Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
12,200 | What is the smallest integer larger than $(\sqrt{5}+\sqrt{3})^4$? | 248 | 16.40625 |
12,201 | Given the function $f(x)=\begin{cases}f(x+1), & x < 4 \\ 2^{x}, & x\geq 4\end{cases}$, evaluate $f(2+\log_{2}3)$. | 24 | 76.5625 |
12,202 | In triangle $\triangle ABC$, with angles A, B, and C, the sides opposite these angles are labeled as $a, b,$ and $c$ respectively. Given the vectors $\overrightarrow{m}=\left( \frac{a}{2}, \frac{c}{2} \right)$ and $\overrightarrow{n}=(\cos C, \cos A)$, it is also known that $\overrightarrow{n} \cdot \overrightarrow{m} ... | 2\sqrt{3} | 24.21875 |
12,203 | Given that $\frac{x}{3} = y^2$ and $\frac{x}{6} = 3y$, solve for the value of $x$. | 108 | 82.03125 |
12,204 | João, Jorge, José, and Jânio are good friends. One time, João was out of money, but his friends had some. So Jorge gave João a fifth of his money, José gave João a fourth of his money, and Jânio gave João a third of his money. If all of them gave the same amount of money to João, what fraction of the group's money did ... | 1/4 | 83.59375 |
12,205 | Let $D$ be the circle with equation $x^2 - 10y - 7 = -y^2 - 8x + 4$. Find the center $(a, b)$ and radius $r$ of $D$, and determine the value of $a + b + r$. | 1 + 2\sqrt{13} | 93.75 |
12,206 | Let $x$ , $y$ , $z$ be positive integers satisfying $x<y<z$ and $x+xy+xyz=37$ . Find the greatest possible value of $x+y+z$ . | 20 | 51.5625 |
12,207 | Given \(\lg 2 = 0.30103\), calculate the number of digits in \( M = 1 + 10^4 + \frac{10^4 (10^4 - 1)}{1 \cdot 2} + \frac{10^4 (10^4-1)(10^4-2)}{1 \cdot 2 \cdot 3} + \cdots + \frac{10^4 (10^4 - 1)}{1 \cdot 2} + 10^4 + 1\). | 3011 | 78.125 |
12,208 | Simplify \(\left(\cos 42^{\circ}+\cos 102^{\circ}+\cos 114^{\circ}+\cos 174^{\circ}\right)^{2}\) into a rational number. | \frac{3}{4} | 25.78125 |
12,209 | The line passing through the point $(0,-2)$ intersects the parabola $y^{2}=16x$ at two points $A(x_1,y_1)$ and $B(x_2,y_2)$, with $y_1^2-y_2^2=1$. Calculate the area of the triangle $\triangle OAB$, where $O$ is the origin. | \frac{1}{16} | 14.84375 |
12,210 | There are very many symmetrical dice. They are thrown simultaneously. With a certain probability \( p > 0 \), it is possible to get a sum of 2022 points. What is the smallest sum of points that can fall with the same probability \( p \)? | 337 | 24.21875 |
12,211 | Let \( OP \) be the diameter of the circle \( \Omega \), and \( \omega \) be a circle with center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). The chord \( OB \) of the circle \( \Omega \) intersects the second circle... | \sqrt{5} | 76.5625 |
12,212 | In the diagram, \(BD\) is perpendicular to \(BC\) and to \(AD\). If \(AB = 52\), \(BC = 21\), and \(AD = 48\), what is the length of \(DC\)? | 29 | 18.75 |
12,213 | A new model car travels 4.2 kilometers more per liter of gasoline than an old model car. Additionally, the fuel consumption for the new model is 2 liters less per 100 km. How many liters of gasoline per 100 km does the new car consume? Round your answer to the nearest hundredth if necessary. | 5.97 | 46.875 |
12,214 | Find the smallest 6-digit palindrome in base 2, that can be expressed as a 4-digit palindrome in a different base. Provide your response in base 2. | 100001_2 | 17.96875 |
12,215 | There are 49 ones written on a board. Each minute, Karlson erases any two numbers and writes their sum on the board, then he eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 49 minutes? | 1176 | 82.8125 |
12,216 | How many triangles are there with all sides being integers and the longest side being 11? | 36 | 77.34375 |
12,217 | Given the function $f(x)=4\sin x\sin \left(x+ \frac {\pi}{3}\right)-1$.
$(1)$ Calculate the value of $f\left( \frac {5\pi}{6}\right)$:
$(2)$ Let $A$ be the smallest angle in $\triangle ABC$, and $f(A)= \frac {8}{5}$, find the value of $f\left(A+ \frac {\pi}{4}\right)$. | \frac {6}{5} | 2.34375 |
12,218 | Let \( P \) be the parabola with equation \( y = x^2 \) and let \( Q = (10, 6) \). There are real numbers \( r \) and \( s \) such that the line through \( Q \) with slope \( m \) does not intersect \( P \) if and only if \( r < m < s \). What is \( r + s \)? | 40 | 99.21875 |
12,219 | If \\(\alpha \in \left( 0, \frac{\pi}{2} \right)\\), and \\(\cos \left( \frac{\pi}{4} - \alpha \right) = 2 \sqrt{2} \cos 2\alpha\\), then \\(\sin 2\alpha = \)_______. | \frac{15}{16} | 28.90625 |
12,220 | Find \( k \) such that, for all \( n \), the following expression is a perfect square:
$$
4 n^{2} + k n + 9
$$ | 12 | 64.84375 |
12,221 | Let $f(x) = \sqrt{-x^2 + 5x + 6}$.
$(1)$ Find the domain of $f(x)$.
$(2)$ Determine the intervals where $f(x)$ is increasing or decreasing.
$(3)$ Find the maximum and minimum values of $f(x)$ on the interval $[1,5]$. | \sqrt{6} | 64.84375 |
12,222 | Given the function \( f(x) = \frac{2+x}{1+x} \), let \( f(1) + f(2) + \cdots + f(1000) = m \) and \( f\left(\frac{1}{2}\right) + f\left(\frac{1}{3}\right) + \cdots + f\left(\frac{1}{1000}\right) = n \). What is the value of \( m + n \)? | 2998.5 | 14.0625 |
12,223 | 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 |
12,224 | Given \(\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4} \in \mathbf{R}^{+}\) and \(\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}=\pi\), find the minimum value of \(\left(2 \sin ^{2} \theta_{1}+\frac{1}{\sin ^{2} \theta_{1}}\right)\left(2 \sin ^{2} \theta_{2}+\frac{1}{\sin ^{2} \theta_{2}}\right)\left(2 \sin ^{2} \theta_{3... | 81 | 64.84375 |
12,225 | Express $0.3\overline{206}$ as a common fraction. | \frac{5057}{9990} | 0 |
12,226 | The ratio of the length to the width of a rectangle is $5$ : $2$. If the rectangle has a diagonal of length $13$ units, find the constant $k$ such that the area of the rectangle can be expressed as $kd^2$. | \frac{10}{29} | 78.125 |
12,227 | Given the following four propositions:
(1) Two lines parallel to the same plane are parallel to each other.
(2) Two lines perpendicular to the same line are parallel to each other.
(3) Through a point outside a known plane, there exists exactly one plane parallel to the given plane.
(4) Through a line outside a... | (3) | 0 |
12,228 | 150 people were surveyed and asked: "Do you think teal is more green or blue?" Of them, 90 believe teal is "more green," and 50 believe it's "more blue." Additionally, 40 believe it's both "more green" and "more blue." Another 20 think teal is neither "more green" nor "more blue."
How many of those 150 people believe ... | 80 | 43.75 |
12,229 | Find the sum of all positive integers \( p \) such that the expression \((x-p)(x-13)+4\) can be expressed in the form \((x+q)(x+r)\) for distinct integers \( q \) and \( r \). | 26 | 5.46875 |
12,230 | Given the parabola $y^{2}=2px(p > 0)$ and the hyperbola $\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a > 0,b > 0)$ have the same focus $F$, and point $A$ is an intersection point of the two curves, and $AF$ is perpendicular to the x-axis, calculate the eccentricity of the hyperbola. | \sqrt {2} + 1 | 0 |
12,231 | Given that Gill leaves Lille at 09:00, the train travels the first 27 km at 96 km/h and then stops at Lens for 3 minutes before traveling the final 29 km to Lillers at 96 km/h, calculate the arrival time at Lillers. | 09:38 | 15.625 |
12,232 | In a row of 10 chairs, one of which is broken and cannot be used, Mary and James randomly select their seats. What is the probability that they do not sit next to each other? | \frac{7}{9} | 63.28125 |
12,233 | Among the rye seeds, there are $0.4\%$ weed seeds. What is the probability of detecting 5 weed seeds when randomly selecting 5000 seeds? | 0.000055 | 0 |
12,234 | The ratio of the sums of the first \( n \) terms of two arithmetic sequences is \(\frac{9n+2}{n+7}\). Find the ratio of their 5th terms. | \frac{83}{16} | 54.6875 |
12,235 | 35 times 61,000 unit cubes are combined to form a large cube with an edge length of 10 units. After being painted, the large cube is then separated back into the original unit cubes. How many of these unit cubes have at least one face painted? | 488 | 71.09375 |
12,236 | In a competition with five participants A, B, C, D, and E, determine the probability that neither B nor C appears adjacent to A. | \frac{3}{10} | 10.9375 |
12,237 | There are 203 students in the third grade, which is 125 fewer than the fourth grade. How many students are there in total in the third and fourth grades? | 531 | 91.40625 |
12,238 | There is a reservoir A and a town B connected by a river. When the reservoir does not release water, the water in the river is stationary; when the reservoir releases water, the water in the river flows at a constant speed. When the reservoir was not releasing water, speedboat M traveled for 50 minutes from A towards B... | 100/3 | 34.375 |
12,239 | Among five numbers, if we take the average of any four numbers and add the remaining number, the sums will be 74, 80, 98, 116, and 128, respectively. By how much is the smallest number less than the largest number among these five numbers? | 72 | 35.15625 |
12,240 | Consider the sequence of six real numbers 60, 10, 100, 150, 30, and $x$ . The average (arithmetic mean) of this sequence is equal to the median of the sequence. What is the sum of all the possible values of $x$ ? (The median of a sequence of six real numbers is the average of the two middle numbers after all the n... | 135 | 49.21875 |
12,241 | Let $[x]$ be the greatest integer less than or equal to the real number $x$. Given the sequence $\left\{a_{n}\right\}$ which satisfies $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+3 a_{n}+1$ for $n \in N^{*}$, find the value of $\left[\sum_{k=1}^{2017} \frac{a_{k}}{a_{k}+2}\right]$. | 2015 | 42.96875 |
12,242 | Given vectors $\overrightarrow{m}=(\cos \alpha- \frac{\sqrt{2}}{3},-1)$ and $\overrightarrow{n}=(\sin \alpha,1)$, vectors $\overrightarrow{m}$ and $\overrightarrow{n}$ are collinear, and $\alpha \in [-\frac{\pi}{2},0]$.
$(1)$ Find the value of $\sin \alpha + \cos \alpha$;
$(2)$ Find the value of $\frac{\sin 2\alpha... | \frac{7}{12} | 90.625 |
12,243 | Given: $A=2a^{2}-5ab+3b$, $B=4a^{2}+6ab+8a$.
$(1)$ Simplify: $2A-B$;
$(2)$ If $a=-2$, $b=1$, find the value of $2A-B$;
$(3)$ If the value of the algebraic expression $2A-B$ is independent of $a$, find the value of $b$. | -\frac{1}{2} | 89.84375 |
12,244 | Let \( a, b \in \{2, 3, \cdots, 8\} \). Find the maximum value of \(\frac{a}{10b + a} + \frac{b}{10a + b}\). | \frac{89}{287} | 0 |
12,245 | Given that \(\alpha, \beta \in \left(\frac{3\pi}{4}, \pi \right)\), \(\cos (\alpha + \beta) = \frac{4}{5}\), and \(\sin \left(\alpha - \frac{\pi}{4}\right) = \frac{12}{13}\), find \(\cos \left(\beta + \frac{\pi}{4}\right)\). | -\frac{56}{65} | 73.4375 |
12,246 | In the Cartesian coordinate system $xOy$, the curve $C$ is given by the parametric equations $ \begin{cases} x=3\cos \alpha \\ y=\sin \alpha \end{cases} $ (where $\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive x-axis as the polar axis, the polar equation of line $... | \frac {18 \sqrt {2}}{5} | 0 |
12,247 | Given the ellipse \( x^{2}+ \frac {y^{2}}{b^{2}+1}=1(b > 0) \) has an eccentricity of \( \frac {\sqrt {10}}{10} \), determine the value of \( b \). | \frac{1}{3} | 74.21875 |
12,248 | The $10\times15$ rectangle $EFGH$ is cut into two congruent pentagons, which are repositioned to form a square. Determine the length $z$ of one side of the pentagons that aligns with one side of the square.
A) $5\sqrt{2}$
B) $5\sqrt{3}$
C) $10\sqrt{2}$
D) $10\sqrt{3}$ | 5\sqrt{3} | 10.9375 |
12,249 | In a particular game, each of $4$ players rolls a standard $8$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again continuing until one player wins. Hugo is one of the players. What is the probability that Hugo's first roll ... | \frac{27}{128} | 55.46875 |
12,250 | Given the set \( S = \left\{ z \mid |z - 7 - 8i| = |z_1^4 + 1 - 2z_1^2| ; z, z_1 \in \mathbb{C}, |z_1| = 1 \right\} \), find the area of the region corresponding to \( S \) in the complex plane. | 16\pi | 60.15625 |
12,251 | A wire has a length of 6 meters and has 5 nodes that divide the wire into 6 equal parts. If a node is randomly selected to cut the wire, what is the probability that both resulting pieces will have lengths not less than 2 meters? | \frac{3}{5} | 40.625 |
12,252 | Given a regular hexagon \( A B C D E F \) with a side length of 1, calculate \((\overrightarrow{A B}+\overrightarrow{D C}) \cdot(\overrightarrow{A D}+\overrightarrow{B E})\). | -3 | 28.125 |
12,253 | How many positive multiples of 3 that are less than 150 have a units digit of 3 or 6? | 10 | 1.5625 |
12,254 | In the book "Nine Chapters on the Mathematical Art," a tetrahedron with all four faces being right-angled triangles is called a "biēnào." Given that tetrahedron $ABCD$ is a "biēnào," $AB\bot $ plane $BCD$, $BC\bot CD$, and $AB=\frac{1}{2}BC=\frac{1}{3}CD$. If the volume of this tetrahedron is $1$, then the surface area... | 14\pi | 49.21875 |
12,255 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\vec{m}=(a,c)$ and $\vec{n}=(\cos C,\cos A)$.
1. If $\vec{m}\parallel \vec{n}$ and $a= \sqrt {3}c$, find angle $A$;
2. If $\vec{m}\cdot \vec{n}=3b\sin B$ and $\cos A= \frac {3}{5}$, find the value of $\... | \frac {4-6 \sqrt {2}}{15} | 0 |
12,256 | Find all 4-digit numbers $n$ , such that $n=pqr$ , where $p<q<r$ are distinct primes, such that $p+q=r-q$ and $p+q+r=s^2$ , where $s$ is a prime number. | 2015 | 92.1875 |
12,257 | When two numbers are sequentially and randomly picked from the set {1, 2, 3, 4}, what is the probability that the product of the two picked numbers is even? | \frac{5}{6} | 12.5 |
12,258 | Jenna rolls a fair icosahedral die with numbers $1,2,3,...,20$ on its faces. What is the expected number of digits in the number she obtains? | 1.55 | 35.15625 |
12,259 | From post office $A$, a car leaves heading towards post office $B$. After 20 minutes, a motorcyclist departs in pursuit of the car, traveling at a speed of 60 km/h. Upon catching up with the car, the motorcyclist delivers a package to the driver's cab and immediately turns back. The car reaches $B$ at the moment when t... | 45 | 28.90625 |
12,260 | Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, a line passing through its right focus $F$ and parallel to the asymptote $y = -\frac{b}{a}x$ intersects the right branch of the hyperbola and the other asymptote at points $A$ and $B$ respectively, with $\overrightarrow{FA} = \overright... | \sqrt{2} | 56.25 |
12,261 | Given 500 points inside a convex 1000-sided polygon, along with the polygon's vertices (a total of 1500 points), none of which are collinear, the polygon is divided into triangles with these 1500 points as the vertices of the triangles. There are no other vertices apart from these. How many triangles is the convex 1000... | 1998 | 64.84375 |
12,262 | Calculate the limit of the function:
\[ \lim _{x \rightarrow \frac{\pi}{3}} \frac{1-2 \cos x}{\sin (\pi-3 x)} \] | -\frac{\sqrt{3}}{3} | 82.8125 |
12,263 | On a square \(ABCD\), a line segment \(BE\) is drawn such that point \(E\) lies on the side \(CD\). The perimeter of triangle \(BCE\) is three-quarters of the perimeter of the square \(ABCD\). The ratio of lengths \(CE : CD\) is \(\lambda : 1\). What is the value of \(960 \times \lambda\)? | 720 | 63.28125 |
12,264 | Consider that for integers from 1 to 1500, $x_1+2=x_2+4=x_3+6=\cdots=x_{1500}+3000=\sum_{n=1}^{1500}x_n + 3001$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{1500}x_n$. | 1500 | 18.75 |
12,265 | The cross-section of a sphere passing through points $A$, $B$, and $C$, whose distance from the center of the sphere is equal to half the radius, and $AB \perp BC$, $AB=1$, $BC=\sqrt{2}$. Calculate the surface area of the sphere. | 4\pi | 72.65625 |
12,266 | Given a parabola $C$ that passes through the point $(4,4)$ and its focus lies on the $x$-axis.
$(1)$ Find the standard equation of parabola $C$.
$(2)$ Let $P$ be any point on parabola $C$. Find the minimum distance between point $P$ and the line $x - y + 4 = 0$. | \frac{3\sqrt{2}}{2} | 94.53125 |
12,267 | In the convex quadrilateral \(ABCD\), the intersection point of its diagonals is \(O\). What is the minimum area of the quadrilateral if the area of triangle \(AOB\) is \(4 \mathrm{~cm}^2\) and the area of triangle \(COD\) is \(9 \mathrm{~cm}^2\)? | 25 | 50.78125 |
12,268 | A teacher received letters on Monday through Friday with counts of $10$, $6$, $8$, $5$, $6$ respectively. Calculate the variance (${s^{2}} =$) of this data set. | 3.2 | 39.0625 |
12,269 | Find the value of: \(\frac{\left(\sqrt{3} \cdot \tan 12^{\circ} - 3\right) \cdot \csc 12^{\circ}}{4 \cos ^{2} 12^{\circ} - 2}\). | -4 \sqrt{3} | 30.46875 |
12,270 | The roots of the equation $x^2 + kx + 8 = 0$ differ by 10. Find the greatest possible value of $k$. | 2\sqrt{33} | 95.3125 |
12,271 | Two people, A and B, alternately pick distinct numbers from the set \(\{0, 1, \cdots, 81\}\). A picks first, and each person picks one number per turn. After all 82 numbers are picked, let \(S_A\) and \(S_B\) be the sums of the numbers chosen by A and B, respectively. During the selection process, A wants to maximize t... | 41 | 47.65625 |
12,272 | The probability that the distance between any two points selected from the four vertices and the center of a square is not less than the side length of the square is $\boxed{\text{answer}}$. | \frac{3}{5} | 35.15625 |
12,273 | In the number \(2016 * * * * 02 *\), each of the 5 asterisks needs to be replaced with any of the digits \(0, 2, 4, 7, 8, 9\) (digits can be repeated) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done? | 1728 | 25 |
12,274 |
In a rectangular parallelepiped $ABCDEFGH$, the edge lengths are given as $AB = 30$, $AD = 32$, and $AA_1 = 20$. Point $E$ is marked at the midpoint of edge $A_1B_1$, and point $F$ is marked at the midpoint of edge $B_1C_1$. Find the distance between the lines $AE$ and $BF$. | 19.2 | 44.53125 |
12,275 | Points \( M, N, P, Q \) are taken on the diagonals \( D_1A, A_1B, B_1C, C_1D \) of the faces of cube \( ABCD A_1B_1C_1D_1 \) respectively, such that:
\[ D_1M: D_1A = BA_1: BN = B_1P: B_1C = DQ: DC_1 = \mu, \]
and the lines \( MN \) and \( PQ \) are mutually perpendicular. Find \( \mu \). | \frac{1}{\sqrt{2}} | 0.78125 |
12,276 | For any two positive integers m and n, define an operation "※" as follows: when m and n are both positive even numbers or positive odd numbers, m※n=m+n; when one of m and n is a positive even number and the other is a positive odd number, m※n=mn. Determine the number of elements in the set M={(a,b)|a※b=12, a∈ℕ∗, b∈ℕ∗}. | 15 | 32.03125 |
12,277 | Elisa creates a sequence in a manner similar to Jacob but starts with the first term as 10. Each succeeding term depends on the outcome of flipping a fair coin: If it lands heads, the next term is obtained by doubling the previous term and subtracting 1; if it lands tails, the next term is half of the previous term, su... | \frac{1}{2} | 46.875 |
12,278 | The value of $3x + 15$ is one third of the value of $6x + 45$. After finding $x$, subtract 5 from the result. What is the final value? | -5 | 81.25 |
12,279 | In the Cartesian coordinate system $xOy$, it is known that the line $l_1$ is defined by the parametric equations $\begin{cases}x=t\cos \alpha\\y=t\sin \alpha\end{cases}$ (where $t$ is the parameter), and the line $l_2$ by $\begin{cases}x=t\cos(\alpha + \frac{\pi}{4})\\y=t\sin(\alpha + \frac{\pi}{4})\end{cases}$ (where ... | 2\sqrt{2} | 72.65625 |
12,280 | On an island, there are red, yellow, green, and blue chameleons.
- On a cloudy day, either one red chameleon changes its color to yellow, or one green chameleon changes its color to blue.
- On a sunny day, either one red chameleon changes its color to green, or one yellow chameleon changes its color to blue.
In Septe... | 11 | 13.28125 |
12,281 | There are 200 matches. How many ways are there to form, using all the matches, a square and (separately) an equilateral triangle? (Different ways are distinguished by the sizes of the square and the triangle). | 16 | 65.625 |
12,282 | If \( x = \frac{2}{3} \) and \( y = \frac{3}{2} \), find the value of \( \frac{1}{3}x^8y^9 \). | \frac{1}{2} | 63.28125 |
12,283 | In a rectangular parallelepiped \( ABCD A_{1} B_{1} C_{1} D_{1} \), the edge lengths are given by \( AB = 42 \), \( AD = 126 \), and \( AA_{1} = 42 \). Point \( E \) is marked at the midpoint of edge \( A_{1}B_{1} \), and point \( F \) is marked at the midpoint of edge \( B_{1}C_{1} \). Find the distance between the li... | 36 | 75 |
12,284 | Let $P(x) = b_0 + b_1x + \dots + b_nx^n$ be a polynomial with integer coefficients, and $0 \le b_i < 5$ for all $0 \le i \le n$.
Given that $P(\sqrt{5}) = 40 + 31\sqrt{5}$, compute $P(3)$. | 381 | 4.6875 |
12,285 | The integer parts of two finite decimals are 7 and 10, respectively. How many possible values are there for the integer part of the product of these two finite decimals? | 18 | 44.53125 |
12,286 | Numbering the pages of an encyclopedia required 6869 digits. How many pages does it contain? | 1994 | 16.40625 |
12,287 | Each of the lateral edges of a pyramid is equal to 269/32. The base of the pyramid is a triangle with sides 13, 14, 15. Find the volume of the pyramid. | 483/8 | 56.25 |
12,288 | A column of infantry stretched out over 1 km. Sergeant Kim, riding on a hoverboard from the end of the column, reached its front and returned to the end. During this time, the infantrymen walked 2 km 400 meters. What distance did the sergeant cover during this time? | 3.6 | 10.15625 |
12,289 | Let \( A = \{1, 2, 3, 4, 5, 6\} \). Find the number of distinct functions \( f: A \rightarrow A \) such that \( f(f(f(n))) = n \) for all \( n \in A \). | 81 | 37.5 |
12,290 | The Rotokas alphabet has twelve letters: A, E, G, I, K, O, P, R, S, T, U, and V. Design license plates of five letters using only these letters where the license plate ends with either G or K, starts with S, cannot contain T, and where no letters repeat. How many such license plates are possible? | 1008 | 28.90625 |
12,291 | Let \( a, b, c \) be the side lengths of a right triangle, with \( a \leqslant b < c \). Determine the maximum constant \( k \) such that \( a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \geqslant k a b c \) holds for all right triangles, and identify when equality occurs. | 2 + 3\sqrt{2} | 8.59375 |
12,292 | Given that a regular tetrahedron has all edge lengths of $\sqrt {2}$, and all four vertices are on the same spherical surface, find the surface area of this sphere. | 3\pi | 100 |
12,293 | Real numbers \( x, y, z \) satisfy \( x \geq y \geq z \geq 0 \) and \( 6x + 5y + 4z = 120 \). Find the sum of the maximum and minimum values of \( x + y + z \). | 44 | 60.9375 |
12,294 | What is the smallest natural number that is divisible by 2022 and starts with 2023? | 20230110 | 0 |
12,295 | A number is formed using the digits \(1, 2, \ldots, 9\). Any digit can be used more than once, but adjacent digits cannot be the same. Once a pair of adjacent digits has occurred, that pair, in that order, cannot be used again. How many digits are in the largest such number? | 73 | 1.5625 |
12,296 | Given the ellipse $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a＞b＞0)$$ with eccentricity $$e= \frac { \sqrt {3}}{2}$$, A and B are the left and right vertices of the ellipse, respectively, and P is a point on the ellipse different from A and B. The angles of inclination of lines PA and PB are $\alpha$ and $\beta$, r... | \frac {3}{5} | 44.53125 |
12,297 | The height $BL$ of the rhombus $ABCD$, dropped perpendicular to the side $AD$, intersects the diagonal $AC$ at point $E$. Find $AE$ if $BL = 8$ and $AL:LD = 3:2$. | 3\sqrt{5} | 0 |
12,298 | Defined on $\mathbf{R}$, the function $f$ satisfies
$$
f(1+x)=f(9-x)=f(9+x).
$$
Given $f(0)=0$, and $f(x)=0$ has $n$ roots in the interval $[-4020, 4020]$, find the minimum value of $n$. | 2010 | 0.78125 |
12,299 | In $\Delta ABC$, $c=2a$, $B={120}^{\circ}$, and the area of $\Delta ABC$ is $\frac{\sqrt{3}}{2}$.
(I) Find the value of $b$;
(II) Find the value of $\tan A$. | \frac{\sqrt{3}}{5} | 85.15625 |
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