Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
11,500 | A new model car travels 4.4 kilometers more per liter of gasoline than an old model car. Additionally, the fuel consumption per 100 km for the new model is 2 liters less than that of the old model. How many liters of gasoline does the new model car consume per 100 km? Round the answer to the nearest hundredth if necess... | 5.82 | 7.03125 |
11,501 | The cost prices of three types of clothing, A, B, and C, are respectively 20 yuan, 30 yuan, and 40 yuan. Their selling prices are 24 yuan, m yuan, and 52 yuan, respectively. After calculation, the total profit from the three types of clothing is the same, and the sum of the sales volumes of clothing A and B is four tim... | 42 | 85.9375 |
11,502 | In a trapezoid, the lengths of the diagonals are known to be 6 and 8, and the length of the midline is 5. Find the height of the trapezoid. | 4.8 | 7.8125 |
11,503 | Given a polynomial of degree 2022 with integer coefficients and a leading coefficient of 1, what is the maximum number of roots it can have within the interval \( (0,1) \)? | 2021 | 22.65625 |
11,504 | A number $x$ is equal to $6 \cdot 18 \cdot 42$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube? | 441 | 93.75 |
11,505 | Let $a$, $b$, $c$ be the sides of a triangle, and let $\alpha$, $\beta$, $\gamma$ be the angles opposite them respectively. If $a^2+b^2=2c^2$, find the value of
\[
\frac{\cot \gamma}{\cot \alpha + \cot \beta}.
\] | \frac{1}{2} | 82.8125 |
11,506 | In the number $2016 * * * * 02 *$, you need to replace each of the 5 asterisks with any of the digits $0, 2, 4, 6, 7, 8$ (digits may repeat) so that the resulting 11-digit number is divisible by 6. How many ways can this be done? | 2160 | 14.0625 |
11,507 | Numbers from 1 to 6 are written on the faces of a gaming die. However, the weight of the die is distributed unevenly, and the probability of landing on number $k$ is directly proportional to $k$. The die is rolled two times in a row. What is the probability that the sum of the rolled numbers will be 7? If necessary, ro... | 0.13 | 90.625 |
11,508 | Given vectors \( \boldsymbol{a} \), \( \boldsymbol{b} \), and \( \boldsymbol{c} \) satisfying
$$
|\boldsymbol{a}|:|\boldsymbol{b}|:|\boldsymbol{c}|=1: k: 3 \quad \left(k \in \mathbf{Z}_{+}\right),
$$
and \( \boldsymbol{b} - \boldsymbol{a} = 2(\boldsymbol{c} - \boldsymbol{b}) \),
if \( \alpha \) is the angle between \( ... | -\frac{1}{12} | 39.84375 |
11,509 | Suppose that \(A\) and \(B\) are digits such that:
\[ \begin{array}{r}
AAA \\
AAB \\
ABB \\
+\ BBB \\
\hline
1503 \\
\end{array} \]
What is the value of \(A^3 + B^2\)? | 57 | 3.90625 |
11,510 | In a $3 \times 4$ grid, you need to place 4 crosses so that there is at least one cross in each row and each column. How many ways are there to do this? | 36 | 38.28125 |
11,511 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $4b\sin A = \sqrt{7}a$.
(1) Find the value of $\sin B$;
(2) If $a$, $b$, and $c$ form an arithmetic sequence with a positive common difference, find the value of $\cos A - \cos C$. | \frac{\sqrt{7}}{2} | 0 |
11,512 | A chocolate bar consists of \(5 \times 8\) square pieces. The bar is broken along lines dividing the pieces until there are 40 individual pieces. How many times must the bar be broken? | 39 | 96.875 |
11,513 | In the Westeros Empire, there were 1000 cities and 2017 roads (each road connects some two cities). From each city, it was possible to travel to any other city. One day, an evil wizard cursed $N$ roads, making them impassable. As a result, 7 kingdoms formed, such that within each kingdom, it is possible to travel from ... | 1024 | 83.59375 |
11,514 | In triangle \(ABC\), the angle bisector \(AD\) divides side \(BC\) in the ratio \(BD : DC = 2 : 1\). In what ratio does the median from vertex \(C\) divide this angle bisector? | 3:1 | 13.28125 |
11,515 | How many three-digit numbers are there in which each digit is greater than the digit to its right? | 84 | 85.15625 |
11,516 | Quarter circles of radius 1' form a pattern as shown below. What is the area, in square feet, of the shaded region in a 3-foot length of this pattern? The quarter circles' flat sides face outward, alternating top and bottom along the length. | \frac{3}{4}\pi | 0 |
11,517 | In a survey of 500 students at a different school, it was found that 75 students own cats and 125 students own dogs. What percent of the students own cats? Also, what percent of the students own dogs? | 25\% | 69.53125 |
11,518 | What is the largest $2$-digit prime factor of the integer $n = {300\choose 150}$? | 97 | 10.9375 |
11,519 | Given the set $P={x|1≦x≦6,x∈N}$, for its non-empty subset $A$, multiply each element $k$ in $A$ by $(-1)^k$ and then sum them up. (For example, if $A={1,3,6}$, the sum would be $(-1)⋅1+(-1)^3⋅3+(-1)^6⋅6=2$.) The total sum of these sums for all non-empty subsets of $M$ is \_\_\_\_\_\_. | 96 | 56.25 |
11,520 | Given a cone with vertex $S$, and generatrices $SA$, $SB$ perpendicular to each other, and the angle between $SA$ and the base of the cone is $30^{\circ}$. If the area of $\triangle SAB$ is $8$, then the volume of this cone is ______. | 8\pi | 64.0625 |
11,521 | Consider the line $15x + 6y = 90$ which forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
A) $21$
B) $35$
C) $41$
D) $21 + 10\sqrt{\frac{1}{29}}$ | 21 + 10\sqrt{\frac{1}{29}} | 50 |
11,522 | Solve the inequality:
\[ 2 \sqrt{(4 x-9)^{2}}+\sqrt[4]{\sqrt{3 x^{2}+6 x+7}+\sqrt{5 x^{2}+10 x+14}+x^{2}+2 x-4} \leq 18-8 x \] | -1 | 10.15625 |
11,523 | Define the sequence of positive integers $a_n$ recursively by $a_1=7$ and $a_n=7^{a_{n-1}}$ for all $n\geq 2$ . Determine the last two digits of $a_{2007}$ . | 43 | 97.65625 |
11,524 | In a revised game of Deal or No Deal, participants choose a box at random from a set of $30$, each containing one of the following values:
\[
\begin{array}{|c|c|}
\hline
\$0.50 & \$50,000 \\
\hline
\$5 & \$100,000 \\
\hline
\$20 & \$150,000 \\
\hline
\$50 & \$200,000 \\
\hline
\$100 & \$250,000 \\
\hline
\$250 & \$300,... | 20 | 6.25 |
11,525 | Given $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, $\overrightarrow{a}\cdot (\overrightarrow{b}-\overrightarrow{a})=0$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} | 100 |
11,526 | For how many integers \( n \) between 1 and 15 (inclusive) is \(\frac{n}{18}\) a repeating decimal? | 10 | 65.625 |
11,527 | Suppose $X$ is a discrete random variable, $P(X=x_{1})= \frac {2}{3},P(X=x_{2})= \frac {1}{3}$, and $x_{1} < x_{2}$, it is also known that $EX= \frac {4}{9}$, $DX=2$, calculate the sum of $x_{1}$ and $x_{2}$. | \frac{17}{9} | 24.21875 |
11,528 | Given the sequence \( a_{1}, a_{2}, \cdots, a_{n}, \cdots \) that satisfies \( a_{1}=a_{2}=1 \) and \( a_{3}=2 \), and for any natural number \( n \), \( a_{n} a_{n+1} a_{n+2} \neq 1 \), and \( a_{n} a_{n+1} a_{n+2} a_{n+3}=a_{n}+a_{n+1}+a_{n+2}+a_{n+3} \), find the value of \( a_{1}+a_{2}+\cdots+a_{100} \). | 200 | 60.9375 |
11,529 | Given a point \\(P(m,-\sqrt{3})\\) (\\(m \neq 0\\)) on the terminal side of angle \\(\alpha\\), and \\(\cos \alpha = \frac{\sqrt{2}m}{4}\\),
\\((i)\\) find the value of \\(m\\);
\\((ii)\\) calculate \\(\sin \alpha\\) and \\(\tan \alpha\\). | -\frac{\sqrt{15}}{5} | 0.78125 |
11,530 | In the diagram, \(PQRS\) is a rectangle with \(SR = 15\). Point \(T\) is above \(PS\) and point \(U\) is on \(PS\) so that \(TU\) is perpendicular to \(PS\). If \(PT = 10\) and \(US = 4\) and the area of \(PQRS\) is 180, what is the area of \(\triangle PTS\)? | 36 | 0.78125 |
11,531 | Simplify $\dfrac{123}{999} \cdot 27.$ | \dfrac{123}{37} | 8.59375 |
11,532 | Given that \( f(x) \) and \( g(x) \) are two quadratic functions both with a leading coefficient of 1, where \( g(6) = 35 \) and \( \frac{f(-1)}{g(-1)} = \frac{f(1)}{g(1)} = \frac{21}{20} \), what is \( f(6) \)? | 35 | 31.25 |
11,533 | Given that \(ABCD\) is a square, points \(E\) and \(F\) lie on the side \(BC\) and \(CD\) respectively, such that \(BE = CF = \frac{1}{3} AB\). \(G\) is the intersection of \(BF\) and \(DE\). If
\[
\frac{\text{Area of } ABGD}{\text{Area of } ABCD} = \frac{m}{n}
\]
is in its lowest terms, find the value of \(m+n\). | 23 | 35.9375 |
11,534 | A public bus departs on schedule at 6:30, 7:00, and 7:30. Student Xiao Ming arrives at the station between 6:50 and 7:30 to catch the bus, and his time of arrival is random. The probability that his waiting time is no more than 10 minutes is ______. | \frac{1}{2} | 25.78125 |
11,535 | A whole number was increased by 2, and its square decreased by 2016. What was the number initially (before the increase)? | -505 | 78.125 |
11,536 | For each positive integer $m$ and $n$ define function $f(m, n)$ by $f(1, 1) = 1$ , $f(m+ 1, n) = f(m, n) +m$ and $f(m, n + 1) = f(m, n) - n$ . Find the sum of all the values of $p$ such that $f(p, q) = 2004$ for some $q$ . | 3007 | 14.0625 |
11,537 | People are standing in a circle - there are liars, who always lie, and knights, who always tell the truth. Each of them said that among the people standing next to them, there is an equal number of liars and knights. How many people are there in total if there are 48 knights? | 72 | 0 |
11,538 | Brachycephalus frogs have three toes on each foot and two fingers on each hand. The common frog has five toes on each foot and four fingers on each hand. Some Brachycephalus and common frogs are in a bucket. Each frog has all its fingers and toes. Between them they have 122 toes and 92 fingers. How many frogs are in th... | 15 | 66.40625 |
11,539 | Find the remainder when \( 8x^4 - 18x^3 - 6x^2 + 4x - 30 \) is divided by \( 2x - 8 \). | 786 | 0 |
11,540 | Every day, the ram learns the same number of languages. By the evening of his birthday, he knew 1000 languages. On the first day of the same month, he knew 820 languages by evening, and on the last day of that month, he knew 1100 languages. When is the ram's birthday? | 19 | 12.5 |
11,541 | A square \(ABCD\) has a side-length of 2, and \(M\) is the midpoint of \(BC\). The circle \(S\) inside the quadrilateral \(AMCD\) touches the three sides \(AM\), \(CD\), and \(DA\). What is its radius? | 3 - \sqrt{5} | 21.09375 |
11,542 | The numbers from 1 to 8 are arranged at the vertices of a cube in such a way that the sum of the numbers at any three vertices on the same face is at least 10. What is the minimum possible sum of the numbers on the vertices of one face? | 16 | 82.03125 |
11,543 | How many unordered pairs of edges in a regular tetrahedron determine a plane? | 12 | 22.65625 |
11,544 | In the diagram, \( Z \) lies on \( XY \) and the three circles have diameters \( XZ \), \( ZY \), and \( XY \). If \( XZ = 12 \) and \( ZY = 8 \), then the ratio of the area of the shaded region to the area of the unshaded region is | 12:13 | 0 |
11,545 | 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 | 52.34375 |
11,546 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, with $c=5$ and $b(2\sin B+\sin A)+(2a+b)\sin A=2c\sin C$.
(1) Find the value of $C$.
(2) If $\cos A= \frac {4}{5}$, find the value of $b$. | 4- \sqrt {3} | 0 |
11,547 | Given the line $y=x+1$ intersects with the ellipse $mx^2+my^2=1(m > n > 0)$ at points $A$ and $B$, where the x-coordinate of the midpoint of the chord $AB$ is equal to $-\frac{1}{3}$, find the eccentricity of the hyperbola $\frac{y^2}{m^2}-\frac{x^2}{n^2}=1$. | \frac{\sqrt{5}}{2} | 4.6875 |
11,548 | Let $F$ be the set of functions from $\mathbb{R}^{+}$ to $\mathbb{R}^{+}$ such that $f(3x) \geq f(f(2x)) + x$. Maximize $\alpha$ such that $\forall x \geq 0, \forall f \in F, f(x) \geq \alpha x$. | \frac{1}{2} | 53.90625 |
11,549 | In a square $\mathrm{ABCD}$, point $\mathrm{E}$ is on $\mathrm{BC}$ with $\mathrm{BE} = 2$ and $\mathrm{CE} = 1$. Point $\mathrm{P}$ moves along $\mathrm{BD}$. What is the minimum value of $\mathrm{PE} + \mathrm{PC}$? | \sqrt{13} | 60.9375 |
11,550 | In a redesign of his company's logo, Wei decided to use a larger square and more circles. Each circle is still tangent to two sides of the square and its adjacent circles, but now there are nine circles arranged in a 3x3 grid instead of a 2x2 grid. If each side of the new square measures 36 inches, calculate the total ... | 1296 - 324\pi | 74.21875 |
11,551 | From 125 sugar cubes, a $5 \times 5 \times 5$ cube was made. Ponchik picked all the cubes that have an odd number of neighbors and ate them (neighbors are those cubes that share a face). How many cubes did Ponchik eat in total? | 62 | 64.84375 |
11,552 | Compute the smallest base-10 positive integer greater than 15 that is a palindrome both in base 2 and base 4. | 17 | 27.34375 |
11,553 | The polynomial \( g(x) = x^4 + ax^3 + bx^2 + cx + d \) has real coefficients, with the roots \( 3i \) and \( 1+2i \). Calculate the sum of the coefficients \( a + b + c + d \). | 39 | 25.78125 |
11,554 | Given there are 2, 1, 3, and 4 paths leading to the top of the mountain from the east, west, south, and north sides, respectively, calculate the maximum number of ways to ascend from one side and descend from any other side. | 24 | 32.8125 |
11,555 | Given the numbers: $8, a, b, 26, x$, where each of the first four numbers is the average of the two adjacent numbers, find the value of $x$. | 32 | 70.3125 |
11,556 | Three frogs in a swamp jumped one after another. Each one landed exactly in the middle of the segment between the other two. The jump length of the second frog is 60 cm. Find the jump length of the third frog. | 30 | 32.03125 |
11,557 | For how many numbers $n$ does $2017$ divided by $n$ have a remainder of either $1$ or $2$ ? | 43 | 33.59375 |
11,558 | Calculate the expression $(-2)^4 + (-2)^3 + (-2)^2 + (-2)^1 + 2^1 + 2^2 + 2^3 + 2^4$. | 40 | 93.75 |
11,559 | How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once? | 42 | 26.5625 |
11,560 | Point P is the intersection of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ($a > 0, b > 0$) and the circle $x^2+y^2=a^2+b^2$ in the first quadrant. $F_1$ and $F_2$ are the left and right foci of the hyperbola, respectively, and $|PF_1|=3|PF_2|$. Calculate the eccentricity of the hyperbola. | \frac{\sqrt{10}}{2} | 20.3125 |
11,561 | Suppose the mean of one set consisting of seven numbers is 18, and the mean of another set consisting of eight numbers is 16. What is the mean of all fifteen numbers combined? | \frac{254}{15} | 3.125 |
11,562 | Let $a, b, c, d$ be real numbers such that $a + b + c + d = 10$ and $ab + ac + ad + bc + bd + cd = 20$. Find the largest possible value of $d$. | \frac{5 + \sqrt{105}}{2} | 18.75 |
11,563 | Simplify and then evaluate the expression: $$(1- \frac {1}{a-2})÷ \frac {a^{2}-6a+9}{2a-4}$$, where $$a=2 \sqrt {3}+3$$ | \frac{\sqrt{3}}{3} | 96.875 |
11,564 | If two stagecoaches travel daily from Bratislava to Brașov, and likewise, two stagecoaches travel daily from Brașov to Bratislava, and considering that the journey takes ten days, how many stagecoaches will you encounter on your way when traveling by stagecoach from Bratislava to Brașov? | 20 | 18.75 |
11,565 | In triangle $ΔABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $a=\sqrt{3}$, $b=\sqrt{2}$, and $A=\frac{\pi}{3}$, find the value of $B=$ _______; and the area of $ΔABC=S_{ΔABC}=$ _______. | \frac{3+ \sqrt{3}}{4} | 38.28125 |
11,566 | Find the integer $n,$ $-180 \le n \le 180,$ such that $\cos n^\circ = \cos 430^\circ.$ | -70 | 61.71875 |
11,567 | OKRA is a trapezoid with OK parallel to RA. If OK = 12 and RA is a positive integer, how many integer values can be taken on by the length of the segment in the trapezoid, parallel to OK, through the intersection of the diagonals? | 10 | 28.90625 |
11,568 | The union of sets \( A \) and \( B \), \( A \cup B = \{a_1, a_2, a_3\} \). When \( A \neq B \), pairs \((A, B)\) and \((B, A)\) are considered different. How many such pairs \((A, B)\) are there? | 27 | 1.5625 |
11,569 | There are two islands, A and B, that are 20 nautical miles apart. When viewing Island C from Island A, the angle between Island B and Island C is 60°. When viewing Island C from Island B, the angle between Island A and Island C is 75°. Find the distance between Island B and Island C. | 10\sqrt{6} | 75 |
11,570 | The shortest distance from a moving point P on circle C: $\rho = -4\sin\theta$ to the line $l: \rho\sin(\theta + \frac{\pi}{4}) = \sqrt{2}$ is ______. | 2\sqrt{2} - 2 | 86.71875 |
11,571 | During the Double 11 shopping festival, a certain online store purchased two types of toys, $A$ and $B$, directly from the factory. The purchase price and selling price are as shown in the table: (Note: Profit = Selling Price - Purchase Price)
| | $A$ toy | $B$ toy |
|----------|---------|---------|
| Purchas... | 36 | 52.34375 |
11,572 | Given that point \( P(x, y) \) satisfies \( |x| + |y| \leq 2 \), find the probability for point \( P \) to have a distance \( d \leq 1 \) from the \( x \)-axis. | 3/4 | 28.125 |
11,573 | A triangle \( ABC \) is given. It is known that \( AB=4 \), \( AC=2 \), and \( BC=3 \). The angle bisector of \( \angle BAC \) intersects side \( BC \) at point \( K \). A line passing through point \( B \) parallel to \( AC \) intersects the extension of the bisector \( AK \) at point \( M \). Find \( KM \). | 2 \sqrt{6} | 3.90625 |
11,574 | Calculate the area of the parallelogram formed by the vectors \( a \) and \( b \).
Given:
\[ a = 3p - 4q \]
\[ b = p + 3q \]
\[ |p| = 2 \]
\[ |q| = 3 \]
\[ \text{Angle between } p \text{ and } q \text{ is } \frac{\pi}{4} \] | 39\sqrt{2} | 76.5625 |
11,575 | Calculate the value of the function $f(x)=3x^{6}-2x^{5}+x^{3}+1$ at $x=2$ using the Horner's method (also known as the Qin Jiushao algorithm) to determine the value of $v_{4}$. | 34 | 22.65625 |
11,576 | Mr. Lee V. Soon starts his morning commute at 7:00 AM to arrive at work by 8:00 AM. If he drives at an average speed of 30 miles per hour, he is late by 5 minutes, and if he drives at an average speed of 70 miles per hour, he is early by 4 minutes. Find the speed he needs to maintain to arrive exactly at 8:00 AM. | 32.5 | 6.25 |
11,577 | 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 | 60.15625 |
11,578 | Points $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C$: $\frac{x^{2}}{2}+y^{2}=1$, respectively. Point $N$ is the top vertex of the ellipse $C$. If a moving point $M$ satisfies $|\overrightarrow{MN}|^{2}=2\overrightarrow{MF_{1}}\cdot\overrightarrow{MF_{2}}$, then the maximum value of $|\overrightarro... | 6+\sqrt{10} | 7.8125 |
11,579 | A line segment is divided into three parts, $x$, $y$, and $z$, such that, $x < y < z$ and $x$ to $y$ is as $y$ to $z$. If all three parts combined form a segment of length $s$, and $x + y = z$, determine the value of the ratio $\frac{x}{y}$.
A) $\frac{-1 - \sqrt{5}}{2}$
B) $\frac{-1 + \sqrt{5}}{2}$
C) $-\frac{1 + \sqrt... | \frac{-1 + \sqrt{5}}{2} | 77.34375 |
11,580 | A certain agency has 18 elderly, 12 middle-aged, and 6 young individuals. When drawing a sample of size X using systematic sampling and stratified sampling, there is no need to discard any individuals. However, if the sample size is increased by 1, then using systematic sampling requires the removal of 1 individual fro... | X = 6 | 42.96875 |
11,581 | Given the function $f(x) = \cos x \cdot \sin\left(x + \frac{\pi}{3}\right) - \sqrt{3}\cos^2x + \frac{\sqrt{3}}{4}$, where $x \in \mathbb{R}$.
(1) Find the interval of monotonic increase for $f(x)$.
(2) In an acute triangle $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respec... | \frac{3\sqrt{3}}{4} | 75.78125 |
11,582 | Calculate the result of the expression \( 2015 \frac{1999}{2015} \times \frac{1}{4} - \frac{2011}{2015} \). | 503 | 59.375 |
11,583 | Suppose you have $6$ red shirts, $7$ green shirts, $9$ pairs of pants, $10$ blue hats, and $10$ red hats, all distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat, if neither the hat nor the pants can match the shirt in color? | 1170 | 33.59375 |
11,584 | Let \( x \) and \( y \) be positive numbers, and let \( s \) be the smallest of the numbers \( x \), \( y + \frac{1}{x} \), and \( \frac{1}{y} \). Find the maximum possible value of \( s \). For which values of \( x \) and \( y \) is it achieved? | \sqrt{2} | 91.40625 |
11,585 | Given the polar equation of curve $C_{1}$ is $\rho=2\sin \theta$, and the polar equation of curve $C_{2}$ is $\theta= \frac {\pi}{3}$ ($\rho\in\mathbb{R}$), curves $C_{1}$ and $C_{2}$ intersect at points $M$ and $N$, then the length of chord $MN$ is ______. | \sqrt {3} | 0 |
11,586 | A sphere is inscribed in a cube. The edge of the cube is 10 inches. Calculate both the volume and the surface area of the sphere. Express your answer for the volume in terms of \(\pi\). | 100\pi | 99.21875 |
11,587 | Three numbers are stored in a computer's memory. Every second, the following operation is performed: each number in this triplet is replaced by the sum of the other two numbers. For example, the triplet \((1; 3; 7)\) becomes \((10; 8; 4)\). What will be the difference between the largest and the smallest number in the ... | 19 | 45.3125 |
11,588 | In the set of positive integers from 1 to \( n \), the numbers that have the most divisors are called the "wang numbers" of these \( n \) positive integers. For example, in the set of positive integers from 1 to 20, the numbers with the most divisors are 12, 18, and 20. Therefore, 12, 18, and 20 are all wang numbers in... | 10080 | 77.34375 |
11,589 | Given that sinα + cosα = $\frac{7}{5}$, find the value of tanα. | \frac{3}{4} | 28.125 |
11,590 | Determine all real numbers $x$ for which the function $$g(x) = \frac{1}{2+\frac{1}{1+\frac{1}{x-1}}}$$ is undefined, and find their sum. | \frac{4}{3} | 52.34375 |
11,591 | Points $A$ and $B$ are 600 kilometers apart. Two people, A and B, start bicycling from point $A$ to point $B$ simultaneously. Person A bicycles 40 kilometers every day, while person B bicycles 60 kilometers every other day, resting every other day. At the end of a certain day, the distance person B has left to cover to... | 12 | 42.1875 |
11,592 | If the ratio of the legs of a right triangle is $3: 4$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is:
A) $\frac{16}{9}$
B) $\frac{9}{16}$
C) $\frac{3}{4}$
D) $\frac{4}{3}$ | \frac{16}{9} | 6.25 |
11,593 | We are given a cone with height 6, whose base is a circle with radius $\sqrt{2}$ . Inside the cone, there is an inscribed cube: Its bottom face on the base of the cone, and all of its top vertices lie on the cone. What is the length of the cube's edge?
 | \frac{3}{2} | 47.65625 |
11,594 | Two circles touch each other internally at point K. The chord \(A B\) of the larger circle is tangent to the smaller circle at point \(L\), and \(A L = 10\). Find \(B L\) if \(A K: B K = 2: 5\). | 25 | 7.8125 |
11,595 | A finite increasing sequence \(a_{1}, a_{2}, \ldots, a_{n}\) of natural numbers is given, where \(n \geq 3\), and for all \(k \leq n-2\) the following equality holds: \(a_{k+2} = 3a_{k+1} - 2a_{k} - 2\). The sequence must include \(a_{k} = 2022\). Determine the maximum number of three-digit numbers, divisible by 4, tha... | 225 | 3.90625 |
11,596 | In how many ways can four black balls, four white balls, and four blue balls be distributed into six different boxes? | 2000376 | 46.875 |
11,597 | In the Cartesian coordinate plane \( xOy \), the coordinates of point \( F \) are \((1,0)\), and points \( A \) and \( B \) lie on the parabola \( y^2 = 4x \). It is given that \( \overrightarrow{OA} \cdot \overrightarrow{OB} = -4 \) and \( |\overrightarrow{FA}| - |\overrightarrow{FB}| = 4\sqrt{3} \). Find the value of... | -11 | 15.625 |
11,598 | Let $m,n$ be natural numbers such that $\hspace{2cm} m+3n-5=2LCM(m,n)-11GCD(m,n).$ Find the maximum possible value of $m+n$ . | 70 | 3.125 |
11,599 | For how many integers \( n \) between 1 and 15 (inclusive) is \(\frac{n}{18}\) a repeating decimal? | 10 | 59.375 |
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