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 |
|---|---|---|---|---|---|---|
Recall that the conjugate of the complex number $w = a + bi$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$, is the complex number $\overline{w} = a - bi$. For any complex number $z$, let $f(z) = 4i\overline{z}$. The polynomial $P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1$ has four complex roots: $z_1$, $z_2$, $z_3$, an... | 208 | 1. **Understanding the function and polynomial transformation**:
Given a complex number $w = a + bi$, its conjugate is $\overline{w} = a - bi$. The function $f(z) = 4i\overline{z}$ transforms $z$ into $4i\overline{z}$. We are given a polynomial $P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1$ with roots $z_1, z_2, z_3, z_4$. We ... | 0.1875 | 7,767.125 | 6,044.666667 | 8,164.615385 |
Mary told John her score on the American High School Mathematics Examination (AHSME), which was over $80$. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over $80$, John could not have determined this. What was Mary's score? (Recall that... | 119 | 0.125 | 8,140.75 | 7,782 | 8,192 | |
The equation $x^{x^{x^{.^{.^.}}}}=2$ is satisfied when $x$ is equal to: | \sqrt{2} | 1. **Define the Infinite Power Tower**: Let's consider the infinite power tower $x^{x^{x^{.^{.^.}}}}$. We denote this expression by $y$, so we have:
\[
y = x^{x^{x^{.^{.^.}}}}
\]
2. **Substitute and Simplify**: Given that $y = 2$, we substitute this into the equation:
\[
x^y = x^{x^{x^{.^{.^.}}}} = 2
... | 0.6875 | 6,060.4375 | 5,091.545455 | 8,192 |
Find the area of the region of the \( xy \)-plane defined by the inequality \( |x|+|y|+|x+y| \leq 1 \). | 3/4 | 0 | 8,178.375 | -1 | 8,178.375 | |
How many three-digit numbers are multiples of neither 5 nor 7? | 618 | 0.875 | 4,140.3125 | 3,561.5 | 8,192 | |
Distribute 5 students into dormitories A, B, and C, with each dormitory having at least 1 and at most 2 students. Among these, the number of different ways to distribute them without student A going to dormitory A is \_\_\_\_\_\_. | 60 | 0 | 8,098 | -1 | 8,098 | |
Find the least integer value of $x$ for which $3|x| - 2 > 13$. | -6 | 1 | 2,097.3125 | 2,097.3125 | -1 | |
Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the ... | 12 | We are given a triangle $T$ with vertices at $(0,0), (4,0),$ and $(0,3)$ and asked to determine how many sequences of three transformations from the set of rotations by $90^\circ, 180^\circ, 270^\circ$ counterclockwise around the origin, and reflections across the $x$-axis and $y$-axis, will return $T$ to its original ... | 0 | 8,191.375 | -1 | 8,191.375 |
There are $8$ balls of the same size, including $4$ different black balls, $2$ different red balls, and $2$ different yellow balls.<br/>$(1)$ Arrange these $8$ balls in a line, with the black balls together, the 2 red balls adjacent, and the 2 yellow balls not adjacent. Find the number of ways to arrange them;<br/>$(2)... | 490 | 0.0625 | 8,146.125 | 7,458 | 8,192 | |
A freight train was delayed on its route for 12 minutes. Then, over a distance of 60 km, it made up for the lost time by increasing its speed by 15 km/h. Find the original speed of the train. | 39.375 | 0 | 3,158.25 | -1 | 3,158.25 | |
Given that there is a gathering attended by 1982 people, and among any group of 4 people, at least 1 person knows the other 3. How many people, at minimum, must know all the attendees at this gathering? | 1979 | 0.0625 | 7,765.0625 | 3,899 | 8,022.8 | |
There are 6 people including A, B, and C standing in a row for a photo, where A cannot stand at either end, and B and C must stand next to each other. How many such arrangements are there? | 144 | 0.375 | 7,283.3125 | 5,768.833333 | 8,192 | |
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \(12^{2}\) and \(13^{2}\)? | 165 | 0.625 | 3,932.125 | 3,636.6 | 4,424.666667 | |
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits? | 45 | 0.6875 | 6,429.5625 | 5,628.454545 | 8,192 | |
If \( p(x) = x^4 - 4x^2 + 3x + 1 \), then find the coefficient of the \( x^3 \) term in the polynomial \( (p(x))^3 \). | 27 | 0 | 7,512.5 | -1 | 7,512.5 | |
In the plane rectangular coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system with the same unit of length. The parametric equation of line $l$ is $\begin{cases}x=2+\frac{\sqrt{2}}{2}t\\y=1+\frac{\sqrt{2}}{2}t\end{cases}$, and th... | \sqrt{2} | 0.9375 | 5,164.0625 | 4,962.2 | 8,192 | |
A supermarket purchases two types of goods, $A$ and $B$. Buying 4 items of type $A$ costs $10$ yuan less than buying 5 items of type $B$. Buying 20 items of type $A$ and 10 items of type $B$ costs a total of $160$ yuan.
$(1)$ Find the cost price per item of goods $A$ and $B$ respectively.
$(2)$ If the store purchas... | 100 | 0.3125 | 6,403.25 | 4,405.4 | 7,311.363636 | |
When plotted in the standard rectangular coordinate system, trapezoid $EFGH$ has vertices $E(2, -3)$, $F(2, 2)$, $G(7, 8)$, and $H(7, 3)$. What is the area of trapezoid $EFGH$? | 25 | 0.9375 | 3,623.75 | 3,319.2 | 8,192 | |
Let $A B C D E$ be a convex pentagon such that $$\begin{aligned} & A B+B C+C D+D E+E A=64 \text { and } \\ & A C+C E+E B+B D+D A=72 \end{aligned}$$ Compute the perimeter of the convex pentagon whose vertices are the midpoints of the sides of $A B C D E$. | 36 | By the midsegment theorem on triangles $A B C, B C D, \ldots, D E A$, the side lengths of the said pentagons are $A C / 2, B D / 2, C E / 2, D A / 2$, and $E B / 2$. Thus, the answer is $$\frac{A C+B D+C E+D A+E B}{2}=\frac{72}{2}=36$$ | 0.75 | 5,137.375 | 4,464.166667 | 7,157 |
How many whole numbers lie in the interval between $\frac{5}{3}$ and $2\pi$ ? | 5 | 1 | 2,844 | 2,844 | -1 | |
Let \( A \) be a point on the parabola \( y = x^2 - 4x \), and let \( B \) be a point on the line \( y = 2x - 3 \). Find the shortest possible distance \( AB \). | \frac{6\sqrt{5}}{5} | 0 | 7,906 | -1 | 7,906 | |
Let $a, b, c,$ and $d$ be real numbers that satisfy the system of equations \begin{align*} a + b &= -3, \\ ab + bc + ca &= -4, \\ abc + bcd + cda + dab &= 14, \\ abcd &= 30. \end{align*} There exist relatively prime positive integers $m$ and $n$ such that \[a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.\]Find $m + n$. | 145 | From the fourth equation we get $d=\frac{30}{abc}.$ substitute this into the third equation and you get $abc + \frac{30(ab + bc + ca)}{abc} = abc - \frac{120}{abc} = 14$. Hence $(abc)^2 - 14(abc)-120 = 0$. Solving we get $abc = -6$ or $abc = 20$. From the first and second equation we get $ab + bc + ca = ab-3c = -4 \Lon... | 0.25 | 7,782.4375 | 6,553.75 | 8,192 |
Solve the following equations using appropriate methods:
(1) $x^2=49$;
(2) $(2x+3)^2=4(2x+3)$;
(3) $2x^2+4x-3=0$ (using the formula method);
(4) $(x+8)(x+1)=-12$. | -5 | 1 | 3,144.3125 | 3,144.3125 | -1 | |
Given sets $A=\{x|x^{2}+2x-3=0,x\in R\}$ and $B=\{x|x^{2}-\left(a+1\right)x+a=0,x\in R\}$.<br/>$(1)$ When $a=2$, find $A\cap C_{R}B$;<br/>$(2)$ If $A\cup B=A$, find the set of real numbers for $a$. | \{1\} | 0 | 4,326.75 | -1 | 4,326.75 | |
An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$.
| 144 | 1 | 2,658.125 | 2,658.125 | -1 | |
On the banks of an island, which has the shape of a circle (viewed from above), there are the cities $A, B, C,$ and $D$. A straight asphalt road $AC$ divides the island into two equal halves. A straight asphalt road $BD$ is shorter than road $AC$ and intersects it. The speed of a cyclist on any asphalt road is 15 km/h.... | 450 | 0 | 8,182.4375 | -1 | 8,182.4375 | |
Use each of the digits 3, 4, 6, 8 and 9 exactly once to create the greatest possible five-digit multiple of 6. What is that multiple of 6? | 98,634 | 0 | 7,842.5 | -1 | 7,842.5 | |
In the diagram, $ABCD$ is a square with side length $8$, and $WXYZ$ is a rectangle with $ZY=12$ and $XY=4$. Additionally, $AD$ and $WX$ are perpendicular. If the shaded area equals three-quarters of the area of $WXYZ$, what is the length of $DP$? | 4.5 | 0 | 8,192 | -1 | 8,192 | |
Point \( A \) lies on the line \( y = \frac{5}{12} x - 11 \), and point \( B \) lies on the parabola \( y = x^{2} \). What is the minimum length of segment \( AB \)? | 6311/624 | 0 | 8,192 | -1 | 8,192 | |
Within a triangular piece of paper, there are 100 points, along with the 3 vertices of the triangle, making it a total of 103 points, and no three of these points are collinear. If these points are used as vertices to create triangles, and the paper is cut into small triangles, then the number of such small triangles i... | 201 | 0.8125 | 5,242.0625 | 4,719.923077 | 7,504.666667 | |
Read the material: Calculate $\frac{1}{30}÷(\frac{2}{3}-\frac{1}{10}+\frac{1}{6}-\frac{2}{5})$. Analysis: It is very cumbersome to calculate the result of $\frac{2}{3}-\frac{1}{10}+\frac{1}{6}-\frac{2}{5}$ using a common denominator. The following method can be used for calculation. Solution: The reciprocal of the orig... | -\frac{2}{29} | 0.9375 | 4,724.8125 | 4,493.666667 | 8,192 | |
We are laying railway tracks that are $15 \text{ meters }$ long at a temperature of $t = -8^{\circ} \text{C}$. What gap should we leave between each rail if the maximum expected temperature is $t = 60^{\circ} \text{C}$? The coefficient of expansion for the rail is $\lambda = 0.000012$. | 12.24 | 0.0625 | 6,089.3125 | 2,646 | 6,318.866667 | |
Inside a square, 100 points are marked. The square is divided into triangles such that the vertices of the triangles are only the marked 100 points and the vertices of the square, and for each triangle in the division, each marked point either lies outside the triangle or is a vertex of that triangle (such divisions ar... | 202 | 0.625 | 5,531.1875 | 3,974.7 | 8,125.333333 | |
A box with a volume of 16 $\text{cm}^3$ can hold 50 paperclips. How many paperclips could a box with a volume of 48 $\text{cm}^3$ hold? | 150 | 1 | 1,710.8125 | 1,710.8125 | -1 | |
Let \(a_{1}, a_{2}, \ldots\) be an infinite sequence of integers such that \(a_{i}\) divides \(a_{i+1}\) for all \(i \geq 1\), and let \(b_{i}\) be the remainder when \(a_{i}\) is divided by 210. What is the maximal number of distinct terms in the sequence \(b_{1}, b_{2}, \ldots\)? | 127 | It is clear that the sequence \(\{a_{i}\}\) will be a concatenation of sequences of the form \(\{v_{i}\}_{i=1}^{N_{0}},\{w_{i} \cdot p_{1}\}_{i=1}^{N_{1}},\{x_{i} \cdot p_{1} p_{2}\}_{i=1}^{N_{2}},\{y_{i} \cdot p_{1} p_{2} p_{3}\}_{i=1}^{N_{3}}\), and \(\{z_{i} \cdot p_{1} p_{2} p_{3} p_{4}\}_{i=1}^{N_{4}}\), for some ... | 0 | 8,192 | -1 | 8,192 |
In the expansion of $(a + b)^n$ there are $n + 1$ dissimilar terms. The number of dissimilar terms in the expansion of $(a + b + c)^{10}$ is: | 66 | To find the number of dissimilar terms in the expansion of $(a + b + c)^{10}$, we can use the concept of combinations to determine the number of distinct terms generated by the expansion.
1. **Understanding the Expansion**:
The expression $(a + b + c)^{10}$ can be expanded using the multinomial theorem. Each term i... | 1 | 1,778.5 | 1,778.5 | -1 |
Let $\pi$ be a randomly chosen permutation of the numbers from 1 through 2012. Find the probability that $\pi(\pi(2012))=2012$. | \frac{1}{1006} | There are two possibilities: either $\pi(2012)=2012$ or $\pi(2012)=i$ and $\pi(i)=2012$ for $i \neq 2012$. The first case occurs with probability $2011!/ 2012!=1 / 2012$, since any permutation on the remaining 2011 elements is possible. Similarly, for any fixed $i$, the second case occurs with probability $2010!/ 2012!... | 0.6875 | 5,634.6875 | 4,571.636364 | 7,973.4 |
How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999?$ For example, both $121$ and $211$ have this property.
$\mathrm{\textbf{(A)} \ }226\qquad \mathrm{\textbf{(B)} \ } 243 \qquad \mathrm{\textbf{(C)} \ } 270 \qquad \mat... | 226 | 0 | 8,192 | -1 | 8,192 | |
A right triangle has sides of lengths 5 cm and 11 cm. Calculate the length of the remaining side if the side of length 5 cm is a leg of the triangle. Provide your answer as an exact value and as a decimal rounded to two decimal places. | 9.80 | 0.1875 | 580.5625 | 676.666667 | 558.384615 | |
What common fraction is exactly half-way between $\frac{2}{3}$ and $\frac{4}{5}$? | \frac{11}{15} | 1 | 2,169.375 | 2,169.375 | -1 | |
Given real numbers \(a, b, c\), the polynomial
$$
g(x) = x^{3} + a x^{2} + x + 10
$$
has three distinct roots, and these three roots are also roots of the polynomial
$$
f(x) = x^{4} + x^{3} + b x^{2} + 100 x + c.
$$
Evaluate the value of \(f(1)\). | -7007 | 0.875 | 4,317 | 3,763.428571 | 8,192 | |
Which of the following divisions is not equal to a whole number: $\frac{60}{12}$, $\frac{60}{8}$, $\frac{60}{5}$, $\frac{60}{4}$, $\frac{60}{3}$? | 7.5 | Since $\frac{60}{8}=60 \div 8=7.5$, then this choice is not equal to a whole number. Note as well that $\frac{60}{12}=5, \frac{60}{5}=12, \frac{60}{4}=15$, and $\frac{60}{3}=20$ are all whole numbers. | 0 | 483.75 | -1 | 483.75 |
How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits? | 90 | 1 | 2,702.625 | 2,702.625 | -1 | |
If $\frac{8^x}{4^{x+y}}=16$ and $\frac{16^{x+y}}{4^{7y}}=1024$, find $x+y$. | 13 | 0.6875 | 2,197.1875 | 2,176.545455 | 2,242.6 | |
The diagonal $KM$ of trapezoid $KLMN$ is 3 times the length of segment $KP$ on this diagonal. The base $KN$ of the trapezoid is 3 times the length of the base $LM$. Find the ratio of the area of trapezoid $KLMN$ to the area of triangle $KPR$, where $R$ is the point of intersection of line $PN$ and side $KL$. | 32/3 | 0 | 7,309.125 | -1 | 7,309.125 | |
Points $E, F, G, H$ are chosen on segments $A B, B C, C D, D A$, respectively, of square $A B C D$. Given that segment $E G$ has length 7 , segment $F H$ has length 8 , and that $E G$ and $F H$ intersect inside $A B C D$ at an acute angle of $30^{\circ}$, then compute the area of square $A B C D$. | \frac{784}{19} | Rotate $E G$ by $90^{\circ}$ about the center of the square to $E^{\prime} G^{\prime}$ with $E^{\prime} \in A D$ and $G^{\prime} \in B C$. Now $E^{\prime} G^{\prime}$ and $F H$ intersect at an angle of $60^{\circ}$. Then consider the translation which takes $E^{\prime}$ to $H$ and $G^{\prime}$ to $I$. Triangle $F H I$ ... | 0 | 8,192 | -1 | 8,192 |
What is the length of $SR$ if in $\triangle PQR$, $PS$ is perpendicular to $QR$, $RT$ is perpendicular to $PQ$, $PT=1$, $TQ=4$, and $QS=3$? | \frac{11}{3} | Since $PT=1$ and $TQ=4$, then $PQ=PT+TQ=1+4=5$. $\triangle PSQ$ is right-angled at $S$ and has hypotenuse $PQ$. By the Pythagorean Theorem, $PS^{2}=PQ^{2}-QS^{2}=5^{2}-3^{2}=16$. Since $PS>0$, then $PS=4$. Consider $\triangle PSQ$ and $\triangle RTQ$. These triangles are similar, so $\frac{PQ}{QS}=\frac{QR}{TQ}$. Thus,... | 0.5625 | 6,070.4375 | 4,908.222222 | 7,564.714286 |
In a gumball machine containing 13 red, 5 blue, 1 white, and 9 green gumballs, what is the least number of gumballs that must be bought to guarantee receiving 3 gumballs of the same color? | 8 | It is possible that after buying 7 gumballs, Wally has received 2 red, 2 blue, 1 white, and 2 green gumballs. This is the largest number of each color that he could receive without having three gumballs of any one color. If Wally buys another gumball, he will receive a blue or a green or a red gumball. In each of these... | 0.9375 | 4,363.625 | 4,560.733333 | 1,407 |
In the plane rectangular coordinate system xOy, the origin is taken as the pole, the positive semi-axis of the x-axis is taken as the polar axis, and the polar coordinate system is established. The same unit length is adopted in both coordinate systems. The polar coordinate equation of the curve C is ρ = 4$\sqrt {2}$co... | \frac { \sqrt {31}}{7} | 0 | 6,577.25 | -1 | 6,577.25 | |
Given that Crystal runs due north for 2 miles, then northwest for 1 mile, and southwest for 1 mile, find the distance of the last portion of her run that returns her directly to her starting point. | \sqrt{6} | 0.8125 | 4,880.3125 | 4,367.923077 | 7,100.666667 | |
Let $ S $ be the set of all sides and diagonals of a regular hexagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? | \frac{17}{35} | 0 | 4,926 | -1 | 4,926 | |
Given the numbers 1, 2, 3, 4, find the probability that $\frac{a}{b}$ is not an integer, where $a$ and $b$ are randomly selected numbers from the set $\{1, 2, 3, 4\}$. | \frac{2}{3} | 0.125 | 5,581.1875 | 4,937.5 | 5,673.142857 | |
Evaluate $|2-4i| + |2+4i|.$ | 4\sqrt{5} | 1 | 1,961.625 | 1,961.625 | -1 | |
Express this sum as a common fraction: $0.\overline{7} + 0.\overline{13}$ | \frac{10}{11} | 0.9375 | 2,451.8125 | 2,341.933333 | 4,100 | |
A swimming pool is in the shape of a circle with diameter 60 ft. The depth varies linearly along the east-west direction from 3 ft at the shallow end in the east to 15 ft at the diving end in the west but does not vary at all along the north-south direction. What is the volume of the pool, in cubic feet (ft³)? | 8100 \pi | 0.5 | 6,116.125 | 4,093.75 | 8,138.5 | |
A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 5-primable positive integers are there that are less than 500? | 17 | 0 | 7,215.3125 | -1 | 7,215.3125 | |
In triangle $A B C$ with $A B=8$ and $A C=10$, the incenter $I$ is reflected across side $A B$ to point $X$ and across side $A C$ to point $Y$. Given that segment $X Y$ bisects $A I$, compute $B C^{2}$. | 84 | Let $E, F$ be the tangency points of the incircle to sides $A C, A B$, respectively. Due to symmetry around line $A I, A X I Y$ is a rhombus. Therefore $$\angle X A I=2 \angle E A I=2\left(90^{\circ}-\angle E I A\right)=180^{\circ}-2 \angle X A I$$ which implies that $60^{\circ}=\angle X A I=2 \angle E A I=\angle B A C... | 0.0625 | 8,192 | 8,192 | 8,192 |
A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then $N=$ | 20 | 1. **Understanding the problem**: We are given a cube with an edge length of $3$ cm, and it is divided into smaller cubes with whole number edge lengths. We need to determine the number of smaller cubes, $N$.
2. **Analyzing possible edge lengths of smaller cubes**: Since the edge length of the original cube is $3$ cm,... | 0.25 | 7,941.125 | 7,247.25 | 8,172.416667 |
In isosceles trapezoid $ABCD$ where $AB$ (shorter base) is 10 and $CD$ (longer base) is 20. The non-parallel sides $AD$ and $BC$ are extended to meet at point $E$. What is the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$? | \frac{1}{3} | 0.375 | 7,056.1875 | 5,163.166667 | 8,192 | |
A company needs to transport two types of products, $A$ and $B$, with the following volumes and masses per unit as shown in the table:
| | Volume $(m^{3}/$unit) | Mass (tons$/$unit) |
|----------|-----------------------|--------------------|
| $A$ type | $0.8$ | $0.5$ |
| $B$ type... | 2100 | 0.5 | 6,524.4375 | 5,003.5 | 8,045.375 | |
What is the earliest row in which the number 2004 may appear? | 12 | By the previous problem, it cannot appear before row 12. By starting off the table as shown above, we see that row 12 is possible, so this is the answer. | 0.0625 | 7,751.5625 | 7,514 | 7,767.4 |
Quadrilateral $ABCD$ is inscribed in a circle with segment $AC$ a diameter of the circle. If $m\angle DAC = 30^\circ$ and $m\angle BAC = 45^\circ$, the ratio of the area of $ABCD$ to the area of the circle can be expressed as a common fraction in simplest radical form in terms of $\pi$ as $\frac{a+\sqrt{b}}{c\pi}$, whe... | 7 | 0.375 | 6,793.5 | 4,462.666667 | 8,192 | |
A rental company owns 100 cars. When the monthly rent for each car is 3000 yuan, all of them can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that is not rented out. The maintenance cost for each rented car is 150 yuan per month, and for each car that is not rented o... | 303000 | 0.3125 | 6,624.75 | 4,973.4 | 7,375.363636 | |
Find the inverse of the matrix
\[\begin{pmatrix} 9 & 18 \\ -6 & -12 \end{pmatrix}.\]If the inverse does not exist, then enter the zero matrix. | \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} | 1 | 1,221.375 | 1,221.375 | -1 | |
Calculate the limit
$$
\lim _{x \rightarrow 0}\left(\frac{1+x^{2} 2^{x}}{1+x^{2} 5^{x}}\right)^{1 / \sin ^{3} x}
$$ | \frac{2}{5} | 0.3125 | 7,767.25 | 6,832.8 | 8,192 | |
Find $x$ such that $\lceil x \rceil \cdot x = 210$. Express $x$ as a decimal. | 14.0 | 0 | 8,192 | -1 | 8,192 | |
How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x, y)$ satisfying $1 \le x \le 5$ and $1 \le y \le 5$? | 2160 | 0 | 8,192 | -1 | 8,192 | |
A solid rectangular prism has dimensions 4 by 2 by 2. A 1 by 1 by 1 cube is cut out of the corner creating the new solid shown. What is the surface area of the new solid? | 40 | The original prism has four faces that are 4 by 2 rectangles, and two faces that are 2 by 2 rectangles. Thus, the surface area of the original prism is \( 4(4 \cdot 2)+2(2 \cdot 2)=32+8=40 \). When a 1 by 1 by cube is cut out, a 1 by 1 square is removed from each of three faces of the prism, but three new 1 by 1 square... | 0.5 | 7,044 | 5,896 | 8,192 |
Two concentric circles have radii of 15 meters and 30 meters. An aardvark starts at point $A$ on the smaller circle and runs along the path that includes half the circumference of each circle and each of the two straight segments that connect the circumferences directly (radial segments). Calculate the total distance t... | 45\pi + 30 | 0.1875 | 6,275.6875 | 6,467 | 6,231.538462 | |
Given that the first term of a geometric sequence $\{{a_n}\}$ is $\frac{3}{2}$ and the sum of its first $n$ terms is $S_n$ $(n \in \mathbb{N^*})$, and $-2S_2, S_3, 4S_4$ form an arithmetic sequence.
(I) Find the general term formula of the sequence $\{{a_n}\}$.
(II) Find the maximum and minimum values of $S_n$ $(n \i... | \frac{3}{4} | 0.25 | 8,070.4375 | 7,705.75 | 8,192 | |
Six horizontal lines and five vertical lines are drawn in a plane. If a specific point, say (3, 4), exists in the coordinate plane, in how many ways can four lines be chosen such that a rectangular region enclosing the point (3, 4) is formed? | 24 | 0.0625 | 7,301 | 6,770 | 7,336.4 | |
Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$ . Find the measure of the angle $\angle PBC$ . | 15 | 0.625 | 5,793.25 | 4,354 | 8,192 | |
Let the coefficient of $x^{-4}$ in the expansion of $\left(1- \frac {1}{x^{2}}\right)^{n}$ (where $n\in\mathbb{N}_{+}$) be denoted as $a_{n}$. Calculate the value of $$\frac {1}{a_{2}}+ \frac {1}{a_{3}}+…+ \frac {1}{a_{2015}}$$. | \frac {4028}{2015} | 1 | 4,489.6875 | 4,489.6875 | -1 | |
A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\times5$ grid. What fraction of the grid is covered by the triangle? | \frac{1}{6} |
To find the fraction of the grid covered by the triangle with vertices $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$, we first need to calculate the area of the triangle and then compare it to the area of the grid.
#### Step 1: Calculate the area of the triangle using the Shoelace Theorem
The Shoelace Theorem provides a formul... | 1 | 3,752.1875 | 3,752.1875 | -1 |
Four friends came back from fishing. Each pair of them counted the sum of their catches. They obtained six numbers: $7, 9, 14, 14, 19, 21$. How many fish were caught in total? | 28 | 0.9375 | 2,828 | 2,470.4 | 8,192 | |
In $\triangle ABC$ , three lines are drawn parallel to side $BC$ dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$ , what is the area of the whole $\triangle ABC$ ?
[asy]
defaultpen(linewidth(0.7)); size(120);
pair B = (0,0), C = (1,0), A = (0.7,1); pair[]... | 560/3 | 0 | 4,309.875 | -1 | 4,309.875 | |
Several energy-saving devices with a total weight of 120 kg were delivered to the factory. It is known that the total weight of the three lightest devices is 31 kg, and the total weight of the three heaviest devices is 41 kg. How many energy-saving devices were delivered to the factory if the weights of any two devices... | 10 | 0.1875 | 7,347.375 | 4,203.666667 | 8,072.846154 | |
The café has enough chairs to seat $312_8$ people. If $3$ people are supposed to sit at one table, how many tables does the café have? | 67 | 0.0625 | 2,693.375 | 2,757 | 2,689.133333 | |
In triangle $ABC$, angle $ACB$ is 50 degrees, and angle $CBA$ is 70 degrees. Let $D$ be the foot of the perpendicular from $A$ to $BC$, $O$ the center of the circle circumscribed about triangle $ABC$, and $E$ the other end of the diameter which goes through $A$. Find the angle $DAE$, in degrees.
[asy]
unitsize(1.5 ... | 20^\circ | 0.3125 | 7,936.9375 | 7,375.8 | 8,192 | |
All positive integers whose digits add up to 12 are listed in increasing order: $39, 48, 57, ...$. What is the twelfth number in that list? | 165 | 0.375 | 7,521 | 6,402.666667 | 8,192 | |
The product of two consecutive page numbers is $20{,}412$. What is the sum of these two page numbers? | 285 | 0 | 8,192 | -1 | 8,192 | |
Acme Corporation has released an alphabet soup in which each of the vowels (A, E, I, O, U) of the English alphabet appears five times (and the consonants do not appear at all). How many five-letter words can be formed from a bowl of Acme Vowel Soup? (Note: The words do not have to be actual words in English!) | 3125 | 0.8125 | 4,679.9375 | 3,869.461538 | 8,192 | |
Given the sequence $\{a\_n\}$, where $a\_n= \sqrt {5n-1}$, $n\in\mathbb{N}^*$, arrange the integer terms of the sequence $\{a\_n\}$ in their original order to form a new sequence $\{b\_n\}$. Find the value of $b_{2015}$. | 5037 | 0.375 | 6,582.25 | 5,917.166667 | 6,981.3 | |
Let P be a moving point on the line $3x+4y+3=0$, and through point P, two tangents are drawn to the circle $C: x^2+y^2-2x-2y+1=0$, with the points of tangency being A and B, respectively. Find the minimum value of the area of quadrilateral PACB. | \sqrt{3} | 0.1875 | 7,949.3125 | 6,998.333333 | 8,168.769231 | |
Let $A_1,B_1,C_1,D_1$ be the midpoints of the sides of a convex quadrilateral $ABCD$ and let $A_2, B_2, C_2, D_2$ be the midpoints of the sides of the quadrilateral $A_1B_1C_1D_1$ . If $A_2B_2C_2D_2$ is a rectangle with sides $4$ and $6$ , then what is the product of the lengths of the diagonals of $ABCD$ ... | 96 | 0 | 7,676.25 | -1 | 7,676.25 | |
Given that $\log_{10} \sin x + \log_{10} \cos x = -1$ and that $\log_{10} (\sin x + \cos x) = \frac{1}{2} (\log_{10} n - 1),$ find $n.$
| 12 | 1 | 2,791.1875 | 2,791.1875 | -1 | |
Sixty percent of a plane's passengers are women and ten percent of those women are in first class. What is the number of women in first class if the plane is carrying 200 passengers? | 12 | 0.9375 | 2,083.6875 | 1,676.466667 | 8,192 | |
Given that $a$ and $b$ are positive real numbers satisfying $a + 2b = 1$, find the minimum value of $a^2 + 4b^2 + \frac{1}{ab}$. | \frac{17}{2} | 0.625 | 7,166.875 | 6,551.8 | 8,192 | |
In the Cartesian coordinate system $xOy$, the graph of the linear function $y=kx+b+2$ ($k \neq 0$) intersects the positive half of the x-axis at point A and the positive half of the y-axis at point B.
(1) Express the area $S_{\triangle AOB}$ of triangle $AOB$ in terms of $b$ and $k$.
(2) If the area $S_{\triangle AOB} ... | 7 + 2\sqrt{10} | 0.625 | 6,763.75 | 6,169.7 | 7,753.833333 | |
Find the coefficient of $x^3$ when $3(x^2 - x^3+x) +3(x +2x^3- 3x^2 + 3x^5+x^3) -5(1+x-4x^3 - x^2)$ is simplified. | 26 | 0.9375 | 4,116.25 | 3,844.533333 | 8,192 | |
What is the value of $\sqrt{36 \times \sqrt{16}}$? | 12 | 1 | 3,526.375 | 3,526.375 | -1 | |
A natural number's proper divisors are defined as positive integer divisors other than 1 and the number itself. A natural number greater than 1 is called "good" if it is equal to the product of all its distinct proper divisors. What is the sum of the first 10 "good" natural numbers? | 182 | 0.6875 | 5,128.6875 | 5,068.909091 | 5,260.2 | |
Find the number of ordered pairs $(A, B)$ such that the following conditions hold: $A$ and $B$ are disjoint subsets of $\{1,2, \ldots, 50\}$, $|A|=|B|=25$, and the median of $B$ is 1 more than the median of $A$. | \binom{24}{12}^{2} | The median of both sets, which we will call $a$ and $b$ respectively, are more than exactly 12 of the members in their own set. Since $a$ and $b$ are consecutive, they must also be higher than the lower half of the other set and lower than the higher half of the other set, meaning that they are both higher than exactly... | 0 | 8,192 | -1 | 8,192 |
Let $\overrightarrow{a}=(\sin x, \frac{3}{4})$, $\overrightarrow{b}=( \frac{1}{3}, \frac{1}{2}\cos x )$, and $\overrightarrow{a} \parallel \overrightarrow{b}$. Find the acute angle $x$. | \frac{\pi}{4} | 1 | 2,599.875 | 2,599.875 | -1 | |
How many positive integers at most 420 leave different remainders when divided by each of 5, 6, and 7? | 250 | Note that $210=5 \cdot 6 \cdot 7$ and $5,6,7$ are pairwise relatively prime. So, by the Chinese Remainder Theorem, we can just consider the remainders $n$ leaves when divided by each of $5,6,7$. To construct an $n$ that leaves distinct remainders, first choose its remainder modulo 5, then modulo 6, then modulo 7. We ha... | 0.0625 | 7,881.3125 | 7,743 | 7,890.533333 |
If $\tan x+\tan y=25$ and $\cot x + \cot y=30$, what is $\tan(x+y)$? | 150 | 0.75 | 3,646.0625 | 2,130.75 | 8,192 | |
Four carpenters were hired by a guest to build a yard. The first carpenter said: "If only I alone were to build the yard, I would complete it in one year." The second carpenter said: "If only I alone were to build the yard, I would complete it in two years." The third carpenter said: "If only I alone were to build the ... | 175.2 | 0 | 2,899.5 | -1 | 2,899.5 | |
A club consists of three board members and a certain number of regular members. Every year, the board members retire and are not replaced. Each regular member recruits one new person to join as a regular member. Initially, there are nine people in the club total. How many people total will be in the club after four yea... | 96 | 0.3125 | 4,786.6875 | 4,892.8 | 4,738.454545 | |
Four friends — Alex, Betty, Clara, and Dave — participated in a relay race by running in pairs, with one pair sitting out each race. Dave ran in 8 races, which was more than any other friend, and Betty ran in 3 races, which was fewer than any other friend. Determine the total number of races those pairs completed. | 10 | 0.125 | 8,129.625 | 7,693 | 8,192 | |
If the direction vectors of two skew lines $l_{1}$ and $l_{2}$ are $\overrightarrow{a}=(0,-1,-2)$ and $\overrightarrow{b}=(4,0,2)$, then the cosine value of the angle between the two skew lines $l_{1}$ and $l_{2}$ is ______. | \frac{2}{5} | 0.875 | 3,277.6875 | 2,768.285714 | 6,843.5 |
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