prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. The probability that no two ants arrive at the same vertex is given in the f... | 261 |
On the AMC $8$ contest Billy answers $13$ questions correctly, answers $7$ questions incorrectly and doesn't answer the last $5$. What is his score? | 13 |
Elisa swims laps in the pool. When she first started, she completed $10$ laps in $25$ minutes. Now, she can finish $12$ laps in $24$ minutes. By how many minutes has she improved her lap time? Provide your answer in the form of a simplified fraction \(\frac{k}{m}\), and calculate the value of \(k + m\). | 3 |
A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display? | 23 |
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 is the percentage of the $20.00$ that she will receive in change, rounded to the nearest integer? | 5 |
Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there was an additional cost of $2$ dollars for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. ... | 4 |
A $2 \times 3$ rectangle and a $3 \times 4$ rectangle are contained within a square without overlapping at any point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square? | 25 |
What non-zero real value for \(x\) satisfies \((7x)^{14}=(14x)^7\)? Provide your answer in the form of \(\frac{a}{b}\), where \(a\) and \(b\) are integers. Please find the value of \(a + b\). | 9 |
A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $\frac{2}{\pi}$, as shown. What is the perimeter of this region? | 4 |
The $8\times18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is the value of $y$? | 6 |
Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water to make lemonade. There are $25$ calories in $100$ grams of lemon juice and $386$ calories in $100$ grams of sugar. Water contains no calories. How many calories are in $200$ grams of her lemonade? | 137 |
The letter T is formed by placing two $2 \times 4$ inch rectangles next to each other, as shown. What is the perimeter of the T, in inches? Provide your answer as an integer. | 20 |
A square of area $40$ is inscribed in a semicircle as shown. If the area of the semicircle is $x\pi$, what is the value of $x$? | 25 |
How many sets of two or more consecutive positive integers have a sum of $15$? Provide your answer as an integer. | 3 |
Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water to make lemonade. There are $25$ calories in $100$ grams of lemon juice and $386$ calories in $100$ grams of sugar. Water contains no calories. How many calories are in $200$ grams of her lemonade? Provide the number of calories a... | 137 |
How many even three-digit integers have the property that their digits, all read from left to right, are in strictly increasing order? Provide your answer as an integer. | 34 |
For how many real values of $x$ is $\sqrt{120-\sqrt{x}}$ an integer? Provide the number of such real values. | 11 |
For how many real values of $x$ is $\sqrt{120-\sqrt{x}}$ an integer? | 11 |
Joe and JoAnn each bought $12$ ounces of coffee in a $16$-ounce cup. Joe drank $2$ ounces of his coffee and then added $2$ ounces of cream. JoAnn added $2$ ounces of cream, stirred the coffee well, and then drank $2$ ounces. The resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee is in the ... | 13 |
The lines \(x=\frac{1}{4}y+a\) and \(y=\frac{1}{4}x+b\) intersect at the point \((1,2)\). The original answer is in the form \(\frac{k}{m}\). Find the value of \(k+m\). | 13 |
Antonette gets $70 \%$ on a $10$-problem test, $80 \%$ on a $20$-problem test and $90 \%$ on a $30$-problem test. If the three tests are combined into one $60$-problem test, what percent is closest to her overall score? | 83 |
A player pays $\textdollar 5$ to play a game. A die is rolled. If the number on the die is odd, the game is lost. If the number on the die is even, the die is rolled again. In this case, the player wins if the second number matches the first and loses otherwise. How much should the player win if the game is fair? (In a... | 60 |
Joe and JoAnn each bought $12$ ounces of coffee in a $16$ ounce cup. Joe drank $2$ ounces of his coffee and then added $2$ ounces of cream. JoAnn added $2$ ounces of cream, stirred the coffee well, and then drank $2$ ounces. Find the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee. The ... | 13 |
Rhombus $ABCD$ is similar to rhombus $BFDE$. The area of rhombus $ABCD$ is $24$, and $\angle BAD = 60^\circ$. What is the area of rhombus $BFDE$? | 8 |
Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what tim... | 11 |
Let \(a\) and \(b\) be the roots of the equation \(x^2-mx+2=0\). Suppose that \(a+\frac{1}{b}\) and \(b+\frac{1}{a}\) are the roots of the equation \(x^2-px+q=0\). The original answer for the value of \(q\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\). | 11 |
Two farmers agree that pigs are worth $300$ dollars and that goats are worth $210$ dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a $390$ dollar debt could be paid with two pigs, with one goat received in ... | 30 |
Odell and Kershaw run for $30$ minutes on a circular track. Odell runs clockwise at $250 \, \text{m/min}$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $300 \, \text{m/min}$ and uses the outer lane with a radius of $60$ meters, starting on the same radial line as Odell. How many... | 47 |
Rhombus $ABCD$ is similar to rhombus $BFDE$. The area of rhombus $ABCD$ is $24$ and $\angle BAD = 60^\circ$. What is the area of rhombus $BFDE$? | 8 |
Suppose \(\cos x=0\) and \(\cos (x+z)=\frac{1}{2}\). What is the smallest possible positive value of \(z\) in terms of \(\pi\)? The answer should be in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of \(k + m\). | 7 |
Circles with centers \(A\) and \(B\) have radius \(3\) and \(8\), respectively. A common internal tangent intersects the circles at \(C\) and \(D\), respectively. Lines \(AB\) and \(CD\) intersect at \(E\), and \(AE=5\). Find the length of \(CD\). The original answer is in \(\frac{k}{m}\) format, please give the value ... | 47 |
Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. Find its area in the form of $x\sqrt{3}$. What is the value of $x$? | 25 |
In rectangle $ADEH$, points $B$ and $C$ trisect $\overline{AD}$, and points $G$ and $F$ trisect $\overline{HE}$. In addition, $AH=AC=2$, and $AD=3$. What is the area of quadrilateral $WXYZ$ shown in the figure? The original answer is in \frac{k}{m} format, please give the value of k + m. | 3 |
Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the cont... | 4 |
Square $ABCD$ has side length $s$, a circle centered at $E$ has radius $r$, and $r$ and $s$ are both rational. 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 $AF=\sqrt{9+5\sqrt{2}}$. The... | 8 |
For a particular peculiar pair of dice, the probabilities of rolling $1$, $2$, $3$, $4$, $5$ and $6$ on each die are in the ratio $1:2:3:4:5:6$. What is the probability of rolling a total of $7$ on the two dice? Provide your answer as a fraction in the form \(\frac{k}{m}\) and calculate the value of \(k + m\). | 71 |
A license plate in a certain state consists of $4$ digits, not necessarily distinct, and $2$ letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible? The original answer is in the form... | 11 |
An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, find the number of different points that could be the final point. | 121 |
A cube with $3$-inch edges is made using $27$ cubes with $1$-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white? The answer should be in the form \frac{k}{m}, where ... | 14 |
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and th... | 5 |
How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression? Provide your answer as an integer. | 59 |
A circle of radius \(2\) is centered at \(O\). Square \(OABC\) has side length \(1\). Sides \(AB\) and \(CB\) are extended past \(B\) to meet the circle at \(D\) and \(E\), respectively. Find the area of the shaded region in the figure, which is bounded by \(BD\), \(BE\), and the minor arc connecting \(D\) and \(E\). T... | 8 |
Triangle $ABC$ is an isosceles triangle with $\overline{AB}=\overline{BC}$. Point $D$ is the midpoint of both $\overline{BC}$ and $\overline{AE}$, and $\overline{CE}$ is $11$ units long. Triangle $ABD$ is congruent to triangle $ECD$. If the length of $\overline{BD}$ is $x$ units, what is the value of $10x$? | 55 |
Six distinct positive integers are randomly chosen between $1$ and $2006$, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of $5$? Express your answer as a fraction. | 1 |
In rectangle $ABCD$, we have $A=(6,-22)$, $B=(2006,178)$, $D=(8,y)$, for some integer $y$. What is the area of rectangle $ABCD$? Provide your answer as an integer. | 40400 |
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex, the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability tha... | 245 |
Let $x$ be chosen at random from the interval $(0,1)$. Find the probability that $\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0$. Here $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$. The answer is in the form of $\frac{k}{m}$, please provide the value of k + m. | 7 |
How many four-digit positive integers have at least one digit that is a $2$ or a $3$? | 5416 |
For a particular peculiar pair of dice, the probabilities of rolling $1$, $2$, $3$, $4$, $5$, and $6$ on each die are in the ratio $1:2:3:4:5:6$. Find the probability of rolling a total of $7$ on the two dice. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 71 |
Rectangle $ABCD$ has area $2006$. An ellipse with area $2006\pi$ passes through $A$ and $C$ and has foci at $B$ and $D$. Find the perimeter of the rectangle. The answer is in the form $k\sqrt{m}$, please provide the value of $k + m$. | 1011 |
Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. Find the difference between the largest and smallest numbers possible in t... | 26 |
Circles with centers \(A\) and \(B\) have radius 3 and 8, respectively. A common internal tangent intersects the circles at \(C\) and \(D\), respectively. Lines \(AB\) and \(CD\) intersect at \(E\), and \(AE=5\). Find the length of \(CD\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m... | 47 |
Given a finite sequence $S=(a_1,a_2,\cdots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence $\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\cdots ,\frac{a_{n-1}+a_n}{2}\right)$ of $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\le m\le n-1$, define $A^m(S)=A(A^{m-1}(S))$. Suppose $x>0$, and let... | 3 |
Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron? Provide your answer in the form \(\frac{k}{m}\), where k and m are integers, and give the value of k + m. | 7 |
Circles with centers \( O \) and \( P \) have radii \( 2 \) and \( 4 \), respectively, and are externally tangent. Points \( A \) and \( B \) on the circle with center \( O \) and points \( C \) and \( D \) on the circle with center \( P \) are such that \( AD \) and \( BC \) are common external tangents to the circles... | 26 |
The expression $$(x+y+z)^{2006}+(x-y-z)^{2006}$$ is simplified by expanding it and combining like terms. How many terms are in the simplified expression? Please give the result of the answer multiplied by 1. | 1008016 |
Let \(S\) be the set of all points \((x,y)\) in the coordinate plane such that \(0 \le x \le \frac{\pi}{2}\) and \(0 \le y \le \frac{\pi}{2}\). Find the area of the subset of \(S\) for which \[ \sin^2x-\sin x \sin y + \sin^2y \le \frac{3}{4}. \] The answer is in the form \(\frac{k\pi^2}{m}\). Please calculate the value... | 7 |
In the multiplication problem below, $A$, $B$, $C$, $D$ are different digits. What is the value of $A+B$? $$\begin{array}{cccc}& A & B & A\\ \times & & C & D\\ \hline C & D & C & D\\ \end{array}$$ | 1 |
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is $9$, spots a license plate with a $4$-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "an... | 5 |
How many non-empty subsets $S$ of $\{1,2,3,\ldots ,15\}$ have the following two properties? $(1)$ No two consecutive integers belong to $S$. $(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$. Provide the number of such subsets. | 405 |
Let \( x \) be the number \( 0.\underbrace{0000\ldots0000}_{1996\text{ zeros}}1, \) where there are \( 1996 \) zeros after the decimal point. Find the largest value of the following expressions: \( 3+x, \ 3-x, \ 3\cdot x, \ \frac{3}{x}, \ \frac{x}{3} \). The original answer is in the form \(\frac{k}{x}\). Please provid... | 3 |
In the number $74982.1035$, determine how many times greater the value of the place occupied by the digit $9$ is compared to the value of the place occupied by the digit $3$. Please provide the answer as an integer value representing the multiple. | 100000 |
All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A= \angle B = 90^\circ.$ If the answer is $x^\circ$, what is the degree measure of $\angle E?$ | 150 |
All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A = \angle B = 90^{\circ}$. If the answer is $x^\circ$, what is the value of $x$ for the degree measure of $\angle E$? | 150 |
On the trip home from the meeting where this $AMC~10$ was constructed, the Contest Chair noted that his airport parking receipt had digits of the form $bbcac,$ where $0 \le a < b < c \le 9,$ and $b$ was the average of $a$ and $c.$ How many different five-digit numbers satisfy all these properties? | 20 |
In trapezoid $ABCD$, $\overline{AD}$ is perpendicular to $\overline{DC}$, $AD = AB = 3$, and $DC = 6$. In addition, $E$ is on $\overline{DC}$, and $\overline{BE}$ is parallel to $\overline{AD}$. If the area of $\triangle BEC$ is $x$, find the value of $10x$. | 45 |
To complete the grid below, each of the digits $1$ through $4$ must occur once in each row and once in each column. Determine the number that will occupy the lower right-hand square. $$ \begin{array}{|c|c|c|c|}\hline1&&2&\\\hline2&3&&\\\hline&&&4\\\hline\\\hline\end{array} $$ | 2 |
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might be... | 37 |
Integers $a, b, c,$ and $d$, not necessarily distinct, are chosen independently and at random from $0$ to $2007$, inclusive. Find the probability that $ad-bc$ is even. The original answer is in \frac{k}{m} format, please give the value of k + m. | 13 |
Two circles of radius $2$ are centered at $(2,0)$ and at $(0,2).$ What is the area of the intersection of the interiors of the two circles? Express your answer in the form k(\pi - m), and find the value of k + m. | 4 |
A traffic light runs repeatedly through the following cycle: green for \(30\) seconds, then yellow for \(3\) seconds, and then red for \(30\) seconds. Leah picks a random three-second time interval to watch the light. What is the probability, expressed as a fraction in simplest form \(\frac{k}{m}\), that the color chan... | 8 |
Four circles of radius \(1\) are each tangent to two sides of a square and externally tangent to a circle of radius \(2\), as shown. The area of the square is expressed in the form \(k + m\sqrt{2}\), where \(k\) and \(m\) are integers. Find the value of \(k + m\). | 34 |
The geometric series \(a+ar+ar^2+\ldots\) has a sum of \(7\), and the terms involving odd powers of \(r\) have a sum of \(3\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 7 |
Suppose that $m$ and $n$ are positive integers such that $75m = n^{3}$. What is the minimum possible value of $m + n$? | 60 |
The polynomial $f(x) = x^{4} + ax^{3} + bx^{2} + cx + d$ has real coefficients, and $f(2i) = f(2 + i) = 0.$ Find the value of $a + b + c + d$. | 9 |
A circle of radius $1$ is surrounded by $4$ circles of radius $r$ as shown. The original answer is in the form $k+\sqrt{m}$, please give the value of $k + m$. | 3 |
Let $a$, $b$, and $c$ be digits with $a\ne 0$. The three-digit integer $abc$ lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer $acb$ lies two thirds of the way between the same two squares. Find the value of $a+b+c$. | 16 |
The product of the two $99$-digit numbers $303,030,303,...,030,303$ and $505,050,505,...,050,505$ has thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$? | 8 |
The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by $4,$ and the second number is divided by $5.$ The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. Find the probability that the p... | 3 |
Rhombus $ABCD$, with side length $6$, is rolled to form a cylinder of volume $6$ by taping $\overline{AB}$ to $\overline{DC}$. What is the value of $\sin(\angle ABC)$? Provide your answer in the form \(\frac{k\pi}{m}\), where \(k\) and \(m\) are integers. Calculate the sum of \(k + m\). | 10 |
A set of 25 square blocks is arranged into a \(5 \times 5\) square. Find the number of different combinations of 3 blocks that can be selected from that set so that no two are in the same row or column. | 600 |
Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra? Express your answer in the form \(\frac{k-m\sqrt{2}}{n}\), where the fraction is in simplest form. Please find the value of \(k + m + n\). | 20 |
The parallelogram bounded by the lines $y=ax+c$, $y=ax+d$, $y=bx+c$, and $y=bx+d$ has area $18$. The parallelogram bounded by the lines $y=ax+c$, $y=ax-d$, $y=bx+c$, and $y=bx-d$ has area $72$. Given that $a$, $b$, $c$, and $d$ are positive integers, find the smallest possible value of $a+b+c+d$. | 16 |
The first $2007$ positive integers are each written in base $3$. How many of these base-$3$ representations are palindromes? | 100 |
Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair? The original answer is in \(\frac{k}{m}\) format, please give the value of k... | 11 |
For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n.$ Find the number of values of $n$ such that $n + S(n) + S(S(n)) = 2007.$ | 4 |
A lemming sits at a corner of a square with side length $10$ meters. The lemming runs $6.2$ meters along a diagonal toward the opposite corner. It stops, makes a $90^{\circ}$ right turn and runs $2$ more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the aver... | 5 |
A pyramid with a square base is cut by a plane that is parallel to its base and $2$ units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. The original answer is in the form k + m\sqrt{2}. Please provide the value of k + m, where k and m a... | 6 |
Square \(ABCD\) has area \(36,\) and \(\overline{AB}\) is parallel to the \(x\)-axis. Vertices \(A,\) \(B\), and \(C\) are on the graphs of \(y = \log_{a}x,\) \(y = 2\log_{a}x,\) and \(y = 3\log_{a}x,\) respectively. The original answer is in the form \(\sqrt[k]{3}\). Please determine the value of \(k\). | 6 |
What is the area of the shaded pinwheel shown in the $5 \times 5$ grid? Provide your answer as an integer. | 6 |
Let $n$ denote the smallest positive integer that is divisible by both $4$ and $9,$ and whose base-$10$ representation consists of only $4$'s and $9$'s, with at least one of each. What are the last four digits of $n?$ | 4944 |
How many pairs of positive integers $(a,b)$ are there such that ${gcd}(a,b)=1$ and $\frac{a}{b} + \frac{14b}{9a}$ is an integer? | 4 |
How many pairs of positive integers $(a,b)$ are there such that $a$ and $b$ have no common factors greater than $1$ and: $\frac{a}{b} + \frac{14b}{9a}$ is an integer? Provide the number of such pairs. | 4 |
Call a set of integers spacy if it contains no more than one out of any three consecutive integers. How many subsets of $\{1,2,3,\cdots,12\},$ including the empty set, are spacy? | 129 |
A bakery owner turns on his doughnut machine at $8:30$ AM. At $11:10$ AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the entire job? Write your answer in the format "HH:MM AM/PM", and then provide the sum of HH and MM. | 34 |
A $4\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums? $\begin{array}{|c|c|c|c|}\hline1&2&3&4\\\hline8&9&10&11\\\hline15&16&17&18\\\h... | 4 |
A $4\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums? | 4 |
The ten-letter code $BEST ~OF ~LUCK$ represents the ten digits $0-9$, in order. What $4$-digit number is represented by the code word $CLUE$? | 8671 |
On circle \(O\), points \(C\) and \(D\) are on the same side of diameter \(\overline{AB}\), \(\angle AOC = 30^\circ\), and \(\angle DOB = 45^\circ\). Find the ratio of the area of the smaller sector \(COD\) to the area of the circle. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 31 |
In the figure, the outer equilateral triangle has area $16$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids? | 5 |
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