problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
In a regular nonagon (a 9-sided polygon), two diagonals are chosen. What is the probability that they intersect inside the nonagon?
|
\frac{14}{39}
| 0.583333 |
A regular hexagon $ABCDEF$ serves as the base of right pyramid $\allowbreak PABCDEF$. Assume triangle $PAD$ is an isosceles right triangle with the equal sides each of length 10 units. Determine the volume of the pyramid.
|
125\sqrt{6}
| 0.083333 |
What is the median number of moons per celestial body? Consider this updated data about several planets and the dwarf planet Pluto:
\begin{tabular}{c|c}
Celestial Body & $\#$ of Moons\\
\hline
Mercury & 0 \\
Venus & 0 \\
Earth & 1 \\
Mars & 2 \\
Jupiter & 67 \\
Saturn & 82 \\
Uranus & 27 \\
Neptune & 14 \\
Pluto & 5 \\
Ceres & 0 \\
\end{tabular}
|
3.5
| 0.75 |
A maintenance team needs to replace tiles at the center $C$ of a round patio. They lay a 20-foot plank from $A$ to $B$ and then a 15-foot plank from $D$ to $C$, where $D$ is the midpoint of $\overline{AB}$. Calculate the area of the circular patio. Express your answer in terms of $\pi$.
|
325 \pi \text{ square feet}
| 0.5 |
Each vertex of this parallelogram has integer coordinates. The perimeter of this parallelogram is $p$ units, while the area is $a$ square units. Determine the value of the sum $p + a$ given the parallelogram with vertices at $(1,1)$, $(6, 3)$, $(9, 3)$, and $(4,1)$.
|
2\sqrt{29} + 12
| 0.916667 |
A function $f$ has domain $[0,2]$ and range $[0,1]$. Define another function by
\[ g(x) = 1 - f(x^2 + 1) \]
Determine the domain and range of $g(x)$ and provide them as an ordered quadruple $(a, b, c, d)$.
|
(-1, 1, 0, 1)
| 0.916667 |
In $\triangle ABC$, $AC = BC$, $AB = BC$, $m\angle ACB = 60^{\circ}$, and $CD$ is the altitude from point $C$ to line $AB$. Point $E$ is the midpoint of $CD$. What is the number of degrees in $m\angle ECB$?
[asy]
pair A,B,C,D,E;
A = (-1,0);
B = (1,0);
C = (0, 1.732/2);
D = (0, 0);
E = (0, 1.732/4);
draw(A--B--C--cycle);
draw(C--D);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, N);
dot("$D$", D, S);
dot("$E$", E, E);
[/asy]
|
30^{\circ}
| 0.416667 |
Let \( z \) and \( w \) be complex numbers such that \( |z| = 2 \) and \( |w| = 4 \). If \( |z+w| = 3 \), what is \( \left | \frac{1}{z} + \frac{1}{w} \right| \)?
|
\frac{3}{8}
| 0.583333 |
Let $\mathbf{N}$ be a matrix such that
\[
\mathbf{N} \begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} \quad \text{and} \quad \mathbf{N} \begin{pmatrix} -4 \\ 6 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}.
\]
Compute $\mathbf{N} \begin{pmatrix} 7 \\ 2 \end{pmatrix}$.
|
\begin{pmatrix} 24 \\ 11 \end{pmatrix}
| 0.833333 |
If $x$ and $y$ are positive real numbers such that $8x^2 + 16xy = 2x^3 + 4x^2y,$ find the value of $x$.
|
4
| 0.916667 |
Define $A \diamond B$ as $A \diamond B = \frac{A^2 + B^2}{5}$. What is the value of $(3 \diamond 7) \diamond 4$?
|
30.112
| 0.916667 |
The sum of Alice's weight and Clara's weight is 240 pounds. If you subtract Alice's weight from Clara's weight, you get one third of Alice's weight. How many pounds does Clara weigh?
|
\frac{960}{7}
| 0.416667 |
Six scores are distributed between Alice and Bob. These scores are 68, 72, 76, 80, 85, and 90. Alice's scores are in ascending order. If the mean score of Alice is 75, what is the mean score of Bob?
|
82
| 0.5 |
If \(a\), \(b\), and \(c\) are positive numbers with \(ab = 24\sqrt[3]{3}\), \(ac = 40\sqrt[3]{3}\), and \(bc = 15\sqrt[3]{3}\), find the value of \(abc\).
|
120\sqrt{3}
| 0.166667 |
Find the number of $x$-intercepts on the graph of $y = \sin \frac{1}{x}$ in the interval $(0.0001, 0.0005).$
|
2547
| 0.166667 |
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=101$. What is $x$?
|
50
| 0.916667 |
Calculate $\frac{4 \cdot 7! + 28 \cdot 6!}{8!}$
|
1
| 0.833333 |
How many real numbers \(x\) are solutions to the equation \[ |x-2| = |x-3| + |x-4| \] ?
|
2
| 0.833333 |
The sequence described in rows is as follows:
- Row 1: $2, 2$
- Row 2: $4, 4, 4, 4$
- Row 3: $6, 6, 6, 6, 6, 6$
- Row 4: $8, 8, 8, 8, 8, 8, 8, 8$
Continuing this pattern, what would be the value of the $60^{\mathrm{th}}$ number if the arrangement were continued?
|
16
| 0.583333 |
Tamika selects two different numbers at random from the set $\{5, 6, 7\}$ and adds them. Carlos takes two different numbers at random from the set $\{2, 4, 8\}$ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? Express your answer as a common fraction.
|
\frac{1}{3}
| 0.25 |
Eight 6-sided dice are rolled. What is the probability that exactly four of these dice show the number 3?
|
\frac{43750}{1679616}
| 0.166667 |
The time right now is 3 o'clock. On a 12-hour clock, what time will it be 2517 hours from now?
|
12
| 0.5 |
Let $b_1, b_2, \ldots$ be a sequence defined by:
(i) $b_1 = 1$, and
(ii) $b_{3n}=n^2\cdot b_n$ for any positive integer $n$.
What is the value of $b_{3^{100}}$?
|
3^{9900}
| 0.333333 |
Find the number of 8-tuples $(x_1, x_2, \dots, x_8)$ of real numbers such that
\[(1 - x_1)^2 + (x_1 - x_2)^2 + (x_2 - x_3)^2 + \dots + (x_7 - x_8)^2 + x_8^2 = \frac{1}{9}.\]
|
1
| 0.416667 |
Eric is now building a larger pyramid with a height of 15 inches and a square base measuring 12 inches on each side. For extra safety, he desires a clearance of at least 3 inches above the pyramid when placed in a cube-shaped box. What is the minimum volume of the cube-shaped box required to hold the pyramid with the additional clearance?
|
5832
| 0.916667 |
What is the sum of the digits of the base-2 expression for $315_{10}$?
|
6
| 0.916667 |
Points $P$, $Q$, $R$, and $S$ are in space such that each of $\overline{SP}$, $\overline{SQ}$, and $\overline{SR}$ is perpendicular to the other two. If $SP = SQ = 12$ and $SR = 11$, then what is the volume of pyramid $SPQR$?
|
264 \text{ cubic units}
| 0.916667 |
If $(4, 20)$ and $(12, -6)$ are the coordinates of two opposite vertices of a rectangle, what is the sum of the $y$-coordinates of the other two vertices?
|
14
| 0.916667 |
When $5000^{150}$ is expanded out, the result is what digit followed by how many zeros?
|
450
| 0.083333 |
From a standard deck of 52 cards, three cards are drawn consecutively without replacement. What is the probability that the first card is a spade, the second card is a heart, and the third card is a diamond?
|
\frac{2197}{132600}
| 0.333333 |
In the six-digit integer $4B4,\!832$, what is the largest digit $B$ so that the six-digit integer will be divisible by both 3 and 4?
|
9
| 0.75 |
In a press conference before a major league game, ten players from four different teams will be taking questions. Three are from the Cubs, three are from the Red Sox, two are from the Yankees, and two are from the Dodgers. If teammates insist on sitting together, how many ways can the ten players be seated in a row?
|
3456
| 0.916667 |
A function \(g\) is defined on the complex numbers by \(g(z) = (c+di)z,\) where \(c\) and \(d\) are real numbers. This function has the property that for each complex number \(z\), \(g(z)\) is equidistant from both \(z\) and the origin. Given that \(|c+di| = 7\), find \(d^2.\)
|
\frac{195}{4}
| 0.916667 |
Find the radius of the circle with equation $x^2 + 8x + y^2 - 10y + 41 = 0$.
|
0
| 0.166667 |
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 |
What is the sum of the digits of the decimal representation of $2^{2010} \times 5^{2012} \times 7$?
|
13
| 0.916667 |
Consider the first $30$ rows of Pascal's Triangle, from the $0$th row to the $29$th row. Calculate the sum of the binomial coefficients in the $10$th row.
|
1024
| 0.916667 |
Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\]
|
75
| 0.916667 |
The Big Eighteen Soccer League has three divisions, with six teams in each division. Each team plays each of the other teams in its own division three times and every team in the other two divisions twice. How many league games are scheduled?
|
351
| 0.5 |
What is the smallest whole number $b$ such that 120 can be expressed in base $b$ using only three digits?
|
5
| 0.916667 |
The exact amount of fencing that enclosed six congruent equilateral triangular corrals is reused to form one large equilateral triangular corral. Given that the side length of each small corral is 10 units, calculate the ratio of the total area of the six small corrals to the area of the new large corral. Express your answer as a common fraction.
|
\frac{1}{6}
| 0.916667 |
In triangle $ABC$, $AC = \sqrt{17}$ and $AB = 4$. What is $\tan A$?
[asy]
pair A,B,C;
B = (0,0);
A = (0,4);
C = (1,0);
draw(A--B--C--A);
draw(rightanglemark(A,B,C,7));
label("$B$",B,SW);
label("$A$",A,NW);
label("$C$",C,SE);
label("$4$",(A+B)/2,NW);
label("$\sqrt{17}$",(A+C)/2,NE);
[/asy]
|
\frac{1}{4}
| 0.916667 |
If $9:x^2 = x:25$, what is the value of $x$?
|
\sqrt[3]{225}
| 0.166667 |
The Bank of New Town offers a Special Interest Account that compounds annually at a rate of two percent. If Mark invests $1500 in this account, how much interest will he earn after eight years?
|
\$257.49
| 0.916667 |
The projection of $\begin{pmatrix} 5 \\ c \end{pmatrix}$ onto $\begin{pmatrix} -3 \\ 2 \end{pmatrix}$ is
\[5\begin{pmatrix} -3 \\ 2 \end{pmatrix}.\] Find $c.$
|
40
| 0.833333 |
For any positive integer $n$, the value of $n!$ is the product of the first $n$ positive integers. Calculate the greatest common divisor of $8!$ and $10!$.
|
40320
| 0.75 |
Two rectangles have integer dimensions, and both have a perimeter of 200 cm. One dimension of one of the rectangles is fixed at 30 cm. What is the smallest possible area of this rectangle?
|
2100 \text{ square cm}
| 0.916667 |
Let \( \mathbf{B} = \begin{pmatrix} a & 2 \\ -3 & d \end{pmatrix} \) for some real numbers \( a \) and \( d \). If
\[ \mathbf{B} + \mathbf{B}^{-1} = \mathbf{0}, \] determine \( \det \mathbf{B} \).
|
1
| 0.916667 |
Let $A=(0,15)$ and $B=(0,18)$. Points $A'$ and $B'$ are on the line $y=x$, and $\overline{AA'}$ and $\overline{BB'}$ intersect at $C=(4,10)$. What is the length of $\overline{A'B'}$?
|
\frac{2\sqrt{2}}{3}
| 0.583333 |
Given the points (2, 15), (9, 30), (15, 50), (21, 55), and (25, 60), what is the sum of the $x$-coordinates of the points that are situated above the line $y = 3x + 5$ in the coordinate plane?
|
2
| 0.916667 |
The value of $3x + 15$ is one third of the value of $6x + 45$. After finding $x$, subtract 5 from the result. What is the final value?
|
-5
| 0.25 |
The numbers from 1 to 200, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is neither a perfect square nor a perfect cube? Express your answer as a common fraction.
|
\frac{183}{200}
| 0.916667 |
What is the digit in the ten-thousandths place of the decimal equivalent of $\frac{5}{32}$?
|
2
| 0.666667 |
What is the smallest positive integer $x$ that, when multiplied by $450$, produces a product that is a multiple of $675$?
|
3
| 0.833333 |
Let $\bold{v} = \begin{pmatrix} 6 \\ -4 \end{pmatrix}$ and $\bold{w} = \begin{pmatrix} 8 \\ -1 \end{pmatrix}$. Find the area of the parallelogram with vertices $\bold{0}$, $\bold{v}$, $2\bold{w}$, and $\bold{v} + 2\bold{w}$.
|
52
| 0.833333 |
What is the remainder when the sum $1^2 + 2^2 + 3^2 + \dots + 15^2$ is divided by 16?
|
8
| 0.833333 |
Find the positive value of $x$ that satisfies $cd = x + 6i$ given $|c|=3$ and $|d|=5$.
|
3\sqrt{21}
| 0.916667 |
Express $2.\overline{06}$ as a reduced fraction, given that $0.\overline{02}$ is $\frac{2}{99}$.
|
\frac{68}{33}
| 0.916667 |
Find the sum: $(-51) + (-49) + \cdots + (1)$. Additionally, identify the middle term of the series.
|
-25
| 0.083333 |
Consider the polynomial equation
\[ z^6 - z^4 + z^2 - 1 = 0. \]
Among all the roots of this polynomial, find the root with the maximum imaginary part, expressed as $\sin \theta$, where $-90^\circ \leq \theta \leq 90^\circ$. Determine $\theta$.
|
45^\circ
| 0.916667 |
Find the smallest positive integer $n$ such that
\[\left(\begin{pmatrix} \cos 120^\circ & -\sin 120^\circ \\ \sin 120^\circ & \cos 120^\circ \end{pmatrix}^2\right)^n = \mathbf{I}.\]
|
3
| 0.916667 |
How many positive multiples of 6 that are less than 150 have a units digit of 6?
|
5
| 0.333333 |
Tamika selects two different numbers at random from the set $\{7, 8, 10\}$ and adds them. Carlos takes two different numbers at random from the set $\{2, 4, 6\}$ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? Express your answer as a common fraction.
|
\frac{6}{9} = \frac{2}{3}
| 0.666667 |
What is the greatest integer value of $b$ so that $5$ is not in the range of the function $y = x^2 + bx + 20$?
|
7
| 0.916667 |
In triangle $ABC$, we have $\angle A = 90^\circ$, $BC = 10$, and $\tan C = 3\cos B$. What is $AB$?
|
\frac{20\sqrt{2}}{3}
| 0.5 |
What is the simplified value of the sum: \( -1^{2022} + (-1)^{2023} + 1^{2024} - 1^{2025} \)?
|
-2
| 0.916667 |
Let
\[
\mathbf{B} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}.
\]
Compute $\mathbf{B}^{93}$.
|
\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}
| 0.666667 |
In the diagram below, we have $AB = 30$ and $\angle ADB = 90^\circ$. If $\sin A = \frac45$ and $\sin C = \frac12$, then what is $DC$?
[asy]
pair A,B,C,D;
A = (0,0);
B = (10*sqrt(5),24);
D = (10*sqrt(5),0);
C = (10*sqrt(5) + 24*sqrt(2),0);
draw(D--B--A--C--B);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,S);
draw(rightanglemark(B,D,A,63));
[/asy]
|
24\sqrt{3}
| 0.083333 |
A zoo houses five different pairs of animals, each pair consisting of one male and one female. To maintain a feeding order by gender alternation, if the initial animal fed is a male lion, how many distinct sequences can the zookeeper follow to feed all the animals?
|
2880
| 0.25 |
A high school basketball team consists of 18 players, including a set of twins: Bob and Bill. In how many ways can we select 8 players for a game if both twins have to be in the lineup?
|
8008
| 0.833333 |
Six couples are at a gathering. Each person shakes hands with everyone else except for their spouse and the first new person they meet. How many handshakes are exchanged?
|
54
| 0.583333 |
The graph of the function $y=g(x)$ is given. For all $x > 5$, it holds that $g(x) > 0.5$. The function $g(x)$ is defined as $g(x) = \frac{x^2}{Ax^2 + Bx + C}$ where $A$, $B$, and $C$ are integers. The vertical asymptotes of $g$ are at $x = -3$ and $x = 4$, and the horizontal asymptote is such that $y = 1/A < 1$. Find $A + B + C$.
|
-24
| 0.833333 |
Suppose I randomly pick an integer \( p \) between 1 and 20 inclusive. What is the probability that I choose a \( p \) such that there exists an integer \( q \) so that \( p \) and \( q \) satisfy the equation \( pq - 6p - 3q = 3 \)? Express your answer as a common fraction.
|
\frac{4}{20} = \frac{1}{5}
| 0.75 |
A ball bounces back up $\frac{3}{4}$ of the height from which it falls. If the ball is dropped from a height of $400$ cm, after how many bounces does the ball first rise less than $50$ cm?
|
8
| 0.916667 |
What is the digit in the thousandths place of the decimal equivalent of $\frac{7}{32}$?
|
8
| 0.916667 |
Let $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{d}$ be vectors such that $\|\mathbf{a}\| = 2$, $\|\mathbf{b}\| = 4$, $\|\mathbf{d}\| = 6$, and
\[
\mathbf{a} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{d}.
\]
If $\phi$ is the angle between $\mathbf{a}$ and $\mathbf{b}$, find $\sin \phi$.
|
\frac{3}{8}
| 0.75 |
Compute
$$\sum_{k=1}^{500} k(\lceil \log_{3}{k}\rceil- \lfloor\log_{3}{k} \rfloor).$$
|
124886
| 0.916667 |
In the seven-term sequence $P, Q, R, S, T, U, V$, the value of $S$ is $7$ and the sum of any three consecutive terms is $21$. What is $P+V$?
|
14
| 0.916667 |
Determine the range of the function $g(x) = \cos^4 x - \sin x \cos x + \sin^4 x$.
|
\left[0, \frac{9}{8}\right]
| 0.75 |
Let $g$ be defined by
\[g(x) = \left\{
\begin{array}{cl}
3-x & \text{ if } x \leq 2, \\
3x-x^2 & \text{ if } x>2.
\end{array}
\right.\]
Calculate $g^{-1}(-4)+g^{-1}(0)+g^{-1}(5)$.
|
5
| 0.916667 |
An ellipse has its foci at $(1, 2)$ and $(4, 2)$. Given that it passes through the point $(-1, 5)$, its equation can be written in the form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\] where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $h+k.$
|
4.5
| 0.083333 |
Suppose the function $g(x)$ is defined on the domain $\{x_1, x_2, x_3\}$ such that the graph of $y = g(x)$ consists of just three points. These three points form a triangle of area $45$. What is the area of the triangle formed by the points on the graph of $y = 3g(3x)$?
|
45
| 0.916667 |
Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes ``$AB$'' and ``$CD$''. Moreover, both these two-digit primes are greater than 50 but less than 100. Find the total number of possible house numbers for Dr. Math.
|
90
| 0.583333 |
How many positive integers $N$ less than $2000$ are there such that the equation $x^{\lfloor x\rfloor} = N$ has a solution for $x$?
|
412
| 0.666667 |
In a bag, there are 6 blue chips, 5 red chips, and 4 yellow chips. One chip is drawn from the bag, then put back in, and a second chip is drawn. Determine the probability that the two selected chips are of different colors. Express your answer as a common fraction.
|
\frac{148}{225}
| 0.916667 |
Thirty tiles are numbered from 5 through 34 and are placed into box $A$. Another thirty tiles numbered 15 to 44 are placed into box $B$. One tile is drawn randomly from each box. What is the probability that the tile from box $A$ is less than 20 and the tile from box $B$ is either odd or greater than 40? Express your answer as a common fraction.
|
\frac{17}{60}
| 0.583333 |
In a certain sequence the first term is \( a_1 = 2010 \) and the second term is \( a_2 = 2011 \). Further, the values of the remaining terms are chosen so that \( 2(a_n + a_{n+1} + a_{n+2}) = 2n + 3 \) for all \( n \geq 1 \). Determine \( a_{1000} \).
|
2343
| 0.166667 |
Find the coefficient of the $x^2$ term in the expansion of the product $(2x^3 + 5x^2 - 3x)(3x^2 - 5x + 1)$.
|
20
| 0.833333 |
Let $T$ be the sum of all the real coefficients of the expansion of $(1+ix)^{2024}$. What is $\log_2(T)$?
|
1012
| 0.5 |
In a triangle with integer side lengths, one side is twice as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle?
|
77
| 0.833333 |
Circle $P$ is on the coordinate plane with its center at $(1, 2)$. One endpoint of a diameter is at $(4, 6)$. What are the coordinates of the other endpoint of this diameter? Express your answer as an ordered pair.
|
(-2, -2)
| 0.916667 |
In the World Series of baseball, the Astros are playing against the Dodgers. To win the World Series, a team must win 4 games before the other team does. If the Dodgers win each game with a probability of $\dfrac{3}{4}$ and there are no ties, what is the probability that the Astros will win the World Series but that the contest will need all seven games to be decided?
|
\frac{135}{4096}
| 0.916667 |
The longest side of a right triangle is 13 meters and one of the other sides is 5 meters. What is the area and the perimeter of the triangle?
|
30
| 0.083333 |
Sides $\overline{AB}$ and $\overline{GH}$ of a regular octagon $ABCDEFGH$ are extended to meet at point $Q$. What is the degree measure of angle $Q$?
|
90^\circ
| 0.416667 |
When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that the product of these numbers $abc = 8$.
|
\frac{7}{216}
| 0.583333 |
How many two-digit numbers are composed of two distinct digits such that one digit is twice the other?
|
8
| 0.666667 |
There are positive integers $a$ and $b$ that satisfy the system of equations\begin{align*}
\log_{10} a + 2 \log_{10} (\text{gcd}(a,b)) &= 12\\
\log_{10} b + 2 \log_{10} (\text{lcm}(a,b)) &= 42.
\end{align*}
Let $p$ be the number of (not necessarily distinct) prime factors in the prime factorization of $a$, and let $q$ be the number of (not necessarily distinct) prime factors in the prime factorization of $b$. Find $3p+2q$.
|
80
| 0.416667 |
What is the greatest common divisor of $118^2 + 227^2 + 341^2$ and $119^2 + 226^2 + 340^2$?
|
3
| 0.25 |
Let $a, b, c$ be non-negative integers less than $5$ satisfying the equations:
\begin{align*}
a + 2b + 3c &\equiv 3 \pmod{5}, \\
2a + 3b + c &\equiv 2 \pmod{5}, \\
3a + b + 2c &\equiv 1 \pmod{5}.
\end{align*}
Find the remainder when $abc$ is divided by $5$.
|
3
| 0.666667 |
Three of the following test scores are Jake's and the other three are Linda's: 80, 86, 90, 92, 95, 97. Jake's mean score is 89. What is Linda's mean score?
|
91
| 0.833333 |
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