prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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A pyramid has a square base with side of length \(1\) and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. The volume of this cube is in the form \(k\sqrt{2} -... | 12 |
At a competition with $N$ players, the number of players given elite status is equal to $2^{1+\lfloor \log_{2} (N-1) \rfloor}-N$. Suppose that $19$ players are given elite status. Find the sum of the two smallest possible values of $N$. | 154 |
A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 2$ passes through no lattice point with $0 < x \leq 100$ for all $m$ such that $\frac{1}{2} < m < a$. The original answer is in \frac{k}{m} format, please give the value of k + m. What is the m... | 149 |
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 region $R$ is given in the form \(\frac{k\sqrt{m}}{n}\). Please find the value of \(k + m + n\). | 8 |
Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50<f(7)<60$, $70<f(8)<80$, $5000k<f(100)<5000(k+1)$ for some integer $k$. Find the value of $k$. | 3 |
Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid? | 750 |
Let \(f_{1}(x)=\sqrt{1-x}\), and for integers \(n \geq 2\), let \(f_{n}(x)=f_{n-1}(\sqrt{n^2 - x})\). If \(N\) is the largest value of \(n\) for which the domain of \(f_{n}\) is nonempty, the domain of \(f_{N}\) is \([c]\). Find the value of \(N+c\). | -226 |
The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is the absolute value of the difference $|x - y|$? | 66 |
Students guess that Norb's age is $24, 28, 30, 32, 36, 38, 41, 44, 47$, and $49$. Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb? | 37 |
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. How many different colorings are possible? | 3120 |
A pyramid has a square base with sides of length \(1\) and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. The volume of this cube can be expressed in the for... | 12 |
Let \(T_1\) be a triangle with side lengths \(2011\), \(2012\), and \(2013\). For \(n \geq 1\), if \(T_n = \triangle ABC\) and \(D, E\), and \(F\) are the points of tangency of the incircle of \(\triangle ABC\) to the sides \(AB\), \(BC\), and \(AC\), respectively, then \(T_{n+1}\) is a triangle with side lengths \(AD,... | 1637 |
Seven students count from $1$ to $1000$ as follows: Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says $1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000$. Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in... | 365 |
What is the hundreds digit of \(2011^{2011}\)? | 6 |
Let \(f(z)= \frac{z+a}{z+b}\) and \(g(z)=f(f(z))\), where \(a\) and \(b\) are complex numbers. Suppose that \(\left| a \right| = 1\) and \(g(g(z))=z\) for all \(z\) for which \(g(g(z))\) is defined. Find the difference between the largest and smallest possible values of \(\left| b \right|\). The original answer is in t... | 4 |
How many $4$-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of $5$, and $5$ is the largest digit? | 84 |
A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 2$ passes through no lattice point with $0 < x \le 100$ for all $m$ such that $\frac{1}{2} < m < a$. What is the maximum possible value of $a$? The original answer is in \(\frac{k}{m}\) format,... | 149 |
Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, and $DA=12$. Find the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral. The original answer is in the form $k\sqrt{m}$, please give the value of k + m. | 8 |
Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. Find the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$. The original answer is in the form of $k\sqrt{m}$, please give the value of k + m. | 10 |
Let \(T_1\) be a triangle with side lengths \(2011, 2012,\) and \(2013\). For \(n \ge 1\), if \(T_n = \triangle ABC\) and \(D, E,\) and \(F\) are the points of tangency of the incircle of \(\triangle ABC\) to the sides \(AB, BC\), and \(AC,\) respectively, then \(T_{n+1}\) is a triangle with side lengths \(AD, BE,\) an... | 1637 |
In a bag of marbles, \(\frac{3}{5}\) of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? Express your answer as a fraction in simplest form. The original answer is in the form \(\frac{k}{m}\). Plea... | 11 |
Suppose that the euro is worth $1.3$ dollars. If Diana has $500$ dollars and Etienne has $400$ euros, by what percent is the value of Etienne's money greater than the value of Diana's money? Provide your answer as a percentage. | 4 |
Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat? Express your answer as a fraction in simplest form, in the format \(\frac{k}{m}\), and give the value of k + m. | 9 |
Two integers have a sum of $26$. When two more integers are added to the first two, the sum is $41$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum is $57$. Find the minimum number of even integers among the $6$ integers. | 1 |
In a bag of marbles, \(\frac{3}{5}\) of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 11 |
Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle? | 8 |
Small lights are hung on a string $6$ inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of $2$ red lights followed by $3$ green lights. How many feet separate the 3rd red light and the 21st red light? If the answer is $x$ feet, what is the value of... | 225 |
An iterative average of the numbers \(1, 2, 3, 4\), and \(5\) is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number.... | 25 |
$A$ pair of six-sided dice are labeled so that one die has only even numbers (two each of $2, 4$, and $6$), and the other die has only odd numbers (two of each $1, 3$, and $5$). The pair of dice is rolled. Find the probability that the sum of the numbers on the tops of the two dice is $7$. Express your answer in the fo... | 4 |
A triangle has an area of $30$, one side of length $10$, and the median to that side has a length of $9$. Let $\theta$ be the acute angle formed by that side and the median. The original answer for $\sin{\theta}$ is in the form \frac{k}{m}. Please provide the value of k + m. | 5 |
What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 = 25$ and $(x-4)^2 + 9y^2 = 81$? Provide your answer as an integer. | 27 |
Externally tangent circles with centers at points $A$ and $B$ have radii of lengths $5$ and $3$, respectively. A line externally tangent to both circles intersects ray $AB$ at point $C$. What is the length of $BC$? | 12 |
Alex, Mel, and Chelsea play a game that has $6$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round, the probability that Alex wins is $\frac{1}{2}$, and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two... | 41 |
Point $B$ is due east of point $A$. Point $C$ is due north of point $B$. The distance between points $A$ and $C$ is $10\sqrt{2}$, and $\angle BAC = 45^\circ$. Point $D$ is $20$ meters due north of point $C$. Calculate the distance $AD$ and determine the interval in which this distance falls, expressed as (a, b) where a... | 63 |
A square region \(ABCD\) is externally tangent to the circle with equation \(x^2+y^2=1\) at the point \((0,1)\) on the side \(CD\). Vertices \(A\) and \(B\) are on the circle with equation \(x^2+y^2=4\). The side length of this square can be expressed in the form \(\frac{k\sqrt{m}-n}{p}\), where \(\sqrt{m}\) is a squar... | 30 |
How many sequences of zeros and ones of length $20$ have all the zeros consecutive, or all the ones consecutive, or both? | 380 |
Two parabolas have equations \(y= x^2 + ax + b\) and \(y= x^2 + cx + d\), where \(a, b, c,\) and \(d\) are integers, each chosen independently by rolling a fair six-sided die. Find the probability that the parabolas will have at least one point in common. The original answer is in \(\frac{k}{m}\) format, please give th... | 67 |
Chubby makes nonstandard checkerboards that have $31$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard? | 481 |
Two equilateral triangles are contained in a square whose side length is \(2\sqrt{3}\). The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. The area of the rhombus is in the form \(k\sqrt{3} - m\). Please find the value of k + m. | 20 |
Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last ... | 7 |
A \(3 \times 3\) square is partitioned into \(9\) unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated \(90\,^{\circ}\) clockwise about its center, and every white square in a position formerly occupied by a ... | 561 |
Jesse cuts a circular disk of radius \(12\), along \(2\) radii to form \(2\) sectors, one with a central angle of \(120^\circ\). He makes two circular cones using each sector to form the lateral surface of each cone. The ratio of the volume of the smaller cone to the larger cone is in the form \(\frac{\sqrt{k}}{n}\). P... | 20 |
Three circles with radius $2$ are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure? Express your answer in the form $a\pi + b\sqrt{3}$, where $a$ and $b$ are constants. Please find the value of $a + b$. | 14 |
Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. The radius of circle $C_1$ is in the form \(\sqrt{k}\). Find the value of \(k\). | 30 |
Jesse cuts a circular paper disk of radius \(12\) along two radii to form two sectors, the smaller having a central angle of \(120\) degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger? Express your answ... | 20 |
Triangle $ABC$ has $AB=27$, $AC=26$, and $BC=25$. Let $I$ be the intersection of the internal angle bisectors of $\triangle ABC$. What is the length of $BI$? | 15 |
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than $50$? | 3127 |
In rectangle \(ABCD\), \(AB=6\), \(AD=30\), and \(G\) is the midpoint of \(\overline{AD}\). Segment \(AB\) is extended \(2\) units beyond \(B\) to point \(E\), and \(F\) is the intersection of \(\overline{ED}\) and \(\overline{BC}\). The area of quadrilateral \(BFDG\) is originally given in the form \(\frac{k}{m}\). Pl... | 137 |
A unit cube has vertices \(P_1,P_2,P_3,P_4,P_1',P_2',P_3',\) and \(P_4'\). Vertices \(P_2\), \(P_3\), and \(P_4\) are adjacent to \(P_1\), and for \(1\le i\le 4,\) vertices \(P_i\) and \(P_i'\) are opposite to each other. A regular octahedron has one vertex in each of the segments \(P_1P_2\), \(P_1P_3\), \(P_1P_4\), \(... | 9 |
Consider the polynomial \[ P(x)=\prod\limits_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024) \] The coefficient of \( x^{2012} \) is equal to \( 2^a \). What is the value of \( a \)? | 6 |
Let points $A$ = $(0 ,0 ,0)$, $B$ = $(1, 0, 0)$, $C$ = $(0, 2, 0)$, and $D$ = $(0, 0, 3)$. Points $E$, $F$, $G$, and $H$ are midpoints of line segments $\overline{BD},\text{ } \overline{AB}, \text{ } \overline {AC},$ and $\overline{DC}$ respectively. Find the area of quadrilateral $EFGH$. The original answer is in the ... | 12 |
Let $a$, $b$, and $c$ be positive integers with $a \ge b \ge c$ such that $a^2-b^2-c^2+ab=2011$ and $a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$. What is the value of $a$? | 253 |
The sum of the first $m$ positive odd integers is $212$ more than the sum of the first $n$ positive even integers. Find the sum of all possible values of $n$. | 255 |
Let $R$ be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$? | 7 |
A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface fa... | 8 |
A circle of radius \(2\) is cut into four congruent arcs. The four arcs are joined to form the star figure shown. Find the ratio of the area of the star figure to the area of the original circle. The original answer is in the form \(\frac{k-\pi}{m\pi}\). Please provide the value of \(k + m\). | 5 |
A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. The original answer is in the form \frac{k}{m}, wher... | 3 |
When counting from $3$ to $201$, $53$ is the $51^{st}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{th}$ number counted. What is the value of $n$? | 149 |
$A$ softball team played ten games, scoring $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$ runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? | 45 |
Joey and his five brothers are ages $3$, $5$, $7$, $9$, $11$, and $13$. One afternoon two of his brothers whose ages sum to $16$ went to the movies, two brothers younger than $10$ went to play baseball, and Joey and the $5$-year-old stayed home. How old is Joey? | 11 |
The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6\times5$. What is the missing number in the top row? [asy] unitsize(0.8cm); draw((-1,0)--(1,0)--(1,-2)--(-1,-2)--cycle); draw((-2,0)--(0,0)--(0,2)--(-2,2)--cycle); draw((0,0)--(2,0)--(2,2)--(0... | 4 |
Six points are equally spaced around a circle of radius \(1\). Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle? The original answer is in \(\frac{\sqrt{k}}{m}\) format, please give the value of k + m. | 5 |
Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying $``1"$, so Blair follows by saying $``1, 2"$. Jo then says $``1, 2, 3"$, and so on. What is the $53^{{rd}}$ number said? | 8 |
Line \(l_1\) has equation \(3x - 2y = 1\) and goes through \(A = (-1, -2)\). Line \(l_2\) has equation \(y = 1\) and meets line \(l_1\) at point \(B\). Line \(l_3\) has positive slope, goes through point \(A\), and meets \(l_2\) at point \(C\). The area of \(\triangle ABC\) is \(3\). The original answer for the slope o... | 7 |
In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$? | 56 |
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$? | 8 |
$A$ flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What is the percentage of the flowers that are carnations? Provide your answer as a percentage. | 70 |
Let $S$ be the set of positive integers $n$ for which $\frac{1}{n}$ has the repeating decimal representation $0.\overline{ab} = 0.ababab\cdots,$ with $a$ and $b$ different digits. Find the sum of the elements of $S$. | 143 |
Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges... | 103 |
Triangle \(ABC\) is equilateral with \(AB=1\). Points \(E\) and \(G\) are on \(\overline{AC}\) and points \(D\) and \(F\) are on \(\overline{AB}\) such that both \(\overline{DE}\) and \(\overline{FG}\) are parallel to \(\overline{BC}\). Furthermore, triangle \(ADE\) and trapezoids \(DFGE\) and \(FBCG\) all have the sam... | 34 |
Ted's grandfather used his treadmill on $3$ days this week. He went $2$ miles each day. On Monday he jogged at a speed of $5$ miles per hour. He walked at the rate of $3$ miles per hour on Wednesday and at $4$ miles per hour on Friday. If Grandfather had always walked at $4$ miles per hour, he would have spent less tim... | 4 |
The angles in a particular triangle are in arithmetic progression, and the side lengths are $4,5,x$. The sum of the possible values of $x$ equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is the value of $a+b+c$? | 36 |
In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. Find the perimeter of parallelogram $ADEF$. | 56 |
Let \(S\) be the set of sides and diagonals of a regular pentagon. A pair of elements of \(S\) are selected at random without replacement. Find the probability that the two chosen segments have the same length. The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is the simplified form of the probabi... | 13 |
Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, and $\widetilde{DE}$. Find the number of different routes from $A$ to $B$ that use each road exactly once. (Such a route will necessarily visit some citie... | 16 |
At the $2013$ Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $＄50$, you get a second pair at a $40\%$ discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What is t... | 30 |
How many three-digit numbers are not divisible by $5$, have digits that sum to less than $20$, and have the first digit equal to the third digit? | 60 |
Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying “$1$”, so Blair follows by saying “$1 , 2$”. Jo then says “$1,2,3$”, and so on. What is the $53^{rd}$ number said? | 8 |
Let points $A = (0,0) , \ B = (1,2), \ C = (3,3),$ and $D = (4,0)$. Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\left (\frac{p}{q}, \frac{r}{s} \right )$, where these fractions are in lowest terms. Find the value of $p + q + r + s$. | 58 |
The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two... | 240 |
The sequence $\log_{12}{162}$, $\log_{12}{x}$, $\log_{12}{y}$, $\log_{12}{z}$, $\log_{12}{1250}$ is an arithmetic progression. What is the value of $x$? | 270 |
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N$. Find the smallest possible value of $N$. | 104 |
A wire is cut into two pieces, one of length \(a\) and the other of length \(b\). The piece of length \(a\) is bent to form an equilateral triangle, and the piece of length \(b\) is bent to form a regular hexagon. The triangle and the hexagon have equal area. The original answer is in the form \(\frac{\sqrt{p}}{n}\) wh... | 8 |
Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done? | 204 |
In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. If the area of $AEDC$ is $x$, what is the value of $10x$? | 135 |
$A$, $B$, $C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40$ pounds, the mean weight of the rocks in $B$ is $50$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $43$ pounds, and the mean weight of the rocks in the combined piles $A$ and $C$ is $44$ pounds. What is the gr... | 59 |
Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next $365$-day period will exactly two friends visit her? | 54 |
Let \(a, b,\) and \(c\) be real numbers such that \(a+b+c=2,\) and \(a^2+b^2+c^2=12\) Find the difference between the maximum and minimum possible values of \(c\). The original answer is in \(\frac{m}{n}\) format, please give the value of \(m+n\). | 19 |
Let points $A = (0, 0)$, $B = (1, 2)$, $C=(3, 3)$, and $D = (4, 0)$. Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\left(\frac{p}{q}, \frac{r}{s}\right)$, where these fractions are in lowest terms. Find the value of $p+q+r+s$. | 58 |
The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. Find how many integers less than $2013$ but greater than $1000$ have this property. | 46 |
Six spheres of radius \(1\) are positioned so that their centers are at the vertices of a regular hexagon of side length \(2\). The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to... | 5 |
Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain? | 280 |
In base $10$, the number $2013$ ends in the digit $3$. In base $9$, on the other hand, the same number is written as $(2676)_{9}$ and ends in the digit $6$. For how many positive integers $b$ does the base-$b$-representation of $2013$ end in the digit $3$? | 13 |
The real numbers \(c, b, a\) form an arithmetic sequence with \(a \geq b \geq c \geq 0\). The quadratic \(ax^2+bx+c\) has exactly one root. The original answer is in the form \(-k+\sqrt{m}\). Please give the value of \(k + m\). | 5 |
In $\bigtriangleup ABC$, $AB = 86$, and $AC = 97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover, $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is the length of $BC$? | 61 |
In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline... | 21 |
The number $2013$ is expressed in the form $2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$, where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. Find the value of $|a_1 - b_1|$. | 2 |
A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to rec... | 1925 |
Samantha lives $2$ blocks west and $1$ block south of the southwest corner of City Park. Her school is $2$ blocks east and $2$ blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner,... | 18 |
Six spheres of radius \(1\) are positioned so that their centers are at the vertices of a regular hexagon of side length \(2\). The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to... | 5 |
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