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|---|---|---|
Suppose that \(x^2\) varies inversely with \(y^4\). If \(x = 10\) when \(y = 2\), find the value of \(x\) when \(y = \sqrt{8}\).
|
5
| 0.916667 |
Solve for \(x\) in the equation
\[3^{(9^x)} = 27^{(3^x)}.\]
|
1
| 0.75 |
The sequence $\{b_n\}$ is defined such that $b_1 = 2$ and $3^{b_{n + 1} - b_n} - 1 = \frac{1}{n + \frac{1}{4}}$ for $n \geq 1$. Find the least integer $k$ greater than $1$ for which $b_k$ is an integer.
|
11
| 0.583333 |
The fourth, fifth, and sixth terms of an arithmetic sequence are 3, 7, and 11, respectively. What is the sum of the first five terms of the sequence?
|
-5
| 0.166667 |
Compute $(\cos 215^\circ + i \sin 215^\circ)^{72}$.
|
1
| 0.916667 |
Given that \(\frac{1}{n} - \frac{1}{n+2} < \frac{1}{15}\), what is the least possible positive integer value of \( n \)?
|
5
| 0.916667 |
Find the point on the line
\[ y = \frac{x + 5}{2} \]
that is closest to the point $(7,2)$.
|
\left( \frac{27}{5}, \frac{26}{5} \right)
| 0.916667 |
An equilateral triangle has an area of $144\sqrt{3}$ cm². If each side of the triangle is decreased by 5 cm, by how many square centimeters is the area decreased?
|
53.75\sqrt{3} \text{ cm}^2
| 0.083333 |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots?
|
100
| 0.333333 |
Evaluate $\log_{\sqrt{10}} (1000\sqrt{10})$.
|
7
| 0.916667 |
Compute
$$\sum_{k=1}^{500} k(\lceil \log_{\sqrt{3}}{k}\rceil- \lfloor\log_{\sqrt{3}}{k} \rfloor).$$
|
124886
| 0.833333 |
Calculate the product: $\frac{8}{4} \times \frac{10}{5} \times \frac{21}{14} \times \frac{16}{8} \times \frac{45}{15} \times \frac{30}{10} \times \frac{49}{35} \times \frac{32}{16}$.
|
302.4
| 0.5 |
Allison, Derek, and Sophie each have a 6-sided cube. All the faces on Allison's cube have a 4. Derek's cube is numbered from 1 to 6 uniformly. On Sophie's cube, four of the faces have a 3, and two of the faces have a 5. All three cubes are rolled simultaneously. What is the probability that Allison's roll is greater than each of Derek's and Sophie's? Express your answer as a common fraction.
|
\frac{1}{3}
| 0.75 |
Find the remainder when \(x^{55} + x^{44} + x^{33} + x^{22} + x^{11} + 1\) is divided by \(x^5 + x^4 + x^3 + x^2 + x + 1.\)
|
0
| 0.583333 |
How many distinct triangles can be constructed by connecting three different vertices of a regular octahedron? (Two triangles are distinct if they have different locations in space.)
|
20
| 0.416667 |
Compute the smallest base-10 positive integer greater than 15 that is a palindrome when written in both base 2 and 4.
|
17
| 0.833333 |
Gretchen now has ten socks, two of each color: magenta, cyan, black, white, and purple. She randomly draws five socks this time. What is the probability that she has exactly one pair of socks with the same color and the rest are all different colors?
|
\frac{40}{63}
| 0.916667 |
What is the sum of the digits of the base-2 expression for $315_{10}$?
|
6
| 0.916667 |
Find the area of the triangle with vertices $(2, 3),$ $(0, 7),$ and $(5, 0).$
|
3
| 0.5 |
A frustum of a right circular cone is formed by slicing off a smaller cone from a larger cone. Given that the frustum has a height of $30$ centimeters, the area of its lower base is $400\pi$ sq cm, and the area of its upper base is $100\pi$ sq cm, determine the height of the small cone that was removed.
|
30 \text{ cm}
| 0.833333 |
The graph of the line $x-y=7$ is a perpendicular bisector of the line segment from $(2,4)$ to $(10,-6)$. What is the value of $b$ in the equation $x-y=b$?
|
7
| 0.916667 |
How many three-digit whole numbers have no digits that are 0, 3, 5, 7, or 9?
|
125
| 0.916667 |
A circle with center $O$ has radius $10$ units and circle $P$ has radius $4$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is the common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively. What is the length of segment $OS$? Express your answer in simplest radical form.
|
2\sqrt{65}
| 0.416667 |
What is the result when we compute the sum $$1^3 + 2^3 + 3^3 + 4^3 + \dots + 49^3+50^3$$ and the sum $$(-1)^3 + (-2)^3 + (-3)^3 + (-4)^3 + \dots + (-49)^3+(-50)^3,$$ then multiply the result of these two sums by the cube of 25?
|
0
| 0.583333 |
Twelve people sit down at random seats around a circular table. Five of them are computer scientists, four others are chemistry majors, and the remaining three are history majors. What is the probability that all five computer scientists sit in consecutive seats?
|
\frac{1}{66}
| 0.833333 |
Twelve congruent pentagonal faces, each of a different color, are used to construct a regular dodecahedron. How many distinguishable ways are there to construct this dodecahedron? (Two colored dodecahedrons are distinguishable if neither can be rotated to look just like the other.)
|
7983360
| 0.916667 |
What is $\log_{8}{1600}$ rounded to the nearest integer?
|
4
| 0.916667 |
The expression $x^2 + 18x + 77$ can be written as $(x + d)(x + e)$, and the expression $x^2 - 19x + 88$ can be written as $(x - e)(x - f)$, where $d$, $e$, and $f$ are integers. What is the value of $d + e + f$?
|
26
| 0.833333 |
Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[
g((x + y)^2) = g(x)^2 + 2xy + y^2
\]
for all real numbers \( x \) and \( y \). Determine the number, \( m \), of possible values for \( g(1) \) and their sum, \( t \). Calculate \( m \times t \).
|
1
| 0.416667 |
Simplify $\tan \frac{\pi}{8} + \tan \frac{3\pi}{8}$.
|
2\sqrt{2}
| 0.416667 |
Five years ago, there were 25 trailer homes on Maple Street with an average age of 12 years. Since then, a group of brand new trailer homes was added, and 5 old trailer homes were removed. Today, the average age of all the trailer homes on Maple Street is 11 years. How many new trailer homes were added five years ago?
|
20
| 0.25 |
A convex polyhedron \( Q \) has \( 30 \) vertices, \( 72 \) edges, and \( 44 \) faces, \( 30 \) of which are triangular and \( 14 \) of which are quadrilaterals. Determine how many space diagonals \( Q \) has.
|
335
| 0.833333 |
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 |
A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?
|
2\sqrt{61}
| 0.833333 |
How many even integers between 8000 and 9000 have four different digits, where the last digit is also a prime number?
|
56
| 0.416667 |
Determine how many trailing zeroes are at the end of the number $500!$, and additionally, find out how many times the number $500!$ is divisible by $3$.
|
247
| 0.833333 |
For a nonnegative integer $n$, let $r_9(n)$ be the remainder when $n$ is divided by $9$. Consider all nonnegative integers $n$ for which $r_9(7n) \leq 5$. Find the $15^{\text{th}}$ entry in an ordered list of all such $n$.
|
21
| 0.25 |
What are the rightmost three digits of $7^{1983} + 123$?
|
466
| 0.083333 |
Let $f(x) = x^2 - x + 2008$. Compute the greatest common divisor of $f(102)$ and $f(103)$.
|
2
| 0.916667 |
The function $f(x)$ satisfies
\[ f(xy) = \frac{f(x)}{y^2} \]
for all positive real numbers $x$ and $y$. If $f(40) = 50$, find $f(80)$.
|
f(80) = 12.5
| 0.833333 |
Find the mean of all solutions for the equation $x^3 + 5x^2 - 2x - 8 = 0$.
|
-\frac{5}{3}
| 0.916667 |
David works at a widget factory. On Monday, David produces $w$ widgets every hour for $t$ hours, knowing that $w = 2t$. Feeling more motivated on Tuesday, he works for one fewer hour and increases his production rate by $5$ widgets per hour. Determine how many more widgets David produces on Monday than he does on Tuesday.
|
-3t + 5
| 0.25 |
Find the vector $\mathbf{u}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 1 \\ 2 \end{pmatrix}} \mathbf{u} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}\]and
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ 1 \end{pmatrix}} \mathbf{u} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}.\]
|
\begin{pmatrix} 6 \\ 2 \end{pmatrix}
| 0.916667 |
Find the distance between the planes \(x + 2y - z = 3\) and \(2x + 4y - 2z = 6\).
|
0
| 0.916667 |
Three dice, each with eight faces numbered 1 through 8, are tossed. What is the probability that the sum of the numbers shown on the top faces is even? Express your answer as a common fraction.
|
\frac{1}{2}
| 0.916667 |
Find the remainder when $(abc+abd+acd+bcd)(abcd)^{-1}$ is divided by $10$, where $a, b, c, d$ are distinct positive integers less than $10$ and are invertible modulo $10$.
|
0
| 0.916667 |
What is the median number of moons per celestial body? (Include Ceres, a dwarf planet, along with the traditional planets and Pluto.) Use the following data:
\begin{tabular}{c|c}
Celestial Body & $\#$ of Moons \\
\hline
Mercury & 0 \\
Venus & 0 \\
Earth & 1 \\
Mars & 2 \\
Jupiter & 20 \\
Saturn & 25 \\
Uranus & 17 \\
Neptune & 3 \\
Pluto & 5 \\
Ceres & 0 \\
\end{tabular}
|
2.5
| 0.5 |
Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles? Express your answer in fully expanded form in terms of $\pi$.
|
\frac{9\pi}{2} - 9
| 0.916667 |
When two fair 9-sided dice are tossed, each with faces showing numbers from 1 to 9, the numbers $a$ and $b$ are obtained. What is the probability that the two-digit number $ab$ (where $a$ and $b$ are digits) and both $a$ and $b$ are divisible by 4?
|
\frac{4}{81}
| 0.666667 |
Let $\mathbf{A}$ be a $2 \times 2$ matrix with real entries such that $\mathbf{A}^4 = \mathbf{0}$. Determine the number of different possible matrices that $\mathbf{A}^2$ can be. If you think the answer is infinite, then enter "infinite".
|
1
| 0.916667 |
What is the total number of digits used when the first 2500 positive even integers are written?
|
9448
| 0.416667 |
The value of $z$ varies inversely as $x^2$ and when $x=3$, $z=2$. What is $x$ when $z=8$?
|
\frac{3}{2}
| 0.333333 |
Let $g(x) = 3x^6 - 2x^4 + 5x^2 - 7.$ If $g(-3) = 9,$ find $g(3)$.
|
9
| 0.916667 |
Berengere and her American friend Emily are at a café in Paris trying to buy a pastry. The pastry costs 8 euros. Emily has seven American dollars with her. How many euros must Berengere contribute if the exchange rate is now 1 euro = 1.1 USD?
|
1.64\text{ euros}
| 0.833333 |
Let
\[g(x) =
\begin{cases}
\frac{x}{3} &\quad \text{if } x \text{ is divisible by } 3, \\
x^2 + 2 &\quad \text{if } x \text{ is not divisible by } 3.
\end{cases}
\]
What is $g(g(g(g(3))))$?
|
3
| 0.75 |
The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $25$. After increasing each vertex's $x$-coordinate by $10\%$, find the sum of the $x$-coordinates of the midpoints of the sides of the triangle.
|
27.5
| 0.666667 |
Evaluate the polynomial
\[ x^3 - 3x^2 - 9x + 7, \]
where $x$ is the positive number such that $x^2 - 3x - 9 = 0$.
|
7
| 0.916667 |
Jane has received the following scores on her quizzes: 98, 97, 92, 85, 93, 88, and 82. What is her mean score now?
|
90.71
| 0.916667 |
Evaluate $\lceil-3.7\rceil$.
|
-3
| 0.916667 |
A pyramid with a volume of 60 cubic inches has a square base. If the side length of the base is tripled and the height is decreased by $25\%$, what is the volume of the modified pyramid, in cubic inches?
|
405
| 0.5 |
Lark has forgotten her bike lock combination. It’s a sequence of four numbers, each in the range from 1 to 40, inclusive. She remembers the following: the first number is odd, the second number is a prime number less than 30, the third number is a multiple of 4, and the fourth number is a perfect square. How many combinations could possibly be Lark's?
|
12000
| 0.5 |
Calculate the value of the expression $(45 + 18)^2 - (45^2 + 18^2 + 10)$.
|
1610
| 0.666667 |
What is the sum of all two-digit primes that are greater than 20 but less than 99, and are still prime when their two digits are interchanged?
|
388
| 0.25 |
Graphs of $$y \equiv 8x + 3 \pmod{20}$$ and $$y \equiv 14x + 15 \pmod{20}$$ on modulo $20$ graph paper intersect at some points. Determine the sum of all distinct $x$-coordinates of these points.
|
26
| 0.416667 |
In Pascal's Triangle, if the sum of the interior numbers in the sixth row is 30, what is the sum of the interior numbers in the eighth row?
|
126
| 0.416667 |
Let $r,$ $s,$ and $t$ be the roots of the equation $x^3 - 15x^2 + 26x - 8 = 0.$ Find the value of $(1+r)(1+s)(1+t).$
|
50
| 0.666667 |
If $m$ and $n$ are positive 3-digit integers such that $\gcd(m,n)=5$, what is the smallest possible value for $\mathop{\text{lcm}}[m,n]$?
|
2100
| 0.833333 |
What number must we add to $5 - 3i$ after multiplying it by 2, to get $4 + 11i$?
|
-6 + 17i
| 0.583333 |
When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of seven consecutive positive integers all of which are nonprime?
|
97
| 0.333333 |
What is the smallest number of whole 3-by-4 rectangles needed to completely cover a square region exactly, with no overlaps and without any portion of the square uncovered?
|
12
| 0.333333 |
A bubble assumes a semi-spherical shape when it rests on a wet, horizontal table. The volume of the hemispherical bubble is the same as the original spherical bubble. If the hemisphere's radius is \( 5\sqrt[3]{2} \) cm after it settles, calculate the radius of the original spherical bubble.
|
5
| 0.916667 |
What is the greatest common divisor (gcd) of $9125$, $4257$, and $2349$?
|
1
| 0.916667 |
A curious historian is hosting a puzzle where participants must guess the year a certain artifact was discovered. The year is made up of the six digits 1, 1, 1, 5, 8, and 9, and the year must start with an even digit. How many different possibilities are there for the artifact's discovery year?
|
20
| 0.5 |
Let there exist vectors $\mathbf{u}$ and $\mathbf{v}$, and a scalar $d$ such that
\[\mathbf{i} \times ((\mathbf{u} + \mathbf{v}) \times \mathbf{i}) + \mathbf{j} \times ((\mathbf{u} + \mathbf{v}) \times \mathbf{j}) + \mathbf{k} \times ((\mathbf{u} + \mathbf{v}) \times \mathbf{k}) = d (\mathbf{u} + \mathbf{v}).\]
Determine the value of $d.$ Assume $\mathbf{u}$ is a constant vector and $\mathbf{v}$ can be any vector.
|
2
| 0.333333 |
Find the matrix $\mathbf{Q}$ such that:
\[\mathbf{Q} \mathbf{v} = -3 \mathbf{v}\]
for all vectors $\mathbf{v}.$
|
\begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix}
| 0.333333 |
Po is challenged with a new equation: $$64x^2+80x-72 = 0.$$ He aims to rewrite the equation in the form: $$(ax + b)^2 = c,$$ where $a$, $b$, and $c$ are integers and $a > 0$. Determine the value of $a + b + c$.
|
110
| 0.583333 |
Jenny places a total of 20 red Easter eggs in several green baskets and a total of 30 orange Easter eggs in some blue baskets. Each basket contains the same number of eggs and there are at least 5 eggs in each basket. How many eggs did Jenny put in each basket?
|
5
| 0.083333 |
How many positive factors of 120 that are perfect cubes?
|
2
| 0.75 |
Suppose I have 8 shirts, 5 pairs of pants, 4 ties, and 2 hats. An outfit requires one shirt and one pair of pants. Additionally, the outfitter can choose to wear one of the ties or no tie at all, and can also select whether to wear one of the hats or not. How many different outfits can I create?
|
600
| 0.833333 |
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. 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, and the center triangle must not be red, how many distinguishable large equilateral triangles can be constructed?
|
840
| 0.166667 |
Maria borrowed money from her friend to buy a bicycle. Her friend has agreed to let her work off her debt by washing cars under the following conditions: her first hour of work is worth $\$2$, the second hour worth $\$4$, the third hour $\$6$, the fourth hour $\$8$, the fifth hour $\$10$, the sixth hour $\$12$, the seventh hour $\$2$, the eighth hour $\$4$, etc. If she repays her debt by working for 48 hours, how many dollars did she borrow?
|
\$336
| 0.75 |
What multiple of 18 is closest to 2500?
|
2502
| 0.75 |
Determine the sum of all real numbers \(x\) satisfying
\[(x^2 - 6x + 5)^{x^2 - 8x + 12} = 1.\]
|
14
| 0.75 |
Multiply $(3x^4 - 4z^3)(9x^8 + 12x^4z^3 + 16z^6)$.
|
27x^{12} - 64z^9
| 0.833333 |
A $180\times 360\times 450$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes?
|
720
| 0.166667 |
Let $x$ and $y$ be two positive real numbers such that $x + y = 50.$ Find the ordered pair $(x,y)$ for which $x^6 y^3$ is maximized.
|
\left(\frac{100}{3}, \frac{50}{3}\right)
| 0.583333 |
If $\sqrt{8 + x} + \sqrt{25 - x} = 8$, what is the value of $(8 + x)(25 - x)$?
|
\frac{961}{4}
| 0.666667 |
Let $p$ be the smallest positive, four-digit integer congruent to 3 (mod 13). Let $q$ be the smallest positive, four-digit integer congruent to 3 (mod 7). Compute the absolute difference $|p - q|$.
|
0
| 0.916667 |
How many positive four-digit integers of the form $\_\_25$ are divisible by 25?
|
90
| 0.833333 |
In a tennis best-of-five series, a player needs to win 3 matches to claim the series. Assume Player A wins each match with a probability of $\frac{3}{4}$, and there are no draws. What is the probability that Player B will win the series, but the series will go to the full five matches?
|
\frac{27}{512}
| 0.833333 |
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 formed by $\mathbf{a},$ $\mathbf{b} - \mathbf{a} \times \mathbf{b},$ and $\mathbf{b}.$
|
\frac{1}{2}
| 0.75 |
Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles? Express your answer in fully expanded form in terms of $\pi$.
|
\frac{9\pi}{2} - 9
| 0.916667 |
Roberto has four pairs of trousers, eight shirts, three jackets, and five pairs of shoes. He can wear any jacket with any shirt but has specific preferences for jackets with trousers: each jacket can only be worn with two specific pairs of trousers. How many different outfits can Roberto put together if an outfit consists of a pair of trousers, a shirt, a jacket, and a pair of shoes?
|
240
| 0.833333 |
The polynomial $h(x) = x^3 - 2x^2 + 4x - 1$ has three roots. Let $j(x) = x^3 + px^2 + qx + r$ be a cubic polynomial with a leading coefficient of $1$ such that the roots of $j(x)$ are two less than the roots of $h(x)$. Find the ordered triple $(p, q, r)$.
|
(4, 8, 7)
| 0.25 |
The Maplewood Youth Soccer Team consists of 25 players, including four who are goalies. During a special drill, each goalie takes a turn in the net while the other 24 players (including the remaining goalies) attempt penalty kicks.
Calculate the total number of penalty kicks that must be taken so each player has the opportunity to take a penalty kick against each goalie.
|
96
| 0.583333 |
Define a $\emph{brilliant integer}$ as an even integer that is greater than 20, less than 120, and such that the sum of its digits is 10. What fraction of all brilliant integers is divisible by 15? Express your answer as a common fraction.
|
0
| 0.75 |
A whole number is called ''12-heavy'' if the remainder when the number is divided by 12 is greater than 8. What is the least three-digit 12-heavy whole number?
|
105
| 0.916667 |
Sheila has been invited to a picnic tomorrow. The picnic will occur, rain or shine. If it rains, there is a 30% probability that Sheila will decide to go, but if it is sunny, there is a 70% probability that she will decide to go. Additionally, Sheila can only go if her friend agrees to drive. There is a 50% chance her friend will agree to drive. The forecast for tomorrow states that there is a 50% chance of rain. What is the probability that Sheila will attend the picnic? Express your answer as a percent.
|
25\%
| 0.75 |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?
|
\sqrt{55}
| 0.166667 |
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 |
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