problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
Let \( A B C D E F \) be a regular hexagon, and let \( P \) be a point inside quadrilateral \( A B C D \). If the area of triangle \( P B C \) is 20, and the area of triangle \( P A D \) is 23, compute the area of hexagon \( A B C D E F \). | 189 | 0.5 | 6,728.8125 | 6,158 | 7,299.625 | |
Given the sequence $\{a\_n\}$, if $a\_1=0$ and $a\_i=k^2$ ($i \in \mathbb{N}^*, 2^k \leqslant i < 2^{k+1}, k=1,2,3,...$), find the smallest value of $i$ that satisfies $a\_i + a_{2i} \geq 100$. | 128 | 0.125 | 7,871.8125 | 6,296 | 8,096.928571 | |
If for any three distinct numbers $a$, $b$, and $c$ we define $f(a,b,c)=\frac{c+a}{c-b}$, then $f(1,-2,-3)$ is | 2 | 1. We are given the function \( f(a, b, c) = \frac{c+a}{c-b} \) and need to evaluate \( f(1, -2, -3) \).
2. Substitute \( a = 1 \), \( b = -2 \), and \( c = -3 \) into the function:
\[
f(1, -2, -3) = \frac{-3 + 1}{-3 - (-2)}.
\]
3. Simplify the numerator and the denominator:
\[
\frac{-3 + 1}{-3 + 2} = ... | 1 | 1,763.6875 | 1,763.6875 | -1 |
Given \( x = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{10^{6}}} \), calculate the value of \([x]\). | 1998 | 0.75 | 7,358.4375 | 7,080.583333 | 8,192 | |
When Frederick was born, his grandparents gave him a gift of $\$2000$, which was invested at an interest rate of $5\%$ per year, compounded yearly. How much money will Frederick have when he collects the money at the age of $18$? Give your answer to the nearest hundredth of a dollar. | \$4813.24 | 0 | 8,107.375 | -1 | 8,107.375 | |
Given $\tan \theta =2$, find $\cos 2\theta =$____ and $\tan (\theta -\frac{π}{4})=$____. | \frac{1}{3} | 1 | 2,183.875 | 2,183.875 | -1 | |
If the following instructions are carried out by a computer, what value of \(X\) will be printed because of instruction \(5\)?
1. START \(X\) AT \(3\) AND \(S\) AT \(0\).
2. INCREASE THE VALUE OF \(X\) BY \(2\).
3. INCREASE THE VALUE OF \(S\) BY THE VALUE OF \(X\).
4. IF \(S\) IS AT LEAST \(10000\),
TH... | 23 | 1. **Initialization**: The program starts with $X = 3$ and $S = 0$.
2. **Loop Execution**:
- **Instruction 2**: Increase the value of $X$ by $2$.
- **Instruction 3**: Increase the value of $S$ by the current value of $X$.
- **Instruction 4**: Check if $S \geq 10000$. If true, go to instruction 5; otherwise, ... | 0 | 5,269.5 | -1 | 5,269.5 |
Given triangle ABC, the sides opposite to angles A, B, and C are denoted as a, b, and c, respectively, and a = 6. Find the maximum value of the area of triangle ABC given that $\sqrt{7}bcosA = 3asinB$. | 9\sqrt{7} | 0.375 | 7,273.8125 | 5,743.5 | 8,192 | |
Team X and team Y play a series where the first team to win four games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team Y wins the third game and team X wins the series, what is the probability that team Y wins the first ... | \frac{5}{12} | 0 | 8,192 | -1 | 8,192 | |
What is the mean value of the measures of the three interior angles of any triangle? | 60^\circ | 1 | 1,228.125 | 1,228.125 | -1 | |
How many sequences of 6 digits $x_1, x_2, \ldots, x_6$ can we form, given the condition that no two adjacent $x_i$ have the same parity? Leading zeroes are allowed. (Parity means 'odd' or 'even'; so, for example, $x_2$ and $x_3$ cannot both be odd or both be even.) | 31,250 | 0 | 5,707.9375 | -1 | 5,707.9375 | |
Find the largest odd natural number that cannot be expressed as the sum of three distinct composite numbers. | 17 | 0 | 8,192 | -1 | 8,192 | |
A positive integer \( n \) cannot be divided by \( 2 \) or \( 3 \), and there do not exist non-negative integers \( a \) and \( b \) such that \( |2^a - 3^b| = n \). Find the smallest value of \( n \). | 35 | 0.0625 | 8,061.625 | 6,106 | 8,192 | |
How many integers are between $(10.2)^3$ and $(10.3)^3$? | 31 | 1 | 3,771.125 | 3,771.125 | -1 | |
Rectangle $ABCD$ has area $2006.$ An ellipse with area $2006\pi$ passes through $A$ and $C$ and has foci at $B$ and $D$. What is the perimeter of the rectangle? | 8\sqrt{1003} | 0.625 | 5,932.625 | 4,577 | 8,192 | |
What is the sum of the largest three-digit multiple of 4 and the smallest four-digit multiple of 3? | 1998 | 0.6875 | 1,273.25 | 1,477.545455 | 823.8 | |
The Happy Valley Kennel has 4 chickens, 2 dogs, and 5 cats. (Some people in Happy Valley like to keep chickens as pets!) In how many ways can the 11 animals be placed in a row of 11 cages, such that all of the animals of each type are in adjacent cages? (Two animals of the same species are considered distinguishable... | 34,\!560 | 0 | 4,868.4375 | -1 | 4,868.4375 | |
Given a regular tetrahedron with vertices $A, B, C$, and $D$ and edge length 1, where point $P$ lies on edge $AC$ one third of the way from vertex $A$ to vertex $C$, and point $Q$ lies on edge $BD$ one third of the way from vertex $B$ to vertex $D$, calculate the least possible distance between points $P$ and $Q$. | \frac{2}{3} | 0 | 8,004.1875 | -1 | 8,004.1875 | |
The lengths of the sides of a triangle are integers, and its area is also an integer.
One side is $21$ and the perimeter is $48$. The shortest side is: | 10 | 1. **Assign Variables to Sides and Perimeter**:
Let the sides of the triangle be $a = 21$, $b$, and $c$. Given that the perimeter is $48$, we have:
\[
a + b + c = 48
\]
Substituting $a = 21$ into the equation, we get:
\[
21 + b + c = 48 \implies b + c = 27
\]
Therefore, we can express $c$ in ... | 0.5625 | 6,855.1875 | 5,815.444444 | 8,192 |
Michael read on average 30 pages each day for the first two days, then increased his average to 50 pages each day for the next four days, and finally read 70 pages on the last day. Calculate the total number of pages in the book. | 330 | 0.9375 | 387.0625 | 388.8 | 361 | |
Compute: $${0.027}^{− \frac{1}{3}}−{(− \frac{1}{7})}^{−2}+{256}^{ \frac{3}{4}}−{3}^{−1}+{( \sqrt{2}−1)}^{0} = $$ \_\_\_\_\_\_. | 19 | 1 | 2,963.625 | 2,963.625 | -1 | |
Doug constructs a square window using $8$ equal-size panes of glass. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window? | 26 | 1. **Identify the dimensions of each pane**: Given that the ratio of the height to width for each pane is $5:2$, let the width of each pane be $2x$ and the height be $5x$.
2. **Calculate the total width and height of the window**: The window is composed of $4$ panes across and $2$ panes vertically, with borders of $2$... | 0.8125 | 3,880 | 3,621.076923 | 5,002 |
Given $(2x-1)^{5}=a\_{0}+a\_{1}x+a\_{2}x^{2}+…+a\_{5}x^{5}$, find the values of:
(1) $a\_{0}+a\_{1}+a\_{2}+a\_{3}+a\_{4}$,
(2) $|a\_{0}|+|a\_{1}|+|a\_{2}|+|a\_{3}|+|a\_{4}|+|a\_{5}|$,
(3) $a\_{1}+a\_{3}+a\_{5}$,
(4) $(a\_{0}+a\_{2}+a\_{4})^{2}-(a\_{1}+a\_{3}+a\_{5})^{2}$. | -243 | 0.625 | 5,624.8125 | 4,285.2 | 7,857.5 | |
A parking lot has 20 spaces in a row. Fifteen cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Uncle Ben arrives in his RV, which requires 3 adjacent spaces. What is the probability that he is able to park?
A) $\frac{273}{969}$
B) $\frac{2... | \frac{232}{323} | 0 | 8,192 | -1 | 8,192 | |
Find the maximum value of
\[y = \tan \left( x + \frac{2 \pi}{3} \right) - \tan \left( x + \frac{\pi}{6} \right) + \cos \left( x + \frac{\pi}{6} \right)\]for $-\frac{5 \pi}{12} \le x \le -\frac{\pi}{3}.$ | \frac{11 \sqrt{3}}{6} | 0 | 7,573.8125 | -1 | 7,573.8125 | |
On circle $O$, points $C$ and $D$ are on the same side of diameter $\overline{AB}$, $\angle AOC = 30^\circ$, and $\angle DOB = 45^\circ$. What is the ratio of the area of the smaller sector $COD$ to the area of the circle? | \frac{7}{24} | 1. **Identify the angles at the center involving points $C$ and $D$:**
- $\angle AOC = 30^\circ$
- $\angle DOB = 45^\circ$
2. **Calculate $\angle AOB$:**
Since $AB$ is a diameter, $\angle AOB$ spans a semicircle, hence $\angle AOB = 180^\circ$.
3. **Determine $\angle COD$:**
- $\angle COD$ can be calculat... | 1 | 4,438.125 | 4,438.125 | -1 |
(This question has a total of 10 points)
From a group consisting of 5 male doctors and 4 female doctors, select 3 doctors to form a medical team. The requirement is that the team must include both male and female doctors. How many different team formation plans are there? | 70 | 1 | 1,667.1875 | 1,667.1875 | -1 | |
Point $(x, y)$ is randomly picked from the rectangular region with vertices at $(0, 0), (3030, 0), (3030, 3031), and (0, 3031)$. What is the probability that $x > 3y$? Express your answer as a common fraction. | \frac{505}{3031} | 0.5625 | 6,835.0625 | 5,823.888889 | 8,135.142857 | |
Among the following four propositions:
(1) The domain of the function $y=\tan (x+ \frac {π}{4})$ is $\{x|x\neq \frac {π}{4}+kπ,k\in Z\}$;
(2) Given $\sin α= \frac {1}{2}$, and $α\in[0,2π]$, the set of values for $α$ is $\{\frac {π}{6}\}$;
(3) The graph of the function $f(x)=\sin 2x+a\cos 2x$ is symmetric about the l... | (1)(3)(4) | 0 | 4,587.6875 | -1 | 4,587.6875 | |
A company has 45 male employees and 15 female employees. A 4-person research and development team was formed using stratified sampling.
(1) Calculate the probability of an employee being selected and the number of male and female employees in the research and development team;
(2) After a month of learning and disc... | \frac {1}{2} | 0.375 | 4,952.9375 | 5,440.333333 | 4,660.5 | |
In triangle $\triangle ABC$, $a=7$, $b=8$, $A=\frac{\pi}{3}$.
1. Find the value of $\sin B$.
2. If $\triangle ABC$ is an obtuse triangle, find the height on side $BC$. | \frac{12\sqrt{3}}{7} | 0 | 7,071.6875 | -1 | 7,071.6875 | |
Given a circle $C: (x-1)^{2} + (y-2)^{2} = 25$ and a line $l: (2m+1)x + (m+1)y - 7m-4 = 0$, where $m \in \mathbb{R}$. Find the minimum value of the chord length $|AB|$ cut by line $l$ on circle $C$. | 4\sqrt{5} | 0.8125 | 6,485.125 | 6,091.230769 | 8,192 | |
Given that the function $f(x)=\sin x+a\cos x$ has a symmetry axis on $x=\frac{5π}{3}$, determine the maximum value of the function $g(x)=a\sin x+\cos x$. | \frac {2\sqrt {3}}{3} | 0 | 4,083.3125 | -1 | 4,083.3125 | |
For how many integers $n$ with $1 \le n \le 2012$ is the product
\[
\prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]equal to zero? | 335 | 0.125 | 7,949.25 | 6,250 | 8,192 | |
Given the parabola $y^2 = 4x$ whose directrix intersects the x-axis at point $P$, draw line $l$ through point $P$ with the slope $k (k > 0)$, intersecting the parabola at points $A$ and $B$. Let $F$ be the focus of the parabola. If $|FB| = 2|FA|$, then calculate the length of segment $AB$. | \frac{\sqrt{17}}{2} | 0 | 7,500.1875 | -1 | 7,500.1875 | |
100 \times 19.98 \times 1.998 \times 1000= | (1998)^2 | 1. **Calculate the product of $19.98$ and $100$:**
\[
19.98 \times 100 = 1998
\]
This step involves multiplying $19.98$ by $100$, which effectively shifts the decimal point two places to the right.
2. **Calculate the product of $1.998$ and $1000$:**
\[
1.998 \times 1000 = 1998
\]
Similarly, mul... | 0 | 391.9375 | -1 | 391.9375 |
Two parabolas are the graphs of the equations $y=2x^2-7x+1$ and $y=8x^2+5x+1$. Give all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. | (0, 1) | 0 | 2,502.375 | -1 | 2,502.375 | |
Given that the two asymptotes of the hyperbola $\dfrac{y^2}{4}-x^2=1$ intersect with the directrix of the parabola $y^2=2px(p > 0)$ at points $A$ and $B$, and $O$ is the origin, determine the value of $p$ given that the area of $\Delta OAB$ is $1$. | \sqrt{2} | 0.9375 | 3,774.375 | 3,813.8 | 3,183 | |
From the set of integers $\{1,2,3,\dots,2009\}$, choose $k$ pairs $\{a_i,b_i\}$ with $a_i<b_i$ so that no two pairs have a common element. Suppose that all the sums $a_i+b_i$ are distinct and less than or equal to $2009$. Find the maximum possible value of $k$. | 803 | 0 | 8,017.75 | -1 | 8,017.75 | |
Find $ [\sqrt{19992000}]$ where $ [x]$ is the greatest integer less than or equal to $ x$ . | 4471 | 0.75 | 6,600.6875 | 6,070.25 | 8,192 | |
Find the number of functions of the form $f(x) = ax^2 + bx + c$ such that
\[f(x) f(-x) = f(x^2).\] | 8 | 0.75 | 6,175.375 | 5,503.166667 | 8,192 | |
Given the function $f(x)=\ln x+ax^{2}+(a+2)x+1$, where $a\in R$.
$(I)$ Find the monotonic interval of the function $f(x)$;
$(II)$ Let $a\in Z$. If $f(x)\leqslant 0$ holds for all $x \gt 0$, find the maximum value of $a$. | -2 | 0.25 | 7,981.9375 | 7,351.75 | 8,192 | |
There is a grid of height 2 stretching infinitely in one direction. Between any two edge-adjacent cells of the grid, there is a door that is locked with probability $\frac{1}{2}$ independent of all other doors. Philip starts in a corner of the grid (in the starred cell). Compute the expected number of cells that Philip... | \frac{32}{7} | For clarity, we will number our grid, with $(0,0)$ being the corner that Philip starts in, and the grid stretching in the positive $x$ direction, i.e. all elements of the grid are of the form $(x, y)$, with $y \in\{0,1\}$ and $x \in \mathbb{N}$. We will use recursion and casework. Let $A$ be the expected number of reac... | 0 | 7,724 | -1 | 7,724 |
$101$ people, sitting at a round table in any order, had $1,2,... , 101$ cards, respectively.
A transfer is someone give one card to one of the two people adjacent to him.
Find the smallest positive integer $k$ such that there always can through no more than $ k $ times transfer, each person hold c... | 42925 |
Given 101 people sitting at a round table, each holding a unique card numbered from 1 to 101, we need to determine the smallest positive integer \( k \) such that through no more than \( k \) transfers, each person can hold the same number of cards, regardless of the initial sitting order.
To find the smallest \( k \... | 0 | 8,182.5 | -1 | 8,182.5 |
Jo, Blair, and Parker take turns counting from 1, increasing by one more than the last number said by the previous person. What is the $100^{\text{th}}$ number said? | 100 | 0.5625 | 6,571.8125 | 5,732.888889 | 7,650.428571 | |
Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\]
[i] | {f(x) = \frac{1}{x}} |
To solve the functional equation for all functions \( f : \mathbb{Q}^+ \to \mathbb{Q}^+ \) such that for all \( x, y \in \mathbb{Q}^+ \),
\[
f(f(x)^2 y) = x^3 f(xy),
\]
we proceed with the following steps:
**Step 1: Simplify the equation using a special substitution.**
First, consider setting \( y = 1 \). The equa... | 0 | 7,972.4375 | -1 | 7,972.4375 |
Compute \[\sum_{n=1}^{500} \frac{1}{n^2 + 2n}.\] | \frac{1499}{2008} | 0 | 8,192 | -1 | 8,192 | |
In the central cell of a $21 \times 21$ board, there is a piece. In one move, the piece can be moved to an adjacent cell sharing a side. Alina made 10 moves. How many different cells can the piece end up in? | 121 | 0.0625 | 7,663.875 | 8,192 | 7,628.666667 | |
What non-zero real value for $x$ satisfies $(7x)^{14}=(14x)^7$? | \frac{2}{7} | 1. **Start with the given equation:**
\[
(7x)^{14} = (14x)^7
\]
2. **Take the seventh root of both sides:**
\[
\sqrt[7]{(7x)^{14}} = \sqrt[7]{(14x)^7}
\]
Simplifying each side, we get:
\[
(7x)^2 = 14x
\]
3. **Expand and simplify the equation:**
\[
(7x)^2 = 49x^2
\]
Setting th... | 1 | 2,442.1875 | 2,442.1875 | -1 |
The distance between two vectors is the magnitude of their difference. Find the value of $t$ for which the vector
\[\bold{v} = \begin{pmatrix} 2 \\ -3 \\ -3 \end{pmatrix} + t \begin{pmatrix} 7 \\ 5 \\ -1 \end{pmatrix}\]is closest to
\[\bold{a} = \begin{pmatrix} 4 \\ 4 \\ 5 \end{pmatrix}.\] | \frac{41}{75} | 0.8125 | 5,134.3125 | 4,428.692308 | 8,192 | |
Compute the value of $k$ such that the equation
\[\frac{x + 2}{kx - 1} = x\]has exactly one solution. | 0 | 0.5 | 7,570.75 | 6,949.5 | 8,192 | |
Turbo the snail plays a game on a board with $2024$ rows and $2023$ columns. There are hidden monsters in $2022$ of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the last row, and that each column contains at ... | 3 |
To solve this problem, we will analyze the board's structure and derive a strategy for Turbo to ensure he reaches the last row in a guaranteed number of attempts. We'll consider the distribution of monsters and Turbo's possible paths.
Given:
- The board has 2024 rows and 2023 columns.
- There is exactly one monster i... | 0 | 7,981.0625 | -1 | 7,981.0625 |
Which number is greater than 0.7? | 0.8 | Each of \( 0.07, -0.41, 0.35, \) and \(-0.9\) is less than 0.7. The number 0.8 is greater than 0.7. | 0.1875 | 483.125 | 387.333333 | 505.230769 |
Two numbers are independently selected from the set of positive integers less than or equal to 6. Exactly one of the numbers must be even. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{2}{3} | 0.1875 | 7,709.125 | 5,616.666667 | 8,192 | |
Define $a@b = ab - b^{2}$ and $a\#b = a + b - ab^{2}$. What is $\frac {6@2}{6\#2}$? | -\frac{1}{2} | 1. **Calculate $6@2$:**
Using the operation defined as $a@b = ab - b^2$, we substitute $a = 6$ and $b = 2$:
\[
6@2 = 6 \cdot 2 - 2^2 = 12 - 4 = 8.
\]
2. **Calculate $6\#2$:**
Using the operation defined as $a\#b = a + b - ab^2$, we substitute $a = 6$ and $b = 2$:
\[
6\#2 = 6 + 2 - 6 \cdot 2^2 = 8 ... | 1 | 1,471.9375 | 1,471.9375 | -1 |
Elisa creates a sequence in a manner similar to Jacob but starts with the first term as 10. Each succeeding term depends on the outcome of flipping a fair coin: If it lands heads, the next term is obtained by doubling the previous term and subtracting 1; if it lands tails, the next term is half of the previous term, su... | \frac{1}{2} | 0 | 7,527.5 | -1 | 7,527.5 | |
Determine the number of ways to arrange the letters of the word SUCCESS. | 420 | 0.1875 | 2,275.75 | 1,752.333333 | 2,396.538462 | |
For each positive integer \( n \), define \( A_{n} = \frac{20^{n} + 11^{n}}{n!} \), where \( n! = 1 \times 2 \times \cdots \times n \). Find the value of \( n \) that maximizes \( A_{n} \). | 19 | 0 | 8,192 | -1 | 8,192 | |
The lateral face of a regular triangular pyramid \( SABC \) is inclined to the base \( ABC \) at an angle \(\alpha = \arctan \frac{3}{4}\). Points \( M, N, K \) are the midpoints of the sides of the base \( ABC \). The triangle \( MNK \) forms the lower base of a right prism. The edges of the upper base of the prism in... | 16 | 0 | 8,192 | -1 | 8,192 | |
1. Given non-negative real numbers \( x, y, z \) satisfying \( x^{2} + y^{2} + z^{2} + x + 2y + 3z = \frac{13}{4} \), determine the maximum value of \( x + y + z \).
2. Given \( f(x) \) is an odd function defined on \( \mathbb{R} \) with a period of 3, and when \( x \in \left(0, \frac{3}{2} \right) \), \( f(x) = \ln ... | \frac{3}{2} | 0 | 8,192 | -1 | 8,192 | |
The numbers \(a, b, c, d\) belong to the interval \([-5.5, 5.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 132 | 0.3125 | 7,716 | 6,668.8 | 8,192 | |
In the sequence \(1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \cdots, 200, 200, \cdots, 200\), each number \(n\) appears \(n\) times consecutively, where \(n \in \mathbf{N}\) and \(1 \leq n \leq 200\). What is the median of this sequence? | 142 | 0.9375 | 4,636.875 | 4,399.866667 | 8,192 | |
Given a geometric sequence $\{a_n\}$ with all positive terms and $\lg=6$, calculate the value of $a_1 \cdot a_{15}$. | 10^4 | 0 | 5,448.1875 | -1 | 5,448.1875 | |
If $x + \frac{1}{x} = 3$, what is $x^6 + \frac{1}{x^6}$? | 322 | 1 | 3,481.9375 | 3,481.9375 | -1 | |
Find the number of triples of natural numbers \( m, n, k \) that are solutions to the equation \( m + \sqrt{n+\sqrt{k}} = 2023 \). | 27575680773 | 0 | 7,618.4375 | -1 | 7,618.4375 | |
For non-negative integers \( x \), the function \( f(x) \) is defined as follows:
$$
f(0) = 0, \quad f(x) = f\left(\left\lfloor \frac{x}{10} \right\rfloor\right) + \left\lfloor \log_{10} \left( \frac{10}{x - 10\left\lfloor \frac{x-1}{10} \right\rfloor} \right) \right\rfloor .
$$
For \( 0 \leqslant x \leqslant 2006 \), ... | 1111 | 0 | 8,192 | -1 | 8,192 | |
The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \begin{tabular}{|l|c|} \hline
The Parabola & 165 feet \\ \hline
The G Force & 119 feet \\ \hline
The Me... | 19 | 1 | 1,694.4375 | 1,694.4375 | -1 | |
Given that points A and B lie on the graph of y = \frac{1}{x} in the first quadrant, ∠OAB = 90°, and AO = AB, find the area of the isosceles right triangle ∆OAB. | \frac{\sqrt{5}}{2} | 0 | 8,059.125 | -1 | 8,059.125 | |
Given \(\omega = -\frac{1}{2} + \frac{1}{2}i\sqrt{3}\), representing a cube root of unity, specifically \(\omega = e^{2\pi i / 3}\). Let \(T\) denote all points in the complex plane of the form \(a + b\omega + c\omega^2\), where \(0 \leq a \leq 2\), \(0 \leq b \leq 1\), and \(0 \leq c \leq 1\). Determine the area of \(... | 2\sqrt{3} | 0.0625 | 8,011.3125 | 7,660 | 8,034.733333 | |
Petya places "+" and "-" signs in all possible ways into the expression $1 * 2 * 3 * 4 * 5 * 6$ at the positions of the asterisks. For each arrangement of signs, he calculates the resulting value and writes it on the board. Some numbers may appear on the board multiple times. Petya then sums all the numbers on the boar... | 32 | 0.25 | 6,456.6875 | 2,781.5 | 7,681.75 | |
A right circular cone and a sphere possess the same radius, $r$. If the volume of the cone is one-third of the volume of the sphere, determine the ratio of the height of the cone to the radius $r$. | \frac{4}{3} | 1 | 1,637.875 | 1,637.875 | -1 | |
Given 5 people stand in a row, and there is exactly 1 person between person A and person B, determine the total number of possible arrangements. | 36 | 0.3125 | 7,278.875 | 5,842.8 | 7,931.636364 | |
Find the greatest common divisor of 12 and 20. | 4 | 1 | 1,055 | 1,055 | -1 | |
The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | \frac{1}{2} | 0 | 6,134.3125 | -1 | 6,134.3125 | |
If $15\%$ of $N$ is $45\%$ of 2003, what is the value of $N$? | 6009 | 1 | 2,741.25 | 2,741.25 | -1 | |
Upon cutting a certain rectangle in half, you obtain two rectangles that are scaled down versions of the original. What is the ratio of the longer side length to the shorter side length? | \sqrt{2} | 0.8125 | 4,358.5 | 3,473.846154 | 8,192 | |
Find $ax^5 + by^5$ if the real numbers $a,b,x,$ and $y$ satisfy the equations \begin{align*} ax + by &= 3, \\ ax^2 + by^2 &= 7, \\ ax^3 + by^3 &= 16, \\ ax^4 + by^4 &= 42. \end{align*} | 20 | A recurrence of the form $T_n=AT_{n-1}+BT_{n-2}$ will have the closed form $T_n=ax^n+by^n$, where $x,y$ are the values of the starting term that make the sequence geometric, and $a,b$ are the appropriately chosen constants such that those special starting terms linearly combine to form the actual starting terms.
Suppo... | 0.5625 | 7,040 | 6,144 | 8,192 |
Find the area of the triangle with vertices $(3,-5),$ $(-2,0),$ and $(1,-6).$ | \frac{15}{2} | 0.8125 | 4,298.75 | 4,173.769231 | 4,840.333333 | |
In the diagram, each of the three identical circles touch the other two. The circumference of each circle is 36. What is the perimeter of the shaded region? | 18 | 1 | 3,022.375 | 3,022.375 | -1 | |
What are the last 8 digits of $$11 \times 101 \times 1001 \times 10001 \times 100001 \times 1000001 \times 111 ?$$ | 19754321 | Multiply terms in a clever order. $$\begin{aligned} 11 \cdot 101 \cdot 10001 & =11,111,111 \\ 111 \cdot 1001 \cdot 1000001 & =111,111,111,111 \end{aligned}$$ The last eight digits of $11,111,111 \cdot 111,111,111,111$ are 87654321. We then just need to compute the last 8 digits of $87654321 \cdot 100001=87654321+\ldots... | 0.0625 | 7,949.5625 | 4,313 | 8,192 |
Given a list of $3000$ positive integers with a unique mode occurring exactly $12$ times, calculate the least number of distinct values that can occur in the list. | 273 | 0.5 | 6,568.375 | 5,213 | 7,923.75 | |
What is the largest prime factor of $5! + 6!$? | 7 | 1 | 2,943.375 | 2,943.375 | -1 | |
Find: $\frac{12}{30}-\frac{1}{7}$ | \frac{9}{35} | 1 | 2,602.9375 | 2,602.9375 | -1 | |
The area of a trapezoid is 1. What is the minimum possible length of the longest diagonal of this trapezoid? | \sqrt{2} | 0.3125 | 7,484.25 | 5,927.2 | 8,192 | |
When an integer is divided by 15, the remainder is 7. Find the sum of the remainders when the same integer is divided by 3 and by 5. | 3 | 1 | 1,667.5625 | 1,667.5625 | -1 | |
In the ancient Chinese mathematical masterpiece "Nine Chapters on the Mathematical Art," there is a problem called "Division of Grains in a Granary": When a granary is opened to receive grain, a person brings 1534 stones of rice. Upon inspection, it is found that there are grains of wheat mixed in the rice. After takin... | 169.1 | 0.5 | 6,191.5 | 4,706.5 | 7,676.5 | |
679 contestants participated in HMMT February 2017. Let \(N\) be the number of these contestants who performed at or above the median score in at least one of the three individual tests. Estimate \(N\). An estimate of \(E\) earns \(\left\lfloor 20-\frac{|E-N|}{2}\right\rfloor\) or 0 points, whichever is greater. | 516 | Out of the 679 total contestants at HMMT February 2017, 188 contestants scored at least the median on all three tests, 159 contestants scored at least the median on two tests, and 169 contestants scored at least the median on one test, giving a total of 516 contestants | 0 | 7,632.5625 | -1 | 7,632.5625 |
The sum of the first $n$ terms in the infinite geometric sequence $\left\{\frac{1}{4},\frac{1}{8},\frac{1}{16},\dots \right\}$ is $\frac{63}{128}$. Find $n$. | 6 | 0.9375 | 2,500.6875 | 2,121.266667 | 8,192 | |
In an isosceles triangle \(ABC\) with base \(AC\), point \(D\) divides side \(BC\) in the ratio \(2:1\) from vertex \(B\), and point \(E\) is the midpoint of side \(AB\). It is known that the median \(CQ\) of triangle \(CED\) is equal to \(\frac{\sqrt{23}}{2}\), and \(DE = \frac{\sqrt{23}}{2}\). Find the radius of the ... | 12/5 | 0.5625 | 7,228.3125 | 6,751.444444 | 7,841.428571 | |
A circle with center $D$ and radius four feet is tangent at $E$ to a circle with center $F$, as shown. If point $F$ is on the small circle, what is the area of the shaded region? Express your answer in terms of $\pi$.
[asy]
filldraw(circle((0,0),8),gray,linewidth(2));
filldraw(circle(4dir(-30),4),white,linewidth(2));
... | 48\pi | 1 | 2,192.375 | 2,192.375 | -1 | |
Let $a\star b = a^b+ab$. If $a$ and $b$ are positive integers greater than or equal to 2 and $a\star b =15$, find $a+b$. | 5 | 1 | 2,596 | 2,596 | -1 | |
Esther and Frida are supposed to fill a rectangular array of 16 columns and 10 rows with the numbers 1 to 160. Esther fills it row-wise so that the first row is numbered 1, 2, ..., 16 and the second row is 17, 18, ..., 32 and so on. Frida fills it column-wise, so that her first column has 1, 2, ..., 10, and the second ... | 322 | 0.8125 | 3,841.3125 | 3,132.461538 | 6,913 | |
In trapezoid \(ABCD\), \(AD\) is parallel to \(BC\) and \(BC : AD = 5 : 7\). Point \(F\) lies on \(AD\) and point \(E\) lies on \(DC\) such that \(AF : FD = 4 : 3\) and \(CE : ED = 2 : 3\). If the area of quadrilateral \(ABEF\) is 123, determine the area of trapezoid \(ABCD\). | 180 | 0.375 | 7,454.3125 | 6,224.833333 | 8,192 | |
Given \( x, y, z \in \mathbb{R}^{+} \) and \( x + 2y + 3z = 1 \), find the minimum value of \( \frac{16}{x^{3}}+\frac{81}{8y^{3}}+\frac{1}{27z^{3}} \). | 1296 | 0.25 | 8,027.0625 | 7,532.25 | 8,192 | |
Given a randomly selected number $x$ in the interval $[0,\pi]$, determine the probability of the event "$-1 \leqslant \tan x \leqslant \sqrt {3}$". | \dfrac{7}{12} | 0.8125 | 5,330.375 | 5,212.153846 | 5,842.666667 | |
Each vertex of this parallelogram has integer coordinates. The perimeter of this parallelogram is $p$ units, while the area is $a$ square units. What is the value of the sum $p + a$?
[asy]
size(5cm,5cm);
draw((-2,0)--(11,0));
draw((0,-1)--(0,6));
draw((0,0)--(3,4)--(10,4)--(7,0)--cycle);
label("$(3,4)$",(3,4),NW);
lab... | 52 | 1 | 3,561.6875 | 3,561.6875 | -1 | |
A block of wood has the shape of a right circular cylinder with a radius of $8$ and a height of $10$. The entire surface of the block is painted red. Points $P$ and $Q$ are chosen on the edge of one of the circular faces such that the arc $\overarc{PQ}$ measures $180^\text{o}$. The block is then sliced in half along th... | 193 | 0 | 8,125.0625 | -1 | 8,125.0625 | |
A cube is inscribed in a regular octahedron in such a way that its vertices lie on the edges of the octahedron. By what factor is the surface area of the octahedron greater than the surface area of the inscribed cube? | \frac{2\sqrt{3}}{3} | 0 | 7,849.75 | -1 | 7,849.75 | |
Compute $\tan 60^\circ$. | \sqrt{3} | 1 | 2,658.9375 | 2,658.9375 | -1 | |
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move... | $(6,2,1)$ | 1. **Understanding the Game**: In this game, Arjun and Beth take turns removing either one brick or two adjacent bricks from a set of walls. The player who removes the last brick wins. The game can be analyzed using the concept of nim-values from combinatorial game theory.
2. **Nim-Values**: The nim-value of a configu... | 0 | 8,138.6875 | -1 | 8,138.6875 |
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