problem
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Consider a sequence of positive real numbers where \( a_1, a_2, \dots \) satisfy
\[ a_n = 9a_{n-1} - n \]
for all \( n > 1 \). Find the smallest possible value of \( a_1 \).
|
\frac{17}{64}
| 0.416667 |
Find the number of intersection points for the graphs of the equations:
\[
y = |3x + 6|,
\]
\[
y = -|4x - 3|
\]
|
0
| 0.75 |
Isabelle has $n$ candies, where $n$ is a three-digit positive integer. If she buys 7 more, she will have an amount that is divisible by 9. If she loses 9 candies, she will have an amount divisible by 7. What is the smallest possible value of $n$?
|
128
| 0.75 |
When a class of music students arranged themselves for a group photo, they noticed that when they tried to form rows of 3, there were 2 students left over. When they attempted to form rows of 5, there were 3 students left over, and when they organized themselves in rows of 8, they had 5 students left over. What is the smallest possible number of students in the class?
|
53
| 0.75 |
Compute
$$\frac{1722^2 - 1715^2}{1731^2 - 1708^2}.$$
|
\frac{7 \times 3437}{23 \times 3439}
| 0.083333 |
Given that $$(x+y+z)(xy+xz+yz)=24$$ and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)+xyz=15$$ for real numbers $x$, $y$, and $z$, what is the value of $xyz$?
|
\frac{9}{2}
| 0.583333 |
Determine the area of the circle described by the equation \(3x^2 + 3y^2 - 15x + 9y + 27 = 0\) in terms of \(\pi\).
|
\frac{\pi}{2}
| 0.583333 |
The nine horizontal and nine vertical lines on a $6\times6$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
|
76
| 0.666667 |
It takes Jana 30 minutes to walk one mile at a steady pace. However, today she starts walking at half her usual pace for the first half of her journey (15 minutes) and doubles her usual pace for the second half (15 minutes). How far will she travel in the first 20 minutes of her walk today?
|
\frac{7}{12} \text{ miles}
| 0.75 |
In the diagram provided, three semicircles are drawn on the straight line segment $AC$ of length 6 units, where $B$ is the midpoint of $AC$. Arc $ADB$ and arc $BEC$ have a radius of 2 units each. Point $D$, point $E$, and point $F$ are the midpoints of arc $ADB$, arc $BEC$, and arc $DFE$ respectively. If arc $DFE$ is also a semicircle with a radius of 1 unit, and a square $DPEF$ inscribes within arc $DFE$ such that $P$ lies at the intersection of lines $DE$ and $BF$, calculate the area of the square $DPEF$.
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|
1
| 0.666667 |
Tom flips an unfair coin 10 times. The coin has a $\frac{1}{3}$ probability of coming up heads and a $\frac{2}{3}$ probability of coming up tails. What is the probability that he flips exactly 4 heads?
|
\frac{13440}{59049}
| 0.833333 |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(xf(y) + y) = xy^2 + f(x)\] for all $x,y \in \mathbb{R}.$
Find the number of possible values of $f(3)$, denote it by $n$. Let $s$ be the sum of all possible values of $f(3)$. Compute $n \times s.$
|
0
| 0.5 |
Given that
\[\frac{\sin^4 \beta}{\sin^2 \alpha} + \frac{\cos^4 \beta}{\cos^2 \alpha} = 1,\]
find the sum of all possible values of
\[\frac{\cos^4 \alpha}{\cos^2 \beta} + \frac{\sin^4 \alpha}{\sin^2 \beta}.\]
|
1
| 0.833333 |
In rectangle $ABCD$, $AB = 6$ and $BC = 4$. Points $F$ and $G$ are on $\overline{CD}$ such that $DF = 2$ and $GC = 1.5$. Lines $AF$ and $BG$ intersect at $E$. Calculate the area of $\triangle AEB$.
|
\frac{144}{7}
| 0.333333 |
Factor $x^2 - 6x + 9 - 49x^4$ into two quadratic polynomials with integer coefficients. Submit your answer in the form $(ax^2+bx+c)(dx^2+ex+f)$, with $a<d$.
|
(-7x^2+x-3)(7x^2+x-3)
| 0.583333 |
Calculate the result of $655_6 - 222_6 + 111_6$ in base-6.
|
544_6
| 0.416667 |
If $w$ is a complex number such that
\[
w + w^{-1} = -\sqrt{3},
\]
what is the value of
\[
w^{2011} + w^{-2011} \, ?
\]
|
\sqrt{3}
| 0.583333 |
Calculate the average value of the series $4z$, $6z$, $9z$, $13.5z$, and $20.25z$.
|
10.55z
| 0.833333 |
A rectangular prism box has dimensions length $x+3$, width $x-3$, and height $x^2+9$. Find the number of positive integer values of $x$ for which the volume of the box is less than 500 units.
|
1
| 0.25 |
How many even integers between 5000 and 9000 have all unique digits?
|
1008
| 0.5 |
Calculate the value closest to a whole number for the ratio $\frac{10^{3000} + 10^{3003}}{10^{3001} + 10^{3002}}$.
|
9
| 0.916667 |
A 12-foot by 15-foot floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. Calculate the total area of the floor that is shaded.
|
180 - 45\pi \text{ square feet}
| 0.916667 |
Let $a$, $b$, and $c$ be positive integers such that $\gcd(a,b) = 120$ and $\gcd(a,c) = 1001$. If $b = 120x$ and $c = 1001y$ for integers $x$ and $y$, what is the smallest possible value of $\gcd(b,c)$?
|
1
| 0.75 |
Circle $C$ has a circumference of $18\pi$ meters, and segment $AB$ is a diameter. If the measure of angle $CAB$ is $60^{\circ}$, what is the length, in meters, of segment $AC$?
|
9
| 0.916667 |
Triangles $ABC$ and $AFG$ have areas $3009$ and $9003,$ respectively, with $B=(0,0), C=(331,0), F=(800,450),$ and $G=(813,463).$ What is the sum of all possible $x$-coordinates of $A$?
|
1400
| 0.166667 |
Given that $b$ is an even multiple of $1177$, find the greatest common divisor of $3b^2 + 34b + 76$ and $b + 14$.
|
2
| 0.75 |
Let
\[\mathbf{C} = \begin{pmatrix} 3 & 1 \\ -4 & -1 \end{pmatrix}.\]
Compute $\mathbf{C}^{50}.$
|
\begin{pmatrix} 101 & 50 \\ -200 & -99 \end{pmatrix}
| 0.083333 |
One line is parameterized by
\[\begin{pmatrix} -2 + s \\ 4 - ks \\ 2 + ks \end{pmatrix}.\]
Another line is parameterized by
\[\begin{pmatrix} t/3 \\ 2 + t \\ 3 - t \end{pmatrix}.\]
Determine the value of \(k\) for which the lines are coplanar.
|
-3
| 0.833333 |
A triangle is made of wood sticks of lengths 12, 25, and 30 inches joined end-to-end. Pieces of the same integral length are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. What is the length of the smallest piece that can be cut from each of the three sticks to make this happen?
|
7
| 0.583333 |
Andrew's grandfather is fifteen times older than Andrew. If Andrew's grandfather was 60 years old when Andrew was born, how old is Andrew now?
|
\frac{30}{7}
| 0.833333 |
Add $537_{8} + 246_{8}$. Express your answer in base $8$, and then convert your answer to base $16$, using alphabetic representation for numbers $10$ to $15$ (e.g., $A$ for $10$, $B$ for $11$, etc.).
|
205_{16}
| 0.583333 |
Define a $\textit{better word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ — some of these letters may not appear in the sequence — where $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is never immediately followed by $A$. How many seven-letter $\textit{better words}$ are there?
|
2916
| 0.833333 |
Let $a$, $b$, and $c$ be solutions of the equation $x^3 - 6x^2 + 11x = 12$.
Compute $\frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b}$.
|
-\frac{23}{12}
| 0.583333 |
Let \(A\) and \(B\) be two points on the parabola \(y = 4x^2\), such that the tangents at \(A\) and \(B\) intersect at an angle of \(45^\circ\). Determine the \(y\)-coordinate of their point of intersection, \(P\).
|
-\frac{1}{16}
| 0.25 |
Determine an expression for the area of $\triangle QDA$ in terms of $p$ and $q$. Given the points $Q(0, 15)$, $A(q, 15)$, $B(15, 0)$, $O(0, 0)$, and $D(0, p)$, where $q > 0$ and $p < 15$. [asy]
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|
\frac{1}{2}q(15-p)
| 0.75 |
Let $a, b, t,$ and $p$ be real numbers such that $a + b = t$ and $ab = p.$ Find the minimum value of $a^2 + ab + b^2$ given these conditions.
|
\frac{3t^2}{4}
| 0.5 |
Find the number of 8-tuples \((y_1, y_2, \dots, y_8)\) of real numbers such that
\[
(2 - y_1)^2 + (y_1 - y_2)^2 + (y_2 - y_3)^2 + \dots + (y_7 - y_8)^2 + y_8^2 = \frac{4}{9}.
\]
|
1
| 0.666667 |
Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+15|$ and $y=8-x$.
|
7
| 0.916667 |
What is the remainder when $1582 \cdot 2031$ is divided by $600$?
|
42
| 0.25 |
The Grunters play the Screamers 5 times. The Grunters are the much better team, and are $60\%$ likely to win any given game. What is the probability that the Grunters will win at least 4 games? Express your answer as a common fraction.
|
\frac{1053}{3125}
| 0.75 |
In how many different ways can five students stand in a straight line if two of the students refuse to stand next to each other, and one student must always stand at either end of the line?
|
24
| 0.416667 |
In a geometric diagram, a regular hexagon has an equilateral triangle inside it, sharing one side with the hexagon. Determine the degree measure of $\angle XYZ$, where $X$ is a vertex of the hexagon, $Y$ is a shared vertex between the triangle and hexagon, and $Z$ is the next vertex of the triangle clockwise from $Y$.
|
60^\circ
| 0.75 |
Alan, Maria, and Nick are going on a bicycle ride. Alan rides at a rate of 6 miles per hour. If Maria rides $\frac{3}{4}$ as fast as Alan, and Nick rides $\frac{4}{3}$ as fast as Maria, how fast does Nick ride?
|
6 \text{ miles per hour}
| 0.916667 |
The product of two positive integers plus their sum equals 131. The integers are relatively prime, and each is less than 25. What is the sum of these two integers?
|
21
| 0.583333 |
What is the eighth term in the arithmetic sequence $\frac{1}{2}, \frac{4}{3}, \frac{7}{6}, \dots$?
|
\frac{19}{3}
| 0.75 |
The Screamers are coached by Coach Yellsalot. The team has expanded to 15 players, including Bob, Yogi, and Moe. No starting lineup of 5 players can contain any two among Bob, Yogi, and Moe. How many possible starting lineups can Coach Yellsalot assemble under these new conditions?
|
2277
| 0.833333 |
Let \( h(x) \) be a monic quintic polynomial such that \( h(-2) = -4 \), \( h(1) = -1 \), \( h(-3) = -9 \), \( h(3) = -9 \), and \( h(5) = -25 \). Find \( h(0) \).
|
-90
| 0.833333 |
Find all positive integer values of $k$ for which the equation ${k}x^2 + 24x + 4k = 0$ has rational solutions.
|
6
| 0.666667 |
**
Triangle $DEF$ has vertices $D(0, 10)$, $E(4, 0)$, and $F(10, 0)$. A line through $E$ cuts the area of $\triangle DEF$ in half. Find the sum of the slope and $y$-intercept of this line.
**
|
-15
| 0.25 |
Evaluate the expression $81^{\frac{1}{4}} + 256^{\frac{1}{2}} - 49^{\frac{1}{2}}$.
|
12
| 0.833333 |
If $y < 0$, find the range of all possible values of $y$ such that $\lceil y \rceil \cdot \lfloor y \rfloor = 132$. Express your answer using interval notation.
|
(-12, -11)
| 0.083333 |
The base of a triangular sheet of paper $ABC$ is $15 \text{ cm}$ long. The triangle is folded over its base, with the crease $DE$ parallel to the base, resulting in a portion projecting below the base. The area of this projected triangle is $25\%$ of the area of the original triangle $ABC$. Determine the length of $DE$.
|
7.5 \text{ cm}
| 0.75 |
If $24^a = 2$ and $24^b = 3$, find $8^{(1-a-b)/(2(1-b))}$.
|
2
| 0.916667 |
On the Cartesian plane where each unit is one meter, a dog is tied to a post at the point $(5,5)$ with a $12$ meter rope. How far can the dog move from the origin?
|
12 + 5\sqrt{2} \text{ meters}
| 0.166667 |
Find the least positive integer such that when its last digit is deleted, the resulting integer is 1/17 of the original integer.
|
17
| 0.666667 |
The number \( m \) is a four-digit positive integer and is the product of the three distinct prime factors \( x \), \( y \), and \( 10x + y \), where \( x \) and \( y \) are prime numbers less than 10, and \( x > y \). What is the largest possible value of \( m \)?
|
1533
| 0.583333 |
Clark writes down all the six-digit numbers that contain each of the digits 1, 2, 3, 4, 5, and 6 exactly once. What is the smallest number in Clark's list that is divisible by 6?
|
123456
| 0.583333 |
Find the area contained by the graph of
\[|x + y| + |x - y| \le 6.\]
|
36
| 0.833333 |
If $a$ is a multiple of $2142$, find the greatest common divisor of $a^2+11a+28$ and $a+6$.
|
2
| 0.833333 |
An ellipse has foci at $(1, 1)$ and $(7, 1)$, and it passes through the point $(0, 8)$. Given this, we can write the equation of the ellipse in standard form as
\[
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,
\]
where \(a, b, h, k\) are constants, and \(a\) and \(b\) are positive. Find the ordered quadruple \((a, b, h, k)\).
|
(6\sqrt{2}, 3\sqrt{7}, 4, 1)
| 0.75 |
A, B, C, D, and E are collinear in that order such that \( AB = BC = 2, CD = 3, \) and \( DE = 5 \). If P can be any point in space, what is the smallest possible value of \( AP^2 + BP^2 + CP^2 + DP^2 + EP^2 \)?
|
88
| 0.833333 |
For what base is the representation of $157_{10}$ a four-digit number whose final digit is odd?
|
4
| 0.583333 |
Cynthia has deposited $\$9,\!000$ into an account that pays $5\%$ interest compounded annually.
David has deposited $\$12,\!000$ into an account that pays $4\%$ simple annual interest.
After 25 years, Cynthia and David compare their respective balances. To the nearest dollar, what is the positive difference between their balances?
|
\$6,\!477
| 0.083333 |
Compute
\[
\sum_{j = 0}^\infty \sum_{k = 0}^\infty 2^{-3k - j - (k + j)^2 - 2}.
\]
|
\frac{1}{3}
| 0.083333 |
Let $k$ and $m$ be real numbers, and suppose that the roots of the equation \[x^3 - 9x^2 + kx - m = 0\] are three distinct positive integers. Compute $k + m.$
|
50
| 0.333333 |
Our club has 24 members, 12 boys and 12 girls. In how many ways can we choose a president, a vice-president, and a secretary if the president must be a girl, the vice-president must be a boy, and the secretary must be of different gender from the vice-president?
|
1584
| 0.333333 |
If $f^{-1}(g(x))=x^4-1$ and $g$ has an inverse, find $g^{-1}(f(15))$.
|
2
| 0.833333 |
In a circle with center $O$, $AD$ is a diameter, $ABC$ is a chord, $BO = 6$, and $\angle ABO = \text{arc } CD = 45^\circ$. Find the length of $BC$.
|
6
| 0.416667 |
In the given 4x4 square board, some squares are divided into triangles. Specifically, the top left 2x2 region is composed of four unit squares, each diagonally divided into two triangles, and the bottom right 2x2 region is divided the same way. One half of each divided square in the top left and bottom right squares is shaded. Calculate the fraction of the entire board that is shaded.
|
\frac{1}{4}
| 0.583333 |
John ran a total of $1732_7$ miles. Convert this distance from base seven to base ten.
|
709
| 0.916667 |
Find the minimum value of \[\frac{4x^2}{x - 10}\] for \(x > 10\) where \(x\) is an integer.
|
160
| 0.916667 |
In a pentagon $FGHIJ$, two interior angles $F$ and $G$ are $90^\circ$ and $70^\circ$, respectively. Angles $H$ and $I$ are equal, and the fifth angle $J$ is $20^\circ$ more than twice $H$. Find the measure of the largest angle.
|
200^\circ
| 0.75 |
A circle is tangent to the lines $5x - 2y = 20$ and $5x - 2y = -40.$ The center of the circle lies on the line $3x + y = 0.$ Find the center of the circle.
|
\left(-\frac{10}{11}, \frac{30}{11}\right)
| 0.583333 |
For all real numbers \(x\) except \(x = 0\) and \(x = 2\), the function \(g(x)\) is defined by
\[g\left(\frac{x}{x - 2}\right) = \frac{1}{x}.\]
Suppose \(0 \leq t \leq \frac{\pi}{2}\). What is the value of \(g(\csc^2 t)\)?
|
\frac{\cos^2 t}{2}
| 0.416667 |
Let $p,$ $q,$ $r,$ $s$ be real numbers such that
\[\frac{(p - q)(r - s)}{(q - r)(s - p)} = \frac{3}{7}.\]Find the sum of all possible values of
\[\frac{(p - r)(q - s)}{(p - q)(r - s)}.\]
|
-\frac{4}{3}
| 0.083333 |
Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-4x+3|$ and $y=\frac{25}{4}-x$.
|
3
| 0.916667 |
In the given diagram, lines \( p \) and \( q \) are parallel, and the transversal \( r \) intersects them. If angle \( a \) formed by \( r \) and \( q \) is \( 45^\circ \) and the angle \( b \) immediately adjacent to it on the same side of the transversal is \( 120^\circ \), find the measure of angle \( x \) which is formed on the inside of lines \( p \) and \( q \) and opposite to \( b \) across line \( p \). Assume all intersections form as shown in the following diagram:
```text
|p
|
| / r
| /
| /
| /
| /b
|--------x---------
|q /a
```
|
120^\circ
| 0.416667 |
What is the smallest positive integer $n$ such that $5n \equiv 1846 \pmod{26}?$
|
26
| 0.833333 |
The function $f$ satisfies
\[
f(x) + f(3x+y) + 7xy = f(4x - y) + 3x^2 + 2y + 3
\]
for all real numbers $x, y$. Determine the value of $f(10)$.
|
-37
| 0.083333 |
If $f^{-1}(g(x)) = x^2 - 2$ and $g$ has an inverse, find $g^{-1}(f(10))$.
|
2\sqrt{3}
| 0.5 |
If $a = \log 25$ and $b = \log 36,$ compute
\[5^{a/b} + 6^{b/a}.\]
|
11
| 0.833333 |
Let \( A \) equal the number of four-digit odd numbers divisible by 3, and \( B \) equal the number of four-digit multiples of 7. Find \( A+B \).
|
2786
| 0.666667 |
The graph of $y = f(x)$ is described as follows:
1. From $x = -3$ to $x = 0$, $y = -2 - x$.
2. From $x = 0$ to $x = 2$, $y = \sqrt{4 - (x - 2)^2} - 2$.
3. From $x = 2$ to $x = 3$, $y = 2(x - 2)$.
Consider the graph of $y = g(x)$ obtained by reflecting $f(x)$ across the $y$-axis and shifting it 4 units to the right and 3 units up. Define $g(x)$ in terms of $f(x)$.
|
g(x) = f(4 - x) + 3
| 0.333333 |
How many ways are there to arrange the letters of the word $\text{ZOO}_1\text{M}_1\text{O}_2\text{M}_2\text{O}_3$, in which the three O's and the two M's are considered distinct?
|
5040
| 0.083333 |
Calculate the product of $0.\overline{08}$ and $0.\overline{36}$ and express it as a common fraction.
|
\frac{32}{1089}
| 0.75 |
A hexagon is constructed by connecting the points (0,0), (1,4), (3,4), (4,0), (3,-4), and (1,-4) on a coordinate plane. Calculate the area of this hexagon.
|
24
| 0.333333 |
Let $b_n = \frac{7^n - 1}{6} $. Define $e_n$ as the greatest common divisor of $b_n$ and $b_{n+1}$. What is the maximum possible value that $e_n$ can take on?
|
1
| 0.666667 |
A puppy, two cats, and a rabbit together weigh 40 pounds. The sum of the squares of the weights of the puppy and the larger cat is four times the weight of the smaller cat, and the sum of the squares of the weights of the puppy and the smaller cat equals the square of the weight of the larger cat. How much does the puppy weigh?
|
\sqrt{2}
| 0.25 |
Suppose that a real number $y$ satisfies \[\sqrt{64-y^2}-\sqrt{36-y^2}=4.\] What is the value of $\sqrt{64-y^2}+\sqrt{36-y^2}$?
|
7
| 0.833333 |
A ball bounces back up $\frac{3}{4}$ of the height from which it falls. If the ball is dropped from a height of $256$ cm, after how many bounces does the ball first rise less than $30$ cm?
|
8
| 0.5 |
The side of a square has the length $(x-4)$, while a rectangle has a length of $(x-5)$ and a width of $(x+6)$. If the area of the rectangle is three times the area of the square, what is the sum of the possible values of $x$?
|
12.5
| 0.166667 |
What is the greatest common divisor of $114^2 + 226^2 + 338^2$ and $113^2 + 225^2 + 339^2$?
|
1
| 0.583333 |
The expansion of $(x+1)^n$ has 3 consecutive terms with coefficients in the ratio $2:3:4$ that can be expressed in the form \[{n\choose k} : {n\choose k+1} : {n \choose k+2}\]. Find the sum of all possible values of $n+k$.
|
47
| 0.833333 |
Let $f(x) = \displaystyle \frac{1}{2ax + 3b}$ where $a$ and $b$ are nonzero constants. Find all solutions to $f^{-1}(x) = -1$. Express your answer in terms of $a$ and/or $b$.
|
x = \frac{1}{-2a + 3b}
| 0.083333 |
Convert $312_{10}$ to base 3. Let $x$ be the number of zeros, $y$ be the number of ones, and $z$ be the number of twos in base 3. What is the value of $z-y+x$?
|
2
| 0.416667 |
Suppose in a right triangle \(PQR\) with \(\angle PQR = 90^\circ\), it is given that \(\cos Q = \frac{3}{5}\). What is the length of \(PR\) if \(QR = 3\)?
|
5
| 0.416667 |
Calculate the mean height of the players on Oak Ridge High School boys' basketball team presented in a modified stem and leaf plot. (Note: “6|2” represents 62 inches.)
Height of the Players on the Basketball Team (inches)
$5|8\; 9$
$6|0\;1\;2\;3\;5\;5\;8$
$7|0\;1\;4\;6\;8\;9$
$8|1\;3\;5\;6$
|
70.74 \text{ inches}
| 0.166667 |
Fifty slips are placed into a hat, each bearing a number from 1 to 10, with each number entered on five slips. Five slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all five slips bear the same number. Let $q$ be the probability that three slips bear a number $a$ and the other two bear a number $b\ne a$. What is the value of $q/p$?
|
450
| 0.333333 |
A cake is cut into thirds and stored. On each trip to the pantry, you eat half of the current cake portion left. How much of the cake is left after 4 trips?
|
\frac{1}{16}
| 0.083333 |
Two circles of radius 3 and 4 are internally tangent to a larger circle. The larger circle circumscribes both the smaller circles. Find the area of the shaded region surrounding the two smaller circles within the larger circle. Express your answer in terms of \(\pi\).
|
24\pi
| 0.333333 |
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