meetforever/GRPO_CL huggingface dataset:
user: At most, how many elements can a set have such that all elements are less than $102$ and the set does not contain the sum of any two of its elements? Provide your answer as an integer.
Answer: 51
user: In a bag of marbles, \(\frac{3}{5}\) of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? The original answer is in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please provide the value of \(k + m\).
Answer: 11
user: Find the number of noncongruent integer-sided triangles with positive area and perimeter less than $15$ that are neither equilateral, isosceles, nor right triangles.
Answer: 5
user: For what value of $k$ does the pair of equations $y = x^2$ and $y = 3x + k$ have two identical solutions? Provide your answer as a fraction in the form $-\frac{m}{n}$, where m and n are integers. Please find the value of m + n.
Answer: 13
user: Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other.
Answer: 20
user: The number $2013$ has the property that it includes four consecutive digits ($0$, $1$, $2$, and $3$). Find how many $4$-digit numbers include $4$ consecutive digits.
Answer: 162
user: If the sum of digits of only $m$ and $m+n$ from the numbers $m$, $m+1$, \ldots, $m+n$ are divisible by $8$ where $m$ and $n$ are positive integers, what is the largest possible value of $n$?
Answer: 15
user: In triangle $ABC$, $D$ is the midpoint of $AB$; $E$ is the midpoint of $DB$; and $F$ is the midpoint of $BC$. If the area of $\triangle ABC$ is $96$, then what is the area of $\triangle AEF$?
Answer: 36
user: Let $\overline{AB}$ be a diameter in a circle of radius $5\sqrt{2}$. Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt{5}$ and $\angle AEC = 45^{\circ}$. Find the value of $CE^2+DE^2$.
Answer: 100
user: Find the product of uncommon real roots of the two polynomials $x^4 + 2x^3 - 8x^2 - 6x + 15$ and $x^3 + 4x^2 - x - 10$.
Answer: 6
user: Find the sum of the $2$ smallest prime factors of $2^{1024} - 1$.
Answer: 8
user: What is the least positive integer $k$ such that $n+k \in S$ for every $n \in S$, where $S = \{ n : n3^n + (2n+1)5^n \equiv 0 \pmod{7} \}$?
Answer: 21
user: If $x_1=5, x_2=401$, and \[ x_n=x_{n-2}-\frac{1}{x_{n-1}} \] for every $3\leq n \leq m$, find the largest value of $m$.
Answer: 2007
user: A sequence of complex numbers $z_0, z_1, z_2, \ldots$ is defined by the rule \[ z_{n+1} = \frac{i z_n}{\overline{z_n}} \] where $\overline{z_n}$ is the complex conjugate of $z_n$ and $i^2 = -1$. Suppose that $|z_0| = 1$ and $z_{2005} = 1$. Find the number of possible values for $z_0$. The original answer is in the form of $x * 2^{y}$, please give the value of $x + y$.
Answer: 2006
user: Let $N$ be the second smallest positive integer that is divisible by every positive integer less than 7. What is the sum of the digits of $N$?
Answer: 3
user: A painting $18'' \times 24''$ is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, find the ratio of the smaller to the larger dimension of the framed painting in the form $a:b$. Then, calculate the value of $a + b$.
Answer: 5
user: A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are \((3,17)\) and \((48,281)\)? Include both endpoints of the segment in your count.
Answer: 4
user: Two men at points $R$ and $S$, 76 miles apart, set out at the same time to walk towards each other. The man at $R$ walks uniformly at the rate of $\frac{9}{2}$ miles per hour; the man at $S$ walks at the constant rate of $\frac{13}{4}$ miles per hour for the first hour, $\frac{15}{4}$ miles per hour for the second hour, and so on, in arithmetic progression. If the men meet $x$ miles nearer $R$ than $S$ in an integral number of hours, find the value of $x$.
Answer: 4
user: Calculate the product \(\frac{1}{3} \times \frac{2}{4} \times \frac{3}{5} \times \cdots \times \frac{18}{20} \times \frac{19}{21}\). What is the value of this product in its simplest form? The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 211
user: The parallelogram bounded by the lines $y = ax + c$, $y = ax + d$, $y = bx + c$, and $y = bx + d$ has an area of 18. The parallelogram bounded by the lines $y = ax + c$, $y = ax - d$, $y = bx + c$, and $y = bx - d$ has an area of 72. Given that $a$, $b$, $c$, and $d$ are positive integers, find the smallest possible value of $a + b + c + d$.
Answer: 16
user: Let $D$ be a point on side $[AB]$ of triangle $ABC$ with $|AB| = |AC|$, such that $[CD]$ is an angle bisector and $m(\widehat{ABC}) = 40^\circ$. Let $F$ be a point on the extension of $[AB]$ beyond $B$ such that $|BC| = |AF|$. Let $E$ be the midpoint of $[CF]$. If $G$ is the intersection of lines $ED$ and $AC$, what is the value of $x$ if $m(\widehat{FBG}) = x^\circ$?
Answer: 150
user: Find the number of prime numbers $p$ such that $2p^4 - 7p^2 + 1$ is equal to the square of an integer.
Answer: 1
user: If the perimeter of a rectangle is \(p\) and its diagonal is \(d\), find the difference between the length and width of the rectangle. The original answer is in the form \(\frac{\sqrt{n \cdot d^2 - p^2}}{m}\). Please calculate the value of \(k + m + n\).
Answer: 11
user: A list of integers has mode $32$ and mean $22$. The smallest number in the list is $10$. The median $m$ of the list is a member of the list. If the list member $m$ were replaced by $m + 10$, the mean and median of the new list would be $24$ and $m + 10$, respectively. If $m$ were instead replaced by $m - 8$, the median of the new list would be $m - 4$. What is the value of $m$?
Answer: 20
user: Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ such that $\overline{DE} \perp \overline{BC}$. Find the ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 4
user: How many functions $f : A \rightarrow A$ are there satisfying $f(f(a)) = a$ for every $a \in A=\{1,2,3,4,5,6,7\}$?
Answer: 232
user: The two-digit integers from $19$ to $92$ are written consecutively to form the larger integer $N = 19202122\ldots909192$. If $3^{k}$ is the highest power of $3$ that is a factor of $N$, what is the value of $k$?
Answer: 1
user: In order to pass $B$ going $40$ mph on a two-lane highway, $A$, going $50$ mph, must gain $30$ feet. Meanwhile, $C$, who is $210$ feet from $A$, is headed toward him at $50$ mph. If $B$ and $C$ maintain their speeds, then, in order to pass safely, $A$ must increase his speed by how many mph?
Answer: 5
user: The probability that event $A$ occurs is \(\frac{3}{4}\); the probability that event $B$ occurs is \(\frac{2}{3}\). Let \(p\) be the probability that both \(A\) and \(B\) occur. Find the smallest interval necessarily containing \(p\) and express it in the form \(\left[\frac{x}{y}, \frac{c}{d}\right]\), where \(x, y, c,\) and \(d\) are integers. Please give the value of \(x + y + c + d\).
Answer: 22
user: A truck is initially moving at velocity \(v\). The driver presses the brake to slow the truck to a stop. The brake applies a constant force \(F\) to the truck. The truck rolls a distance \(x\) before coming to a stop, and the time it takes to stop is \(t\).

Find the expression that represents the initial momentum of the truck (i.e., the momentum before the driver starts braking) in terms of \(F\), \(x\), and \(v\). The original answer is in the form \(\frac{kF \cdot x}{mv}\). Please give the value of \(k + m\).
Answer: 3
user: How many ways are there to partition $7$ students into the groups of $2$ or $3$? Provide your answer as an integer.
Answer: 105
user: Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. Find the probability that the four chords are the sides of a convex quadrilateral. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 92
user: Asli will distribute $100$ candies among her brother and $18$ friends of his. Asli splits the friends of her brother into several groups and distributes all the candies into these groups. In each group, the candies are shared in a fair way such that each child in a group takes the same number of candies, and this number is the largest possible. Then, Asli's brother takes the remaining candies of each group. At most, how many candies can Asli's brother have?
Answer: 16
user: What is the largest real number $C$ that satisfies the inequality $x^2 \geq C \lfloor x \rfloor (x-\lfloor x \rfloor)$ for every real $x$?
Answer: 4
user: The sum of the largest number and the smallest number of a triple of positive integers $(x,y,z)$ is called the power of the triple. Find the sum of powers of all triples $(x,y,z)$ where $x,y,z \leq 9$.
Answer: 7290
user: Three mutually tangent spheres of radius \(1\) rest on a horizontal plane. A sphere of radius \(2\) rests on them. What is the distance from the plane to the top of the larger sphere? Express your answer in the form \(3 + \frac{\sqrt{n}}{m}\) where \(n\) and \(m\) are integers. Please give the value of \(n + m\).
Answer: 72
user: Find the integer that divides $3^{3n+1} + 5^{3n+2} + 7^{3n+3}$ for every positive integer $n$.
Answer: 7
user: Find the radius of the inscribed circle of a triangle with sides of length \(13\), \(30\), and \(37\). 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: 11
user: In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. What is the common ratio? Provide your answer in the form of \(\frac{\sqrt{k} - m}{n}\) and find the value of \(k + m + n\).
Answer: 8
user: Call a number "prime-looking" if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000? Provide your answer as an integer.
Answer: 100
user: Calculate the value of \(\cot 10 + \tan 5\) and express your answer in terms of a trigonometric function. If the answer is \(\csc x\), what is the value of \(x\)?
Answer: 10
user: A positive number is mistakenly divided by $6$ instead of being multiplied by $6$. Calculate the error committed, and provide the answer to the nearest percent.
Answer: 97
user: Find the sum of all prime numbers between $1$ and $100$ that are simultaneously $1$ greater than a multiple of $4$ and $1$ less than a multiple of $5$. What is the sum?
Answer: 118
user: Let $P$ equal the product of $3,659,893,456,789,325,678$ and $342,973,489,379,256$. Find the number of digits in $P$.
Answer: 34
user: Find the number of real values of $x$ satisfying the equation $2^{2x^2 - 7x + 5} = 1$.
Answer: 2
user: The arithmetic mean of the nine numbers in the set $\{9, 99, 999, 9999, \ldots, 999999999\}$ is a 9-digit number $M$, all of whose digits are distinct. Find the digit that is not present in the number $M$.
Answer: 0
user: Let $u$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y)$ satisfy all of the following conditions:

1. $\frac{a}{2} \le x \le 2a$
2. $\frac{a}{2} \le y \le 2a$
3. $x + y \ge a$
4. $x + a \ge y$
5. $y + a \ge x$

The boundary of set $S$ is a polygon with how many sides?
Answer: 6
user: A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13, 19, 20, 25$, and $31$, although this is not necessarily their order around the pentagon. What is the area of the pentagon?
Answer: 745
user: A charity sells 140 benefit tickets for a total of \$2001. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets? If the answer is \$x, what is the value of x?
Answer: 782
user: Let \(ABC\) be a triangle such that \(|AB| = 7\), \(|BC| = 8\), \(|AC| = 6\). Let \(D\) be the midpoint of side \([BC]\). If the circle through \(A\), \(B\), and \(D\) cuts \(AC\) at \(A\) and \(E\), find the length of \(|AE|\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 5
user: The numbers \(1, 2, \ldots, 9\) are arranged so that the \(1\)st term is not \(1\) and the \(9\)th term is not \(9\). Find the probability that the third term is \(3\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 442
user: Find the odd integer that contains the digit 5, is divisible by 3, and lies between $12^2$ and $13^2$.
Answer: 165
user: The function $f$ satisfies the functional equation \[ f(x) + f(y) = f(x + y) - xy - 1 \] for every pair $x, y$ of real numbers. If $f(1) = 1$, find the number of integers $n \neq 1$ for which $f(n) = n$.
Answer: 1
user: How many integers $n$ satisfy $|n| < 2020$ and the equation $11^3 \mid n^3 + 3n^2 - 107n + 1$? Provide your answer as an integer.
Answer: 0
user: Let $m = (abab)$ and $n = (cdcd)$ be four-digit numbers in the decimal system. If $m + n$ is a perfect square, find the largest value of $a \cdot b \cdot c \cdot d$.
Answer: 600
user: The number $10!$ (where $10$ is written in base $10$), when converted to the base $12$ system, ends in exactly $k$ zeroes. What is the value of $k$?
Answer: 4
user: Suppose that $m$ and $n$ are positive integers such that $m < n$, the geometric mean of $m$ and $n$ is greater than $2007$, and the arithmetic mean of $m$ and $n$ is less than $2007$. Find the number of pairs $(m, n)$ that satisfy these conditions.
Answer: 0
user: Eight points are equally spaced around a circle of radius $r$. If we draw a circle of radius $1$ centered at each of the eight points, then each of these circles will be tangent to two of the other eight circles that are next to it. If $r^2 = a + b\sqrt{2}$, where $a$ and $b$ are integers, find the value of $a + b$.
Answer: 6
user: How many different prime numbers are factors of $N$ if 
\[ \log_2 (\log_3 (\log_5 (\log_7 N))) = 11? \]
Answer: 1
user: Real numbers \(x\) and \(y\) are chosen independently and uniformly at random from the interval \((0,1)\). Find the probability that \(\lfloor \log_2{x} \rfloor = \lfloor \log_2{y} \rfloor\), where \(\lfloor r \rfloor\) denotes the greatest integer less than or equal to the real number \(r\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 4
user: Charles has $5q + 1$ quarters and Richard has $q + 5$ quarters. Find the difference in their money in dimes. The answer should be in the format $k(q - m)$. Please calculate the value of $k + m$.
Answer: 11
user: On a map with a scale of \( 400 \) miles to an inch and a half, a certain estate is represented by a rhombus with a \( 60^{\circ} \) angle. The diagonal opposite the \( 60^{\circ} \) angle is \( \frac{3}{16} \) inches. Calculate the area of the estate in square miles. The original answer is in the form of \( k\sqrt{m} \). Please provide the value of \( k + m \).
Answer: 1253
user: What is the maximum area of a triangle that can be inscribed in an ellipse with semi-axes $a$ and $b$? Express your answer in terms of $a$ and $b$. The original answer is in the form $ab\frac{k\sqrt{3}}{m}$, where $k$ and $m$ are integers. Please give the value of $k + m$.
Answer: 7
user: Lori makes a list of all the numbers between $1$ and $999$ inclusive. She first colors all the multiples of $5$ red. Then she colors blue every number which is adjacent to a red number. How many numbers in her list are left uncolored?
Answer: 402
user: How many integer triples $(x,y,z)$ are there such that:
\[
\begin{array}{rcl}
 x - yz^2 & \equiv & 1 \pmod{13} \\
 xz + y & \equiv & 4 \pmod{13} \\
\end{array}
\]
where $0 \leq x < 13$, $0 \leq y < 13$, and $0 \leq z < 13$? Provide your answer as an integer.
Answer: 23
user: In a right triangle $ABC$, let $D$ be the midpoint of the hypotenuse $AB$. The circumradius of the triangle is $\frac{5}{2}$ and $|BC| = 3$. Find the distance between the circumcenter of $\triangle ACD$ and the incenter of $\triangle BCD$. The answer is in the form \(\frac{k\sqrt{m}}{n}\), please provide the value of \(k + m + n\).
Answer: 51
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: Find the number of positive integral divisors of 2006.
Answer: 8
user: 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
user: $ABC$ 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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