prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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A particular $12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$, it mistakenly displays a $9$. For example, when it is $1:16$ PM the clock incorrectly shows $9:96$ PM. Find the fraction of the day during which the clock shows the correct time. The orig... | 3 |
Two angles of an isosceles triangle measure $70^\circ$ and $x^\circ$. What is the sum of the three possible values of $x$? | 165 |
Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is the value of $AE$? | 6 |
A convex polyhedron $Q$ has vertices $V_1,V_2,\cdots,V_n$, and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\cdots,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. ... | 300 |
What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by $8$? Fill in the blank with your answer. | 4 |
Let $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are complex numbers. Suppose that $$p(2009 + 9002\pi i) = p(2009) = p(9002) = 0$$ Find the number of nonreal zeros of $x^{12} + ax^8 + bx^4 + c$. | 8 |
Andy and Bethany have a rectangular array of numbers with $40$ rows and $75$ columns. Andy adds the numbers in each row. The average of his $40$ sums is $A$. Bethany adds the numbers in each column. The average of her $75$ sums is $B$. Find the value of \(\frac{A}{B}\). The answer should be in the form \(\frac{k}{m}\),... | 23 |
Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probabil... | 13 |
Parallelogram $ABCD$ has area $1,000,000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = x$ and $y = kx$ for some integer $k > 1$, respectively. How many such parallelograms are there? | 784 |
Find the number of whole numbers between 1 and 1000 that do not contain the digit 1. | 728 |
Functions $f$ and $g$ are quadratic, $g(x) = - f(100 - x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 150$. Then $x_4 - x_1 = m + n\sqrt p$, where $m$, $n$, and $p$ ar... | 752 |
Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is the length of $AE$? | 6 |
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every \(90\) seconds, and Robert runs clockwise and completes a lap every \(80\) seconds. Both start from the same line at the same time. At some random time between \(10\) minutes and \(11\) minutes after they begin to run, a p... | 19 |
A region \( S \) in the complex plane is defined by \( S = \{x + iy: -1 \le x \le 1, -1 \le y \le 1\} \). A complex number \( z = x + iy \) is chosen uniformly at random from \( S \). Find the probability that \(\left(\frac{3}{4} + \frac{3}{4}i\right)z\) is also in \( S \). The original answer is in \(\frac{k}{m}\) for... | 16 |
Three distinct vertices of a cube are chosen at random. What is the probability, in simplest form, that the plane determined by these three vertices contains points inside the cube? The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is the simplest form of the probability. Please give the value of ... | 11 |
The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle measure in degrees of the larger interior a... | 100 |
The letters $A$, $B$, $C$ and $D$ represent digits. If $\begin{array}{rrr}&\mathrm{A}&\mathrm{B}\\+&\mathrm{C}&\mathrm{A}\\\hline&\mathrm{D}&\mathrm{A}\\\end{array}$ and $\begin{array}{ccc}&\mathrm{A}&\mathrm{B}\\-&\mathrm{C}&\mathrm{A}\\\hline&&\mathrm{A}\\\end{array}$, what digit does $D$ represent? | 9 |
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? Provide your answer as a simplified... | 19 |
A ticket to a school play costs $x$ dollars, where $x$ is a whole number. A group of $9$th graders buys tickets costing a total of $＄48$, and a group of $10$th graders buys tickets costing a total of $＄64$. How many values for $x$ are possible? | 5 |
A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month? | 3 |
Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for $a$, $b$, $c$, and $d$ were... | 3 |
A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month? | 3 |
Halfway through a $100$-shot archery tournament, Chelsea leads by $50$ points. For each shot, a bullseye scores $10$ points, with other possible scores being $8, 4, 2$, and $0$ points. Chelsea always scores at least $4$ points on each shot. If Chelsea's next $n$ shots are bullseyes, she will be guaranteed victory. What... | 42 |
A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers \( x \) and \( x+32 \) are three-digit and four-digit palindromes, respectively. What is the sum of the digits of \( x \)? | 24 |
At the beginning of the school year, $50\%$ of all students in Mr. Well's class answered "Yes" to the question "Do you love math", and $50\%$ answered "No." At the end of the school year, $70\%$ answered "Yes" and $30\%$ answered "No." Altogether, $x\%$ of the students gave a different answer at the beginning and end o... | 60 |
Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How m... | 23 |
A ${palindrome}$, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$? | 24 |
Let \( n \) be the smallest positive integer such that \( n \) is divisible by 20, \( n^2 \) is a perfect cube, and \( n^3 \) is a perfect square. Find the number of digits of \( n \). | 7 |
The average of the numbers \(1, 2, 3, \ldots, 98, 99,\) and \(x\) is \(100x\). The answer is in the form of \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Please find the value of \(k + m\). | 151 |
The solution of the equation \(7^{x+7} = 8^x\) can be expressed in the form \(x = \log_b 7^7\). The original answer is in the format \(\frac{k}{m}\). Please give the value of \(k + m\). | 15 |
A palindrome between $1000$ and $10,000$ is chosen at random. Find the probability that it is divisible by $7$. The original answer is in \frac{k}{m} format, please give the value of k + m. | 6 |
Angelina drove at an average rate of $80$ kmh and then stopped $20$ minutes for gas. After the stop, she drove at an average rate of $100$ kmh. Altogether she drove $250$ km in a total trip time of $3$ hours including the stop. Write the equation that could be used to solve for the time $t$ in hours that she drove befo... | 191 |
The average of the numbers $1, 2, 3, \cdots, 98, 99,$ and $x$ is $100x$. The original answer is in the form of \frac{k}{m}, where k and m are integers. Find the value of k + m. | 151 |
Nondegenerate $\triangle ABC$ has integer side lengths, $\overline{BD}$ is an angle bisector, $AD = 3$, and $DC=8$. What is the smallest possible value of the perimeter of $\triangle ABC$? | 33 |
Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2010$ and let $M$ be the largest of the sums $a+b$, $b+c$, $c+d$, and $d+e$. What is the smallest possible value of $M$? | 671 |
On a $50$-question multiple choice math contest, students receive $4$ points for a correct answer, $0$ points for an answer left blank, and $-1$ point for an incorrect answer. Jesse’s total score on the contest was $99$. What is the maximum number of questions that Jesse could have answered correctly? | 29 |
A coin is altered so that the probability that it lands on heads is less than \(\frac{1}{2}\) and when the coin is flipped four times, the probability of an equal number of heads and tails is \(\frac{1}{6}\). What is the probability that the coin lands on heads? Express your answer in the form \(\frac{k-\sqrt{m}}{n}\),... | 12 |
For how many ordered triples $(x,y,z)$ of nonnegative integers less than $20$ are there exactly two distinct elements in the set $\{i^x, (1+i)^y, z\}$, where $i=\sqrt{-1}$? | 225 |
A square of side length $1$ and a circle of radius $\dfrac{\sqrt{3}}{3}$ share the same center. Find the area inside the circle, but outside the square. The original answer is in the form \frac{k\pi}{m} - \frac{\sqrt{n}}{p}, please give the value of k + m + n + p. | 17 |
Bernardo randomly picks $3$ distinct numbers from the set \(\{1,2,3,\cdots,7,8,9\}\) and arranges them in descending order to form a $3$-digit number. Silvia randomly picks $3$ distinct numbers from the set \(\{1,2,3,\cdots,6,7,8\}\) and also arranges them in descending order to form a $3$-digit number. What is the pro... | 93 |
Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2010\}$. Find the probability that $abc + ab + a$ is divisible by $3$. The original answer is in the form of a simplified fraction $\frac{m}{n}$, please give the value of $m + n$. | 40 |
A solid cube has side length $3$ inches. A $2$-inch by $2$-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid? | 7 |
Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How m... | 23 |
Equiangular hexagon $ABCDEF$ has side lengths $AB=CD=EF=1$ and $BC=DE=FA=r$. The area of $\triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $r$? | 6 |
The diagram shows an octagon consisting of $10$ unit squares. The portion below $\overline{PQ}$ is a unit square and a triangle with base $5$. If $\overline{PQ}$ bisects the area of the octagon, find the ratio \(\frac{XQ}{QY}\). The original answer is in the form \(\frac{k}{m}\), please provide the value of \(k + m\). | 5 |
Bernardo randomly picks $3$ distinct numbers from the set $\{1,2,3,...,7,8,9\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks $3$ distinct numbers from the set $\{1,2,3,...,6,7,8\}$ and also arranges them in descending order to form a $3$-digit number. Find the probability that B... | 93 |
Positive integers \(a\), \(b\), and \(c\) are randomly and independently selected with replacement from the set \(\{1, 2, 3, \cdots, 2010\}\). Find the probability that \(abc + ab + a\) is divisible by \(3\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 40 |
A $16$-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by $1$. Find how many such paths stay outside or on the boundary of the square $-2 \le x \le 2$, $-2 \le y \le 2$ at each step. | 1698 |
A frog makes $3$ jumps, each exactly $1$ meter long. The directions of the jumps are chosen independently at random. Find the probability that the frog's final position is no more than $1$ meter from its starting position. The original answer is in the form \(\frac{k}{m}\). Please give the value of \(k + m\). | 5 |
Each of $2010$ boxes in a line contains a single red marble, and for $1 \le k \le 2010$, the box in the $k{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be ... | 45 |
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arit... | 34 |
A fly trapped inside a cubical box with side length \(1\) meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. The maxi... | 8 |
Arithmetic sequences $\left(a_n\right)$ and $\left(b_n\right)$ have integer terms with $a_1=b_1=1<a_2 \le b_2$ and $a_n b_n = 2010$ for some $n$. What is the largest possible value of $n$? | 8 |
A geometric sequence $(a_n)$ has $a_1=\sin x$, $a_2=\cos x$, and $a_3= \tan x$ for some real number $x$. Find the value of $n$ such that $a_n=1+\cos x$. | 8 |
The polynomial $x^3-ax^2+bx-2010$ has three positive integer roots. What is the smallest possible value of $a$? | 78 |
The graph of $y=x^6-10x^5+29x^4-4x^3+ax^2$ lies above the line $y=bx+c$ except at three values of $x$, where the graph and the line intersect. What is the largest of these values of $x$? | 4 |
Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that $P(1) = P(3) = P(5) = P(7) = a,$ and $P(2) = P(4) = P(6) = P(8) = -a.$ What is the smallest possible value of $a$? | 315 |
Let \(ABCD\) be a cyclic quadrilateral. The side lengths of \(ABCD\) are distinct integers less than \(15\) such that \(BC\cdot CD=AB\cdot DA\). Find the largest possible value of \(BD\) in the form \(\sqrt{\frac{x}{y}}\). Please provide the value of \(x + y\). | 427 |
Each of $2010$ boxes in a line contains a single red marble, and for $1 \le k \le 2010$, the box in the $k{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be ... | 45 |
The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is the value of $n$? | 12 |
Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$. Find the sum of the minimum values of $P(x)$ and $Q(x)$. | -100 |
Semicircles $POQ$ and $ROS$ pass through the center $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$? The answer is in the form of \frac{k}{m}, where k and m are integers. Please provide the value of k + m. | 3 |
$A$ high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing ar... | 34 |
Let $f(x) = \log_{10} \left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)$. The intersection of the domain of $f(x)$ with the interval $[0,1]$ is a union of $n$ disjoint open intervals. What is the value of $n$? | 12 |
The set of real numbers $x$ for which $$\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge1$$ is the union of intervals of the form $a<x\le b$. What is the sum of the lengths of these intervals? | 3 |
Jim starts with a positive integer $n$ and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with $n = 55$, then his sequence contains $5$ numbers: $$\begin{array... | 3 |
Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that $P(1) = P(3) = P(5) = P(7) = a$, and $P(2) = P(4) = P(6) = P(8) = -a$. Find the smallest possible value of $a$. | 315 |
Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to $32$? Provide your answer as an integer. | 568 |
Suppose [$a$ $b$] denotes the average of $a$ and $b$, and {$a$ $b$ $c$} denotes the average of $a$, $b$, and $c$. Calculate the value of $\{\{{1 1 0}\} { [0 1] } 0\}$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 25 |
Extend the square pattern of $8$ black and $17$ white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern? The original answer is in the form of a ratio, $a:b$. Please provide the value of $a + b$. | 49 |
Last summer $30\%$ of the birds living on Town Lake were geese, $25\%$ were swans, $10\%$ were herons, and $35\%$ were ducks. What is the percentage of the birds that were not swans and were geese? If the answer is x%, what is the value of x? | 40 |
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 $N$? | 3 |
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was $... | 13 |
Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2 : 3$. What is the degree measure of $\angle{BAC}$? If the answer is $x^\circ$, what is the value of $x$? | 36 |
A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils ... | 11 |
Let \(x\) and \(y\) be two-digit positive integers with mean \(60\). Find the maximum value of the ratio \(\frac{x}{y}\). The answer is in the form \(\frac{m}{n}\), please provide the value of \(m + n\). | 40 |
In the eight term sequence $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. Find the value of $A+H$. | 25 |
At a twins and triplets convention, there were $9$ sets of twins and $6$ sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many ha... | 441 |
Two real numbers are selected independently and at random from the interval \([-20,10]\). Find the probability that the product of those numbers is greater than zero. Provide your answer as a simplified fraction \(\frac{k}{m}\), and then give the value of \(k + m\). | 14 |
A pair of standard 6-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? Provide your answer as a fraction in simplest form, \(\frac{k}{m}\... | 13 |
Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD=\angle CMD$. If the degree measure of $\angle AMD$ is $x^\circ$, what is the value of $x$? | 75 |
Circles $A, B,$ and $C$ each have radius $1$. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}.$ Find the area inside circle $C$ but outside circle $A$ and circle $B.$ | 2 |
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of \(6\) meters, and it takes her \(36\) seconds longer to walk around the outside edge of the track than around the inside edge. The original answer for... | 4 |
$A$ power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the raf... | 45 |
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. Find the probability that the dart lands within the center square. The original answer is in the form \(\frac{k\sqrt{m} - n}{p}\), please find the value of k + ... | 4 |
Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. Find the probability that Angie and Carlos are seated opposite each other. Express your answer in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please provide the value of \(k + m\). | 4 |
Triangle $ABC$ has side-lengths $AB = 12, BC = 24,$ and $AC = 18.$ The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N.$ What is the perimeter of $\triangle AMN?$ | 30 |
Brian writes down four integers $w > x > y > z$ whose sum is $44$. The pairwise positive differences of these numbers are $1, 3, 4, 5, 6$ and $9$. What is the sum of the possible values of $w$? | 31 |
Two congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown. What is the percentage of the area of rectangle $AQRD$ that is shaded? If the answer is $x\%$, what is the value of $x$? | 20 |
Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. Find the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$. The original answer is in \frac{k}{m} format, please give the value of k + m. | 100 |
A segment through the focus $F$ of a parabola with vertex $V$ is perpendicular to $\overline{FV}$ and intersects the parabola in points $A$ and $B$. Find $\cos\left(\angle AVB\right)$. The original answer is in the form -\frac{k}{m}, where k and m are integers. Please give the value of k + m. | 8 |
The circular base of a hemisphere of radius \(2\) rests on the base of a square pyramid of height \(6\). The hemisphere is tangent to the other four faces of the pyramid. The original answer is in the form of \(\sqrt{k}\). What is the value of \(k\) if it represents the edge-length squared of the base of the pyramid? | 18 |
A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. Find the probability that the dart lands within the center square. The original answer is in the form \(\frac{\sqrt{k} - 1}{m}\). Please give the value of k + m. | 4 |
Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. Find the number of different colorings possible. | 3120 |
Rhombus \(ABCD\) has side length \(2\) and \(\angle B = 120^\circ\). Region \(R\) consists of all points inside the rhombus that are closer to vertex \(B\) than any of the other three vertices. The area of \(R\) is given in the form \(\frac{k\sqrt{n}}{m}\). Please find the value of \(k + m + n\). | 8 |
In the eight term sequence $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. What is $A+H$? | 25 |
In the given circle, the diameter $\overline{EB}$ is parallel to $\overline{DC}$, and $\overline{AB}$ is parallel to $\overline{ED}$. The angles $AEB$ and $ABE$ are in the ratio $4 : 5$. What is the degree measure of angle $BCD$? If the answer is $x^\circ$, what is the value of $x$? | 130 |
Circles with radii \(1\), \(2\), and \(3\) are mutually externally tangent. Find the area of the triangle determined by the points of tangency. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 11 |
Let $f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))$, and $h_n(x) = h_1(h_{n-1}(x))$ for integers $n \geq 2$. Find the sum of the digits of $h_{2011}(1)$. | 16089 |
Circles $A, B,$ and $C$ each have radius $1$. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}.$ Find the area inside circle $C$ but outside circle $A$ and circle $B$. | 2 |
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