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32,600 | Given an ellipse $C:\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\left( a > b > 0 \right)$ with left and right focal points ${F}_{1},{F}_{2}$, and a point $P\left( 1,\frac{\sqrt{2}}{2} \right)$ on the ellipse such that $\vert P{F}_{1}\vert+\vert P{F}_{2}\vert=2 \sqrt{2}$.
(1) Find the standard equation of the ellipse $C$.
(2) A line $l$ passes through ${F}_{2}$ and intersects the ellipse at two points $A$ and $B$. Find the maximum area of $\triangle AOB$. | \frac{\sqrt{2}}{2} | 22.65625 |
32,601 | In triangle $ABC$, where $AB = 50$, $BC = 36$, and $AC = 42$. A line $CX$ from $C$ is perpendicular to $AB$ and intersects $AB$ at point $X$. Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$. Express your answer as a simplified common fraction. | \frac{6}{7} | 19.53125 |
32,602 | Given the function $f(x) = 3x^4 + 6$, find the value of $f^{-1}(150)$. | \sqrt[4]{48} | 3.125 |
32,603 | In triangle $\triangle ABC$, given that $A=60^{\circ}$ and $BC=4$, the diameter of the circumcircle of $\triangle ABC$ is ____. | \frac{8\sqrt{3}}{3} | 89.84375 |
32,604 | The minimum value of the function $y = \sin 2 \cos 2x$ is ______. | - \frac{1}{2} | 17.96875 |
32,605 | Find the integer \( n \), \( -180 \le n \le 180 \), such that \( \cos n^\circ = \cos 745^\circ \). | -25 | 14.0625 |
32,606 | Given a ball with a diameter of 6 inches rolling along the path consisting of four semicircular arcs, with radii $R_1 = 100$ inches, $R_2 = 60$ inches, $R_3 = 80$ inches, and $R_4 = 40$ inches, calculate the distance traveled by the center of the ball from the start to the end of the track. | 280\pi | 0 |
32,607 | Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 100 residents of this island lined up, and each of them said one of the following phrases:
- "To the left of me there are as many liars as knights."
- "To the left of me there is 1 more liar than knights."
- "To the left of me there are 2 more liars than knights."
- "To the left of me there are 99 more liars than knights."
It is known that each phrase was said by exactly one person. What is the minimum number of liars that can be among these 100 residents? | 50 | 34.375 |
32,608 | A positive integer is *bold* iff it has $8$ positive divisors that sum up to $3240$ . For example, $2006$ is bold because its $8$ positive divisors, $1$ , $2$ , $17$ , $34$ , $59$ , $118$ , $1003$ and $2006$ , sum up to $3240$ . Find the smallest positive bold number. | 1614 | 16.40625 |
32,609 | The endpoints of a line segment AB, which has a fixed length of 3, move on the parabola $y^2=x$. If M is the midpoint of the line segment AB, then the minimum distance from M to the y-axis is ______. | \frac{5}{4} | 16.40625 |
32,610 | Given the function \( f(x) = |x-1| + |x-3| + \mathrm{e}^x \) (where \( x \in \mathbf{R} \)), find the minimum value of the function. | 6-2\ln 2 | 28.90625 |
32,611 | (1) If the terminal side of angle $\theta$ passes through $P(-4t, 3t)$ ($t>0$), find the value of $2\sin\theta + \cos\theta$.
(2) Given that a point $P$ on the terminal side of angle $\alpha$ has coordinates $(x, -\sqrt{3})$ ($x\neq 0$), and $\cos\alpha = \frac{\sqrt{2}}{4}x$, find $\sin\alpha$ and $\tan\alpha$. | \frac{2}{5} | 1.5625 |
32,612 | Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins or indefinitely. If Nathaniel goes first, determine the probability that he ends up winning. | 5/11 | 0 |
32,613 | Set \( S \) satisfies the following conditions:
1. The elements of \( S \) are positive integers not exceeding 100.
2. For any \( a, b \in S \) where \( a \neq b \), there exists \( c \in S \) different from \( a \) and \( b \) such that \(\gcd(a + b, c) = 1\).
3. For any \( a, b \in S \) where \( a \neq b \), there exists \( c \in S \) different from \( a \) and \( b \) such that \(\gcd(a + b, c) > 1\).
Determine the maximum value of \( |S| \). | 50 | 81.25 |
32,614 | Given that point $P$ is an intersection point of the ellipse $\frac{x^{2}}{a_{1}^{2}} + \frac{y^{2}}{b_{1}^{2}} = 1 (a_{1} > b_{1} > 0)$ and the hyperbola $\frac{x^{2}}{a_{2}^{2}} - \frac{y^{2}}{b_{2}^{2}} = 1 (a_{2} > 0, b_{2} > 0)$, $F_{1}$, $F_{2}$ are the common foci of the ellipse and hyperbola, $e_{1}$, $e_{2}$ are the eccentricities of the ellipse and hyperbola respectively, and $\angle F_{1}PF_{2} = \frac{2\pi}{3}$, find the maximum value of $\frac{1}{e_{1}} + \frac{1}{e_{2}}$. | \frac{4 \sqrt{3}}{3} | 4.6875 |
32,615 | Complex numbers $p, q, r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48,$ find $|pq + pr + qr|.$ | 768 | 42.96875 |
32,616 | Circles $P$, $Q$, and $R$ are externally tangent to each other and internally tangent to circle $S$. Circles $Q$ and $R$ are congruent. Circle $P$ has radius 2 and passes through the center of $S$. What is the radius of circle $Q$? | \frac{16}{9} | 3.90625 |
32,617 | A Martian traffic light consists of six identical bulbs arranged in two horizontal rows (one below the other) with three bulbs in each row. A rover driver in foggy conditions can distinguish the number and relative positions of the lit bulbs on the traffic light (for example, if two bulbs are lit, whether they are in the same horizontal row or different ones, whether they are in the same vertical column, adjacent vertical columns, or in the two outermost vertical columns). However, the driver cannot distinguish unlit bulbs and the traffic light's frame. Therefore, if only one bulb is lit, it is impossible to determine which of the six bulbs it is. How many different signals from the Martian traffic light can the rover driver distinguish in the fog? If none of the bulbs are lit, the driver cannot see the traffic light. | 44 | 1.5625 |
32,618 | Given that $α∈( \dfrac {π}{2},π)$, and $\sin \dfrac {α}{2}+\cos \dfrac {α}{2}= \dfrac {2 \sqrt {3}}{3}$.
(1) Find the values of $\sin α$ and $\cos α$;
(2) If $\sin (α+β)=- \dfrac {3}{5},β∈(0, \dfrac {π}{2})$, find the value of $\sin β$. | \dfrac {6 \sqrt {2}+4}{15} | 0 |
32,619 |
West, Non-West, Russia:
1st place - Russia: 302790.13 cubic meters/person
2nd place - Non-West: 26848.55 cubic meters/person
3rd place - West: 21428 cubic meters/person | 302790.13 | 3.90625 |
32,620 | A rectangle has length $AC = 40$ and width $AE = 24$. Point $B$ is positioned one-third of the way along $AC$ from $A$ to $C$, and point $F$ is halfway along $AE$. Find the total area enclosed by rectangle $ACDE$ and the semicircle with diameter $AC$ minus the area of quadrilateral $ABDF$. | 800 + 200\pi | 3.90625 |
32,621 | The Crystal Products Factory in the East Sea had an annual output of 100,000 pieces last year, with each crystal product selling for 100 yuan and a fixed cost of 80 yuan. Starting this year, the factory invested 1 million yuan in technology costs and plans to invest an additional 1 million yuan in technology costs each year thereafter. It is expected that the output will increase by 10,000 pieces each year, and the relationship between the fixed cost $g(n)$ of each crystal product and the number of times $n$ technology costs are invested is $g(n) = \frac{80}{\sqrt{n+1}}$. If the selling price of the crystal products remains unchanged, the annual profit after the $n$th investment is $f(n)$ million yuan.
(1) Find the expression for $f(n)$;
(2) Starting from this year, in which year will the profit be the highest? What is the highest profit in million yuan? | 520 | 0 |
32,622 | Given that $0 < x < \frac{1}{2}$, find the minimum and maximum value of the function $x^{2}(1-2x)$. | \frac{1}{27} | 18.75 |
32,623 | Given that the center of the ellipse C is at the origin, its eccentricity is equal to $\frac{1}{2}$, and one of its minor axis endpoints is the focus of the parabola x²=$4\sqrt{3}y$.
1. Find the standard equation of the ellipse C.
2. The left and right foci of the ellipse are $F_1$ and $F_2$, respectively. A line $l$ passes through $F_2$ and intersects the ellipse at two distinct points A and B. Determine whether the area of the inscribed circle of $\triangle{F_1AB}$ has a maximum value. If it exists, find this maximum value and the equation of the line at this time; if not, explain the reason. | \frac{9}{16}\pi | 0 |
32,624 | In the rhombus \(ABCD\), the angle \(\angle ABC = 60^{\circ}\). A circle is tangent to the line \(AD\) at point \(A\), and the center of the circle lies inside the rhombus. Tangents to the circle, drawn from point \(C\), are perpendicular. Find the ratio of the perimeter of the rhombus to the circumference of the circle. | \frac{\sqrt{3} + \sqrt{7}}{\pi} | 0 |
32,625 | A rental company owns 100 cars. When the monthly rent for each car is 3000 yuan, all of them can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that is not rented out. The maintenance cost for each rented car is 150 yuan per month, and for each car not rented out, the maintenance cost is 50 yuan per month.
(1) How many cars can be rented out when the monthly rent for each car is 3600 yuan?
(2) At what monthly rent per car will the rental company's monthly revenue be maximized? What is the maximum monthly revenue? | 307050 | 87.5 |
32,626 | Given $f\left(x\right)=(e^{x-a}-1)\ln \left(x+2a-1\right)$, if $f\left(x\right)\geqslant 0$ always holds for $x\in \left(1-2a,+\infty \right)$, then the real number $a=$____. | \frac{2}{3} | 1.5625 |
32,627 | Find the common ratio of the infinite geometric series: $$\frac{-4}{7}+\frac{14}{3}+\frac{-98}{9} + \dots$$ | -\frac{49}{6} | 70.3125 |
32,628 | In rectangle \(ABCD\), \(A B: AD = 1: 2\). Point \(M\) is the midpoint of \(AB\), and point \(K\) lies on \(AD\), dividing it in the ratio \(3:1\) starting from point \(A\). Find the sum of \(\angle CAD\) and \(\angle AKM\). | 90 | 32.03125 |
32,629 | A parabola has focus $F$ and vertex $V$ , where $VF = 1$ 0. Let $AB$ be a chord of length $100$ that passes through $F$ . Determine the area of $\vartriangle VAB$ .
| 100\sqrt{10} | 8.59375 |
32,630 | For any positive integer $n$, let
\[f(n) =\left\{\begin{matrix}\log_{4}{n}, &\text{if }\log_{4}{n}\text{ is rational,}\\ 0, &\text{otherwise.}\end{matrix}\right.\]
What is $\sum_{n = 1}^{1023}{f(n)}$?
A) $\frac{40}{2}$
B) $\frac{45}{2}$
C) $\frac{21}{2}$
D) $\frac{36}{2}$ | \frac{45}{2} | 7.8125 |
32,631 | In $\triangle ABC$, it is known that $\overrightarrow {AB}\cdot \overrightarrow {AC}=9$ and $\overrightarrow {AB}\cdot \overrightarrow {BC}=-16$. Find:
1. The value of $AB$;
2. The value of $\frac {sin(A-B)}{sinC}$. | \frac{7}{25} | 16.40625 |
32,632 | Given rational numbers $x$ and $y$ satisfy $|x|=9$, $|y|=5$.
$(1)$ If $x \lt 0$, $y \gt 0$, find the value of $x+y$;
$(2)$ If $|x+y|=x+y$, find the value of $x-y$. | 14 | 77.34375 |
32,633 | Given the sequence $1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,2,\cdots$ where the number of 2's between each pair of 1's increases by one each time, find the sum of the first 1234 terms of the sequence. | 2419 | 86.71875 |
32,634 | Compute the number of increasing sequences of positive integers $b_1 \le b_2 \le b_3 \le \cdots \le b_{15} \le 3005$ such that $b_i-i$ is odd for $1\le i \le 15$. Express your answer as ${p \choose q}$ for some integers $p > q$ and find the remainder when $p$ is divided by 1000. | 509 | 21.09375 |
32,635 | Given that $ a,b,c,d$ are rational numbers with $ a>0$ , find the minimal value of $ a$ such that the number $ an^{3} + bn^{2} + cn + d$ is an integer for all integers $ n \ge 0$ . | \frac{1}{6} | 71.09375 |
32,636 | Xiaoming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, due to some reasons, Xiaoming first took the subway and then transferred to the bus, taking 40 minutes to reach the school. The transfer process took 6 minutes. How many minutes did Xiaoming spend on the bus that day? | 10 | 0 |
32,637 | In a plane, there is a point set \( M \) and seven distinct circles \( C_{1}, C_{2}, \ldots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \), circle \( C_{6} \) passes through exactly 6 points in \( M \), and so on, until circle \( C_{1} \) passes through exactly 1 point in \( M \). What is the minimum number of points in \( M \)? | 12 | 13.28125 |
32,638 | For a finite sequence \( B = (b_1, b_2, \dots, b_{150}) \) of numbers, the Cesaro sum of \( B \) is defined to be
\[
\frac{S_1 + \cdots + S_{150}}{150},
\]
where \( S_k = b_1 + \cdots + b_k \) and \( 1 \leq k \leq 150 \).
If the Cesaro sum of the 150-term sequence \( (b_1, \dots, b_{150}) \) is 1200, what is the Cesaro sum of the 151-term sequence \( (2, b_1, \dots, b_{150}) \)? | 1194 | 8.59375 |
32,639 | $\triangle PQR$ is similar to $\triangle STU$. The length of $\overline{PQ}$ is 10 cm, $\overline{QR}$ is 12 cm, and the length of $\overline{ST}$ is 5 cm. Determine the length of $\overline{TU}$ and the perimeter of $\triangle STU$. Express your answer as a decimal. | 17 | 0.78125 |
32,640 | In $\triangle{ABC}$ with side lengths $AB = 15$, $AC = 8$, and $BC = 17$, let $O$ and $I$ denote the circumcenter and incenter, respectively. A circle with center $M$ is tangent to the legs $AC$ and $BC$ and to the circumcircle of $\triangle{ABC}$. What is the area of $\triangle{MOI}$? | 3.4 | 0 |
32,641 | Person A and person B each have a certain number of books. If person A gives 10 books to person B, then the total number of books between the two of them will be equal. If person B gives 10 books to person A, then the number of books person A has will be twice the number of books person B has left. Find out how many books person A and person B originally had. | 50 | 16.40625 |
32,642 | Given a hyperbola with the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, its asymptotes intersect the parabola $y^2 = 4x$ at two points A and B, distinct from the origin O. Let F be the focus of the parabola $y^2 = 4x$. If $\angle AFB = \frac{2\pi}{3}$, then the eccentricity of the hyperbola is ________. | \frac{\sqrt{21}}{3} | 3.125 |
32,643 | Given two circles intersecting at points A(1, 3) and B(m, -1), where the centers of both circles lie on the line $x - y + c = 0$, find the value of $m + c$. | -1 | 3.90625 |
32,644 | Compute the sum of the squares of cosine for the angles progressing by 10 degrees starting from 0 degrees up to 180 degrees:
\[\cos^2 0^\circ + \cos^2 10^\circ + \cos^2 20^\circ + \dots + \cos^2 180^\circ.\] | \frac{19}{2} | 6.25 |
32,645 | Given that there are 21 students in Dr. Smith's physics class, the average score before including Simon's project score was 86. After including Simon's project score, the average for the class rose to 88. Calculate Simon's score on the project. | 128 | 8.59375 |
32,646 | Let \( S = \{(x, y) \mid x, y \in \mathbb{Z}, 0 \leq x, y \leq 2016\} \). Given points \( A = (x_1, y_1), B = (x_2, y_2) \) in \( S \), define
\[ d_{2017}(A, B) = (x_1 - x_2)^2 + (y_1 - y_2)^2 \pmod{2017} \]
The points \( A = (5, 5) \), \( B = (2, 6) \), and \( C = (7, 11) \) all lie in \( S \). There is also a point \( O \in S \) that satisfies
\[ d_{2017}(O, A) = d_{2017}(O, B) = d_{2017}(O, C) \]
Find \( d_{2017}(O, A) \). | 1021 | 0.78125 |
32,647 | What work is required to stretch a spring by \(0.06 \, \text{m}\), if a force of \(1 \, \text{N}\) stretches it by \(0.01 \, \text{m}\)? | 0.18 | 0.78125 |
32,648 | For any two non-zero plane vectors $\overrightarrow \alpha$ and $\overrightarrow \beta$, a new operation $\odot$ is defined as $\overrightarrow \alpha ⊙ \overrightarrow \beta = \frac{\overrightarrow \alpha • \overrightarrow \beta}{\overrightarrow \beta • \overrightarrow \beta}$. Given non-zero plane vectors $\overrightarrow a$ and $\overrightarrow b$ such that $\overrightarrow a ⊙ \overrightarrow b$ and $\overrightarrow b ⊙ \overrightarrow a$ are both in the set $\{x|x=\frac{{\sqrt{3}k}}{3}, k∈{Z}\}$, and $|\overrightarrow a| \geq |\overrightarrow b|$. Let the angle between $\overrightarrow a$ and $\overrightarrow b$ be $θ∈(\frac{π}{6},\frac{π}{4})$, then $(\overrightarrow a ⊙ \overrightarrow b) \sin θ =$ ____. | \frac{2}{3} | 6.25 |
32,649 | 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. | 0.293 | 0 |
32,650 | David has a very unique calculator that performs only two operations, the usual addition $(+)$ and another operation denoted by $*$, which satisfies:
(i) $a * a = a$,
(ii) $a * 0 = 2a$ and
(iii) $(a * b) + (c * d) = (a + c) * (b + d)$,
for any integers $a$ and $b$. What are the results of the operations $(2 * 3) + (0 * 3)$ and $1024 * 48$? | 2000 | 7.03125 |
32,651 | In a right-angled geometric setup, $\angle ABC$ and $\angle ADB$ are both right angles. The lengths of segments are given as $AC = 25$ units and $AD = 7$ units. Determine the length of segment $DB$. | 3\sqrt{14} | 3.125 |
32,652 | Eight distinct points, $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, $P_6$, $P_7$, and $P_8$, are evenly spaced around a circle. If four points are chosen at random from these eight points to form two chords, what is the probability that the chord formed by the first two points chosen intersects the chord formed by the last two points chosen? | \frac{1}{3} | 48.4375 |
32,653 | Determine the remainder when \(1 + 5 + 5^2 + \cdots + 5^{1002}\) is divided by \(500\). | 31 | 60.9375 |
32,654 | How many non-empty subsets $T$ of $\{1,2,3,\ldots,17\}$ have the following two properties?
$(1)$ No two consecutive integers belong to $T$.
$(2)$ If $T$ contains $k$ elements, then $T$ contains no number less than $k+1$. | 594 | 0 |
32,655 | What is the value of $\sqrt[4]{2^3 + 2^4 + 2^5 + 2^6}$? | 2^{3/4} \cdot 15^{1/4} | 0 |
32,656 | In the frequency distribution histogram of a sample, there are a total of $m(m\geqslant 3)$ rectangles, and the sum of the areas of the first $3$ groups of rectangles is equal to $\frac{1}{4}$ of the sum of the areas of the remaining $m-3$ rectangles. The sample size is $120$. If the areas of the first $3$ groups of rectangles, $S_1, S_2, S_3$, form an arithmetic sequence and $S_1=\frac{1}{20}$, then the frequency of the third group is ______. | 10 | 54.6875 |
32,657 | What is the greatest integer less than or equal to $\frac{4^{50}+3^{50}}{4^{47}+3^{47}}$? | 64 | 71.09375 |
32,658 | Given a geometric sequence $\{a_n\}$ with the first term $\frac{3}{2}$ and common ratio $- \frac{1}{2}$, the sum of the first $n$ terms is $S_n$. If for any $n \in N^*$, it holds that $S_n - \frac{1}{S_n} \in [s, t]$, then the minimum value of $t-s$ is \_\_\_\_\_\_. | \frac{17}{12} | 23.4375 |
32,659 | A string has 150 beads of red, blue, and green colors. It is known that among any six consecutive beads, there is at least one green bead, and among any eleven consecutive beads, there is at least one blue bead. What is the maximum number of red beads that can be on the string? | 112 | 13.28125 |
32,660 | Given $a$, $b$, $c \in \{1, 2, 3, 4, 5, 6\}$, if the lengths $a$, $b$, and $c$ can form an isosceles (including equilateral) triangle, then there are \_\_\_\_\_\_ such triangles. | 27 | 8.59375 |
32,661 | Divide 6 volunteers into 4 groups for service at four different venues of the 2012 London Olympics. Among these groups, 2 groups will have 2 people each, and the other 2 groups will have 1 person each. How many different allocation schemes are there? (Answer with a number) | 540 | 4.6875 |
32,662 | If $(x^{2}+1)(2x+1)^{9}=a\_{0}+a\_{1}(x+2)+a\_{2}(x+2)^{2}+...+a\_{11}(x+2)^{11}$, then the value of $a\_{0}+a\_{1}+...+a\_{11}$ is $\boxed{\text{answer}}$. | -2 | 74.21875 |
32,663 | Given that the mean of the data set \\(k\_{1}\\), \\(k\_{2}\\), \\(…\\), \\(k\_{8}\\) is \\(4\\), and the variance is \\(2\\), find the mean and variance of \\(3k\_{1}+2\\), \\(3k\_{2}+2\\), \\(…\\), \\(3k\_{8}+2\\). | 18 | 76.5625 |
32,664 | In the expansion of $(x^2+ \frac{4}{x^2}-4)^3(x+3)$, find the constant term. | -240 | 3.125 |
32,665 | Divers extracted a certain number of pearls, not exceeding 1000. The distribution of the pearls happens as follows: each diver in turn approaches the heap of pearls and takes either exactly half or exactly one-third of the remaining pearls. After all divers have taken their share, the remainder of the pearls is offered to the sea god. What is the maximum number of divers that could have participated in the pearl extraction? | 12 | 1.5625 |
32,666 | Form a six-digit number using the digits 1, 2, 3, 4, 5, 6 without repetition, where both 5 and 6 are on the same side of 3. How many such six-digit numbers are there? | 480 | 18.75 |
32,667 | Given the function $f(x) = \frac{\ln{x}}{x + a}$ where $a \in \mathbb{R}$:
(1) If the tangent to the curve $y = f(x)$ at the point $(1, f(1))$ is perpendicular to the line $x + y + 1 = 0$, find the value of $a$.
(2) Discuss the number of real roots of the equation $f(x) = 1$. | a = 0 | 45.3125 |
32,668 | Let \( a, b, c, d \) be 4 distinct nonzero integers such that \( a + b + c + d = 0 \) and the number \( M = (bc - ad)(ac - bd)(ab - cd) \) lies strictly between 96100 and 98000. Determine the value of \( M \). | 97344 | 71.875 |
32,669 | The diagram shows a large square divided into squares of three different sizes. What percentage of the large square is shaded?
A) 61%
B) 59%
C) 57%
D) 55%
E) 53% | 59\% | 12.5 |
32,670 | How many positive integers less than $200$ are multiples of $5$, but not multiples of either $10$ or $6$? | 20 | 3.125 |
32,671 | A student has five different physics questions numbered 1, 2, 3, 4, and 5, and four different chemistry questions numbered 6, 7, 8, and 9. The student randomly selects two questions, each with an equal probability of being chosen. Let the event `(x, y)` represent "the two questions with numbers x and y are chosen, where x < y."
(1) How many basic events are there? List them out.
(2) What is the probability that the sum of the numbers of the two chosen questions is less than 17 but not less than 11? | \frac{5}{12} | 20.3125 |
32,672 | In how many ways can five girls and five boys be seated around a circular table such that no two people of the same gender sit next to each other? | 28800 | 6.25 |
32,673 | If the integer solutions to the system of inequalities
\[
\begin{cases}
9x - a \geq 0, \\
8x - b < 0
\end{cases}
\]
are only 1, 2, and 3, how many ordered pairs \((a, b)\) of integers satisfy this system? | 72 | 30.46875 |
32,674 | An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 3 \angle D$ and $\angle BAC = m \pi$ in radians, determine the value of $m$. | \frac{5}{11} | 7.8125 |
32,675 | How many natural numbers from 1 to 700, inclusive, contain the digit 6 at least once? | 133 | 0 |
32,676 | The cost of 1 piece of gum is 2 cents. What is the cost of 500 pieces of gum, in cents and in dollars? | 10.00 | 0 |
32,677 | (1) Simplify: $\dfrac{\tan(3\pi-\alpha)\cos(2\pi-\alpha)\sin(-\alpha+\dfrac{3\pi}{2})}{\cos(-\alpha-\pi)\sin(-\pi+\alpha)\cos(\alpha+\dfrac{5\pi}{2})}$;
(2) Given $\tan\alpha=\dfrac{1}{4}$, find the value of $\dfrac{1}{2\cos^{2}\alpha-3\sin\alpha\cos\alpha}$. | \dfrac{17}{20} | 60.9375 |
32,678 | Given that $40\%$ of students initially answered "Yes", $40\%$ answered "No", and $20\%$ were "Undecided", and $60\%$ answered "Yes" after a semester, $30\%$ answered "No", and $10\%$ remained "Undecided", determine the difference between the maximum and minimum possible values of $y\%$ of students who changed their answer. | 40\% | 7.03125 |
32,679 | Person A and person B independently attempt to decrypt a password. Their probabilities of successfully decrypting the password are $\dfrac{1}{3}$ and $\dfrac{1}{4}$, respectively. Calculate:
$(1)$ The probability that exactly one of them decrypts the password.
$(2)$ If the probability of decrypting the password needs to be $\dfrac{99}{100}$, what is the minimum number of people like B required? | 17 | 89.84375 |
32,680 | In trapezoid $EFGH$, the sides $EF$ and $GH$ are equal. The length of $FG$ is 10 and the length of $EH$ is 20. Each of the non-parallel sides makes right triangles with legs 5 and an unknown base segment. What is the perimeter of $EFGH$?
[asy]
pen p = linetype("4 4");
draw((0,0)--(4,5)--(12,5)--(16,0)--cycle);
draw((4,0)--(4,5), p);
draw((3.5,0)--(3.5, 1)--(4.0,1));
label(scale(0.75)*"E", (0,0), W);
label(scale(0.75)*"F", (4,5), NW);
label(scale(0.75)*"G", (12, 5), NE);
label(scale(0.75)*"H", (16, 0), E);
label(scale(0.75)*"10", (8,5), N);
label(scale(0.75)*"20", (8,0), S);
label(scale(0.75)*"5", (4, 2.5), E);
[/asy] | 30 + 10\sqrt{2} | 22.65625 |
32,681 | $13$ fractions are corrected by using each of the numbers $1,2,...,26$ once.**Example:** $\frac{12}{5},\frac{18}{26}.... $ What is the maximum number of fractions which are integers? | 12 | 2.34375 |
32,682 | Let the following system of equations hold for positive numbers \(x, y, z\):
\[ \left\{\begin{array}{l}
x^{2}+x y+y^{2}=48 \\
y^{2}+y z+z^{2}=25 \\
z^{2}+x z+x^{2}=73
\end{array}\right. \]
Find the value of the expression \(x y + y z + x z\). | 40 | 88.28125 |
32,683 | Triangle $PQR$ has positive integer side lengths with $PQ=PR$. Let $J$ be the intersection of the bisectors of $\angle Q$ and $\angle R$. Suppose $QJ=10$. Find the smallest possible perimeter of $\triangle PQR$. | 198 | 0 |
32,684 | Given the function y=cos(2x- π/3), derive the horizontal shift required to transform the graph of the function y=sin 2x. | \frac{\pi}{12} | 26.5625 |
32,685 | Using the numbers from 1 to 22 exactly once each, Antoine writes 11 fractions. For example, he could write the fractions \(\frac{10}{2}, \frac{4}{3}, \frac{15}{5}, \frac{7}{6}, \frac{8}{9}, \frac{11}{19}, \frac{12}{14}, \frac{13}{17}, \frac{22}{21}, \frac{18}{16}, \frac{20}{1}\).
Antoine wants to have as many fractions with integer values as possible among the written fractions. In the previous example, he wrote three fractions with integer values: \(\frac{10}{2}=5\), \(\frac{15}{5}=3\), and \(\frac{20}{1}=20\). What is the maximum number of fractions that can have integer values? | 10 | 45.3125 |
32,686 | Factor the expression $81x^4 - 256y^4$ and find the sum of all integers in its complete factorization $(ax^2 + bxy + cy^2)(dx^2 + exy + fy^2)$. | 31 | 3.125 |
32,687 | What multiple of 15 is closest to 3500? | 3510 | 11.71875 |
32,688 | If the sum of the digits of a positive integer $a$ equals 6, then $a$ is called a "good number" (for example, 6, 24, 2013 are all "good numbers"). List all "good numbers" in ascending order as $a_1$, $a_2$, $a_3$, …, if $a_n = 2013$, then find the value of $n$. | 51 | 1.5625 |
32,689 | Given circle M: $(x+1)^2+y^2=1$, and circle N: $(x-1)^2+y^2=9$, a moving circle P is externally tangent to circle M and internally tangent to circle N. The trajectory of the center of circle P is curve C.
(1) Find the equation of C:
(2) Let $l$ be a line that is tangent to both circle P and circle M, and $l$ intersects curve C at points A and B. When the radius of circle P is the longest, find $|AB|$. | \frac{18}{7} | 0 |
32,690 | A rectangular park is to be fenced on three sides using a 150-meter concrete wall as the fourth side. Fence posts are to be placed every 15 meters along the fence, including at the points where the fence meets the concrete wall. Calculate the minimal number of posts required to fence an area of 45 m by 90 m. | 13 | 39.0625 |
32,691 | If $\cos(α + \frac{π}{3}) = -\frac{\sqrt{3}}{3}$, find the value of $\sin α$. | \frac{\sqrt{6} + 3}{6} | 0 |
32,692 | Point $D$ lies on side $AC$ of isosceles triangle $ABC$ (where $AB = AC$) such that angle $BAC$ is $100^\circ$ and angle $DBC$ is $30^\circ$. Determine the ratio of the area of triangle $ADB$ to the area of triangle $CDB$. | \frac{1}{3} | 4.6875 |
32,693 | Given the function $f(x) = \cos(\omega x + \phi)$ where $\omega > 0$ and $0 < \phi < \pi$. The graph of the function passes through the point $M(\frac{\pi}{6}, -\frac{1}{2})$ and the distance between two adjacent intersections with the x-axis is $\pi$.
(I) Find the analytical expression of $f(x)$;
(II) If $f(\theta + \frac{\pi}{3}) = -\frac{3}{5}$, find the value of $\sin{\theta}$. | \frac{3 + 4\sqrt{3}}{10} | 3.90625 |
32,694 | Consider an equilateral triangle $ABC$ with side length $3$. Right triangle $CBD$ is constructed outwardly on side $BC$ of triangle $ABC$ such that $CB = BD$ and $BCD$ is a right angle at $B$. Find $\sin^2\left(\angle CAD\right)$.
A) $\frac{3}{4}$
B) $\frac{1}{4}$
C) $\frac{1}{2}$
D) $\frac{\sqrt{2}}{2}$
E) $\frac{\sqrt{3}}{2}$ | \frac{1}{2} | 20.3125 |
32,695 | A cuckoo clock produces a number of "cuckoo" sounds equal to the hour it indicates (for example, at 19:00, it sounds "cuckoo" 7 times). One morning, Maxim approaches the clock at 9:05 and starts turning the minute hand forward until the clock shows 7 hours later. How many "cuckoo" sounds are made during this time? | 43 | 0.78125 |
32,696 | Evaluate the sum:
\[
\sum_{n=1}^\infty \frac{n^3 + n^2 - n}{(n+3)!}.
\] | \frac{1}{6} | 3.90625 |
32,697 | Let the area of the regular octagon $A B C D E F G H$ be $n$, and the area of the quadrilateral $A C E G$ be $m$. Calculate the value of $\frac{m}{n}$. | \frac{\sqrt{2}}{2} | 24.21875 |
32,698 | **
Evaluate
\[
\frac{3}{\log_8{3000^4}} + \frac{4}{\log_9{3000^4}},
\]
giving your answer as a fraction in lowest terms.
** | \frac{1}{4} | 7.03125 |
32,699 | Given that the four vertices of the tetrahedron $P-ABC$ are all on the surface of a sphere with radius $3$, and $PA$, $PB$, $PC$ are mutually perpendicular, find the maximum value of the lateral surface area of the tetrahedron $P-ABC$. | 18 | 57.03125 |
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