prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Chloe chooses a real number uniformly at random from the interval \([ 0,2017 ]\). Independently, Laurent chooses a real number uniformly at random from the interval \([ 0 , 4034 ]\). Find the probability that Laurent's number is greater than Chloe's number. The original answer is in \(\frac{k}{m}\) format, please give ... | 7 |
At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What is the value of $x$ if $x\%$ of stu... | 25 |
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle? | 143 |
Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. Find the total number of monotonous positive int... | 1524 |
There are $10$ horses, named Horse $1$, Horse $2, \ldots,$ Horse $10$. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time $0$ all the horses are together at the starting point on the track. The horses start running... | 3 |
What is the sum of the roots of $z^{12}=64$ that have a positive real part? Express your answer in the form $a\sqrt{b}+c\sqrt{d}$, where $a$, $b$, $c$, and $d$ are integers. Please find the value of a + b + c + d. | 11 |
Define a sequence recursively by $F_{0}=0,~F_{1}=1,$ and $F_{n}=$ the remainder when $F_{n-1}+F_{n-2}$ is divided by $3,$ for all $n\geq 2.$ Thus the sequence starts $0,1,1,2,0,2,\cdots$ Find the value of $F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}.$ | 9 |
Peter, Emma, and Kyler played chess with each other. Peter won $4$ games and lost $2$ games. Emma won $3$ games and lost $3$ games. If Kyler lost $3$ games, how many games did he win? Fill in the number of games Kyler won. | 1 |
An integer $N$ is selected at random in the range $1\leq N \leq 2020$. Find the probability that the remainder when $N^{16}$ is divided by $5$ is $1$. Express your answer as a fraction \frac{k}{m} in its simplest form, and then give the value of k + m. | 9 |
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. Find the number of ways for the five of them to sit in a row of $5$ chairs under these conditions. | 28 |
Chloe and Zoe are both students in Ms. Demeanor's math class. Last night, they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only $80\%$ of the problems she solved alone, but overall $88\%$ of her answers were correct. Zoe had c... | 93 |
Chloe chooses a real number uniformly at random from the interval \([0, 2017]\). Independently, Laurent chooses a real number uniformly at random from the interval \([0, 4034]\). Find the probability that Laurent's number is greater than Chloe's number. The original answer is in \(\frac{k}{m}\) format, please give the ... | 7 |
Rectangle \(ABCD\) has \(AB=3\) and \(BC=4\). Point \(E\) is the foot of the perpendicular from \(B\) to diagonal \(\overline{AC}\). What is the area of \(\triangle AED\)? The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 79 |
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3CA$. Find the ratio of the area of $\triangle A'B'... | 37 |
In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path only allows moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture. [asy] fill((0.5, 4.5)-... | 24 |
There are $10$ horses, named Horse $1$, Horse $2$, . . . , Horse $10$. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time $0$ all the horses are together at the starting point on the track. The horses start running... | 3 |
In the figure below, semicircles with centers at \(A\) and \(B\) and with radii \(2\) and \(1\), respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter \(JK\). The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A cir... | 13 |
The number \(21! = 51,090,942,171,709,440,000\) has over \(60,000\) positive integer divisors. One of them is chosen at random. Find the probability that it is odd. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 20 |
In the figure below, choose point \(D\) on \(\overline{BC}\) so that \(\triangle ACD\) and \(\triangle ABD\) have equal perimeters. What is the area of \(\triangle ABD\)? [asy]draw((0,0)--(4,0)--(0,3)--(0,0)); label("$A$", (0,0), SW); label("$B$", (4,0), ESE); label("$C$", (0, 3), N); label("$3$", (0, 1.5), W); label("... | 17 |
Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How many monotonous positive integers are there?... | 1524 |
The diameter \(AB\) of a circle of radius \(2\) is extended to a point \(D\) outside the circle so that \(BD=3\). Point \(E\) is chosen so that \(ED=5\) and line \(ED\) is perpendicular to line \(AD\). Segment \(AE\) intersects the circle at a point \(C\) between \(A\) and \(E\). Find the area of \(\triangle ABC\). The... | 177 |
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? Provide your answer as an integer. | 28 |
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$. Find the ratio of the ar... | 38 |
Let $N=123456789101112\dots4344$ be the $79$-digit number that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$? | 9 |
For any positive integer $M$, the notation $M!$ denotes the product of the integers $1$ through $M$. What is the largest integer $n$ for which $5^n$ is a factor of the sum $98!+99!+100!$? Find the value of $n$. | 26 |
The number $21! = 51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd? Express your answer in the form of \frac{k}{m} and find the value of k + m. | 20 |
How many ordered pairs $(a,b)$ such that $a$ is a positive real number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\log_b a)^{2017}=\log_b(a^{2017})?$ | 597 |
Real numbers \(x\) and \(y\) are chosen independently and uniformly at random from the interval \((0,1)\). Find the probability that \(\lfloor\log_2x\rfloor=\lfloor\log_2y\rfloor\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 4 |
An integer between $1000$ and $9999$, inclusive, is chosen at random. Find the probability that it is an odd integer whose digits are all distinct. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 281 |
A square with side length \(x\) is inscribed in a right triangle with sides of length \(3\), \(4\), and \(5\) so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length \(y\) is inscribed in another right triangle with sides of length \(3\), \(4\), and \(5\) so tha... | 72 |
In \(\triangle ABC\), \(AB=6\), \(AC=8\), \(BC=10\), and \(D\) is the midpoint of \(\overline{BC}\). Find the sum of the radii of the circles inscribed in \(\triangle ADB\) and \(\triangle ADC\). The original answer is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please give the value of... | 23 |
Last year, Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test, she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test? | 100 |
Sides \(\overline{AB}\) and \(\overline{AC}\) of equilateral triangle \(ABC\) are tangent to a circle at points \(B\) and \(C\) respectively. Find the fraction of the area of \(\triangle ABC\) that lies outside the circle. The original answer is in the form \(\frac{k}{m}-\frac{n\sqrt{3}\pi}{p}\). Please give the value ... | 38 |
The diameter $\overline{AB}$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and line $ED$ is perpendicular to line $AD$. Segment $\overline{AE}$ intersects the circle at a point $C$ between $A$ and $E$. Find the area of $\triangle ABC$. The ori... | 177 |
A square is drawn in the Cartesian coordinate plane with vertices at $(2, 2)$, $(-2, 2)$, $(-2, -2)$, $(2, -2)$. A particle starts at $(0,0)$. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous ... | 39 |
How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive? | 2148 |
Let $N=123456789101112\dots4344$ be the $79$-digit number that is formed by writing the integers from $1$ to $44$ in order, one after the other. Find the remainder when $N$ is divided by $45$. | 9 |
The graph of \(y=f(x)\), where \(f(x)\) is a polynomial of degree \(3\), contains points \(A(2,4)\), \(B(3,9)\), and \(C(4,16)\). Lines \(AB\), \(AC\), and \(BC\) intersect the graph again at points \(D\), \(E\), and \(F\), respectively, and the sum of the \(x\)-coordinates of \(D\), \(E\), and \(F\) is \(24\). The ori... | 29 |
Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by $5$ minutes over the preceding day. Each of the four days, her distance trav... | 25 |
For certain real numbers \(a\), \(b\), and \(c\), the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of \(g(x)\) is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is the value of \(f(1)\)? | -7007 |
Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\triangle ABC \sim \triangle BCD$, and $AB > BC$. There is a point $E$ in the interior of $ABCD$ such that $\triangle ABC \sim \triangle CEB$ and the area of $\triangle AED$ is $17$ times the area of $\triangle CEB$. The original answer for \(\tfrac{AB}{BC}\) is in... | 7 |
Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. Find the number of days during the next year when she did not receive a phone call from any of her grandchildren. | 146 |
How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999$? For example, both $121$ and $211$ have this property. Provide your answer as an integer. | 226 |
Last year Isabella took $7$ math tests and received $7$ different scores, each an integer between $91$ and $100$, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was $95$. What was her score on the sixth test? | 100 |
The vertices \(V\) of a centrally symmetric hexagon in the complex plane are given by \[ V=\left\{ \sqrt{2}i,-\sqrt{2}i, \frac{1}{\sqrt{8}}(1+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}. \] For each \(j\), \(1\leq j\leq 12\), an element \(z_j\) is chosen from \(V\) at random, i... | 233 |
What is the value of \(\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?\) Provide your answer in the form of \(\frac{k}{m}\), where the fraction is in the simplest form. Calculate the value of \(k + m\). | 18 |
While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $14 each, $4$-pound rocks worth $11 each, and $1$-pound rocks worth $2 each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave? | 50 |
A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January $1$. On what day does his unit of blood expire? Provide the answer by converting the month to its corresponding number (e.g., January is 1, February is 2, etc.) and the day to a number, then give the... | 14 |
How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.) Find the total number of ways. | 24 |
A three-dimensional rectangular box with dimensions $X$, $Y$, and $Z$ has faces whose surface areas are $24$, $24$, $48$, $48$, $72$, and $72$ square units. What is $X + Y + Z$? | 22 |
A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. Find the probability that $3$ draws are required. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 6 |
For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$. What is the value of $m+n$? | 21 |
For how many (not necessarily positive) integer values of $n$ is the value of $4000\cdot \left(\tfrac{2}{5}\right)^n$ an integer? | 9 |
In the figure below, $N$ congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the semicircles. T... | 19 |
What is the value of $$\log_3 7 \cdot \log_5 9 \cdot \log_7 11 \cdot \log_9 13 \cdots \log_{21} 25 \cdot \log_{23} 27?$$ | 6 |
Line segment $\overline{AB}$ is a diameter of a circle with $AB = 24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle ABC$ traces out a closed curve missing two points. To the nearest positive integer, find the area of the region... | 50 |
The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$. Let $p$ be the probability that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$. What other sum occurs with the same probability as $p$? | 39 |
Monica is tiling the floor of her $12$-foot by $16$-foot living room. She plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will she use in total? | 87 |
In the rectangular parallelepiped shown, $AB = 3$, $BC = 1$, and $CG = 2$. Point $M$ is the midpoint of $\overline{FG}$. Find the volume of the rectangular pyramid with base $BCHE$ and apex $M$. | 2 |
How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? $$\begin{align*} x+3y&=3 \\ \big||x|-|y|\big|&=1 \end{align*}$$ Fill in the number of such ordered pairs. | 3 |
A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list? Provide your answer as an integer. | 225 |
A paper triangle with sides of lengths \(3, 4,\) and \(5\) inches, as shown, is folded so that point \(A\) falls on point \(B\). The original answer for the length of the crease is in the form \(\frac{k}{m}\). Please find the value of \(k + m\). | 23 |
Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown. $$\begin{array}{cccc}\mathrm{X}&&\mathrm{X}&&\mathrm{X}\\\mathrm{X}&&\mathrm{X}&&\mathrm{X}\end{array}$$ If the seating positions are assigned randomly, find the probability that Abby and Bridget are adjacent ... | 22 |
How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? \[ x+3y=3 \] \[ \big||x|-|y|\big|=1 \] | 3 |
Side $\overline{AB}$ of $\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is the value of $m+n$? | 18 |
The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping, he notes that his car clock and his watch (which is accurate) both say $12:00$ noon. When he is done shopping, his watch says $12:30$ and his car clock says $12:35$. Later that day, Sri loses his watch. He looks... | 6 |
A paper triangle with sides of lengths \(3, 4,\) and \(5\) inches, as shown, is folded so that point \(A\) falls on point \(B\). The original answer for the length of the crease is in \(\frac{m}{n}\) format, where \(m\) and \(n\) are integers in simplest form. Please find the value of \(m + n\). | 23 |
How many of the first $2018$ numbers in the sequence $101, 1001, 10001, 100001, \ldots$ are divisible by $101$? Fill in the blank with the correct number. | 505 |
How many nonnegative integers can be written in the form $$a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$$ where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$? Find the total number of such integers. | 3281 |
Square $ABCD$ has side length $30$. Point $P$ lies inside the square so that $AP = 12$ and $BP = 26$. The centroids of $\triangle{ABP}$, $\triangle{BCP}$, $\triangle{CDP}$, and $\triangle{DAP}$ are the vertices of a convex quadrilateral. What is the area of that quadrilateral? | 200 |
Lailai took five math tests, each worth a maximum of 100 points. Lailai's score on each test was an integer between 0 and 100, inclusive. Lailai received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for... | 4 |
What is the greatest integer less than or equal to $$\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?$$ Find this integer. | 80 |
The solution to the equation $\log_{3x} 4 = \log_{2x} 8$, where $x$ is a positive real number other than $\frac{1}{3}$ or $\frac{1}{2}$, can be written as $\frac {p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$? | 31 |
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is $1$ year older than Chloe, and Zoe is exactly $1$ year old today. Today is the first of the $9$ birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age i... | 11 |
Let $N$ be the greatest five-digit number whose digits have a product of $120$. What is the sum of the digits of $N$? | 18 |
Two circles of radius $5$ are externally tangent to each other and are internally tangent to a circle of radius $13$ at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\frac{k}{l}$, where $k$ and $l$ are relatively prime positive integers. Find the value of $k+l$. | 69 |
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the side... | 4 |
A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called $\textit{symmetric}$ if its look does not change when the entire square is rotated by a m... | 1022 |
How many odd positive $3$-digit integers are divisible by $3$ but do not contain the digit $3$? Provide your answer as an integer. | 96 |
Let $a_1,a_2,\dots,a_{2018}$ be a strictly increasing sequence of positive integers such that \[a_1+a_2+\cdots+a_{2018}=2018^{2018}.\] Find the remainder when $a_1^3+a_2^3+\cdots+a_{2018}^3$ is divided by $6$. | 4 |
The solutions to the equation \((z+6)^8=81\) are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled \(A,B,\) and \(C\). Find the least possible area of \(\triangle ABC\). The original answer is in the form \(\frac{k}{m}\sqrt{n} - \frac{k}{m}\), where k, m, and n are int... | 7 |
Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. Find the number of ways to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together. | 5760 |
In rectangle $PQRS$, $PQ=8$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon ... | 7 |
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths \(3\) and \(4\) units. In the corner where those sides meet at a right angle, he leaves a small unplanted square \(S\) so that from the air it looks like the right angle symbol. The rest of the field is planted. The s... | 292 |
Let $p$ and $q$ be positive integers such that \[\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\] and $q$ is as small as possible. Find the value of $q-p$. | 7 |
How many nonnegative integers can be written in the form $$a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$$ where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$? | 3281 |
Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. Find the area of quadrilateral $FDBG$. | 75 |
A function $f$ is defined recursively by $f(1)=f(2)=1$ and \[f(n)=f(n-1)-f(n-2)+n\] for all integers $n \geq 3$. Find the value of $f(2018)$. | 2017 |
A number $m$ is randomly selected from the set $\{11,13,15,17,19\}$, and a number $n$ is randomly selected from $\{1999,2000,2001,\cdots,2018\}$. Find the probability that $m^n$ has a units digit of $1$. Express your answer in the form of a fraction \frac{k}{l} in simplest form, and provide the value of k + l. | 7 |
Let $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum \[\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cd... | 19 |
Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,\ldots,\dfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$? | 340 |
In a sign pyramid, a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. Find the number of possible ways to fill the four cells in the bottom row to produce a "+" at the top of the... | 8 |
A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called symmetric if its look does not change when the entire square is rotated by a multiple of ... | 1022 |
Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\overline{BC}$. Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$, the length $CI$ can be wri... | 12 |
Let \(ABCDEF\) be a regular hexagon with side length \(1\). Denote by \(X\), \(Y\), and \(Z\) the midpoints of sides \(\overline{AB}\), \(\overline{CD}\), and \(\overline{EF}\), respectively. Find the area of the convex hexagon whose interior is the intersection of the interiors of \(\triangle ACE\) and \(\triangle XYZ... | 50 |
In \(\triangle ABC\), a point \(E\) is on \(\overline{AB}\) with \(AE=1\) and \(EB=2.\) Point \(D\) is on \(\overline{AC}\) so that \(\overline{DE} \parallel \overline{BC}\) and point \(F\) is on \(\overline{BC}\) so that \(\overline{EF} \parallel \overline{AC}.\) The ratio of the area of \(CDEF\) to the area of \(\tri... | 13 |
In \(\triangle{ABC}\) with side lengths \(AB = 13\), \(AC = 12\), and \(BC = 5\), let \(O\) and \(I\) denote the circumcenter and incenter, respectively. A circle with center \(M\) is tangent to the legs \(AC\) and \(BC\) and to the circumcircle of \(\triangle{ABC}\). Find the area of \(\triangle{MOI}\). The answer is ... | 9 |
How many positive three-digit integers have a remainder of $2$ when divided by $6$, a remainder of $5$ when divided by $9$, and a remainder of $7$ when divided by $11$? | 5 |
Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$? Provide your answer as an integer. | 220 |
How many ordered pairs $(a, b)$ of positive integers satisfy the equation \[a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),\] where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ denotes their least common multiple? | 2 |
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