problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
What is the largest integer less than $\log_3 \frac{3}{2} + \log_3 \frac{4}{3} + \cdots + \log_3 \frac{3010}{3009} + \log_3 \frac{3011}{3010}$?
|
6
| 0.833333 |
Define a new operation $@$ by
\[a @ b = \frac{a + b}{1 + ab}.\]
Compute the value of
\[1 @ (2 @ (3 @ (\dotsb @ (999 @ 1000) \dotsb))).\]
|
1
| 0.583333 |
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/37 of the original integer.
|
925
| 0.916667 |
Let $a$, $b$, $c$, $x$, $y$, and $z$ be real numbers that satisfy the three equations:
\begin{align*}
15x + by + cz &= 0 \\
ax + 25y + cz &= 0 \\
ax + by + 45z &= 0.
\end{align*}
Suppose that $a \neq 15$, $b \neq 25$ and $x \neq 0$. What is the value of
\[ \frac{a}{a - 15} + \frac{b}{b - 25} + \frac{c}{c - 45} \, ?\]
|
1
| 0.25 |
Let \( O \) be the origin and let \( (a, b, c) \) be a fixed point. A plane with the equation \( x + 2y + 3z = 6 \) 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 passing through \( A, B, C, \) and \( O \). Find
\[
\frac{a}{p} + \frac{b}{q} + \frac{c}{r}.
\]
|
2
| 0.25 |
Calculate \(\log_{10} 50 + \log_{10} 20 - \log_{10} 4\).
|
2 + \log_{10} 2.5
| 0.166667 |
Fifty cards are placed into a box, each bearing a number from 1 to 10, with each number appearing on five cards. Five cards are drawn from the box at random and without replacement. Let $p$ be the probability that all five cards bear the same number. Let $q$ be the probability that four of the cards bear a number $a$ and the fifth bears a number $b$ that is not equal to $a$. What is the value of $q/p$?
|
225
| 0.916667 |
What is the total number of digits used when the first 2500 positive even integers are written?
|
9448
| 0.416667 |
The matrix for projecting onto a certain line $\ell,$ which passes through the origin, is given by
\[\renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{9} & \frac{1}{18} & \frac{1}{6} \\ \frac{1}{18} & \frac{1}{36} & \frac{1}{12} \\ \frac{1}{6} & \frac{1}{12} & \frac{1}{4} \end{pmatrix} \renewcommand{\arraystretch}{1}.\]Find the direction vector of line $\ell.$ Enter your answer in the form $\begin{pmatrix} a \\ b \\ c \end{pmatrix},$ where $a,$ $b,$ and $c$ are integers, $a > 0,$ and $\gcd(|a|,|b|,|c|) = 1.$
|
\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}
| 0.166667 |
Determine the positive difference between the roots of the quadratic equation $5x^2 - 9x + 1 = 0$. Express the difference in the form $\frac{\sqrt{p}}{q}$, where $p$ and $q$ are integers with $p$ not divisible by the square of any prime number. Compute $p + q$.
|
66
| 0.75 |
Pirate Pete shares his treasure with Pirate Paul, but this time in a slightly different manner. Pete starts by saying, "One for me, one for you," giving himself one coin and starting Paul's pile with one coin. He then says, "Two for me, and two for you," giving himself two more coins but making Paul's pile two coins in total. The pattern continues with Pete announcing, "Three for me, three for you," and so forth, until Pete gives himself $x$ more coins making Paul’s pile $x$ coins in total. They continue this process until all the coins are distributed, with Pirate Pete having exactly three times as many coins as Pirate Paul. Determine the total number of gold coins they have.
|
20
| 0.166667 |
Find the smallest real number $c$ such that \[|x_1| + |x_2| + \dots + |x_9| \geq c|M|\] whenever $x_1, x_2, \ldots, x_9$ are real numbers such that $x_1+x_2+\cdots+x_9 = 10$ and $M$ is the median of $x_1, x_2, \ldots, x_9$.
|
c = 9
| 0.25 |
A square carpet of side length 12 feet is designed with one large shaded square and twelve smaller, congruent shaded squares. If the ratios $12:\text{S}$ and $\text{S}:\text{T}$ are both equal to 4, and $\text{S}$ and $\text{T}$ are the side lengths of the shaded squares, what is the total shaded area?
|
15.75
| 0.916667 |
Find \(89^{-1} \pmod{90}\), as a residue modulo 90. (Give an answer between 0 and 89, inclusive.)
|
89
| 0.916667 |
Suppose that \( f(x) \) and \( g(x) \) are functions which satisfy \( f(g(x)) = x^3 \) and \( g(f(x)) = x^4 \) for all \( x \ge 1 \). If \( g(81) = 81 \), compute \( [g(3)]^4 \).
|
81
| 0.75 |
Fifty slips are placed into a hat, each bearing a number from 1 to 12, with each number appearing on five slips. Five slips are drawn at random without replacement. Let $p'$ be the probability that all five slips bear the same number. Let $q'$ be the probability that three of the slips bear a number $a$ and the other two bear a number $b$ where $b \ne a$. What is the value of $q'/p'$?
|
550
| 0.416667 |
After replacing both the numerator and the denominator of a fraction with new numbers, a student incremented each by 8, resulting in a fraction value of $\frac{2}{5}$. If the original numerator was 3, what was the original denominator?
|
19.5
| 0.75 |
A region \( R \) in the complex plane is defined by:
\[ R = \{x + iy : -2 \leq x \leq 2, -2 \leq y \leq 2\}. \]
A complex number \( z = x + iy \) is chosen uniformly at random from \( R \). What is the probability that \( \left(\frac12 + \frac12i\right)z \) is also in \( R \)?
|
1
| 0.75 |
Calculate the sum of the squares of the coefficients of the expanded form of $3(x^4 + 2x^3 + 5x^2 + x + 2)$.
|
315
| 0.416667 |
The product of two consecutive page numbers is \(20,412.\) What is the sum of the two page numbers?
|
285
| 0.583333 |
I have 6 marbles numbered 1 through 6 in a bag. Suppose I take out two different marbles at random. What is the expected value of the sum of the numbers on the marbles?
|
7
| 0.916667 |
A function \( g(x) \) is defined for all real numbers \( x \). For all non-zero values \( x \), we have
\[ 3g(x) + g\left(\frac{1}{x}\right) = 7x + 6. \]
Let \( T \) denote the sum of all of the values of \( x \) for which \( g(x) = 2005 \). Compute the integer nearest to \( T \).
|
763
| 0.916667 |
Determine the numerical value of $k$ such that the equations are valid:
\[\frac{9}{x - y} = \frac{k}{x + z} = \frac{16}{z + y}.\]
|
25
| 0.916667 |
Given the system of equations
\[
4x - 2y = a,
\]
\[
6y - 12x = b.
\]
Find \(\frac{a}{b}\), assuming \(b \neq 0\).
|
-\frac{1}{3}
| 0.75 |
When $\sqrt[4]{2^5 \cdot 5^3}$ is fully simplified, the result is $a\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$?
|
252
| 0.916667 |
In quadrilateral $ABCD$, sides $\overline{AB}$ and $\overline{BC}$ both have length 13, sides $\overline{CD}$ and $\overline{DA}$ both have length 20, and the measure of angle $ADC$ is $120^\circ$. What is the length of diagonal $\overline{AC}$?
|
20\sqrt{3}
| 0.833333 |
What is the area enclosed by the graph of $|x| + |3y| = 12$?
|
96
| 0.75 |
For a finite sequence \( B = (b_1, b_2, \dots, b_{199}) \) of numbers, the Cesaro sum of \( B \) is defined to be
\[ \frac{T_1 + \cdots + T_{199}}{199}, \]
where \( T_k = b_1 + \cdots + b_k \) and \( 1 \leq k \leq 199 \).
If the Cesaro sum of the 199-term sequence \( (b_1, \dots, b_{199}) \) is 2000, what is the Cesaro sum of the 200-term sequence \( (2, b_1, \dots, b_{199}) \)?
|
1992
| 0.5 |
Determine all values of $m$ so that the domain of the function
\[ f(x) = \frac{3mx^2 - 4x + 1}{4x^2 - 3x + m} \]
is the set of all real numbers.
|
\left( \frac{9}{16}, \infty \right)
| 0.833333 |
Find the integer $n$, $0 \le n \le 12$, such that \[n \equiv 123456 \pmod{11}.\]
|
3
| 0.5 |
Simplify $(2^8 + 4^5)(2^3 - (-2)^2)^{11}$.
|
5368709120
| 0.166667 |
For the given quadrilateral, calculate the number of possible integer values for the length of the diagonal, represented by the dashed line.
[asy]
draw((0,0)--(4,4)--(10,-1)--(6,-7)--cycle,linewidth(0.7));
draw((0,0)--(10,-1),dashed);
label("6",(2,2),NW);
label("12",(7,1.5),NE);
label("14",(8, -4),SE);
label("10",(3,-3.5),SW);
[/asy]
|
11
| 0.916667 |
Dots are spaced one unit apart, horizontally and vertically. Consider a polygon formed by joining the dots at coordinates (0,0), (3,0), (6,0), (6,3), (9,3), (9,6), (6,6), (6,9), (3,9), (3,6), (0,6), (0,3), and back to (0,0). What is the number of square units enclosed by this polygon?
|
54
| 0.083333 |
The function $g$ defined by $g(x) = \frac{px+q}{rx+s}$, where $p$, $q$, $r$, and $s$ are nonzero real numbers, has the properties $g(13)=13$, $g(61)=61$, and $g(g(x))=x$ for all values except $\frac{-s}{r}$. Find the unique number that is not in the range of $g$.
|
37
| 0.083333 |
What is the base $2$ representation of $123_{10}$?
|
1111011_2
| 0.833333 |
How many solutions does the equation \[\frac{(x-1)(x-2)(x-3)\dotsm(x-120)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-120^2)} = 0\] have for $x$?
|
110
| 0.333333 |
The repeating decimal for $\frac{7}{26}$ is $0.abcdabcdabcd\ldots$ What is the value of the expression $3a - b$?
|
0
| 0.833333 |
In triangle $ABC$, $AB=AC$ and $D$ is a point on $\overline{AC}$ such that $\overline{BD}$ bisects angle $ABC$. Point $E$ is on $\overline{BC}$ such that $\overline{DE}$ bisects $\angle BDC$. Given $BD=BC$, find the measure, in degrees, of angle $BDE$.
|
36^\circ
| 0.583333 |
Let a sequence be given where $b_1 = 3$, $b_2 = 9$, and for $n \geq 3$, $b_{n+1} = b_n b_{n-1}$. Calculate $b_{20}$.
|
3^{10946}
| 0.25 |
How many pairs of positive integers $(x, y)$ satisfy the equation $x^2 - y^2 = 77$?
|
2
| 0.916667 |
Determine how many of the first $25$ rows of Pascal's triangle, excluding row $0$ and row $1$, consist entirely of even numbers.
|
4
| 0.333333 |
Factorize the expression $27x^6 - 512y^6$ and find the sum of all integer coefficients in its factorized form.
|
92
| 0.166667 |
Calculate the last three digits of $11^{30}$.
|
801
| 0.333333 |
Calculate the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$ in the diagram if line $CX$ bisects $\angle ACB$. See the diagram below:
- $BC = 36$ units
- $AC = 45$ units
- The line $CX$ is the angle bisector of $\angle ACB$.
Express your answer as a common fraction.
|
\frac{4}{5}
| 0.916667 |
What is the minimum value of the expression $x^2 + y^2 - 8x + 6y + 20$ for real $x$ and $y$?
|
-5
| 0.666667 |
The sum of two numbers is 60, and their difference is 10. Calculate their product and the square of their sum.
|
3600
| 0.166667 |
Mark has $\frac{4}{5}$ of a dollar, Carolyn has $\frac{2}{5}$ of a dollar, and Dave has $\frac{1}{2}$ of a dollar. How many dollars do they have altogether?
|
\$1.70
| 0.833333 |
Let $p$, $q$, $r$, $s$, and $t$ be positive integers with $p+q+r+s+t=2025$ and let $N$ be the largest of the sums $p+q$, $q+r$, $r+s$, and $s+t$. Determine the smallest possible value of $N$.
|
676
| 0.083333 |
Suppose \( q(x) \) is a polynomial such that \( q(x) + (2x^6 + 4x^4 + 5x^3 + 11x) = (10x^4 + 30x^3 + 40x^2 + 8x + 3) \). Express \( q(x) \) as a polynomial with the degrees of the terms in decreasing order.
|
-2x^6 + 6x^4 + 25x^3 + 40x^2 - 3x + 3
| 0.833333 |
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=101$. What is $x$?
|
50
| 0.916667 |
Given that $x$ is a multiple of $3456$, what is the greatest common divisor of $f(x)=(5x+3)(11x+2)(14x+7)(3x+8)$ and $x$?
|
48
| 0.25 |
Construct a stem-and-leaf plot and find the positive difference between the median and the mode of the data given below:
\begin{tabular}{|c|c|}
\hline
\textbf{Tens} & \textbf{Units} \\
\hline
2 & $0 \hspace{2mm} 0 \hspace{2mm} 1 \hspace{2mm} 1 \hspace{2mm} 1$ \\
\hline
3 & $4 \hspace{2mm} 4 \hspace{2mm} 5 \hspace{2mm} 5 \hspace{2mm} 7 \hspace{2mm} 9$ \\
\hline
4 & $1 \hspace{2mm} 3 \hspace{2mm} 5 \hspace{2mm} 7 \hspace{2mm} 7$ \\
\hline
\end{tabular}
|
14
| 0.833333 |
Find the area of the triangle bounded by the $y$-axis and the lines $y - 4x = -3$ and $4y + x = 16$.
|
\frac{98}{17}
| 0.916667 |
Compute
\[\sin^6 0^\circ + \sin^6 1^\circ + \sin^6 2^\circ + \dots + \sin^6 90^\circ.\]
|
\frac{229}{8}
| 0.25 |
The least common multiple of $x$, $15$, and $21$ is $105$. What is the greatest possible value of $x$?
|
105
| 0.916667 |
If $B$ is an angle such that $\tan B + \sec B = 3,$ find all possible values of $\cos B.$
|
\frac{3}{5}
| 0.916667 |
How many integers from 1 through 9999, inclusive, do not contain any of the digits 2, 3, 4, 5, or 8?
|
624
| 0.916667 |
Find the integer $n,$ $-90 \le n \le 90,$ such that $\sin n^\circ = \cos 390^\circ.$
|
60
| 0.916667 |
An infinite geometric series has common ratio \(\frac{1}{4}\) and sum \(40.\) What is the second term of the sequence?
|
\frac{15}{2}
| 0.666667 |
What is the coefficient of \(a^3b^3\) in \((a+b)^6\left(c+\dfrac{1}{c}\right)^6\)?
|
400
| 0.833333 |
Compute the exact value of the expression $\left|3\pi - | 3\pi - 10 | \right|$. Write your answer using only integers and $\pi$, without any absolute value signs.
|
6\pi - 10
| 0.916667 |
An isosceles trapezoid has bases of lengths 25 units (AB) and 13 units (CD), with the non-parallel sides (AD and BC) both measuring 13 units. How long is the diagonal AC?
|
\sqrt{494}
| 0.333333 |
How many integers from 1 through 999, inclusive, do not contain the digits 1, 2, 4, or 8?
|
215
| 0.25 |
What is the least integer greater than $\sqrt{500}$?
|
23
| 0.916667 |
Find the quadratic polynomial, with real coefficients, which has $3 + 4i$ as a root, and where the coefficient of $x$ is $8$.
|
-\frac{4}{3}x^2 + 8x - \frac{100}{3}
| 0.916667 |
Given that $a+b=2$ and $a^3+b^3=16$, find the value of $ab$.
|
-\frac{4}{3}
| 0.916667 |
What is the smallest possible value of the sum $\lvert x + 3\rvert + \lvert x + 6\rvert + \lvert x + 8\rvert$ as $x$ varies?
|
5
| 0.833333 |
How many digits are located to the right of the decimal point when $\frac{5^7}{10^5 \cdot 8}$ is expressed as a decimal?
|
8
| 0.666667 |
Copying a page costs 3.5 cents. There is a special discount of 5 cents total if you copy at least 400 pages. How many pages can you copy for $\$25$ if you meet the discount requirement?
|
715
| 0.166667 |
The graph of the equation $y = \frac{x}{x^3 + Ax^2 + Bx + C}$, where $A, B, C$ are integers, displays three vertical asymptotes at $x = -3, 0, 4$. Calculate $A + B + C$.
|
-13
| 0.916667 |
You have $7$ red shirts, $5$ blue shirts, $8$ green shirts, $10$ pairs of pants, $10$ green hats, $6$ red hats, and $7$ blue hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat without having the same color of shirts and hats?
|
3030
| 0.25 |
Suppose that $\{b_n\}$ is an arithmetic sequence such that
$$
b_1+b_2+ \cdots +b_{150}=150 \quad \text{and} \quad
b_{151}+b_{152}+ \cdots + b_{300}=450.
$$
What is the value of $b_2 - b_1$? Express your answer as a common fraction.
|
\frac{1}{75}
| 0.333333 |
How many distinct arrangements of the letters in the word "balloon" are there?
|
1260
| 0.666667 |
If $\frac{1}{8}$ of $2^{32}$ is $4^x$, then what is the value of $x$?
|
\frac{29}{2}
| 0.833333 |
$ABCDEFGHIJ$ is a rectangular prism with $AB = 2$, $BC = 3$, $AE = 4$. Find $\sin \angle GAC$.
[asy]
import three;
triple A, B, C, D, EE, F, G, H;
A = (0, 0, 0);
B = (2, 0, 0);
C = (2, 3, 0);
D = (0, 3, 0);
EE = (0, 0, 4);
F = B + EE;
G = C + EE;
H = D + EE;
draw(B--C--D--A--cycle);
draw(EE--F--G--H--EE);
draw(A--EE,dashed);
draw(B--F);
draw(C--G);
draw(D--H);
draw(dashed, G--A--C);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, NE);
label("$D$", D, NW);
label("$E$", EE, S);
label("$F$", F, E);
label("$G$", G, N);
label("$H$", H, W);
[/asy]
|
\frac{4}{\sqrt{29}}
| 0.333333 |
The function \( g(x) \) satisfies
\[
g(3^x) + (x + 1)g(3^{-x}) = 3
\]
for all real numbers \( x \). Find \( g(3) \).
|
-3
| 0.75 |
A circular cylindrical post with a circumference of 6 feet has a string wrapped around it, spiraling from the bottom of the post to the top. The string evenly loops around the post exactly three full times, starting at the bottom edge and finishing at the top edge. The height of the post is 15 feet. What is the length, in feet, of the string?
|
3\sqrt{61}
| 0.833333 |
Consider the 2004th, 2005th, and 2006th rows of Pascal's triangle. Let $(a_i)$, $(b_i)$, and $(c_i)$ represent the sequence, from left to right, of elements in these rows, respectively, with the leftmost element occurring at $i = 0.$ Compute
\[
\sum_{i = 0}^{2005} \frac{b_i}{c_i} - \sum_{i = 0}^{2004} \frac{a_i}{b_i}.
\]
|
0.5
| 0.5 |
Determine how many integers between 1 and 1200 inclusive can be expressed as the difference of the squares of two nonnegative integers.
|
900
| 0.666667 |
An ellipse has foci at \((8, 1)\) and \((8, 9)\), and it passes through the point \((17, 5)\). Write the equation of this ellipse in the standard form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,\] where \(a\), \(b\), \(h\), and \(k\) are constants, with \(a\) and \(b\) positive. Find the ordered quadruple \((a, b, h, k)\).
|
(9, \sqrt{97}, 8, 5)
| 0.25 |
Find the largest constant $m,$ so that for any positive real numbers $a,$ $b,$ $c,$ $d,$ and $e,$
\[\sqrt{\frac{a}{b + c + d + e}} + \sqrt{\frac{b}{a + c + d + e}} + \sqrt{\frac{c}{a + b + d + e}} + \sqrt{\frac{d}{a + b + c + e}} + \sqrt{\frac{e}{a + b + c + d}} > m.\]
|
2
| 0.25 |
The sum of two numbers is $24$ and their difference is $8$. What is the sum of the squares of their sum and difference?
|
640
| 0.916667 |
Find the maximum value of the expression:
\[
\cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_3 + \cos \theta_3 \sin \theta_4 + \cos \theta_4 \sin \theta_5 + \cos \theta_5 \sin \theta_6 + \cos \theta_6 \sin \theta_7 + \cos \theta_7 \sin \theta_1,
\]
over all real numbers $\theta_1,\theta_2,\theta_3,\theta_4,\theta_5, \theta_6, \theta_7$.
|
\frac{7}{2}
| 0.583333 |
The quadratic equation $x^2 + mx + n = 0$ has roots that are three times those of $x^2 + px + m = 0$, and none of $m, n, p$ is zero. Determine the value of $\frac{n}{p}$.
|
27
| 0.916667 |
Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^4$ for all $x \ge 1.$ If $g(64) = 64,$ then compute $[g(8)]^4.$
|
64
| 0.166667 |
A convex polyhedron $Q$ has $30$ vertices, $72$ edges, and $44$ faces, of which $32$ are triangular and $12$ are quadrilaterals. Determine the number of space diagonals inside $Q$.
|
339
| 0.916667 |
Alice and Bob play a similar game with a basketball. On each turn, if Alice has the ball, there is a 2/3 chance that she will toss it to Bob and a 1/3 chance that she will keep the ball. If Bob has the ball, there is a 1/4 chance that he will toss it to Alice, and a 3/4 chance that he keeps it. Alice starts with the ball. What is the probability that Alice has the ball again after two turns?
|
\frac{5}{18}
| 0.75 |
In a conference room, each row can seat either 9 or 10 people. Fifty-four people need seating with every seat filled. How many rows must seat exactly 10 people?
|
0
| 0.416667 |
In how many ways can the digits of $85,\!550$ be arranged to form a 5-digit number? (Remember, numbers cannot begin with 0.)
|
16
| 0.916667 |
What is the greatest integer less than 200 for which the greatest common factor of that integer and 72 is 9?
|
189
| 0.5 |
A box contains 8 white balls and 7 black balls. Seven balls are drawn out of the box at random. What is the probability that they all are white?
|
\dfrac{8}{6435}
| 0.666667 |
Compute
\[
\left\lfloor \frac{2011! + 2008!}{2010! + 2009!} \right\rfloor.
\]
|
2010
| 0.75 |
If $f(x) = ax + b$ and $f^{-1}(x) = bx + a + c$ where $c$ is an additional constant and $a, b, c$ are real numbers, what is the value of $a + b + c$?
|
0
| 0.333333 |
Simplify $(6^8 - 4^7) (2^3 - (-2)^3)^{10}$.
|
1663232 \cdot 16^{10}
| 0.083333 |
Let $b_1, b_2, b_3,\dots$ be an increasing arithmetic sequence of integers. If $b_4b_5 = 21$, what is $b_3b_6$?
|
-11
| 0.75 |
Find the integer \( n \), \( -180 < n < 180 \), such that \( \tan n^\circ = \tan 1500^\circ \).
|
n = 60
| 0.833333 |
In the diagram, $PQRS$ is a trapezoid with an area of $18.$ $RS$ is three times the length of $PQ.$ What is the area of $\triangle PQS?$
[asy]
draw((0,0)--(1,4)--(4,4)--(12,0)--cycle);
draw((4,4)--(0,0));
label("$S$",(0,0),W);
label("$P$",(1,4),NW);
label("$Q$",(4,4),NE);
label("$R$",(12,0),E);
[/asy]
|
4.5
| 0.583333 |
After collecting coins for a year, Sarah has a jar containing 157 quarters and 342 dimes. Mark has a jar containing 211 quarters and 438 dimes. They decide to combine their coins and roll them into rolls containing 25 quarters and 40 dimes each. Calculate the total value, in dollars, of the quarters and dimes that cannot be rolled.
|
\$6.50
| 0.916667 |
Points $A$, $B$, $C$, and $D$ are located on $\overline{AB}$ such that $AB = 4AD = 8BC$. If a point is selected at random on $\overline{AB}$, what is the probability that it is between $C$ and $D$?
|
\frac{5}{8}
| 0.916667 |
I take variable $c$, triple it, and add six. I subtract $6c$ from this new expression, and divide the resulting difference by three. What is my final expression in simplest form?
|
-c + 2
| 0.916667 |
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