problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
What is the base ten equivalent of $54321_7$?
|
13539
| 0.166667 |
Consider the ellipse \[16(x+2)^2 + 4y^2 = 64.\] Let \( C \) be one of the endpoints of its major axis, and let \( D \) be one of the endpoints of its minor axis. Find the distance \( CD. \)
|
2\sqrt{5}
| 0.916667 |
In a right triangle, the two legs have lengths of 48 inches and 55 inches. Calculate both the area of the triangle and its hypotenuse.
|
73 \text{ inches}
| 0.25 |
A school has eight identical copies of a specific textbook. At any given time, some of these copies are in the school's storage and some are distributed to students. How many different ways are there for some of the books to be in storage and the rest to be distributed to students if at least one book is in storage and at least one is distributed?
|
7
| 0.083333 |
A 4x4x4 cube is assembled from 64 unit cubes. Only one unit square on each of the six faces of the cube is painted. How many of the 64 unit cubes have no paint on them?
|
58
| 0.916667 |
Determine the number of complex numbers \( z \) that satisfy \( |z| = 1 \) and
\[
\left| \frac{z^2}{\overline{z}^2} + \frac{\overline{z}^2}{z^2} \right| = 1.
\]
|
16
| 0.083333 |
We are allowed to remove exactly one integer from the list $$-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,$$ and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 12?
|
6
| 0.5 |
Let $a$, $b$, $c$ be integers, and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1$. Also, let $a = 2b - c$. Find the smallest possible value of
\[
|a + b\omega + c\omega^3|.
\]
|
0
| 0.916667 |
Compute the number of degrees in the smallest positive angle $y$ such that
\[6 \sin y \cos^3 y - 6 \sin^3 y \cos y = \frac{3}{2}.\]
|
22.5^\circ
| 0.916667 |
A line is parameterized by a parameter $t,$ so that the vector on the line at $t = 1$ is $\begin{pmatrix} 4 \\ 5 \end{pmatrix},$ and the vector on the line at $t = 5$ is $\begin{pmatrix} 12 \\ -11 \end{pmatrix}.$ Find the vector on the line at $t = -3.$
|
\begin{pmatrix} -4 \\ 21 \end{pmatrix}
| 0.583333 |
Given vectors $\mathbf{a}$ and $\mathbf{b},$ let $\mathbf{p}$ be a vector such that
\[\|\mathbf{p} - \mathbf{b}\| = 3 \|\mathbf{p} - \mathbf{a}\|.\] Among all such vectors $\mathbf{p},$ there exists constants $t$ and $u$ such that $\mathbf{p}$ is at a fixed distance from $t \mathbf{a} + u \mathbf{b}.$ Enter the ordered pair $(t,u).$
|
\left(\frac{9}{8}, -\frac{1}{8}\right)
| 0.5 |
Let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be vectors such that \(\|\mathbf{a}\| = 3\), \(\|\mathbf{b}\| = 2\), \(\|\mathbf{c}\| = 5\), and
\[
\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}.
\]
Find the smallest possible angle between \(\mathbf{a}\) and \(\mathbf{c}\) in degrees.
|
180^\circ
| 0.5 |
Find the coefficient of $x$ when the expression $5(2x - 5) + 3(6 - 3x^2 + 4x) - 7(3x - 2)$ is simplified.
|
1
| 0.916667 |
There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles as shown. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. Given that there are eight different colors of triangles from which to choose, how many distinguishable large equilateral triangles can be constructed?
|
960
| 0.083333 |
What is the smallest positive integer $n$ such that $\frac{n}{n+53}$ is equal to a terminating decimal?
|
11
| 0.75 |
The equation $x^2 - 6x + 11 = 23$ has two solutions, $c$ and $d$, where $c \geq d$. Determine the value of $3c + 2d$.
|
15 + \sqrt{21}
| 0.25 |
What is the sum of the digits of the base $8$ representation of $888_{10}$?
|
13
| 0.833333 |
A pet store has 20 puppies, 10 kittens, 12 hamsters, and 5 birds. Alice, Bob, Charlie, and Dana each want to buy a pet. For variety, they each want a different kind of pet. How many ways can Alice, Bob, Charlie, and Dana buy pets and leave the store satisfied?
|
288000
| 0.416667 |
Simplify $\sqrt{50y^3} \cdot \sqrt{18y} \cdot \sqrt{98y^5}$.
|
210y^4\sqrt{2y}
| 0.666667 |
Find the value of $k$ so that
\[5 + \frac{5 + k}{5} + \frac{5 + 2k}{5^2} + \frac{5 + 3k}{5^3} + \dotsb = 10.\]
|
12
| 0.916667 |
Points $C$ and $D$ are on the parabola $y = -3x^2 + 2x + 5$, and the origin is the midpoint of $\overline{CD}$. Find the square of the length $\overline{CD}$.
|
\frac{100}{3}
| 0.666667 |
The line \( y = 4x - 7 \) is parameterized by the form
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} s \\ -3 \end{pmatrix} + t \begin{pmatrix} 3 \\ m \end{pmatrix}.
\]
Determine the ordered pair \( (s, m) \).
|
(1, 12)
| 0.916667 |
Let $Q(x)$ be a polynomial such that when $Q(x)$ is divided by $x-17$, the remainder is $15$, and when $Q(x)$ is divided by $x-13$, the remainder is $8$. What is the remainder when $Q(x)$ is divided by $(x-17)(x-13)$?
|
\frac{7}{4}x - \frac{59}{4}
| 0.916667 |
A ball is dropped from 20 feet, bouncing back three-quarters of the distance it just fell each time. Determine the number of bounces needed for the ball to first reach a height less than 2 feet.
|
9
| 0.666667 |
The positive five-digit integers that use each of the digits $1,$ $2,$ $3,$ $4,$ and $5$ exactly once are ordered from least to greatest. What is the $50^{\text{th}}$ integer in the list?
|
31254
| 0.166667 |
Let $P$ be the parabola with equation $y = x^2 + 3x + 1$ and let $Q = (10, 50)$. Determine the values of $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s$. What is $r + s$?
|
46
| 0.916667 |
A piece of string fits exactly around the perimeter of a rectangle whose area is 180. The length of one side of the rectangle is three times the length of the other side. Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string?
|
306
| 0.666667 |
Evaluate \(2i^{13} - 3i^{18} + 4i^{23} - 5i^{28} + 6i^{33}\).
|
4i - 2
| 0.333333 |
Of the numbers 1, 2, 3, ..., 20, which number less than 20 has the greatest number of divisors and is also a composite number?
|
18
| 0.166667 |
At CleverCat Academy, cats are trained to perform three tricks: jump, fetch, and spin. Here is the available data:
\begin{tabular}{l@{\qquad}l}
60 cats can jump & 25 cats can jump and fetch \\
35 cats can fetch & 18 cats can fetch and spin \\
40 cats can spin & 20 cats can jump and spin \\
12 cats can do all three & 15 cats can do none
\end{tabular}
How many cats are in the academy?
|
99
| 0.583333 |
How many positive three-digit integers with each digit greater than 6 are divisible by 12?
|
1
| 0.333333 |
Let \( f(x) = x^3 - x^2 + 2x + 2000 \). What is the greatest common divisor of \( f(50) \) and \( f(51) \)?
|
8
| 0.833333 |
A square with sides 8 inches is illustrated. If $Q$ is a point such that the segments $\overline{QA}$, $\overline{QB}$, and $\overline{QC}$ are equal in length, and segment $\overline{QC}$ is perpendicular to segment $\overline{FD}$, find the area, in square inches, of triangle $AQB$. [asy]
pair A, B, C, D, F, Q;
A = (0,0); B= (8,0); C = (4,2); D = (8,8); F = (0,8); Q = (4,4);
draw(A--B--D--F--cycle);
draw(C--Q); draw(Q--A); draw(Q--B);
label("$A$",A,SW); label("$B$",B,SE);label("$C$",C,N);label("$D$",D,NE);label("$Q$",Q,NW);label("$F$",F,NW);
label("$8''$",(4,0),S);
[/asy]
|
12
| 0.083333 |
Let \( x = 1 + \frac{\sqrt{3}}{1 + \frac{\sqrt{3}}{1 + \dots}} \). Find the value of \( \frac{1}{(x+2)(x-3)} \). When your answer is in the form \( \frac{A+\sqrt{B}}{C} \), where \( A \), \( B \), and \( C \) are integers, and \( B \) is not divisible by the square of a prime, what is \( |A|+|B|+|C| \)?
|
42
| 0.25 |
In the diagram, the area of triangle $ABC$ is 36 square units. What is the area of triangle $BCD$?
[asy]
draw((0,0)--(40,0)--(12,18)--(0,0));
dot((0,0));
label("$A$",(0,0),SW);
label("8",(4,0),S);
dot((8,0));
label("$C$",(8,0),S);
label("32",(24,0),S);
dot((40,0));
label("$D$",(40,0),SE);
dot((12,18));
label("$B$",(12,18),N);
draw((8,0)--(12,18));
[/asy]
|
144
| 0.75 |
How many positive divisors of $180$ are not divisible by 2?
|
6
| 0.916667 |
In triangle $ABC$, $\angle C = 90^\circ$, $AB = 13$, and $BC = 5$. What is $\cos B$?
|
\cos B = \frac{5}{13}
| 0.083333 |
The sum of two numbers is 10. The difference of their squares is 24. What is the positive difference of the two numbers?
|
\frac{12}{5}
| 0.25 |
The lattice shown is continued for $12$ rows, each containing $7$ consecutive numbers. The first number in each row starts from $1$ in Row 1, and increases by $8$ as the row number increases. What will be the fifth number in the $12$th row?
|
93
| 0.916667 |
What is the domain of the function $g(x) = \log_3(\log_4(\log_5(\log_6x)))$?
|
(7776, \infty)
| 0.5 |
For some constants $a$ and $b,$ let \[g(x) = \left\{
\begin{array}{cl}
ax + b & \text{if } x < 3, \\
9 - 2x & \text{if } x \ge 3.
\end{array}
\right.\]
The function $g$ has the property that $g(g(x)) = x$ for all $x.$ Additionally, $g(x)$ is continuous at $x = 3$. What is $a + b?$
|
4
| 0.333333 |
In the NBA finals between the Warriors and the Nets, a team is declared the champion after winning 4 games. If the Warriors win each game with a probability of $\frac{1}{4}$ and there are no ties, what is the probability that the Nets will win the NBA finals but that the series will need all seven games to be decided? Express your answer as a fraction.
|
\frac{405}{4096}
| 0.75 |
Given that $21^{-1} \equiv 17 \pmod{53}$, find $32^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.)
|
36
| 0.666667 |
The deli has five kinds of bread, seven kinds of meat, and six kinds of cheese. A sandwich includes one type of bread, one type of meat, and one type of cheese. Turkey, roast beef, Swiss cheese, and rye bread are each available at the deli. If Al never orders a sandwich with a turkey/Swiss cheese combination nor a sandwich with a rye bread/roast beef combination, how many different sandwiches could Al order?
|
199
| 0.833333 |
A traffic light follows a cycle of green for 45 seconds, yellow for 5 seconds, and red for 40 seconds. Sam observes the light for a random five-second interval. What is the probability that the light changes from one color to another during his observation?
|
\frac{1}{6}
| 0.25 |
Let $\mathbf{a} = \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}$. Find the vector $\mathbf{v}$ that satisfies $2\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $3\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}$.
|
\begin{pmatrix} \frac{7}{6} \\ -\frac{2}{3} \\ \frac{7}{6} \end{pmatrix}
| 0.166667 |
How many positive divisors does \( 8! \) have?
|
96
| 0.916667 |
The marble statue of George Washington in a museum has a height of 120 feet. A scale model of the statue has a height of 6 inches. How many feet of the statue does one inch of the model represent?
|
20
| 0.916667 |
In how many ways is it possible to arrange the digits of 11120 to get a four-digit multiple of 5?
|
4
| 0.333333 |
How many positive integers smaller than $1{,}000{,}000$ are powers of $3$, but are not powers of $9$?
|
6
| 0.833333 |
In parallelogram $ABCD$, angle $B$ measures $135^\circ$. What is the number of degrees in the measure of angles $C$ and $A$?
|
45^\circ
| 0.916667 |
What is the $156$th digit after the decimal point when $\frac{37}{740}$ is expressed as a decimal?
|
0
| 0.416667 |
The sequence 2,000,000; 1,000,000; 500,000 and so on, is made by repeatedly dividing by 2. What is the last integer in this sequence?
|
15625
| 0.5 |
How many four-digit positive integers have at least one digit that is a 5 or a 7?
|
5416
| 0.916667 |
A jar contains $35\frac{2}{3}$ tablespoons of peanut butter and $18\frac{1}{2}$ tablespoons of jelly. If one serving of peanut butter is $2\frac{1}{6}$ tablespoons and one serving of jelly is 1 tablespoon, how many servings of each does the jar contain? Express your answer as mixed numbers.
|
18\frac{1}{2}
| 0.166667 |
Compute $\tan 30^\circ + 4 \cos 30^\circ.$
|
\frac{7\sqrt{3}}{3}
| 0.916667 |
If $7a+3b=0$ and $a$ is three less than $b$, what is $8b$?
|
16.8
| 0.333333 |
A student, Jim, needs to earn 30 biology exercise points. For the first 6 points, each point requires 1 exercise. For the next 6 points, each point needs 2 exercises. After this, for each subsequent group of 6 points, the number of exercises required per point increases by 1. How many exercises in total must Jim complete to earn all 30 points?
|
90
| 0.666667 |
What is the 12th term in the geometric sequence starting with 5 and each subsequent term being multiplied by $\frac{2}{5}$?
|
\frac{10240}{48828125}
| 0.833333 |
What is the smallest positive integer $n$ such that all the roots of $z^6 - z^3 + 1 = 0$ are $n^{\text{th}}$ roots of unity?
|
18
| 0.5 |
Let $\theta$ be the angle between the planes $2x - y + 3z - 4 = 0$ and $4x + 3y - z + 2 = 0.$ Find $\cos \theta.$
|
\frac{1}{\sqrt{91}}
| 0.083333 |
A quadratic is given as $x^2+bx+\frac16$, where $b$ is a negative number. It can be rewritten using the completion of the square as $(x+p)^2+\frac{1}{18}$. Determine the value of $b$.
|
-\frac{2}{3}
| 0.833333 |
Find the smallest positive integer \(b\) for which \(x^2 + bx + 4032\) factors into a product of two binomials, each having integer coefficients.
|
127
| 0.583333 |
How many distinct, positive factors does $1320$ have?
|
32
| 0.916667 |
The quadratic equation $ax^2 + 30x + c = 0$ has exactly one solution. If $a+c=35$, and $a < c$, find the ordered pair $(a, c)$.
|
\left(\frac{35 - 5\sqrt{13}}{2}, \frac{35 + 5\sqrt{13}}{2}\right)
| 0.416667 |
The integer $x$ has 18 positive factors. The numbers 18 and 24 are factors of $x$. What is the smallest possible value of $x$?
|
288
| 0.333333 |
Peter borrows $2000$ dollars from John, who charges an interest of $6\%$ per month (which compounds monthly). What is the least integer number of months after which Peter will owe more than three times as much as he borrowed?
|
19
| 0.916667 |
Let $p, q, r, s, t, u, v,$ and $w$ be distinct elements in the set
\[
\{-6, -4, -1, 0, 3, 5, 7, 10\}.
\]
What is the minimum possible value of
\[
(p+q+r+s)^{2} + (t+u+v+w)^{2}?
\]
|
98
| 0.166667 |
Yan is at a location between his home and the market. He can walk directly to the market, or he can walk back home and then ride his bicycle to the market. Yan rides 9 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the market?
|
\frac{4}{5}
| 0.75 |
Let $O$ be the origin, and let $(a,b,c)$ be a fixed point. A plane passes through $(a,b,c)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$ Let $(p,q,r)$ be the center of the sphere shifted by $(s,t,u)$ that passes through $A,$ $B,$ $C,$ and $O.$ If $(s,t,u) = (1,1,1)$ find:
\[\frac{a}{p} + \frac{b}{q} + \frac{c}{r}.\]
|
2
| 0.583333 |
Jeff continues to play with the Magic 8 Ball, this time asking it 7 questions. Each time he asks a question, the probability of receiving a positive answer is still 2/5. What is the probability that he receives exactly 3 positive answers?
|
\frac{22680}{78125}
| 0.833333 |
Completely factor the following expression: \[(16y^6 + 36y^4 - 9) - (4y^6 - 9y^4 + 9).\]
|
3(4y^6 + 15y^4 - 6)
| 0.166667 |
When the base-8 number $12345_8$ is written in base 10, what is the decimal equivalent?
|
5349
| 0.833333 |
A cube with an edge length of 5 units has the same volume as a square-based pyramid with base edge lengths of 10 units and a height of $h$ units. What is the value of $h$?
|
3.75
| 0.583333 |
Many states have started using a sequence of three letters followed by a sequence of four digits as their standard license-plate pattern. Calculate the probability that such a license plate will contain at least one palindrome (either the three-letter arrangement reads the same left-to-right and right-to-left, or the four-digit arrangement reads the same left-to-right and right-to-left). Express your answer as a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and find $m+n$.
|
109
| 0.083333 |
For how many digits $C$ is the positive four-digit number $1C34$ a multiple of 4?
|
0
| 0.333333 |
What is the value of the expression $(25 + 9)^2 - (25^2 + 9^2)$?
|
450
| 0.916667 |
The equation $x^2 + bx = -21$ has only integer solutions for $x$. If $b$ is a positive integer, what is the greatest possible value of $b$?
|
22
| 0.833333 |
Let $f(n)$ be the sum of the positive integer divisors of $n$. For how many values of $n$, where $1 \le n \le 50$, is $f(n)$ prime?
|
5
| 0.833333 |
The letters of the alphabet are each assigned a random integer value, and $T=15$. The value of a word comes from the sum of its letters' values. If $BALL$ is 40 points, $LAB$ is 25 points, and $ALL$ is 30 points, what is the value of $B$?
|
10
| 0.916667 |
What is the measure, in degrees, of the angle formed by the hour hand and the minute hand of a 12-hour clock at 3:25?
|
47.5^\circ
| 0.916667 |
Let $a$ and $b$ be nonzero real numbers such that $a^2 + b^2 = 10ab$. Find the value of $\left|\frac{a+b}{a-b}\right|$.
|
\frac{\sqrt{6}}{2}
| 0.916667 |
In a triangle, two of the side lengths are 10 and 12, and the angle between them is $150^\circ$. Find the length of the third side.
|
c = \sqrt{244 + 120\sqrt{3}}
| 0.5 |
The coefficients of the polynomial
\[a_{12} x^{12} + a_{11} x^{11} + \dots + a_1 x + a_0 = 0\]
are all integers, and its roots $r_1, r_2, \dots, r_{12}$ are all integers. Furthermore, the roots of the polynomial
\[a_0 x^{12} + a_1 x^{11} + \dots + a_{11} x + a_{12} = 0\]
are also $r_1, r_2, \dots, r_{12}$. Find the number of possible multisets $S = \{r_1, r_2, \dots, r_{12}\}$.
|
13
| 0.416667 |
One of the asymptotes of a hyperbola has equation $y = -2x + 4$. The foci of the hyperbola have the same $x$-coordinate, which is $-3$. Find the equation of the other asymptote of the hyperbola, giving your answer in the form "$y = mx + b$".
|
y = 2x + 16
| 0.416667 |
Find all real numbers $x$ such that \[2 \le \frac{x}{2x-5} < 7.\](Give your answer in interval notation.)
|
(\frac{35}{13}, \frac{10}{3}]
| 0.25 |
Calculate the product of $0.\overline{4567}$ and $11$, and express your result as a fraction in simplified form.
|
\frac{50237}{9999}
| 0.5 |
Susan wants to determine the average and median number of candies in a carton. She buys 9 cartons of candies, opens them, and counts the number of candies in each one. She finds that the cartons contain 5, 7, 8, 10, 12, 14, 16, 18, and 20 candies. What are the average and median number of candies per carton?
|
12
| 0.083333 |
The positive difference between the two roots of the quadratic equation $5x^2 - 2x - 15 = 0$ can be written as $\frac{\sqrt{p}}{q}$, where $q$ is an integer and $p$ is an integer not divisible by the square of any prime number. Find $p + q$.
|
309
| 0.25 |
Find the distance from the point $(2,3,1)$ to the line described by:
\[
\begin{pmatrix} 8 \\ 10 \\ 12 \end{pmatrix} + s \begin{pmatrix} 2 \\ 3 \\ -3 \end{pmatrix}.
\]
|
\sqrt{206}
| 0.5 |
Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^2}$ is a real number and $|\alpha - \beta| = 4 \sqrt{3}.$ Find $|\alpha|.$
|
4
| 0.75 |
Consider the following table showing the salaries of employees at a particular company:
\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
CEO & 1 & \$150{,}000 \\\hline
General Manager & 3 & \$100{,}000 \\\hline
Manager & 12 & \$80{,}000 \\\hline
Supervisor & 8 & \$55{,}000 \\\hline
Worker & 35 & \$30{,}000 \\\hline
\end{tabular}
According to the table, how many dollars are in the median value of the 59 salaries paid to this company's employees?
|
\$30,000
| 0.416667 |
In triangle $ABC,$ $AB = 4,$ $AC = 7,$ $BC = 9,$ and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC.$ Find $\cos \angle BAD.$
|
\frac{\sqrt{70}}{14}
| 0.333333 |
Find the smallest positive four-digit number that is divisible by each of the five smallest prime numbers.
|
2310
| 0.916667 |
A \(90^\circ\) rotation around the origin in the clockwise direction is applied to \(4 + 2i\). What is the resulting complex number?
|
2 - 4i
| 0.833333 |
Compute \[\left\lfloor \dfrac{101^3}{99 \cdot 100} - \dfrac{99^3}{100 \cdot 101} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$
|
8
| 0.416667 |
Sixty cards are placed into a box, each bearing a number 1 through 15, with each number represented on four cards. Four cards are drawn from the box at random without replacement. Let \(p\) be the probability that all four cards bear the same number. Let \(q\) be the probability that three of the cards bear a number \(a\) and the other bears a number \(b\) that is not equal to \(a\). What is the value of \(q/p\)?
|
224
| 0.833333 |
Suppose that $a,b,$ and $c$ are positive integers satisfying $(a+b+c)^3 - a^3 - b^3 - c^3 = 270$. Find $a+b+c$.
|
7
| 0.916667 |
Let \(S\) be the set of all nonzero real numbers. Define a function \(f : S \to S\) such that for all \(x, y \in S\) with \(x + y = 1\),
\[ f(x) + f(y) = f(xyf(x + y)) \]
Determine the number of possible values of \(f(3)\) and the sum of all possible values of \(f(3)\). Calculate \(n \times s\) where \(n\) is the number of possible values and \(s\) is the sum of those values.
|
\frac{1}{3}
| 0.916667 |
Evaluate the polynomial \[ p(x) = x^4 - 3x^3 - 9x^2 + 27x - 8, \] where $x$ is a positive number such that $x^2 - 3x - 9 = 0$.
|
\frac{65 + 81\sqrt{5}}{2}
| 0.083333 |
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