meetforever/GRPO_CL huggingface dataset:
user: 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
user: 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
user: 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
user: 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
user: 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
user: For how many positive integers $m$ is \[ \frac{2002}{m^2-2} \] a positive integer? Write your answer as a number.
Answer: 3
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: What are the last two digits of the decimal representation of \(9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}\)?
Answer: 21
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: How many positive integers less than $1000$ are $6$ times the sum of their digits?
Answer: 1
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: What is the smallest integer $m$ such that $\frac{10!}{m}$ is a perfect square? Provide your answer as an integer.
Answer: 7
user: 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
user: 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
user: Find the number of roots satisfying the equation $\sqrt{5 - x} = x\sqrt{5 - x}$.
Answer: 2
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: How many $10$-digit positive integers can be written by using four $0$s, five $1$s, and one $2$?
Answer: 756
user: An uncrossed belt is fitted without slack around two circular pulleys with radii of $14$ inches and $4$ inches. If the distance between the points of contact of the belt with the pulleys is $24$ inches, find the distance between the centers of the pulleys in inches.
Answer: 26
user: What is the minimum value of \( \sqrt{x^2 - 4x + 7 - 2\sqrt{2}} + \sqrt{x^2 - 8x + 27 - 6\sqrt{2}} \) where \( x \) is a real number? The original answer is in the form \( k\sqrt{2} \), please find the value of \( k \).
Answer: 2
user: The fraction \(\frac{\sqrt{a^2+x^2}-(x^2-a^2)/\sqrt{a^2+x^2}}{a^2+x^2}\) reduces to a simpler form. The reduced expression is in the form \(\frac{k a^2}{(a^2 + x^2)^{\frac{m}{n}}}\). Please find the value of \(k + m + n + p\).
Answer: 7
user: The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n \geq 3$, $F_n$ is constructed from $F_{n-1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n-1}$ had on each side of its outside square. For example, figure $F_3$ has $13$ diamonds. How many diamonds are there in figure $F_{20}$?
Answer: 761
user: What is the least real number $C$ that satisfies $\sin x \cos x \leq C(\sin^6x+\cos^6x)$ for every real number $x$?
Answer: 2
user: In quadrilateral \(ABCD\), \(m\angle B = m\angle C = 120^\circ\), \(AB = 3\), \(BC = 4\), and \(CD = 5\). Find the area of \(ABCD\) in the form \(\frac{k\sqrt{3}}{m}\), where \(k\) and \(m\) are integers. Please provide the value of \(k + m\).
Answer: 51
user: A positive integer $n$ not exceeding $100$ is chosen such that if $n \leq 50$, the probability of choosing $n$ is $p$, and if $n > 50$, the probability of choosing $n$ is $3p$. What is the value of the probability that a perfect square is chosen multiplied by 100?
Answer: 8
user: A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. Find the number of dates in $2023$ for which each digit appears an even number of times in the 8-digit display for that date.
Answer: 9
user: A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $\$17.71$. What was the cost of a pencil in cents?
Answer: 11
user: Let $a, b, c, d, e, f, g,$ and $h$ be distinct elements in the set:
\[
\{-7, -5, -3, -2, 2, 4, 6, 13\}.
\]
Find the minimum possible value of:
\[
(a + b + c + d)^2 + (e + f + g + h)^2
\]
Answer: 34
user: What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 = 25$ and $(x-4)^2 + 9y^2 = 81$?
Answer: 27
user: A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$. Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set $\{V, W, X, Y, Z\}$. Using this correspondence, the cryptographer finds that three consecutive integers in increasing order are coded as $VYZ$, $VYX$, $VVW$, respectively. What is the base-10 expression for the integer coded as $XYZ$?
Answer: 108
user: Let $ABC$ be a triangle and $AD$ be the bisector of the triangle ($D \in (BC)$). Assume that $AB = 14$ cm, $AC = 35$ cm, and $AD = 12$ cm. What is the area of triangle $ABC$ in cm$^2$? The original answer is in \frac{k}{m} format, please give the value of k + m.
Answer: 1181
user: A $4 \times 4$ block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?
Answer: 4
user: In triangle $ABC$ shown in the adjoining figure, $M$ is the midpoint of side $BC$, $AB=12$ and $AC=16$. Points $E$ and $F$ are taken on $AC$ and $AB$, respectively, and lines $EF$ and $AM$ intersect at $G$. If $AE=2AF$, find the value of \(\frac{EG}{GF}\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 5
user: Two congruent squares are packed into an isosceles right triangle as shown below. Each of the squares has an area of 10. What is the area of the triangle?

```asy
\draw((0,0) -- (3\sqrt{10}, 0) -- (0, 3\sqrt{10}) -- cycle);
\draw((0,0) -- (2\sqrt{10}, 0) -- (2\sqrt{10}, \sqrt{10}) -- (0, \sqrt{10}));
\draw((\sqrt{10}, \sqrt{10}) -- (\sqrt{10}, 0));
```
Answer: 45
user: An engineer said he could finish a highway section in $3$ days with his present supply of a certain type of machine. However, with $3$ more of these machines, the job could be done in $2$ days. If the machines all work at the same rate, how many days would it take to do the job with one machine?
Answer: 18
user: There are exactly $N$ distinct rational numbers $k$ such that $|k| < 200$ and \[5x^2 + kx + 12 = 0\] has at least one integer solution for $x$. What is the value of $N$?
Answer: 78
user: What is the area of the region defined by the inequality $|3x-18|+|2y+7| \leq 3$?
Answer: 3
user: Kyle found the sum of the digits of $2014^{2014}$. Then, Shannon found the sum of the digits of Kyle's result. Finally, James found the sum of the digits of Shannon's result. What number did James find?
Answer: 7
user: In how many ways can $7$ red and $7$ white balls be distributed into $7$ boxes such that every box contains exactly $2$ balls?
Answer: 393
user: Let $f$ be a function from the non-negative integers to the non-negative integers such that $f(nm) = n f(m) + m f(n)$, with the following conditions: $f(10) = 19$, $f(12) = 52$, and $f(15) = 26$. What is the value of $f(8)$?
Answer: 36
user: A plane contains points \(A\) and \(B\) with \(AB = 1\). Let \(S\) be the union of all disks of radius \(1\) in the plane that cover \(\overline{AB}\). The original answer is in the form \(k\pi - \frac{\sqrt{m}}{n}\). Please find the value of \(k + m + n\).
Answer: 8
user: How many integers $n$ are there such that $n^3 + 8$ has at most 3 positive divisors? Provide your answer as an integer.
Answer: 3
user: Lines $L_1, L_2, \dots, L_{100}$ are distinct. All lines $L_{4n}$, where $n$ is a positive integer, are parallel to each other. All lines $L_{4n-3}$, where $n$ is a positive integer, pass through a given point $A$. Find the maximum number of points of intersection of pairs of lines from the complete set $\{L_1, L_2, \dots, L_{100}\}$.
Answer: 4351
user: The ratio of the radii of two concentric circles is $1:3$. If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB = 12$, find the radius of the larger circle.
Answer: 18
user: Sequence $(a_n)$ is defined as $a_{n+1} - 2a_n + a_{n-1} = 7$ for every $n \geq 2$, where $a_1 = 1$, $a_2 = 5$. Find the value of $a_{17}$.
Answer: 905
user: Given points \(P(-1,-2)\) and \(Q(4,2)\) in the \(xy\)-plane; point \(R(1,m)\) is taken so that \(PR+RQ\) is a minimum. The original answer is in the form \(-\frac{a}{b}\), where a and b are integers and the fraction is in its simplest form. Please find the value of \(a + b\).
Answer: 7
user: What is the value of 100 times the expression $\sqrt{65} - \sqrt{63}$ rounded to the nearest integer?
Answer: 13