plane a parabola passes through the points $(7,6)$, $(7,12)$, $(18,19)$, and $(18,48)$. The axis of symmetry of the parabola is a line with slope $\tfrac{r}{s}$ where r and s are relatively prime positive integers. Find $r + s$.

Answer: 71
user: There are exactly two four-digit numbers that are multiples of three where their first digit is double their second digit, their third digit is three more than their fourth digit, and their second digit is $2$ less than their fourth digit. Find the difference of these two numbers.

Answer: 6333
user: The sum $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}=\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Answer: 69
user: What percent of the numbers $1, 2, 3, ... 1000$ are divisible by exactly one of the numbers $4$ and $5?$

Answer: 35
user: Find the least positive integer $k$ so that $k + 25973$ is a palindrome (a number which reads the same forward and backwards).

Answer: 89
user: Half the volume of a 12 foot high cone-shaped pile is grade A ore while the other half is grade B ore. The pile is worth \$62. One-third of the volume of a similarly shaped 18 foot pile is grade A ore while the other two-thirds is grade B ore. The second pile is worth \$162. Two-thirds of the volume of a similarly shaped 24 foot pile is grade A ore while the other one-third is grade B ore. What is the value in dollars (\$) of the 24 foot pile?

Answer: 608
user: There are two rows of seats with three side-by-side seats in each row. Two little boys, two little girls, and two adults sit in the six seats so that neither little boy sits to the side of either little girl. In how many different ways can these six people be seated?

Answer: 176
user: Let $x$ and $y$ be real numbers satisfying \[(x^2+x-1)(x^2-x+1)=2(y^3-2\sqrt{5}-1)\] and \[(y^2+y-1)(y^2-y+1)=2(x^3+2\sqrt{5}-1)\] Find $8x^2+4y^3$.

Answer: 20
user: The centers of the three circles A, B, and C are collinear with the center of circle B lying between the centers of circles A and C. Circles A and C are both externally tangent to circle B, and the three circles share a common tangent line. Given that circle A has radius $12$ and circle B has radius $42,$ find the radius of circle C.

Answer: 147
user: Let $a$ be the sum of the numbers:

$99 \times 0.9$

$999 \times 0.9$

$9999 \times 0.9$

$\vdots$

$999\cdots 9 \times 0.9$

where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$.
Find the sum of the digits in the number $a$.

Answer: 891
user: Find the number of sets $A$ that satisfy the three conditions:
$\star$ $A$ is a set of two positive integers
$\star$ each of the numbers in $A$ is at least $22$ percent the size of the other number
$\star$ $A$ contains the number $30.$

Answer: 129
user: Evaluate the sum $1 + 2 - 3 + 4 + 5 - 6 + 7 + 8 - 9 \cdots + 208 + 209 - 210.$

Answer: 7245
user: Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$. There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$. Find $m+n$.

Answer: 10
user: Five congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two neighbors. A sixth circle (shaded in the diagram below) congruent to the other five is placed tangent to two of the five. If this sixth circle is allowed to roll without slipping around the exterior of the figure formed by the other five circles, then it will turn through an angle of $k$ degrees before it returns to its starting position. Find $k$.

Answer: 1320
user: Find the remainder when $A=3^3\cdot 33^{33}\cdot 333^{333}\cdot 3333^{3333}$ is divided by $100$.

Answer: 19
user: Let $x$ be a real number in the interval $\left(0,\dfrac{\pi}{2}\right)$ such that $\dfrac{1}{\sin x \cos x}+2\cot 2x=\dfrac{1}{2}$. Evaluate $\dfrac{1}{\sin x \cos x}-2\cot 2x$.

Answer: 8
user: A $3$ by $3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant.

Answer: 405
user: How many numbers are there that appear both in the arithmetic sequence $10,
16, 22, 28, ... 1000$ and the arithmetic sequence $10, 21, 32, 43, ..., 1000?$

Answer: 16
user: Find the smallest prime number that does not divide \[9+9^2+9^3+\cdots+9^{2010}.\]

Answer: 17
user: Let $a$ be a positive real number such that $\tfrac{a^2}{a^4-a^2+1}=\tfrac{4}{37}$. Then $\tfrac{a^3}{a^6-a^3+1}=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 259
user: Find the area of the region in the coordinate plane satisfying the three conditions

$\star$  $x \le 2y$
$\star$ $y \le 2x$
$\star$ $x + y \le 60.$

Answer: 600
user: Five rays $\overrightarrow{OA}$,$\overrightarrow{OB}$, $\overrightarrow{OC}$, $\overrightarrow{OD}$, and $\overrightarrow{OE}$ radiate in a clockwise order from $O$ forming four non-overlapping angles such that $\angle EOD = 2\angle COB$, $\angle COB = 2\angle BOA$, while $\angle DOC = 3\angle BOA$. If $E$, $O$, $A$ are collinear with $O$ between $A$ and $E$, what is the degree measure of $\angle DOB?$

Answer: 90
user: Six distinct positive integers are randomly chosen between $1$ and $2011;$ inclusive. The probability that some pair of the six chosen integers has a difference that is a multiple of $5 $ is $n$ percent. Find $n.$

Answer: 100
user: Find the number of ordered quadruples $(a, b, c, d)$ where each of $a, b, c,$ and $d$ are (not necessarily distinct) elements of $\{1, 2, 3, 4, 5, 6, 7\}$ and $3abc + 4abd + 5bcd$ is even. For example, $(2, 2, 5, 1)$ and $(3, 1, 4, 6)$ satisfy the conditions.

Answer: 2017
user: The diagram below shows two parallel rows with seven points in the upper row and nine points in the lower row. The points in each row are spaced one unit apart, and the two rows are two units apart. How many trapezoids which are not parallelograms have vertices in this set of $16$ points and have area of at least six square units?

Answer: 361
user: In how many distinguishable rearrangements of the letters ABCCDEEF does the A precede both C's, the F appears between the 2 C's, and the D appears after the F?

Answer: 336
user: The diagram below shows a regular hexagon with an inscribed square where two sides of the square are parallel to two sides of the hexagon. There are positive integers $m$, $n$, and $p$ such that the ratio of the area of the hexagon to the area of the square can be written as $\tfrac{m+\sqrt{n}}{p}$ where $m$ and $p$ are relatively prime. Find $m + n + p$.

Answer: 19
user: Four congruent spheres are stacked so that each is tangent to the other three. A larger sphere, $R$, contains the four congruent spheres so that all four are internally tangent to $R$. A smaller sphere, $S$, sits in the space between the four congruent spheres so that all four are externally tangent to $S$. The ratio of the surface area of $R$ to the surface area of $S$ can be written $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.

Answer: 2449
user: How many ordered pairs of sets $(A, B)$ have the properties:

$ A\subseteq \{1, 2, 3, 4, 5, 6\} $
$ B\subseteq\{2, 3, 4, 5, 6, 7, 8\} $
$ A\cap B $ has exactly $3$ elements.

Answer: 720
user: Let $a$ and $b$ be nonzero real numbers such that $\tfrac{1}{3a}+\tfrac{1}{b}=2011$ and $\tfrac{1}{a}+\tfrac{1}{3b}=1$. What is the quotient when $a+b$ is divided by $ab$?

Answer: 1509
user: Let $V$ be the set of vertices of a regular $25$ sided polygon with center at point $C.$ How many triangles have vertices in $ V$ and contain the point $C$ in the interior of the triangle?

Answer: 650
user: The diagram below shows a $12$-sided figure made up of three congruent squares. The figure has total perimeter $60$. Find its area.

Answer: 75
user: A jar contains one quarter red jelly beans and three quarters blue jelly beans. If Chris removes three quarters of the red jelly beans and one quarter of the blue jelly beans, what percent of the jelly beans remaining in the jar will be
red?

Answer: 10
user: Evaluate $1^3-2^3+3^3-4^3+5^3-\cdots+101^3$.

Answer: 522801
user: Points $A$ and $B$ are the endpoints of a diameter of a circle with center $C$. Points $D$ and $E$ lie on the same diameter so that $C$ bisects segment $\overline{DE}$. Let $F$ be a randomly chosen point within the circle. The probability that $\triangle DEF$ has a perimeter less than the length of the diameter of the circle is $\tfrac{17}{128}$. There are relatively prime positive integers m and n so that the ratio of $DE$ to $AB$ is $\tfrac{m}{n}.$ Find $m + n$.

Answer: 47
user: When $126$ is added to its reversal, $621,$ the sum is $126 + 621 = 747.$ Find the greatest integer which when added to its reversal yields $1211.$

Answer: 952
user: Jerry buys a bottle of 150 pills. Using a standard 12 hour clock, he sees that the clock reads exactly 12 when he takes the first pill. If he takes one pill every five hours, what hour will the clock read when he takes the last pill in
the bottle?

Answer: 1
user: Let $S$ be a randomly selected four-element subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Let $m$ and $n$ be relatively prime positive integers so that the expected value of the maximum element in $S$ is $\dfrac{m}{n}$. Find $m + n$.

Answer: 41
user: How many positive integer solutions are there to $w+x+y+z=20$ where $w+x\ge 5$ and $y+z\ge 5$?

Answer: 781
user: Find the smallest positive integer whose remainder when divided by $5$ is $3$, when divided by $7$ is $5$, and when divided by $9$ is $1$.

Answer: 208
user: Consider a sequence of eleven squares that have side lengths $3, 6, 9, 12,\ldots, 33$. Eleven copies of a single square each with area $A$ have the same total area as the total area of the eleven squares of the sequence. Find $A$.

Answer: 414
user: In the tribe of Zimmer, being able to hike long distances and knowing the roads through the forest are both extremely important, so a boy who reaches the age of manhood is not designated as a man by the tribe until he completes an interesting rite of passage. The man must go on a sequence of hikes. The first hike is a $5$ kilometer hike down the main road. The second hike is a $5\frac{1}{4}$ kilometer hike down a secondary road. Each hike goes down a different road and is a quarter kilometer longer than the previous hike. The rite of passage is completed at the end of the hike where the cumulative distance walked by the man on all his hikes exceeds $1000$ kilometers. So in the tribe of Zimmer, how many roads must a man walk down, before you call him a man?

Answer: 73
user: A bag contains $8$ green candies and $4$ red candies. You randomly select one candy at a time to eat. If you eat five candies, there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that you do not eat a green candy after you eat a red candy. Find $m+n$.

Answer: 6
user: Let $A=\{1, 3, 5, 7, 9\}$ and $B=\{2, 4, 6, 8, 10\}$. Let $f$ be a randomly chosen function from the set $A\cup B$ into itself. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probablity that $f$ is a one-to-one function on $A\cup B$ given that it maps $A$ one-to-one into $A\cup B$ and it maps $B$ one-to-one into $A\cup B$. Find $m+n$.

Answer: 253
user: Find the value of $x$ that satisfies $\log_{3}(\log_9x)=\log_9(\log_3x)$

Answer: 81
user: Ted flips seven fair coins. there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that Ted flips at least two heads given that he flips at least three tails. Find $m+n$.

Answer: 190
user: Find the remainder when $2^{5^9}+5^{9^2}+9^{2^5}$ is divided by $11$.

Answer: 8
user: Find the number of three-digit numbers such that its first two digits are each divisible by its third digit.

Answer: 138
user: The following sequence lists all the positive rational numbers that do not exceed $\frac12$ by first listing the fraction with denominator 2, followed by the one with denominator 3, followed by the two fractions with denominator 4 in increasing order, and so forth so that the sequence is
\[
\frac12,\frac13,\frac14,\frac24,\frac15,\frac25,\frac16,\frac26,\frac36,\frac17,\frac27,\frac37,\cdots.
\]
Let $m$ and $n$ be relatively prime positive integers so that the $2012^{\text{th}}$ fraction in the list is equal to $\frac{m}{n}$. Find $m+n$.

Answer: 61
user: Seven boys and three girls are playing basketball. I how many different ways can they make two teams of five players so that both teams have at least one girl?

Answer: 105
user: Meredith drives 5 miles to the northeast, then 15 miles to the southeast, then 25 miles to the southwest, then 35 miles to the northwest, and finally 20 miles to the northeast. How many miles is Meredith from where she started?

Answer: 20
user: In the following addition, different letters represent different non-zero digits. What is the 5-digit number $ABCDE$?

$
\begin{array}{ccccccc}
A&B&C&D&E&D&B\\
&B&C&D&E&D&B\\
&&C&D&E&D&B\\
&&&D&E&D&B\\
&&&&E&D&B\\
&&&&&D&B\\
+&&&&&&B\\ \hline
A&A&A&A&A&A&A
\end{array}
$

Answer: 84269
user: Points $E$ and $F$ lie inside rectangle $ABCD$ with $AE=DE=BF=CF=EF$. If $AB=11$ and $BC=8$, find the area of the quadrilateral $AEFB$.

Answer: 32
user: Find the least positive multiple of 999 that does not have a 9 as a digit.

Answer: 111888
user: Find the largest prime that divides $1\cdot 2\cdot 3+2\cdot 3\cdot 4+\cdots +44\cdot 45\cdot 46$

Answer: 47
user: A teacher suggests four possible books for students to read. Each of six students selects one of the four books. How many ways can these selections be made if each of the books is read by at least one student?

Answer: 1560
user: Each time you click a toggle switch, the switch either turns from [i]off[/i] to [i]on[/i] or from [i]on[/i] to [i]off[/i]. Suppose that you start with three toggle switches with one of them [i]on[/i] and two of them [i]off[/i]. On each move you randomly select one of the three switches and click it. Let $m$ and $n$ be relatively prime positive integers so that $\frac{m}{n}$ is the probability that after four such clicks, one switch will be [i]on[/i] and two of them will be [i]off[/i]. Find $m+n$.

Answer: 142
user: Find the least positive integer $n$ so that both $n$ and $n+1$ have prime factorizations with exactly four (not necessarily distinct) prime factors.

Answer: 135
user: For some integers $a$ and $b$ the function $f(x)=ax+b$ has the properties that $f(f(0))=0$ and $f(f(f(4)))=9$. Find $f(f(f(f(10))))$.

Answer: 10
user: A circle in the first quadrant with center on the curve $y=2x^2-27$ is tangent to the $y$-axis and the line $4x=3y$. The radius of the circle is $\frac{m}{n}$ where $M$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 11
user: At the 4 PM show, all the seats in the theater were taken, and 65 percent of the audience was children. At the 6 PM show, again, all the seats were taken, but this time only 50 percent of the audience was children. Of all the people who attended either of the shows, 57 percent were children although there were 12 adults and 28 children who attended both shows. How many people does the theater seat?

Answer: 520
user: A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$.

Answer: 203
user: Floyd looked at a standard $12$ hour analogue clock at $2\!:\!36$. When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved added to $247$ degrees. How many minutes after $3\!:\!00$ was that?

Answer: 14
user: Suppose $x,y$ and $z$ are integers that satisfy the system of equations \[x^2y+y^2z+z^2x=2186\] \[xy^2+yz^2+zx^2=2188.\] Evaluate $x^2+y^2+z^2.$

Answer: 245
user: Let $z$ be a complex number satisfying $(z+\tfrac{1}{z})(z+\tfrac{1}{z}+1)=1$. Evaluate $(3z^{100}+\tfrac{2}{z^{100}}+1)(z^{100}+\tfrac{2}{z^{100}}+3)$.

Answer: 36
user: A picture with an area of $160$ square inches is surrounded by a $2$ inch border. The picture with its border is a rectangle twice as long as it is wide. How many inches long is that rectangle?

Answer: 24
user: Six children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that no two children who stood next to each other in the line end up sitting next to each other in the circle. Find $m + n$.

Answer: 13
user: There is a pile of eggs. Joan counted the eggs, but her count was way off by $1$ in the $1$'s place. Tom counted in the eggs, but his count was off by $1$ in the $10$'s place. Raoul counted the eggs, but his count was off by $1$ in the $100$'s place. Sasha, Jose, Peter, and Morris all counted the eggs and got the correct count. When these seven people added their counts together, the sum was $3162$. How many eggs were in the pile?

Answer: 439
user: Find the least six-digit palindrome that is a multiple of $45$. Note that a palindrome is a number that reads the same forward and backwards such as $1441$ or $35253$.

Answer: 504405
user: How many ordered triples $(a, b, c)$ of positive integers satisfy $a \le b  \le c$ and $a  \cdot b\cdot  c = 1000$?

Answer: 19
user: In how many rearrangements of the numbers $1, \ 2, \ 3, \ 4, \ 5,\ 6, \ 7, \ 8,\ 9$ do the numbers form a $\textit{hill}$, that is, the numbers form an increasing sequence at the beginning up to a peak, and then form a decreasing sequence to the end such as in $129876543$ or $258976431$?

Answer: 254
user: In the following diagram two sides of a square are tangent to a circle with diameter $8$. One corner of the square lies on the circle. There are positive integers $m$ and $n$ so that the area of the square is $m +\sqrt{n}$. Find $m + n$.
import graph;
size(4.4cm); 
real labelscalefactor = 0.5;
pen dotstyle = black;
filldraw((-0.707106781,0.707106781)--(-0.707106781,-1)--(1,-1)--(1,0.707106781)--cycle,gray, linewidth(1.4));
draw(circle((0,0),1), linewidth(1.4));
[/asy]

Answer: 134
user: For positive integers $m$ and $n$, the decimal representation for the fraction $\tfrac{m}{n}$ begins $0.711$ followed by other digits. Find the least possible value for $n$.

Answer: 45
user: Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits.

Answer: 397
user: How many seven-digit positive integers do not either start or end with $7$?

Answer: 7200000