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 the Cartesian coordinate plane, point $P$ is a moving point on the line $x=-1$, point $F(1,0)$, point $Q$ is the midpoint of $PF$, point $M$ satisfies $MQ \perp PF$ and $\overrightarrow{MP}=\lambda \overrightarrow{OF}$, and the tangent line is drawn through point $M$ on the circle $(x-3)^{2}+y^{2}=2$ with tangent po... | \sqrt{6} | 0.125 | 7,729.6875 | 6,119 | 7,959.785714 | |
Let the domain of the function $f(x)$ be $R$. $f(x+1)$ is an odd function, and $f(x+2)$ is an even function. When $x\in [1,2]$, $f(x)=ax^{2}+b$. If $f(0)+f(3)=6$, calculate $f(\frac{9}{2})$. | \frac{5}{2} | 0.125 | 7,617.4375 | 5,139.5 | 7,971.428571 | |
Given a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for any real numbers \( x \) and \( y \), \( f(2x) + f(2y) = f(x+y) f(x-y) \). Additionally, \( f(\pi) = 0 \) and \( f(x) \) is not identically zero. What is the period of \( f(x) \)? | 4\pi | 0 | 8,192 | -1 | 8,192 | |
Given that \(a - b = 2 + \sqrt{3}\) and \(b - c = 2 - \sqrt{3}\), find the value of \(a^2 + b^2 + c^2 - ab - bc - ca\). | 15 | 0.8125 | 4,508.625 | 4,164.384615 | 6,000.333333 | |
Employees from department X are 30, while the employees from department Y are 20. Since employees from the same department do not interact, the number of employees from department X that will shake hands with the employees from department Y equals 30, and the number of employees from department Y that will shake hands ... | 600 | 0.9375 | 740.8125 | 744.733333 | 682 | |
The areas of two squares are in the ratio $25:36$. What is the ratio of their perimeters? Express your answer in the form $a:b$. | 5:6 | 1 | 1,280.5 | 1,280.5 | -1 | |
In the rectangular coordinate system, a polar coordinate system is established with the origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. Given the curve $C:ρ\sin^2θ=2a\cos θ (a > 0)$, the line $l:\begin{cases}x=-2+\frac{\sqrt{2}}{2}t\\y=-4+\frac{\sqrt{2}}{2}t\end{cases} (t$ is the parame... | a=1 | 0.875 | 5,391 | 5,378.714286 | 5,477 | |
Given \(\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j)\), find the value of the expression. | 1010000 | 0.9375 | 4,010.8125 | 3,732.066667 | 8,192 | |
What is the sum of all the positive two-digit integers divisible by both the sum and product of their digits? | 72 | 0.0625 | 8,140.5 | 7,368 | 8,192 | |
Find the minimum value of the expression
$$
\sqrt{x^{2}-2 \sqrt{3} \cdot|x|+4}+\sqrt{x^{2}+2 \sqrt{3} \cdot|x|+12}
$$
as well as the values of $x$ at which it is achieved. | 2 \sqrt{7} | 0.4375 | 6,860.625 | 6,933 | 6,804.333333 | |
How many ordered pairs of integers $(a, b)$ satisfy all of the following inequalities?
\[ \begin{aligned}
a^2 + b^2 &< 25 \\
a^2 + b^2 &< 8a + 4 \\
a^2 + b^2 &< 8b + 4
\end{aligned} \] | 14 | 0 | 7,955.9375 | -1 | 7,955.9375 | |
Using the vertices of a cube as vertices, how many triangular pyramids can you form? | 58 | 0.375 | 7,222 | 5,605.333333 | 8,192 | |
Let \( XYZ \) be a triangle with \( \angle X = 60^\circ \) and \( \angle Y = 45^\circ \). A circle with center \( P \) passes through points \( A \) and \( B \) on side \( XY \), \( C \) and \( D \) on side \( YZ \), and \( E \) and \( F \) on side \( ZX \). Suppose \( AB = CD = EF \). Find \( \angle XPY \) in degrees. | 120 | 0 | 8,192 | -1 | 8,192 | |
Let $\ell_A$ and $\ell_B$ be two distinct perpendicular lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m$ lie on $\ell_A$, and distinct points $B_1, B_2, B_3, \ldots, B_n$ lie on $\ell_B$. Additionally, when segments $\overline{A_iB_j}$ are drawn ... | 244 | We want to derive a general function $f(m,n)$ that indicates the number of bounded regions. Observing symmetry, we know this is a symmetric function about $m$ and $n$. Now let's focus on $f(m+1, n)-f(m, n)$, which is the difference caused by adding one point to the existing $m$ points of line $\ell_A$. This new point, ... | 0 | 8,167.875 | -1 | 8,167.875 |
In square $ABCD$, a point $P$ is chosen at random. The probability that $\angle APB < 90^{\circ}$ is ______. | 1 - \frac{\pi}{8} | 0.25 | 7,854.1875 | 6,987.25 | 8,143.166667 | |
For a finite sequence \(P = \left(p_{1}, p_{2}, \cdots, p_{n}\right)\), the Caesar sum (named after a mathematician Caesar) is defined as \(\frac{s_{1}+s_{2}+\cdots+s_{n}}{n}\), where \(s_{k} = p_{1} + p_{2} + \cdots + p_{k}\) for \(1 \leq k \leq n\). If a sequence of 99 terms \(\left(p_{1}, p_{2}, \cdots, p_{99}\right... | 991 | 0.625 | 5,836.875 | 4,866.7 | 7,453.833333 | |
From an arbitrary tetrahedron, four smaller tetrahedra are separated by four planes passing through the midpoints of the edges emanating from each vertex. Calculate the ratio of the volume of the remaining body to the volume of the original tetrahedron. | 1/2 | 0.0625 | 8,192 | 8,192 | 8,192 | |
(1) Given $$x^{ \frac {1}{2}}+x^{- \frac {1}{2}}=3$$, find the value of $x+x^{-1}$;
(2) Calculate $$( \frac {1}{8})^{- \frac {1}{3}}-3^{\log_{3}2}(\log_{3}4)\cdot (\log_{8}27)+2\log_{ \frac {1}{6}} \sqrt {3}-\log_{6}2$$. | -3 | 0.4375 | 5,148.5625 | 4,002.571429 | 6,039.888889 | |
A jar contains quarters (worth $\$0.25$ each), nickels (worth $\$0.05$ each) and pennies (worth $\$0.01$ each). The value of the quarters is $\$10.00.$ The value of the nickels is $\$10.00.$ The value of the pennies is $\$10.00.$ If Judith randomly chooses one coin from the jar, what is the probability that it is a qua... | \dfrac{1}{31} | 1 | 2,119.3125 | 2,119.3125 | -1 | |
A board of size \(2022 \times 2022\) is given. Lisa and Varya take turns painting \(2 \times 2\) squares on the board with red and blue colors. They agreed that each cell can be painted no more than once in blue and no more than once in red. Cells that are painted blue and then red (or vice versa) become purple. Once a... | 2021 \cdot 2020 | 0 | 8,192 | -1 | 8,192 | |
Under normal circumstances, for people aged between 18 and 38 years old, the regression equation for weight $y$ (kg) based on height $x$ (cm) is $y=0.72x-58.5$. Zhang Honghong, who is neither fat nor thin, has a height of 1.78 meters. His weight should be around \_\_\_\_\_ kg. | 70 | 0.0625 | 756.1875 | 5,028 | 471.4 | |
Using the seven digits $1, 2, 3, 4, 5, 6, 7$ to appropriately arrange them into a 7-digit number so that it is a multiple of 11, how many such numbers can be formed? | 576 | 0.875 | 5,945.3125 | 5,760.5 | 7,239 | |
Given the function $f(x)=\frac{cos2x+a}{sinx}$, if $|f(x)|\leqslant 3$ holds for any $x\in \left(0,\pi \right)$, then the set of possible values for $a$ is ______. | \{-1\} | 0.0625 | 8,032.875 | 6,772 | 8,116.933333 | |
Alice and Bob have an $8 \times 8$ chessboard in front of them. Initially, all the squares are white. Each turn, Alice selects a white square and colors it black. Bob then chooses to color one of the neighboring squares (sharing an edge) black or does nothing. Alice can stop the game whenever she wants. Her goal is to ... | 16 | 0 | 7,572.1875 | -1 | 7,572.1875 | |
Given vectors $\overrightarrow{a} = (\sin x, \cos x)$, $\overrightarrow{b} = (\sin x, \sin x)$, and $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$
(1) If $x \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$, find the range of the function $f(x)$.
(2) Let the sides opposite the acute angles $A$, $B$, and $C$ of tr... | \frac{\sqrt{6} + \sqrt{2}}{2} | 0 | 7,793.4375 | -1 | 7,793.4375 | |
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}| = |\overrightarrow{b}| = |\overrightarrow{a} + \overrightarrow{b}|$, the cosine of the angle between $\overrightarrow{a}$ and $2\overrightarrow{a} - \overrightarrow{b}$ is ______. | \frac{5\sqrt{7}}{14} | 0 | 4,052.0625 | -1 | 4,052.0625 | |
Given the function $y=4\cos (2x+\frac{\pi}{4})$, determine the direction and magnitude of horizontal shift required to obtain the graph of the function $y=4\cos 2x$. | \frac{\pi}{8} | 0.625 | 5,062.875 | 3,548.7 | 7,586.5 | |
The line $10x + 8y = 80$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
A) $\frac{18\sqrt{41} + 40}{\sqrt{41}}$
B) $\frac{360}{17}$
C) $\frac{107}{5}$
D) $\frac{43}{2}$
E) $\frac{281}{13}$ | \frac{18\sqrt{41} + 40}{\sqrt{41}} | 0 | 5,389.25 | -1 | 5,389.25 | |
Roberto has four pairs of trousers, seven shirts, and three jackets. How many different outfits can he put together if an outfit consists of a pair of trousers, a shirt, and a jacket? | 84 | 1 | 1,100.8125 | 1,100.8125 | -1 | |
In tetrahedron \( ABCD \), the dihedral angle between face \( ABC \) and face \( BCD \) is \( 60^\circ \). Vertex \( A \)'s projection onto face \( BCD \) is point \( H \), which is the orthocenter of \( \triangle BCD \). Point \( G \) is the centroid of \( \triangle ABC \). Given that \( AH = 4 \) and \( AB = AC \), f... | \frac{4\sqrt{21}}{9} | 0 | 8,044.1875 | -1 | 8,044.1875 | |
Points \( E, F, M \) are located on the sides \( AB, BC, \) and \( AC \) of triangle \( ABC \), respectively. The segment \( AE \) is one third of side \( AB \), the segment \( BF \) is one sixth of side \( BC \), and the segment \( AM \) is two fifths of side \( AC \). Find the ratio of the area of triangle \( EFM \) ... | 23/90 | 0.6875 | 7,167.8125 | 6,702.272727 | 8,192 | |
The number of 4-digit integers with distinct digits, whose first and last digits' absolute difference is 2, is between 1000 and 9999. | 840 | 0.25 | 7,649.5625 | 6,022.25 | 8,192 | |
Andrea flips a fair coin repeatedly, continuing until she either flips two heads in a row (the sequence $H H$ ) or flips tails followed by heads (the sequence $T H$ ). What is the probability that she will stop after flipping $H H$ ? | 1/4 | The only way that Andrea can ever flip $H H$ is if she never flips $T$, in which case she must flip two heads immediately at the beginning. This happens with probability $\frac{1}{4}$. | 0.1875 | 7,578 | 6,927 | 7,728.230769 |
Given the function $f(x) = 2\sin\omega x\cos\omega x + 2\sqrt{3}\sin^2\omega x - \sqrt{3}$ ($\omega > 0$) has the smallest positive period of $\pi$.
(1) Find the interval of monotonic increase for the function $f(x)$;
(2) The graph of $f(x)$ is obtained by translating the graph of $y=\sin x$ in what way;
(3) If th... | \frac{59\pi}{12} | 0 | 8,192 | -1 | 8,192 | |
From June to August 1861, a total of 1026 inches of rain fell in Cherrapunji, India, heavily influenced by the monsoon season, and the total duration of these summer months is 92 days and 24 hours per day. Calculate the average rainfall in inches per hour. | \frac{1026}{2208} | 0 | 887.0625 | -1 | 887.0625 | |
Let $a$ and $b$ be the numbers obtained by rolling a pair of dice twice. The probability that the equation $x^{2}-ax+2b=0$ has two distinct real roots is $\_\_\_\_\_\_$. | \frac{1}{4} | 0.375 | 6,537.75 | 6,148.833333 | 6,771.1 | |
Three aluminum cans can be recycled to make a new can. How many new cans can eventually be made from 243 aluminum cans? (Remember that the first new cans that are made can then be recycled into even newer cans!) Do not include the original 243 cans in your count. | 121 | 1 | 3,774.75 | 3,774.75 | -1 | |
Let
\[f(x) = \left\{
\begin{array}{cl}
x^2 + 3 & \text{if $x < 15$}, \\
3x - 2 & \text{if $x \ge 15$}.
\end{array}
\right.\]
Find $f^{-1}(10) + f^{-1}(49).$ | \sqrt{7} + 17 | 0 | 6,769.75 | -1 | 6,769.75 | |
What is the ratio of the volume of a cube with edge length four inches to the volume of a cube with edge length two feet? Additionally, calculate the ratio of their surface areas. | \frac{1}{36} | 0.9375 | 2,232.5625 | 2,215.2 | 2,493 | |
Two circles are externally tangent. Lines $\overline{PAB}$ and $\overline{PA'B'}$ are common tangents with points $A$, $A'$ on the smaller circle and $B$, $B'$ on the larger circle. If $PA=AB=5$ and the radius of the larger circle is 3 times the radius of the smaller circle, find the area of the smaller circle. | 5\pi | 0 | 7,479.8125 | -1 | 7,479.8125 | |
Given an odd function $f(x)$ defined on $R$ satisfying $f(x+1)=f(1-x)$, when $0\leqslant x\leqslant 1$, $f(x)=x^{3}$, then $f(2019)=\_\_\_\_\_\_$. | -1 | 0.75 | 5,165.4375 | 4,156.583333 | 8,192 | |
The shaded region formed by the two intersecting perpendicular rectangles, in square units, is | 38 | 1. **Identify the areas of the individual rectangles**:
- The first rectangle has dimensions $2 \times 10$. Therefore, its area is:
\[
2 \cdot 10 = 20 \text{ square units}
\]
- The second rectangle has dimensions $3 \times 8$. Therefore, its area is:
\[
3 \cdot 8 = 24 \text{ square units... | 0 | 4,342.1875 | -1 | 4,342.1875 |
A five-digit integer is chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit is an even number and less than 6? Express your answer as a common fraction. | \frac{3}{10} | 1 | 2,145.5625 | 2,145.5625 | -1 | |
In how many ways can 10 people be seated in a row of chairs if four of the people, Alice, Bob, Cindy, and Dave, refuse to sit in four consecutive seats? | 3507840 | 0.8125 | 3,617.1875 | 2,561.461538 | 8,192 | |
Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that for all $x,y\in{{\mathbb{R}}}$ holds
$f(x^2)+f(2y^2)=(f(x+y)+f(y))(f(x-y)+f(y))$ | $f(x) = \frac{1}{2},f(x) = 0,f(x) = x^2$ |
To solve the functional equation
\[
f(x^2) + f(2y^2) = (f(x+y) + f(y))(f(x-y) + f(y))
\]
for all functions \( f: \mathbb{R} \to \mathbb{R} \), we will analyze the equation under specific substitutions and deduce the form of \( f(x) \).
### Step 1: Substitution and Exploration
1. **Substituting \( x = 0 \):**
\... | 0 | 7,974 | -1 | 7,974 |
\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}+\sqrt{5}} equals | \sqrt{2}+\sqrt{3}-\sqrt{5} | To simplify the expression $\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$, we will use the technique of rationalizing the denominator.
1. **Rationalize the Denominator**:
Multiply the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{2} - (\sqrt{3} + \sqrt{5})$:
\[
\frac{2\sqr... | 0.875 | 4,618.5625 | 4,108.071429 | 8,192 |
If for any ${x}_{1},{x}_{2}∈[1,\frac{π}{2}]$, $x_{1} \lt x_{2}$, $\frac{{x}_{2}sin{x}_{1}-{x}_{1}sin{x}_{2}}{{x}_{1}-{x}_{2}}＞a$ always holds, then the maximum value of the real number $a$ is ______. | -1 | 0.125 | 8,131.4375 | 7,707.5 | 8,192 | |
Suppose a function $f(x)$ is defined on the domain $[-8,4]$. If we define a new function $g(x)$ by $$g(x) = f(-2x),$$ then what is the domain of $g(x)$? Express your answer in interval notation. | [-2,4] | 1 | 3,038.5 | 3,038.5 | -1 | |
What is the constant term of the expansion of $\left(5x + \dfrac{1}{3x}\right)^8$? | \frac{43750}{81} | 0.875 | 5,876.8125 | 5,546.071429 | 8,192 | |
What is the value of $\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}$? | \frac{5}{3} | To solve the problem, we start by simplifying the expression \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}.\]
1. **Factor out common terms**:
\[
\frac{2^{2014} + 2^{2012}}{2^{2014} - 2^{2012}} = \frac{2^{2012}(2^2 + 1)}{2^{2012}(2^2 - 1)}
\]
Here, we factor out $2^{2012}$ from both the numerator and the de... | 0.9375 | 1,887.6875 | 1,934 | 1,193 |
What is the value of $x$ for which $|3x+5|$ is not positive? Express your answer as a common fraction. | -\frac{5}{3} | 1 | 990.6875 | 990.6875 | -1 | |
Given the nine-sided regular polygon $A_1A_2A_3A_4A_5A_6A_7A_8A_9$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $\{A_1, A_2, \ldots A_9\}$? | 66 | 0.0625 | 8,095.8125 | 6,653 | 8,192 | |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $|\overrightarrow{a}-\overrightarrow{b}|=\sqrt{7}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 0.125 | 1,675.75 | 1,825.5 | 1,654.357143 | |
During the FIFA World Cup in Russia, a certain store sells a batch of football commemorative books. The cost price of each book is $40$ yuan, and the selling price is set not less than $44$ yuan, with a profit margin not exceeding $30\%$. It was found during the sales period that when the selling price is set at $44$ y... | 2640 | 0.625 | 4,941.3125 | 4,015.6 | 6,484.166667 | |
Given a circle $C$: $x^{2}+y^{2}-2x-2ay+a^{2}-24=0$ ($a\in\mathbb{R}$) whose center lies on the line $2x-y=0$.
$(1)$ Find the value of the real number $a$;
$(2)$ Find the minimum length of the chord formed by the intersection of circle $C$ and line $l$: $(2m+1)x+(m+1)y-7m-4=0$ ($m\in\mathbb{R}$). | 4\sqrt{5} | 0.6875 | 6,307.0625 | 5,450.272727 | 8,192 | |
On a busy afternoon, David decides to drink a cup of water every 20 minutes to stay hydrated. Assuming he maintains this pace, how many cups of water does David drink in 3 hours and 45 minutes? | 11.25 | 0.125 | 398.0625 | 396 | 398.357143 | |
What is the sum of the solutions to the equation $\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}$? | 337 | Let $y = \sqrt[4]{x}$. Then we have $y(7 - y) = 12$, or, by simplifying, \[y^2 - 7y + 12 = (y - 3)(y - 4) = 0.\]
This means that $\sqrt[4]{x} = y = 3$ or $4$.
Thus the sum of the possible solutions for $x$ is $4^4 + 3^4 = \boxed{337}$. | 1 | 1,608.5 | 1,608.5 | -1 |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing 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\% | 0.5625 | 3,279 | 2,740.777778 | 3,971 | |
Juan rolls a fair regular dodecahedral die marked with the numbers 1 through 12. Then Amal rolls a fair eight-sided die. What is the probability that the product of the two rolls is a multiple of 4? | \frac{7}{16} | 0.0625 | 7,822.75 | 2,284 | 8,192 | |
Let \( A = \{1, 2, \cdots, 10\} \). If the equation \( x^2 - bx - c = 0 \) satisfies \( b, c \in A \) and the equation has at least one root \( a \in A \), then the equation is called a "beautiful equation". Find the number of "beautiful equations". | 12 | 0.0625 | 8,109.375 | 6,870 | 8,192 | |
In an equilateral triangle $ABC$ with side length $6$, point $D$ is the midpoint of $BC$. Calculate $\tan{\angle BAD}$. | \frac{1}{\sqrt{3}} | 0 | 4,086.1875 | -1 | 4,086.1875 | |
An octopus told me that his underwater cave is $567_{8}$ years old. How many years is this in base ten? | 375 | 1 | 1,487.5625 | 1,487.5625 | -1 | |
If the complex number $a^2 - 1 + (a - 1)i$ (where $i$ is the imaginary unit) is purely imaginary, calculate the real number $a$. | -1 | 0.625 | 3,892.75 | 2,607.6 | 6,034.666667 | |
In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $A$ and $C$ coincide, forming the pentagon $ABEFD$. What is the length of segment $EF$? Express your answer in simplest radical form. | 2\sqrt{5} | 0.5 | 5,923.8125 | 4,947.25 | 6,900.375 | |
A math test consists of 12 multiple-choice questions, each worth 5 points. It is known that a student is confident in correctly answering 6 of these questions. For another three questions, the student can eliminate one incorrect option. For two questions, the student can eliminate two incorrect options. For the last qu... | 41.25 | 0.0625 | 6,236.5625 | 7,639 | 6,143.066667 | |
Given the function $f(x) = 2\sin^2x + \cos\left(\frac{\pi}{3} - 2x\right)$.
(1) Find the decreasing interval of $f(x)$ on $[0, \pi]$.
(2) Let $\triangle ABC$ have internal angles $A$, $B$, $C$ opposite sides $a$, $b$, $c$ respectively. If $f(A) = 2$, and the vector $\overrightarrow{m} = (1, 2)$ is collinear with th... | \sqrt{3} | 0.8125 | 6,233.0625 | 5,781 | 8,192 | |
Let \( ABC \) be a triangle. The midpoints of the sides \( BC \), \( AC \), and \( AB \) are denoted by \( D \), \( E \), and \( F \) respectively.
The two medians \( AD \) and \( BE \) are perpendicular to each other and their lengths are \(\overline{AD} = 18\) and \(\overline{BE} = 13.5\).
Calculate the length of t... | 22.5 | 0.1875 | 7,055 | 5,011 | 7,526.692308 | |
Julie works for 48 hours per week for 12 weeks during the summer, making $\$5000$. If she works for 48 weeks during the school year at the same rate of pay and needs to make another $\$5000$, how many hours per week must she work? | 12 | 0.3125 | 6,539.375 | 4,608.8 | 7,416.909091 | |
Let $a, b, c$, and $d$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
a^2+b^2 &=& c^2+d^2 &=& 2016, \\
ac &=& bd &=& 1024.
\end{array}
\]
If $S = a + b + c + d$, compute the value of $\lfloor S \rfloor$. | 127 | 0.25 | 8,041.4375 | 7,589.75 | 8,192 | |
Given that $\binom{18}{8}=31824$, $\binom{18}{9}=48620$, and $\binom{18}{10}=43758$, calculate $\binom{20}{10}$. | 172822 | 0 | 8,192 | -1 | 8,192 | |
At a bus stop near Absent-Minded Scientist's house, two bus routes stop: #152 and #251. Both go to the subway station. The interval between bus #152 is exactly 5 minutes, and the interval between bus #251 is exactly 7 minutes. The intervals are strictly observed, but these two routes are not coordinated with each other... | 5/14 | 0.375 | 7,680.4375 | 6,827.833333 | 8,192 | |
Given $(625^{\log_2 250})^{\frac{1}{4}}$, find the value of the expression. | 250 | 0.0625 | 7,975 | 4,720 | 8,192 | |
Calculate the definite integral:
$$
\int_{\pi / 2}^{2 \pi} 2^{8} \cdot \cos ^{8} x \, dx
$$ | 105\pi | 0.3125 | 6,989.9375 | 6,037.6 | 7,422.818182 | |
In a particular sequence, the first term is $a_1=1007$ and the second term is $a_2=1008$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = 2n$ for all $n \geq 1$. Determine $a_{500}$. | 1339 | 0 | 7,953.5 | -1 | 7,953.5 | |
Eight distinct integers are picked at random from $\{1,2,3,\ldots,15\}$. What is the probability that, among those selected, the third smallest is $5$? | \frac{4}{17} | 0.3125 | 4,808.0625 | 3,909.2 | 5,216.636364 | |
It is known that
\[
\begin{array}{l}
1 = 1^2 \\
1 + 3 = 2^2 \\
1 + 3 + 5 = 3^2 \\
1 + 3 + 5 + 7 = 4^2
\end{array}
\]
If \(1 + 3 + 5 + \ldots + n = 20^2\), find \(n\). | 39 | 1 | 2,843.4375 | 2,843.4375 | -1 | |
In $\triangle ABC$, the length of the side opposite to angle $A$ is $2$, and the vectors $\overrightarrow{m} = (2, 2\cos^2\frac{B+C}{2} - 1)$ and $\overrightarrow{n} = (\sin\frac{A}{2}, -1)$.
1. Find the value of angle $A$ when the dot product $\overrightarrow{m} \cdot \overrightarrow{n}$ is at its maximum.
2. Under t... | \sqrt{3} | 1 | 5,247.5625 | 5,247.5625 | -1 | |
The sum of Alice's weight and Clara's weight is 220 pounds. If you subtract Alice's weight from Clara's weight, you get one-third of Clara's weight. How many pounds does Clara weigh? | 88 | 0 | 1,666.25 | -1 | 1,666.25 | |
\(\log_{5} x + \log_{25} x = \log_{1/5} \sqrt{3}\). | \frac{1}{\sqrt[3]{3}} | 0 | 3,497.25 | -1 | 3,497.25 | |
Given $0 < \beta < \alpha < \frac{\pi}{2}$, point $P(1,4 \sqrt{3})$ is a point on the terminal side of angle $\alpha$, and $\sin \alpha \sin \left(\frac{\pi}{2}-\beta \right)+\cos \alpha \cos \left(\frac{\pi}{2}+\beta \right)= \frac{3 \sqrt{3}}{14}$, calculate the value of angle $\beta$. | \frac{\pi}{3} | 0.75 | 6,241.375 | 5,591.166667 | 8,192 | |
Given $0 \leq x_0 < 1$, for all integers $n > 0$, let
$$
x_n = \begin{cases}
2x_{n-1}, & \text{if } 2x_{n-1} < 1,\\
2x_{n-1} - 1, & \text{if } 2x_{n-1} \geq 1.
\end{cases}
$$
Find the number of initial values of $x_0$ such that $x_0 = x_6$. | 64 | 0 | 7,716.875 | -1 | 7,716.875 | |
Given that the probability that a ball is tossed into bin k is 3^(-k) for k = 1,2,3,..., find the probability that the blue ball is tossed into a higher-numbered bin than the yellow ball. | \frac{7}{16} | 0 | 7,939.75 | -1 | 7,939.75 | |
The 2-digit integers from 19 to 92 are written consecutively to form the integer \(N=192021\cdots9192\). Suppose that \(3^k\) is the highest power of 3 that is a factor of \(N\). What is \(k\)? | 1 | To find the highest power of 3 that divides $N$, we need to determine the divisibility of $N$ by powers of 3. We can use the fact that a number is divisible by 3 if the sum of its digits is divisible by 3, and it is divisible by 9 if the sum of its digits is divisible by 9.
1. **Summing the digits of $N$:**
- **Uni... | 0.25 | 7,949.3125 | 7,487.75 | 8,103.166667 |
Find the number of six-digit palindromes. | 900 | 0.75 | 3,070.25 | 1,892.25 | 6,604.25 | |
An acute triangle $ABC$ is inscribed in a circle of radius 1 with centre $O;$ all the angles of $ABC$ are greater than $45^\circ.$
$B_{1}$ is the foot of perpendicular from $B$ to $CO,$ $B_{2}$ is the foot of perpendicular from $B_{1}$ to $AC.$
Similarly, $C_{1}$ is the foot of perpendicular from $C$ to $BO,$ $C_{2}$... | \frac{1}{2} |
Given an acute triangle \( ABC \) inscribed in a circle with radius 1 and center \( O \), where all angles of \( \triangle ABC \) are greater than \( 45^\circ \), we are tasked with finding the circumradius of triangle \( A_3B_3C_3 \). The construction is defined as follows:
- \( B_1 \): foot of the perpendicular fro... | 0 | 8,192 | -1 | 8,192 |
A loonie is a $\$ 1$ coin and a dime is a $\$ 0.10$ coin. One loonie has the same mass as 4 dimes. A bag of dimes has the same mass as a bag of loonies. The coins in the bag of loonies are worth $\$ 400$ in total. How much are the coins in the bag of dimes worth? | 160 | Since the coins in the bag of loonies are worth $\$ 400$, then there are 400 coins in the bag. Since 1 loonie has the same mass as 4 dimes, then 400 loonies have the same mass as $4(400)$ or 1600 dimes. Therefore, the bag of dimes contains 1600 dimes, and so the coins in this bag are worth $\$ 160$. | 1 | 1,921.875 | 1,921.875 | -1 |
Choose any four distinct digits $w, x, y, z$ and form the four-digit numbers $wxyz$ and $zyxw$. What is the greatest common divisor of the numbers of the form $wxyz + zyxw + wxyz \cdot zyxw$? | 11 | 0 | 8,179.1875 | -1 | 8,179.1875 | |
What is the coefficient of $x^3$ when $$x^4-3x^3 + 5x^2-6x + 1$$is multiplied by $$2x^3 - 3x^2 + 4x + 7$$and the like terms are combined? | 19 | 0.5625 | 5,395.875 | 3,934.666667 | 7,274.571429 | |
10 - 1.05 ÷ [5.2 × 14.6 - (9.2 × 5.2 + 5.4 × 3.7 - 4.6 × 1.5)] = ? | 9.93 | 0.375 | 919.6875 | 1,260.5 | 715.2 | |
Find the focus of the parabola $y = 4x^2 - 3.$ | \left( 0, -\frac{47}{16} \right) | 1 | 3,225.4375 | 3,225.4375 | -1 | |
How many four-digit positive integers $x$ satisfy $3874x + 481 \equiv 1205 \pmod{31}$? | 290 | 0.8125 | 4,249.1875 | 3,604.307692 | 7,043.666667 | |
Given a geometric sequence $\{a_{n}\}$ with the sum of the first $n$ terms denoted as $S_{n}$, satisfying $S_{n} = 2^{n} + r$ (where $r$ is a constant). Define $b_{n} = 2\left(1 + \log_{2} a_{n}\right)$ for $n \in \mathbf{N}^{*}$.
(1) Find the sum of the first $n$ terms of the sequence $\{a_{n} b_{n}\}$, denoted as $T... | \frac{3 \sqrt{2}}{4} | 0 | 8,175.0625 | -1 | 8,175.0625 | |
Convert the point $(\rho,\theta,\phi) = \left( 12, \frac{7 \pi}{6}, \frac{\pi}{3} \right)$ in spherical coordinates to rectangular coordinates. | (-9, -3 \sqrt{3}, 6) | 0.9375 | 2,452.75 | 2,455.6 | 2,410 | |
Let the positive divisors of \( 2014^{2} \) be \( d_{1}, d_{2}, \cdots, d_{k} \). Then,
$$
\frac{1}{d_{1}+2014}+\frac{1}{d_{2}+2014}+\cdots+\frac{1}{d_{k}+2014} \quad = \quad ? | \frac{27}{2014} | 0 | 7,702.8125 | -1 | 7,702.8125 | |
Given that the hyperbola $C_2$ and the ellipse $C_1$: $$\frac {x^{2}}{4} + \frac {y^{2}}{3} = 1$$ have the same foci, the eccentricity of the hyperbola $C_2$ when the area of the quadrilateral formed by their four intersection points is maximized is ______. | \sqrt {2} | 0 | 6,746.125 | -1 | 6,746.125 | |
Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\begin{align*}
a + b + c + d &= 6, \\
a^2 + b^2 + c^2 + d^2 &= 12.
\end{align*}Let $m$ and $M$ denote minimum and maximum values of
\[4(a^3 + b^3 + c^3 + d^3) - (a^4 + b^4 + c^4 + d^4),\]respectively. Find $m + M.$ | 84 | 0 | 8,092.3125 | -1 | 8,092.3125 | |
$(1)$ Solve the fractional equation: $\frac{x}{x+1}=\frac{2x}{3x+3}+1$;<br/>$(2)$ Simplify first ($\frac{x+2}{{x}^{2}-2x}-\frac{x-1}{{x}^{2}-4x+4})\div \frac{x+2}{{x}^{3}-4x}$, then choose a suitable number from $2$, $0$, $-1$ to substitute into the simplified result for evaluation. | \frac{5}{3} | 0.8125 | 4,586.6875 | 3,754.692308 | 8,192 | |
Given the function $f(2x+1)=x^{2}-2x$, determine the value of $f(\sqrt{2})$. | \frac{5-4\sqrt{2}}{4} | 0 | 3,368.3125 | -1 | 3,368.3125 | |
Convert the binary number $101101_2$ to an octal number. The result is | 55_8 | 0.4375 | 3,531.4375 | 3,158.714286 | 3,821.333333 | |
How many 5 digit positive integers are there such that each of its digits, except for the last one, is greater than or equal to the next digit? | 715 | 0 | 7,651.1875 | -1 | 7,651.1875 |
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