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22,200 | Compute the square of 1033 without a calculator. | 1067089 | 52.34375 |
22,201 | In chess tournaments, each victory is worth 1 point, each draw is worth 0.5 points, and each loss is worth zero points. In the "Magistral Championship" of chess, only International Masters (IMs) and Grand Masters (GMs) participated. The number of GMs was ten times the number of IMs. Each player played only once against all other opponents, and thus, if \( n \) is the number of players, then there were \( \frac{n(n-1)}{2} \) games. The sum of the points of all GMs was 4.5 times the sum of all IMs' points. Therefore, the questions are:
a) How many International Masters participated in this competition?
b) How many Grand Masters participated in this championship?
c) How many games were played in the tournament? | 55 | 85.15625 |
22,202 | Convert the fraction $\frac{123456789}{2^{26} \times 5^{4}}$ to a decimal. How many decimal places does the resulting number have? | 26 | 32.8125 |
22,203 | How many four-digit numbers have at least one 6 or at least one 8 as digits? | 5416 | 97.65625 |
22,204 | Express this sum as a common fraction: $0.\overline{7} + 0.\overline{13}$ | \frac{10}{11} | 96.09375 |
22,205 | A certain brand of computers has a warranty period of $1$ year. Based on a large amount of repair record data, the maximum number of repairs for this brand of computers within one year is $3$ times, with $15\%$ needing $1$ repair, $6\%$ needing $2$ repairs, and $4\%$ needing $3$ repairs. <br/>$(1)$ If a person buys $1$ of this brand of computer, find the probabilities of the following events: $A=$"needs repair within the warranty period"; $B=$"does not need more than $1$ repair within the warranty period"; <br/>$(2)$ If a person buys $2$ of this brand of computers, and the need for repair within the warranty period for the $2$ computers is independent, find the probability that the total number of repairs for these $2$ computers within the warranty period does not exceed $2$ times. | 0.9 | 0.78125 |
22,206 | A natural number plus 13 is a multiple of 5, and its difference with 13 is a multiple of 6. What is the smallest natural number that satisfies these conditions? | 37 | 0 |
22,207 | Find the smallest positive real number $c$ such that for all nonnegative real numbers $x, y,$ and $z$, the following inequality holds:
\[\sqrt[3]{xyz} + c |x - y + z| \ge \frac{x + y + z}{3}.\] | \frac{1}{3} | 84.375 |
22,208 | Let $f(x)$ be a function defined on $\mathbb{R}$ with a period of 2. On the interval $[-1,1)$, $f(x)$ is given by
$$
f(x) = \begin{cases}
x+a & \text{for } -1 \leq x < 0,\\
\left| \frac{2}{5} - x \right| & \text{for } 0 \leq x < 1,
\end{cases}
$$
where $a \in \mathbb{R}$. If $f\left(-\frac{5}{2}\right) = f\left(\frac{9}{2}\right)$, find the value of $f(5a)$. | -\frac{2}{5} | 32.8125 |
22,209 | Given that $n$ is a positive integer, find the minimum value of $|n-1| + |n-2| + \cdots + |n-100|$. | 2500 | 95.3125 |
22,210 | Given the function $f\left(x\right)=x^{2}-2$, find $\lim_{{Δx→0}}\frac{{f(3)-f({3-2Δx})}}{{Δx}}$. | 12 | 99.21875 |
22,211 | Let \( S \) be the set comprising all sides and diagonals of a regular hexagon. A pair of elements from \( S \) are selected randomly without replacement. What is the probability that the two chosen segments have the same length? | \frac{11}{35} | 67.1875 |
22,212 | Ninety-nine children are standing in a circle, each initially holding a ball. Every minute, each child with a ball throws their ball to one of their two neighbors. If two balls end up with the same child, one of these balls is irrevocably lost. What is the minimum time required for the children to have only one ball left? | 98 | 60.15625 |
22,213 | In the diagram, \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \). If \( PQ=4 \), \( QR=8 \), \( RS=8 \), and \( ST=3 \), then the distance from \( P \) to \( T \) is | 13 | 9.375 |
22,214 | Given the volume of the right prism $ABCD-A_{1}B_{1}C_{1}D_{1}$ is equal to the volume of the cylinder with the circumscribed circle of square $ABCD$ as its base, calculate the ratio of the lateral area of the right prism to that of the cylinder. | \sqrt{2} | 40.625 |
22,215 | Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 10x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$ | 37 | 35.9375 |
22,216 | Given that $x = \frac{3}{5}$ is a solution to the equation $30x^2 + 13 = 47x - 2$, find the other value of $x$ that will solve the equation. Express your answer as a common fraction. | \frac{5}{6} | 77.34375 |
22,217 | For any $n\in\mathbb N$ , denote by $a_n$ the sum $2+22+222+\cdots+22\ldots2$ , where the last summand consists of $n$ digits of $2$ . Determine the greatest $n$ for which $a_n$ contains exactly $222$ digits of $2$ . | 222 | 63.28125 |
22,218 | The mean (average), the median, and the mode of the five numbers \( 12, 9, 11, 16, x \) are all equal. What is the value of \( x \)? | 12 | 76.5625 |
22,219 | A shop sells two kinds of products $A$ and $B$ at the price $\$ 2100$. Product $A$ makes a profit of $20\%$, while product $B$ makes a loss of $20\%$. Calculate the net profit or loss resulting from this deal. | -175 | 30.46875 |
22,220 | Calculate the value of $\left(\left((4-1)^{-1} - 1\right)^{-1} - 1\right)^{-1} - 1$. | $\frac{-7}{5}$ | 0 |
22,221 | Calculate the probability that the numbers 1, 1, 2, 2, 3, 3 can be arranged into two rows and three columns such that no two identical numbers appear in the same row or column. | \frac{2}{15} | 6.25 |
22,222 | Let \(S\) be the set of all nonzero real numbers. Let \(f : S \to S\) be a function such that
\[f(x) + f(y) = cf(xyf(x + y))\]
for all \(x, y \in S\) such that \(x + y \neq 0\) and for some nonzero constant \(c\). Determine all possible functions \(f\) that satisfy this equation and calculate \(f(5)\). | \frac{1}{5} | 64.0625 |
22,223 | In a paralleogram $ABCD$ , a point $P$ on the segment $AB$ is taken such that $\frac{AP}{AB}=\frac{61}{2022}$ and a point $Q$ on the segment $AD$ is taken such that $\frac{AQ}{AD}=\frac{61}{2065}$ .If $PQ$ intersects $AC$ at $T$ , find $\frac{AC}{AT}$ to the nearest integer | 67 | 82.8125 |
22,224 | The MathMatters competition consists of 10 players $P_1$ , $P_2$ , $\dots$ , $P_{10}$ competing in a ladder-style tournament. Player $P_{10}$ plays a game with $P_9$ : the loser is ranked 10th, while the winner plays $P_8$ . The loser of that game is ranked 9th, while the winner plays $P_7$ . They keep repeating this process until someone plays $P_1$ : the loser of that final game is ranked 2nd, while the winner is ranked 1st. How many different rankings of the players are possible? | 512 | 78.90625 |
22,225 | Determine the number of ways to arrange the letters of the word "SUCCESS". | 420 | 14.0625 |
22,226 | 1. Given that ${(3x-2)^{6}}={a_{0}}+{a_{1}}(2x-1)+{a_{2}}{(2x-1)^{2}}+ \cdots +{a_{6}}{(2x-1)^{6}}$, find the value of $\dfrac{{a_{1}}+{a_{3}}+{a_{5}}}{{a_{0}}+{a_{2}}+{a_{4}}+{a_{6}}}$.
2. A group of 6 volunteers is to be divided into 4 teams, with 2 teams of 2 people and the other 2 teams of 1 person each, to be sent to 4 different schools for teaching. How many different allocation plans are there in total? (Answer with a number)
3. A straight line $l$ passes through the point $P(1,1)$ with an angle of inclination $\alpha = \dfrac{\pi}{6}$. The line $l$ intersects the curve $\begin{cases} x=2\cos \theta \\ y=2\sin \theta \end{cases}$ ($\theta$ is a parameter) at points $A$ and $B$. Find the value of $|PA| + |PB|$.
4. Find the distance between the centers of the two circles with polar equations $\rho = \cos \theta$ and $\rho = \sin \theta$ respectively. | \dfrac{ \sqrt{2}}{2} | 57.03125 |
22,227 | Determine the value of $b$ that satisfies the equation $295_{b} + 467_{b} = 762_{b}$. | 10 | 94.53125 |
22,228 | In the diagram, $CE$ and $DE$ are two equal chords of circle $O$. The arc $\widehat{AB}$ is $\frac{1}{4}$ of the circumference. Find the ratio of the area of $\triangle CED$ to the area of $\triangle AOB$. | 2: 1 | 0 |
22,229 | Given a triangle \(ABC\) with sides \(AB = 13\), \(BC = 14\), and \(AC = 15\). Point \(K\) is marked on side \(AB\), point \(L\) is marked on side \(BC\), and point \(N\) is marked on side \(AC\). It is known that \(BK = \frac{14}{13}\), \(AN = 10\), and \(BL = 1\). A line is drawn through point \(N\) parallel to \(KL\) which intersects side \(BC\) at point \(M\). Find the area of the quadrilateral \(KLMN\).
Answer: \(\frac{36503}{1183}=30 \frac{1013}{1183}\). | \frac{36503}{1183} | 42.1875 |
22,230 | Let $a$, $b$, and $c$ be three positive real numbers such that $a(a+b+c)=bc$. Determine the maximum value of $\frac{a}{b+c}$. | \frac{\sqrt{2}-1}{2} | 39.0625 |
22,231 | If the inequality system $\left\{\begin{array}{l}{x-m＞0}\\{x-2＜0}\end{array}\right.$ has only one positive integer solution, then write down a value of $m$ that satisfies the condition: ______. | 0.5 | 37.5 |
22,232 | A writer composed a series of essays totaling 60,000 words over a period of 150 hours. However, during the first 50 hours, she was exceptionally productive and wrote half of the total words. Calculate the average words per hour for the entire 150 hours and separately for the first 50 hours. | 600 | 75 |
22,233 | In triangle \(PQR\), the point \(S\) is on \(PQ\) so that the ratio of the length of \(PS\) to the length of \(SQ\) is \(2: 3\). The point \(T\) lies on \(SR\) so that the area of triangle \(PTR\) is 20 and the area of triangle \(SQT\) is 18. What is the area of triangle \(PQR\)? | 80 | 1.5625 |
22,234 | In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a^2 + c^2 - b^2 = ac$, $c=2$, and point $G$ satisfies $|\overrightarrow{BG}| = \frac{\sqrt{19}}{3}$ and $\overrightarrow{BG} = \frac{1}{3}(\overrightarrow{BA} + \overrightarrow{BC})$, find $\sin A$. | \frac{3 \sqrt{21}}{14} | 72.65625 |
22,235 | What is the sum of the proper divisors of $600$? Additionally, determine if $600$ is a perfect square. | 1260 | 0 |
22,236 | What is the area enclosed by the graph of the equation $(x - 1)^2 + (y - 1)^2 = |x - 1| + |y - 1|$?
A) $\frac{\pi}{4}$
B) $\frac{\pi}{2}$
C) $\frac{\pi}{3}$
D) $\pi$ | \frac{\pi}{2} | 57.8125 |
22,237 | Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\pi$. | 40\pi | 19.53125 |
22,238 | Let $a_n$ be the closest to $\sqrt n$ integer.
Find the sum $1/a_1 + 1/a_2 + ... + 1/a_{1980}$ . | 88 | 91.40625 |
22,239 | Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $s$ be a real number. Two of the roots of $f(x)$ are $s + 2$ and $s + 8$. Two of the roots of $g(x)$ are $s + 5$ and $s + 11$, and
\[f(x) - g(x) = 2s\] for all real numbers $x$. Find $s$. | \frac{81}{4} | 2.34375 |
22,240 | The value of \(0.001 + 1.01 + 0.11\) is | 1.121 | 84.375 |
22,241 | Let positive numbers $x$ and $y$ satisfy: $x > y$, $x+2y=3$. Find the minimum value of $\frac{1}{x-y} + \frac{9}{x+5y}$. | \frac{8}{3} | 47.65625 |
22,242 | A cylindrical water tank, placed horizontally, has an interior length of 15 feet and an interior diameter of 8 feet. If the surface area of the water exposed is 60 square feet, find the depth of the water in the tank. | 4 - 2\sqrt{3} | 20.3125 |
22,243 | The product of two 2-digit numbers is $5488$. What is the smaller of the two numbers? | 56 | 67.96875 |
22,244 | Find the minimum value of the function $f(x)=\sum_{n=1}^{19}{|x-n|}$. | 90 | 49.21875 |
22,245 | In the sequence $\{a_n\}$, if for any $n$, $a_n + a_{n+1} + a_{n+2}$ is a constant value (where $n \in \mathbb{N}^*$), and $a_7 = 2$, $a_9 = 3$, $a_{98} = 4$, then the sum of the first 100 terms of this sequence, $S_{100}$, equals to ______. | 299 | 73.4375 |
22,246 | Given $\tan \left(α- \frac {π}{4}\right)=2$, find the value of $\sin \left(2α- \frac {π}{4}\right)$. | \frac {\sqrt {2}}{10} | 0 |
22,247 | Consider the sequence $\{a\_n\}(n\geqslant 1,n\in\mathbb{N})$ that satisfies $a\_1=2$, $a\_2=6$, and $a\_{n+2}-2a\_{n+1}+a\_n=2$. If $[x]$ represents the largest integer not exceeding $x$, then the value of $\left[\frac{2017}{a\_1}+\frac{2017}{a\_2}+...+\frac{2017}{a\_{2017}}\right]$ is \_\_\_\_\_\_. | 2016 | 39.84375 |
22,248 | Given that the prime factorization of a positive integer \( A \) can be written as \( A = 2^{\alpha} \times 3^{\beta} \times 5^{\gamma} \), where \( \alpha, \beta, \gamma \) are natural numbers. If half of \( A \) is a perfect square, one-third of \( A \) is a perfect cube, and one-fifth of \( A \) is a fifth power of some natural number, determine the minimum value of \( \alpha + \beta + \gamma \). | 31 | 63.28125 |
22,249 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = \sqrt{3}$, $|\overrightarrow{b}| = 2$, and $|\overrightarrow{a} + 2\overrightarrow{b}| = \sqrt{7}$.
(I) Find $\overrightarrow{a} \cdot \overrightarrow{b}$;
(II) If vector $λ\overrightarrow{a} + 2\overrightarrow{b}$ is perpendicular to vector $2\overrightarrow{a} - \overrightarrow{b}$, find the value of the real number $λ$. | \frac{20}{9} | 95.3125 |
22,250 | Two positive integers \( x \) and \( y \) are such that:
\[ \frac{2010}{2011} < \frac{x}{y} < \frac{2011}{2012} \]
Find the smallest possible value for the sum \( x + y \). | 8044 | 47.65625 |
22,251 | In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} x=1+\cos \alpha \\ y=\sin \alpha\end{cases}$ ($\alpha$ is the parameter), and 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 $l$ is $\rho\sin (\theta+ \dfrac {\pi}{4})=2 \sqrt {2}$.
(Ⅰ) Convert the parametric equation of curve $C$ and the polar equation of line $l$ into ordinary equations in the Cartesian coordinate system;
(Ⅱ) A moving point $A$ is on curve $C$, a moving point $B$ is on line $l$, and a fixed point $P$ has coordinates $(-2,2)$. Find the minimum value of $|PB|+|AB|$. | \sqrt {37}-1 | 0 |
22,252 | How many unordered pairs of edges of a given square pyramid determine a plane? | 22 | 7.03125 |
22,253 | A green chameleon always tells the truth, while a brown chameleon lies and immediately turns green after lying. In a group of 2019 chameleons (both green and brown), each chameleon, in turn, answered the question, "How many of them are green right now?" The answers were the numbers $1,2,3, \ldots, 2019$ (in some order, not necessarily in the given sequence). What is the maximum number of green chameleons that could have been present initially? | 1010 | 27.34375 |
22,254 | Tom, Dick, and Harry each flip a fair coin repeatedly until they get their first tail. Calculate the probability that all three flip their coins an even number of times and they all get their first tail on the same flip. | \frac{1}{63} | 31.25 |
22,255 | Let $x$, $y$, and $z$ be nonnegative real numbers such that $x + y + z = 8$. Find the maximum value of
\[
\sqrt{3x + 2} + \sqrt{3y + 2} + \sqrt{3z + 2}.
\] | 3\sqrt{10} | 89.0625 |
22,256 | The numbers \( a, b, c, \) and \( d \) are distinct positive integers chosen from 1 to 10 inclusive. What is the least possible value \(\frac{a}{b}+\frac{c}{d}\) could have?
A) \(\frac{2}{10}\)
B) \(\frac{3}{19}\)
C) \(\frac{14}{45}\)
D) \(\frac{29}{90}\)
E) \(\frac{25}{72}\) | \frac{14}{45} | 52.34375 |
22,257 | How many 0.1s are there in 1.9? How many 0.01s are there in 0.8? | 80 | 99.21875 |
22,258 | Given the function \( y = \frac{1}{2}\left(x^{2}-100x+196+\left|x^{2}-100x+196\right|\right) \), calculate the sum of the function values when the variable \( x \) takes on the 100 natural numbers \( 1, 2, 3, \ldots, 100 \). | 390 | 25 |
22,259 | From 8 female students and 4 male students, 3 students are to be selected to participate in a TV program. Determine the number of different selection methods when the selection is stratified by gender. | 112 | 0 |
22,260 | An equilateral triangle \( ABC \) is inscribed in the ellipse \( \frac{x^2}{p^2} + \frac{y^2}{q^2} = 1 \), such that vertex \( B \) is at \( (0, q) \), and \( \overline{AC} \) is parallel to the \( x \)-axis. The foci \( F_1 \) and \( F_2 \) of the ellipse lie on sides \( \overline{BC} \) and \( \overline{AB} \), respectively. Given \( F_1 F_2 = 2 \), find the ratio \( \frac{AB}{F_1 F_2} \). | \frac{8}{5} | 0.78125 |
22,261 | Given $\sin x = \frac{3}{5}$, with $x \in \left( \frac{\pi}{2}, \pi \right)$, find the values of $\cos 2x$ and $\tan\left( x + \frac{\pi}{4} \right)$. | \frac{1}{7} | 67.1875 |
22,262 | Add $45.23$ to $78.569$ and round your answer to the nearest tenth. | 123.8 | 98.4375 |
22,263 | In the rectangular coordinate system $xoy$, the parametric equations of the curve $C$ are $x=3\cos \alpha$ and $y=\sin \alpha$ ($\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive semi-axis of $x$ as the polar axis, the polar equation of the line $l$ is $\rho \sin (\theta -\frac{\pi }{4})=\sqrt{2}$.
1. Find the ordinary equation of the curve $C$ and the rectangular coordinate equation of the line $l$.
2. Let point $P(0, 2)$. The line $l$ intersects the curve $C$ at points $A$ and $B$. Find the value of $|PA|+|PB|$. | \frac{18\sqrt{2}}{5} | 15.625 |
22,264 | Given the coordinates of $A$, $B$, and $C$ are $A(4,0)$, $B(0,4)$, and $C(3\cos \alpha,3\sin \alpha)$ respectively:
$(1)$ If $\alpha \in (-\pi,0)$ and $|\overrightarrow{AC}|=|\overrightarrow{BC}|$, find the value of $\alpha$;
$(2)$ If $\overrightarrow{AC} \cdot \overrightarrow{BC}=0$, find the value of $\frac{2\sin^2\alpha+2\sin\alpha\cos\alpha}{1+\tan\alpha}$. | -\frac{7}{16} | 78.90625 |
22,265 | Given that $\sin x + \cos x = \frac{1}{2}$, where $x \in [0, \pi]$, find the value of $\sin x - \cos x$. | \frac{\sqrt{7}}{2} | 82.03125 |
22,266 | A fair coin is flipped 8 times. What is the probability that exactly 6 of the flips come up heads? | \frac{7}{64} | 53.125 |
22,267 | What is the maximum number of kings, not attacking each other, that can be placed on a standard $8 \times 8$ chessboard? | 16 | 3.90625 |
22,268 | In the coordinate plane, let $A = (-8, 0)$ , $B = (8, 0)$ , and $C = (t, 6)$ . What is the maximum value of $\sin m\angle CAB \cdot \sin m\angle CBA$ , over all real numbers $t$ ? | 3/8 | 24.21875 |
22,269 | Choose $3$ different numbers from the $5$ numbers $0$, $1$, $2$, $3$, $4$ to form a three-digit even number. | 30 | 87.5 |
22,270 | Given $f(x)$ be a differentiable function, satisfying $\lim_{x \to 0} \frac{f(1)-f(1-x)}{2x} = -1$, find the slope of the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$. | -2 | 75.78125 |
22,271 | In the diagram below, $WXYZ$ is a trapezoid where $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$. If $YZ = 20$, $\tan Z = 2$, and $\tan X = 2.5$, then what is the length of $XY$? | 4\sqrt{116} | 0 |
22,272 | Let's call a number palindromic if it reads the same left to right as it does right to left. For example, the number 12321 is palindromic.
a) Write down any five-digit palindromic number that is divisible by 5.
b) How many five-digit palindromic numbers are there that are divisible by 5? | 100 | 92.96875 |
22,273 | A company has 200 employees randomly assigned numbers from 1 to 200, and they are evenly divided into 40 groups. If employee number 22 is selected from the 5th group, find the number of the employee to be selected from the 10th group. | 47 | 33.59375 |
22,274 | In triangle \( \triangle ABC \), if \( \sin A = 2 \sin C \) and the three sides \( a, b, c \) form a geometric sequence, find the value of \( \cos A \). | -\frac{\sqrt{2}}{4} | 79.6875 |
22,275 | Given the triangular pyramid P-ABC, PA is perpendicular to the base ABC, AB=2, AC=AP, BC is perpendicular to CA. If the surface area of the circumscribed sphere of the triangular pyramid P-ABC is $5\pi$, find the value of BC. | \sqrt{3} | 20.3125 |
22,276 | Given that $(2-x)^{5}=a\_{0}+a\_{1}x+a\_{2}x^{2}+…+a\_{5}x^{5}$, find the value of $\frac{a\_0+a\_2+a\_4}{a\_1+a\_3}$. | -\frac{61}{60} | 67.1875 |
22,277 | The number of two-digit numbers that can be formed using the digits 0, 1, 2, 3, 4 without repeating any digit must be calculated. | 16 | 75 |
22,278 | Given sets $A=\{1,2,3,4,5\}$, $B=\{0,1,2,3,4\}$, and a point $P$ with coordinates $(m,n)$, where $m\in A$ and $n\in B$, find the probability that point $P$ lies below the line $x+y=5$. | \dfrac{2}{5} | 57.8125 |
22,279 | Let $U = \{2, 4, 3-a^2\}$ and $P = \{2, a^2+2-a\}$. Given that the complement of $P$ in $U$, denoted as $\complement_U P$, is $\{-1\}$, find the value of $a$. | a = 2 | 56.25 |
22,280 | A box contains 6 cards numbered 1, 2, ..., 6. A card is randomly drawn from the box, and its number is denoted as $a$. The box is then adjusted to retain only the cards with numbers greater than $a$. A second draw is made, and the probability that the first draw is an odd number and the second draw is an even number is to be determined. | \frac{17}{45} | 26.5625 |
22,281 | Given that $x \in (0, \frac{1}{2})$, find the minimum value of $\frac{2}{x} + \frac{9}{1-2x}$. | 25 | 97.65625 |
22,282 | Let $x$ and $y$ be real numbers satisfying $3 \leqslant xy^2 \leqslant 8$ and $4 \leqslant \frac{x^2}{y} \leqslant 9$. Find the maximum value of $\frac{x^3}{y^4}$. | 27 | 56.25 |
22,283 | When a die is thrown twice in succession, the numbers obtained are recorded as $a$ and $b$, respectively. The probability that the line $ax+by=0$ and the circle $(x-3)^2+y^2=3$ have no points in common is ______. | \frac{2}{3} | 25.78125 |
22,284 | Multiply \(333\) by \(111\) and express the result. | 36963 | 0.78125 |
22,285 | John places a total of 15 red Easter eggs in several green baskets and a total of 30 blue Easter eggs in some yellow baskets. Each basket contains the same number of eggs, and there are at least 3 eggs in each basket. How many eggs did John put in each basket? | 15 | 14.84375 |
22,286 | A natural number of five digits is called *Ecuadorian*if it satisfies the following conditions: $\bullet$ All its digits are different. $\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = 1 + 3 + 5 + 0$ , but $54210$ is not since $5 \ne 4 + 2 + 1 + 0$ .
Find how many Ecuadorian numbers exist. | 168 | 96.09375 |
22,287 | Given that the lengths of the three sides of $\triangle ABC$ form an arithmetic sequence with a common difference of 2, and the sine of the largest angle is $\frac{\sqrt{3}}{2}$, the sine of the smallest angle of this triangle is \_\_\_\_\_\_. | \frac{3\sqrt{3}}{14} | 71.875 |
22,288 | Given that $α,β∈(0,π), \text{and } cosα= \frac {3}{5}, cosβ=- \frac {12}{13}$.
(1) Find the value of $cos2α$,
(2) Find the value of $sin(2α-β)$. | - \frac{253}{325} | 68.75 |
22,289 | 0.8 + 0.02 | 0.82 | 89.84375 |
22,290 | Find the largest value of $n$ such that $5x^2 + nx + 90$ can be factored as the product of two linear factors with integer coefficients. | 451 | 73.4375 |
22,291 | In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle? | 50 | 79.6875 |
22,292 | Use the six digits 0, 1, 2, 3, 4, 5 to form four-digit even numbers without repeating any digit. Calculate the total number of such numbers that can be formed. | 156 | 85.15625 |
22,293 | Given the sets $M=\{2, 4, 6, 8\}$, $N=\{1, 2\}$, $P=\left\{x \mid x= \frac{a}{b}, a \in M, b \in N\right\}$, determine the number of proper subsets of set $P$. | 63 | 90.625 |
22,294 | Calculate and simplify
(1) $(1\frac{1}{2})^0 - (1-0.5^{-2}) \div \left(\frac{27}{8}\right)^{\frac{2}{3}}$
(2) $\sqrt{2\sqrt{2\sqrt{2}}}$ | 2^{\frac{7}{8}} | 33.59375 |
22,295 | Given six balls numbered 1, 2, 3, 4, 5, 6 and boxes A, B, C, D, each to be filled with one ball, with the conditions that ball 2 cannot be placed in box B and ball 4 cannot be placed in box D, determine the number of different ways to place the balls into the boxes. | 252 | 12.5 |
22,296 | A science student is asked to find the coefficient of the $x^2$ term in the expansion of $(x^2-3x+2)^4$. The coefficient is \_\_\_\_\_\_. (Answer with a number) | 248 | 74.21875 |
22,297 | Let $p,q,r$ be the roots of $x^3 - 6x^2 + 8x - 1 = 0$, and let $t = \sqrt{p} + \sqrt{q} + \sqrt{r}$. Find $t^4 - 12t^2 - 8t$. | -4 | 69.53125 |
22,298 | Consider the system of equations:
\[
8x - 6y = a,
\]
\[
12y - 18x = b.
\]
If there's a solution $(x, y)$ where both $x$ and $y$ are nonzero, determine $\frac{a}{b}$, assuming $b$ is nonzero. | -\frac{4}{9} | 15.625 |
22,299 | In $\triangle ABC$, $A=\frac{\pi}{4}, B=\frac{\pi}{3}, BC=2$.
(I) Find the length of $AC$;
(II) Find the length of $AB$. | 1+ \sqrt{3} | 35.9375 |
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