ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: There are relatively prime positive integers $m$ and $n$ so that the parabola with equation $y = 4x^2$ is tangent to the parabola with equation $x = y^2 + \frac{m}{n}$ . Find $m + n$.
Answer: 19
Problem: Each of the $48$ faces of eight $1\times 1\times 1$ cubes is randomly painted either blue or green. The probability that these eight cubes can then be assembled into a $2\times 2\times 2$ cube in a way so that its surface is solid green can be written $\frac{p^m}{q^n}$ , where $p$ and $q$ are prime numbers and $m$ and $n$ are positive integers. Find $p + q + m + n$.
Answer: 77
Problem: Binhao has a fair coin. He writes the number $+1$ on a blackboard. Then he flips the coin. If it comes up heads (H), he writes $+\frac12$ , and otherwise, if he flips tails (T), he writes $-\frac12$ . Then he flips the coin again. If it comes up heads, he writes $+\frac14$ , and otherwise he writes $-\frac14$ . Binhao continues to flip the coin, and on the nth flip, if he flips heads, he writes $+ \frac{1}{2n}$ , and otherwise he writes $- \frac{1}{2n}$ . For example, if Binhao flips HHTHTHT, he writes $1 + \frac12 + \frac14 - \frac18 + \frac{1}{16} -\frac{1}{32} + \frac{1}{64} -\frac{1}{128}$ . The probability that Binhao will generate a series whose sum is greater than $\frac17$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + 10q$.
Answer: 153
Problem: Find the number of positive integers less than $2019$ that are neither multiples of $3$ nor have any digits that are multiples of $3$.
Answer: 321
Problem: A [i]derangement [/i] of the letters $ABCDEF$ is a permutation of these letters so that no letter ends up in the position it began such as $BDECFA$. An [i]inversion [/i] in a permutation is a pair of letters $xy$ where $x$ appears before $y$ in the original order of the letters, but $y$ appears before $x$ in the permutation. For example, the derangement $BDECFA$ has seven inversions: $AB, AC, AD, AE, AF, CD$, and $CE$. Find the total number of inversions that appear in all the derangements of $ABCDEF$.
Answer: 2275
Problem: In a right circular cone, $A$ is the vertex, $B$ is the center of the base, and $C$ is a point on the circumference of the base with $BC = 1$ and $AB = 4$. There is a trapezoid $ABCD$ with $\overline{AB} \parallel \overline{CD}$. A right circular cylinder whose surface contains the points $A, C$, and $D$ intersects the cone such that its axis of symmetry is perpendicular to the plane of the trapezoid, and $\overline{CD}$ is a diameter of the cylinder. A sphere radius $r$ lies inside the cone and inside the cylinder. The greatest possible value of $r$ is $\frac{a\sqrt{b}-c}{d}$ , where $a, b, c$, and $d$ are positive integers, $a$ and $d$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a + b + c + d$.
Answer: 113
Problem: Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m -3n)^2 = 2019$. Find $3m + n$.
Answer: 46
Problem: A $12$-sided polygon is inscribed in a circle with radius $r$. The polygon has six sides of length $6\sqrt3$ that alternate with six sides of length $2$. Find $r^2$.
Answer: 148
Problem: In the diagram below, points $D, E$, and $F$ are on the inside of equilateral $\vartriangle ABC$ such that $D$ is on $\overline{AE}, E$ is on $\overline{CF}, F$ is on $\overline{BD}$, and the triangles $\vartriangle AEC, \vartriangle BDA$, and $\vartriangle CFB$ are congruent. Given that $AB = 10$ and $DE = 6$, the perimeter of $\vartriangle BDA$ is $\frac{a+b\sqrt{c}}{d}$, where $a, b, c$, and $d$ are positive integers, $b$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.
[img]https://cdn.artofproblemsolving.com/attachments/8/6/98da82fc1c26fa13883a47ba6d45a015622b20.png[/img]
Answer: 308
Problem: Four small spheres each with radius $6$ are each internally tangent to a large sphere with radius $17$. The four small spheres form a ring with each of the four spheres externally tangent to its two neighboring small spheres. A sixth intermediately sized sphere is internally tangent to the large sphere and externally tangent to each of the four small spheres. Its radius is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$.
[img]https://cdn.artofproblemsolving.com/attachments/7/2/25955cd6f22bc85f2f3c5ba8cd1ee0821c9d50.png[/img]
Answer: 56
Problem: Find the number of permutations of the letters $AAAABBBCC$ where no letter is next to another letter of the same type. For example, count $ABCABCABA$ and $ABABCABCA$ but not $ABCCBABAA$.
Answer: 79
Problem: Let $a, b$, and $c$ be real numbers such that $3^a = 125$,  $5^b = 49$,and $7^c = 8$1.
Find the product $abc$.
Answer: 24
Problem: In $\vartriangle ABC, \angle A = 52^o$ and $\angle B = 57^o$. One circle passes through the points $B, C$, and the incenter of $\vartriangle ABC$, and a second circle passes through the points $A, C$, and the circumcenter of $\vartriangle ABC$. Find the degree measure of the acute angle at which the two circles intersect.
Answer: 59
Problem: Let $P$ be the set of positive integers that are prime numbers. Find the number of subsets of $P$ that have the property that the sum of their elements is $34$ such as $\{3, 31\}$.
Answer: 9
Problem: Let $x$ be a real number such that $3 \sin^4 x -2 \cos^6 x = -\frac{17}{25}$ . Then $3 \cos^4 x - 2 \sin^6 x = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $10m + n$.
Answer: 445
Problem: An ant starts at vertex $A$ in equilateral triangle $\triangle ABC$ and walks around the perimeter of the triangle from $A$ to $B$ to $C$ and back to $A$. When the ant is $42$ percent of its way around the triangle, it stops for a rest. Find the percent of the way from $B$ to $C$ the ant is at that point
Answer: 26
Problem: There are two distinct pairs of positive integers $a_1 < b_1$ and $a_2 < b_2$ such that both $(a_1 + ib_1)(b_1 - ia_1) $ and $(a_2 + ib_2)(b_2 - ia_2)$ equal $2020$, where $i =\sqrt{-1}$. Find $a_1 + b_1 + a_2 + b_2$.
Answer: 120
Problem: Points $E$ and $F$ lie on diagonal $\overline{AC}$ of square $ABCD$ with side length $24$, such that $AE = CF = 3\sqrt2$. An ellipse with foci at $E$ and $F$ is tangent to the sides of the square. Find the sum of the distances from any point on the ellipse to the two foci.
Answer: 30
Problem: There is a real number $x$ between $0$ and $\frac{\pi}{2}$ such that $$\frac{\sin^3 x + \cos^3 x}{\sin^5 x + \cos^5 x}=\frac{12}{11}$$ and $\sin x + \cos x =\frac{\sqrt{m}}{n}$ , where $m$ and $n$ are positive integers, and $m$ is not divisible by the square of any prime. Find $m + n$.
Answer: 8
Problem: There is a complex number $K$ such that the quadratic polynomial $7x^2 +Kx + 12 - 5i$ has exactly one root, where $i =\sqrt{-1}$. Find $|K|^2$.
Answer: 364
Problem: There are relatively prime positive integers $s$ and $t$ such that $$\sum_{n=2}^{100}\left(\frac{n}{n^2-1}- \frac{1}{n}\right)=\frac{s}{t}$$ Find $s + t$.
Answer: 25249
Problem: A deck of eight cards has cards numbered $1, 2, 3, 4, 5, 6, 7, 8$, in that order, and a deck of five cards has cards numbered $1, 2, 3, 4, 5$, in that order. The two decks are riffle-shuffled together to form a deck with $13$ cards with the cards from each deck in the same order as they were originally. Thus, numbers on the cards might end up in the order $1122334455678$ or $1234512345678$ but not $1223144553678$. Find the number of possible sequences of the $13$ numbers.
Answer: 572
Problem: Find the least prime number greater than $1000$ that divides $2^{1010} \cdot  23^{2020} + 1$.
Answer: 1013
Problem: Six different small books and three different large books sit on a shelf. Three children may each take either  two small books or one large book. Find the number of ways the three children can select their books.
Answer: 1176
Problem: A storage depot is a pyramid with height $30$ and a square base with side length $40$. Determine how many cubical $3\times 3\times 3$ boxes can be stored in this depot if the boxes are always packed so that each of their edges is parallel to either an edge of the base or the altitude of the pyramid.
Answer: 471
Problem: 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
Problem: 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
Problem: 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
Problem: What percentage of twenty thousand is a quarter of a million?
Answer: 1250
Problem: 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
Problem: Let $p, q$, and $r$ be prime numbers such that $2pqr + p + q + r = 2020$. Find $pq + qr + rp$.
Answer: 585
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the sum of all positive integers $x$ such that there is a positive integer $y$ satisfying $9x^2 - 4y^2 = 2021$.
Answer: 352
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the minimum possible value of |m -n|, where $m$ and $n$ are integers satisfying $m + n = mn - 2021$.
Answer: 331
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the greatest prime number $p$ such that $p^3$ divides $$\frac{122!}{121}+ 123!:$$
Answer: 61
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the least odd positive integer that is the middle number of five consecutive integers that are all composite.
Answer: 93
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the number of positive divisors of $20^{22}$ that are perfect squares or perfect cubes.
Answer: 364
Problem: 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
Problem: 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
Problem: 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