prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
Bertha has $6$ daughters and no sons. Some of her daughters have $6$ daughters, and the rest have none. Bertha has a total of $30$ daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters? | 26 |
An integer \(x\), with \(10 \leq x \leq 99\), is to be chosen. If all choices are equally likely, what is the probability that at least one digit of \(x\) is a \(7\)? Express your answer as a fraction in simplest form \(\frac{a}{b}\), and provide the value of \(a + b\). | 6 |
A set of three points is randomly chosen from the grid shown. Each three-point set has the same probability of being chosen. What is the probability that the points lie on the same straight line? Express your answer in the form \(\frac{k}{m}\), where \(k\) and \(m\) are integers, and find the value of \(k + m\). | 23 |
Let $U=2\cdot 2004^{2005}$, $V=2004^{2005}$, $W=2003\cdot 2004^{2004}$, $X=2\cdot 2004^{2004}$, $Y=2004^{2004}$ and $Z=2004^{2003}$. Find the largest value among the expressions $U-V$, $V-W$, $W-X$, $X-Y$, and $Y-Z$. The expression $U-V$ can be written in the form $k\cdot 2004^{m}$. Please find the value of $k + m$. | 2006 |
A grocer stacks oranges in a pyramid-like stack whose rectangular base is $5$ oranges by $8$ oranges. Each orange above the first level rests in a pocket formed by four oranges below. The stack is completed by a single row of oranges. How many oranges are in the stack? | 100 |
A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players $A$, $B$, and $C$ start with $15$, $14$, and $13$ tokens, re... | 37 |
A square has sides of length $10$, and a circle centered at one of its vertices has radius $10$. What is the area of the union of the regions enclosed by the square and the circle? Express the area in the form $a + b\pi$, where $a$ and $b$ are constants. Please find the value of x + y, where a = x and b = y. | 175 |
Minneapolis-St. Paul International Airport is $8$ miles southwest of downtown St. Paul and $10$ miles southeast of downtown Minneapolis. Find the number of miles between downtown St. Paul and downtown Minneapolis that is closest to the actual distance. | 13 |
Find the number of two-digit positive integers whose digits total $7$. | 7 |
A square has sides of length $10$, and a circle centered at one of its vertices has radius $10$. Find the area of the union of the regions enclosed by the square and the circle. The original answer is in the form of k + m\pi, please give the value of k + m. | 175 |
Coin $A$ is flipped three times and coin $B$ is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same? Express your answer as a fraction in simplest form, in the format \(\frac{k}{m}\). Please give the value of \(k + m\). | 163 |
The sum of $49$ consecutive integers is $7^5$. What is their median? Provide your answer in the form of $7^x$, where $x$ is an integer. Please give the value of $x$. | 3 |
A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars is increased by $25\%$ without altering the volume, by what percent must the height be decreased? Provide your answer as a percentage. | 36 |
Two eight-sided dice each have faces numbered $1$ through $8$. When the dice are rolled, each face has an equal probability of appearing on the top. Find the probability that the product of the two top numbers is greater than their sum. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 7 |
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $20$ cents. If she had one more quarter, the average value would be $21$ cents. How many dimes does she have in her purse? | 0 |
All the students in an algebra class took a $100$-point test. Five students scored $100$, each student scored at least $60$, and the mean score was $76$. What is the smallest possible number of students in the class? | 13 |
The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules: $1$. The largest isn't first, but it is in one of the first three places. $2$. The smallest isn't last, but it is in one of the last three places. $3$. The median isn't first or last. If the average of the first and last numbers is $x$, what is... | 65 |
Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for $24$ hours. If she is using it constantly, the battery will last for only $3$ hours. Since the last recharge, her phone has been on $9$ hours, and during that time she has used it for $60$ minutes.... | 8 |
In the United States, coins have the following thicknesses: penny, $1.55 \, \text{mm}$; nickel, $1.95 \, \text{mm}$; dime, $1.35 \, \text{mm}$; quarter, $1.75 \, \text{mm}$. If a stack of these coins is exactly $14 \, \text{mm}$ high, how many coins are in the stack? | 8 |
Let $S$ be the set of points $(a,b)$ in the coordinate plane, where each of $a$ and $b$ may be $-1$, $0$, or $1$. Find the number of distinct lines that pass through at least two members of $S$. | 20 |
If $f(x) = ax+b$ and $f^{-1}(x) = bx+a$ with $a$ and $b$ real, find the value of $a+b$. | -2 |
A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only \(\frac{1}{3}\) of the marbles in the bag are blue. Then yellow marbles are added to the bag until only \(\frac{1}{5}\) of the marbles in the bag are blue. Finally, the number of blue marbles... | 4 |
In $\triangle ABC$, $AB=13$, $AC=5$, and $BC=12$. Points $M$ and $N$ lie on $AC$ and $BC$, respectively, with $CM=CN=4$. Points $J$ and $K$ are on $AB$ so that $MJ$ and $NK$ are perpendicular to $AB$. Find the area of pentagon $CMJKN$. The original answer is in \frac{m}{n} format, please give the value of m + n. | 253 |
Given that $-4\leq x\leq-2$ and $2\leq y\leq4$, find the largest possible value of $\frac{x+y}{x}$. The original answer is in \frac{k}{m} format, please give the value of k + m. | 3 |
Patty has $20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $70$ cents more. How much are her coins worth in dollars? Provide your answer as a numerical value, and then multiply it by 100 to give your final answer. | 115 |
The $5\times 5$ grid shown contains a collection of squares with sizes from $1\times 1$ to $5\times 5$. How many of these squares contain the black center square? | 19 |
Three circles of radius \(1\) are externally tangent to each other and internally tangent to a larger circle. The radius of the large circle can be expressed in the form \(\frac{k + 2\sqrt{m}}{n}\), where the expression is in its simplest form. Find the value of \(k + m + n\). | 9 |
The set of all real numbers \(x\) for which \[ \log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x}))) \] is defined is \(\{x\mid x > c\}\). The original answer is in the form \(a^b\). What is the value of \(a+b\)? | 4003 |
In the right triangle $\triangle ACE$, we have $AC=12$, $CE=16$, and $EA=20$. Points $B$, $D$, and $F$ are located on $AC$, $CE$, and $EA$, respectively, so that $AB=3$, $CD=4$, and $EF=5$. Find the ratio of the area of $\triangle DBF$ to that of $\triangle ACE$. The answer is in the form \frac{k}{m}, please provide th... | 23 |
Square $ABCD$ has side length $2$. A semicircle with diameter $\overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $\overline{AD}$ at $E$. Find the length of $\overline{CE}$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 7 |
A white cylindrical silo has a diameter of $30$ feet and a height of $80$ feet. A red stripe with a horizontal width of $3$ feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet? | 240 |
In the sequence $2001$, $2002$, $2003$, $\cdots$, each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $2001 + 2002 - 2003 = 2000$. What is the $2004^{th}$ term in this sequence? | 0 |
In \(\triangle ABC\), points \(D\) and \(E\) lie on \(BC\) and \(AC\), respectively. If \(AD\) and \(BE\) intersect at \(T\) so that \(\frac{AT}{DT}=3\) and \(\frac{BT}{ET}=4\), find the value of \(\frac{CD}{BD}\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 15 |
Select numbers $a$ and $b$ between $0$ and $1$ independently and at random, and let $c$ be their sum. Let $A, B$ and $C$ be the results when $a, b$ and $c$, respectively, are rounded to the nearest integer. Find the probability that $A + B = C$. The original answer is in \frac{k}{m} format, please give the value of k +... | 7 |
Each face of a cube is painted either red or blue, each with probability $1/2$. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? The original answer is in \frac{k}{m} format, pl... | 21 |
Let $1; 4; 7; \cdots$ and $9; 16; 23; \cdots$ be two arithmetic progressions. The set $S$ is the union of the first $2004$ terms of each sequence. How many distinct numbers are in $S$? | 3722 |
Square $ABCD$ has side length $2$. A semicircle with diameter $\overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $\overline{AD}$ at $E$. What is the length of $\overline{CE}$? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 7 |
A triangle with sides of \(5, 12,\) and \(13\) has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles? Provide your answer in the form \(\frac{\sqrt{k}}{m}\), and calculate the value of \(k + m\). | 67 |
Three mutually tangent spheres of radius \(1\) rest on a horizontal plane. A sphere of radius \(2\) rests on them. Find the distance from the plane to the top of the larger sphere, expressed in the form \(3 + \frac{\sqrt{x}}{y}\). Please find the value of \(x + y\). | 72 |
The square $$ \begin{array}{|c|c|c|}\hline50&b&c\\\hline d&e&f\\\hline g&h&2\\\hline\end{array} $$ is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries are positive integers, find the sum of the possible values of $g$. | 35 |
Each face of a cube is painted either red or blue, each with probability \( \frac{1}{2} \). The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? The original answer is in the form \... | 21 |
The polynomial $x^3 - 2004 x^2 + mx + n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. If the number of possible values of $n$ is denoted as $k$, find the value of $k$. | 250500 |
In triangle \(ABC\) we have \(AB=7\), \(AC=8\), \(BC=9\). Point \(D\) is on the circumscribed circle of the triangle so that \(AD\) bisects angle \(BAC\). The value of \(\frac{AD}{CD}\) is given in the form \(\frac{k}{m}\), where k and m are coprime integers. Please find the value of \(k + m\). | 8 |
In \(\triangle ABC\), \(AB = BC\), and \(\overline{BD}\) is an altitude. Point \(E\) is on the extension of \(\overline{AC}\) such that \(BE = 10\). The values of \(\tan \angle CBE\), \(\tan \angle DBE\), and \(\tan \angle ABE\) form a geometric progression, and the values of \(\cot \angle DBE,\) \(\cot \angle CBE,\) \... | 53 |
Three mutually tangent spheres of radius \(1\) rest on a horizontal plane. A sphere of radius \(2\) rests on them. Find the distance from the plane to the top of the larger sphere. The answer is in the form \(k + \frac{\sqrt{m}}{n}\), where \(n\) is not divisible by any square number other than \(1\). Please calculate ... | 75 |
A circle of radius \(1\) is internally tangent to two circles of radius \(2\) at points \(A\) and \(B\), where \(AB\) is a diameter of the smaller circle. Find the area of the region, shaded in the picture, that is outside the smaller circle and inside each of the two larger circles. The original answer is in the form ... | 13 |
Given that $2^{2004}$ is a $604$-digit number whose first digit is $1$, find how many elements of the set $S = \{2^0,2^1,2^2,\cdots ,2^{2003}\}$ have a first digit of $4$. | 195 |
Two \(4 \times 4\) squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. Find the area of the shaded region created by removing the circle from the squares. The answer is in the form \(k - m\pi\), please provide the ... | 30 |
Let $f(t) = \frac{t}{1-t}$, where $t \neq 1$. If $y = f(x)$, express $x$ in terms of $y$. The original answer is in the form \frac{m}{n}, please give the value of m + n. | 2 |
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? Provide your answer in the form of \(\frac{k}{m}\), where the fraction is in simplest... | 7 |
A store normally sells windows at $100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? | 100 |
In $\triangle ABC$, we have $AC=BC=7$ and $AB=2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=8$. What is the length of $BD$? | 3 |
Suppose $d$ is a digit. For how many values of $d$ is $2.00d5 > 2.005$? Provide your answer as an integer. | 5 |
A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. Find the ratio of the area of the smallest circle to the area of the largest square. The original answer is in \(\frac{k\pi}{m}\) format, please give the value of k + m. | 9 |
Square $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\sqrt{50}$ and $BE = 1$. What is the area of the inner square $EFGH$? | 36 |
What is the area enclosed by the graph of $|3x|+|4y|=12$? Provide your answer as an integer. | 24 |
Three tiles are marked \(X\) and two other tiles are marked \(O\). The five tiles are randomly arranged in a row. Find the probability that the arrangement reads \(XOXOX\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 11 |
One fair die has faces \(1, 1, 2, 2, 3, 3\) and another has faces \(4, 4, 5, 5, 6, 6\). The dice are rolled and the numbers on the top faces are added. Find the probability that the sum will be odd. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 14 |
There are two values of \(a\) for which the equation \(4x^2 + ax + 8x + 9 = 0\) has only one solution for \(x\). The discriminant of the quadratic equation can be expressed in the form \(m + n = 0\). Please find the sum of these values of \(m\) and \(n\). | -16 |
On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points. Find the difference between the mean and the median score on this exam. | 1 |
An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $＄$ $20$ or more? Express your answer in the form of a simplified fraction \(\frac{k}{m}\), and provide the value of \(k + m\). | 3 |
The sales tax rate in Rubenenkoville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up $90.00 and adds $6%$ sales tax, then subtracts $20%$ from this total. Jill rings up $90.00$, sub... | 0 |
Twelve fair dice are rolled. Find the probability that the product of the numbers on the top faces is prime. If the probability can be expressed as \(\left(\frac{1}{6}\right)^n\), what is the value of \(n\)? | 10 |
A line passes through $A\ (1,1)$ and $B\ (100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$? | 8 |
How many numbers between $1$ and $2005$ are integer multiples of $3$ or $4$ but not $12$? | 835 |
In the five-sided star shown, the letters $A$, $B$, $C$, $D$ and $E$ are replaced by the numbers $3, 5, 6, 7$ and $9,$ although not necessarily in that order. The sums of the numbers at the ends of the line segments $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DE}$, and $\overline{EA}$ form an arithmet... | 12 |
Suppose that \(4^{x_1}=5\), \(5^{x_2}=6\), \(6^{x_3}=7\), ... , \(127^{x_{124}}=128\). The original answer is in the form \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Please find the value of \(k + m\). | 9 |
The area of polygon $ABCDEF$ is 52 with $AB=8$, $BC=9$ and $FA=5$. What is the value of $DE+EF$? | 9 |
Equilateral \(\triangle ABC\) has side length \(2\), \(M\) is the midpoint of \(\overline{AC}\), and \(C\) is the midpoint of \(\overline{BD}\). Find the area of \(\triangle CDM\). The original answer is in the format \(\frac{\sqrt{n}}{m}\), please give the value of n + m. | 5 |
On a standard die, one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots? Express your answer as a fraction in simplest form, \(\frac{k}{m}\). Please provide the value of \(k + m\). | 32 |
How many positive cubes divide $3! \cdot 5! \cdot 7!$? | 6 |
An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $20$ or more? Express your answer in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest form. Please give the value of k + m. | 3 |
Let $\overline{AB}$ be a diameter of a circle and $C$ be a point on $\overline{AB}$ with $2 \cdot AC = BC$. Let $D$ and $E$ be points on the circle such that $\overline{DC} \perp \overline{AB}$ and $\overline{DE}$ is a second diameter. The ratio of the area of $\triangle DCE$ to the area of $\triangle ABD$ is in the fo... | 4 |
The sum of four two-digit numbers is $221$. None of the eight digits is $0$ and no two of them are the same. Determine which digit is not included among the eight digits. | 4 |
In the five-sided star shown, the letters $A, B, C, D,$ and $E$ are replaced by the numbers $3, 5, 6, 7,$ and $9$, although not necessarily in this order. The sums of the numbers at the ends of the line segments $AB$, $BC$, $CD$, $DE$, and $EA$ form an arithmetic sequence, although not necessarily in that order. What i... | 12 |
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure $1$. The cube is then cut in the same manner along the dashed lines shown in Figure $2$. This creates nine pieces. Find the volume of the piece that contains vertex $W$. The original answer is in \frac{k}{m} format,... | 13 |
How many distinct four-tuples $(a,b,c,d)$ of rational numbers are there with $$a\cdot\log_{10}2+b\cdot\log_{10}3+c\cdot\log_{10}5+d\cdot\log_{10}7=2005?$$ | 1 |
Team $A$ and team $B$ play a series. The first team to win three games wins the series. Before each game, each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If it turns out that team $B$ won the second game and team $A$ won the series, what is the ... | 6 |
All of David's telephone numbers have the form $555-abc-defg$, where $a$, $b$, $c$, $d$, $e$, $f$, and $g$ are distinct digits and in increasing order, and none is either $0$ or $1$. How many different telephone numbers can David have? | 8 |
Call a number prime-looking if it is composite but not divisible by $2, 3,$ or $5.$ The three smallest prime-looking numbers are $49, 77$, and $91$. There are $168$ prime numbers less than $1000$. How many prime-looking numbers are there less than $1000$? | 100 |
Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point \(B\) from the line on which the bases of the original s... | 6 |
On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam? | 1 |
A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled? | 1462 |
An equilangular octagon has four sides of length $1$ and four sides of length $\frac{\sqrt{2}}{2}$, arranged so that no two consecutive sides have the same length. What is the area of the octagon? The original answer is in \frac{k}{m} format, please give the value of k + m. | 9 |
What is the average (mean) of all 5-digit numbers that can be formed by using each of the digits 1, 3, 5, 7, and 8 exactly once? If the average is x, find the value of 10x. | 533328 |
For each $x$ in $[0,1]$, define $$\begin{array}{clr} f(x) & = 2x, &\mathrm { if } 0 \leq x \leq \frac {1}{2}; \\ f(x) & = 2 - 2x, & \mathrm{ if } \frac {1}{2} < x \leq 1. \end{array}$$ Let $f^{[2]}(x) = f(f(x))$, and $f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $n \geq 2$. Find the number of values of $x$ in $[0,1... | 2007 |
Let $a,b,c,d,e,f,g$ and $h$ be distinct elements in the set $\{-7,-5,-3,-2,2,4,6,13\}.$ Find the minimum possible value of $(a+b+c+d)^{2}+(e+f+g+h)^{2}.$ | 34 |
Alice and Bob play a game involving a circle whose circumference is divided by $12$ equally-spaced points. The points are numbered clockwise, from $1$ to $12$. Both start on point $12$. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves $5$ points clockwise and Bob moves $9$ points coun... | 6 |
Find the number of ordered triples of integers \((a, b, c)\), with \(a \ge 2\), \(b \ge 1\), and \(c \ge 0\), that satisfy both \(\log_a b = c^{2005}\) and \(a + b + c = 2005\). | 2 |
Forty slips are placed into a hat, each bearing a number $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, or $10$, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all four slips bear the same number. Let $q$ be the probability that... | 162 |
For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \cdots + n$? Find the number of such integers. | 16 |
A sequence of complex numbers $z_{0}, z_{1}, z_{2}, \ldots$ is defined by the rule $$z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},$$ where $\overline {z_{n}}$ is the complex conjugate of $z_{n}$ and $i^{2}=-1$. Suppose that $|z_{0}|=1$ and $z_{2005}=1$. Find the number of possible values for $z_{0}$. The original answer... | 2007 |
Let \(AB\) be a diameter of a circle and let \(C\) be a point on \(AB\) with \(2 \cdot AC = BC\). Let \(D\) and \(E\) be points on the circle such that \(DC \perp AB\) and \(DE\) is a second diameter. What is the ratio of the area of \(\triangle DCE\) to the area of \(\triangle ABD\)? The original answer is in \(\frac{... | 4 |
Two distinct numbers \(a\) and \(b\) are chosen randomly from the set \(\{2, 2^2, 2^3, \cdots, 2^{25}\}\). Find the probability that \(\log_a b\) is an integer. The original answer is in \(\frac{k}{m}\) format, where the fraction is in simplest form. Please provide the value of \(k + m\). | 331 |
Let \(S\) be the set of ordered triples \((x,y,z)\) of real numbers for which \(\log_{10}(x+y) = z\) and \(\log_{10}(x^{2}+y^{2}) = z+1.\) There are real numbers \(a\) and \(b\) such that for all ordered triples \((x,y,z)\) in \(S\) we have \(x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.\) The original answer is in \(... | 31 |
For each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. Find the number of positive integers $n$ for which both $P(n) = \sqrt{n}$ and $P(n+48) = \sqrt{n+48}$ hold true. | 1 |
Let $P(x)=(x-1)(x-2)(x-3)$. Determine the number of polynomials $Q(x)$ for which there exists a polynomial $R(x)$ of degree 3 such that $P(Q(x))=P(x) \cdot R(x)$. | 22 |
All three vertices of an equilateral triangle are on the parabola $y = x^2$, and one of its sides has a slope of $2$. The $x$-coordinates of the three vertices have a sum of $m/n$, where $m$ and $n$ are relatively prime positive integers. What is the value of $m + n$? | 14 |
In \(\triangle ABC\), we have \(AB = 25\), \(BC = 39\), and \(AC=42\). Points \(D\) and \(E\) are on \(AB\) and \(AC\) respectively, with \(AD = 19\) and \(AE = 14\). Find the ratio of the area of triangle \(ADE\) to the area of the quadrilateral \(BCED\). The original answer is in \(\frac{k}{m}\) format, please give t... | 75 |
A subset $B$ of the set of integers from $1$ to $100$, inclusive, has the property that no two elements of $B$ sum to $125$. What is the maximum possible number of elements in $B$? | 62 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.