meetforever/GRPO_CL huggingface dataset:
user: In isosceles $\vartriangle ABC, AB = AC, \angle BAC$ is obtuse, and points $E$ and $F$ lie on sides $AB$ and $AC$, respectively, so that $AE = 10, AF = 15$. The area of $\vartriangle AEF$ is $60$, and the area of quadrilateral $BEFC$ is $102$. Find $BC$.
Answer: 36
user: Wendy randomly chooses a positive integer less than or equal to $2020$. The probability that the digits in Wendy's number add up to $10$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 107
user: Three doctors, four nurses, and three patients stand in a line in random order. The probability that there is at least one doctor and at least one nurse between each pair of patients is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 153
user: What percentage of twenty thousand is a quarter of a million?
Answer: 1250
user: Find the sum of all values of $x$ such that the set $\{107, 122,127, 137, 152,x\}$ has a mean that is equal to its median.
Answer: 381
user: Let $p, q$, and $r$ be prime numbers such that $2pqr + p + q + r = 2020$. Find $pq + qr + rp$.
Answer: 585
user: Find the number of three-digit palindromes that are divisible by $3$. Recall that a palindrome is a number that reads the same forward and backward like $727$ or $905509$.
Answer: 30
user: Given that $a, b$, and $c$ are distinct positive integers such that $a \cdot b \cdot  c = 2020$, the minimum possible positive value of $\frac{1}{a}-\frac{1}{b}-\frac{1}{c}$, is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 2101
user: The diagram shows a semicircle with diameter $20$ and the circle with greatest diameter that fits inside the semicircle. The area of the shaded region is $N\pi$, where $N$ is a positive integer. Find $N$.
[img]https://cdn.artofproblemsolving.com/attachments/1/b/456f1281041cc28a95e0c9f592c4a40b64413b.png[/img]
Answer: 25
user: Let $M$ and $m$ be, respectively, the greatest and the least ten-digit numbers that are rearrangements of the digits $0$ through $9$ such that no two adjacent digits are consecutive. Find $M - m$.
Answer: 8456173452
user: For some integers $u$,$ v$, and $w$, the equation
$$26ab - 51bc + 74ca = 12(a^2 + b^2 + c^2)$$
holds for all real numbers a, b, and c that satisfy
$$au + bv + cw = 0$$
Find the minimum possible value of $u^2 + v^2 + w^2$.
Answer: 53
user: Find the value of $x$ where the graph of $$y=\log_3(\sqrt{x^2+729}+x)-2\log_3(\sqrt{x^2+729}-x)$$ crosses the $x$-axis.
Answer: 36
user: The area of the triangle whose altitudes have lengths $36.4$, $39$, and $42$ can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 3553
user: Find the sum of all positive integers $x$ such that there is a positive integer $y$ satisfying $9x^2 - 4y^2 = 2021$.
Answer: 352
user: Let $ABCD$ be a convex quadrilateral with positive integer side lengths, $\angle{A} = \angle{B} = 120^{\circ}, |AD - BC| = 42,$ and $CD = 98$. Find the maximum possible value of $AB$.
Answer: 69
user: Points $X$ and $Y$ lie on side $\overline{AB}$ of $\vartriangle ABC$ such that $AX = 20$, $AY = 28$, and $AB = 42$. Suppose $XC = 26$ and $Y C = 30$. Find $AC + BC$.
Answer: 66
user: Let $L_1$ and $L_2$ be perpendicular lines, and let $F$ be a point at a distance $18$ from line $L_1$ and a distance $25$ from line $L_2$. There are two distinct points, $P$ and $Q$, that are each equidistant from $F$, from line $L_1$, and from line $L_2$. Find the area of $\triangle{FPQ}$.
Answer: 210
user: For real numbers $x$ let $$f(x)=\frac{4^x}{25^{x+1}}+\frac{5^x}{2^{x+1}}.$$ Then $f\left(\frac{1}{1-\log_{10}4}\right)=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 15001
user: Among the $100$ constants $a_1,a_2,a_3,...,a_{100}$, there are $39$ equal to $-1$ and $61$ equal to $+1$. Find the sum of all the products $a_ia_j$, where $1\le i < j \le 100$.
Answer: 192
user: The diagram shows a regular pentagon $ABCDE$ and a square $ABFG$. Find the degree measure of $\angle FAD$.
[img]https://cdn.artofproblemsolving.com/attachments/7/5/d90827028f426f2d2772f7d7b875eea4909211.png[/img]
Answer: 27
user: Let $z_1$, $z_2$, $z_3$, $\cdots$, $z_{2021}$ be the roots of the polynomial $z^{2021}+z-1$. Evaluate $$\frac{z_1^3}{z_{1}+1}+\frac{z_2^3}{z_{2}+1}+\frac{z_3^3}{z_{3}+1}+\cdots+\frac{z_{2021}^3}{z_{2021}+1}.$$
Answer: 1347
user: The diagram below shows two concentric circles whose areas are $7$ and $53$ and a pair of perpendicular lines where one line contains diameters of both circles and the other is tangent to the smaller circle. Find the area of the shaded region.
[img]https://cdn.artofproblemsolving.com/attachments/3/b/87cbb97a799686cf5dbec9dcd79b6b03e1995c.png[/img]
Answer: 23
user: Square $ABCD$ with side length $2$ begins in position $1$ with side $AD$ horizontal and vertex $A$ in the lower right corner. The square is rotated $90^o$ clockwise about vertex $ A$ into position $2$ so that vertex $D$ ends up where vertex $B$ was in position $1$. Then the square is rotated $90^o$ clockwise about vertex $C$ into position $3$ so that vertex $B$ ends up where vertex $D$ was in position $2$ and vertex $B$ was in position $1$, as shown below. The area of the region of points in the plane that were covered by the square at some time during its rotations can be written $\frac{p\pi + \sqrt{q} + r}{s}$, where $p, q, r,$ and $s$ are positive integers, and $p$ and $s$ are relatively prime. Find $p + q + r + s$.
[img]https://cdn.artofproblemsolving.com/attachments/9/2/cb15769c30018545abfa82a9f922201c4ae830.png[/img]
Answer: 133
user: The product $$\left(\frac{1}{2^3-1}+\frac12\right)\left(\frac{1}{3^3-1}+\frac12\right)\left(\frac{1}{4^3-1}+\frac12\right)\cdots\left(\frac{1}{100^3-1}+\frac12\right)$$ can be written as $\frac{r}{s2^t}$ where $r$, $s$, and $t$ are positive integers and $r$ and $s$ are odd and relatively prime. Find $r+s+t$.
Answer: 5990
user: A farmer wants to create a rectangular plot along the side of a barn where the barn forms one side of the rectangle and a fence forms the other three sides. The farmer will build the fence by tting together $75$ straight sections of fence which are each $4$ feet long. The farmer will build the fence to maximize the area of the rectangular plot. Find the length in feet along the side of the barn of this rectangular plot.
Answer: 148
user: Find the number of distinguishable groupings into which you can place $3$ indistinguishable red balls and $3$ indistinguishable blue balls. Here the groupings $RR-BR-B-B$ and $B-RB-B-RR$ are indistinguishable because the groupings are merely rearranged, but $RRB-BR-B$ is distinguishable from $RBB-BR-R$.
Answer: 31
user: The diagram shows two intersecting line segments that form some of the sides of two squares with side lengths $3$ and $6$. Two line segments join vertices of these squares. Find the area of the region enclosed by the squares and segments.
[img]https://cdn.artofproblemsolving.com/attachments/8/8/d2873dbad87ac923335f1f1389ab5f9b7a6055.png[/img]
Answer: 63
user: Let $a, b, c, d$ be an increasing arithmetic sequence of positive real numbers with common difference $\sqrt2$. Given that the product $abcd = 2021$, $d$ can be written as $\frac{m+\sqrt{n}}{\sqrt{p}}$ , where $m, n,$ and $p$ are positive integers not divisible by the square of any prime. Find $m + n + p$.
Answer: 100
user: Let $x$ be a real number such that $$4^{2x}+2^{-x}+1=(129+8\sqrt2)(4^{x}+2^{-x}-2^{x}).$$ Find $10x$.
Answer: 35
user: A semicircle has diameter $AB$ with $AB = 100$. Points $C$ and $D$ lie on the semicircle such that $AC = 28$ and $BD = 60$. Find $CD$.
Answer: 60
user: A building contractor needs to pay his $108$ workers $\$200$ each. He is carrying $122$ one hundred dollar bills and $188$ fifty dollar bills. Only $45$ workers get paid with two $\$100$ bills. Find the number of workers who get paid with four $\$50$ bills.
Answer: 31
user: Two cubes with edge length $3$ and two cubes with edge length $4$ sit on plane $P$ so that the four cubes share a vertex, and the two larger cubes share no faces with each other as shown below. The cube vertices that do not touch $P$ or any of the other cubes are labeled $A$, $B$, $C$, $D$, $E$, $F$, $G$, and $H$. The four cubes lie inside a right rectangular pyramid whose base is on $P$ and whose triangular sides touch the labeled vertices with one side containing vertices $A$, $B$, and $C$, another side containing vertices $D$, $E$, and $F$, and the two other sides each contain one of $G$ and $H$. Find the volume of the pyramid.
[img]https://cdn.artofproblemsolving.com/attachments/c/7/dd0a0f54d62ad0961416a605f3bd8d789008b3.png[/img]
Answer: 1152
user: In base ten, the number $100! = 100 \cdot 99 \cdot  98 \cdot  97... 2 \cdot 1$ has $158$ digits, and the last $24$ digits are all zeros. Find the number of zeros there are at the end of this same number when it is written in base $24$.
Answer: 32
user: Find the minimum possible value of |m -n|, where $m$ and $n$ are integers satisfying $m + n = mn - 2021$.
Answer: 331
user: Fiona had a solid rectangular block of cheese that measured $6$ centimeters from left to right, $5$ centimeters from front to back, and $4$ centimeters from top to bottom. Fiona took a sharp knife and sliced off a $1$ centimeter thick slice from the left side of the block and a $1$ centimeter slice from the right side of the block. After that, she sliced off a $1$ centimeter thick slice from the front side of the remaining block and a $1$ centimeter slice from the back side of the remaining block. Finally, Fiona sliced off a $1$ centimeter slice from the top of the remaining block and a $1$ centimeter slice from the bottom of the remaining block. Fiona now has $7$ blocks of cheese. Find the total surface area of those seven blocks of cheese measured in square centimeters.
Answer: 340
user: Let $a$, $b$, and $c$ be real numbers satisfying the equations $$a^3+abc=26$$ $$b^3+abc=78$$ $$c^3-abc=104.$$ Find $a^3+b^3+c^3$.
Answer: 184
user: At one school, $85$ percent of the students are taking mathematics courses, $55$ percent of the students are taking history courses, and $7$ percent of the students are taking neither mathematics nor history courses. Find the percent of the students who are taking both mathematics and history courses.
Answer: 47
user: The least positive angle $\alpha$ for which $$\left(\frac34-\sin^2(\alpha)\right)\left(\frac34-\sin^2(3\alpha)\right)\left(\frac34-\sin^2(3^2\alpha)\right)\left(\frac34-\sin^2(3^3\alpha)\right)=\frac1{256}$$ has a degree measure of $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 131
user: Find the greatest prime number $p$ such that $p^3$ divides $$\frac{122!}{121}+ 123!:$$
Answer: 61
user: Find the least integer $n > 60$ so that when $3n$ is divided by $4$, the remainder is $2$ and when $4n$ is divided by $5$, the remainder is $1$.
Answer: 74
user: The sum $$\sum_{k=3}^{\infty} \frac{1}{k(k^4-5k^2+4)^2}$$ is equal to $\frac{m^2}{2n^2}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 49
user: Let $a$ and $b$ be positive real numbers satisfying
$$a -12b = 11  -\frac{100}{a} \,\,\,\,and \,\,\,\, a -\frac{12}{b}= 4 -\frac{100}{a}.$$
Then $a + b = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 47
user: Let $m$ and $n$ be positive integers such that
$$(m^3 -27)(n^3- 27) = 27(m^2n^2 + 27):$$
Find the maximum possible value of $m^3 + n^3$.
Answer: 1792
user: For positive integer $k$, define $x_k=3k+\sqrt{k^2-1}-2(\sqrt{k^2-k}+\sqrt{k^2+k})$. Then $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_{1681}}=\sqrt{m}-n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 911
user: The side lengths of a scalene triangle are roots of the polynomial $$x^3-20x^2+131x-281.3.$$ Find the square of the area of the triangle.
Answer: 287
user: Three red books, three white books, and three blue books are randomly stacked to form three piles of three books each. The probability that no book is the same color as the book immediately on top of it is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 6
user: Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\tfrac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$.
Answer: 446
user: A furniture store set the sticker price of a table $40$ percent higher than the wholesale price that the store paid for the table. During a special sale, the table sold for $35$ percent less than this sticker price. Find the percent the final sale price was of the original wholesale price of the table.
Answer: 91
user: Paula rolls three standard fair dice. The probability that the three numbers rolled on the dice are the side lengths of a triangle with positive area is $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 109
user: At Ignus School there are $425$ students. Of these students $351$ study mathematics, $71$ study Latin, and $203$ study chemistry. There are $199$ students who study more than one of these subjects, and $8$ students who do not study any of these subjects. Find the number of students who study all three of these subjects.
Answer: 9
user: Find the least odd positive integer that is the middle number of five consecutive integers that are all composite.
Answer: 93
user: Let $ABCD$ be a convex quadrilateral inscribed in a circle with $AC = 7$, $AB = 3$, $CD = 5$, and $AD - BC = 3$. Then $BD = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 62
user: For positive integer $n$ let $z_n=\sqrt{\frac{3}{n}}+i$, where $i=\sqrt{-1}$. Find $|z_1 \cdot z_2 \cdot z_3 \cdots z_{47}|$.
Answer: 140
user: In quadrilateral $ABCD$, let $AB = 7$, $BC = 11$, $CD = 3$, $DA = 9$, $\angle BAD = \angle BCD = 90^o$, and diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. The ratio $\frac{BE}{DE} = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 104
user: Call a date mm/dd/yy $\textit{multiplicative}$ if its month number times its day number is a two-digit integer equal to its year expressed as a two-digit year. For example, $01/21/21$, $03/07/21$, and $07/03/21$ are multiplicative. Find the number of dates between January 1, 2022 and December 31, 2030 that are multiplicative.
Answer: 29
user: Find the number of positive divisors of $20^{22}$ that are perfect squares or perfect cubes.
Answer: 364
user: Let $ABCD$ be a parallelogram with diagonal $AC = 10$ such that the distance from $A$ to line $CD$ is $6$ and the distance from $A$ to line $BC$ is $7$. There are two non-congruent configurations of $ABCD$ that satisfy these conditions. The sum of the areas of these two parallelograms is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 4717
user: The product
$$\left(\frac{1+1}{1^2+1}+\frac{1}{4}\right)\left(\frac{2+1}{2^2+1}+\frac{1}{4}\right)\left(\frac{3+1}{3^2+1}+\frac{1}{4}\right)\cdots\left(\frac{2022+1}{2022^2+1}+\frac{1}{4}\right)$$
can be written as $\frac{q}{2^r\cdot s}$, where $r$ is a positive integer, and $q$ and $s$ are relatively prime odd positive integers. Find $s$.
Answer: 1
user: Six gamers play a round-robin tournament where each gamer plays one game against each of the other five gamers. In each game there is one winner and one loser where each player is equally likely to win that game, and the result of each game is independent of the results of the other games. The probability that the tournament will end with exactly one gamer scoring more wins than any other player is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 833
user: Let $a$ be a positive real number such that $$4a^2+\frac{1}{a^2}=117.$$ Find $$8a^3+\frac{1}{a^3}.$$
Answer: 1265
user: A rectangular box has width $12$ inches, length $16$ inches, and height $\tfrac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three rectangular sides of the box meet at a corner of the box. The center points of those three rectangular sides are the vertices of a triangle with area $30$ square inches. Find $m + n.$
Answer: 41
user: Starting at $12:00:00$ AM on January $1,$ $2022,$ after $13!$ seconds it will be $y$ years (including leap years) and $d$ days later, where $d < 365.$ Find $y + d.$
Answer: 317
user: Sphere $S$ with radius $100$ has diameter $\overline{AB}$ and center $C$. Four small spheres all with radius $17$ have centers that lie in a plane perpendicular to $\overline{AB}$ such that each of the four spheres is internally tangent to $S$ and externally tangent to two of the other small spheres. Find the radius of the smallest sphere that is both externally tangent to two of the four spheres with radius $17$ and internally tangent to $S$ at a point in the plane perpendicular to $\overline{AB}$ at $C$.
Answer: 66
user: A rectangle with width $30$ inches has the property that all points in the rectangle are within $12$ inches of at least one of the diagonals of the rectangle. Find the maximum possible length for the rectangle in inches.
Answer: 40
user: There is a positive integer s such that there are s solutions to the equation
$64sin^2(2x)+tan^2(x)+cot^2(x)=46$ in the interval $(0,\frac{\pi}{2})$ all of the form $\frac{m_k}{n_k}\pi$
where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, . . . , s$. Find $(m_1 + n_1) + (m_2 + n_2) + (m_3 + n_3) + · · · + (m_s + n_s)$.
Answer: 100
user: Find the number of permutations of the letters $AAABBBCCC$ where no letter appears in a position that originally contained that letter. For example, count the permutations $BBBCCCAAA$ and $CBCAACBBA$ but not the permutation $CABCACBAB$.
Answer: 56
user: Find the number of rearrangements of the nine letters $\text{AAABBBCCC}$ where no three consecutive letters are the same. For example, count $\text{AABBCCABC}$ and $\text{ACABBCCAB}$ but not $\text{ABABCCCBA}.$
Answer: 1314
user: Find the least positive integer with the property that if its digits are reversed and then $450$ is added to this reversal, the sum is the original number. For example, $621$ is not the answer because it is not true that $621 = 126 + 450$.
Answer: 1501
user: Circle $\omega_1$ has radius $7$ and center $C_1$. Circle $\omega_2$ has radius $23$ and center $C_2$ with $C_1C_2 = 34$. Let a common internal tangent of $\omega_1$ and $\omega_2$ pass through $A_1$ on $\omega_1$ and $A_2$ on $\omega_2$, and let a common external tangent of $\omega_1$ and $\omega_2$ pass through $B_1$ on $\omega_1$ and $B_2$ on $\omega_2$ such that $A_1$ and $B_1$ lie on the same side of the line $C_1C_2$. Let $P$ be the intersection of lines $A_1A_2$ and $B_1B_2$. Find the area of quadrilateral $PC_1A_2C_2$.
Answer: 345
user: Given that $a_1, a_2, a_3, . . . , a_{99}$ is a permutation of $1, 2, 3, . . . , 99,$ find the maximum possible value of
$$|a_1 - 1| + |a_2 - 2| + |a_3 - 3| + \dots + |a_{99} - 99|.$$
Answer: 4900
user: Let $a_1=2021$ and for $n \ge 1$ let $a_{n+1}=\sqrt{4+a_n}$. Then $a_5$ can be written as $$\sqrt{\frac{m+\sqrt{n}}{2}}+\sqrt{\frac{m-\sqrt{n}}{2}},$$ where $m$ and $n$ are positive integers. Find $10m+n$.
Answer: 45
user: For integer $k \ge 1$, let $a_k =\frac{k}{4k^4 + 1}$. Find the least integer $n$ such that $a_1 + a_2 + a_3 + ... + a_n > \frac{505.45}{2022}$.
Answer: 71
user: In $\vartriangle ABC$, let $D$ be on $BC$ such that $\overline{AD} \perp \overline{BC}$. Suppose also that $\tan B = 4 \sin C$, $AB^2 +CD^2 = 17$, and $AC^2 + BC^2 = 21$. Find the measure of $\angle C$ in degrees between $0^o$ and $180^o$ .
Answer: 72
user: The value of
$$\left(1-\frac{1}{2^2-1}\right)\left(1-\frac{1}{2^3-1}\right)\left(1-\frac{1}{2^4-1}\right)\dots\left(1-\frac{1}{2^{29}-1}\right)$$
can be written as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $2m - n.$
Answer: 1
user: There are real numbers $x, y,$ and $z$ such that the value of $$x+y+z-\left(\frac{x^2}{5}+\frac{y^2}{6}+\frac{z^2}{7}\right)$$ reaches its maximum of $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n + x + y + z.$
Answer: 20
user: Let $\mathcal{S}$ be a sphere with radius $2.$ There are $8$ congruent spheres whose centers are at the vertices of a cube, each has radius $x,$ each is externally tangent to $3$ of the other $7$ spheres with radius $x,$ and each is internally tangent to $\mathcal{S}.$ There is a sphere with radius $y$ that is the smallest sphere internally tangent to $\mathcal{S}$ and externally tangent to $4$ spheres with radius $x.$ There is a sphere with radius $z$ centered at the center of $\mathcal{S}$ that is externally tangent to all $8$ of the spheres with radius $x.$ Find $18x + 5y + 4z.$
Answer: 18
user: Let x be a real number such that  $(\sqrt{6})^x -3^x = 2^{x-2}$. Evaluate $\frac{4^{x+1}}{9^{x-1}}$ .
Answer: 576
user: Antonio plays a game where he continually flips a fair coin to see the sequence of heads ($H$) and tails ($T$) that he flips. Antonio wins the game if he sees on four consecutive flips the sequence $TTHT$ before he sees the sequence $HTTH$. The probability that Antonio wins the game is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 17
user: In $\triangle ABC$ let point $D$ be the foot of the altitude from $A$ to $BC.$ Suppose that $\angle A = 90^{\circ}, AB - AC = 5,$ and $BD - CD = 7.$ Find the area of $\triangle ABC.$
Answer: 150
user: Let $A_1, A_2, A_3, \ldots , A_{12}$ be the vertices of a regular $12-$gon (dodecagon). Find the number of points in the plane that are equidistant to at least $3$ distinct vertices of this $12-$gon.
Answer: 1
user: Each different letter in the following addition represents a different decimal digit. The sum is a six-digit integer whose digits are all equal.
$$\begin{tabular}{ccccccc}
 & P  & U & R & P & L  &  E\\
+ &  & C & O & M & E & T \\
\hline
\\
\end{tabular}$$
Find the greatest possible value that the five-digit number $COMET$ could represent.
Answer: 98057
user: Find the number of sequences of 10 letters where all the letters are either $A$ or $B$, the first letter is $A$, the last letter is $B$, and the sequence contains no three consecutive letters reading $ABA$. For example, count $AAABBABBAB$ and $ABBBBBBBAB$ but not $AABBAABABB$ or $AAAABBBBBA$.
Answer: 86
user: There are prime numbers $a$, $b$, and $c$ such that the system of equations
$$a \cdot x - 3  \cdot y + 6  \cdot z = 8$$
$$b  \cdot x + 3\frac12  \cdot y + 2\frac13  \cdot z = -28$$
$$c  \cdot x - 5\frac12  \cdot y + 18\frac13  \cdot z = 0$$
has infinitely many solutions for $(x, y, z)$. Find the product $a \cdot b \cdot c$.
Answer: 154
user: A jar contains red, blue, and yellow candies. There are $14\%$ more yellow candies than blue candies, and $14\%$ fewer red candies than blue candies. Find the percent of candies in the jar that are yellow.
Answer: 38
user: Let $a$ be a real number such that $$5 \sin^4 \left( \frac{a}{2} \right)+ 12 \cos a = 5 cos^4 \left( \frac{a}{2} \right)+ 12 \sin a.$$ There are relatively prime positive integers $m$ and $n$ such that $\tan a = \frac{m}{n}$ . Find $10m + n$.
Answer: 82
user: For positive integer $n,$ let $s(n)$ be the sum of the digits of n when n is expressed in base ten. For example, $s(2022) = 2 + 0 + 2 + 2 = 6.$ Find the sum of the two solutions to the equation $n - 3s(n) = 2022.$
Answer: 4107
user: Of the integers $a$, $b$, and $c$ that satisfy $0 < c < b < a$ and
$$a^3 - b^3 - c^3 - abc + 1 = 2022,$$
let $c'$ be the least value of $ c$ appearing in any solution, let $a'$ be the least value of $a$ appearing in any solution with $c = c'$, and let $b'$ be the value of $b$ in the solution where $c = c'$ and $a = a'$. Find $a' + b' + c'$.
Answer: 19
user: Let $a$ and $b$ be positive real numbers satisfying
$$\frac{a}{b} \left( \frac{a}{b}+ 2 \right) + \frac{b}{a} \left(  \frac{b}{a}+ 2 \right)= 2022.$$
Find the positive integer $n$ such that $$\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}=\sqrt{n}.$$
Answer: 46