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 |
|---|---|---|---|---|---|---|
In $\triangle ABC$, $AB=1$, $BC=2$, $\angle B=\frac{\pi}{3}$, let $\overrightarrow{AB}=\overrightarrow{a}$, $\overrightarrow{BC}= \overrightarrow{b}$.
(I) Find the value of $(2\overrightarrow{a}-3\overrightarrow{b})\cdot(4\overrightarrow{a}+\overrightarrow{b})$;
(II) Find the value of $|2\overrightarrow{a}-\overrightar... | 2 \sqrt{3} | 0.375 | 3,965 | 3,676.333333 | 4,138.2 | |
The real number \( a \) makes the equation \( 4^{x} - 4^{-x} = 2 \cos(ax) \) have exactly 2015 solutions. For this \( a \), how many solutions does the equation \( 4^{x} + 4^{-x} = 2 \cos(ax) + 4 \) have? | 4030 | 0 | 8,192 | -1 | 8,192 | |
$O$ is the origin, and $F$ is the focus of the parabola $C:y^{2}=4x$. A line passing through $F$ intersects $C$ at points $A$ and $B$, and $\overrightarrow{FA}=2\overrightarrow{BF}$. Find the area of $\triangle OAB$. | \dfrac{3\sqrt{2}}{2} | 0 | 6,808.375 | -1 | 6,808.375 | |
A right triangle has integer side lengths. One of its legs is 1575 units shorter than its hypotenuse, and the other leg is less than 1991 units. Find the length of the hypotenuse of this right triangle. | 1799 | 0.25 | 7,867.6875 | 6,894.75 | 8,192 | |
Let $f(x)$ be an odd function. Is $f(f(f(x)))$ even, odd, or neither?
Enter "odd", "even", or "neither". | \text{odd} | 0.75 | 2,737.8125 | 2,689.916667 | 2,881.5 | |
There are red and blue socks in a drawer, with a total number not exceeding 2017. If two socks are randomly drawn, the probability that they are of the same color is $\frac{1}{2}$. What is the maximum number of red socks in the drawer? | 990 | 0.6875 | 6,422.75 | 5,618.545455 | 8,192 | |
Among the following propositions, the true one is __________
(1) In a plane, the locus of points whose sum of distances from two fixed points $F_{1}$ and $F_{2}$ is a constant is an ellipse;
(2) If vectors $\overrightarrow{e_{1}}$, $\overrightarrow{e_{2}}$, $\overrightarrow{e_{3}}$ are three non-collinear vectors... | (3) | 0 | 6,355.3125 | -1 | 6,355.3125 | |
A rectangular table measures $12'$ in length and $9'$ in width and is currently placed against one side of a rectangular room. The owners desire to move the table to lay diagonally in the room. Determine the minimum length of the shorter side of the room, denoted as $S$, in feet, for the table to fit without tilting or... | 15' | 0 | 6,231.5625 | -1 | 6,231.5625 | |
The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$? | 90 | 1. **Given Information**: The mean, median, and mode of the data set $60, 100, x, 40, 50, 200, 90$ are all equal to $x$.
2. **Calculating the Mean**:
The mean of the data set is given by:
\[
\text{Mean} = \frac{60 + 100 + x + 40 + 50 + 200 + 90}{7} = \frac{540 + x}{7}
\]
Since the mean is equal to $x$, ... | 1 | 4,661.875 | 4,661.875 | -1 |
The degree of $(x^2+1)^4 (x^3+1)^3$ as a polynomial in $x$ is | 17 | To find the degree of the polynomial $(x^2+1)^4 (x^3+1)^3$, we need to consider the highest degree terms from each factor when expanded.
1. **Analyzing the first factor $(x^2+1)^4$:**
- The highest degree term in $x^2+1$ is $x^2$.
- When raised to the fourth power, the highest degree term in $(x^2+1)^4$ is $(x^2... | 1 | 1,831.6875 | 1,831.6875 | -1 |
The vertices of a triangle have coordinates \(A(1 ; 3.5)\), \(B(13.5 ; 3.5)\), and \(C(11 ; 16)\). We consider horizontal lines defined by the equations \(y=n\), where \(n\) is an integer. Find the sum of the lengths of the segments cut by these lines on the sides of the triangle. | 78 | 0.5 | 5,950.3125 | 4,182 | 7,718.625 | |
I have 8 unit cubes of different colors, which I want to glue together into a $2 \times 2 \times 2$ cube. How many distinct $2 \times 2 \times 2$ cubes can I make? Rotations of the same cube are not considered distinct, but reflections are. | 1680 | Our goal is to first pin down the cube, so it can't rotate. Without loss of generality, suppose one of the unit cubes is purple, and let the purple cube be in the top left front position. Now, look at the three positions that share a face with the purple cube. There are $\binom{7}{3}$ ways to pick the three cubes that ... | 0.6875 | 5,822.9375 | 4,746.090909 | 8,192 |
In rectangle $ABCD$, $AB=100$. Let $E$ be the midpoint of $\overline{AD}$. Given that line $AC$ and line $BE$ are perpendicular, find the greatest integer less than $AD$.
| 141 | 1 | 2,431.625 | 2,431.625 | -1 | |
For positive integer $k>1$, let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$, as $12$ can be factored in these $4$ ways: $12,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3$.
Prove: If $n$ is a positive integer greater... | \frac{n}{p} |
For a positive integer \( k > 1 \), let \( f(k) \) represent the number of ways to factor \( k \) into a product of positive integers greater than 1. For example, \( f(12) = 4 \) because 12 can be factored in these 4 ways: \( 12 \), \( 2 \cdot 6 \), \( 3 \cdot 4 \), and \( 2 \cdot 2 \cdot 3 \).
We aim to prove that i... | 0 | 8,192 | -1 | 8,192 |
Find the sum of the distinct prime factors of $7^7 - 7^4$. | 24 | 0.0625 | 2,028.0625 | 2,178 | 2,018.066667 | |
What is the difference between the sum of the first $2003$ even counting numbers and the sum of the first $2003$ odd counting numbers? | 2003 | To find the difference between the sum of the first $2003$ even counting numbers and the sum of the first $2003$ odd counting numbers, we first need to calculate each sum separately.
1. **Sum of the first $2003$ odd counting numbers:**
The sequence of the first $2003$ odd numbers is $1, 3, 5, \ldots, 4005$. This is... | 0.9375 | 2,393.8125 | 2,007.266667 | 8,192 |
Find the coefficient of $\frac{1}{x}$ in the expansion of $((1-x^{2})^{4}(\frac{x+1}{x})^{5})$. | -29 | 0.25 | 5,889.5625 | 4,433.75 | 6,374.833333 | |
Calculate the coefficient of the term containing $x^4$ in the expansion of $(x-1)(x-2)(x-3)(x-4)(x-5)$. | -15 | 0.875 | 4,347.25 | 3,798 | 8,192 | |
The graphs of $y=|x|$ and $y=-x^2-3x-2$ are drawn. For every $x$, a vertical segment connecting these two graphs can be drawn as well. Find the smallest possible length of one of these vertical segments. | 1 | 0.875 | 4,709.9375 | 4,212.5 | 8,192 | |
With the popularity of cars, the "driver's license" has become one of the essential documents for modern people. If someone signs up for a driver's license exam, they need to pass four subjects to successfully obtain the license, with subject two being the field test. In each registration, each student has 5 chances to... | \frac{1}{9} | 0 | 6,262 | -1 | 6,262 | |
If $\tan \alpha = -\frac{4}{3}$, then the value of $\sin^2\alpha + 2\sin \alpha \cos \alpha$ is ______. | -\frac{8}{25} | 0.9375 | 4,930.375 | 4,712.933333 | 8,192 | |
You have a square with vertices at $(2,1)$, $(5,1)$, $(2,4)$, and $(5,4)$. A line joining $(2,1)$ and $(5,3)$ divides the square into two regions. What fraction of the area of the square is above this line? | \frac{2}{3} | 0.5 | 7,076.5625 | 5,961.125 | 8,192 | |
Determine the volume of the region in space defined by
\[|x + y + z| + |x + y - z| \le 12\]
and \(x, y, z \ge 0.\) | 108 | 0.375 | 7,490.4375 | 6,321.166667 | 8,192 | |
What is the remainder when $2024 \cdot 3047$ is divided by $800$? | 728 | 0.5625 | 6,873.5625 | 5,848.111111 | 8,192 | |
The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number? | \frac{3}{2} | 1. Let $a$ and $b$ be two positive numbers such that $a > b$. According to the problem, the sum of these two numbers is $5$ times their difference. This can be expressed as:
\[
a + b = 5(a - b)
\]
2. Expanding and rearranging the equation:
\[
a + b = 5a - 5b
\]
\[
a + b - 5a + 5b = 0
\]
\... | 1 | 1,286 | 1,286 | -1 |
Let $[x]$ denote the greatest integer less than or equal to the real number $x$,
$$
\begin{array}{c}
S=\left[\frac{1}{1}\right]+\left[\frac{2}{1}\right]+\left[\frac{1}{2}\right]+\left[\frac{2}{2}\right]+\left[\frac{3}{2}\right]+ \\
{\left[\frac{4}{2}\right]+\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{... | 1078 | 0.4375 | 6,814.375 | 5,680.714286 | 7,696.111111 | |
Given a positive integer \( N \) that has exactly nine positive divisors, with three of these divisors \( a, b, \) and \( c \) satisfying
\[ a + b + c = 2017 \]
and
\[ ac = b^2. \]
Find the value of \( N \). | 82369 | 0.375 | 7,774.5625 | 7,078.833333 | 8,192 | |
Let the set \( S = \{1,2,3, \cdots, 10\} \). Given that subset \( A \) of \( S \) satisfies \( A \cap \{1,2,3\} \neq \varnothing \) and \( A \cup \{4,5,6\} \neq S \), determine the number of such subsets \( A \).
(Note: The original problem includes determining the count of subset \( A \) that meets the given conditi... | 888 | 0 | 7,870.5625 | -1 | 7,870.5625 | |
Seven little children sit in a circle. The teacher distributes pieces of candy to the children in such a way that the following conditions hold. - Every little child gets at least one piece of candy. - No two little children have the same number of pieces of candy. - The numbers of candy pieces given to any two adjacen... | 44 | An optimal arrangement is 2-6-3-9-12-4-8. Note that at least two prime factors must appear. In addition, any prime factor that appears must appear in at least two non-prime powers unless it is not used as a common factor between any two adjacent little children. Thus with the distinctness condition we easily see that, ... | 0 | 8,192 | -1 | 8,192 |
Let \( (a_1, a_2, \dots, a_{12}) \) be a list of the first 12 positive integers such that for each \( 2 \le i \le 12 \), either \( a_i+1 \) or \( a_i-1 \) or both appear somewhere before \( a_i \) in the list. Determine the number of such lists. | 2048 | 0.0625 | 8,159.75 | 8,192 | 8,157.6 | |
The amount of algae covering the Smith's backyard pond doubled every day until it was completely covered in algae on day $30$ of the month. On what day of that month was $75\%$ of the pond algae-free? | 28 | 0.9375 | 3,531.75 | 3,447.466667 | 4,796 | |
Let $P$ be a plane passing through the origin. When $\begin{pmatrix} 5 \\ 3 \\ 5 \end{pmatrix}$ is projected onto plane $P,$ the result is $\begin{pmatrix} 3 \\ 5 \\ 1 \end{pmatrix}.$ When $\begin{pmatrix} 4 \\ 0 \\ 7 \end{pmatrix}$ is projected onto plane $P,$ what is the result? | \begin{pmatrix} 1 \\ 3 \\ 1 \end{pmatrix} | 0.75 | 4,598.8125 | 3,401.083333 | 8,192 | |
Triangle $ABC$ with right angle at $C$, $\angle BAC < 45^\circ$ and $AB = 4$. Point $P$ on $\overline{AB}$ is chosen such that $\angle APC = 2\angle ACP$ and $CP = 1$. The ratio $\frac{AP}{BP}$ can be represented in the form $p + q\sqrt{r}$, where $p$, $q$, $r$ are positive integers and $r$ is not divisible by the squa... | 7 | Let $\angle{ACP}$ be equal to $x$. Then by Law of Sines, $PB = -\frac{\cos{x}}{\cos{3x}}$ and $AP = \frac{\sin{x}}{\sin{3x}}$. We then obtain $\cos{3x} = 4\cos^3{x} - 3\cos{x}$ and $\sin{3x} = 3\sin{x} - 4\sin^3{x}$. Solving, we determine that $\sin^2{x} = \frac{4 \pm \sqrt{2}}{8}$. Plugging this in gives that $\frac{A... | 0.0625 | 8,126.125 | 7,138 | 8,192 |
29 boys and 15 girls attended a ball. Some boys danced with some of the girls (no more than once with each pair). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have mentioned? | 29 | 0 | 8,042.125 | -1 | 8,042.125 | |
Several young men and women are seated around a round table. It is known that to the left of exactly 7 women, there are women, and to the left of 12 women, there are men. It is also known that for 75% of the young men, there are women to their right. How many people are seated at the table? | 35 | 0 | 8,192 | -1 | 8,192 | |
Points \( M \) and \( N \) are the midpoints of sides \( BC \) and \( AD \) of quadrilateral \( ABCD \). It is known that \(\angle B = 150^\circ\), \(\angle C = 90^\circ\), and \(AB = CD\). Find the angle between the lines \(MN\) and \(BC\). | 60 | 0.75 | 6,431.25 | 6,147.916667 | 7,281.25 | |
In parallelogram $EFGH$, point $Q$ is on $\overline{EF}$ such that $\frac{EQ}{EF} = \frac{1}{8}$, and point $R$ is on $\overline{EH}$ such that $\frac{ER}{EH} = \frac{1}{9}$. Let $S$ be the point of intersection of $\overline{EG}$ and $\overline{QR}$. Find the ratio $\frac{ES}{EG}$. | \frac{1}{9} | 0 | 3,700.625 | -1 | 3,700.625 | |
At the mall's food court, Crystal wants to buy a meal consisting of one entree, one drink and one dessert. The table below lists Crystal's favorite foods in the food court. How many distinct possible meals can she buy from these options?
\begin{tabular}{ |c | c | c | }
\hline \textbf{Entrees} & \textbf{Drinks}&\textbf... | 16 | 0.1875 | 1,614 | 976.666667 | 1,761.076923 | |
The ellipse $x^2+4y^2=4$ and the hyperbola $x^2-m(y+2)^2 = 1$ are tangent. Compute $m.$ | \frac{12}{13} | 0.8125 | 4,702 | 3,896.615385 | 8,192 | |
Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions into which they divide the plane is: | 22 |
#### Detailed Analysis:
1. **Understanding the Problem:**
We are given six straight lines in a plane, with the condition that no two lines are parallel and no three lines are concurrent. We need to determine how many regions these lines divide the plane into.
2. **Using Incremental Line Addition:**
- **First L... | 1 | 3,272 | 3,272 | -1 |
For real numbers $s,$ the intersection points of the lines $2x - 3y = 4s + 6$ and $2x + y = 3s + 1$ are plotted. All these points lie on a particular line. Determine the slope of this line. | -\frac{2}{13} | 0.9375 | 4,789.4375 | 4,562.6 | 8,192 | |
What is the quotient when $8x^3+16x^2-7x+4$ is divided by $2x+5$? | 4x^2 -2x + \frac{3}{2} | 0.875 | 5,474 | 5,085.714286 | 8,192 | |
A total of $n$ points are equally spaced around a circle and are labelled with the integers 1 to $n$, in order. Two points are called diametrically opposite if the line segment joining them is a diameter of the circle. If the points labelled 7 and 35 are diametrically opposite, then what is the value of $n$? | 56 | The number of points on the circle equals the number of spaces between the points around the circle. Moving from the point labelled 7 to the point labelled 35 requires moving $35-7=28$ points and so 28 spaces around the circle. Since the points labelled 7 and 35 are diametrically opposite, then moving along the circle ... | 0.9375 | 2,716.875 | 2,351.866667 | 8,192 |
For real numbers $t,$ the point
\[(x,y) = \left( \frac{1 - t^2}{1 + t^2}, \frac{2t}{1 + t^2} \right)\]is plotted. All the plotted points lie on what kind of curve?
(A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola
Enter the letter of the correct option. | \text{(B)} | 0 | 2,501.5625 | -1 | 2,501.5625 | |
Find the non-zero value of $c$ for which there is exactly one positive value of $b$ for which there is one solution to the equation $x^2 + \left(b + \frac 1b\right)x + c = 0$. | 1 | 0.875 | 3,816.5625 | 3,773.571429 | 4,117.5 | |
If \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), where \(a, b\), and \(c\) are positive integers, the number of different ordered triples \((a, b, c)\) such that \(a+2b+c \leq 40\) is: | 42 | 0.125 | 8,020 | 6,816 | 8,192 | |
Let $a_i,b_i,i=1,\cdots,n$ are nonnegitive numbers,and $n\ge 4$,such that $a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n>0$.
Find the maximum of $\frac{\sum_{i=1}^n a_i(a_i+b_i)}{\sum_{i=1}^n b_i(a_i+b_i)}$ | n - 1 |
Let \( a_i, b_i \) for \( i = 1, \ldots, n \) be nonnegative numbers, and let \( n \geq 4 \) such that \( \sum_{i=1}^n a_i = \sum_{i=1}^n b_i > 0 \).
We aim to find the maximum value of the expression:
\[
\frac{\sum_{i=1}^n a_i(a_i + b_i)}{\sum_{i=1}^n b_i(a_i + b_i)}.
\]
We will prove that for \( n \geq 4 \), the m... | 0 | 8,192 | -1 | 8,192 |
Compute without using a calculator: $9!/8!$ | 9 | 1 | 1,211.5625 | 1,211.5625 | -1 | |
Given the system of equations for the positive numbers \(x, y, z\):
$$
\left\{\begin{array}{l}
x^{2}+xy+y^{2}=108 \\
y^{2}+yz+z^{2}=16 \\
z^{2}+xz+x^{2}=124
\end{array}\right.
$$
Find the value of the expression \(xy + yz + xz\). | 48 | 0.375 | 7,457.5625 | 6,233.5 | 8,192 | |
The slant height of a cone forms an angle $\alpha$ with the plane of its base, where $\cos \alpha = \frac{1}{4}$. A sphere is inscribed in the cone, and a plane is drawn through the circle of tangency of the sphere and the lateral surface of the cone. The volume of the part of the cone enclosed between this plane and t... | 37 | 0 | 8,015.375 | -1 | 8,015.375 | |
A circle has a radius of 6. What is the area of the smallest square that can entirely contain this circle, and what is the circumference of the circle? | 12\pi | 0.4375 | 1,426.75 | 1,191.285714 | 1,609.888889 | |
A cell phone plan costs $20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay? | 28.00 | 1. **Calculate the base cost of the plan:**
The base price of Michelle's cell phone plan is $20$ dollars.
2. **Calculate the cost for text messages:**
Michelle sent $100$ text messages, and each text message costs $5$ cents. Therefore, the total cost for text messages is:
\[
100 \text{ texts} \times 5 \tex... | 0 | 1,364.4375 | -1 | 1,364.4375 |
There are 15 different-colored crayons in a box. Karl wants to first select three crayons for his art project and then select four crayons for his friend's project. How many ways can Karl select these seven crayons if the order of selection does not matter for each set? | 225225 | 0.5625 | 1,571.75 | 2,374.222222 | 540 | |
Svitlana writes the number 147 on a blackboard. Then, at any point, if the number on the blackboard is $n$, she can perform one of the following three operations: - if $n$ is even, she can replace $n$ with $\frac{n}{2}$; - if $n$ is odd, she can replace $n$ with $\frac{n+255}{2}$; and - if $n \geq 64$, she can replace ... | 163 | The answer is $163=\sum_{i=0}^{4}\binom{8}{i}$. This is because we can obtain any integer less than $2^{8}$ with less than or equal to 4 ones in its binary representation. Note that $147=2^{7}+2^{4}+2^{1}+2^{0}$. We work in binary. Firstly, no operation can increase the number of ones in $n$'s binary representation. Th... | 0 | 8,072.8125 | -1 | 8,072.8125 |
The projection of $\begin{pmatrix} 0 \\ 3 \\ z \end{pmatrix}$ onto $\begin{pmatrix} -3 \\ 5 \\ -1 \end{pmatrix}$ is
\[\frac{12}{35} \begin{pmatrix} -3 \\ 5 \\ -1 \end{pmatrix}.\]Find $z.$ | 3 | 0.9375 | 2,762.9375 | 2,401 | 8,192 | |
Find the smallest three-digit palindrome whose product with 101 is not a five-digit palindrome. | 505 | 0.3125 | 7,799.5 | 7,043.6 | 8,143.090909 | |
Xiao Zhang and Xiao Zhao can only take on the first two roles, while the other three can take on any of the four roles, calculate the total number of different selection schemes. | 48 | 0 | 6,607.5 | -1 | 6,607.5 | |
The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y| \leq 1000$? | 40 | 1. **Identify the axis of symmetry**: Given the focus of the parabola $P$ at $(0,0)$ and points $(4,3)$ and $(-4,-3)$ on $P$, we observe that the line connecting these points has a slope of $\frac{3 - (-3)}{4 - (-4)} = \frac{6}{8} = \frac{3}{4}$. This suggests that the axis of symmetry of the parabola makes an angle $\... | 0 | 8,192 | -1 | 8,192 |
The principal of a certain school decided to take a photo of the graduating class of 2008. He arranged the students in parallel rows, all with the same number of students, but this arrangement was too wide for the field of view of his camera. To solve this problem, the principal decided to take one student from each ro... | 24 | 0 | 8,192 | -1 | 8,192 | |
Find the sum of all four-digit numbers in which the digits $0, 4, 5, 9$ are absent. | 6479352 | 0.5625 | 5,550.75 | 3,705.888889 | 7,922.714286 | |
Let $\mathbf{a} = \begin{pmatrix} -3 \\ 10 \\ 1 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} 5 \\ \pi \\ 0 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} -2 \\ -2 \\ 7 \end{pmatrix}.$ Compute
\[(\mathbf{a} - \mathbf{b}) \cdot [(\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a})].\] | 0 | 0.875 | 4,537.25 | 4,015.142857 | 8,192 | |
Compute the value of the infinite series \[
\sum_{n=2}^{\infty} \frac{n^4+3n^2+10n+10}{2^n \cdot \left(n^4+4\right)}
\] | \frac{11}{10} | 0.0625 | 8,028.875 | 8,015 | 8,029.8 | |
The first number in the following sequence is $1$ . It is followed by two $1$ 's and two $2$ 's. This is followed by three $1$ 's, three $2$ 's, and three $3$ 's. The sequence continues in this fashion.
\[1,1,1,2,2,1,1,1,2,2,2,3,3,3,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,\dots.\]
Find the $2014$ th number in this sequ... | 13 | 0.5625 | 6,813.4375 | 5,758 | 8,170.428571 | |
In a right triangle $\triangle STU$, where $\angle S = 90^\circ$, suppose $\sin T = \frac{3}{5}$. If the length of $SU$ is 15, find the length of $ST$. | 12 | 0 | 1,699.6875 | -1 | 1,699.6875 | |
There are numbers $1, 2, \cdots, 36$ to be filled into a $6 \times 6$ grid, with each cell containing one number. Each row must be in increasing order from left to right. What is the minimum sum of the six numbers in the third column? | 63 | 0 | 8,105.125 | -1 | 8,105.125 | |
Evaluate the sum:
\[
\sum_{n=1}^\infty \frac{n^3 + n^2 - n}{(n+3)!}.
\] | \frac{1}{6} | 0 | 8,094.875 | -1 | 8,094.875 | |
Find the value of $t$ that satisfies $\frac{1}{t+2} + \frac{2t}{t+2} - \frac{3}{t+2} = 3$. | -8 | 1 | 1,763.5625 | 1,763.5625 | -1 | |
Compute $97^2$ in your head. | 9409 | 0.9375 | 2,855.25 | 2,499.466667 | 8,192 | |
How many pages does the book "Folk Tales" have if from the first page to the last page, a total of 357 digits were used to number the pages? | 155 | 0.9375 | 1,786.8125 | 1,845.8 | 902 | |
Consider the following transformation of the Cartesian plane: choose a lattice point and rotate the plane $90^\circ$ counterclockwise about that lattice point. Is it possible, through a sequence of such transformations, to take the triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$ to the triangle with vertices $(0,0)... | $\text { No }$ |
To determine if it is possible to transform the triangle with vertices \((0,0)\), \((1,0)\), and \((0,1)\) into the triangle with vertices \((0,0)\), \((1,0)\), and \((1,1)\) through a sequence of 90° counterclockwise rotations about lattice points, we analyze the effects of such rotations on the plane.
### Step 1: U... | 0 | 7,668.875 | -1 | 7,668.875 |
What is the largest number of acute angles that a convex hexagon can have? | 3 | 1. **Calculate the sum of interior angles of a hexagon**:
The sum of the interior angles of a polygon with $n$ sides is given by the formula $(n-2) \times 180^\circ$. For a hexagon ($n=6$), this sum is:
\[
(6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.
\]
2. **Consider the case of six acute angl... | 0.8125 | 5,625.625 | 5,033.384615 | 8,192 |
Mr. Fat needs 20 minutes to eat a pound of cereal, while Mr. Thin needs 30 minutes. If they eat together, how long does it take for them to finish off three pounds of cereal? Express your answer in minutes. | 36 | 1 | 2,020.625 | 2,020.625 | -1 | |
In this version of SHORT BINGO, a $5\times5$ card is again filled by marking the middle square as WILD and placing 24 other numbers in the remaining 24 squares. Now, the card is made by placing 5 distinct numbers from the set $1-15$ in the first column, 5 distinct numbers from $11-25$ in the second column, 4 distinct n... | 360360 | 0.125 | 2,472.625 | 1,682.5 | 2,585.5 | |
Given vectors $\overrightarrow{a}=(-3,1)$, $\overrightarrow{b}=(1,-2)$, and $\overrightarrow{n}=\overrightarrow{a}+k\overrightarrow{b}$ ($k\in\mathbb{R}$).
$(1)$ If $\overrightarrow{n}$ is perpendicular to the vector $2\overrightarrow{a}-\overrightarrow{b}$, find the value of the real number $k$;
$(2)$ If vector $\ov... | -\frac {1}{3} | 0.9375 | 3,433.5 | 3,116.266667 | 8,192 | |
For each value of $x,$ $g(x)$ is defined to be the minimum value of the three numbers $3x + 3,$ $\frac{1}{3} x + 2,$ and $-\frac{1}{2} x + 8.$ Find the maximum value of $g(x).$ | \frac{22}{5} | 0.5625 | 6,880.6875 | 6,429.555556 | 7,460.714286 | |
Define the polynomials $P_0, P_1, P_2 \cdots$ by:
\[ P_0(x)=x^3+213x^2-67x-2000 \]
\[ P_n(x)=P_{n-1}(x-n), n \in N \]
Find the coefficient of $x$ in $P_{21}(x)$. | 61610 |
To find the coefficient of \( x \) in \( P_{21}(x) \), we need to evaluate the transformation of the polynomial \( P_0(x) \) through a series of substitutions as defined by the recurrence relation \( P_n(x) = P_{n-1}(x-n) \).
Initially, we have:
\[ P_0(x) = x^3 + 213x^2 - 67x - 2000. \]
### Step-by-Step Transformati... | 0.25 | 7,950.25 | 7,225 | 8,192 |
Let $f(x)=x^4+14x^3+52x^2+56x+16$. Let $z_1,z_2,z_3,z_4$ be the four roots of $f$. Find the smallest possible value of $|z_{a}z_{b}+z_{c}z_{d}|$ where $\{a,b,c,d\}=\{1,2,3,4\}$. | 8 | 0 | 8,192 | -1 | 8,192 | |
Given that $x^{2}+y^{2}=1$, determine the maximum and minimum values of $x+y$. | -\sqrt{2} | 1 | 3,329.0625 | 3,329.0625 | -1 | |
If the maximum value of the function $f(x)=a^{x} (a > 0, a \neq 1)$ on $[-2,1]$ is $4$, and the minimum value is $m$, what is the value of $m$? | \frac{1}{2} | 0.375 | 7,950.6875 | 7,639.333333 | 8,137.5 | |
In the quadrilateral \(ABCD\), it is known that \(\angle ABD = \angle ACD = 45^\circ\), \(\angle BAC = 30^\circ\), and \(BC = 1\). Find \(AD\). | \sqrt{2} | 0.0625 | 8,132.1875 | 7,235 | 8,192 | |
Let \( p \) and \( q \) be positive integers such that \( \frac{5}{8}<\frac{p}{q}<\frac{7}{8} \). What is the smallest value of \( p \) such that \( p+q=2005 \)? | 772 | 0.625 | 6,501.1875 | 5,486.7 | 8,192 | |
Five consecutive two-digit positive integers, each less than 50, are not prime. What is the largest of these five integers? | 36 | 0.25 | 7,302.375 | 5,314 | 7,965.166667 | |
Five people, named A, B, C, D, and E, stand in a row. If A and B must be adjacent, and B must be to the left of A, what is the total number of different arrangements? | 24 | 0.5625 | 5,902.25 | 4,273.555556 | 7,996.285714 | |
A rectangular room measures 15-feet by 8-feet and has a triangular extension with a base of 8-feet and a height of 5-feet. How many square yards of carpet are needed to cover the entire floor of the room, including the triangular extension? | 16 | 0 | 3,123.5 | -1 | 3,123.5 | |
Let $x$ and $y$ be distinct real numbers such that
\[
\begin{vmatrix} 2 & 5 & 10 \\ 4 & x & y \\ 4 & y & x \end{vmatrix}
= 0.\]Find $x + y.$ | 30 | 1 | 2,707.75 | 2,707.75 | -1 | |
Triangle $PQR$ is a right triangle with legs $PQ$ and $PR$. Points $U$ and $V$ are on legs $PQ$ and $PR$, respectively so that $PU:UQ = PV:VR = 1:3$. If $QU = 18$ units, and $RV = 45$ units, what is the length of hypotenuse $PQ$? Express your answer in simplest radical form. | 12\sqrt{29} | 0.5625 | 5,485.625 | 4,488.777778 | 6,767.285714 | |
Let $A B C$ be a triangle with $A B=8, B C=15$, and $A C=17$. Point $X$ is chosen at random on line segment $A B$. Point $Y$ is chosen at random on line segment $B C$. Point $Z$ is chosen at random on line segment $C A$. What is the expected area of triangle $X Y Z$ ? | 15 | Let $\mathbb{E}(X)$ denote the expected value of $X$, and let $[S]$ denote the area of $S$. Then $$\begin{aligned} \mathbb{E}([\triangle X Y Z]) & =\mathbb{E}([\triangle A B C]-[\triangle X Y B]-[\triangle Z Y C]-[\triangle X B Z]) \\ & =[\triangle A B C]-\mathbb{E}([\triangle X Y B])-\mathbb{E}([\triangle Z Y C])-[\tr... | 0.3125 | 7,584.1875 | 6,528.8 | 8,063.909091 |
In the diagram, $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{BC}$ respectively. Determine the sum of the $x$ and $y$ coordinates of $F$, the point of intersection of $\overline{AE}$ and $\overline{CD}$. [asy]
size(180); defaultpen(linewidth(.7pt)+fontsize(10pt));
pair A, B, C, D, E, F;
A=(0,6);
B=(0... | \frac{14}{3} | 1 | 2,770.1875 | 2,770.1875 | -1 | |
Grandpa is twice as strong as Grandma, Grandma is three times as strong as Granddaughter, Granddaughter is four times as strong as Doggie, Doggie is five times as strong as Cat, and Cat is six times as strong as Mouse. Grandpa, Grandma, Granddaughter, Doggie, and Cat together with Mouse can pull up the Turnip, but with... | 1237 | 0.5 | 6,898.6875 | 5,827.875 | 7,969.5 | |
Given that the decomposition rate $v$ of a certain type of garbage satisfies the function relationship $v=a\cdot b^{t}$ (where $a$ and $b$ are non-zero constants) with time $t$ (unit: months), and after $6$ months, the decomposition rate is $5\%$, and after $12$ months, the decomposition rate is $10\%$, determine how m... | 32 | 0.5 | 7,388.625 | 6,603.875 | 8,173.375 | |
A rectangular flag is divided into four triangles, labelled Left, Right, Top, and Bottom. Each triangle is to be colored one of red, white, blue, green, and purple such that no two triangles that share an edge are the same color. How many different flags can be made? | 260 | 0.375 | 7,507.0625 | 6,365.5 | 8,192 | |
Given an arithmetic sequence $\{a\_n\}$, where $a\_1=\tan 225^{\circ}$ and $a\_5=13a\_1$, let $S\_n$ denote the sum of the first $n$ terms of the sequence $\{(-1)^na\_n\}$. Determine the value of $S\_{2015}$. | -3022 | 0.5625 | 6,467 | 5,125.333333 | 8,192 | |
Triangle $ABC$ has $AB = 13, BC = 14$, and $AC = 15$. The points $D, E$, and $F$ are the midpoints of $\overline{AB}, \overline{BC}$, and $\overline{AC}$ respectively. Let $X \neq E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA + XB + XC$? | \frac{195}{8} | 1. **Identify the Midpoints and Congruent Triangles**:
- Given $D, E, F$ are midpoints of $\overline{AB}, \overline{BC}, \overline{AC}$ respectively.
- The segments $AD = DB = \frac{13}{2}$, $BE = EC = 7$, and $CF = FA = \frac{15}{2}$.
2. **Congruence of Triangles**:
- $\triangle BDE$ and $\triangle CEF$ are ... | 0.8125 | 6,608.625 | 6,243.230769 | 8,192 |
What is the value of $\sqrt{3! \cdot 3!}$ expressed as a positive integer? | 6 | 1 | 1,494.625 | 1,494.625 | -1 | |
Recall that a perfect square is the square of some integer. How many perfect squares less than 10,000 can be represented as the difference of two consecutive perfect squares? | 50 | 0.875 | 3,460.5625 | 2,784.642857 | 8,192 | |
For nonnegative integers $a$ and $b$ with $a + b \leq 6$, let $T(a, b) = \binom{6}{a} \binom{6}{b} \binom{6}{a + b}$. Let $S$ denote the sum of all $T(a, b)$, where $a$ and $b$ are nonnegative integers with $a + b \leq 6$. Find the remainder when $S$ is divided by $1000$.
Major Note
Most solutions use committee formin... | 564 | Let $c=6-(a+b)$, and note that $\binom{6}{a + b}=\binom{6}{c}$. The problem thus asks for the sum $\binom{6}{a} \binom{6}{b} \binom{6}{c}$ over all $a,b,c$ such that $a+b+c=6$. Consider an array of 18 dots, with 3 columns of 6 dots each. The desired expression counts the total number of ways to select 6 dots by conside... | 0.4375 | 7,270.9375 | 6,086.714286 | 8,192 |
Given a regular tetrahedron $P-ABC$, where points $P$, $A$, $B$, and $C$ are all on the surface of a sphere with radius $\sqrt{3}$. If $PA$, $PB$, and $PC$ are mutually perpendicular, calculate the distance from the center of the sphere to the plane $ABC$. | \dfrac{\sqrt{3}}{3} | 0 | 7,662.8125 | -1 | 7,662.8125 | |
In an isosceles trapezoid, the bases are 40 and 24, and its diagonals are mutually perpendicular. Find the area of the trapezoid. | 1024 | 0.5625 | 6,177.4375 | 4,610.555556 | 8,192 | |
What is the maximum area of a triangle if none of its side lengths exceed 2? | \sqrt{3} | 0.75 | 7,217.875 | 6,893.166667 | 8,192 | |
Given the function $y=\cos({2x+\frac{π}{3}})$, determine the horizontal shift of the graph of the function $y=\sin 2x$. | \frac{5\pi}{12} | 0.375 | 6,720.8125 | 5,340.5 | 7,549 |
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