prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Inside a right circular cone with base radius \(5\) and height \(12\) are three congruent spheres with radius \(r\). Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. Find the radius \(r\) of each sphere. The original answer is in the form \(\frac{k-n\sqrt{3}}{m}\). Plea... | 141 |
Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the n... | 13 |
Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is \(\fra... | 471 |
For each integer $n\geq 2$, let $S_n$ be the sum of all products $jk$, where $j$ and $k$ are integers and $1\leq j<k\leq n$. Find the sum of the $10$ least values of $n$ such that $S_n$ is divisible by $3$. | 197 |
Azar and Carl play a game of tic-tac-toe. Azar places an $X$ in one of the boxes in a $3$-by-$3$ array of boxes, then Carl places an $O$ in one of the remaining boxes. After that, Azar places an $X$ in one of the remaining boxes, and so on until all boxes are filled or one of the players has three of their symbols in a... | 148 |
Right triangle \(ABC\) has side lengths \(BC=6\), \(AC=8\), and \(AB=10\). A circle centered at \(O\) is tangent to line \(BC\) at \(B\) and passes through \(A\). A circle centered at \(P\) is tangent to line \(AC\) at \(A\) and passes through \(B\). The original answer is in \(\frac{k}{m}\) format, please give the val... | 47 |
Frieda the frog begins a sequence of hops on a \(3 \times 3\) grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite ed... | 57 |
Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,....$ More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\frac{p... | 55 |
Each of the $5$ sides and the $5$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color? Provide your answer as ... | 509 |
A quadratic polynomial with real coefficients and leading coefficient \(1\) is called \(\emph{disrespectful}\) if the equation \(p(p(x))=0\) is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial \(\tilde{p}(x)\) for which the sum of the roots is... | 21 |
What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{ 1, 2, 3, \ldots, 30\}$? (For example the set $\{1, 17, 18, 19, 30\}$ has $2$ pairs of consecutive integers.) Find the average number of pairs of consecutive integers in such a subse... | 5 |
Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points. [asy] size(350); de... | 81 |
Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$, and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$. The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $B$ and $F \ne B$. What is the length of... | 30 |
Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac{a}{b},$ where $a$ and $b$ are relatively prime pos... | 85 |
Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the divisor function.) Let\[f(n)=\frac{d(n)}{\sqrt [3]n}.\]There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$. Wh... | 9 |
Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1\le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial \[q(x) = c_3x^3+c_2x^2+c_1x+c_0\]such that $D(m) = q(m)$ for all odd integers $m\ge 5$. Find the val... | 11 |
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$ | 2 |
What is the value of $$3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}?$$ The answer is in the form \(\frac{k}{m}\). Please find the value of \(k + m\). | 142 |
What is the value of \[3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}?\] The original answer is in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\). | 142 |
In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ so that $\overline{BP}$ $\perp$ $\overline{AD}$, $AP = 3$, and $PD = 2$. Find the area of $ABCD$. (Note: The figure is not drawn to scale.) | 20 |
How many three-digit positive integers have an odd number of even digits? Provide your answer as an integer. | 450 |
When three positive integers $a$, $b$, and $c$ are multiplied together, their product is $100$. Suppose $a < b < c$. In how many ways can the numbers be chosen? | 4 |
A donkey suffers an attack of hiccups and the first hiccup happens at $4:00$ one afternoon. Suppose that the donkey hiccups regularly every $5$ seconds. Find the number of seconds after $4:58$ when the donkey’s $700$th hiccup occurs. | 15 |
The least common multiple of a positive integer $n$ and $18$ is $180$, and the greatest common divisor of $n$ and $45$ is $15$. What is the sum of the digits of $n$? | 6 |
For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots? Provide the number of such values. | 8 |
Square \(ABCD\) has side length \(1\). Points \(P\), \(Q\), \(R\), and \(S\) each lie on a side of \(ABCD\) such that \(APQCRS\) is an equilateral convex hexagon with side length \(s\). The original answer is in the form \(a - \sqrt{b}\). Please give the value of \(a + b\). | 4 |
The taxicab distance between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is given by \[|x_1 - x_2| + |y_1 - y_2|.\] Find the number of points $P$ with integer coordinates such that the taxicab distance between $P$ and the origin is less than or equal to $20$. | 841 |
Consider the following $100$ sets of $10$ elements each: \begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21,22,23,\ldots,30\},\\ &\vdots\\ &\{991,992,993,\ldots,1000\}. \end{align*} Find how many of these sets contain exactly two multiples of $7$. | 42 |
What is the value of \(\frac{1}{3}\cdot\frac{2}{4}\cdot\frac{3}{5}\cdots\frac{18}{20}\cdot\frac{19}{21}\cdot\frac{20}{22}?\) The answer is in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\). | 232 |
On Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent state... | 7 |
The sequence $a_0,a_1,a_2,\cdots$ is a strictly increasing arithmetic sequence of positive integers such that $$2^{a_7}=2^{27} \cdot a_7.$$ Find the minimum possible value of $a_2$. | 12 |
Regular hexagon \(ABCDEF\) has side length \(2\). Let \(G\) be the midpoint of \(\overline{AB}\), and let \(H\) be the midpoint of \(\overline{DE}\). Find the perimeter of quadrilateral \(GCHF\). The original answer is in the form \(k\sqrt{m}\). Please provide the value of \(k + m\). | 11 |
Henry the donkey has a very long piece of pasta. He takes a number of bites of pasta, each time eating $3$ inches of pasta from the middle of one piece. In the end, he has $10$ pieces of pasta whose total length is $17$ inches. How long, in inches, was the piece of pasta he started with? | 44 |
On Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent state... | 7 |
Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. Find the value of $\cos(\angle CMD)$. The original answer is in the form of \frac{k}{m}, where k and m are integers in simplest form. Please provide the value of k + m. | 4 |
Kayla rolls four fair $6$-sided dice. Find the probability that at least one of the numbers Kayla rolls is greater than $4$ and at least two of the numbers she rolls are greater than $2$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 142 |
Let $\triangle ABC$ be a scalene triangle. Point $P$ lies on $\overline{BC}$ so that $\overline{AP}$ bisects $\angle BAC.$ The line through $B$ perpendicular to $\overline{AP}$ intersects the line through $A$ parallel to $\overline{BC}$ at point $D.$ Suppose $BP=2$ and $PC=3.$ Find the length of $AD.$ | 10 |
Let $\mathcal{R}$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the sum of complex numbers $z_1$ and $z_2$, where $z_1$ lies on the segment with endpoints $3$ and $4i$, and $z_2$ has magnitude at most $1$. Find the integer closest to the area of $\mathcal{R}$. | 13 |
How many positive integers can fill the blank in the sentence below? “One positive integer is _____ more than twice another, and the sum of the two numbers is $28$.” Provide your answer as an integer. | 9 |
What is the integer value of the expression: $$(\log 5)^{3}+(\log 20)^{3}+(\log 8)(\log 0.25)$$ where $\log$ denotes the base-ten logarithm? | 2 |
The graph of $y=x^2+2x-15$ intersects the $x$-axis at points $A$ and $C$ and the $y$-axis at point $B$. Find $\tan(\angle ABC)$. The original answer is in the form \frac{k}{m}, please give the value of k + m. | 11 |
Quadrilateral $ABCD$ with side lengths $AB=7, BC=24, CD=20, DA=15$ is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form $\frac{a\pi-b}{c},$ where $a,b,$ and $c$ are positive integers such that $a$ and $c$ have no common prime factor. Find the value of $a... | 1565 |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence that has a common difference of $2$. The quotient $\frac{S_{3n}}{S_n}$ does not depend on $n$. What is the value of $S_{20}$? | 400 |
The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box? | 30 |
The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. Find the area of the region inside both the square and the rectangle. The original answer is in \frac{k}{m} format, please give the v... | 133 |
A $\emph{triangular number}$ is a positive integer that can be expressed in the form $t_n = 1+2+3+\cdots+n$, for some positive integer $n$. The three smallest triangular numbers that are also perfect squares are $t_1 = 1 = 1^2$, $t_8 = 36 = 6^2$, and $t_{49} = 1225 = 35^2$. What is the sum of the digits of the fourth s... | 18 |
Suppose \(x\) and \(y\) are positive real numbers such that \(x^y=2^{64}\) and \((\log_2{x})^{\log_2{y}}=2^{7}.\) The original answer is in the form \(k+\sqrt{m}\). Please find the value of \(k + m\). | 5 |
How many three-digit positive integers $\underline{a} \ \underline{b} \ \underline{c}$ are there whose nonzero digits $a,b,$ and $c$ satisfy $$0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?$$ (The bar indicates repetition, thus $0.\overline{\unde... | 13 |
Suppose \(a\) is a real number such that the equation \[ a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)} \] has more than one solution in the interval \((0, \pi)\). The set of all such \(a\) can be written in the form \((p,q) \cup (q,r),\) where \(p, q,\) and \(r\) are real numbers with \(p < q < r\). What is the value of \(p+... | -4 |
Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. Find the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \cdots, T_n$ returns the point $(1... | 359 |
The midpoints of the four sides of a rectangle are $(-3,0), (2,0), (5,4),$ and $(0,4).$ What is the area of the rectangle? | 40 |
Define $L_n$ as the least common multiple of all the integers from $1$ to $n$ inclusive. There is a unique integer $h$ such that $$\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{17}=\frac{h}{L_{17}}$$ What is the remainder when $h$ is divided by $17$? | 5 |
Mr. Ramos gave a test to his class of $20$ students. The dot plot below shows the distribution of test scores. Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students $5$ extra points, which increased the median test score to $85$. What is ... | 4 |
A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are $57$, $60$, and $91$. Find the fourth term of this sequence. | 206 |
Let $ABCD$ be a rhombus with $\angle ADC = 46^\circ$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle BFC$? If the answer is $x^\circ$, what is the value of $x$? | 113 |
Isosceles trapezoid \(ABCD\) has parallel sides \(\overline{AD}\) and \(\overline{BC},\) with \(BC < AD\) and \(AB = CD.\) There is a point \(P\) in the plane such that \(PA=1, PB=2, PC=3,\) and \(PD=4.\) Find the value of \(\frac{BC}{AD}.\) The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is in ... | 4 |
Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x+2$, and when $P(x)$ is divided by the polynomial $x^2+1$, the remainder is $2x+1$. There is a unique polynomial of least degree with these two properties. Find the sum of the squar... | 23 |
The grid below is to be filled with integers in such a way that the sum of the numbers in each row and the sum of the numbers in each column are the same. Four numbers are missing. The number $x$ in the lower left corner is larger than the other three missing numbers. What is the smallest possible value of $x$? $\begin... | 8 |
Let $P(x) = x^{2022} + x^{1011} + 1.$ Find a polynomial that is a factor of $P(x)$. The original answer is in the form ax^6 + bx^3 + c. Please provide the value of a + b + c. | 3 |
Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. If the sum of the areas of all circles in $S$ is $A\pi$, what is the value of $A$? | 136 |
Steph scored $15$ baskets out of $20$ attempts in the first half of a game, and $10$ baskets out of $10$ attempts in the second half. Candace took $12$ attempts in the first half and $18$ attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the sam... | 9 |
Ant Amelia starts on the number line at \(0\) and crawls in the following manner. For \(n=1,2,3,\) Amelia chooses a time duration \(t_n\) and an increment \(x_n\) independently and uniformly at random from the interval \((0,1).\) During the \(n\)th step of the process, Amelia moves \(x_n\) units in the positive directi... | 5 |
A bus takes $2$ minutes to drive from one stop to the next, and waits $1$ minute at each stop to let passengers board. Zia takes $5$ minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwi... | 17 |
Let $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $n$, define $$S_n = \sum_{k=0}^{n-1} x_k 2^k$$ Suppose $7S_n \equiv 1 \pmod{2^n}$ for all $n \geq 1$. Find the value of the sum $$x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}.$$ | 6 |
Consider functions $f$ that satisfy \[|f(x)-f(y)|\leq \frac{1}{2}|x-y|\] for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300) = f(900)$, find the greatest possible value of \[f(f(800))-f(f(400))?\] | 50 |
The figure below depicts a regular $7$-gon inscribed in a unit circle. [asy] import geometry; unitsize(3cm); draw(circle((0,0),1),linewidth(1.5)); for (int i = 0; i < 7; ++i) { for (int j = 0; j < i; ++j) { draw(dir(i * 360/7) -- dir(j * 360/7),linewidth(1.5)); } } for(int i = 0; i < 7; ++i) { dot(dir(i * 360/7),5+blac... | 147 |
Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the $x$-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\frac{... | 337 |
Let $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $n$, define $S_n = \sum\limits_{k=0}^{n-1} x_k 2^k$ Suppose $7S_n \equiv 1 \pmod{2^n}$ for all $n \geq 1$. Find the value of the sum $x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}.$ | 6 |
A cricket randomly hops between $4$ leaves, on each turn hopping to one of the other $3$ leaves with equal probability. After $4$ hops, what is the probability that the cricket has returned to the leaf where it started? Express your answer in the form \(\frac{k}{m}\) where the fraction is in simplest form, and find the... | 34 |
Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only \(\frac{1}{3}\) full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all fo... | 7 |
How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$? Find the number of digits. | 18 |
Jackson's paintbrush makes a narrow strip with a width of \(6.5\) millimeters. Jackson has enough paint to make a strip \(25\) meters long. How many square centimeters of paper could Jackson cover with paint? The original answer is in the form \(a \times b\). Please calculate the value of \(a + b\). | 90 |
The numbers from $1$ to $49$ are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number $7.$ How many of these four numbers are prime? Provid... | 3 |
Janet rolls a standard $6$-sided die $4$ times and keeps a running total of the numbers she rolls. Find the probability that at some point, her running total will equal $3$. Express your answer in the form \(\frac{k}{m}\) where the fraction is in simplest form, and then give the value of \(k + m\). | 265 |
An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppos... | 126 |
When the roots of the polynomial \[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}\] are removed from the number line, what remains is the union of $11$ disjoint open intervals. Find the number of these intervals on which \( P(x) \) is positive. | 6 |
A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-d... | 9 |
For how many integers $n$ does the expression \[ \sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}} \] represent a real number, where log denotes the base $10$ logarithm? Provide your answer as an integer. | 901 |
A rectangle, with sides parallel to the $x$-axis and $y$-axis, has opposite vertices located at $(15, 3)$ and $(16, 5)$. A line is drawn through points $A(0, 0)$ and $B(3, 1)$. Another line is drawn through points $C(0, 10)$ and $D(2, 9)$. How many points on the rectangle lie on at least one of the two lines? | 1 |
How many nonempty subsets $B$ of $\{0, 1, 2, 3, \cdots, 12\}$ have the property that the number of elements in $B$ is equal to the least element of $B$? For example, $B = \{4, 6, 8, 11\}$ satisfies the condition. Find the number of such subsets. | 144 |
The numbers $16$ and $25$ are a pair of consecutive positive squares whose difference is $9$. Find how many pairs of consecutive positive perfect squares have a difference of less than or equal to $2023$. | 1011 |
A square of area \(2\) is inscribed in a square of area \(3\), creating four congruent triangles, as shown below. Find the ratio of the shorter leg to the longer leg in the shaded right triangle. The answer is in the form \(k-\sqrt{m}\). Please find the value of \(k+m\). | 5 |
In the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origin, and a circle with radius $10$ with center on the positive $y$-axis is tangent to the $x$-axis at the origin. Find the slope of the line passing through the two points at which these circles intersect... | 7 |
Harold made a plum pie to take on a picnic. He was able to eat only \(\frac{1}{4}\) of the pie, and he left the rest for his friends. A moose came by and ate \(\frac{1}{3}\) of what Harold left behind. After that, a porcupine ate \(\frac{1}{3}\) of what the moose left behind. How much of the original pie still remained... | 4 |
A square of area \(2\) is inscribed in a square of area \(3\), creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle? The original answer is in the form \(k - \sqrt{m}\). Please provide the value of \(k + m\). | 5 |
What is the maximum area of an isosceles trapezoid that has legs of length \(1\) and one base twice as long as the other? Express your answer in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of \(k + m\). | 5 |
How many three-digit positive integers $N$ satisfy the following properties? The number $N$ is divisible by $7$. The number formed by reversing the digits of $N$ is divisible by $5$. Provide the number of such integers. | 14 |
What is the value of \[2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?\] | 3159 |
For complex number $u = a+bi$ and $v = c+di$ (where $i=\sqrt{-1}$), define the binary operation $u \otimes v = ac + bdi$. Suppose $z$ is a complex number such that $z\otimes z = z^{2}+40$. The original answer for $|z|$ is in the form $k\sqrt{m}$. Please find the value of $k + m$. | 7 |
Abdul and Chiang are standing $48$ feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures $60^\circ$. What is the square of the distance (in feet) between Abdul and Bharat? | 3072 |
In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambid... | 36 |
Along the route of a bicycle race, $7$ water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also $2$ repair stations evenly spaced between the start and finish lines. The $3$rd water station is located $2$ miles after the $1$st repair station. How long is the race... | 48 |
A number is chosen at random from among the first 100 positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by 11? Express your answer as a fraction in simplest form, \(\frac{m}{n}\). Please provide the value of m + n. | 209 |
How many ordered pairs of integers $(m, n)$ satisfy the equation $m^2+mn+n^2 = m^2n^2$? | 3 |
How many complex numbers satisfy the equation $z^5=\overline{z}$, where $\overline{z}$ is the conjugate of the complex number $z$? Fill in the number of such complex numbers. | 7 |
For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots? | 5 |
Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of $5$-cent, $10$-cent, and $25$-cent stamps, with exactly $20$ of each type. What is the greatest number of sta... | 55 |
An even number of circles are nested, starting with a radius of $1$ and increasing by $1$ each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius $2$ but outside the circle of radius $1.$ An example showing $8$ circles is displayed bel... | 64 |
What is the least positive integer $m$ such that $m\cdot2!\cdot3!\cdot4!\cdot5!\cdots16!$ is a perfect square? Find the value of $m$. | 70 |
Usain is walking for exercise by zigzagging across a \(100\)-meter by \(30\)-meter rectangular field, beginning at point \(A\) and ending on the segment \(\overline{BC}\). He wants to increase the distance walked by zigzagging as shown in the figure below \((APQRS)\). What angle \(\theta = \angle PAB=\angle QPC=\angle ... | 11 |
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