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 |
|---|---|---|---|---|---|---|
Given the function $f(x) = A\sin(x + \varphi)$ ($A > 0$, $0 < \varphi < \pi$) has a maximum value of 1, and its graph passes through point M ($\frac{\pi}{3}$, $\frac{1}{2}$), then $f(\frac{3\pi}{4})$ = \_\_\_\_\_\_. | -\frac{\sqrt{2}}{2} | 0 | 2,898.6875 | -1 | 2,898.6875 | |
Find the product of all positive integral values of $x$ such that $x^2 - 40x + 399 = q$ for some prime number $q$. Note that there must be at least one such $x$. | 396 | 0.8125 | 5,590.1875 | 4,989.769231 | 8,192 | |
Suppose $xy-5x+2y=30$ , where $x$ and $y$ are positive integers. Find the sum of all possible values of $x$ | 31 | 0.9375 | 3,489.6875 | 3,176.2 | 8,192 | |
Convert the binary number $110101_{(2)}$ to decimal. | 53 | 0.9375 | 2,765.8125 | 2,404.066667 | 8,192 | |
Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = -x^{2} + 4x$.
- (Ⅰ) Find the analytical expression of the function $f(x)$.
- (Ⅱ) Find the minimum value of the function $f(x)$ on the interval $\left[-2,a\right]$ where $\left(a > -2\right)$. | -4 | 0.3125 | 7,050.25 | 6,300 | 7,391.272727 | |
What is the smallest positive integer with exactly 12 positive integer divisors? | 96 | 0 | 4,485.625 | -1 | 4,485.625 | |
Let \( M = \{1, 2, \cdots, 2005\} \), and \( A \) be a subset of \( M \). If for any \( a_i, a_j \in A \) with \( a_i \neq a_j \), an isosceles triangle can be uniquely determined with \( a_i \) and \( a_j \) as side lengths, find the maximum value of \( |A| \). | 11 | 0 | 8,192 | -1 | 8,192 | |
We roll a fair 6-sided die 5 times. What is the probability that we get an odd number in exactly 4 of the 5 rolls? | \dfrac{5}{32} | 1 | 2,145.5625 | 2,145.5625 | -1 | |
If $a$ and $b$ are two unequal positive numbers, then: | \frac {a + b}{2} > \sqrt {ab} > \frac {2ab}{a + b} | To solve this problem, we will use the inequalities between the arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM) of two positive numbers $a$ and $b$. These inequalities are known as the AM-GM-HM inequalities.
1. **Arithmetic Mean (AM)**:
\[
\text{AM} = \frac{a+b}{2}
\]
2. **Geometric Mean (G... | 0 | 7,248.6875 | -1 | 7,248.6875 |
Given that the function $f(x+1)$ is an odd function, $f(x-1)$ is an even function, and $f(0)=2$, find $f(4)=$ | -2 | 0.4375 | 6,337.1875 | 3,952.428571 | 8,192 | |
For any positive integer $n$, let
$f(n) =\begin{cases}\log_{8}{n}, &\text{if }\log_{8}{n}\text{ is rational,}\\ 0, &\text{otherwise.}\end{cases}$
What is $\sum_{n = 1}^{1997}{f(n)}$? | \frac{55}{3} | 1. **Identify when $\log_8 n$ is rational**:
For $\log_8 n$ to be rational, $n$ must be a power of $8$'s base, which is $2$. Thus, $n = 2^k$ for some integer $k$. This is because $\log_8 2^k = k \log_8 2 = \frac{k}{3}$, which is rational.
2. **Determine the range of $k$**:
We need $2^k \leq 1997$. The largest $... | 0.875 | 6,279.25 | 6,154 | 7,156 |
A right triangle \(ABC\) is inscribed in a circle. A chord \(CM\) is drawn from the vertex \(C\) of the right angle, intersecting the hypotenuse at point \(K\). Find the area of triangle \(ABM\) if \(AK : AB = 1 : 4\), \(BC = \sqrt{2}\), and \(AC = 2\). | \frac{9}{19} \sqrt{2} | 0 | 6,691 | -1 | 6,691 | |
Find the largest real number \( C \) such that for any \( n \in \mathbf{Z}_{+} \) and any sequence \(\{x_{k}\}\) satisfying \(0 = x_{0} < x_{1} < x_{2} < \cdots < x_{n} = 1\), the following inequality holds:
\[
\sum_{k=1}^{n} x_{k}^{2} \left(x_{k} - x_{k-1}\right) > C.
\] | 1/3 | 0.4375 | 7,315.1875 | 6,187.857143 | 8,192 | |
A \(101 \times 101\) grid is given, where all cells are initially colored white. You are allowed to choose several rows and paint all the cells in those rows black. Then, choose exactly the same number of columns and invert the color of all cells in those columns (i.e., change white cells to black and black cells to wh... | 5100 | 0.3125 | 7,301.25 | 5,341.6 | 8,192 | |
The operation $\nabla$ is defined by $g \nabla h=g^{2}-h^{2}$. If $g>0$ and $g \nabla 6=45$, what is the value of $g$? | 9 | Using the definition of the operation, $g \nabla 6=45$ gives $g^{2}-6^{2}=45$. Thus, $g^{2}=45+36=81$. Since $g>0$, then $g=\sqrt{81}=9$. | 1 | 725.625 | 725.625 | -1 |
The factors of $x^4+64$ are: | (x^2-4x+8)(x^2+4x+8) | 1. **Identify the Expression**: We start with the expression \(x^4 + 64\).
2. **Recognize the Form**: Notice that \(64\) can be rewritten as \(2^6\), so we have:
\[
x^4 + 64 = x^4 + 2^6
\]
3. **Apply the Sophie Germain Identity**: The Sophie Germain Identity states that \(a^4 + 4b^4\) can be factored as \((a... | 0 | 3,830.8125 | -1 | 3,830.8125 |
For any three real numbers $a$, $b$, and $c$, with $b\neq c$, the operation $\otimes$ is defined by:
\[\otimes(a,b,c)=\frac{a}{b-c}\]
What is $\otimes(\otimes(1,2,3),\otimes(2,3,1),\otimes(3,1,2))$? | -\frac{1}{4} | 1. **Calculate each $\otimes$ operation inside the main operation**:
- $\otimes(1,2,3) = \frac{1}{2-3} = \frac{1}{-1} = -1$
- $\otimes(2,3,1) = \frac{2}{3-1} = \frac{2}{2} = 1$
- $\otimes(3,1,2) = \frac{3}{1-2} = \frac{3}{-1} = -3$
2. **Substitute these values into the main $\otimes$ operation**:
- $\otime... | 1 | 3,007.5 | 3,007.5 | -1 |
What is the positive integer $N$ for which $$ 22^2 \times 55^2 = 10^2 \times N^2 \ ? $$ | 121 | 1 | 1,808.375 | 1,808.375 | -1 | |
A certain set of integers is assigned to the letters of the alphabet such that $H=10$. The value of a word is the sum of its assigned letter values. Given that $THIS=50$, $HIT=35$ and $SIT=40$, find the value of $I$. | 15 | 0 | 7,018.625 | -1 | 7,018.625 | |
Find all pairs of real numbers $(x,y)$ satisfying the system of equations
\begin{align*}
\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2) \\
\frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4).
\end{align*} | x = (3^{1/5}+1)/2, y = (3^{1/5}-1)/2 | By adding and subtracting the two given equations, we obtain the equivalent pair of equations
\begin{align*}
2/x &= x^4 + 10x^2y^2 + 5y^4 \\
1/y &= 5x^4 + 10x^2y^2 + y^4.
\end{align*}
Multiplying the former by $x$ and the latter by $y$, then adding and subtracting the two resulting equations, we obtain another pair of ... | 0 | 8,192 | -1 | 8,192 |
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} | 0 | 2,997.4375 | -1 | 2,997.4375 | |
For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \cdots + n?$ | 16 | 1. **Sum of the first $n$ positive integers**: The sum of the first $n$ positive integers is given by the formula:
\[
1 + 2 + \cdots + n = \frac{n(n+1)}{2}
\]
This formula can be derived by pairing terms from the beginning and end of the sequence, each pair summing to $n+1$, and noting there are $\frac{n}{2... | 0.125 | 8,021.625 | 6,829 | 8,192 |
Add $254_{9} + 627_{9} + 503_{9}$. Express your answer in base 9. | 1485_{9} | 0 | 4,630.6875 | -1 | 4,630.6875 | |
In triangle $ABC,$ $\angle C = \frac{\pi}{2}.$ Find
\[\arctan \left( \frac{a}{b + c} \right) + \arctan \left( \frac{b}{a + c} \right).\] | \frac{\pi}{4} | 1 | 4,486.9375 | 4,486.9375 | -1 | |
Let $d$ be a positive number such that when $145$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$. | 47 | 1 | 4,829.625 | 4,829.625 | -1 | |
If a passenger travels from Moscow to St. Petersburg by a regular train, it will take him 10 hours. If he takes the express train, which he has to wait for more than 2.5 hours, he will arrive 3 hours earlier than the regular train. Find the ratio of the speeds of the express train and the regular train, given that 2 ho... | 2.5 | 0 | 8,007.4375 | -1 | 8,007.4375 | |
$ (a_n)_{n \equal{} 1}^\infty$ is defined on real numbers with $ a_n \not \equal{} 0$ , $ a_na_{n \plus{} 3} \equal{} a_{n \plus{} 2}a_{n \plus{} 5}$ and $ a_1a_2 \plus{} a_3a_4 \plus{} a_5a_6 \equal{} 6$ . So $ a_1a_2 \plus{} a_3a_4 \plus{} \cdots \plus{}a_{41}a_{42} \equal{} ?$ | 42 | 0.1875 | 7,870.75 | 6,478.666667 | 8,192 | |
In a privately-owned company in Wenzhou manufacturing a product, it is known from past data that the fixed daily cost of producing the product is 14,000 RMB. The variable cost increases by 210 RMB for each additional unit produced. The relationship between the daily sales volume $f(x)$ and the production quantity $x$ i... | 30000 | 0.4375 | 7,573.125 | 6,777.428571 | 8,192 | |
Square $EFGH$ has a side length of $40$. Point $Q$ lies inside the square such that $EQ = 16$ and $FQ = 34$. The centroids of $\triangle{EFQ}$, $\triangle{FGQ}$, $\triangle{GHQ}$, and $\triangle{HEQ}$ are the vertices of a convex quadrilateral. Calculate the area of this quadrilateral. | \frac{3200}{9} | 0 | 8,192 | -1 | 8,192 | |
If $\sqrt{\frac{3}{x} + 3} = \frac{5}{3}$, solve for $x$. | -\frac{27}{2} | 1 | 2,348 | 2,348 | -1 | |
Find the greatest common divisor of 91 and 72. | 1 | 0.875 | 1,794.5 | 1,893.428571 | 1,102 | |
Triangle $ABC$ has side-lengths $AB = 12, BC = 24,$ and $AC = 18.$ The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N.$ What is the perimeter of $\triangle AMN?$
$\textbf{(A)}\ 27 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 33 \qqua... | 30 | 0 | 5,939.8125 | -1 | 5,939.8125 | |
Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $(1, \frac{2\sqrt{3}}{3})$, with its left and right foci being $F_1$ and $F_2$ respectively. The chord formed by the intersection of the circle $x^2 + y^2 = 2$ and the line $x + y + b = 0$ has a length of $2$.
(Ⅰ) Find t... | \frac{2\sqrt{3}}{3} | 0 | 8,192 | -1 | 8,192 | |
Find the last two digits of $\tbinom{200}{100}$ . Express the answer as an integer between $0$ and $99$ . (e.g. if the last two digits are $05$ , just write $5$ .) | 20 | 0 | 8,053.5625 | -1 | 8,053.5625 | |
Let $d$ be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than $d$ degrees. What is the minimum possible value for $d$ ? | 120 | 0 | 7,846.4375 | -1 | 7,846.4375 | |
In a circle, there are two mutually perpendicular chords $AB$ and $CD$. Determine the distance between the midpoint of segment $AD$ and the line $BC$, given that $AC=6$, $BC=5$, and $BD=3$. If necessary, round the answer to two decimal places. | 4.24 | 0.125 | 7,619.125 | 6,277.5 | 7,810.785714 | |
In a certain school, 3 teachers are chosen from a group of 6 to give support teaching in 3 remote areas, with each area receiving one teacher. There are restrictions such that teacher A and teacher B cannot go together, and teacher A can only go with teacher C or not go at all. How many different dispatch plans are the... | 42 | 0 | 7,913.5625 | -1 | 7,913.5625 | |
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half dollar. What is the probability that at least 25 cents worth of coins come up heads? | \frac{3}{4} | 0.25 | 7,534.375 | 6,026.5 | 8,037 | |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% | 0.375 | 4,434.3125 | 3,478.666667 | 5,007.7 | |
Given the digits 1, 2, 3, 7, 8, 9, find the smallest sum of two 3-digit numbers that can be obtained by placing each of these digits in one of the six boxes in the given addition problem, with the condition that each number must contain one digit from 1, 2, 3 and one digit from 7, 8, 9. | 417 | 0.125 | 7,949.5 | 6,252 | 8,192 | |
Given the function $f(x)=\sin ^{2}x+a\sin x\cos x-\cos ^{2}x$, and $f(\frac{\pi }{4})=1$.
(1) Find the value of the constant $a$;
(2) Find the smallest positive period and minimum value of $f(x)$. | -\sqrt{2} | 1 | 3,399.75 | 3,399.75 | -1 | |
Mrs. Anna Quick needs to pick up her friend from the airport every Friday. The airport is a certain distance from her home. If she drives at an average speed of 50 miles per hour, she arrives 5 minutes late, and if she drives at an average speed of 75 miles per hour, she arrives 5 minutes early. Determine the speed at ... | 60 | 0.9375 | 3,556.5 | 3,247.466667 | 8,192 | |
Given $sin({α+\frac{π}{4}})=\frac{{12}}{{13}}$, and $\frac{π}{4}＜α＜\frac{{3π}}{4}$, find the value of $\cos \alpha$____. | \frac{7\sqrt{2}}{26} | 0 | 6,857.4375 | -1 | 6,857.4375 | |
In a right triangle, the sides adjacent to the right angle are $a$ and $b$, with the hypotenuse being $c$. A perpendicular from the right angle vertex divides the hypotenuse $c$ into two segments $r$ and $s$, where $r$ is adjacent to $a$ and $s$ is adjacent to $b$. Given that the ratio of $a$ to $b$ is $2 : 5$, find th... | \frac{4}{25} | 0.6875 | 4,830.5625 | 4,237.090909 | 6,136.2 | |
Given two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an acute angle between them, and satisfying $|\overrightarrow{a}|= \frac{8}{\sqrt{15}}$, $|\overrightarrow{b}|= \frac{4}{\sqrt{15}}$. If for any $(x,y)\in\{(x,y)| |x \overrightarrow{a}+y \overrightarrow{b}|=1, xy > 0\}$, it holds that $|x+y|\leqslant ... | \frac{8}{15} | 0.25 | 7,975.6875 | 7,326.75 | 8,192 | |
Find all real numbers $p$ such that the cubic equation
$$
5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p
$$
has two roots that are both natural numbers. | 76 | 0 | 8,192 | -1 | 8,192 | |
(2014•Shanghai) In a certain game, the scores are 1, 2, 3, 4, 5. The random variable $\xi$ represents Xiao Bai's score in this game. If $E(\xi) = 4.2$, then the probability that Xiao Bai scores 5 points is at least ___. | 0.2 | 0.5 | 6,134.125 | 4,765.125 | 7,503.125 | |
Given that $\sin \theta$ and $\cos \theta$ are the two roots of the equation $4x^{2}-4mx+2m-1=0$, and $\frac {3\pi}{2} < \theta < 2\pi$, find the angle $\theta$. | \frac {5\pi}{3} | 0.75 | 5,923.8125 | 5,167.75 | 8,192 | |
Given the function $f(x)=\frac{1}{3}x^{3}+ax^{2}+bx-\frac{2}{3}$, the equation of the tangent line at $x=2$ is $x+y-2=0$.
(I) Find the values of the real numbers $a$ and $b$.
(II) Find the extreme values of the function $f(x)$. | -\frac{2}{3} | 0.875 | 3,358.3125 | 3,330.642857 | 3,552 | |
Add 22 and 62. | 84 | 0.875 | 249.9375 | 254.714286 | 216.5 | |
Calculate the limit of the function:
$\lim _{x \rightarrow \pi} \frac{\sin \left(\frac{x^{2}}{\pi}\right)}{2^{\sqrt{\sin x+1}}-2}$ | \frac{2}{\ln 2} | 0.4375 | 6,937.875 | 5,850.857143 | 7,783.333333 | |
Find all irreducible fractions \( \frac{a}{b} \) that can be represented in the form \( b, a \) (comma separates the decimal representations of natural numbers \( b \) and \( a \)). | \frac{5}{2} | 0 | 8,192 | -1 | 8,192 | |
Given $f(α) = \frac{\sin(2π-α)\cos(π+α)\cos\left(\frac{π}{2}+α\right)\cos\left(\frac{11π}{2}-α\right)}{2\sin(3π + α)\sin(-π - α)\sin\left(\frac{9π}{2} + α\right)}$.
(1) Simplify $f(α)$;
(2) If $α = -\frac{25}{4}π$, find the value of $f(α)$. | -\frac{\sqrt{2}}{4} | 0 | 5,046.6875 | -1 | 5,046.6875 | |
If $x=\frac{1-i\sqrt{3}}{2}$ where $i=\sqrt{-1}$, then $\frac{1}{x^2-x}$ is equal to | -1 | To solve for $\dfrac{1}{x^2-x}$ given $x=\dfrac{1-i\sqrt{3}}{2}$, we first need to compute $x^2$ and $x^2 - x$.
1. **Calculate $x^2$:**
\[
x^2 = \left(\frac{1-i\sqrt{3}}{2}\right)^2 = \frac{(1-i\sqrt{3})^2}{4} = \frac{1 - 2i\sqrt{3} - 3}{4} = \frac{-2 - 2i\sqrt{3}}{4} = -\frac{1}{2} - \frac{i\sqrt{3}}{2}
\]
... | 0.8125 | 4,987.5 | 4,248 | 8,192 |
Consider an octagonal lattice where each vertex is evenly spaced and one unit from its nearest neighbor. How many equilateral triangles have all three vertices in this lattice? Every side of the octagon is extended one unit outward with a single point placed at each extension, keeping the uniform distance of one unit b... | 24 | 0 | 8,192 | -1 | 8,192 | |
What is \[2-4x-6x^2+8+10x-12x^2-14+16x+18x^2\] in terms of $x$? | 22x-4 | 0.9375 | 1,951.1875 | 2,004.066667 | 1,158 | |
To encourage residents to conserve water, a city charges residents for domestic water use in a tiered pricing system. The table below shows partial information on the tiered pricing for domestic water use for residents in the city, each with their own water meter:
| Water Sales Price | Sewage Treatment Price |
|------... | 35 | 0 | 6,819.75 | -1 | 6,819.75 | |
If $n$ is a positive integer such that $n^{3}+2 n^{2}+9 n+8$ is the cube of an integer, find $n$. | 7 | Since $n^{3}<n^{3}+2 n^{2}+9 n+8<(n+2)^{3}$, we must have $n^{3}+2 n^{2}+9 n+8=(n+1)^{3}$. Thus $n^{2}=6 n+7$, so $n=7$. | 0.5 | 6,842.1875 | 5,492.375 | 8,192 |
The value of $\log_{10}{28471}$ is between the consecutive integers $a$ and $b$. Find $a+b$. | 9 | 1 | 5,011.5 | 5,011.5 | -1 | |
Let $x<y$ be positive real numbers such that $\sqrt{x}+\sqrt{y}=4$ and $\sqrt{x+2}+\sqrt{y+2}=5$. Compute $x$. | \frac{49}{36} | Adding and subtracting both equations gives $$\begin{aligned} & \sqrt{x+2}+\sqrt{x}+\sqrt{y+2}+\sqrt{y}=9 \\ & \sqrt{x+2}-\sqrt{x}+\sqrt{y+2}-\sqrt{y}=1 \end{aligned}$$ Substitute $a=\sqrt{x}+\sqrt{x+2}$ and $b=\sqrt{y}+\sqrt{y+2}$. Then since $(\sqrt{x+2}+\sqrt{x})(\sqrt{x+2}-\sqrt{x})=2$, we have $$\begin{gathered} a... | 0.8125 | 5,198.6875 | 4,621.153846 | 7,701.333333 |
A construction company purchased a piece of land for 80 million yuan. They plan to build a building with at least 12 floors on this land, with each floor having an area of 4000 square meters. Based on preliminary estimates, if the building is constructed with x floors (where x is greater than or equal to 12 and x is a ... | 5000 | 0.5625 | 5,913.375 | 4,141.111111 | 8,192 | |
Person A and Person B start simultaneously from points A and B respectively, walking towards each other. Person A starts from point A, and their speed is 4 times that of Person B. The distance between points A and B is \( S \) kilometers, where \( S \) is a positive integer with 8 factors. The first time they meet at p... | 105 | 0 | 7,948.1875 | -1 | 7,948.1875 | |
Find all reals $ k$ such that
\[ a^3 \plus{} b^3 \plus{} c^3 \plus{} d^3 \plus{} 1\geq k(a \plus{} b \plus{} c \plus{} d)
\]
holds for all $ a,b,c,d\geq \minus{} 1$ .
*Edited by orl.* | \frac{3}{4} | 0 | 8,192 | -1 | 8,192 | |
(1) Evaluate the expression $$(\frac{\sqrt{121}}{2018} - 5)^0 + 2^{-2} \cdot (2\frac{1}{4})^{-\frac{1}{2}} - \log_4 3 \cdot \log_3 \sqrt{8}$$;
(2) The function $f(x) = x^{2-m}$ is an odd function defined on $[-3-m, m^2-m]$. Find the value of $f(m)$. | -1 | 0.6875 | 6,748 | 6,098.727273 | 8,176.4 | |
Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after the first step has six octagonal faces an... | 974 | Notice that the number of vertices and edges triple with each step. We always have 3 edges meeting at one vertex, and slicing off a pyramid doesn't change this (we make new vertices from which one edge from the previous step and two of the pyramid edges emanate). So at each step we replace the sliced-off vertex with th... | 0.0625 | 7,972.125 | 4,674 | 8,192 |
Isabella's house has $3$ bedrooms. Each bedroom is $12$ feet long, $10$ feet wide, and $8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $60$ square feet in each bedroom. How many square feet of walls must be painted? | 876 | 1. **Calculate the area of the walls in one bedroom**:
Each bedroom has four walls, two pairs of opposite walls. Each pair consists of:
- Two walls of dimensions $12$ feet (length) by $8$ feet (height).
- Two walls of dimensions $10$ feet (width) by $8$ feet (height).
The area of the walls for each dimens... | 1 | 2,148.9375 | 2,148.9375 | -1 |
In the Cartesian coordinate system, establish a polar coordinate system with the coordinate origin as the pole and the non-negative semi-axis of the $x$-axis as the polar axis. Given that point $A$ has polar coordinates $(\sqrt{2}, \frac{\pi}{4})$, and the parametric equations of line $l$ are $\begin{cases} x = \frac{3... | \frac{4\sqrt{2}}{5} | 0 | 5,875.875 | -1 | 5,875.875 | |
A gold coin is worth $x\%$ more than a silver coin. The silver coin is worth $y\%$ less than the gold coin. Both $x$ and $y$ are positive integers. How many possible values for $x$ are there? | 12 | 0.1875 | 7,692.125 | 6,626.666667 | 7,938 | |
Solve the equations:<br/>$(1)x^{2}-10x-10=0$;<br/>$(2)3\left(x-5\right)^{2}=2\left(5-x\right)$. | \frac{13}{3} | 0.3125 | 2,181.875 | 2,766.6 | 1,916.090909 | |
Uncle Wang has some chickens, rabbits, and geese. Geese stand on two legs during the day and on one leg at night; chickens tuck their heads under their wings when sleeping. Careful Yue Yue discovered that the difference between the number of legs and the number of heads is always the same, regardless of whether it is d... | 14 | 0 | 7,574.3125 | -1 | 7,574.3125 | |
$ABCD$ is a convex quadrilateral such that $AB=2$, $BC=3$, $CD=7$, and $AD=6$. It also has an incircle. Given that $\angle ABC$ is right, determine the radius of this incircle. | \frac{1+\sqrt{13}}{3} | 0 | 7,490.75 | -1 | 7,490.75 | |
Express $\frac{0.\overline{666}}{1.\overline{333}}$ as a common fraction. | \frac{1}{2} | 0.9375 | 3,414.75 | 3,096.266667 | 8,192 | |
Given $f(\alpha)= \dfrac {\sin (\pi-\alpha)\cos (2\pi-\alpha)\cos (-\alpha+ \dfrac {3}{2}\pi)}{\cos ( \dfrac {\pi}{2}-\alpha)\sin (-\pi-\alpha)}$
$(1)$ Simplify $f(\alpha)$;
$(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos (\alpha- \dfrac {3}{2}\pi)= \dfrac {1}{5}$, find the value of $f(\alpha)$;
$(... | -\dfrac {1}{2} | 0.6875 | 4,933.5625 | 4,534.454545 | 5,811.6 | |
The 15th number in a regularly arranged sequence of numbers 2, 1, 4, 3, 6, 5, 8, 7, … is 16. What is the sum of the first 15 numbers? | 121 | 0.875 | 4,291.5625 | 4,161.142857 | 5,204.5 | |
For each real number $x$, let $\lfloor x \rfloor$ denote the greatest integer that does not exceed x. For how many positive integers $n$ is it true that $n<1000$ and that $\lfloor \log_{2} n \rfloor$ is a positive even integer? | 340 | For integers $k$, we want $\lfloor \log_2 n\rfloor = 2k$, or $2k \le \log_2 n < 2k+1 \Longrightarrow 2^{2k} \le n < 2^{2k+1}$. Thus, $n$ must satisfy these inequalities (since $n < 1000$):
$4\leq n <8$
$16\leq n<32$
$64\leq n<128$
$256\leq n<512$
There are $4$ for the first inequality, $16$ for the second, $64$ for th... | 0.8125 | 5,382.375 | 4,815.538462 | 7,838.666667 |
The annual incomes of $1,000$ families range from $8200$ dollars to $98,000$ dollars. In error, the largest income was entered on the computer as $980,000$ dollars. The difference between the mean of the incorrect data and the mean of the actual data is | 882 | 1. **Define the variables:**
Let $S$ be the sum of the incomes of all families except the one with the highest income, which is $98,000$.
2. **Calculate the mean of the actual data:**
The mean of the actual data is calculated by adding the highest income ($98,000$) to the sum $S$ and then dividing by the total n... | 1 | 1,863.6875 | 1,863.6875 | -1 |
Through the focus of the parabola $y^{2}=4x$, two chords $AB$ and $CD$ are drawn perpendicular to each other. Calculate $\left( \frac{1}{|AB|}+ \frac{1}{|CD|} \right)$. | \frac{1}{4} | 0.5 | 7,247.25 | 6,302.5 | 8,192 | |
A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point? | $\sqrt{13}$ | 0 | 6,748 | -1 | 6,748 | |
Find the smallest positive angle $\theta,$ in degrees, for which
\[\cos \theta = \sin 60^\circ + \cos 42^\circ - \sin 12^\circ - \cos 6^\circ.\] | 66^\circ | 0.875 | 4,979.75 | 4,520.857143 | 8,192 | |
The 31st World University Summer Games will be held in Chengdu, Sichuan from July 28th to August 8th, 2023. A company decided to evaluate a certain product under its umbrella for bidding for related endorsement activities. The original selling price of the product was $25 per unit, with an annual sales volume of 80,000... | 30 | 0 | 8,114.5625 | -1 | 8,114.5625 | |
A ship sails on a river. After 6 hours, it returns to its starting point, having covered a distance of 36 km according to the map (naturally, the ship had to move in one direction and then in the opposite direction).
What is the speed of the ship if we assume that it did not spend any time turning around and the speed... | 3 + 3\sqrt{2} | 0.125 | 4,968.1875 | 4,829 | 4,988.071429 | |
A right pyramid has a square base where each side measures 15 cm. The height of the pyramid, measured from the center of the base to the peak, is 15 cm. Calculate the total length of all edges of the pyramid. | 60 + 4\sqrt{337.5} | 0 | 4,693.6875 | -1 | 4,693.6875 | |
For how many even positive integers $n$ less than or equal to 800 is $$(\sin t - i\cos t)^n = \sin nt - i\cos nt$$ true for all real $t$? | 200 | 0.0625 | 7,974.875 | 5,343 | 8,150.333333 | |
15 boys and 20 girls sat around a round table. It turned out that the number of pairs of boys sitting next to each other is one and a half times less than the number of pairs of girls sitting next to each other. Find the number of boy-girl pairs sitting next to each other. | 10 | 0.5 | 6,544.4375 | 4,896.875 | 8,192 | |
Four points, $A, B, C$, and $D$, are chosen randomly on the circumference of a circle with independent uniform probability. What is the expected number of sides of triangle $A B C$ for which the projection of $D$ onto the line containing the side lies between the two vertices? | 3/2 | By linearity of expectations, the answer is exactly 3 times the probability that the orthogonal projection of $D$ onto $A B$ lies interior to the segment. This happens exactly when either $\angle D A B$ or $\angle D B A$ is obtuse, which is equivalent to saying that $A$ and $B$ lie on the same side of the diameter thro... | 0 | 7,850.5625 | -1 | 7,850.5625 |
Express $\sqrt{a} \div \sqrt{b}$ as a common fraction, given:
$$\frac{{\left(\frac{1}{3}\right)}^2 + {\left(\frac{1}{4}\right)}^2}{{\left(\frac{1}{5}\right)}^2 + {\left(\frac{1}{6}\right)}^2} = \frac{25a}{61b}$$ | \frac{5}{2} | 0.8125 | 4,645.125 | 4,427.846154 | 5,586.666667 | |
If two identical, uniform tetrahedral dice, each with the numbers $1$, $2$, $3$, $5$ on their faces, are rolled simultaneously and $x$ represents the sum of the numbers on the two faces facing down, determine the probability that $x$ is not less than $6$. | \frac{1}{2} | 0.875 | 5,691.875 | 5,575.928571 | 6,503.5 | |
If the line $2x+my=2m-4$ is parallel to the line $mx+2y=m-2$, find the value of $m$. | -2 | 0.375 | 5,714.625 | 4,096.166667 | 6,685.7 | |
If
\[
x + \sqrt{x^2 - 1} + \frac{1}{x + \sqrt{x^2 - 1}} = 12,
\]
then find the value of
\[
x^3 + \sqrt{x^6 - 1} + \frac{1}{x^3 + \sqrt{x^6 - 1}}.
\] | 432 | 0.8125 | 5,284.875 | 4,614 | 8,192 | |
A natural number of five digits is called *Ecuadorian*if it satisfies the following conditions: $\bullet$ All its digits are different. $\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = 1 + 3 + 5 + 0$ , but $54210$ is not since ... | 168 | 0.4375 | 7,270.4375 | 6,275.571429 | 8,044.222222 | |
An open box is constructed by starting with a rectangular sheet of metal 10 in. by 14 in. and cutting a square of side $x$ inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is: | 140x - 48x^2 + 4x^3 | To find the volume of the box formed by cutting squares from each corner of a rectangular sheet and folding the sides up, we follow these steps:
1. **Identify the dimensions of the original rectangle:**
The original metal sheet is a rectangle with dimensions $10$ inches by $14$ inches.
2. **Determine the size of t... | 0 | 4,541.75 | -1 | 4,541.75 |
What is the greatest three-digit number that is one more than a multiple of 9 and three more than a multiple of 5? | 973 | 0.9375 | 4,636.875 | 4,399.866667 | 8,192 | |
Consider the graph of $y=f(x)$, which consists of five line segments as described below:
- From $(-5, -4)$ to $(-3, 0)$
- From $(-3, 0)$ to $(-1, -1)$
- From $(-1, -1)$ to $(1, 3)$
- From $(1, 3)$ to $(3, 2)$
- From $(3, 2)$ to $(5, 6)$
What is the sum of the $x$-coordinates of all points where $f(x) = 2.3$? | 4.35 | 0.25 | 6,527.375 | 7,126 | 6,327.833333 | |
A 9 by 9 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard? | 91 | 0.3125 | 7,124 | 6,156.2 | 7,563.909091 | |
Nine integers from 1 to 5 are written on a board. It is known that seven of them are at least 2, six are greater than 2, three are at least 4, and one is at least 5. Find the sum of all the numbers. | 26 | 0.6875 | 4,987.125 | 4,342.363636 | 6,405.6 | |
Let $1=d_1<d_2<d_3<\dots<d_k=n$ be the divisors of $n$ . Find all values of $n$ such that $n=d_2^2+d_3^3$ . | 68 | 0.0625 | 8,176 | 7,936 | 8,192 | |
The sum of three numbers $x$, $y$, and $z$ is 150. If we increase $x$ by 7, decrease $y$ by 12 and multiply $z$ by 4, the three resulting numbers are equal. What is the value of $y$? | \frac{688}{9} | 0.9375 | 4,092.375 | 3,819.066667 | 8,192 | |
Given the equation of line $l$ is $y=x+4$, and the parametric equation of circle $C$ is $\begin{cases} x=2\cos \theta \\ y=2+2\sin \theta \end{cases}$ (where $\theta$ is the parameter), with the origin as the pole and the positive half-axis of $x$ as the polar axis. Establish a polar coordinate system.
- (I) Find the p... | \sqrt{2}+2 | 0 | 6,731.3125 | -1 | 6,731.3125 | |
In triangle $ABC$, $\tan B= \sqrt {3}$, $AB=3$, and the area of triangle $ABC$ is $\frac {3 \sqrt {3}}{2}$. Find the length of $AC$. | \sqrt {7} | 0 | 3,902.375 | -1 | 3,902.375 | |
The value of \( 2 \frac{1}{10} + 3 \frac{11}{100} \) can be calculated. | 5.21 | 0.0625 | 1,547.375 | 2,447 | 1,487.4 |
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