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22,600 | The diagram shows an arc \( PQ \) of a circle with center \( O \) and radius 8. Angle \( QOP \) is a right angle, the point \( M \) is the midpoint of \( OP \), and \( N \) lies on the arc \( PQ \) so that \( MN \) is perpendicular to \( OP \). Which of the following is closest to the length of the perimeter of triangle \( PNM \)? | 19 | 69.53125 |
22,601 | A sports league has 20 teams split into four divisions of 5 teams each. How many games are there in a complete season if each team plays every other team in its own division three times and every team in the other divisions once? | 270 | 79.6875 |
22,602 | Determine the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_8 \le 1023$ such that $a_i-i$ is even for $1\le i \le 8$. The answer can be expressed as $\binom{m}{n}$ for some $m > n$. Compute the remainder when $m$ is divided by 1000. | 515 | 38.28125 |
22,603 | Point $C(0,p)$ lies on the $y$-axis between $Q(0,15)$ and $O(0,0)$. Point $B$ has coordinates $(15,0)$. Determine an expression for the area of $\triangle COB$ in terms of $p$, and compute the length of segment $QB$. Your answer should be simplified as much as possible. | 15\sqrt{2} | 92.1875 |
22,604 | In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=t}\\{y=-1+\sqrt{3}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\rho =2\sin \theta +2\cos \theta$.
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;
$(2)$ Let point $P(0,-1)$. If the line $l$ intersects the curve $C$ at points $A$ and $B$, find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$. | \frac{2\sqrt{3}+1}{3} | 17.1875 |
22,605 | Two sides of a right triangle have the lengths 8 and 15. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth. | 215.7 | 9.375 |
22,606 | How many $6$ -tuples $(a, b, c, d, e, f)$ of natural numbers are there for which $a>b>c>d>e>f$ and $a+f=b+e=c+d=30$ are simultaneously true? | 364 | 89.0625 |
22,607 | Given a geometric sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. Given that $a_1 + a_2 + a_3 = 3$ and $a_4 + a_5 + a_6 = 6$, calculate the value of $S_{12}$. | 45 | 46.09375 |
22,608 | Given the function $f(x)=e^{x}(ax+b)-x^{2}-4x$, the equation of the tangent line to the curve $y=f(x)$ at the point $(0, f(0))$ is $y=4x+4$.
(1) Find the values of $a$ and $b$;
(2) Determine the intervals of monotonicity for $f(x)$ and find the maximum value of $f(x)$. | 4(1-e^{-2}) | 0 |
22,609 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $\angle A=45^{\circ}$, $a=6$.
(1) If $\angle C=105^{\circ}$, find $b$;
(2) Find the maximum area of $\triangle ABC$. | 9(1+\sqrt{2}) | 2.34375 |
22,610 | Let $\theta=\frac{2\pi}{2015}$ , and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form $\frac{b}{2^a}$ , where $a$ is a non-negative integer and $b$ is an odd integer (not necessarily positive). Find $a+b$ .
*2017 CCA Math Bonanza Tiebreaker Round #3* | 1441 | 65.625 |
22,611 | A circle has a radius of 3 units. A line segment of length 3 units is tangent to the circle at its midpoint. Determine the area of the region consisting of all such line segments.
A) $1.5\pi$
B) $2.25\pi$
C) $3\pi$
D) $4.5\pi$ | 2.25\pi | 12.5 |
22,612 | Given that 3 females and 2 males participate in a performance sequence, and the 2 males cannot appear consecutively, and female A cannot be the first to appear, determine the total number of different performance sequences. | 60 | 35.15625 |
22,613 | Given an arithmetic sequence $\{a_n\}$ with a positive common difference $d > 0$, and $a_2$, $a_5 - 1$, $a_{10}$ form a geometric sequence. If $a_1 = 5$, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, determine the minimum value of $$\frac{2S_{n} + n + 32}{a_{n} + 1}$$ | \frac{20}{3} | 4.6875 |
22,614 | Which of the following five values of \( n \) is a counterexample to the statement: For a positive integer \( n \), at least one of \( 6n-1 \) and \( 6n+1 \) is prime? | 20 | 2.34375 |
22,615 | In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=t}\\{y=-1+\sqrt{3}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\rho =2\sin \theta +2\cos \theta$.
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;
$(2)$ Let point $P(0,-1)$. If the line $l$ intersects the curve $C$ at points $A$ and $B$, find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$. | \frac{2\sqrt{3} + 1}{3} | 24.21875 |
22,616 | Given triangle ABC, the sides opposite to angles A, B, and C are denoted as a, b, and c, respectively, and a = 6. Find the maximum value of the area of triangle ABC given that $\sqrt{7}bcosA = 3asinB$. | 9\sqrt{7} | 33.59375 |
22,617 | The Stromquist Comet is visible every 61 years. If the comet is visible in 2017, what is the next leap year when the comet will be visible? | 2444 | 1.5625 |
22,618 | Given \( a_{n} = 4^{2n - 1} + 3^{n - 2} \) (for \( n = 1, 2, 3, \cdots \)), where \( p \) is the smallest prime number dividing infinitely many terms of the sequence \( a_{1}, a_{2}, a_{3}, \cdots \), and \( q \) is the smallest prime number dividing every term of the sequence, find the value of \( p \cdot q \). | 5 \times 13 | 0 |
22,619 | In a school, 40 students are enrolled in both the literature and science classes. Ten students received an A in literature and 18 received an A in science, including six who received an A in both subjects. Determine how many students did not receive an A in either subject. | 18 | 93.75 |
22,620 | A 10-digit arrangement $ 0,1,2,3,4,5,6,7,8,9$ is called *beautiful* if (i) when read left to right, $ 0,1,2,3,4$ form an increasing sequence, and $ 5,6,7,8,9$ form a decreasing sequence, and (ii) $ 0$ is not the leftmost digit. For example, $ 9807123654$ is a beautiful arrangement. Determine the number of beautiful arrangements. | 126 | 92.96875 |
22,621 | The inclination angle of the line $x-y+1=0$ can be calculated. | \frac{\pi}{4} | 77.34375 |
22,622 | Let the hyperbola $C:\frac{x^2}{a^2}-y^2=1\;(a>0)$ intersect the line $l:x+y=1$ at two distinct points $A$ and $B$.
$(1)$ Find the range of real numbers for $a$.
$(2)$ If the intersection point of the line $l$ and the $y$-axis is $P$, and $\overrightarrow{PA}=\frac{5}{12}\overrightarrow{PB}$, find the value of the real number $a$. | a = \frac{17}{13} | 29.6875 |
22,623 | Given that $b$ is an odd multiple of 9, find the greatest common divisor of $8b^2 + 81b + 289$ and $4b + 17$. | 17 | 31.25 |
22,624 | Given two non-zero planar vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy the condition: for any $λ∈R$, $| \overrightarrow{a}-λ \overrightarrow{b}|≥slant | \overrightarrow{a}- \frac {1}{2} \overrightarrow{b}|$, then:
$(①)$ If $| \overrightarrow{b}|=4$, then $\overrightarrow{a}· \overrightarrow{b}=$ _______ ;
$(②)$ If the angle between $\overrightarrow{a}, \overrightarrow{b}$ is $\frac {π}{3}$, then the minimum value of $\frac {|2 \overrightarrow{a}-t· \overrightarrow{b}|}{| \overrightarrow{b}|}$ is _______ . | \sqrt {3} | 0 |
22,625 | Let $a$, $b$, $c$ be positive numbers, and $a+b+9c^2=1$. Find the maximum value of $\sqrt{a}+ \sqrt{b}+ \sqrt{3}c$. | \frac{\sqrt{21}}{3} | 10.15625 |
22,626 | Six distinct integers are picked at random from $\{1,2,3,\ldots,10\}$ . What is the probability that, among those selected, the second smallest is $3$ ? | $\frac{1}{3}$ | 0 |
22,627 | Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles? Express your answer in fully expanded form in terms of $\pi$. | \frac{9\pi}{2} - 9 | 82.03125 |
22,628 | Given that $\sin(α + \frac{π}{6}) - \cos α = \frac{1}{3}$, find the value of $2 \sin α \cos(α + \frac{π}{6})$. | \frac{5}{18} | 21.875 |
22,629 | There are 5 different books to be distributed among three people, with each person receiving at least 1 book and at most 2 books. Calculate the total number of different distribution methods. | 90 | 7.8125 |
22,630 | Given the function $f(x)=\sin (\omega x+\varphi)$ with $\omega > 0$ and $|\varphi| < \frac {\pi}{2}$, the function has a minimum period of $4\pi$ and, after being shifted to the right by $\frac {2\pi}{3}$ units, becomes symmetric about the $y$-axis. Determine the value of $\varphi$. | -\frac{\pi}{6} | 45.3125 |
22,631 | Consider a geometric sequence with terms $a$, $a(a-1)$, $a(a-1)^2$, ..., and let the sum of the first $n$ terms be denoted as $S_n$.
(1) Determine the range of the real number $a$ and the expression for $S_n$;
(2) Does there exist a real number $a$ such that $S_1$, $S_3$, $S_2$ form an arithmetic sequence? If it exists, find the value of $a$; if not, explain why. | \frac{1}{2} | 18.75 |
22,632 | In a regular hexagon $A B C D E F$, the diagonals $A C$ and $C E$ are divided by points $M$ and $N$ respectively in the following ratios: $\frac{A M}{A C} = \frac{C N}{C E} = r$. If points $B$, $M$, and $N$ are collinear, determine the ratio $r$. | \frac{\sqrt{3}}{3} | 32.03125 |
22,633 | Let $p$ and $q$ be the roots of the equation $x^2 - 7x + 12 = 0$. Compute the value of:
\[ p^3 + p^4 q^2 + p^2 q^4 + q^3. \] | 3691 | 35.9375 |
22,634 | If the function $f(x)$ is monotonic in its domain $(-\infty, +\infty)$, and for any real number $x$, it satisfies $f(f(x)+e^{x})=1-e$, where $e$ is the base of the natural logarithm, determine the value of $f(\ln 2)$. | -1 | 51.5625 |
22,635 | In an isosceles triangle $ABC$ with $AB = AC = 6$ units and $BC = 5$ units, a point $P$ is randomly selected inside the triangle $ABC$. What is the probability that $P$ is closer to vertex $C$ than to either vertex $A$ or vertex $B$? | \frac{1}{3} | 7.8125 |
22,636 | Given that the function $f(x)$ is defined on $\mathbb{R}$ and has a period of $4$ as an even function. When $x \in [2,4]$, $f(x) =|\log_4 \left(x-\frac{3}{2}\right)|$. Determine the value of $f\left(\frac{1}{2}\right)$. | \frac{1}{2} | 42.96875 |
22,637 | Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $s$ be a real number. Assume two of the roots of $f(x)$ are $s + 2$ and $s + 5,$ and two of the roots of $g(x)$ are $s + 4$ and $s + 8.$ Given that:
\[ f(x) - g(x) = 2s \] for all real numbers $x.$ Find $s.$ | 3.6 | 0.78125 |
22,638 | The letters \( A, J, H, S, M, E \) and the numbers \( 1, 9, 8, 9 \) are "rotated" as follows:
\begin{tabular}{rrr}
AJHSME & 1989 & \\
1. JHSMEA & 9891 & (1st rotation) \\
2. HSMEAJ & 8919 & (2nd rotation) \\
3. SMEAJH & 9198 & (3rd rotation) \\
..... & &
\end{tabular}
To make AJHSME1989 reappear, the minimum number of rotations needed is: | 12 | 42.96875 |
22,639 | In the circle \(x^{2} + y^{2} \leq R^{2}\), the two-dimensional probability density is \(f(x, y) = C\left(R - \sqrt{x^{2} + y^{2}}\right)\); outside the circle \(f(x, y) = 0\). Find: a) the constant \(C\); b) the probability that a random point \((X, Y)\) falls within a circle of radius \(r = 1\) centered at the origin, if \(R = 2\). | \frac{1}{2} | 76.5625 |
22,640 | Given that $\cos \alpha$ is a root of the equation $3x^2 - x - 2 = 0$ and $\alpha$ is an angle in the third quadrant, find the value of $\frac{\sin (-\alpha + \frac{3\pi}{2}) \cos (\frac{3\pi}{2} + \alpha) \tan^2 (\pi - \alpha)}{\cos (\frac{\pi}{2} + \alpha) \sin (\frac{\pi}{2} - \alpha)}$. | \frac{5}{4} | 68.75 |
22,641 | Let \(C\) be the circle with the equation \(x^2 - 4y - 18 = -y^2 + 6x + 26\). Find the center \((a, b)\) and radius \(r\) of the circle, and compute \(a + b + r\). | 5 + \sqrt{57} | 79.6875 |
22,642 | Given two circles $A:(x+4)^2+y^2=25$ and $B:(x-4)^2+y^2=1$, a moving circle $M$ is externally tangent to both fixed circles. Let the locus of the center of moving circle $M$ be curve $C$.
(I) Find the equation of curve $C$;
(II) If line $l$ intersects curve $C$ at points $P$ and $Q$, and $OP \perp OQ$. Is $\frac{1}{|OP|^2}+\frac{1}{|OQ|^2}$ a constant value? If it is, find the value; if not, explain the reason. | \frac{1}{6} | 10.15625 |
22,643 | (1) Given the hyperbola $C$: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, its right vertex is $A$, and a circle $A$ with center $A$ and radius $b$ intersects one of the asymptotes of the hyperbola $C$ at points $M$ and $N$. If $\angle MAN = 60^{\circ}$, then the eccentricity of $C$ is ______.
(2) The equation of one of the asymptotes of the hyperbola $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{9} = 1$ $(a > 0)$ is $y = \dfrac{3}{5}x$, then $a=$ ______.
(3) A tangent line to the circle $x^{2} + y^{2} = \dfrac{1}{4}a^{2}$ passing through the left focus $F$ of the hyperbola $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$ intersects the right branch of the hyperbola at point $P$. If $\overrightarrow{OE} = \dfrac{1}{2}(\overrightarrow{OF} + \overrightarrow{OP})$, then the eccentricity of the hyperbola is ______.
(4) A line passing through the focus $F$ of the parabola $y^{2} = 2px$ $(p > 0)$ with an inclination angle of $\dfrac{\pi}{4}$ intersects the parabola at points $A$ and $B$. If the perpendicular bisector of chord $AB$ passes through point $(0,2)$, then $p=$ ______. | \dfrac{4}{5} | 9.375 |
22,644 | Solve for $x$: $\sqrt[4]{40x + \sqrt[4]{40x + 24}} = 24.$ | 8293.8 | 28.125 |
22,645 | How many distinct arrangements of the letters in the word "balloon" are there, considering the repeated 'l' and 'o'? | 1260 | 92.1875 |
22,646 | Determine with proof the number of positive integers $n$ such that a convex regular polygon with $n$ sides has interior angles whose measures, in degrees, are integers. | 22 | 90.625 |
22,647 | A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 88 and the floor forms a perfect square with an even side length, find the total number of tiles that cover the floor. | 1936 | 68.75 |
22,648 | Given non-zero vectors \\(a\\) and \\(b\\) satisfying \\(|b|=2|a|\\) and \\(a \perp (\sqrt{3}a+b)\\), find the angle between \\(a\\) and \\(b\\). | \dfrac{5\pi}{6} | 98.4375 |
22,649 | Given the sequence $\{a_n\}$ satisfies $a_1=1$, $a_2=2$, $a_{n+2}=(1+\cos ^{2} \frac {nπ}{2})a_{n}+\sin ^{2} \frac {nπ}{2}$, find the sum of the first 12 terms of the sequence. | 147 | 0 |
22,650 | Given quadrilateral $ABCD$ where $AC \perp BD$ and $AC=2$, $BD=3$, find the minimum value of $\overrightarrow{AB} \cdot \overrightarrow{CD}$. | - \dfrac{13}{4} | 48.4375 |
22,651 | A function, defined on the set of positive integers, is such that \( f(x y) = f(x) + f(y) \) for all \( x \) and \( y \). It is known that \( f(10) = 14 \) and \( f(40) = 20 \).
What is the value of \( f(500) \)? | 39 | 96.09375 |
22,652 | Given the lateral area of a cylinder with a square cross-section is $4\pi$, calculate the volume of the cylinder. | 2\pi | 92.96875 |
22,653 | Given the function $f(x)=2\sin x\cos x+2\sqrt{3}\cos^{2}x-\sqrt{3}$, $x\in R$.
(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;
(2) In acute triangle $ABC$, if $f(A)=1$, $\overrightarrow{AB}\cdot\overrightarrow{AC}=\sqrt{2}$, find the area of $\triangle ABC$. | \frac{\sqrt{2}}{2} | 94.53125 |
22,654 | Gavrila is in an elevator cabin which is moving downward with a deceleration of 5 m/s². Find the force with which Gavrila presses on the floor. Gavrila's mass is 70 kg, and the acceleration due to gravity is 10 m/s². Give the answer in newtons, rounding to the nearest whole number if necessary. | 350 | 50.78125 |
22,655 | The famous Italian mathematician Fibonacci, while studying the problem of rabbit population growth, discovered a sequence of numbers: 1, 1, 2, 3, 5, 8, 13, ..., where starting from the third number, each number is equal to the sum of the two numbers preceding it. This sequence of numbers $\{a_n\}$ is known as the "Fibonacci sequence". Determine which term in the Fibonacci sequence is represented by $$\frac { a_{ 1 }^{ 2 }+ a_{ 2 }^{ 2 }+ a_{ 3 }^{ 2 }+…+ a_{ 2015 }^{ 2 }}{a_{2015}}$$. | 2016 | 63.28125 |
22,656 | Given the plane rectangular coordinate system $(xOy)$, with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system. The polar coordinate equation of the curve $C_{1}$ is $\rho = 4\cos \theta$, and the line $l$: $\begin{cases} x = 1 - \frac{2\sqrt{5}}{5}t \\ y = 1 + \frac{\sqrt{5}}{5}t \end{cases}$ ($t$ is a parameter)
(I) Find the rectangular coordinate equation of the curve $C_{1}$ and the general equation of the line $l$;
(II) If the parametric equation of the curve $C_{2}$ is $\begin{cases} x = 2\cos \alpha \\ y = \sin \alpha \end{cases}$ ($\alpha$ is a parameter), $P$ is a point on the curve $C_{1}$ with a polar angle of $\frac{\pi}{4}$, and $Q$ is a moving point on the curve $C_{2}$, find the maximum value of the distance from the midpoint $M$ of $PQ$ to the line $l$. | \frac{\sqrt{10}}{5} | 92.96875 |
22,657 | Let $h(x) = x - 3$ and $k(x) = x / 4$. Also denote the inverses to these functions as $h^{-1}$ and $k^{-1}$. Compute \[h(k^{-1}(h^{-1}(h^{-1}(k(h(27)))))).\] | 45 | 67.96875 |
22,658 | Given $F_{1}$ and $F_{2}$ are the foci of a pair of related curves, and $P$ is the intersection point of the ellipse and the hyperbola in the first quadrant, when $\angle F_{1}PF_{2}=60^{\circ}$, determine the eccentricity of the ellipse in this pair of related curves. | \dfrac{\sqrt{3}}{3} | 1.5625 |
22,659 | Given a sequence \( a_1, a_2, a_3, \ldots, a_n \) of non-zero integers such that the sum of any 7 consecutive terms is positive and the sum of any 11 consecutive terms is negative, what is the largest possible value for \( n \)? | 16 | 57.03125 |
22,660 | Calculate the value of $\frac{2468_{10}}{111_{3}} - 3471_{9} + 1234_{7}$. Express your answer in base 10. | -1919 | 6.25 |
22,661 | Given that the two distinct square roots of a positive number $a$ are $2x-2$ and $6-3x$, find the cube root of $a$. | \sqrt[3]{36} | 41.40625 |
22,662 | A triangle with side lengths 8, 13, and 17 has an incircle. The side length of 8 is divided by the point of tangency into segments \( r \) and \( s \), with \( r < s \). Find the ratio \( r : s \). | 1: 3 | 21.09375 |
22,663 | Let $O$ be the origin, the parabola $C_{1}$: $y^{2}=2px\left(p \gt 0\right)$ and the hyperbola $C_{2}$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\left(a \gt 0,b \gt 0\right)$ have a common focus $F$. The line passing through $F$ and perpendicular to the $x$-axis intersects $C_{1}$ at points $A$ and $B$, and intersects $C_{2}$ in the first quadrant at point $M$. If $\overrightarrow{OM}=m\overrightarrow{OA}+n\overrightarrow{OB}\left(m,n\in R\right)$ and $mn=\frac{1}{8}$, find the eccentricity of the hyperbola $C_{2}$. | \frac{\sqrt{6} + \sqrt{2}}{2} | 3.90625 |
22,664 | What is the sum of the odd integers from 21 through 65, inclusive? | 989 | 49.21875 |
22,665 | Suppose we have 12 dogs and need to divide them into three groups: one with 4 dogs, one with 5 dogs, and one with 3 dogs. Determine how many ways the groups can be formed if Rocky, a notably aggressive dog, must be in the 4-dog group, and Bella must be in the 5-dog group. | 4200 | 86.71875 |
22,666 | If 2023 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 48 | 29.6875 |
22,667 | Given $\sin \left(\frac{3\pi }{2}+\theta \right)=\frac{1}{4}$,
find the value of $\frac{\cos (\pi +\theta )}{\cos \theta [\cos (\pi +\theta )-1]}+\frac{\cos (\theta -2\pi )}{\cos (\theta +2\pi )\cos (\theta +\pi )+\cos (-\theta )}$. | \frac{32}{15} | 78.125 |
22,668 | A piece of wood of uniform density in the shape of a right triangle with base length $3$ inches and hypotenuse $5$ inches weighs $12$ ounces. Another piece of the same type of wood, with the same thickness, also in the shape of a right triangle, has a base length of $5$ inches and a hypotenuse of $7$ inches. Calculate the approximate weight of the second piece. | 24.5 | 50.78125 |
22,669 | In the rectangular coordinate system $xOy$, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of circle $C$ is $\rho^2 - 2m\rho\cos\theta + 4\rho\sin\theta = 1 - 2m$.
(1) Find the rectangular coordinate equation of $C$ and its radius.
(2) When the radius of $C$ is the smallest, the curve $y = \sqrt{3}|x - 1| - 2$ intersects $C$ at points $A$ and $B$, and point $M(1, -4)$. Find the area of $\triangle MAB$. | 2 + \sqrt{3} | 3.125 |
22,670 | Robert read a book for 10 days. He read an average of 25 pages per day for the first 5 days and an average of 40 pages per day for the next 4 days, and read 30 more pages on the last day. Calculate the total number of pages in the book. | 315 | 95.3125 |
22,671 | A two-digit number, when three times the sum of its units and tens digits is subtracted by -2, still results in the original number. Express this two-digit number algebraically. | 28 | 45.3125 |
22,672 | Let $a$ and $b$ be integers such that $ab = 72.$ Find the minimum value of $a + b.$ | -73 | 98.4375 |
22,673 | Let
\[f(x) = \left\{
\begin{array}{cl}
x + 5 & \text{if $x < 15$}, \\
3x - 1 & \text{if $x \ge 15$}.
\end{array}
\right.\]Find $f^{-1}(10) + f^{-1}(50).$ | 22 | 97.65625 |
22,674 | Given that the equation of line $l_{1}$ is $y=x$, and the equation of line $l_{2}$ is $y=kx-k+1$, find the value of $k$ for which the area of triangle $OAB$ is $2$. | \frac{1}{5} | 7.03125 |
22,675 | The digits 1, 3, 4, and 5 are arranged randomly to form a four-digit number. What is the probability that the number is odd? | \frac{3}{4} | 92.1875 |
22,676 | Given vectors $\overrightarrow{a}=(\cos α,\sin α)$ and $\overrightarrow{b}=(-2,2)$.
(1) If $\overrightarrow{a}\cdot \overrightarrow{b}= \frac {14}{5}$, find the value of $(\sin α+\cos α)^{2}$;
(2) If $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $\sin (π-α)\cdot\sin ( \frac {π}{2}+α)$. | -\frac{1}{2} | 89.84375 |
22,677 | Given that $\cos \theta = \frac{12}{13}, \theta \in \left( \pi, 2\pi \right)$, find the values of $\sin \left( \theta - \frac{\pi}{6} \right)$ and $\tan \left( \theta + \frac{\pi}{4} \right)$. | \frac{7}{17} | 90.625 |
22,678 | Find the smallest natural number \( n \) that satisfies the following conditions:
1. The units digit of \( n \) is 6.
2. If the units digit 6 is moved to the front of the number, the new number is 4 times \( n \). | 153846 | 98.4375 |
22,679 | Given that $\tan\left(\alpha + \frac{\pi}{3}\right)=2$, find the value of $\frac{\sin\left(\alpha + \frac{4\pi}{3}\right) + \cos\left(\frac{2\pi}{3} - \alpha\right)}{\cos\left(\frac{\pi}{6} - \alpha\right) - \sin\left(\alpha + \frac{5\pi}{6}\right)}$. | -3 | 57.03125 |
22,680 | Given that $y < 1$ and \[(\log_{10} y)^2 - \log_{10}(y^3) = 75,\] compute the value of \[(\log_{10}y)^3 - \log_{10}(y^4).\] | \frac{2808 - 336\sqrt{309}}{8} - 6 + 2\sqrt{309} | 0 |
22,681 | Given that point \( P \) lies on the hyperbola \(\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1\), and the distance from \( P \) to the right directrix of this hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of this hyperbola, find the x-coordinate of \( P \). | -\frac{64}{5} | 3.125 |
22,682 | Given that $\alpha$ is an angle in the third quadrant, the function $f(\alpha)$ is defined as:
$$f(\alpha) = \frac {\sin(\alpha - \frac {\pi}{2}) \cdot \cos( \frac {3\pi}{2} + \alpha) \cdot \tan(\pi - \alpha)}{\tan(-\alpha - \pi) \cdot \sin(-\alpha - \pi)}.$$
1. Simplify $f(\alpha)$.
2. If $\cos(\alpha - \frac {3\pi}{2}) = \frac {1}{5}$, find $f(\alpha + \frac {\pi}{6})$. | \frac{6\sqrt{2} - 1}{10} | 45.3125 |
22,683 | In a regular triangle $ABC$ with side length $3$, $D$ is a point on side $BC$ such that $\overrightarrow{CD}=2\overrightarrow{DB}$. Calculate the dot product $\overrightarrow{AB} \cdot \overrightarrow{AD}$. | \frac{15}{2} | 72.65625 |
22,684 | Fill the numbers 1, 2, 3 into a 3×3 grid such that each row and each column contains no repeated numbers. How many different ways can this be done? | 12 | 60.15625 |
22,685 | Consider a polynomial with integer coefficients given by:
\[8x^5 + b_4 x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 24 = 0.\]
Find the number of different possible rational roots of this polynomial. | 28 | 0 |
22,686 | Let \( N = 34 \times 34 \times 63 \times 270 \). The ratio of the sum of all odd factors of \( N \) to the sum of all even factors of \( N \) is ( ). | 1: 14 | 0 |
22,687 | A person rides a bicycle from place A to place B. If they increase their speed by 6 km/h, they can arrive 5 minutes earlier; if they decrease their speed by 5 km/h, they will be 6 minutes late. What is the distance between place A and place B in kilometers? | 15 | 53.90625 |
22,688 | In triangle $PQR$, $PQ = 8$, $PR = 17$, and the length of median $PM$ is 12. Additionally, the angle $\angle QPR = 60^\circ$. Find the area of triangle $PQR$. | 34\sqrt{3} | 50.78125 |
22,689 | What is the sum of all two-digit positive integers whose squares end with the digits 25? | 495 | 20.3125 |
22,690 | Find the distance between the vertices of the hyperbola given by the equation $4x^2 + 16x - 9y^2 + 18y - 23 = 0.$ | \sqrt{30} | 57.8125 |
22,691 | Let there be three individuals, labeled A, B, and C, to be allocated across seven laboratories in preparation for experiments. Each laboratory can accommodate no more than two people. Calculate the total number of distinct allocation schemes. | 336 | 25.78125 |
22,692 | Given that $x > 0$, $y > 0$, and $x+y=1$, find the minimum value of $\frac{x^{2}}{x+2}+\frac{y^{2}}{y+1}$. | \frac{1}{4} | 71.09375 |
22,693 | If $a$, $b$, and $c$ are natural numbers, and $a < b$, $a + b = 719$, $c - a = 915$, then the largest possible value of $a + b + c$ is. | 1993 | 94.53125 |
22,694 | A circle is divided into six equal sections. Each section is to be coloured with a single colour so that three sections are red, one is blue, one is green, and one is yellow. Two circles have the same colouring if one can be rotated to match the other. How many different colourings are there for the circle? | 20 | 41.40625 |
22,695 | Given that $\alpha$ is an angle in the third quadrant, $f(\alpha) = \frac {\sin(\pi-\alpha)\cdot \cos(2\pi-\alpha)\cdot \tan(-\alpha-\pi)}{\tan(-\alpha )\cdot \sin(-\pi -\alpha)}$.
1. Simplify $f(\alpha)$;
2. If $\cos\left(\alpha- \frac {3}{2}\pi\right) = \frac {1}{5}$, find the value of $f(\alpha)$;
3. If $\alpha=-1860^\circ$, find the value of $f(\alpha)$. | \frac {1}{2} | 68.75 |
22,696 | A population consists of $20$ individuals numbered $01$, $02$, $\ldots$, $19$, $20$. Using the following random number table, select $5$ individuals. The selection method is to start from the numbers in the first row and first two columns of the random number table, and select two numbers from left to right each time. If the two selected numbers are not within the population, remove them and continue selecting two numbers to the right. Then, the number of the $4$th individual selected is ______.<br/><table><tbody><tr><td width="84" align="center">$7816$</td><td width="84" align="center">$6572$</td><td width="84" align="center">$0802$</td><td width="84" align="center">$6314$</td><td width="84" align="center">$0702$</td><td width="84" align="center">$4369$</td><td width="84" align="center">$9728$</td><td width="84" align="center">$0198$</td></tr><tr><td align="center">$3204$</td><td align="center">$9234$</td><td align="center">$4935$</td><td align="center">$8200$</td><td align="center">$3623$</td><td align="center">$4869$</td><td align="center">$6938$</td><td align="center">$7481$</td></tr></tbody></table> | 14 | 70.3125 |
22,697 | The eccentricity of the ellipse given by the parametric equations $\begin{cases} x=3\cos\theta \\ y=4\sin\theta\end{cases}$ is $\frac{\sqrt{7}}{\sqrt{3^2+4^2}}$, calculate this value. | \frac { \sqrt {7}}{4} | 0 |
22,698 | Calculate the value of $({-\frac{4}{5}})^{2022} \times ({\frac{5}{4}})^{2021}$. | \frac{4}{5} | 95.3125 |
22,699 | Selected Exercise $4-4$: Coordinate Systems and Parametric Equations
In the rectangular coordinate system $xOy$, the parametric equations of the curve $C$ are given by $\begin{cases} & x=\cos \theta \\ & y=\sin \theta \end{cases}$, where $\theta$ is the parameter. In the polar coordinate system with the same unit length as the rectangular coordinate system $xOy$, taking the origin $O$ as the pole and the non-negative half of the $x$-axis as the polar axis, the equation of the line $l$ is given by $\sqrt{2}p \sin (\theta - \frac{\pi}{4}) = 3$.
(I) Find the Cartesian equation of the curve $C$ and the equation of the line $l$ in rectangular coordinates.
(II) Let $P$ be any point on the curve $C$. Find the maximum distance from the point $P$ to the line $l$. | \frac{3\sqrt{2}}{2} + 1 | 12.5 |
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