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 set $\{1,2,3,5,8,13,21,34,55\}$, calculate the number of integers between 3 and 89 that cannot be written as the sum of exactly two elements of the set. | 53 | 0 | 8,026.1875 | -1 | 8,026.1875 | |
Given $F_{1}$ and $F_{2}$ are the foci of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$, a regular triangle $M F_{1} F_{2}$ is constructed with $F_{1} F_{2}$ as one side. If the midpoint of the side $M F_{1}$ lies on the hyperbola, what is the eccentricity of the hyperbola? | $\sqrt{3}+1$ | 0 | 4,664.3125 | -1 | 4,664.3125 | |
$A B C D$ is a parallelogram satisfying $A B=7, B C=2$, and $\angle D A B=120^{\circ}$. Parallelogram $E C F A$ is contained in $A B C D$ and is similar to it. Find the ratio of the area of $E C F A$ to the area of $A B C D$. | \frac{39}{67} | First, note that $B D$ is the long diagonal of $A B C D$, and $A C$ is the long diagonal of $E C F A$. Because the ratio of the areas of similar figures is equal to the square of the ratio of their side lengths, we know that the ratio of the area of $E C F A$ to the area of $A B C D$ is equal to the ratio $\frac{A C^{2... | 0 | 8,192 | -1 | 8,192 |
A four-digit number has the following properties:
(a) It is a perfect square;
(b) Its first two digits are equal
(c) Its last two digits are equal.
Find all such four-digit numbers. | 7744 | 0.6875 | 5,764.0625 | 4,660.454545 | 8,192 | |
For any real number $t$ , let $\lfloor t \rfloor$ denote the largest integer $\le t$ . Suppose that $N$ is the greatest integer such that $$ \left \lfloor \sqrt{\left \lfloor \sqrt{\left \lfloor \sqrt{N} \right \rfloor}\right \rfloor}\right \rfloor = 4 $$ Find the sum of digits of $N$ . | 24 | 0.625 | 6,151.4375 | 4,944.1 | 8,163.666667 | |
If the graph of the function $f(x) = (1-x^2)(x^2+ax+b)$ is symmetric about the line $x = -2$, then the maximum value of $f(x)$ is ______. | 16 | 0.375 | 7,444.1875 | 6,597 | 7,952.5 | |
Two students, A and B, each choose 2 out of 6 extracurricular reading materials. Calculate the number of ways in which the two students choose extracurricular reading materials such that they have exactly 1 material in common. | 120 | 0.5 | 6,407.875 | 4,753.25 | 8,062.5 | |
On the island of Misfortune with a population of 96 people, the government decided to implement five reforms. Each reform is disliked by exactly half of the citizens. A citizen protests if they are dissatisfied with more than half of all the reforms. What is the maximum number of people the government can expect at th... | 80 | 0 | 7,457.6875 | -1 | 7,457.6875 | |
Given the areas of three squares in the diagram, find the area of the triangle formed. The triangle shares one side with each of two squares and the hypotenuse with the third square.
[asy]
/* Modified AMC8-like Problem */
draw((0,0)--(10,0)--(10,10)--cycle);
draw((10,0)--(20,0)--(20,10)--(10,10));
draw((0,0)--(0,-10)--... | 50 | 0 | 8,192 | -1 | 8,192 | |
Given that $C_{n}^{4}$, $C_{n}^{5}$, and $C_{n}^{6}$ form an arithmetic sequence, find the value of $C_{n}^{12}$. | 91 | 0.625 | 6,270.4375 | 5,383.8 | 7,748.166667 | |
A market survey shows that the price $f(x)$ (in yuan) and the sales volume $g(x)$ (in units) of a certain product in the past $20$ days are both functions of time $x$ (in days), and the price satisfies $f(x)=20-\frac{1}{2}|x-10|$, and the sales volume satisfies $g(x)=80-2x$, where $0\leqslant x\leqslant 20$, $x\in N$.
... | 600 | 0.6875 | 6,242.875 | 5,356.909091 | 8,192 | |
Person A arrives between 7:00 and 8:00, while person B arrives between 7:20 and 7:50. The one who arrives first waits for the other for 10 minutes, after which they leave. Calculate the probability that the two people will meet. | \frac{1}{3} | 0.125 | 8,030.6875 | 6,901.5 | 8,192 | |
Given the function $f(x)=\frac{1}{1+{2}^{x}}$, if the inequality $f(ae^{x})\leqslant 1-f\left(\ln a-\ln x\right)$ always holds, then the minimum value of $a$ is ______. | \frac{1}{e} | 0.3125 | 7,637.5 | 6,417.6 | 8,192 | |
In the sequence \(\left\{a_{n}\right\}\), \(a_{1} = -1\), \(a_{2} = 1\), \(a_{3} = -2\). Given that for all \(n \in \mathbf{N}_{+}\), \(a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{n} + a_{n+1} + a_{n+2} + a_{n+3}\), and \(a_{n+1} a_{n+2} a_{n+3} \neq 1\), find the sum of the first 4321 terms of the sequence \(S_{4321}\). | -4321 | 0.125 | 7,746.0625 | 7,117 | 7,835.928571 | |
Find the number of pairs $(a, b)$ of positive integers with the property that the greatest common divisor of $a$ and $ b$ is equal to $1\cdot 2 \cdot 3\cdot ... \cdot50$, and the least common multiple of $a$ and $ b$ is $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$. | 32768 |
To solve this problem, we need to examine the conditions given for the pairs \((a, b)\) of positive integers:
1. The greatest common divisor (GCD) of \(a\) and \(b\) is \(1 \cdot 2 \cdot 3 \cdot \ldots \cdot 50\).
2. The least common multiple (LCM) of \(a\) and \(b\) is \(1^2 \cdot 2^2 \cdot 3^2 \cdot \ldots \cdot 50^... | 0 | 8,192 | -1 | 8,192 |
Cookie Monster now finds a bigger cookie with the boundary described by the equation $x^2 + y^2 - 8 = 2x + 4y$. He wants to know both the radius and the area of this cookie to determine if it's enough for his dessert. | 13\pi | 0.5625 | 1,322.9375 | 1,686.777778 | 855.142857 | |
A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden? | 400 | 1 | 1,345.75 | 1,345.75 | -1 | |
What integer value of $n$ will satisfy $n + 10 > 11$ and $-4n > -12$? | 2 | 1 | 1,161.875 | 1,161.875 | -1 | |
Consider the 100th, 101st, and 102nd rows of Pascal's triangle, denoted as sequences $(p_i)$, $(q_i)$, and $(r_i)$ respectively. Calculate:
\[
\sum_{i = 0}^{100} \frac{q_i}{r_i} - \sum_{i = 0}^{99} \frac{p_i}{q_i}.
\] | \frac{1}{2} | 0.0625 | 8,124.0625 | 7,105 | 8,192 | |
Given \( x^{2} + y^{2} - 2x - 2y + 1 = 0 \) where \( x, y \in \mathbb{R} \), find the minimum value of \( F(x, y) = \frac{x + 1}{y} \). | 3/4 | 0.4375 | 7,447.4375 | 6,490.142857 | 8,192 | |
In a plane, there are 7 points, with no three points being collinear. If 18 line segments are connected between these 7 points, then at most how many triangles can these segments form? | 23 | 0 | 8,181.875 | -1 | 8,181.875 | |
On the hypotenuse \( AB \) of a right triangle \( ABC \), square \( ABDE \) is constructed externally with \( AC=2 \) and \( BC=5 \). In what ratio does the angle bisector of angle \( C \) divide side \( DE \)? | 2 : 5 | 0.4375 | 7,294.25 | 6,592.714286 | 7,839.888889 | |
Given that \( x - \frac{1}{x} = 5 \), find the value of \( x^4 - \frac{1}{x^4} \). | 727 | 0 | 5,619.375 | -1 | 5,619.375 | |
The numbers in the sequence $101$, $104$, $109$, $116$,$\ldots$ are of the form $a_n=100+n^2$, where $n=1,2,3,\ldots$ For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n+1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.
| 401 | 0.9375 | 4,803.625 | 4,577.733333 | 8,192 | |
In the Cartesian coordinate system \(xOy\), the equation of the ellipse \(C\) is given by the parametric form:
\[
\begin{cases}
x=5\cos\varphi \\
y=3\sin\varphi
\end{cases}
\]
where \(\varphi\) is the parameter.
(I) Find the general equation of the straight line \(l\) that passes through the right focus of the ellipse ... | 30 | 0.75 | 5,687.5 | 4,852.666667 | 8,192 | |
Determine how many prime dates occurred in 2008, a leap year. A "prime date" is when both the month and the day are prime numbers. | 53 | 0.4375 | 5,974.8125 | 4,988.142857 | 6,742.222222 | |
A semicircle of diameter 1 sits at the top of a semicircle of diameter 2, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a $\textit{lune}$. Determine the area of this lune. Express your answer in terms of $\pi$ and in simplest radical form.
[asy]
fill((0,2.73)..(1,1... | \frac{\sqrt{3}}{4} - \frac{1}{24}\pi | 0 | 8,192 | -1 | 8,192 | |
In a right triangle $PQR$, medians are drawn from $P$ and $Q$ to divide segments $\overline{QR}$ and $\overline{PR}$ in half, respectively. If the length of the median from $P$ to the midpoint of $QR$ is $8$ units, and the median from $Q$ to the midpoint of $PR$ is $4\sqrt{5}$ units, find the length of segment $\overli... | 16 | 0 | 5,699.4375 | -1 | 5,699.4375 | |
Given the function $f(x)=\ln x+ \frac {1}{2}ax^{2}-x-m$ ($m\in\mathbb{Z}$).
(I) If $f(x)$ is an increasing function, find the range of values for $a$;
(II) If $a < 0$, and $f(x) < 0$ always holds, find the minimum value of $m$. | -1 | 0 | 8,140.3125 | -1 | 8,140.3125 | |
Find the limit, when $n$ tends to the infinity, of $$ \frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k} $$ | \sqrt{3} | 0.5 | 6,927.75 | 5,663.5 | 8,192 | |
Given that the domains of functions f(x) and g(x) are both R, and f(x) + g(2-x) = 5, g(x) - f(x-4) = 7. If the graph of y = g(x) is symmetric about the line x = 2, g(2) = 4, determine the value of \sum _{k=1}^{22}f(k). | -24 | 0.125 | 7,702.25 | 5,806.5 | 7,973.071429 | |
Given that the function $f(x)$ and its derivative $f'(x)$ have a domain of $R$, and $f(x+2)$ is an odd function, ${f'}(2-x)+{f'}(x)=2$, ${f'}(2)=2$, then $\sum_{i=1}^{50}{{f'}}(i)=$____. | 51 | 0.25 | 6,657.8125 | 5,920.5 | 6,903.583333 | |
Find the number of four-digit numbers with distinct digits, formed using the digits 0, 1, 2, ..., 9, such that the absolute difference between the units and hundreds digit is 8. | 154 | 0 | 7,075.6875 | -1 | 7,075.6875 | |
A man chooses two positive integers \( m \) and \( n \). He defines a positive integer \( k \) to be good if a triangle with side lengths \( \log m \), \( \log n \), and \( \log k \) exists. He finds that there are exactly 100 good numbers. Find the maximum possible value of \( mn \). | 134 | 0 | 8,192 | -1 | 8,192 | |
What is the smallest natural number whose digits in decimal representation are either 0 or 1 and which is divisible by 225? (China Junior High School Mathematics League, 1989) | 11111111100 | 0.375 | 7,313.9375 | 5,850.5 | 8,192 | |
Square the numbers \( a = 101 \) and \( b = 10101 \). Find the square root of the number \( c = 102030405060504030201 \). | 10101010101 | 0.0625 | 7,745.125 | 5,236 | 7,912.4 | |
In a physical education class, students line up in four rows to do exercises. One particular class has over 30 students, with three rows having the same number of students and one row having one more student than the other three rows. What is the smallest possible class size for this physical education class? | 33 | 1 | 1,721 | 1,721 | -1 | |
Given the function $f(x)=a\cos (x+\frac{\pi }{6})$, its graph passes through the point $(\frac{\pi }{2}, -\frac{1}{2})$.
(1) Find the value of $a$;
(2) If $\sin \theta =\frac{1}{3}, 0 < \theta < \frac{\pi }{2}$, find $f(\theta ).$ | \frac{2\sqrt{6}-1}{6} | 0 | 2,563.0625 | -1 | 2,563.0625 | |
Draw a $2004 \times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array? | 561 |
To determine the largest integer \( n \) for which it is possible to draw a convex \( n \)-gon whose vertices are chosen from the points in a \( 2004 \times 2004 \) array, we need to consider the properties of the convex hull and the arrangement of points.
Given the array of points, the problem can be approached by c... | 0 | 7,681.5625 | -1 | 7,681.5625 |
Let $x$ be a positive real number. Find the minimum value of $4x^5 + 5x^{-4}.$ | 9 | 0.9375 | 3,453.8125 | 3,137.933333 | 8,192 | |
Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively.
A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of $\triangle APR$ is | 40 | 1. **Identify Tangent Properties**: From the problem, we know that $AB$ and $AC$ are tangents from point $A$ to the circle, touching the circle at points $B$ and $C$ respectively. By the tangent properties, $AB = AC$.
2. **Tangent Lengths**: Since $AB = 20$, we also have $AC = 20$.
3. **Third Tangent Properties**: Th... | 0.3125 | 7,421.6875 | 5,727 | 8,192 |
What is the smallest positive integer with exactly 18 positive divisors? | 288 | 0 | 5,984.0625 | -1 | 5,984.0625 | |
Farmer James wishes to cover a circle with circumference $10 \pi$ with six different types of colored arcs. Each type of arc has radius 5, has length either $\pi$ or $2 \pi$, and is colored either red, green, or blue. He has an unlimited number of each of the six arc types. He wishes to completely cover his circle with... | 93 | Fix an orientation of the circle, and observe that the problem is equivalent to finding the number of ways to color ten equal arcs of the circle such that each arc is one of three different colors, and any two arcs which are separated by exactly one arc are of different colors. We can consider every other arc, so we ar... | 0 | 8,192 | -1 | 8,192 |
Given a circle \\(O: x^2 + y^2 = 2\\) and a line \\(l: y = kx - 2\\).
\\((1)\\) If line \\(l\\) intersects circle \\(O\\) at two distinct points \\(A\\) and \\(B\\), and \\(\angle AOB = \frac{\pi}{2}\\), find the value of \\(k\\).
\\((2)\\) If \\(EF\\) and \\(GH\\) are two perpendicular chords of the circle \\(O: x^2... | \frac{5}{2} | 0.0625 | 7,917.8125 | 5,569 | 8,074.4 | |
A tetrahedron \(ABCD\) has six edges with lengths \(7, 13, 18, 27, 36, 41\) units. If the length of \(AB\) is 41 units, then the length of \(CD\) is | 27 | 0 | 7,566.75 | -1 | 7,566.75 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $a > c$. Given that $\overrightarrow{BA} \cdot \overrightarrow{BC} = 2$ and $\cos B = \frac{1}{3}$, with $b=3$, find:
$(1)$ The values of $a$ and $c$; $(2)$ The value of $\cos(B - C)$. | \frac{23}{27} | 1 | 4,271.4375 | 4,271.4375 | -1 | |
The sum of the ages of three people A, B, and C, denoted as \(x, y, z\), is 120, with \(x, y, z \in (20, 60)\). How many ordered triples \((x, y, z)\) satisfy this condition? | 1198 | 0.0625 | 6,984.0625 | 8,192 | 6,903.533333 | |
In triangle \(ABC\), side \(AB = 6\), \(\angle BAC = 30^\circ\), and the radius of the circumscribed circle is 5. Find side \(AC\). | 3\sqrt{3} + 4 | 0.0625 | 7,249.0625 | 4,838 | 7,409.8 | |
How many numbers are in the list $-50, -44, -38, \ldots, 68, 74$? | 22 | 0 | 7,437 | -1 | 7,437 | |
Find the number of square units in the area of the triangle.
[asy]size(125);
draw( (-10,-2) -- (2,10), Arrows);
draw( (0,-2)-- (0,10) ,Arrows);
draw( (5,0) -- (-10,0),Arrows);
label("$l$",(2,10), NE);
label("$x$", (5,0) , E);
label("$y$", (0,-2) , S);
filldraw( (-8,0) -- (0,8) -- (0,0) -- cycle, lightgray);
dot( (-2,... | 32 | 0.9375 | 3,783.9375 | 3,844.466667 | 2,876 | |
In $\triangle ABC$, $\angle A = 60^\circ$, $AB > AC$, point $O$ is the circumcenter, and the altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, such that $BM = CN$. Find the value of $\frac{MH + NH}{OH}$. | \sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Let $M$ be the number of positive integers that are less than or equal to $2050$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $M$ is divided by $1000$. | 374 | 0 | 8,192 | -1 | 8,192 | |
In the expansion of $(x-y)^{8}(x+y)$, the coefficient of $x^{7}y^{2}$ is ____. | 20 | 0.875 | 5,886.75 | 5,557.428571 | 8,192 | |
What is the number of centimeters in the length of $EF$ if $AB\parallel CD\parallel EF$?
[asy]
size(4cm,4cm);
pair A,B,C,D,E,F,X;
A=(0,1);
B=(1,1);
C=(1,0);
X=(0,0);
D=(1/3)*C+(2/3)*X;
draw (A--B--C--D);
draw(D--B);
draw(A--C);
E=(0.6,0.4);
F=(1,0.4);
draw(E--F);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$"... | 60 | 0.4375 | 7,021 | 6,519 | 7,411.444444 | |
Suppose $x$ is in the interval $[0, \pi/2]$ and $\log_{24\sin x} (24\cos x)=\frac{3}{2}$. Find $24\cot^2 x$. | 192 | Like Solution 1, we can rewrite the given expression as \[24\sin^3x=\cos^2x\] Divide both sides by $\sin^3x$. \[24 = \cot^2x\csc x\] Square both sides. \[576 = \cot^4x\csc^2x\] Substitute the identity $\csc^2x = \cot^2x + 1$. \[576 = \cot^4x(\cot^2x + 1)\] Let $a = \cot^2x$. Then \[576 = a^3 + a^2\]. Since $\sqrt[3]{57... | 1 | 3,182.125 | 3,182.125 | -1 |
Expand the product ${(x+2)(x+5)}$. | x^2 + 7x + 10 | 1 | 1,467.75 | 1,467.75 | -1 | |
Given that the monogram of a person's first, middle, and last initials is in alphabetical order with no letter repeated, and the last initial is 'M', calculate the number of possible monograms. | 66 | 0.5625 | 3,626.375 | 3,690.777778 | 3,543.571429 | |
A trainee cook took two buckets of unpeeled potatoes and peeled everything in an hour. Meanwhile, $25\%$ of the potatoes went into peels. How much time did it take for him to collect exactly one bucket of peeled potatoes? | 40 | 0.375 | 4,867.125 | 4,703.166667 | 4,965.5 | |
Brian has a 20-sided die with faces numbered from 1 to 20, and George has three 6-sided dice with faces numbered from 1 to 6. Brian and George simultaneously roll all their dice. What is the probability that the number on Brian's die is larger than the sum of the numbers on George's dice? | \frac{19}{40} | Let Brian's roll be $d$ and let George's rolls be $x, y, z$. By pairing the situation $d, x, y, z$ with $21-d, 7-x, 7-y, 7-z$, we see that the probability that Brian rolls higher is the same as the probability that George rolls higher. Given any of George's rolls $x, y, z$, there is exactly one number Brian can roll wh... | 0.125 | 7,947.8125 | 6,238.5 | 8,192 |
Alexio now has 150 cards numbered from 1 to 150, inclusive, and places them in a box. He chooses a card at random. What is the probability that the number on the card he picks is a multiple of 4, 5 or 6? Express your answer as a reduced fraction. | \frac{7}{15} | 0.9375 | 4,003.9375 | 4,044.266667 | 3,399 | |
Two distinct positive integers $a$ and $b$ are factors of 48. If $a\cdot b$ is not a factor of 48, what is the smallest possible value of $a\cdot b$? | 32 | 0 | 7,665.375 | -1 | 7,665.375 | |
To factorize the quadratic trinomial $x^{2}+4x-5$, we can first add $4$ to $x^{2}+4x$ to make it a perfect square trinomial. Then, subtract $4$ so that the value of the entire expression remains unchanged. Therefore, we have: $x^{2}+4x-5=x^{2}+4x+4-4-5=\left(x+2\right)^{2}-9=\left(x+2+3\right)\left(x+2-3\right)=\left(x... | -24 | 0.9375 | 2,726.875 | 2,362.533333 | 8,192 | |
Let $f(n)$ be the base-10 logarithm of the sum of the elements of the $n$th row in Pascal's triangle. Express $\frac{f(n)}{\log_{10} 2}$ in terms of $n$. Recall that Pascal's triangle begins
\begin{tabular}{rccccccccc}
$n=0$:& & & & & 1\\\noalign{\smallskip\smallskip}
$n=1$:& & & & 1 & & 1\\\noalign{\smallskip\smallsk... | n | 1 | 1,253.625 | 1,253.625 | -1 | |
A company offers its employees a salary increase, provided they increase their work productivity by 2% per week. If the company operates 5 days a week, by what percentage per day must employees increase their productivity to achieve the desired goal? | 0.4 | 0.6875 | 4,753.1875 | 4,910.181818 | 4,407.8 | |
Xiaoming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, due to some reasons, Xiaoming first took the subway and then transferred to the bus, taking 40 minutes to reach the school. The transfer process took 6 minutes. How many minutes did Xiaoming spend on the bus that day? | 10 | 0.125 | 452.3125 | 412.5 | 458 | |
At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts? | 32 | 1. **Total number of students and pairs:**
Given that there are $57$ students wearing blue shirts and $75$ students wearing yellow shirts, the total number of students is $57 + 75 = 132$. These students are divided into $66$ pairs.
2. **Pairs with both students wearing blue shirts:**
It is given that in $23$ pai... | 0.9375 | 3,239.25 | 2,909.066667 | 8,192 |
Given that $\cos(x - \frac{\pi}{4}) = \frac{\sqrt{2}}{10}$, with $x \in (\frac{\pi}{2}, \frac{3\pi}{4})$.
(1) Find the value of $\sin x$;
(2) Find the value of $\cos(2x - \frac{\pi}{3})$. | -\frac{7 + 24\sqrt{3}}{50} | 0 | 4,370.0625 | -1 | 4,370.0625 | |
Let $x$ be inversely proportional to $y$. If $x = 4$ when $y = 2$, find the value of $x$ when $y = -3$ and when $y = 6$. | \frac{4}{3} | 0.9375 | 1,495.5625 | 1,393.866667 | 3,021 | |
In the Cartesian coordinate system $xOy$, the curve $C_{1}$ is defined by $\begin{cases} x=-2+\cos \alpha \\ y=-1+\sin \alpha \end{cases}$ (where $\alpha$ is a parameter). In the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the curve $C_{2}$ is defined by ... | \frac {1}{2} | 0.5 | 6,731.1875 | 5,676 | 7,786.375 | |
Given that Joy has 40 thin rods, one each of every integer length from 1 cm through 40 cm, with rods of lengths 4 cm, 9 cm, and 18 cm already placed on a table, determine how many of the remaining rods can be chosen as the fourth rod to form a quadrilateral with positive area. | 22 | 0 | 6,609.25 | -1 | 6,609.25 | |
Given set $A=\{a-2, 12, 2a^2+5a\}$, and $-3$ belongs to $A$, find the value of $a$. | -\frac{3}{2} | 0.75 | 5,700.5 | 5,691.666667 | 5,727 | |
Two circles are centered at the origin, as shown. The point $P(8,6)$ is on the larger circle and the point $S(0,k)$ is on the smaller circle. If $QR=3$, what is the value of $k$?
[asy]
unitsize(0.2 cm);
defaultpen(linewidth(.7pt)+fontsize(10pt));
dotfactor=4;
draw(Circle((0,0),7)); draw(Circle((0,0),10));
dot((0,0)... | 7 | 0.4375 | 4,701.625 | 2,937.428571 | 6,073.777778 | |
A sample size of 100 is divided into 10 groups with a class interval of 10. In the corresponding frequency distribution histogram, a certain rectangle has a height of 0.03. What is the frequency of that group? | 30 | 0 | 4,718.3125 | -1 | 4,718.3125 | |
If $x$ and $y$ are positive integers with $3^{x} 5^{y} = 225$, what is the value of $x + y$? | 4 | Since $15^{2}=225$ and $15=3 \cdot 5$, then $225=15^{2}=(3 \cdot 5)^{2}=3^{2} \cdot 5^{2}$. Therefore, $x=2$ and $y=2$, so $x+y=4$. | 1 | 1,637.625 | 1,637.625 | -1 |
A boy is riding a scooter from one bus stop to another and looking in the mirror to see if a bus appears behind him. As soon as the boy notices the bus, he can change the direction of his movement. What is the maximum distance between the bus stops so that the boy is guaranteed not to miss the bus, given that he rides ... | 1.5 | 0 | 7,471.8125 | -1 | 7,471.8125 | |
The sum of all three-digit numbers that, when divided by 7 give a remainder of 5, when divided by 5 give a remainder of 2, and when divided by 3 give a remainder of 1 is | 4436 | 0.9375 | 3,632.625 | 3,595.333333 | 4,192 | |
In recent years, China's scientific and technological achievements have been remarkable. The Beidou-3 global satellite navigation system has been operational for many years. The Beidou-3 global satellite navigation system consists of 24 medium Earth orbit satellites, 3 geostationary Earth orbit satellites, and 3 inclin... | \frac{4}{5} | 0.3125 | 3,299.5625 | 1,855 | 3,956.181818 | |
The sum of the coefficients of the expanded form of $(x+ \frac{a}{x})(2x- \frac{1}{x})^{5}$ is 2. Find the constant term in the expanded form. | 40 | 0.75 | 6,045.375 | 5,329.833333 | 8,192 | |
It is known that \(4 \operatorname{tg}^{2} Y + 4 \operatorname{ctg}^{2} Y - \frac{1}{\sin ^{2} \gamma} - \frac{1}{\cos ^{2} \gamma} = 17\). Find the value of the expression \(\cos ^{2} Y - \cos ^{4} \gamma\). | \frac{3}{25} | 0.1875 | 7,951.25 | 6,908 | 8,192 | |
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[17]{3}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer? | 11 | 0 | 7,442.9375 | -1 | 7,442.9375 | |
Consider a revised dataset given in the following stem-and-leaf plot, where $7|1$ represents 71:
\begin{tabular}{|c|c|}\hline
\textbf{Tens} & \textbf{Units} \\ \hline
2 & $0 \hspace{2mm} 0 \hspace{2mm} 1 \hspace{2mm} 1 \hspace{2mm} 2$ \\ \hline
3 & $3 \hspace{2mm} 6 \hspace{2mm} 6 \hspace{2mm} 7$ \\ \hline
4 & $3 \hsp... | 23 | 0 | 1,703.875 | -1 | 1,703.875 | |
A local government intends to encourage entrepreneurship by rewarding newly established small and micro enterprises with an annual output value between 500,000 and 5,000,000 RMB. The reward scheme follows these principles: The bonus amount $y$ (in ten thousand RMB) increases with the yearly output value $x$ (in ten tho... | 315 | 0 | 8,147.8125 | -1 | 8,147.8125 | |
What integer is closest to the value of $\sqrt[3]{6^3+8^3}$? | 9 | 0.8125 | 5,422 | 4,782.769231 | 8,192 | |
Given a pyramid-like structure with a rectangular base consisting of $4$ apples by $7$ apples, each apple above the first level resting in a pocket formed by four apples below, and the stack topped off with a single row of apples, determine the total number of apples in the stack. | 60 | 0.375 | 5,706.5625 | 5,890.666667 | 5,596.1 | |
If $x$ is a number less than -2, which of the following expressions has the least value: $x$, $x+2$, $\frac{1}{2}x$, $x-2$, or $2x$? | 2x | For any negative real number $x$, the value of $2x$ will be less than the value of $\frac{1}{2}x$. Therefore, $\frac{1}{2}x$ cannot be the least of the five values. Thus, the least of the five values is either $x-2$ or $2x$. When $x < -2$, we know that $2x - (x-2) = x + 2 < 0$. Since the difference between $2x$ and $x-... | 0.75 | 4,786.6875 | 4,533.833333 | 5,545.25 |
How many distinct diagonals of a convex heptagon (7-sided polygon) can be drawn? | 14 | 1 | 1,781.25 | 1,781.25 | -1 | |
What is the probability that the arrow stops on a shaded region if a circular spinner is divided into six regions, four regions each have a central angle of $x^{\circ}$, and the remaining regions have central angles of $20^{\circ}$ and $140^{\circ}$? | \frac{2}{3} | The six angles around the centre of the spinner add to $360^{\circ}$.
Thus, $140^{\circ}+20^{\circ}+4x^{\circ}=360^{\circ}$ or $4x=360-140-20=200$, and so $x=50$.
Therefore, the sum of the central angles of the shaded regions is $140^{\circ}+50^{\circ}+50^{\circ}=240^{\circ}$.
The probability that the spinner lan... | 0.1875 | 5,448.0625 | 6,729 | 5,152.461538 |
Given that point $P$ is on curve $C_1: y^2 = 8x$ and point $Q$ is on curve $C_2: (x-2)^2 + y^2 = 1$. If $O$ is the coordinate origin, find the maximum value of $\frac{|OP|}{|PQ|}$. | \frac{4\sqrt{7}}{7} | 0 | 7,064.6875 | -1 | 7,064.6875 | |
What is the smallest positive integer that is both a multiple of $7$ and a multiple of $4$? | 28 | 1 | 1,549.1875 | 1,549.1875 | -1 | |
An eight-sided die is rolled seven times. Find the probability of rolling at least a seven at least six times. | \frac{11}{2048} | 0 | 3,802.375 | -1 | 3,802.375 | |
Given the function $f(x) = \frac{x^2}{1+x^2}$.
$(1)$ Calculate the values of $f(2) + f\left(\frac{1}{2}\right)$, $f(3) + f\left(\frac{1}{3}\right)$, $f(4) + f\left(\frac{1}{4}\right)$, and infer a general conclusion (proof not required);
$(2)$ Calculate the value of $2f(2) + 2f(3) + \ldots + 2f(2017) + f\left(\frac{1... | 4032 | 0.5625 | 6,621 | 5,399.111111 | 8,192 | |
Suppose that $a$ and $b$ are positive integers for which $a$ has $3$ factors and $b$ has $a$ factors. If $b$ is divisible by $a$, then what is the least possible value of $b?$ | 8 | 0.5625 | 6,344.9375 | 4,908.333333 | 8,192 | |
What integer $n$ satisfies $0 \leq n < 151$ and $$100n \equiv 93 \pmod {151}~?$$ | 29 | 0 | 4,545 | -1 | 4,545 | |
The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 120^{\circ}$ and $\angle AOB =
140^{\circ}$, as shown. What is the degree measure of $\angle
ABC$?
[asy]
pair A,B,C;
draw(Circle((0,0),20),linewidth(0.7));
label("$O$",(0,0),S);
A=(-16,-12);
C=(16,-12);
B=(3,19.7);
draw(A... | 50^{\circ} | 1 | 2,469.9375 | 2,469.9375 | -1 | |
The quadratic $x^2-3x+9=x+41$ has two solutions. What is the positive difference between these solutions? | 12 | 1 | 2,062.4375 | 2,062.4375 | -1 | |
Evaluate the expression:
\[4(1+4(1+4(1+4(1+4(1+4(1+4(1+4(1))))))))\] | 87380 | 0 | 5,263.3125 | -1 | 5,263.3125 | |
Let $s(n)$ be the number of 1's in the binary representation of $n$ . Find the number of ordered pairs of integers $(a,b)$ with $0 \leq a < 64, 0 \leq b < 64$ and $s(a+b) = s(a) + s(b) - 1$ .
*Author:Anderson Wang* | 1458 | 0 | 8,192 | -1 | 8,192 | |
Given a circle $C: (x-3)^{2}+(y-4)^{2}=1$, and points $A(-1,0)$, $B(1,0)$, let $P$ be a moving point on the circle, then the maximum and minimum values of $d=|PA|^{2}+|PB|^{2}$ are \_\_\_\_\_\_ and \_\_\_\_\_\_ respectively. | 34 | 0.875 | 5,692.4375 | 5,391.5 | 7,799 | |
Solve the following equation and provide its root. If the equation has multiple roots, provide their product.
\[ \sqrt{2 x^{2} + 8 x + 1} - x = 3 \] | -8 | 0.0625 | 3,854.5 | 5,688 | 3,732.266667 | |
Compute $\dbinom{5}{3}$. | 10 | 1 | 2,223.4375 | 2,223.4375 | -1 |
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