problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Let $p(x)$ be a monic polynomial of degree 4, such that $p(1) = 24,$ $p(2) = 48,$ $p(3) = 72,$ and $p(4) = 96.$ Find $p(0) + p(5).$
|
168
| 0.833333 |
Calculate $8 \cdot 9\frac{2}{5}$.
|
75\frac{1}{5}
| 0.583333 |
What common fraction is equivalent to $0.4\overline{56}$?
|
\frac{226}{495}
| 0.75 |
Jane's graduating class has 360 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 8 rows and at least 12 students in each row, then there can be $x$ students in each row. What is the sum of all possible values of $x$?
|
12 + 15 + 18 + 20 + 24 + 30 + 36 + 40 + 45 = 240
| 0.5 |
In the expression $c \cdot b^a - d$, the values of $a$, $b$, $c$, and $d$ are 0, 1, 2, and 3, although not necessarily in that order, with the restriction that $d = 0$. What is the maximum possible value of this expression?
|
9
| 0.333333 |
Let $b_1$, $b_2$, $b_3$, $c_1$, $c_2$, and $c_3$ be real numbers such that for every real number $x$, we have
\[
x^6 - 2x^5 + 3x^4 - 3x^3 + 3x^2 - 2x + 1 = (x^2 + b_1 x + c_1)(x^2 + b_2 x + c_2)(x^2 + b_3 x + c_3).
\]
Compute $b_1 c_1 + b_2 c_2 + b_3 c_3$.
|
-2
| 0.333333 |
A $\textit{composite number}$ is a number that has two or more prime factors. The number 105 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers?
|
3
| 0.75 |
Determine the ordered pair $(x,y)$ that satisfies the following system of equations:
\[
x + 2y = (7 - x) + (3 - 2y),
\]
\[
x - 3y = (x + 2) - (y - 2).
\]
|
(9, -2)
| 0.75 |
Let $x$ and $y$ be positive real numbers such that $5x + 6y < 90$. Find the maximum value of
\[xy (90 - 5x - 6y).\]
|
900
| 0.833333 |
What are the last two digits in the sum of the factorials of the first 15 positive integers?
|
13
| 0.75 |
In triangle $\triangle XYZ$, the medians $\overline{XM}$ and $\overline{YN}$ are perpendicular. If $XM=12$ and $YN=18$, then what is the area of $\triangle XYZ$?
|
144
| 0.416667 |
Given real numbers $t$, consider the point of intersection of the lines $3x + 4y = 12t + 6$ and $2x + 3y = 8t - 1$. All plotted points from various values of $t$ lie on a line. Determine the slope of this line.
|
0
| 0.583333 |
The values of \( a \), \( b \), \( c \), and \( d \) are 1, 3, 5, and 7, but not necessarily in that order. What is the smallest possible value of the sum of the four products \( ab \), \( bc \), \( cd \), and \( da \)?
|
48
| 0.666667 |
In a similar factor tree, each value is the product of the two values below it, unless a value is a prime number already. What is the value of \( X \) on the factor tree shown?
[asy]
draw((-1,-.3)--(0,0)--(1,-.3),linewidth(1));
draw((-2,-1.3)--(-1.5,-.8)--(-1,-1.3),linewidth(1));
draw((1,-1.3)--(1.5,-.8)--(2,-1.3),linewidth(1));
label("X",(0,0),N);
label("Y",(-1.5,-.8),N);
label("7",(-2,-1.3),S);
label("Z",(1.5,-.8),N);
label("F",(-1,-1.3),S);
label("11",(1,-1.3),S);
label("G",(2,-1.3),S);
draw((-1.5,-2.3)--(-1,-1.8)--(-.5,-2.3),linewidth(1));
draw((1.5,-2.3)--(2,-1.8)--(2.5,-2.3),linewidth(1));
label("2",(-1.5,-2.3),S);
label("5",(-.5,-2.3),S);
label("7",(1.5,-2.3),S);
label("3",(2.5,-2.3),S);
[/asy]
|
16170
| 0.416667 |
In triangle $\triangle JKL$, where $\triangle JKL$ is a right triangle at $J$, $\tan K = \frac{4}{3}$. If the length $JK = 3$, what is the length of $KL$?
|
5
| 0.916667 |
Let $g(x) = 3x^5 - 2x^4 + dx - 8$. If $g(-2) = 4$, find $g(2)$.
|
-84
| 0.916667 |
How many ways are there to arrange the letters of the word $\text{B}_1\text{B}_2\text{A}_1\text{N}_1\text{A}_2\text{N}_2\text{A}_3$, in which the three A's are considered different, the two B's are identical, and the two N's are different?
|
2520
| 0.333333 |
The polynomial \(x^{104} + Cx + D\) is divisible by \(x^2 + x + 1\) for some real numbers \(C\) and \(D.\) Find \(C + D.\)
|
2
| 0.833333 |
What is $0.05 \div 0.002$?
|
25
| 0.916667 |
What is the coefficient of $a^2b^3$ in $(a+b)^5\left(c^2+\dfrac{1}{c^2}\right)^3$?
|
0
| 0.166667 |
What is the minimum value of the expression $x^2 + y^2 - 8x - 6y + 20$ for real $x$ and $y$?
|
-5
| 0.916667 |
Integers $a$ and $b$ with $a > b > 0$ satisfy $a + b + ab = 119$. What is $a$?
|
59
| 0.416667 |
Compute
\[\cos^2 0^\circ + \cos^2 1^\circ + \cos^2 2^\circ + \dots + \cos^2 180^\circ.\]
|
91
| 0.166667 |
The force required to loosen bolts varies inversely with the length of the wrench handle used. Using a wrench with a handle length of 10 inches requires 300 pounds of force to loosen Bolt A. To loosen Bolt B, which is tighter and requires 400 pounds of force with a 10-inch handle, how many pounds of force will be needed using a 20-inch wrench handle?
|
200
| 0.333333 |
A point has rectangular coordinates $(3, 8, -6)$ and spherical coordinates $(\rho, \theta, \phi)$. Find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, -\phi)$.
|
(-3,-8,-6)
| 0.333333 |
Let \(a,\) \(b,\) \(c,\) and \(d\) be nonzero real numbers, and let:
\[
x = \frac{b+d}{c+d} + \frac{c+d}{b+d}, \quad y = \frac{a+d}{c+d} + \frac{c+d}{a+d}, \quad z = \frac{a+d}{b+d} + \frac{b+d}{a+d}.
\]
Simplify \(x^2 + y^2 + z^2 - xyz.\)
|
4
| 0.833333 |
Trisha invests \$2,000 in a savings account at an interest rate of 5% per annum, compounded annually. Additionally, she deposits \$300 at the end of each year into the account. How much total interest will Trisha have earned by the end of 5 years?
|
\$710.25
| 0.833333 |
Find $3^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 27^{\frac{1}{27}} \cdot 81^{\frac{1}{81}} \dotsm.$
|
\sqrt[4]{27}
| 0.166667 |
Bertha has 8 daughters and no sons. Each of her daughters has 4 daughters. Bertha has a total of 40 daughters and granddaughters. How many of Bertha's daughters and granddaughters have no daughters?
|
32
| 0.25 |
Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 317^\circ.$
|
43
| 0.5 |
Point $A$ has coordinates $(x, 7)$. When Point $A$ is reflected over the x-axis, it lands on Point $B$. What is the sum of the four coordinate values of points $A$ and $B$?
|
2x
| 0.833333 |
The length of the shortest trip from $A$ to $B$ along the edges of the cube shown is the length of 4 edges. How many different 4-edge trips are there from $A$ to $B$?
[asy]
size(4cm,4cm);
pair a1, b1, c1, d1;
a1=(1,1);
b1=(0,1);
c1=(1.6,1.4);
d1=(1,0);
pair e1, f1, g1, h1;
e1=(0,0);
f1=c1-(a1-d1);
g1=b1+(c1-a1);
h1=e1+(g1-b1);
draw(a1--d1--e1--b1--a1);
draw(b1--g1--c1--a1);
draw(c1--f1--d1);
draw(g1--h1--e1,dotted+1pt);
draw(h1--f1,dotted+1pt);
label("$A$",e1,SW);
label("$B$",c1,NE);
[/asy]
|
6
| 0.083333 |
Mr. and Mrs. Lopez, their two children, and their babysitter are going on a trip in their family car, which has a driver's seat, one front passenger seat, and three back seats. Either Mr. or Mrs. Lopez must be the driver. How many different seating arrangements are possible under these conditions?
|
48
| 0.666667 |
Let
\[
\mathbf{C} = \begin{pmatrix} 3 & 1 \\ -8 & -3 \end{pmatrix}.
\]
Compute $\mathbf{C}^{50}$.
|
\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
| 0.583333 |
Compute $i^{603} + i^{602} + \cdots + i + 1 + 3$, where $i^2 = -1$.
|
3
| 0.5 |
How many positive integer multiples of $1001^2$ can be expressed in the form $10^{j} - 10^{i}$, where $i$ and $j$ are even integers and $1\leq i < j \leq 99$?
|
0
| 0.416667 |
Let $p$ and $q$ be positive integers such that \[\frac{3}{5} < \frac{p}{q} < \frac{5}{8}\] and $q$ is as small as possible. What is $q-p$?
|
5
| 0.833333 |
A triangle in a Cartesian coordinate plane has vertices at (2, 3), (2, -4), and (9, 3). What is the area of this triangle, expressed in square units?
|
24.5
| 0.416667 |
Suppose two distinct numbers are chosen from between 6 and 20, inclusive. What is the probability that their product is even, or exactly one of the numbers is a prime?
|
\frac{94}{105}
| 0.25 |
Find the distance between the vertices of the hyperbola \( 16x^2 + 64x - 4y^2 + 8y + 36 = 0 \).
|
\sqrt{6}
| 0.916667 |
Let $\theta$ be the angle between the planes $3x - 2y + z - 4 = 0$ and $9x - 6y - 4z + 12 = 0$. Find $\cos \theta$.
|
\cos \theta = \frac{35}{\sqrt{1862}}
| 0.25 |
The number $1023$ can be written as $17n + m$ where $n$ and $m$ are positive integers. What is the greatest possible value of $n - m$?
|
57
| 0.666667 |
Find the sum of all possible positive integer values of \( b \) such that the quadratic equation \( 3x^2 + 7x + b = 0 \) has rational roots.
|
6
| 0.833333 |
Suppose $x,$ $y,$ and $z$ are real numbers such that
\[
\frac{xz}{x + y} + \frac{yx}{y + z} + \frac{zy}{z + x} = -5
\]
and
\[
\frac{yz}{x + y} + \frac{zx}{y + z} + \frac{xy}{z + x} = 7.
\]
Compute the value of
\[
x + y + z.
\]
|
2
| 0.916667 |
A point has rectangular coordinates $(8, 6, -3)$ and spherical coordinates $(\rho, \theta, \phi)$. Find the rectangular coordinates of the point with spherical coordinates $(\rho, -\theta, \phi)$.
|
(8, -6, -3)
| 0.833333 |
One of the roots of \( z^2 = -63 + 16i \) is \( 5 + 4i \). What is the other root?
|
-5 - 4i
| 0.833333 |
The length of each of the equal sides of an isosceles triangle is $20$ inches, and the base is $24$ inches. This triangle is circumscribed by a circle. Determine the radius of the circle and the height of the triangle from the base to the apex.
|
16 \text{ inches}
| 0.5 |
What is $3^{-1} + 3^{-2} + 3^{-3} + 3^{-4} \pmod{7}$?
Express your answer as an integer from $0$ to $6$, inclusive.
|
3
| 0.833333 |
Two wheels, one larger outside the other and concentric, with radii $2 \text{ m}$ and $1 \text{ m}$ respectively, are rolled in a straight line on a flat horizontal surface. Each makes one complete revolution. How many meters did the center of each wheel travel horizontally from their starting locations?
|
2 \pi
| 0.083333 |
Find the sum of $1234_6$, $654_6$, and $12_6$ in base $6$.
|
2344_6
| 0.583333 |
Determine how many of the first 15 cumulative sums of Fibonacci numbers are also prime. The Fibonacci sequence starts with 1, 1, 2, 3, 5, etc.
|
2
| 0.75 |
Find the remainder when $5x^5 - 12x^4 + 3x^3 - 7x + 15$ is divided by $3x - 6.$
|
-7
| 0.75 |
Find \( r \) if \( 5(r - 9) = 6(3 - 3r) + 6 \).
|
3
| 0.916667 |
Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=15$. What is the maximum value of \[A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A?\]
|
200
| 0.666667 |
Palindromes are numbers that read the same backwards and forwards, like 1221. What is the least possible four-digit palindrome that is divisible by 5?
|
5005
| 0.666667 |
Let $x_1$, $x_2$, $x_3$, $x_4$, $x_5$, $x_6$ be the roots of the polynomial $f(x) = x^6 + x^3 + 1,$ and let $g(x) = x^2 - 3.$ Find
\[
g(x_1) g(x_2) g(x_3) g(x_4) g(x_5) g(x_6).
\]
|
757
| 0.083333 |
Determine if there is a scalar \(d\) such that
\[\mathbf{i} \times (\mathbf{v} \times \mathbf{i}) + \mathbf{j} \times (\mathbf{v} \times \mathbf{j}) + \mathbf{k} \times (\mathbf{v} \times \mathbf{k}) + \mathbf{v} = d \mathbf{v}\]
for all vectors \(\mathbf{v}\).
|
3
| 0.666667 |
In the given diagram, there are two regular polygons, a pentagon and a square. Find the sum of the measures of angles \(ABC\) and \(ABD\) in degrees.
[asy]
draw(10dir(0)--10dir(72)--10dir(144)--10dir(216)--10dir(288)--cycle,linewidth(2));
draw(10dir(216)--10dir(288)--10dir(288)+(0,-10)--10dir(216)+(0,-10)--10dir(216)--cycle,linewidth(2));
label("A",10dir(216),W);
label("B",10dir(288),E);
label("C",10dir(0),E);
label("D",10dir(288)+(0,-10),E);
[/asy]
|
198^\circ
| 0.833333 |
Let three positive integers be \( a \), \( b \), and \( c \). The product \( N \) of these integers is \( 8 \) times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of \( N \).
|
560
| 0.916667 |
Let $g(x) = 3x^4 - 20x^3 + 37x^2 - 18x - 80$. Find $g(6)$.
|
712
| 0.916667 |
John is training for a duathlon but skips the swimming part. He cycles 15 miles at a speed of $3x - 2$ miles per hour, then takes a 10-minute break to switch gear, and finally runs 3 miles at $x$ miles per hour. His total exercise time, including the break, is 130 minutes. Determine John's running speed, rounding to the nearest hundredth of a mile per hour.
|
4.44 \text{ mph}
| 0.666667 |
Solve the inequality:
\[\frac{x + 5}{x^2 + 3x + 9} \ge 0.\]
Enter your answer using interval notation.
|
[-5, \infty)
| 0.916667 |
The function \( g(x) \) satisfies
\[ g(3^x) + xg(3^{-x}) = 2 \] for all real numbers \( x \). Find \( g(3) \).
|
0
| 0.916667 |
Given that $3+\sqrt{5}$ is a root of the equation \[x^3 + cx^2 + dx + 20 = 0\] and that $c$ and $d$ are rational numbers, compute $d.$
|
-26
| 0.916667 |
If $\mathbf{a}$ and $\mathbf{b}$ are two unit vectors, with an angle of $\frac{\pi}{4}$ between them, compute the volume of the parallelepiped generated by $\mathbf{a}, \mathbf{b} - \mathbf{a} \times \mathbf{b},$ and $\mathbf{b}$.
|
\frac{1}{2}
| 0.5 |
The first term of a geometric sequence is 250. If the sum of the first 50 terms is 625 and the sum of the first 100 terms is 1225, find the sum of the first 150 terms.
|
1801
| 0.25 |
The graph of an equation \[\sqrt{(x-1)^2 + (y-3)^2} + \sqrt{(x+7)^2 + (y+2)^2} = 24.\] is an ellipse. What is the distance between its foci?
|
\sqrt{89}
| 0.833333 |
A triangle has vertices at $(1, 3), (-2, 4), (4, -1)$. Calculate the area of this triangle.
|
4.5
| 0.083333 |
Expand $(4x + 3y + 2)(2x + 5y + 3)$ and find the sum of the coefficients of the terms which contain a nonzero power of $y$.
|
60
| 0.916667 |
The vertical drops of six roller coasters at Fermat Fun World are shown in the table.
\begin{tabular}{|l|c|}
\hline
The Loop & 180 feet \\ \hline
The Cyclone & 150 feet \\ \hline
The Giant Drop & 210 feet \\ \hline
The Sky Scream & 195 feet \\ \hline
The Thunder & 170 feet \\ \hline
The Ultra Twister & 220 feet \\ \hline
\end{tabular}
What is the positive difference between the mean and the median of these values?
|
0
| 0.75 |
What is the simplified form of the expression \[3 - 5x - 7x^2 + 9 + 11x - 13x^2 - 15 + 17x + 19x^2\]?
|
-x^2 + 23x - 3
| 0.916667 |
Let \( a \), \( b \), \( c \), and \( d \) be real numbers such that \( |a-b|=2 \), \( |b-c|=4 \), and \( |c-d|=5 \). Determine the sum of all possible values of \( |a-d| \).
|
22
| 0.916667 |
Half of the people in a room are seated in two-thirds of the chairs. The rest of the people are standing. If there are 8 empty chairs, how many people are in the room?
|
32
| 0.916667 |
Given the function $f(x)$ values in the table below:
\begin{tabular}{|c||c|c|c|c|c|} \hline $x$ & 1 & 2 & 3 & 4 & 5 \\ \hline $f(x)$ & 4 & 3 & 2 & 5 & 1 \\ \hline
\end{tabular}
If $f^{-1}$ exists, then what is $f^{-1}(f^{-1}(f^{-1}(3)))$?
|
2
| 0.75 |
Determine the number of positive integers \( a \) less than \( 14 \) such that the congruence \( ax \equiv 1 \pmod{14} \) has a solution in \( x \).
|
6
| 0.916667 |
Find the largest value of $n$ such that $6x^2 + nx + 72$ can be factored as the product of two linear factors with integer coefficients.
|
433
| 0.5 |
Find the value of $b$ if $b$ is positive, and the numbers $10, b, \frac{2}{3}$ are the first, second, and third terms, respectively, of a geometric sequence.
|
\frac{2\sqrt{15}}{3}
| 0.916667 |
Alice throws five identical darts. Each hits one of three identical dartboards on the wall. After throwing the five darts, she lists the number of darts that hit each board, from greatest to least. How many different lists are possible?
|
5
| 0.75 |
Mrs. Lee has 12 grandchildren. Assuming that each grandchild is male with a probability of $\frac{2}{3}$ and female with a probability of $\frac{1}{3}$, independently of the others, what is the probability that Mrs. Lee has more grandsons than granddaughters or vice versa?
|
\frac{472305}{531441}
| 0.25 |
If $x$ is a real or complex number and $x^3 = 64$, what is the sum of all real values of $x$?
|
4
| 0.833333 |
Six balls are numbered 1 through 6 and placed in a bowl. Sarah will randomly choose a ball from the bowl, look at its number, and then put it back into the bowl. Then Sarah will again randomly choose a ball from the bowl and look at its number. What is the probability that the product of the two numbers will be odd and less than 20? Express your answer as a common fraction.
|
\frac{2}{9}
| 0.916667 |
Determine the number of solutions to
\[3\cos^3 x - 7 \cos^2 x + 3 \cos x = 0\] in the range $0 \le x \le 2\pi.$
|
4
| 0.75 |
Determine the minimum value of $y$ if $y = 5x^2 + 20x + 25$.
|
5
| 0.916667 |
Let $p, q, r, s, t, u, v, w$ be distinct elements in the set \[
\{-6, -4, -3, -1, 1, 3, 5, 7\}.
\] What is the minimum possible value of \[
(p+q+r+s)^{2} + (t+u+v+w)^{2}?
\]
|
2
| 0.25 |
Find the integer $m$, where $0 \le m \le 360$, such that $\cos m^\circ = \cos 970^\circ$.
|
250
| 0.666667 |
For the set $\{1, 2, 3, \ldots, 8\}$ and each of its non-empty subsets, a unique alternating sum is defined similarly as before. Arrange the numbers in the subset in decreasing order and then, starting with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1, 2, 3, 7, 8\}$ is $8-7+3-2+1=3$. Calculate the sum of all such alternating sums for $n=8$.
|
1024
| 0.083333 |
Determine how many distinct prime factors the sum of the positive divisors of $450$ has and find the greatest common divisor (GCD) of this sum with $450$.
|
3
| 0.916667 |
Evaluate the expression: $\sqrt{10 - 2\sqrt{21}} + \sqrt{10 + 2\sqrt{21}}$.
|
2\sqrt{7}
| 0.833333 |
Find the number of permutations \((a_1, a_2, a_3, a_4, a_5, a_6, a_7)\) of \((1,2,3,4,5,6,7)\) that satisfy \[\frac{a_1 + 1}{2} \cdot \frac{a_2 + 2}{2} \cdot \frac{a_3 + 3}{2} \cdot \frac{a_4 + 4}{2} \cdot \frac{a_5 + 5}{2} \cdot \frac{a_6 + 6}{2} \cdot \frac{a_7 + 7}{2} = 7!\] and \(a_7 > a_1\).
|
1
| 0.25 |
Determine the product of all possible values for $c$ if the length of the segment between the points $(3c, c+5)$ and $(1, 4)$ is $5$ units.
|
-2.3
| 0.416667 |
In the diagram, $AB$ is a line segment, and $C$, $D$, and $E$ are points such that $CD$ is perpendicular to $AB$ and $CE$ is a diagonal line making an angle with $CD$. What is the value of $x$ if the angle $DCE$ measures $60^\circ$?
[asy]
draw((0,0)--(10,0),black+linewidth(1));
draw((4,0)--(4,8),black+linewidth(1));
draw((4,0)--(3.5,0)--(3.5,0.5)--(4,0.5)--cycle,black+linewidth(1));
draw((4,0)--(9,7),black+linewidth(1));
label("$A$",(0,0),W);
label("$B$",(10,0),E);
label("$x^\circ$",(4.75,2.25));
label("$60^\circ$",(5.5,0.75));
label("$C$",(4,0),S);
label("$D$",(4,8),N);
label("$E$",(9,7),NE);
[/asy]
|
30^\circ
| 0.833333 |
In the given configuration, a regular pentagon and a regular triangle are attached at one vertex. Find the sum, in degrees, of the measures of angles \(PQR\) and \(PQS\), where \(P\) is the shared vertex, \(Q\) is a vertex on the pentagon, and \(S\) is a vertex on the triangle.
[asy]
draw(10dir(18)--10dir(90)--10dir(162)--10dir(234)--10dir(306)--cycle,linewidth(2)); // Pentagon
draw(10dir(234)--10dir(306)--10dir(306)+(0,-10)--cycle,linewidth(2)); // Triangle
label("P",10dir(234),W);
label("Q",10dir(306),E);
label("R",10dir(18),E);
label("S",10dir(306)+(0,-10),E);
[/asy]
|
168^\circ
| 0.5 |
Altitudes $\overline{AX}$ and $\overline{BY}$ of acute triangle $ABC$ intersect at $H$. If $\angle BAC = 58^\circ$ and $\angle ABC = 69^\circ$, then what is $\angle CHX$?
|
69^\circ
| 0.166667 |
Compute the value of \[M = 120^2 + 119^2 - 118^2 - 117^2 + 116^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2,\]where the additions and subtractions alternate in pairs.
|
14520
| 0.416667 |
A fair 8-sided die is rolled. If I roll \(n\), then I win \(n^3\) dollars. What is the expected value of my win? Express your answer as a dollar value rounded to the nearest cent.
|
\$162.00
| 0.833333 |
Let \( p \), \( q \), and \( r \) be the roots of the cubic equation \( x^3 - 18x^2 + 40x - 15 = 0 \). Compute \[ (p+q)^2 + (q+r)^2 + (r+p)^2. \]
|
568
| 0.916667 |
Each day, Jenny ate 30% of the jellybeans that were in her jar at the beginning of that day. At the end of the third day, 28 remained. How many jellybeans were in the jar originally?
|
82
| 0.75 |
Let $r(x)$ be a monic quartic polynomial such that $r(1) = 0,$ $r(2) = 3,$ $r(3) = 8,$ and $r(4) = 15$. Find $r(5)$.
|
48
| 0.666667 |
Let \( y_1, y_2, \dots, y_{50} \) be real numbers such that \( y_1 + y_2 + \dots + y_{50} = 2 \) and \( \frac{y_1}{1-y_1} + \frac{y_2}{1-y_2} + \dots + \frac{y_{50}{1-y_{50}}} = 2\). Find the value of \( \frac{y_1^2}{1-y_1} + \frac{y_2^2}{1-y_2} + \dots + \frac{y_{50}^2}{1 - y_{50}} \).
|
0
| 0.666667 |
Jenna collects stamps. She puts the same number of stamps on each page and then inserts each page into one of her three stamp books. One of her stamp books has a total of 924 stamps, another has 1260 stamps, and the third has 1386 stamps. What is the largest number of stamps that Jenna could be putting on each page?
|
42
| 0.25 |
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