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31,900 | Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | 24 | 70.3125 |
31,901 | Four red beads, two white beads, and one green bead are placed in a line in random order. What is the probability that no two neighboring beads are the same color?
A) $\frac{1}{15}$
B) $\frac{2}{15}$
C) $\frac{1}{7}$
D) $\frac{1}{30}$
E) $\frac{1}{21}$ | \frac{2}{15} | 11.71875 |
31,902 | From noon till midnight, Clever Cat sleeps under the oak tree and from midnight till noon he is awake telling stories. A poster on the tree above him says "Two hours ago, Clever Cat was doing the same thing as he will be doing in one hour's time". For how many hours a day does the poster tell the truth? | 18 | 7.8125 |
31,903 | Given that the hyperbola $C_2$ and the ellipse $C_1$: $$\frac {x^{2}}{4} + \frac {y^{2}}{3} = 1$$ have the same foci, the eccentricity of the hyperbola $C_2$ when the area of the quadrilateral formed by their four intersection points is maximized is ______. | \sqrt {2} | 0 |
31,904 | Given that the equations of the two asymptotes of a hyperbola are $y = \pm \sqrt{2}x$ and it passes through the point $(3, -2\sqrt{3})$.
(1) Find the equation of the hyperbola;
(2) Let $F$ be the right focus of the hyperbola. A line with a slope angle of $60^{\circ}$ intersects the hyperbola at points $A$ and $B$. Find the length of the segment $|AB|$. | 16 \sqrt{3} | 86.71875 |
31,905 | The median \(AD\) of an acute-angled triangle \(ABC\) is 5. The orthogonal projections of this median onto the sides \(AB\) and \(AC\) are 4 and \(2\sqrt{5}\), respectively. Find the side \(BC\). | 2 \sqrt{10} | 3.125 |
31,906 | A travel agency has 120 rooms, with a daily rental rate of 50 yuan per room, and they are fully booked every day. After renovation, the agency plans to increase the rental rate. Market research indicates that for every 5 yuan increase in the daily rental rate per room, the number of rooms rented out each day will decrease by 6. Without considering other factors, to what amount should the agency increase the daily rental rate per room to maximize the total daily rental income? And how much more will this be compared to the total daily rental income before the renovation? | 750 | 46.09375 |
31,907 | Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.295 | 0 |
31,908 | In the number $56439.2071$, the value of the place occupied by the digit 6 is how many times as great as the value of the place occupied by the digit 2? | 10,000 | 0 |
31,909 | A television station is set to broadcast 6 commercials in a sequence, which includes 3 different business commercials, 2 different World Expo promotional commercials, and 1 public service commercial. The last commercial cannot be a business commercial, and neither the World Expo promotional commercials nor the public service commercial can play consecutively. Furthermore, the two World Expo promotional commercials must also not be consecutive. How many different broadcasting orders are possible? | 36 | 0 |
31,910 | Given \(\frac{\sin (\beta+\gamma) \sin (\gamma+\alpha)}{\cos \alpha \cos \gamma}=\frac{4}{9}\), find the value of \(\frac{\sin (\beta+\gamma) \sin (\gamma+\alpha)}{\cos (\alpha+\beta+\gamma) \cos \gamma}\). | \frac{4}{5} | 0 |
31,911 | In triangle $ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. It is given that $a=b\cos C+c\sin B$.
$(1)$ Find $B$;
$(2)$ If $b=2$, find the maximum area of $\triangle ABC$. | \sqrt {2}+1 | 0 |
31,912 | Given a geometric series \(\left\{a_{n}\right\}\) with the sum of its first \(n\) terms denoted by \(S_{n}\), and satisfying the equation \(S_{n}=\frac{\left(a_{n}+1\right)^{2}}{4}\), find the value of \(S_{20}\). | 400 | 0.78125 |
31,913 | a) Is the equation \( A = B \) true if
\[
A = \frac{1 + \frac{1 + \frac{1 + \frac{1}{2}}{4}}{2}}{2}, \quad B = \frac{1}{1 + \frac{1}{2 + \frac{1}{1 + \frac{1}{2 + \frac{1}{4}}}}}
\]
b) Which is greater between the numbers \( \frac{23}{31} \) and \( \frac{35}{47} \)? Determine a four-digit decimal approximation that deviates by approximately the same amount from these two numbers. | 0.7433 | 78.90625 |
31,914 | Find the natural number $A$ such that there are $A$ integer solutions to $x+y\geq A$ where $0\leq x \leq 6$ and $0\leq y \leq 7$ .
*Proposed by David Tang* | 10 | 93.75 |
31,915 | Given the function $f(x) = \cos(\omega x - \frac{\pi}{3}) - \cos(\omega x)$ $(x \in \mathbb{R}, \omega$ is a constant, and $1 < \omega < 2)$, the graph of function $f(x)$ is symmetric about the line $x = \pi$.
(Ⅰ) Find the smallest positive period of the function $f(x)$.
(Ⅱ) In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a=1$ and $f(\frac{3}{5}A) = \frac{1}{2}$, find the maximum area of $\triangle ABC$. | \frac{\sqrt{3}}{4} | 22.65625 |
31,916 | Let $p>q$ be primes, such that $240 \nmid p^4-q^4$ . Find the maximal value of $\frac{q} {p}$ . | 2/3 | 12.5 |
31,917 | Let $BCDK$ be a convex quadrilateral such that $BC=BK$ and $DC=DK$ . $A$ and $E$ are points such that $ABCDE$ is a convex pentagon such that $AB=BC$ and $DE=DC$ and $K$ lies in the interior of the pentagon $ABCDE$ . If $\angle ABC=120^{\circ}$ and $\angle CDE=60^{\circ}$ and $BD=2$ then determine area of the pentagon $ABCDE$ . | \sqrt{3} | 16.40625 |
31,918 | Given the eccentricity $e= \frac { \sqrt {3}}{2}$ of an ellipse $C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1 (a>b>0)$ with one of its foci at $F( \sqrt {3} , 0)$,
(I) Find the equation of ellipse C;
(II) Let line $l$, passing through the origin O and not perpendicular to the coordinate axes, intersect curve C at points M and N. Additionally, consider point $A(1, \frac {1}{2})$. Determine the maximum area of △MAN. | \sqrt {2} | 0 |
31,919 | Consider a 4-by-4 grid where each of the unit squares can be colored either purple or green. Each color choice is equally likely independent of the others. Compute the probability that the grid does not contain a 3-by-3 grid of squares all colored purple. Express your result in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers and give the value of $m+n$. | 255 | 0.78125 |
31,920 | Let $N$ be the number of ordered pairs of integers $(x, y)$ such that
\[
4x^2 + 9y^2 \le 1000000000.
\]
Let $a$ be the first digit of $N$ (from the left) and let $b$ be the second digit of $N$ . What is the value of $10a + b$ ? | 52 | 72.65625 |
31,921 | The diagram shows a square, its two diagonals, and two line segments, each of which connects two midpoints of the sides of the square. What fraction of the area of the square is shaded?
A) $\frac{1}{8}$
B) $\frac{1}{10}$
C) $\frac{1}{12}$
D) $\frac{1}{16}$
E) $\frac{1}{24}$ | \frac{1}{16} | 25.78125 |
31,922 | Vaccination is one of the important means to protect one's own and others' health and lives. In order to test the immune effect of a certain vaccine on the $C$ virus, researchers used white rabbits as experimental subjects and conducted the following experiments:<br/>Experiment 1: Select 10 healthy white rabbits, numbered 1 to 10, inject them with the vaccine once, and then expose them to an environment containing the $C$ virus. The results of the experiment showed that four of the white rabbits, numbers 2, 6, 7, and 10, were still infected with the $C$ virus, while the others were not infected.<br/>Experiment 2: The vaccine can be injected a second time, but the time interval between the two injections must be more than three weeks. After injecting the vaccine, if antibodies can be produced in the organism, it is considered effective, and the vaccine's protection period is eight months. Researchers injected the first dose of the vaccine to the white rabbits three weeks after the first dose, and whether the vaccine is effective for the white rabbits is not affected by each other.<br/>$(1)$ In "Experiment 1," white rabbits numbered 1 to 5 form Experiment 1 group, and white rabbits numbered 6 to 10 form Experiment 2 group. Researchers first randomly select one group from the two experimental groups, and then randomly select a white rabbit from the selected group for research. Only one white rabbit is taken out each time, and it is not put back. Find the probability that the white rabbit taken out in the first time is infected given that the white rabbit taken out in the second time is not infected;<br/>$(2)$ If the frequency of white rabbits not infected with the $C$ virus in "Experiment 1" is considered as the effectiveness rate of the vaccine, can the effectiveness rate of a white rabbit after two injections of the vaccine reach 96%? If yes, please explain the reason; if not, what is the minimum effectiveness rate of each vaccine injection needed to ensure that the effectiveness rate after two injections of the vaccine is not less than 96%. | 80\% | 0 |
31,923 | Find the least positive integer $n$ satisfying the following statement: for eash pair of positive integers $a$ and $b$ such that $36$ divides $a+b$ and $n$ divides $ab$ it follows that $36$ divides both $a$ and $b$ . | 1296 | 64.0625 |
31,924 | The positive integers are arranged in rows and columns as shown below.
| Row 1 | 1 |
| Row 2 | 2 | 3 |
| Row 3 | 4 | 5 | 6 |
| Row 4 | 7 | 8 | 9 | 10 |
| Row 5 | 11 | 12 | 13 | 14 | 15 |
| Row 6 | 16 | 17 | 18 | 19 | 20 | 21 |
| ... |
More rows continue to list the positive integers in order, with each new row containing one more integer than the previous row. How many integers less than 2000 are in the column that contains the number 2000? | 16 | 0 |
31,925 | Express $0.3\overline{45}$ as a common fraction. | \frac{83}{110} | 0 |
31,926 | Given 5 points \( A, B, C, D, E \) on a plane, with no three points being collinear. How many different ways can one connect these points with 4 segments such that each point is an endpoint of at least one segment? | 135 | 0 |
31,927 | Let $\alpha$ and $\beta$ be acute angles, and $\cos \alpha = \frac{\sqrt{5}}{5}$, $\sin (\alpha + \beta) = \frac{3}{5}$. Find $\cos \beta$. | \frac{2\sqrt{5}}{25} | 17.96875 |
31,928 | Given the following pseudocode:
```
S = 0
i = 1
Do
S = S + i
i = i + 2
Loop while S ≤ 200
n = i - 2
Output n
```
What is the value of the positive integer $n$? | 27 | 0 |
31,929 | Triangle $ABC$ has $AC = 450$ and $BC = 300$ . Points $K$ and $L$ are located on $\overline{AC}$ and $\overline{AB}$ respectively so that $AK = CK$ , and $\overline{CL}$ is the angle bisector of angle $C$ . Let $P$ be the point of intersection of $\overline{BK}$ and $\overline{CL}$ , and let $M$ be the point on line $BK$ for which $K$ is the midpoint of $\overline{PM}$ . If $AM = 180$ , find $LP$ . | 72 | 2.34375 |
31,930 | The average of the numbers $1, 2, 3,\dots, 50,$ and $x$ is $51x$. What is $x$? | \frac{51}{104} | 85.9375 |
31,931 | Six horizontal lines and five vertical lines are drawn in a plane. If a specific point, say (3, 4), exists in the coordinate plane, in how many ways can four lines be chosen such that a rectangular region enclosing the point (3, 4) is formed? | 24 | 41.40625 |
31,932 | Given the parametric equation of circle $C$ as $\begin{cases} x=1+3\cos \theta \\ y=3\sin \theta \end{cases}$ (where $\theta$ is the parameter), and establishing a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of line $l$ is $\theta= \frac {\pi}{4}(\rho\in\mathbb{R})$.
$(1)$ Write the polar coordinates of point $C$ and the polar equation of circle $C$;
$(2)$ Points $A$ and $B$ are respectively on circle $C$ and line $l$, and $\angle ACB= \frac {\pi}{3}$. Find the minimum length of segment $AB$. | \frac {3 \sqrt {3}}{2} | 0 |
31,933 | If 260 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 10 | 9.375 |
31,934 | Petya plans to spend all 90 days of his vacation in the village, swimming in the lake every second day (i.e., every other day), going shopping for groceries every third day, and solving math problems every fifth day. (On the first day, Petya did all three tasks and got very tired.) How many "pleasant" days will Petya have, when he needs to swim but does not need to go shopping or solve math problems? How many "boring" days will he have, when he has no tasks at all? | 24 | 57.03125 |
31,935 | Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | 24 | 67.96875 |
31,936 | Consider the parametric equations for a curve given by
\begin{align*}
x &= \cos t + \frac{t}{3}, \\
y &= \sin t.
\end{align*}
Determine how many times the graph intersects itself between $x = 0$ and $x = 60$. | 28 | 0.78125 |
31,937 | Points $E$ and $F$ lie inside rectangle $ABCD$ with $AE=DE=BF=CF=EF$ . If $AB=11$ and $BC=8$ , find the area of the quadrilateral $AEFB$ . | 32 | 3.90625 |
31,938 | Equilateral $\triangle ABC$ has side length $2$, and shapes $ABDE$, $BCHT$, $CAFG$ are formed outside the triangle such that $ABDE$ and $CAFG$ are squares, and $BCHT$ is an equilateral triangle. What is the area of the geometric shape formed by $DEFGHT$?
A) $3\sqrt{3} - 1$
B) $3\sqrt{3} - 2$
C) $3\sqrt{3} + 2$
D) $4\sqrt{3} - 2$ | 3\sqrt{3} - 2 | 7.03125 |
31,939 | How many positive integers, not exceeding 200, are multiples of 3 or 5 but not 6? | 60 | 0.78125 |
31,940 | Given a magician's hat contains 4 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 4 of the reds are drawn or until both green chips are drawn, calculate the probability that all 4 red chips are drawn before both green chips are drawn. | \frac{1}{3} | 15.625 |
31,941 | Given that the four real roots of the quartic polynomial $f(x)$ form an arithmetic sequence with a common difference of $2$, calculate the difference between the maximum root and the minimum root of $f'(x)$. | 2\sqrt{5} | 11.71875 |
31,942 | In the diagram, rectangle \(P Q R S\) has \(P Q = 30\) and rectangle \(W X Y Z\) has \(Z Y = 15\). If \(S\) is on \(W X\) and \(X\) is on \(S R\), such that \(S X = 10\), then \(W R\) equals: | 35 | 8.59375 |
31,943 | Let \( z = \frac{1+\mathrm{i}}{\sqrt{2}} \). Then calculate the value of \( \left(\sum_{k=1}^{12} z^{k^{2}}\right)\left(\sum_{k=1}^{12} \frac{1}{z^{k^{2}}}\right) \). | 36 | 39.84375 |
31,944 | A right circular cone is inscribed in a right rectangular prism as shown. The base of the prism has dimensions, where one side is exactly twice the length of the other ($a$ and $2a$). The cone's base fits perfectly into the base of the prism making one side of the rectangle the diameter of the cone's base. The height of the prism and the height of the cone are equal. Calculate the ratio of the volume of the cone to the volume of the prism, and express your answer as a common fraction in terms of $\pi$. | \frac{\pi}{24} | 89.84375 |
31,945 | (1) Given a point P(-4, 3) on the terminal side of angle $\alpha$, calculate the value of $$\frac {\cos( \frac {\pi}{2}+\alpha)\sin(-\pi-\alpha)}{\cos( \frac {11\pi}{2}-\alpha )\sin( \frac {9\pi}{2}+\alpha )}$$.
(2) If $\sin x= \frac {m-3}{m+5}$ and $\cos x= \frac {4-2m}{m+5}$, where $x$ is in the interval ($\frac {\pi}{2}$, $\pi$), find $\tan x$. | -\frac {5}{12} | 84.375 |
31,946 | Given a triangle $ABC$ with $\angle B = \frac{\pi}{3}$,
(I) if $AB=8\sqrt{3}$ and $AC=12$, find the area of $\triangle ABC$;
(II) if $AB=4$ and $\vec{BM} = \vec{MN} = \vec{NC}$ with $AN=2\sqrt{3}BM$, find the length of $AM$. | \sqrt{13} | 29.6875 |
31,947 | Given the function f(x) = 2/(x+1) for a positive number x, calculate the sum of f(100) + f(99) + f(98) + ... + f(2) + f(1) + f(1/2) + ... + f(1/98) + f(1/99) + f(1/100). | 199 | 39.0625 |
31,948 | Given the function $f(x) = 2\sin x \cos x + 2\sqrt{3}\cos^2 x$.
(1) Find the smallest positive period of the function $f(x)$;
(2) When $x \in \left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$, find the maximum and minimum values of the function $f(x)$. | 2 + \sqrt{3} | 0 |
31,949 | Given two lines $l_1: y = m$ and $l_2: y = \frac{8}{2m+1}$ ($m > 0$), line $l_1$ intersects the graph of the function $y = |\log_2 x|$ from left to right at points $A$ and $B$, and line $l_2$ intersects the graph of the function $y = |\log_2 x|$ from left to right at points $C$ and $D$. The lengths of the projections of segments $AC$ and $BD$ on the $x$-axis are denoted as $a$ and $b$, respectively. When $m$ varies, the minimum value of $\frac{b}{a}$ is __________. | 8\sqrt{2} | 27.34375 |
31,950 | Find the smallest prime $p$ for which there exist positive integers $a,b$ such that
\[
a^{2} + p^{3} = b^{4}.
\] | 23 | 2.34375 |
31,951 | Let the arithmetic sequences $\{a_n\}$ and $\{b_n\}$ have the sums of their first n terms denoted by $S_n$ and $T_n$, respectively. If $\frac{a_n}{b_n} = \frac{2n-1}{n+1}$, then calculate the value of $\frac{S_{11}}{T_{11}}$. | \frac{11}{7} | 26.5625 |
31,952 | A pentagonal prism is used as the base of a new pyramid. One of the seven faces of this pentagonal prism will be chosen as the base of the pyramid. Calculate the maximum value of the sum of the exterior faces, vertices, and edges of the resulting structure after this pyramid is added. | 42 | 42.96875 |
31,953 | Find the distance between the foci of the ellipse
\[\frac{x^2}{36} + \frac{y^2}{9} = 5.\] | 2\sqrt{5.4} | 0 |
31,954 | Given the function $f(x)=e^{x}-mx^{3}$ ($m$ is a nonzero constant).
$(1)$ If the function $f(x)$ is increasing on $(0,+\infty)$, find the range of real numbers for $m$.
$(2)$ If $f_{n+1}(x)$ ($n\in \mathbb{N}$) represents the derivative of $f_{n}(x)$, where $f_{0}(x)=f(x)$, and when $m=1$, let $g_{n}(x)=f_{2}(x)+f_{3}(x)+\cdots +f_{n}(x)$ ($n\geqslant 2, n\in \mathbb{N}$). If the minimum value of $y=g_{n}(x)$ is always greater than zero, find the minimum value of $n$. | n = 8 | 39.84375 |
31,955 | Liu and Li, each with one child, go to the park together to play. After buying tickets, they line up to enter the park. For safety reasons, the first and last positions must be occupied by fathers, and the two children must stand together. The number of ways for these 6 people to line up is \_\_\_\_\_\_. | 24 | 31.25 |
31,956 | The line $4kx-4y-k=0$ intersects the parabola $y^{2}=x$ at points $A$ and $B$. If the distance between $A$ and $B$ is $|AB|=4$, determine the distance from the midpoint of chord $AB$ to the line $x+\frac{1}{2}=0$. | \frac{9}{4} | 41.40625 |
31,957 | Twelve tiles numbered $1$ through $12$ are turned face down. One tile is turned up at random, and an 8-sided die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
A) $\frac{11}{96}$
B) $\frac{17}{96}$
C) $\frac{21}{96}$
D) $\frac{14}{96}$ | \frac{17}{96} | 80.46875 |
31,958 | Marisela is putting on a juggling show! She starts with $1$ ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute after she starts juggling. | 390 | 3.125 |
31,959 | Given a random variable $\eta = 8 - \xi$, if $\xi \sim B(10, 0.6)$, then calculate $E\eta$ and $D\eta$. | 2.4 | 60.15625 |
31,960 | If \(\lceil \sqrt{x} \rceil = 12\), how many possible integer values of \(x\) are there? | 23 | 92.96875 |
31,961 | The area of triangle $\triangle OFA$, where line $l$ has an inclination angle of $60^\circ$ and passes through the focus $F$ of the parabola $y^2=4x$, and intersects with the part of the parabola that lies on the x-axis at point $A$, is equal to $\frac{1}{2}\cdot OA \cdot\frac{1}{2} \cdot OF \cdot \sin \theta$. Determine the value of this expression. | \sqrt {3} | 0 |
31,962 | Two workers started paving paths in a park simultaneously from point $A$. The first worker paves the section $A-B-C$ and the second worker paves the section $A-D-E-F-C$. They worked at constant speeds and finished at the same time, having spent 9 hours on the job. It is known that the second worker works 1.2 times faster than the first worker. How many minutes did the second worker spend paving the section $DE$? | 45 | 0.78125 |
31,963 | What is the value of $ { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd} $ $ - { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even} $ | 55 | 6.25 |
31,964 | Given the function $y=ax^{2}+bx+c$, where $a$, $b$, $c\in R$.<br/>$(Ⅰ)$ If $a \gt b \gt c$ and $a+b+c=0$, and the distance between the graph of this function and the $x$-axis is $l$, find the range of values for $l$;<br/>$(Ⅱ)$ If $a \lt b$ and the solution set of the inequality $y \lt 0$ is $\varnothing $, find the minimum value of $\frac{{2a+2b+8c}}{{b-a}}$. | 6+4\sqrt{3} | 9.375 |
31,965 | In the diagram, $ABCD$ is a square with side length $8$, and $WXYZ$ is a rectangle with $ZY=12$ and $XY=4$. Additionally, $AD$ and $WX$ are perpendicular. If the shaded area equals three-quarters of the area of $WXYZ$, what is the length of $DP$? | 4.5 | 16.40625 |
31,966 | There are numbers from 1 to 2013 on the blackboard. Each time, two numbers can be erased and replaced with the sum of their digits. This process continues until there are four numbers left, whose product is 27. What is the sum of these four numbers? | 10 | 18.75 |
31,967 | Given the sets $M={x|m\leqslant x\leqslant m+ \frac {7}{10}}$ and $N={x|n- \frac {2}{5}\leqslant x\leqslant n}$, both of which are subsets of ${x|0\leqslant x\leqslant 1}$, find the minimum value of the "length" of the set $M\cap N$. (Note: The "length" of a set ${x|a\leqslant x\leqslant b}$ is defined as $b-a$.) | \frac{1}{10} | 33.59375 |
31,968 | In the Cartesian coordinate plane $(xOy)$, the sum of the distances from point $P$ to the two points $(0,-\sqrt{3})$ and $(0,\sqrt{3})$ is equal to $4$. Let the trajectory of point $P$ be denoted as $C$.
(I) Write the equation of $C$;
(II) If the line $y=kx+1$ intersects $C$ at points $A$ and $B$, for what value of $k$ is $\overrightarrow{OA} \perp \overrightarrow{OB}$? What is the value of $|\overrightarrow{AB}|$ at this time? | \frac{4 \sqrt{65}}{17} | 33.59375 |
31,969 | Given that \( n! \) is evenly divisible by \( 1 + 2 + \cdots + n \), find the number of positive integers \( n \) less than or equal to 50. | 36 | 0 |
31,970 | A TV station is broadcasting 5 advertisements in a row, including 3 different commercial advertisements and 2 different public service advertisements. The last advertisement cannot be a commercial one, and the two public service advertisements cannot be broadcast consecutively. How many different broadcasting methods are there? (Answer with a number). | 36 | 46.09375 |
31,971 | Given the parabola $y^2=2px$ ($p>0$) with focus $F(1,0)$, and the line $l: y=x+m$ intersects the parabola at two distinct points $A$ and $B$. If $0\leq m<1$, determine the maximum area of $\triangle FAB$. | \frac{8\sqrt{6}}{9} | 57.8125 |
31,972 | Given that Charlie estimates 80,000 fans in Chicago, Daisy estimates 70,000 fans in Denver, and Ed estimates 65,000 fans in Edmonton, and given the actual attendance in Chicago is within $12\%$ of Charlie's estimate, Daisy's estimate is within $15\%$ of the actual attendance in Denver, and the actual attendance in Edmonton is exactly as Ed estimated, find the largest possible difference between the numbers attending any two of the three games, to the nearest 1,000. | 29000 | 21.875 |
31,973 | Given the complex number $z$ that satisfies $$z= \frac {1-i}{i}$$ (where $i$ is the imaginary unit), find $z^2$ and $|z|$. | \sqrt {2} | 0 |
31,974 | At 7:10 in the morning, Xiao Ming's mother wakes him up and asks him to get up. However, Xiao Ming sees the time in the mirror and thinks that it is not yet time to get up. He tells his mother, "It's still early!" Xiao Ming mistakenly believes that the time is $\qquad$ hours $\qquad$ minutes. | 4:50 | 16.40625 |
31,975 | Between $5^{5} - 1$ and $5^{10} + 1$, inclusive, calculate the number of perfect cubes. | 199 | 23.4375 |
31,976 | Given a square $ABCD$ . Let $P\in{AB},\ Q\in{BC},\ R\in{CD}\ S\in{DA}$ and $PR\Vert BC,\ SQ\Vert AB$ and let $Z=PR\cap SQ$ . If $BP=7,\ BQ=6,\ DZ=5$ , then find the side length of the square. | 10 | 18.75 |
31,977 | A monkey in Zoo becomes lucky if he eats three different fruits. What is the largest number of monkeys one can make lucky, by having $20$ oranges, $30$ bananas, $40$ peaches and $50$ tangerines? Justify your answer. | 40 | 0.78125 |
31,978 | Given a set $I=\{1,2,3,4,5\}$, select two non-empty subsets $A$ and $B$ such that the largest number in set $A$ is less than the smallest number in set $B$. The total number of different selection methods is $\_\_\_\_\_\_$. | 49 | 15.625 |
31,979 | Given an arithmetic sequence ${a_{n}}$, let $S_{n}$ denote the sum of its first $n$ terms. The first term $a_{1}$ is given as $-20$. The common difference is a real number in the interval $(3,5)$. Determine the probability that the minimum value of $S_{n}$ is only $S_{6}$. | \dfrac{1}{3} | 57.03125 |
31,980 | Let $ f (n) $ be a function that fulfills the following properties: $\bullet$ For each natural $ n $ , $ f (n) $ is an integer greater than or equal to $ 0 $ . $\bullet$ $f (n) = 2010 $ , if $ n $ ends in $ 7 $ . For example, $ f (137) = 2010 $ . $\bullet$ If $ a $ is a divisor of $ b $ , then: $ f \left(\frac {b} {a} \right) = | f (b) -f (a) | $ .
Find $ \displaystyle f (2009 ^ {2009 ^ {2009}}) $ and justify your answer. | 2010 | 31.25 |
31,981 | A store is having a sale for a change of season, offering discounts on a certain type of clothing. If each item is sold at 40% of the marked price, there is a loss of 30 yuan per item, while selling it at 70% of the marked price yields a profit of 60 yuan per item.
Find:
(1) What is the marked price of each item of clothing?
(2) To ensure no loss is incurred, what is the maximum discount that can be offered on this clothing? | 50\% | 67.1875 |
31,982 | On a road of length $A B = 8 \text{ km}$, buses travel in both directions at a speed of $12 \text{ km/h}$. The first bus from each location starts at 6 o'clock, with subsequent buses departing every 10 minutes.
A pedestrian starts walking from $A$ to $B$ at $\frac{81}{4}$ hours; their speed is $4 \text{ km/h}$.
Determine graphically how many oncoming buses the pedestrian will meet, and also when and where these encounters will happen. | 16 | 3.90625 |
31,983 | On a sphere, there are four points A, B, C, and D satisfying $AB=1$, $BC=\sqrt{3}$, $AC=2$. If the maximum volume of tetrahedron D-ABC is $\frac{\sqrt{3}}{2}$, then the surface area of this sphere is _______. | \frac{100\pi}{9} | 6.25 |
31,984 | When the expression $3(x^2 - 3x + 3) - 8(x^3 - 2x^2 + 4x - 1)$ is fully simplified, what is the sum of the squares of the coefficients of the terms? | 2395 | 18.75 |
31,985 | In a modified game, each of 5 players rolls a standard 6-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved must roll again. This continues until only one person has the highest number. If Cecilia is one of the players, what is the probability that Cecilia's first roll was a 4, given that she won the game?
A) $\frac{41}{144}$
B) $\frac{256}{1555}$
C) $\frac{128}{1296}$
D) $\frac{61}{216}$ | \frac{256}{1555} | 18.75 |
31,986 | For real numbers $a,$ $b,$ and $c,$ and a real scalar $\lambda,$ consider the matrix
\[\begin{pmatrix} a + \lambda & b & c \\ b & c + \lambda & a \\ c & a & b + \lambda \end{pmatrix}.\] Determine all possible values of \[\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b}\] assuming the matrix is not invertible. | \frac{3}{2} | 28.125 |
31,987 | Given the coordinates of the three vertices of $\triangle P_{1}P_{2}P_{3}$ are $P_{1}(1,2)$, $P_{2}(4,3)$, and $P_{3}(3,-1)$, the length of the longest edge is ________, and the length of the shortest edge is ________. | \sqrt {10} | 0 |
31,988 | A store prices an item so that when 5% sales tax is added to the price in cents, the total cost rounds naturally to the nearest multiple of 5 dollars. What is the smallest possible integer dollar amount $n$ to which the total cost could round?
A) $50$
B) $55$
C) $60$
D) $65$
E) $70$ | 55 | 36.71875 |
31,989 | Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$. | 15 | 96.875 |
31,990 | Three sectors of a circle are removed from a regular hexagon to form the shaded shape shown. Each sector has a perimeter of 18 mm. What is the perimeter, in mm, of the shaded shape formed? | 54 | 13.28125 |
31,991 | Let $g(x)$ be a third-degree polynomial with real coefficients satisfying \[|g(0)|=|g(1)|=|g(3)|=|g(4)|=|g(5)|=|g(8)|=10.\] Find $|g(2)|$. | 20 | 1.5625 |
31,992 | Write any natural number on a piece of paper, and rotate the paper 180 degrees. If the value remains the same, such as $0$, $11$, $96$, $888$, etc., we call such numbers "神马数" (magical numbers). Among all five-digit numbers, how many different "magical numbers" are there? | 60 | 1.5625 |
31,993 | Given $f(x)=\frac{1}{x}$, calculate the limit of $\frac{f(2+3\Delta x)-f(2)}{\Delta x}$ as $\Delta x$ approaches infinity. | -\frac{3}{4} | 0 |
31,994 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $M$ is the midpoint of $BC$ with $BM = 2$. $AM = c - b$. Find the maximum area of $\triangle ABC$. | 2\sqrt{3} | 7.03125 |
31,995 | On the sides \( BC \) and \( AC \) of triangle \( ABC \), points \( M \) and \( N \) are taken respectively such that \( CM:MB = 1:3 \) and \( AN:NC = 3:2 \). Segments \( AM \) and \( BN \) intersect at point \( K \). Find the area of quadrilateral \( CMKN \), given that the area of triangle \( ABC \) is 1. | 3/20 | 8.59375 |
31,996 | What is the maximum number of rooks that can be placed in an $8 \times 8 \times 8$ cube so that no two rooks attack each other? | 64 | 56.25 |
31,997 | Given a rectangular grid of dots with 5 rows and 6 columns, determine how many different squares can be formed using these dots as vertices. | 40 | 28.125 |
31,998 | The integer 2019 can be formed by placing two consecutive two-digit positive integers, 19 and 20, in decreasing order. What is the sum of all four-digit positive integers greater than 2019 that can be formed in this way? | 478661 | 0 |
31,999 | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 16.40625 |
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