problem
stringlengths 18
4.46k
| answer
stringlengths 1
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| pass_at_n
float64 0.08
0.92
|
|---|---|---|
If $3^{2y} = 16$, evaluate $9^{y+2}$.
|
1296
| 0.916667 |
Let $b > 0$, and let $Q(x)$ be a polynomial with integer coefficients such that
\[Q(2) = Q(4) = Q(6) = Q(8) = b\]and
\[Q(1) = Q(3) = Q(5) = Q(7) = -b.\]
What is the smallest possible value of $b$?
|
315
| 0.083333 |
For how many integers $n$ between 1 and 150 is the greatest common divisor of 18 and $n$ equal to 6?
|
17
| 0.833333 |
What is the area, in square units, of triangle $PQR$ given the coordinates $P(-2, 2)$, $Q(8, 2)$, and $R(6, -4)$?
|
30 \text{ square units}
| 0.916667 |
Determine the real number value of the expression $3 + \frac{5}{2 + \frac{5}{3 + \frac{5}{2 + \cdots}}}$, where the $2$s and $3$s alternate.
|
\frac{3 + \sqrt{39}}{2}
| 0.333333 |
Consider the polynomial $3x^4 + ax^3 + 48x^2 + bx + 12$. It has a factor of $2x^2 - 3x + 2$. Find the ordered pair $(a,b)$.
|
(-26.5, -40)
| 0.083333 |
Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g,$ and the $x$-coordinate of the vertex of $g$ is a root of $f.$ Both $f$ and $g$ have the same minimum value. If the graphs of the two polynomials intersect at the point $(50, -200),$ what is the value of $a + c$?
|
a + c = -200
| 0.833333 |
Find the positive real number $x$ such that $\lfloor x \rfloor \cdot x = 100$. Express $x$ as a decimal.
|
10
| 0.833333 |
The polynomial $g(x)$ satisfies $g(x + 1) - g(x) = 8x + 6.$ Find the leading coefficient of $g(x).$
|
4
| 0.833333 |
Cindy wishes to arrange her coins into \( X \) piles, each consisting of the same number of coins, \( Y \). Each pile will have more than one coin and no pile will have all the coins. If there are 15 possible values for \( Y \) given all of the restrictions, what is the smallest number of coins she could have?
|
65536
| 0.583333 |
What is the largest possible median for the five-number set $\{x, 2x, 4, 1, 7\}$ if $x$ can be any negative integer?
|
1
| 0.75 |
Mary and James are sitting in a row of 10 chairs. They choose their seats randomly. What is the probability that they do not sit next to each other?
|
\frac{4}{5}
| 0.75 |
In the diagram, what is the value of $x$? [asy]
size(120);
draw(Circle((0,0),1));
draw((0,0)--(.5,sqrt(3)/2));
draw((0,0)--(sqrt(3)/2,.5));
draw((0,0)--(sqrt(3)/2,-.5));
draw((0,0)--(-1,0));
label("$6x^\circ$",(0,0),NNW); label("$3x^\circ$",(0,0),SSW);
label("$x^\circ$",(.3,0));label("$x^\circ$",(.3,.3));
label("$4x^\circ$",(-.3,-.3));
[/asy]
|
24
| 0.75 |
Calculate $\sqrt{42q} \cdot \sqrt{7q} \cdot \sqrt{14q}$. Express your answer in simplest radical form in terms of $q$.
|
14q \sqrt{21q}
| 0.5 |
Let $m$ and $n$ be consecutive odd integers, where $n < m$. Determine the largest integer that divides all possible numbers of the form $m^2 - n^2$.
|
8
| 0.916667 |
Kevin Kangaroo starts hopping on a number line at 0 and aims to reach the point 1. However, he can only hop $\frac{1}{4}$ of the distance to 1 in each hop. Each subsequent hop covers $\frac{1}{4}$ of the remaining distance to 1. Calculate how far Kevin has hopped after six hops. Express your answer as a common fraction.
|
\frac{3367}{4096}
| 0.666667 |
A spherical ball of cookie dough with a radius of 3 inches is used to make flat cookies. When rolled out, it covers a circular area of 9 inches in radius. Assuming the density of the cookie dough changes such that the resulting volume is doubled, what is the thickness of this layer of cookie dough expressed as a fraction?
|
\frac{8}{9}
| 0.916667 |
Fourteen girls are standing around a circle. A ball is thrown clockwise around the circle. The first girl, Bella, starts with the ball, skips the next two girls and throws to the fourth girl, who then skips the next two girls and throws the ball to the seventh girl. If the throwing pattern continues, including Bella's initial throw, how many total throws are necessary for the ball to return to Bella?
|
14
| 0.916667 |
Calculate the integer nearest to \(1000\sum_{n=4}^{10005}\frac{1}{n^2-4}\).
|
321
| 0.083333 |
Ray climbs a staircase of $n$ steps in two different ways. When he climbs $6$ steps at a time, he has $4$ steps remaining. When he climbs $7$ steps at a time, there are $3$ steps remaining. Determine the smallest possible value of $n$ that is greater than $15$.
|
52
| 0.833333 |
Consider a grid where each block is 1 unit by 1 unit. You need to walk from point $A$ to point $B$, which are 9 units apart on the grid. The grid is structured such that point $A$ is at the bottom left and point $B$ is located 5 units to the right and 4 units up from $A$. How many different paths can you take assuming that you need to pass through a checkpoint $C$ located 3 units to the right and 2 units up from $A$?
|
60
| 0.75 |
Rational Woman and Irrational Woman decide to drive around two racetracks. Rational Woman drives on a path parameterized by
\[
x = 2 + \cos t, \quad y = \sin t,
\]
and Irrational Woman drives on a path parameterized by
\[
x = 3 + 3 \cos \frac{t}{2}, \quad y = \sin \frac{t}{2}.
\]
Find the smallest possible distance between any two points, each belonging to one of the tracks.
|
0
| 0.166667 |
Ollie's leash is tied to a stake at the center of his yard, which is now in the shape of a square. His leash is exactly long enough to reach the midpoint of each side of his square yard. The leash allows him to cover a circular area centered at the stake where the leash is tied. If the area of Ollie's yard that he is able to reach is expressed as a fraction of the total area of his yard, what is this fraction?
|
\frac{\pi}{4}
| 0.833333 |
Either increasing the radius of a cylinder by 4 inches or increasing the height by 12 inches results in the same volume. The original height of the cylinder is 3 inches. What is the original radius in inches?
|
1 + \sqrt{5}
| 0.833333 |
Determine how many odd whole numbers are factors of 252.
|
6
| 0.916667 |
For which positive integer values of $k$ does $kx^2+16x+k=0$ have rational solutions? Express your answers separated by commas and in increasing order.
|
8
| 0.666667 |
Interior numbers begin in the third row of Pascal's Triangle. The sum of the interior numbers in the fourth row is 6. The sum of the interior numbers of the fifth row is 14. What is the sum of the interior numbers of the ninth row?
|
254
| 0.5 |
Consider a monic polynomial of degree \( n \) with real coefficients, where the first two terms after \( x^n \) are \( b_{n-1}x^{n-1} \) and \( b_{n-2}x^{n-2} \), and it is given that \( b_{n-1} = -2b_{n-2} \). Find the absolute value of the greatest lower bound for the sum of the squares of the roots of this polynomial.
|
\frac{1}{4}
| 0.666667 |
How many positive integers less than $201$ are multiples of either $5$ or $7$, but not both at once?
|
58
| 0.75 |
If $a$ and $b$ are integers with $a \neq b$, what is the smallest possible positive value of $\frac{a^2+b^2}{a^2-b^2} + \frac{a^2-b^2}{a^2+b^2}$?
|
2
| 0.583333 |
For how many values of $k$ is $36^{12}$ the least common multiple of the positive integers $6^6$, $8^8$, $9^9$, and $k$?
|
25
| 0.333333 |
The sum of the first 3000 terms of a geometric sequence is 500. The sum of the first 6000 terms is 950. Find the sum of the first 9000 terms.
|
1355
| 0.75 |
Find the matrix $\mathbf{N}$ that doubles the first row and quadruples the second row of a matrix. Specifically,
\[\mathbf{N} \begin{pmatrix} x & y \\ z & w \end{pmatrix} = \begin{pmatrix} 2x & 2y \\ 4z & 4w \end{pmatrix}.\]
If no such matrix $\mathbf{N}$ exists, then enter the zero matrix.
|
\begin{pmatrix} 2 & 0 \\ 0 & 4 \end{pmatrix}
| 0.5 |
Consider the function $$g(t) = \frac{1}{(t-2)^2 + (t-3)^2 - 2}.$$ What is the domain of the function $g(t)$? Express your answer in interval notation.
|
(-\infty, \frac{5 - \sqrt{3}}{2}) \cup (\frac{5 - \sqrt{3}}{2}, \frac{5 + \sqrt{3}}{2}) \cup (\frac{5 + \sqrt{3}}{2}, \infty)
| 0.75 |
Michael finds another interesting aspect of quadratic polynomials and focuses on the polynomial $x^2 - sx + q$. He realizes that for certain values of $s$ and $q$, the roots $r_1$ and $r_2$ of the polynomial satisfy $r_1 + r_2 = r_1^2 + r_2^2 = r_1^{10} + r_2^{10}$. He wants to find the maximum value of $\dfrac{1}{r_1^{11}} + \dfrac{1}{r_2^{11}}$. Help Michael determine this maximum value.
|
2
| 0.833333 |
Find the least positive integer $x$ that satisfies $x + 7237 \equiv 5017 \pmod{12}$.
|
12
| 0.833333 |
An ellipse has a major axis of length 12 and a minor axis of length 10. Using one focus as the center, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. Compute the radius of the circle.
|
6 - \sqrt{11}
| 0.75 |
For what value of $k$ does the equation $x^2 + 8x + y^2 + 4y - k = 0$ represent a circle of radius 7?
|
29
| 0.916667 |
A student's score on a 150-point test is directly proportional to the hours she studies. If she scores 90 points after studying for 2 hours, what would her score be if she studied for 5 hours?
|
225
| 0.083333 |
In triangle $ABC$, $AB=15$ and $AC=8$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AB$ and $CM$. The ratio $BP$ to $PA$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
|
31
| 0.666667 |
The function $g$ is graphed below. Each small box has width and height 1.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {
import graph;
real i;
if(complexplane) {
label("$\textnormal{Re}$",(xright,0),SE);
label("$\textnormal{Im}$",(0,ytop),NW);
} else {
label("$x$",(xright+0.4,-0.5));
label("$y$",(-0.5,ytop+0.2));
}
ylimits(ybottom,ytop);
xlimits( xleft, xright);
real[] TicksArrx,TicksArry;
for(i=xleft+xstep; i<xright; i+=xstep) {
if(abs(i) >0.1) {
TicksArrx.push(i);
}
}
for(i=ybottom+ystep; i<ytop; i+=ystep) {
if(abs(i) >0.1) {
TicksArry.push(i);
}
}
if(usegrid) {
xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true);
yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows);
}
if(useticks) {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
} else {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));
}
};
rr_cartesian_axes(-1,9,-1,9);
dot((0,0),red+5bp);
dot((2,2),red+5bp);
dot((4,4),red+5bp);
dot((6,6),red+5bp);
dot((8,8),red+5bp);
dot((1,7),red+5bp);
dot((3,5),red+5bp);
dot((5,3),red+5bp);
dot((7,1),red+5bp);
dot((9,0),red+5bp);
[/asy]
Simon writes the number 2 on his thumb. He then applies $g$ to 2 and writes the output on his index finger. If Simon continues this process of applying $g$ and writing the output on a new finger, what number will Simon write on his tenth finger?
|
2
| 0.833333 |
Given that $\log_{10} \sin x + \log_{10} \cos x = -2$ and that $\log_{10} (\sin x + \cos x) = \frac{1}{2} (\log_{10} n - 2),$ find $n.$
|
102
| 0.916667 |
Find $\begin{pmatrix} 5 \\ -3 \end{pmatrix} - (-2) \begin{pmatrix} 3 \\ -4 \end{pmatrix}.$
|
\begin{pmatrix} 11 \\ -11 \end{pmatrix}
| 0.916667 |
If \( 0 \leq p \leq 1 \) and \( 0 \leq q \leq 1 \), define \( H(p, q) \) by
\[
H(p, q) = -3pq + 4p(1-q) + 4(1-p)q - 5(1-p)(1-q).
\]
Define \( J(p) \) to be the maximum of \( H(p, q) \) over all \( q \) (in the interval \( 0 \leq q \leq 1 \)). What is the value of \( p \) (in the interval \( 0 \leq p \leq 1 \)) that minimizes \( J(p) \)?
|
\frac{9}{16}
| 0.916667 |
For what digit $d$ is the five-digit number $5678d$ a multiple of 9?
|
1
| 0.833333 |
Let $a$ and $b$ be nonzero complex numbers such that $a^2 + ab + b^2 = 0.$ Evaluate
\[\frac{a^{15} + b^{15}}{(a + b)^{15}}.\]
|
-2
| 0.833333 |
Factor completely: $y^8 - 4y^6 + 6y^4 - 4y^2 + 1$.
|
(y-1)^4(y+1)^4
| 0.916667 |
What is the smallest positive integer $n$ such that $503n \equiv 1019n \pmod{48}$?
|
4
| 0.916667 |
What is $2^{24} \div 8^3$? Write your answer as an integer.
|
32768
| 0.666667 |
Let \( z \) and \( w \) be complex numbers such that \( |z| = 2 \) and \( |w| = 4 \). If \( |z+w| = 5 \), what is \( \left | \frac{1}{z} + \frac{1}{w} \right| \)?
|
\frac{5}{8}
| 0.916667 |
What is the median number of moons per celestial body? (Include Pluto and the newly discovered dwarf planet Eris, which has 1 moon.) \begin{tabular}{c|c}
Celestial Body & $\#$ of Moons\\
\hline
Mercury & 0\\
Venus & 0\\
Earth & 1\\
Mars & 2\\
Jupiter & 18\\
Saturn & 25\\
Uranus & 17\\
Neptune & 3\\
Pluto & 5\\
Eris & 1\\
\end{tabular}
|
2.5
| 0.416667 |
A woman labels the squares of a very large chessboard from $1$ through $64$. On each square $k$, she puts $2^k$ grains of rice. How many more grains of rice are placed on the $20^{th}$ square than on the first $15$ squares combined?
|
983042
| 0.833333 |
Grace and her sister each spin a spinner once. This spinner has six equal sectors, labeled 1 through 6. If the non-negative difference of their numbers is less than 4, Grace wins. Otherwise, her sister wins. What is the probability that Grace wins? Express your answer as a common fraction.
|
\frac{5}{6}
| 0.416667 |
How many positive integers less than 100 are divisible by 2, 4, and 5?
|
4
| 0.75 |
Four distinct digits $a$, $b$, $c$, and $d$ (none consecutive and all different) are selected to form the four-digit numbers $abcd$ and $dcba$. What is the greatest common divisor (GCD) of all possible numbers of the form $abcd+dcba$, considering $a, b, c, d$ non-consecutive and distinct?
|
11
| 0.916667 |
Define a function $g(x),$ for positive integer values of $x,$ by \[g(x) = \left\{\begin{aligned} \log_3 x & \quad \text{ if } \log_3 x \text{ is an integer} \\ 1 + g(x + 1) & \quad \text{ otherwise}. \end{aligned} \right.\]Compute $g(200).$
|
48
| 0.333333 |
Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a + b + c = 2.$ Find the maximum value of $a + b^2 + c^3.$
|
8
| 0.583333 |
Find all real solutions to $x^3 + (x + 2)^3 + (x + 4)^3 = (x + 6)^3$.
|
6
| 0.916667 |
A farmer has a rectangular field with dimensions \( 3m+8 \) and \( m-3 \). If the field has an area of 85 square units, what is the value of \( m \)?
|
\frac{1 + \sqrt{1309}}{6}
| 0.916667 |
Simplify $(81 \times 10^{12}) \div (9 \times 10^4)$.
|
900,000,000
| 0.083333 |
Given that 12 is the arithmetic mean of the set $\{8, 15, 20, 6, y\}$, what is the value of $y$?
|
11
| 0.083333 |
Acme T-Shirt Company now charges a $75 setup fee plus $8 for each shirt printed. Gamma T-shirt Company has no setup fee but charges $16 per shirt. What is the minimum number of shirts for which a customer saves money by using Acme?
|
10
| 0.916667 |
A lattice point in the $x,y$-plane is a point both of whose coordinates are integers (not necessarily positive). How many lattice points lie on the graph of the equation $x^2-y^2=45$?
|
12
| 0.416667 |
The base-10 numbers 356 and 78 are multiplied. The product is then added to 49, and this sum is written in base-8. What is the units digit of the base-8 representation?
|
1
| 0.666667 |
A function $g(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have
\[3g\left(x\right) + g\left(\frac{1}{x}\right) = 6x + 9\]
Let $T$ denote the sum of all of the values of $x$ for which $g(x) = 3000$. Compute the integer nearest to $T$.
|
1332
| 0.75 |
Let $p$ and $q$ be real numbers. One of the roots of the equation
\[ x^3 + px^2 + qx - 6 = 0 \]
is $2 - i$. Determine the ordered pair $(p, q)$.
|
\left(-\frac{26}{5}, \frac{49}{5}\right)
| 0.833333 |
A sphere intersects the $xy$-plane in a circle centered at $(3,3,0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0,3,-8)$, with radius $r$. Find $r$.
|
\sqrt{59}
| 0.416667 |
The graph of the function $g(x) = ax^4 + bx^3 + cx^2 + dx + e$ is given. The function intersects the x-axis at $x = -1, 0, 1, 2$ and passes through the point $(0,3)$. Determine the coefficient $b$.
|
b = -2
| 0.75 |
Calculate the sum of the squares of the roots of the equation \[x^{1010} + 22x^{1007} + 6x^6 + 808 = 0.\]
|
0
| 0.833333 |
For how many values of the digit $A$ is it true that $49$ is divisible by $A$ and $573{,}4A6$ is divisible by $4$?
|
2
| 0.666667 |
Let $f(x) = x^2 - 5x + 12$ and let $g(f(x)) = 3x + 4$. What is the sum of all possible values of $g(9)$?
|
23
| 0.916667 |
Let the ordered triples $(x,y,z)$ of complex numbers satisfy
\begin{align*}
x + yz &= 9, \\
y + xz &= 14, \\
z + xy &= 14.
\end{align*}
Find the sum of all $x$ values from the solutions $(x_1,y_1,z_1), (x_2,y_2,z_2), \dots, (x_n,y_n,z_n)$.
|
9
| 0.166667 |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 3.$ Find the minimum value of
\[\frac{a + b}{abc}.\]
|
\frac{16}{9}
| 0.583333 |
If $A\ \diamond\ B$ is defined as $A\ \diamond\ B = 4A + B^2 + 7$, calculate the value of $A$ for which $A\ \diamond\ 3 = 85$.
|
17.25
| 0.416667 |
Compute $\arccos (\sin 3)$. All functions are in radians.
|
3 - \frac{\pi}{2}
| 0.25 |
A frequency distribution for a history class is shown. Calculate the percent of the class that scored in the $70\%$-$79\%$ range.
\begin{tabular}{|c|c|}
Test Scores & Frequencies\\
\hline
$90\% - 100\%$ & IIII I\\
$80\% - 89\%$ & IIII III\\
$70\% - 79\%$ & IIII IIII\\
$60\% - 69\%$ & IIII \\
Below $60\%$ & III
\end{tabular}
|
29.63\%
| 0.083333 |
Using only the digits 6, 7, and 8, how many positive nine-digit integers can be created that are palindromes?
|
243
| 0.666667 |
Three fair, standard six-sided dice are rolled. What is the probability that the sum of the numbers on the top faces is 10 and at least one die shows a 5?
|
\frac{1}{18}
| 0.416667 |
In a different ellipse, the center is at $(2, -1)$, one focus is at $(2, -4)$, and one endpoint of a semi-major axis is at $(2, 3)$. Find the semi-minor axis of this ellipse.
|
\sqrt{7}
| 0.916667 |
Express $\cos(a + b) + \cos(a - b)$ as a product of trigonometric functions.
|
2 \cos a \cos b
| 0.916667 |
Abby, Bart, Cindy, and Damon weigh themselves in pairs. Together Abby and Bart weigh 260 pounds, Bart and Cindy weigh 245 pounds, Cindy and Damon weigh 270 pounds, and Abby and Cindy weigh 220 pounds. How many pounds do Abby and Damon weigh together?
|
285
| 0.666667 |
A student is solving the expression $\left(7^2 - 6^2\right)^4$. What is the value of the expression?
|
28561
| 0.916667 |
Let $f(x) = |\{x\}^2 - 2\{x\} + 1.5|$ where $\{x\}$ denotes the fractional part of $x$. The number $n$ is the smallest positive integer such that the equation \[n f(xf(x)) = x\] has at least $2023$ real solutions. What is $n$?
|
45
| 0.083333 |
What is the 30th digit after the decimal point of the sum of the decimal equivalents for the fractions $\frac{1}{11}$ and $\frac{1}{13}$?
|
2
| 0.583333 |
Let $f(x) = -3x^3 - 3x^2 + x - 5$, $g(x) = -6x^2 + 4x - 9$, and $h(x) = 5x^2 + 6x + 2$. Express $f(x) + g(x) + h(x)$ as a single polynomial, with the terms in order by decreasing degree.
|
-3x^3 - 4x^2 + 11x - 12
| 0.916667 |
If
\[1 \cdot 1990 + 2 \cdot 1989 + 3 \cdot 1988 + \dots + 1989 \cdot 2 + 1990 \cdot 1 = 1990 \cdot 995 \cdot y,\]
compute the integer \(y\).
|
664
| 0.333333 |
Jennifer wants to enclose her rectangular vegetable garden using 160 feet of fencing. She has decided that one side of the garden should be exactly 30 feet long. What is the maximum area that she can enclose, assuming the sides of the rectangle are natural numbers?
|
1500
| 0.833333 |
Determine the exact value of the series:
\[
\frac{1}{3 + 1} + \frac{2}{3^2 + 1} + \frac{4}{3^4 + 1} + \frac{8}{3^8 + 1} + \frac{16}{3^{16} + 1} + \dotsb.
\]
|
\frac{1}{2}
| 0.75 |
Simplify $(3x)^5 + (5x)(x^4) - 7x^5$.
|
241x^5
| 0.916667 |
Calculate both the product and the sum of the least common multiple (LCM) and the greatest common divisor (GCD) of $12$ and $15$.
|
63
| 0.333333 |
Let $g(x) = \frac{3x - 2}{x + 4}$. The inverse of $g$ can be written as $g^{-1}(x) = \frac{ax + b}{cx + d}$. Find the value of $\frac{a}{c}$.
|
-4
| 0.833333 |
Express $\sin 3x + \sin 7x$ as a product of trigonometric functions.
|
2 \sin 5x \cos 2x
| 0.916667 |
If 5 knicks = 3 knacks and 4 knacks = 5 knocks, how many knicks are equal to 30 knocks?
|
40
| 0.916667 |
A play has three different male roles, three different female roles, and no roles that can be either gender. Only a man can be assigned to a male role, and only a woman can be assigned to a female role. If five men and six women audition, in how many ways can the six roles be assigned?
|
7200
| 0.75 |
Compute $\arccos(\sin 3)$, where all functions are in radians.
|
3 - \frac{\pi}{2}
| 0.5 |
What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A(3, 0)$ and $B(-4,5)$?
|
\frac{16}{5}
| 0.75 |
In triangle $DEF$, with side lengths $DE=6$, $EF=8$, and $FD=10$, two bugs start simultaneously from point $D$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $G$. What is $EG$?
|
6
| 0.5 |
Determine the number of digits in the value of $2^{15} \times 3^2 \times 5^{10}$.
|
13
| 0.5 |
Find the positive value of $x$ that satisfies $ab = x-6i$ given $|a|=3$ and $|b|=5$.
|
3\sqrt{21}
| 0.916667 |
In how many ways can the digits of $70,\!616$ be arranged to form a 5-digit number? (Note, numbers cannot begin with 0.)
|
48
| 0.916667 |
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