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32,700 | Determine the integer \( m \), \( -180 \leq m \leq 180 \), such that \(\sin m^\circ = \sin 945^\circ.\) | -135 | 16.40625 |
32,701 | Given $(625^{\log_2 250})^{\frac{1}{4}}$, find the value of the expression. | 250 | 14.84375 |
32,702 | Determine the smallest positive integer \( m \) for which \( m^3 - 90 \) is divisible by \( m + 9 \). | 12 | 6.25 |
32,703 | Given an arithmetic sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, and it is known that $a_1 > 0$, $a_3 + a_{10} > 0$, $a_6a_7 < 0$, determine the maximum natural number $n$ that satisfies $S_n > 0$. | 12 | 27.34375 |
32,704 | If $f^{-1}(g(x))=x^2-4$ and $g$ has an inverse, find $g^{-1}(f(10))$. | \sqrt{14} | 63.28125 |
32,705 | Diana drew a rectangular grid of 2 by 1009 squares. Some of the squares were then painted black. In each white square, she wrote the number of black squares that shared an edge with it (a whole edge, not just a vertex). What is the maximum value that she could obtain as the result of the sum of all the numbers in this grid?
A 1262
B 2016
C 2018
D 3025
E 3027 | 3025 | 0 |
32,706 | Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6 and with their centers 13 units apart. If a plane \(\alpha\) is tangent to both spheres and intersects the cylindrical surface forming an ellipse, what is the length of the major axis of this ellipse? | 13 | 20.3125 |
32,707 | Xiao Ming and Xiao Hua are counting picture cards in a box together. Xiao Ming is faster, being able to count 6 cards in the same time it takes Xiao Hua to count 4 cards. When Xiao Hua reached 48 cards, he forgot how many cards he had counted and had to start over. When he counted to 112 cards, there was only 1 card left in the box. How many cards were originally in the box? | 169 | 6.25 |
32,708 | One yuan, two yuan, five yuan, and ten yuan RMB notes, each one piece, can form a total of \_\_\_\_\_ different denominations. (Fill in the number) | 15 | 49.21875 |
32,709 | Given the function $f(x)=\log _{3}(9x) \cdot \log _{3}(3x)$, where $\frac {1}{9}\leqslant x\leqslant 9$.
(I) If $m=\log _{3}x$, find the range of $m$;
(II) Find the maximum and minimum values of $f(x)$, and provide the corresponding $x$ values. | 12 | 14.0625 |
32,710 | In $\triangle ABC$, $A, B, C$ are the three interior angles, and $a, b, c$ are the sides opposite to angles $A, B, C$ respectively. It is given that $2 \sqrt{2}\left(\sin^2 A - \sin^2 C\right) = (a - b) \sin B$, and the radius of the circumcircle of $\triangle ABC$ is $\sqrt{2}$.
(1) Find angle $C$;
(2) Find the maximum area $S$ of $\triangle ABC$. | \frac{3\sqrt{3}}{2} | 70.3125 |
32,711 | In triangle \( DEF \) where \( DE = 5, EF = 12, DF = 13 \), and point \( H \) is the centroid. After rotating triangle \( DEF \) by \( 180^\circ \) around \( H \), vertices \( D', E', F' \) are formed. Calculate the area of the union of triangles \( DEF \) and \( D'E'F' \). | 60 | 64.0625 |
32,712 | Given that point $P$ is on curve $C_1: y^2 = 8x$ and point $Q$ is on curve $C_2: (x-2)^2 + y^2 = 1$. If $O$ is the coordinate origin, find the maximum value of $\frac{|OP|}{|PQ|}$. | \frac{4\sqrt{7}}{7} | 0 |
32,713 | In an isosceles triangle \(ABC\) (\(AC = BC\)), an incircle with radius 3 is inscribed. A line \(l\) is tangent to this incircle and is parallel to the side \(AC\). The distance from point \(B\) to the line \(l\) is 3. Find the distance between the points where the incircle touches the sides \(AC\) and \(BC\). | 3\sqrt{3} | 0 |
32,714 | Given that there are two types of golden triangles, one with a vertex angle of $36^{\circ}$ and the other with a vertex angle of $108^{\circ}$, and the ratio of the side opposite the $36^{\circ}$ angle to the side opposite the $72^{\circ}$ angle in the golden triangle with a vertex angle of $36^{\circ}$ is $\frac{\sqrt{5}-1}{2}$, express $\frac{1-2\sin^{2}27^{\circ}}{2t\sqrt{4-t^{2}}}$ in terms of $t$, where $t=\frac{\sqrt{5}-1}{2}$. | \frac{1}{4} | 15.625 |
32,715 | How many lattice points lie on the graph of the equation $x^2 - y^2 = 45$? | 12 | 77.34375 |
32,716 | Let the natural number $N$ be a perfect square, which has at least three digits, its last two digits are not $00$, and after removing these two digits, the remaining number is still a perfect square. Then, the maximum value of $N$ is ____. | 1681 | 94.53125 |
32,717 | A ship sails eastward at a speed of 15 km/h. At point A, the ship observes a lighthouse B at an angle of 60° northeast. After sailing for 4 hours, the ship reaches point C, where it observes the lighthouse at an angle of 30° northeast. At this time, the distance between the ship and the lighthouse is ______ km. | 60 | 27.34375 |
32,718 | Given the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) with asymptotic equations $y = \pm \sqrt{3}x$, and $O$ as the origin, the point $M(\sqrt{5}, \sqrt{3})$ lies on the hyperbola.
$(1)$ Find the equation of the hyperbola $C$.
$(2)$ If a line $l$ intersects the hyperbola at points $P$ and $Q$, and $\overrightarrow{OP} \cdot \overrightarrow{OQ} = 0$, find the minimum value of $|OP|^2 + |OQ|^2$. | 24 | 28.125 |
32,719 | The equation
\[(x - \sqrt[3]{20})(x - \sqrt[3]{70})(x - \sqrt[3]{120}) = \frac{1}{2}\] has three distinct solutions $u,$ $v,$ and $w.$ Calculate the value of $u^3 + v^3 + w^3.$ | 211.5 | 3.125 |
32,720 | The fictional country of Isoland uses an alphabet with ten unique letters: A, B, D, E, I, L, N, O, R, U. License plates in Isoland are structured with five letters. How many different license plates are possible if they must begin with A or I, end with R, cannot include the letter B, and no letters may repeat? | 420 | 37.5 |
32,721 | Given that tetrahedron P-ABC is a 'Bie'zhi' with PA⊥ plane ABC, PA=AB=2, and AC=4, and all four vertices of the tetrahedron P-ABC lie on the surface of sphere O, calculate the surface area of the sphere O. | 20\pi | 17.96875 |
32,722 | A regular $17$ -gon with vertices $V_1, V_2, . . . , V_{17}$ and sides of length $3$ has a point $ P$ on $V_1V_2$ such that $V_1P = 1$ . A chord that stretches from $V_1$ to $V_2$ containing $ P$ is rotated within the interior of the heptadecagon around $V_2$ such that the chord now stretches from $V_2$ to $V_3$ . The chord then hinges around $V_3$ , then $V_4$ , and so on, continuing until $ P$ is back at its original position. Find the total length traced by $ P$ . | 4\pi | 25.78125 |
32,723 | Given the function $f(x) = 2\sqrt{3}\sin^2(x) + 2\sin(x)\cos(x) - \sqrt{3}$, where $x \in \left[ \frac{\pi}{3}, \frac{11\pi}{24} \right]$:
1. Determine the range of the function $f(x)$.
2. Suppose an acute-angled triangle $ABC$ has two sides of lengths equal to the maximum and minimum values of the function $f(x)$, respectively, and the circumradius of triangle $ABC$ is $\frac{3\sqrt{2}}{4}$. Calculate the area of triangle $ABC$. | \sqrt{2} | 6.25 |
32,724 | Find constants $b_1, b_2, b_3, b_4, b_5, b_6, b_7$ such that
\[
\cos^7 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta + b_4 \cos 4 \theta + b_5 \cos 5 \theta + b_6 \cos 6 \theta + b_7 \cos 7 \theta
\]
for all angles $\theta$, and compute $b_1^2 + b_2^2 + b_3^2 + b_4^2 + b_5^2 + b_6^2 + b_7^2$. | \frac{1716}{4096} | 0 |
32,725 | Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=10,$ find the smallest possible value of $|w^3+z^3|$. | 26 | 53.125 |
32,726 | What is the minimum distance between $(2019, 470)$ and $(21a - 19b, 19b + 21a)$ for $a, b \in Z$ ? | \sqrt{101} | 7.8125 |
32,727 | If the graph of the function $y=3\sin(2x+\phi)$ $(0 < \phi < \pi)$ is symmetric about the point $\left(\frac{\pi}{3},0\right)$, then $\phi=$ ______. | \frac{\pi}{3} | 86.71875 |
32,728 | We will call a two-digit number power-less if neither of its digits can be written as an integer to a power greater than 1. For example, 53 is power-less, but 54 is not power-less since \(4 = 2^{2}\). Which of the following is a common divisor of the smallest and the largest power-less numbers? | 11 | 19.53125 |
32,729 | A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of each square is 24, and the perimeter of each small rectangle is 16. What is the perimeter of the large rectangle? | 52 | 15.625 |
32,730 | At the rally commemorating the 60th anniversary of the Chinese people's victory in the War of Resistance against Japan, two schools each send 3 representatives to speak in turns, criticizing the heinous crimes committed by the Japanese aggressors and praising the heroic deeds of the Chinese people in their struggle against Japan. How many different speaking orders are possible? | 72 | 3.90625 |
32,731 | A class prepared 5 programs to participate in the Xiamen No.1 Middle School Music Square event (this event only has 5 programs), and the order of the programs has the following requirements: Program A must be in the first two positions, Program B cannot be in the first position, and Program C must be in the last position. How many possible arrangements of the program order are there for this event? | 10 | 39.84375 |
32,732 | A capricious mathematician writes a book with pages numbered from $2$ to $400$ . The pages are to be read in the following order. Take the last unread page ( $400$ ), then read (in the usual order) all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. So, the order would be $2, 4, 5, ... , 400, 3, 7, 9, ... , 399, ...$ . What is the last page to be read? | 397 | 7.03125 |
32,733 | A digital 12-hour clock has a malfunction such that every time it should display a "2", it instead shows a "5". For example, when it is 2:27 PM, the clock incorrectly shows 5:57 PM. What fraction of the day will the clock show the correct time? | \frac{5}{8} | 3.90625 |
32,734 | Determine the area of the region of the circle defined by $x^2 + y^2 - 8x + 16 = 0$ that lies below the $x$-axis and to the left of the line $y = x - 4$. | 4\pi | 1.5625 |
32,735 | Convert the binary number $111011_{(2)}$ to a decimal number. | 1 \times 2^{5} + 1 \times 2^{4} + 1 \times 2^{3} + 0 \times 2^{2} + 1 \times 2^{1} + 1 | 0 |
32,736 | Given a triangle $ABC$ with internal angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively, and $A=2C$.
(Ⅰ) If $\triangle ABC$ is an acute triangle, find the range of $\frac{a}{c}$.
(Ⅱ) If $b=1, c=3$, find the area of $\triangle ABC$. | \sqrt{2} | 61.71875 |
32,737 | Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. | \frac{3}{4} | 3.125 |
32,738 | A rectangle with dimensions $8 \times 2 \sqrt{2}$ and a circle with a radius of 2 have a common center. What is the area of their overlapping region? | $2 \pi + 4$ | 0 |
32,739 | Amy writes down four integers \(a > b > c > d\) whose sum is 50. The pairwise positive differences of these numbers are \(2, 3, 4, 5, 7,\) and \(10\). What is the sum of the possible values for \(a\)? | 35 | 3.90625 |
32,740 | Given an arithmetic-geometric sequence {$a_n$} with the first term as $\frac{4}{3}$ and a common ratio of $- \frac{1}{3}$. The sum of its first n terms is represented by $S_n$. If $A ≤ S_{n} - \frac{1}{S_{n}} ≤ B$ holds true for any n∈N*, find the minimum value of B - A. | \frac{59}{72} | 27.34375 |
32,741 | Given that the function $f(x)= \frac{1}{2}(m-2)x^{2}+(n-8)x+1$ is monotonically decreasing in the interval $\left[ \frac{1}{2},2\right]$ where $m\geqslant 0$ and $n\geqslant 0$, determine the maximum value of $mn$. | 18 | 15.625 |
32,742 | Given two functions $f(x) = e^{2x-3}$ and $g(x) = \frac{1}{4} + \ln \frac{x}{2}$, if $f(m) = g(n)$ holds, calculate the minimum value of $n-m$. | \frac{1}{2} + \ln 2 | 3.125 |
32,743 | Given the planar vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $\overrightarrow{a}(\overrightarrow{a}+ \overrightarrow{b})=5$, and $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac{\pi}{3} | 100 |
32,744 | The front wheel of Georgina's bicycle has a diameter of 0.75 metres. She cycled for 6 minutes at a speed of 24 kilometres per hour. The number of complete rotations that the wheel made during this time is closest to: | 1020 | 7.8125 |
32,745 | Arrange 7 staff members to be on duty from May 1st to May 7th. Each person works for one day, among them, person A and person B are not scheduled on May 1st and 2nd. The total number of different arrangements is $\boxed{\text{\_\_\_\_\_\_\_\_}}$. | 2400 | 79.6875 |
32,746 | The function f is defined recursively by f(1)=f(2)=1 and f(n)=f(n-1)-f(n-2)+n for all integers n ≥ 3. Find the value of f(2018). | 2017 | 97.65625 |
32,747 | Given $A=\{a, b, c\}$ and $B=\{0, 1, 2\}$, determine the number of mappings $f: A \to B$ that satisfy the condition $f(a) + f(b) > f(c)$. | 14 | 0 |
32,748 | In the arithmetic sequence $\{a\_n\}$, $a_{66} < 0$, $a_{67} > 0$, and $a_{67} > |a_{66}|$. $S_{n}$ denotes the sum of the first $n$ terms of the sequence. Find the smallest value of $n$ such that $S_{n} > 0$. | 132 | 32.03125 |
32,749 | Let $b > 0$, and let $Q(x)$ be a polynomial with integer coefficients such that
\[Q(2) = Q(4) = Q(6) = Q(8) = b\]and
\[Q(1) = Q(3) = Q(5) = Q(7) = -b.\]
What is the smallest possible value of $b$? | 315 | 35.15625 |
32,750 | Given a moving circle $M$ that passes through the fixed point $F(0,-1)$ and is tangent to the line $y=1$. The trajectory of the circle's center $M$ forms a curve $C$. Let $P$ be a point on the line $l$: $x-y+2=0$. Draw two tangent lines $PA$ and $PB$ from point $P$ to the curve $C$, where $A$ and $B$ are the tangent points.
(I) Find the equation of the curve $C$;
(II) When point $P(x_{0},y_{0})$ is a fixed point on line $l$, find the equation of line $AB$;
(III) When point $P$ moves along line $l$, find the minimum value of $|AF|⋅|BF|$. | \frac{9}{2} | 17.1875 |
32,751 | How many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$ or $d$ divide each other? (Order does not matter.)
*Proposed by Miles Yamner and Andrew Wu*
(Note: wording changed from original to clarify) | 1225 | 48.4375 |
32,752 | If an integer is divisible by $6$ and the sum of its last two digits is $15$, then what is the product of its last two digits? | 54 | 51.5625 |
32,753 | The expression $\cos 2x + \cos 6x + \cos 10x + \cos 14x$ can be written in the equivalent form
\[a \cos bx \cos cx \cos dx\] for some positive integers $a,$ $b,$ $c,$ and $d.$ Find $a + b + c + d.$ | 18 | 84.375 |
32,754 | Triangle \(ABC\) has \(AB = 10\) and \(BC:AC = 35:36\). What is the largest area that this triangle can have? | 1260 | 0.78125 |
32,755 | The sequence $\{a_n\}$ is an arithmetic sequence, and the sequence $\{b_n\}$ satisfies $b_n = a_na_{n+1}a_{n+2} (n \in \mathbb{N}^*)$. Let $S_n$ be the sum of the first $n$ terms of $\{b_n\}$. If $a_{12} = \frac{3}{8}a_5 > 0$, then when $S_n$ reaches its maximum value, the value of $n$ is equal to . | 16 | 17.96875 |
32,756 | A square flag features a symmetric red cross with uniform width and two identical blue squares in the center on a white background. The entire cross, including the red arms and the two blue centers, occupies 40% of the flag's area. Determine what percent of the flag's area is occupied by the two blue squares. | 20\% | 1.5625 |
32,757 | Let \( P \) be a regular 2006-sided polygon. If a diagonal of \( P \), whose endpoints divide the boundary of \( P \) into two parts each containing an odd number of sides, is called a "good diagonal". Note that each side of \( P \) is considered a "good diagonal". Given that 2003 non-intersecting diagonals within \( P \) divide \( P \) into several triangles, determine the maximum number of isosceles triangles with two "good diagonals" produced by this division.
(Problem from the 47th IMO) | 1003 | 81.25 |
32,758 | Given four points O, A, B, C on a plane satisfying OA=4, OB=3, OC=2, and $\overrightarrow{OB} \cdot \overrightarrow{OC} = 3$, find the maximum area of $\triangle ABC$. | 2\sqrt{7} + \frac{3\sqrt{3}}{2} | 0 |
32,759 | The sum of the squares of the first ten binomial coefficients ${C}_{2}^{2}+{C}_{3}^{2}+{C}_{4}^{2}+\cdots +{C}_{10}^{2}$ can be found. | 165 | 16.40625 |
32,760 | Given a rectangular storage with length 20 feet, width 15 feet, and height 10 feet, and with the floor and each of the four walls being two feet thick, calculate the total number of one-foot cubical blocks needed for the construction. | 1592 | 40.625 |
32,761 | Consider the function \( g(x) = \sum_{k=3}^{12} (\lfloor kx \rfloor - k \lfloor x \rfloor) \) where \( \lfloor r \rfloor \) denotes the greatest integer less than or equal to \( r \). Determine how many distinct values \( g(x) \) can take for \( x \ge 0 \).
A) 42
B) 43
C) 44
D) 45
E) 46 | 45 | 25.78125 |
32,762 | Given \\(-\pi < x < 0\\), \\(\sin x + \cos x = \frac{1}{5}\\).
\\((1)\\) Find the value of \\(\sin x - \cos x\\);
\\((2)\\) Find the value of \\(\frac{3\sin^2 \frac{x}{2} - 2\sin \frac{x}{2}\cos \frac{x}{2} + \cos^2 \frac{x}{2}}{\tan x + \frac{1}{\tan x}}\\). | -\frac{108}{125} | 62.5 |
32,763 | The dollar is now worth $\frac{1}{980}$ ounce of gold. After the $n^{th}$ 7001 billion dollars bailout package passed by congress, the dollar gains $\frac{1}{2{}^2{}^{n-1}}$ of its $(n-1)^{th}$ value in gold. After four bank bailouts, the dollar is worth $\frac{1}{b}(1-\frac{1}{2^c})$ in gold, where $b, c$ are positive integers. Find $b + c$ . | 506 | 1.5625 |
32,764 | Mike had a bag of candies, and all candies were whole pieces that cannot be divided. Initially, Mike ate $\frac{1}{4}$ of the candies. Then, he shared $\frac{1}{3}$ of the remaining candies with his sister, Linda. Next, both Mike and his father ate 12 candies each from the remaining candies Mike had. Later, Mike’s sister took between one to four candies, leaving Mike with five candies in the end. Calculate the number of candies Mike started with initially. | 64 | 10.15625 |
32,765 | In recent years, China's scientific and technological achievements have been remarkable. The Beidou-3 global satellite navigation system has been operational for many years. The Beidou-3 global satellite navigation system consists of 24 medium Earth orbit satellites, 3 geostationary Earth orbit satellites, and 3 inclined geosynchronous orbit satellites, totaling 30 satellites. The global positioning accuracy of the Beidou-3 global satellite navigation system is better than 10 meters, and the measured navigation positioning accuracy is 2 to 3 meters. The global service availability is 99%, with better performance in the Asia-Pacific region. Now, two satellites are selected for signal analysis from the geostationary Earth orbit satellites and inclined geosynchronous orbit satellites.<br/>$(1)$ Find the probability of selecting exactly one geostationary Earth orbit satellite and one inclined geosynchronous orbit satellite;<br/>$(2)$ Find the probability of selecting at least one inclined geosynchronous orbit satellite. | \frac{4}{5} | 89.84375 |
32,766 | Find all positive integers $ k$ with the following property: There exists an integer $ a$ so that $ (a\plus{}k)^{3}\minus{}a^{3}$ is a multiple of $ 2007$ . | 669 | 25.78125 |
32,767 | In the rectangular coordinate system on a plane, establish a polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The parametric equations of the curve $C$ are $\begin{cases} x=1+\cos \alpha \\ y=\sin \alpha \end{cases} (\alpha \text{ is the parameter, } \alpha \in \left[ 0,\pi \right])$, and the polar equation of the line $l$ is $\rho = \frac{4}{\sqrt{2}\sin \left( \theta - \frac{\pi }{4} \right)}$.
(I) Write the Cartesian equation of curve $C$ and the polar equation of line $l$.
(II) Let $P$ be any point on curve $C$ and $Q$ be any point on line $l$. Find the minimum value of $|PQ|$. | \frac{5 \sqrt{2}}{2}-1 | 50 |
32,768 | An equilateral triangle with a side length of 1 is cut along a line parallel to one of its sides, resulting in a trapezoid. Let $S = \frac{\text{(perimeter of the trapezoid)}^2}{\text{area of the trapezoid}}$. Find the minimum value of $S$. | \frac{32\sqrt{3}}{3} | 2.34375 |
32,769 | A uniform tetrahedron has its four faces numbered with 1, 2, 3, and 4. It is randomly thrown twice, and the numbers on the bottom face of the tetrahedron are $x_1$ and $x_2$, respectively. Let $t = (x_{1}-3)^{2}+(x_{2}-3)^{2}$.
(1) Calculate the probabilities of $t$ reaching its maximum and minimum values, respectively;
(2) Calculate the probability of $t \geq 4$. | \frac{5}{16} | 12.5 |
32,770 | Given that $x$ is a multiple of $46200$, determine the greatest common divisor of $f(x) = (3x + 5)(5x + 3)(11x + 6)(x + 11)$ and $x$. | 990 | 7.8125 |
32,771 | Let point $P$ be on the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$. A line through $P$ intersects the asymptotes at $P_{1}$ and $P_{2}$, and $\overrightarrow{P_{1} P} \overrightarrow{P P_{2}}$ $=3$. Let $O$ be the origin. Find the area of $\triangle O P_{1} P_{2}$. | 16 | 0 |
32,772 | Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits. | 32768 | 67.96875 |
32,773 | Jo and Blair take turns counting from 1, increasing by 1 each turn. However, if the number is a multiple of 3, they skip it. Jo starts by saying "1." Calculate the 53rd number said. | 80 | 35.15625 |
32,774 | Given a circle with center $C$ and equation $x^{2}+y^{2}+2x-4y+m=0$, which intersects with the line $2x+y-3=0$ at points $A$ and $B$:
$(1)$ If $\triangle ABC$ is an equilateral triangle, find the value of $m$;
$(2)$ Is there a constant $m$ such that the circle with diameter $AB$ passes through the origin? If so, find the value of $m$; if not, explain why. | -\frac{18}{5} | 29.6875 |
32,775 | There are several balls of the same shape and size in a bag, including $a+1$ red balls, $a$ yellow balls, and $1$ blue ball. Now, randomly draw a ball from the bag, with the rule that drawing a red ball earns $1$ point, a yellow ball earns $2$ points, and a blue ball earns $3$ points. If the expected value of the score $X$ obtained from drawing a ball from the bag is $\frac{5}{3}$. <br/>$(1)$ Find the value of the positive integer $a$; <br/>$(2)$ Draw $3$ balls from the bag at once, and find the probability that the sum of the scores obtained is $5$. | \frac{3}{10} | 35.15625 |
32,776 | Two numbers are independently selected from the set of positive integers less than or equal to 7. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{36}{49} | 8.59375 |
32,777 | Given vectors $$\overrightarrow {m}=( \sqrt {3}\sin x+\cos x,1), \overrightarrow {n}=(\cos x,-f(x)), \overrightarrow {m}\perp \overrightarrow {n}$$.
(1) Find the monotonic intervals of $f(x)$;
(2) Given that $A$ is an internal angle of $\triangle ABC$, and $$f\left( \frac {A}{2}\right)= \frac {1}{2}+ \frac { \sqrt {3}}{2},a=1,b= \sqrt {2}$$, find the area of $\triangle ABC$. | \frac { \sqrt {3}-1}{4} | 0 |
32,778 | Suppose $x$, $y$, and $z$ are all positive real numbers, and $x^{2}+y^{2}+z^{2}=1$, then the minimum value of $\frac{(z+1)^{2}}{2xyz}$ is $\_\_\_\_\_\_$. | 3+2 \sqrt{2} | 13.28125 |
32,779 | In how many distinct ways can I arrange my six keys on a keychain, if my house key must be exactly opposite my car key and my office key should be adjacent to my house key? For arrangement purposes, two placements are identical if one can be obtained from the other through rotation or flipping the keychain. | 12 | 18.75 |
32,780 | Given the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>0, b>0)$, with left and right foci $F_{1}$, $F_{2}$, and the origin $O$, a perpendicular line is drawn from $F_{1}$ to a asymptote of $C$, with the foot of the perpendicular being $D$, and $|DF_{2}|=2\sqrt{2}|OD|$. Find the eccentricity of $C$. | \sqrt{5} | 19.53125 |
32,781 | The difference between two positive integers is 8 and their product is 56. What is the sum of these integers? | 12\sqrt{2} | 0 |
32,782 | Find all real numbers \(k\) such that the inequality
\[
a^{3} + b^{3} + c^{3} + d^{3} + 1 \geq k(a + b + c + d)
\]
holds for any \(a, b, c, d \in [-1, +\infty)\). | \frac{3}{4} | 73.4375 |
32,783 | 10 times 10,000 is ; 10 times is 10 million; times 10 million is 100 million. There are 10,000s in 100 million. | 10000 | 57.8125 |
32,784 | Given that $0 \leqslant x \leqslant 2$, find the minimum and maximum values of the function $f(x) = 4^{x - \frac{1}{2}} - 3 \cdot 2^x + 5$. | \frac{5}{2} | 35.9375 |
32,785 | Given that the sequence $\{a\_n\}$ is a monotonically increasing arithmetic sequence, the probability that any three terms taken from $\{a\_1, a\_2, a\_3, a\_4, a\_5, a\_6, a\_7\}$ will leave the remaining four terms as a monotonically increasing arithmetic sequence is $\_\_\_\_\_\_\_\_\_.$ | \frac{1}{7} | 21.875 |
32,786 | The number of integers \(N\) from 1 to 1990 for which \(\frac{N^{2}+7}{N+4}\) is not a reduced fraction is: | 86 | 69.53125 |
32,787 | In the cube $ABCDEFGH$, find $\sin \angle BAC$ ensuring that the angle is uniquely determined and forms a right angle. | \frac{\sqrt{2}}{2} | 65.625 |
32,788 | Given that \( AE \) and \( BD \) are straight lines that intersect at \( C \), \( BD = 16 \), \( AB = 9 \), \( CE = 5 \), and \( DE = 3 \), calculate the length of \( AC \). | 15 | 31.25 |
32,789 | Select 3 people from 3 boys and 2 girls to participate in a speech competition.
(1) Calculate the probability that the 3 selected people are all boys;
(2) Calculate the probability that exactly 1 of the 3 selected people is a girl;
(3) Calculate the probability that at least 1 of the 3 selected people is a girl. | \dfrac{9}{10} | 75 |
32,790 | Given that the moving point $P$ satisfies $|\frac{PA}{PO}|=2$ with two fixed points $O(0,0)$ and $A(3,0)$, let the locus of point $P$ be curve $\Gamma$. The equation of $\Gamma$ is ______; the line $l$ passing through $A$ is tangent to $\Gamma$ at points $M$, where $B$ and $C$ are two points on $\Gamma$ with $|BC|=2\sqrt{3}$, and $N$ is the midpoint of $BC$. The maximum area of triangle $AMN$ is ______. | 3\sqrt{3} | 3.90625 |
32,791 | The positive number $a$ is chosen such that the terms $25, a, b, \frac{1}{25}$ are the first, second, third, and fourth terms, respectively, of a geometric sequence. What is the value of $a$ and $b$? | 25^{-1/3} | 0 |
32,792 | In a round-robin tournament, every team played exactly once against each other. Each team won 9 games and lost 9 games; there were no ties. Find the number of sets of three teams {A, B, C} such that team A beat team B, team B beat team C, and team C beat team A. | 969 | 6.25 |
32,793 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 4$, and $\vec{a}\cdot \vec{b}=2$, determine the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac{\pi }{3} | 97.65625 |
32,794 | On a 4x4 grid (where each unit distance is 1), calculate how many rectangles can be formed where each of the rectangle's vertices is a point on this grid. | 36 | 10.9375 |
32,795 | The parabola \(C_{1}: x^{2}=2 p y\) has a focus at \(F\). The hyperbola \(C_{2}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) has foci \(F_{1}\) and \(F_{2}\). Point \(P\) is a common point of the two curves in the first quadrant. If the points \(P\), \(F\), and \(F_{1}\) are collinear and there is a common tangent to \(C_{1}\) and \(C_{2}\) at \(P\), find the eccentricity of \(C_{2}\). | \sqrt{2} | 42.96875 |
32,796 | There are 3 teachers who have all assigned homework. Determine the number of possible situations where 4 students are doing homework at the same time. | 3^{4} | 0 |
32,797 | Given the functions $f(x)= \frac {\ln x}{x}$, $g(x)=kx(k > 0)$, and the function $F(x)=\max\{f(x),g(x)\}$, where $\max\{a,b\}= \begin{cases} a, & \text{if } a\geqslant b\\ b, & \text{if } a < b \end{cases}$
$(I)$ Find the extreme value of $f(x)$;
$(2)$ Find the maximum value of $F(x)$ on the interval $[1,e]$ ($e$ is the base of the natural logarithm). | \frac {1}{e} | 8.59375 |
32,798 | What is the smallest natural number whose digits in decimal representation are either 0 or 1 and which is divisible by 225? (China Junior High School Mathematics League, 1989) | 11111111100 | 96.875 |
32,799 | Given an ellipse M: $$\frac {y^{2}}{a^{2}}+ \frac {x^{2}}{b^{2}}=1$$ (where $a>b>0$) whose eccentricity is the reciprocal of the eccentricity of the hyperbola $x^{2}-y^{2}=1$, and the major axis of the ellipse is 4.
(1) Find the equation of ellipse M;
(2) If the line $y= \sqrt {2}x+m$ intersects ellipse M at points A and B, and P$(1, \sqrt {2})$ is a point on ellipse M, find the maximum area of $\triangle PAB$. | \sqrt {2} | 0 |
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