ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. The original answer is in the form \frac{k}{m}, where \frac{k}{m} is a fully reduced fraction. Please find the value of k + m.
Answer: 3
Problem: When counting from $3$ to $201$, $53$ is the $51^{st}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{th}$ number counted. What is the value of $n$?
Answer: 149
Problem: $A$ softball team played ten games, scoring $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$ runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?
Answer: 45
Problem: Joey and his five brothers are ages $3$, $5$, $7$, $9$, $11$, and $13$. One afternoon two of his brothers whose ages sum to $16$ went to the movies, two brothers younger than $10$ went to play baseball, and Joey and the $5$-year-old stayed home. How old is Joey?
Answer: 11
Problem: The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6\times5$. What is the missing number in the top row?

[asy] unitsize(0.8cm); draw((-1,0)--(1,0)--(1,-2)--(-1,-2)--cycle); draw((-2,0)--(0,0)--(0,2)--(-2,2)--cycle); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((-3,2)--(-1,2)--(-1,4)--(-3,4)--cycle); draw((-1,2)--(1,2)--(1,4)--(-1,4)--cycle); draw((1,2)--(1,4)--(3,4)--(3,2)--cycle); label("600",(0,-1)); label("30",(-1,1)); label("6",(-2,3)); label("5",(0,3)); [/asy]

Find the missing number in the top row.
Answer: 4
Problem: A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?
Answer: 9
Problem: Six points are equally spaced around a circle of radius \(1\). Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle? The original answer is in \(\frac{\sqrt{k}}{m}\) format, please give the value of k + m.
Answer: 5
Problem: Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying $``1"$, so Blair follows by saying $``1, 2"$. Jo then says $``1, 2, 3"$, and so on. What is the $53^{{rd}}$ number said?
Answer: 8
Problem: Line \(l_1\) has equation \(3x - 2y = 1\) and goes through \(A = (-1, -2)\). Line \(l_2\) has equation \(y = 1\) and meets line \(l_1\) at point \(B\). Line \(l_3\) has positive slope, goes through point \(A\), and meets \(l_2\) at point \(C\). The area of \(\triangle ABC\) is \(3\). The original answer for the slope of \(l_3\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 7
Problem: In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$?
Answer: 56
Problem: What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$?
Answer: 8
Problem: $A$ flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What is the percentage of the flowers that are carnations? Provide your answer as a percentage.
Answer: 70
Problem: Let $S$ be the set of positive integers $n$ for which $\frac{1}{n}$ has the repeating decimal representation $0.\overline{ab} = 0.ababab\cdots,$ with $a$ and $b$ different digits. Find the sum of the elements of $S$.
Answer: 143
Problem: Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?
Answer: 103
Problem: Triangle \(ABC\) is equilateral with \(AB=1\). Points \(E\) and \(G\) are on \(\overline{AC}\) and points \(D\) and \(F\) are on \(\overline{AB}\) such that both \(\overline{DE}\) and \(\overline{FG}\) are parallel to \(\overline{BC}\). Furthermore, triangle \(ADE\) and trapezoids \(DFGE\) and \(FBCG\) all have the same perimeter. Find the value of \(DE+FG\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 34
Problem: Ted's grandfather used his treadmill on $3$ days this week. He went $2$ miles each day. On Monday he jogged at a speed of $5$ miles per hour. He walked at the rate of $3$ miles per hour on Wednesday and at $4$ miles per hour on Friday. If Grandfather had always walked at $4$ miles per hour, he would have spent less time on the treadmill. How many minutes less did he spend?
Answer: 4
Problem: The angles in a particular triangle are in arithmetic progression, and the side lengths are $4,5,x$. The sum of the possible values of $x$ equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is the value of $a+b+c$?
Answer: 36
Problem: In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. Find the perimeter of parallelogram $ADEF$.
Answer: 56
Problem: Let \(S\) be the set of sides and diagonals of a regular pentagon. A pair of elements of \(S\) are selected at random without replacement. Find the probability that the two chosen segments have the same length. The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is the simplified form of the probability. Please give the value of \(k + m\).
Answer: 13
Problem: Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, and $\widetilde{DE}$. Find the number of different routes from $A$ to $B$ that use each road exactly once. (Such a route will necessarily visit some cities more than once.)
Answer: 16
Problem: At the $2013$ Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $＄50$, you get a second pair at a $40\%$ discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What is the percentage of the $＄150$ regular price that he saved? Provide your answer as a percentage.
Answer: 30
Problem: How many three-digit numbers are not divisible by $5$, have digits that sum to less than $20$, and have the first digit equal to the third digit?
Answer: 60
Problem: Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying “$1$”, so Blair follows by saying “$1 , 2$”. Jo then says “$1,2,3$”, and so on. What is the $53^{rd}$ number said?
Answer: 8
Problem: Let points $A = (0,0) , \ B = (1,2), \ C = (3,3),$ and $D = (4,0)$. Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\left (\frac{p}{q}, \frac{r}{s} \right )$, where these fractions are in lowest terms. Find the value of $p + q + r + s$.
Answer: 58
Problem: The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$? If the answer is \(x^\circ\), what is the value of \(x\)?
Answer: 240
Problem: The sequence    
$\log_{12}{162}$, $\log_{12}{x}$, $\log_{12}{y}$, $\log_{12}{z}$, $\log_{12}{1250}$   
is an arithmetic progression. What is the value of $x$?
Answer: 270
Problem: Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N$. Find the smallest possible value of $N$.
Answer: 104
Problem: A wire is cut into two pieces, one of length \(a\) and the other of length \(b\). The piece of length \(a\) is bent to form an equilateral triangle, and the piece of length \(b\) is bent to form a regular hexagon. The triangle and the hexagon have equal area. The original answer is in the form \(\frac{\sqrt{p}}{n}\) where the fraction is in its simplest form. Please give the value of \(p + n\).
Answer: 8
Problem: Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?
Answer: 204
Problem: In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. If the area of $AEDC$ is $x$, what is the value of $10x$?
Answer: 135
Problem: $A$, $B$, $C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40$ pounds, the mean weight of the rocks in $B$ is $50$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $43$ pounds, and the mean weight of the rocks in the combined piles $A$ and $C$ is $44$ pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles $B$ and $C$?
Answer: 59
Problem: Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next $365$-day period will exactly two friends visit her?
Answer: 54
Problem: Let \(a, b,\) and \(c\) be real numbers such that  
  
\(a+b+c=2,\) and  
\(a^2+b^2+c^2=12\)  
  
Find the difference between the maximum and minimum possible values of \(c\). The original answer is in \(\frac{m}{n}\) format, please give the value of \(m+n\).
Answer: 19
Problem: Let points $A = (0, 0)$, $B = (1, 2)$, $C=(3, 3)$, and $D = (4, 0)$. Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\left(\frac{p}{q}, \frac{r}{s}\right)$, where these fractions are in lowest terms. Find the value of $p+q+r+s$.
Answer: 58
Problem: The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. Find how many integers less than $2013$ but greater than $1000$ have this property.
Answer: 46
Problem: Six spheres of radius \(1\) are positioned so that their centers are at the vertices of a regular hexagon of side length \(2\). The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 5
Problem: Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?
Answer: 280
Problem: In base $10$, the number $2013$ ends in the digit $3$. In base $9$, on the other hand, the same number is written as $(2676)_{9}$ and ends in the digit $6$. For how many positive integers $b$ does the base-$b$-representation of $2013$ end in the digit $3$?
Answer: 13
Problem: The real numbers \(c, b, a\) form an arithmetic sequence with \(a \geq b \geq c \geq 0\). The quadratic \(ax^2+bx+c\) has exactly one root. The original answer is in the form \(-k+\sqrt{m}\). Please give the value of \(k + m\).
Answer: 5
Problem: In $\bigtriangleup ABC$, $AB = 86$, and $AC = 97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover, $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is the length of $BC$?
Answer: 61
Problem: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 21
Problem: The number $2013$ is expressed in the form    
$2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,    
where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. Find the value of $|a_1 - b_1|$.
Answer: 2
Problem: A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $12^{{th}}$ pirate receive?
Answer: 1925
Problem: Samantha lives $2$ blocks west and $1$ block south of the southwest corner of City Park. Her school is $2$ blocks east and $2$ blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?
Answer: 18
Problem: Six spheres of radius \(1\) are positioned so that their centers are at the vertices of a regular hexagon of side length \(2\). The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. The radius of this eighth sphere can be expressed in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest terms. Please find the value of \(k + m\).
Answer: 5
Problem: The regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are all equal. In how many ways can this be done?
Answer: 1152
Problem: A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base $10$ with no leading zeros. A $6$-digit palindrome $n$ is chosen uniformly at random. Find the probability that \(\frac{n}{11}\) is also a palindrome. The original answer is in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please give the value of \(k + m\).
Answer: 41
Problem: Toothpicks are used to make a grid that is $60$ toothpicks long and $32$ toothpicks wide. How many toothpicks are used altogether?
Answer: 3932
Problem: In $\triangle ABC$, $AB = 86$, and $AC=97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover, $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is the integer length of $BC$?
Answer: 61
Problem: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$.
Answer: 21
Problem: A positive integer $n$ is nice if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$) such that the sum of the four divisors is equal to $n$. Find how many numbers in the set $\{ 2010,2011,2012,\cdots,2019 \}$ are nice.
Answer: 1
Problem: A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are \(R_1 = 100\) inches, \(R_2 = 60\) inches, and \(R_3 = 80\) inches, respectively. The ball always remains in contact with the track and does not slip. If the answer is \(x\pi\), what is the value of \(x\) for the distance the center of the ball travels over the course from A to B?
Answer: 238
Problem: For certain real numbers \(a\), \(b\), and \(c\), the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of \(g(x)\) is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] Find the value of \(f(1)\). The original answer is in the form \(m - n\), please give the value of \(m + n\).
Answer: 7007
Problem: Three numbers $a, b, c$, none zero, form an arithmetic progression. Increasing $a$ by $1$ or increasing $c$ by $2$ results in a geometric progression. Find the value of $b$.
Answer: 12
Problem: Square $ABCD$ has side length $s$, and a circle centered at $E$ has radius $r$. Both $r$ and $s$ are rational numbers. The circle passes through $D$, and $D$ lies on $\overline{BE}$. Point $F$ lies on the circle, on the same side of $\overline{BE}$ as $A$. Segment $AF$ is tangent to the circle, and $AF = \sqrt{9 + 5\sqrt{2}}$. Find the ratio \(\frac{r}{s}\) in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of \(k + m\).
Answer: 14
Problem: Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x + 2$, and when $P(x)$ is divided by the polynomial $x^2 + 1$, the remainder is $2x + 1$. There is a unique polynomial of least degree with these two properties. Find the sum of the squares of the coefficients of that polynomial.
Answer: 23
Problem: Given $A = \{1, 2, 3, 5, 8, 13, 21, 34, 55\}$, find how many of the numbers between $3$ and $89$ cannot be written as the sum of two elements of set $A$.
Answer: 51
Problem: In \(\triangle ABC\) shown in the figure, \(AB=7\), \(BC=8\), \(CA=9\), and \(\overline{AH}\) is an altitude. Points \(D\) and \(E\) lie on sides \(\overline{AC}\) and \(\overline{AB}\), respectively, so that \(\overline{BD}\) and \(\overline{CE}\) are angle bisectors, intersecting \(\overline{AH}\) at \(Q\) and \(P\), respectively. Find the length of \(PQ\). The answer is in the form \(\frac{k}{m}\sqrt{n}\), please provide the value of \(k + m + n\).
Answer: 28
Problem: Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the positive direction; when he gets tails, he moves 1 unit in the negative direction. The probability that he reaches 4 at some time during this process is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is $a+b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH$.)
Answer: 151
Problem: In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D$, $E$, and $F$ lie on sides $BC$, $AC$, and $AB$, respectively, and $CE=CD$ and $BF=BD$, then if $\measuredangle EDF$ equals $x^\circ$, what is the value of $x$?
Answer: 50
Problem: The solutions of the equation $z^4 + 4z^3i - 6z^2 - 4zi - i = 0$ are the vertices of a convex polygon in the complex plane. What is the area of the polygon? The original answer is in the form $2^{x/y}$, where $x$ and $y$ are integers, and $y$ cannot be zero. Please give the value of $x + y$.
Answer: 9
Problem: The sides of a right triangle are $a$ and $b$, and the hypotenuse is $c$. A perpendicular from the vertex divides $c$ into segments $r$ and $s$, adjacent respectively to $a$ and $b$. If $a : b = 1 : 3$, find the ratio of $r$ to $s$. The original answer is in the format of k:m, please give the value of k + m.
Answer: 10
Problem: If 64 is divided into three parts proportional to 2, 4, and 6, find the smallest part and express your answer in the form \frac{x}{y}, where \frac{x}{y} is in simplest terms. Please provide the value of x + y.
Answer: 35
Problem: A crude approximation is that the Earth travels in a circular orbit about the Sun at constant speed, at a distance of $150,000,000\text{ km}$ from the Sun. What is the closest value for the acceleration of the Earth in this orbit? Provide your answer as a number without including the unit, and multiply the original answer by 1000.
Answer: 6
Problem: For $x^2 + 2x + 5$ to be a factor of $x^4 + px^2 + q$, the original answer can be expressed as two integers $m$ and $n$ for the values of $p$ and $q$, respectively. Please find the value of $m + n$.
Answer: 31
Problem: Edward is trying to spell the word "CAT". He has an equal chance of spelling the word in any order of letters (e.g., TAC or TCA). What is the probability that he spells "CAT" incorrectly? Provide your answer as a simplified fraction in the form \(\frac{k}{m}\), and then calculate the value of k + m.
Answer: 11
Problem: The area of a circle inscribed in a regular hexagon is \(100\pi\). What is the area of the hexagon? The original answer is in the form of \(k\sqrt{m}\), please provide the value of k + m.
Answer: 203
Problem: Mary is about to pay for five items at the grocery store. The prices of the items are $7.99, $4.99, $2.99, $1.99, and $0.99. Mary will pay with a twenty-dollar bill. What percentage of the $20.00 will she receive in change? Provide your answer as an integer percentage.
Answer: 5
Problem: A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? Provide your answer as an integer.
Answer: 100
Problem: The graphs of $y = -|x - a| + b$ and $y = |x - c| + d$ intersect at points $(2,5)$ and $(8,3)$. Find the value of $a + c$.
Answer: 10
Problem: How many unique $3$-letter sequences with no spaces can be made using the letters in "AUGUSTIN LOUIS CAUCHY", which contains $19$ letters? For example, "GAA" is one acceptable sequence, but "GGA" is not an acceptable sequence because there is only one G available. The original ordering of the letters does not have to be preserved.
Answer: 1486
Problem: The coordinates of \(A\), \(B\), and \(C\) are \((5,5)\), \((2,1)\), and \((0,k)\) respectively. Find the value of \(k\) that makes \(\overline{AC} + \overline{BC}\) as small as possible. The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\).
Answer: 22
Problem: Point $P$ is $9$ units from the center of a circle with a radius of $15$. How many different chords of the circle contain $P$ and have integer lengths?
Answer: 12
Problem: When \(p = \sum_{k=1}^{6} k \ln{k}\), the number \(e^p\) is an integer. What is the largest power of \(2\) that is a factor of \(e^p\)? If the answer is \(2^x\), what is the value of \(x\)?
Answer: 16
Problem: Let \(ABCD\) be a trapezoid with \(AB \parallel CD\). Let \(a = AB\) and \(b = CD\). For \(MN \parallel AB\) such that \(M\) lies on \(AD\), \(N\) lies on \(BC\), and the trapezoids \(ABNM\) and \(MNCD\) have the same area, find the length of \(MN\). The original answer is in the form \(\sqrt{\frac{ka^2 + mb^2}{n}}\). Please find the value of \(k + m + n\).
Answer: 4
Problem: If $R_n=\frac{1}{2}(a^n+b^n)$ where $a=3+2\sqrt{2}$, $b=3-2\sqrt{2}$, and $n=0,1,2, \ldots$, then $R_{12345}$ is an integer. Find the units digit of $R_{12345}$.
Answer: 9