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22,000 | Quadrilateral $EFGH$ has right angles at $F$ and $H$, and $EG = 5$. If $EFGH$ has two sides with distinct integer lengths, and each side length is greater than 1, what is the area of $EFGH$? Express your answer in simplest radical form. | 12 | 30.46875 |
22,001 | Find $B^2$, where $B$ is the sum of the absolute values of all roots of the equation:
\[x = \sqrt{26} + \frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{x}}}}}}}}}.\] | 502 | 85.9375 |
22,002 | Subtract $256.26$ from $512.52$ and then multiply the result by $3$. Express the final result as a decimal to the nearest hundredth. | 768.78 | 73.4375 |
22,003 | Given two numbers, a and b, are randomly selected within the interval (-π, π), determine the probability that the function f(x) = x^2 + 2ax - b^2 + π has a root. | \dfrac{3}{4} | 87.5 |
22,004 | Two distinct positive integers $a$ and $b$ are factors of 48. If $a\cdot b$ is not a factor of 48, what is the smallest possible value of $a\cdot b$? | 18 | 0 |
22,005 | James wrote a different integer from 1 to 9 in each cell of a table. He then calculated the sum of the integers in each of the rows and in each of the columns of the table. Five of his answers were 12, 13, 15, 16, and 17, in some order. What was his sixth answer? | 17 | 12.5 |
22,006 | Given vectors $\overrightarrow {m}$=(cosx, sinx) and $\overrightarrow {n}$=(cosx, $\sqrt {3}$cosx), where x∈R, define the function f(x) = $\overrightarrow {m}$$\cdot \overrightarrow {n}$+ $\frac {1}{2}$.
(1) Find the analytical expression and the interval where the function is strictly increasing;
(2) Let a, b, and c be the sides opposite to angles A, B, and C of △ABC, respectively. If f(A)=2, b+c=$2 \sqrt {2}$, and the area of △ABC is $\frac {1}{2}$, find the value of a. | \sqrt {3}-1 | 0 |
22,007 | Observe the following equations:
\\(① \dfrac {1}{ \sqrt {2}+1}= \dfrac { \sqrt {2}-1}{( \sqrt {2}+1)( \sqrt {2}-1)}= \sqrt {2}-1\\);
\\(② \dfrac {1}{ \sqrt {3}+ \sqrt {2}}= \dfrac { \sqrt {3}- \sqrt {2}}{( \sqrt {3}+ \sqrt {2})( \sqrt {3}- \sqrt {2})}= \sqrt {3}- \sqrt {2}\\);
\\(③ \dfrac {1}{ \sqrt {4}+ \sqrt {3}}= \dfrac { \sqrt {4}- \sqrt {3}}{( \sqrt {4}+ \sqrt {3})( \sqrt {4}- \sqrt {3})}= \sqrt {4}- \sqrt {3}\\);\\(…\\)
Answer the following questions:
\\((1)\\) Following the pattern of the equations above, write the \\(n\\)th equation: \_\_\_\_\_\_ ;
\\((2)\\) Using the pattern you observed, simplify: \\( \dfrac {1}{ \sqrt {8}+ \sqrt {7}}\\);
\\((3)\\) Calculate: \\( \dfrac {1}{1+ \sqrt {2}}+ \dfrac {1}{ \sqrt {2}+ \sqrt {3}}+ \dfrac {1}{ \sqrt {3}+2}+…+ \dfrac {1}{3+ \sqrt {10}}\\). | \sqrt {10}-1 | 2.34375 |
22,008 | In \\(\Delta ABC\\), given that \\(a= \sqrt{3}, b= \sqrt{2}, B=45^{\circ}\\), find \\(A, C\\) and \\(c\\). | \frac{\sqrt{6}- \sqrt{2}}{2} | 0 |
22,009 | What's the largest number of elements that a set of positive integers between $1$ and $100$ inclusive can have if it has the property that none of them is divisible by another? | 50 | 94.53125 |
22,010 | Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, perpendicular lines to the $x$-axis are drawn through points $F\_1$ and $F\_2$ intersecting the ellipse at four points to form a square, determine the eccentricity $e$ of the ellipse. | \frac{\sqrt{5} - 1}{2} | 72.65625 |
22,011 | Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = -x^{2} + 4x$.
- (Ⅰ) Find the analytical expression of the function $f(x)$.
- (Ⅱ) Find the minimum value of the function $f(x)$ on the interval $\left[-2,a\right]$ where $\left(a > -2\right)$. | -4 | 35.9375 |
22,012 | In triangle $PQR$, $PQ = 12$, $QR = 16$, and $PR = 20$. Point $X$ is on $\overline{PQ}$, $Y$ is on $\overline{QR}$, and $Z$ is on $\overline{PR}$. Let $PX = u \cdot PQ$, $QY = v \cdot QR$, and $RZ = w \cdot PR$, where $u$, $v$, and $w$ are positive and satisfy $u+v+w=3/4$ and $u^2+v^2+w^2=1/2$. The ratio of the area of triangle $XYZ$ to the area of triangle $PQR$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 41 | 43.75 |
22,013 | What is the smallest positive integer that has eight positive odd integer divisors and sixteen positive even integer divisors? | 420 | 42.96875 |
22,014 | Given that $\cos(75^\circ + \alpha) = \frac{1}{3}$, where $\alpha$ is an angle in the third quadrant, find the value of $\cos(105^\circ - \alpha) + \sin(\alpha - 105^\circ)$. | \frac{2\sqrt{2} - 1}{3} | 86.71875 |
22,015 | Find the minimum value of $m$ such that any $m$ -element subset of the set of integers $\{1,2,...,2016\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\le 3$ . | 505 | 82.8125 |
22,016 | Given that α and β are acute angles, and $\tan \alpha = \frac{2}{t}$, $\tan \beta = \frac{t}{15}$. When $10\tan \alpha + 3\tan \beta$ reaches its minimum value, the value of $\alpha + \beta$ is \_\_\_\_\_\_. | \frac{\pi}{4} | 99.21875 |
22,017 | A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and consists only of prime numbers? Express your answer as a common fraction. | \frac{1}{6561} | 60.15625 |
22,018 | Let $a_n = -n^2 + 10n + 11$, then find the value of $n$ for which the sum of the sequence $\{a_n\}$ from the first term to the nth term is maximized. | 11 | 12.5 |
22,019 | Given a moving point $E$ such that the product of the slopes of the lines from $E$ to points $A(2,0)$ and $B(-2,0)$ is $- \frac {1}{4}$, and the trajectory of point $E$ is curve $C$.
$(1)$ Find the equation of curve $C$;
$(2)$ Draw a line $l$ through point $D(1,0)$ that intersects curve $C$ at points $P$ and $Q$. Find the maximum value of $\overrightarrow{OP} \cdot \overrightarrow{OQ}$. | \frac {1}{4} | 57.8125 |
22,020 | A deck consists of six red cards and six green cards, each with labels $A$, $B$, $C$, $D$, $E$ corresponding to each color. Two cards are dealt from this deck. A winning pair consists of cards that either share the same color or the same label. Calculate the probability of drawing a winning pair.
A) $\frac{1}{2}$
B) $\frac{10}{33}$
C) $\frac{30}{66}$
D) $\frac{35}{66}$
E) $\frac{40}{66}$ | \frac{35}{66} | 38.28125 |
22,021 | Given the set of integers $\{1, 2, 3, \dots, 9\}$, from which three distinct numbers are arbitrarily selected as the coefficients of the quadratic function $f_{(x)} = ax^2 + bx + c$, determine the total number of functions $f_{(x)}$ that satisfy $\frac{f(1)}{2} \in \mathbb{Z}$. | 264 | 46.09375 |
22,022 | What is the value of $\left(\left((3+2)^{-1}-1\right)^{-1}-1\right)^{-1}-1$? | -\frac{13}{9} | 39.84375 |
22,023 | Given $f\left(\alpha \right)=\frac{\mathrm{sin}\left(\pi -\alpha \right)\mathrm{cos}\left(2\pi -\alpha \right)\mathrm{cos}\left(-\alpha +\frac{3\pi }{2}\right)}{\mathrm{cos}\left(\frac{\pi }{2}-\alpha \right)\mathrm{sin}\left(-\pi -\alpha \right)}$.
(1) Simplify $f(\alpha )$.
(2) If $\alpha$ is an angle in the third quadrant and $\mathrm{cos}(\alpha -\frac{3\pi }{2})=\frac{1}{5}$, find the value of $f(\alpha )$. | \frac{2\sqrt{6}}{5} | 96.875 |
22,024 | A standard die is rolled consecutively two times. Calculate the probability that the face-up numbers are adjacent natural numbers. | \frac{5}{18} | 42.96875 |
22,025 | Given that $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $C: \frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, point $P$ is on the hyperbola $C$, $PF_{2}$ is perpendicular to the x-axis, and $\sin \angle PF_{1}F_{2} = \frac {1}{3}$, determine the eccentricity of the hyperbola $C$. | \sqrt{2} | 88.28125 |
22,026 | Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 10.\] | \frac{131}{11} | 4.6875 |
22,027 | From the five numbers \\(1, 2, 3, 4, 5\\), select any \\(3\\) to form a three-digit number without repeating digits. When the three digits include both \\(2\\) and \\(3\\), \\(2\\) must be placed before \\(3\\) (not necessarily adjacent). How many such three-digit numbers are there? | 51 | 0 |
22,028 | When the numbers \(\sqrt{5}, 2.1, \frac{7}{3}, 2.0 \overline{5}, 2 \frac{1}{5}\) are arranged in order from smallest to largest, the middle number is: | 2 \frac{1}{5} | 64.0625 |
22,029 | Sabrina has a fair tetrahedral die whose faces are numbered 1, 2, 3, and 4, respectively. She creates a sequence by rolling the die and recording the number on its bottom face. However, she discards (without recording) any roll such that appending its number to the sequence would result in two consecutive terms that sum to 5. Sabrina stops the moment that all four numbers appear in the sequence. Find the expected (average) number of terms in Sabrina's sequence. | 10 | 43.75 |
22,030 | In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=3+t\cos \alpha \\ y=1+t\sin \alpha\end{cases}$ (where $t$ is the parameter), in the polar coordinate system (with the same unit length as the Cartesian coordinate system $xOy$, and the origin $O$ as the pole, and the non-negative half-axis of $x$ as the polar axis), the equation of curve $C$ is $\rho=4\cos \theta$.
$(1)$ Find the equation of curve $C$ in the Cartesian coordinate system;
$(2)$ If point $P(3,1)$, suppose circle $C$ intersects line $l$ at points $A$ and $B$, find the minimum value of $|PA|+|PB|$. | 2 \sqrt {2} | 0 |
22,031 | Two tangents are drawn to a circle from an exterior point A; they touch the circle at points B and C respectively. A third tangent intersects segment AB in P and AC in R, and touches the circle at Q. If AB = 24, and the lengths BP = PQ = x and QR = CR = y with x + y = 12, find the perimeter of triangle APR. | 48 | 83.59375 |
22,032 | Given a function defined on $\mathbb{R}$, $f(x)=A\sin (\omega x+\varphi)$ where $A > 0$, $\omega > 0$, and $|\varphi| \leqslant \frac {\pi}{2}$, the minimum value of the function is $-2$, and the distance between two adjacent axes of symmetry is $\frac {\pi}{2}$. After the graph of the function is shifted to the left by $\frac {\pi}{12}$ units, the resulting graph corresponds to an even function.
$(1)$ Find the expression for the function $f(x)$.
$(2)$ If $f\left( \frac {x_{0}}{2}\right)=- \frac {3}{8}$, and $x_{0}\in\left[ \frac {\pi}{2},\pi\right]$, find the value of $\cos \left(x_{0}+ \frac {\pi}{6}\right)$. | - \frac { \sqrt {741}}{32}- \frac {3}{32} | 0 |
22,033 | The shortest distance from a point on the curve $y=\ln x$ to the line $y=x+2$ is what value? | \frac{3\sqrt{2}}{2} | 65.625 |
22,034 | A regular hexagon's center and vertices together make 7 points. Calculate the number of triangles that can be formed using any 3 of these points as vertices. | 32 | 17.96875 |
22,035 | Suppose that \(g(x)\) is a function such that
\[g(xy) + 2x = xg(y) + g(x)\] for all real numbers \(x\) and \(y.\) If \(g(-1) = 3\) and \(g(1) = 1\), then compute \(g(-101).\) | 103 | 27.34375 |
22,036 | If $(2,12)$ and $(8,3)$ are the coordinates of two opposite vertices of a rectangle, what is the sum of the $x$-coordinates of the other two vertices? | 10 | 97.65625 |
22,037 | Describe how to place the vertices of a triangle in the faces of a cube in such a way that the shortest side of the triangle is the biggest possible. | \sqrt{2} | 15.625 |
22,038 | Given the function $f(x)=\sin({ωx+φ})$ $({ω＞0,|φ|≤\frac{π}{2}})$, $f(0)=\frac{{\sqrt{2}}}{2}$, and the function $f\left(x\right)$ is monotonically decreasing on the interval $({\frac{π}{{16}},\frac{π}{8}})$, then the maximum value of $\omega$ is ______. | 10 | 52.34375 |
22,039 | Suppose $x$ and $y$ satisfy the system of inequalities $\begin{cases} & x-y \geqslant 0 \\ & x+y-2 \geqslant 0 \\ & x \leqslant 2 \end{cases}$, calculate the minimum value of $x^2+y^2-2x$. | -\dfrac{1}{2} | 0.78125 |
22,040 | Given the function $f(x)=\sin(\omega x+\varphi)$ is monotonically increasing on the interval ($\frac{π}{6}$,$\frac{{2π}}{3}$), and the lines $x=\frac{π}{6}$ and $x=\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, determine the value of $f(-\frac{{5π}}{{12}})$. | \frac{\sqrt{3}}{2} | 21.09375 |
22,041 | How many ways are there to allocate a group of 8 friends among a basketball team, a soccer team, a track team, and the option of not participating in any sports? Each team, including the non-participant group, could have anywhere from 0 to 8 members. Assume the friends are distinguishable. | 65536 | 94.53125 |
22,042 | From a large sheet of aluminum, triangular sheets (with each cell side equal to 1) are cut with vertices at marked points. What is the minimum area of the triangle that can be obtained? | $\frac{1}{2}$ | 0 |
22,043 | Seven students stand in a row for a photo, among them, students A and B must stand next to each other, and students C and D must not stand next to each other. The total number of different arrangements is. | 960 | 81.25 |
22,044 | Determine how many ordered pairs of positive integers $(x, y)$, where $x < y$, have a harmonic mean of $5^{20}$. | 20 | 75 |
22,045 | The function $g$ is defined on the set of integers and satisfies \[g(n)= \begin{cases} n-5 & \mbox{if }n\ge 1200 \\ g(g(n+7)) & \mbox{if }n<1200. \end{cases}\] Find $g(70)$. | 1195 | 38.28125 |
22,046 | Given the function $f(x) = \sin x \cos x - \sqrt{3} \cos (x+\pi) \cos x, x \in \mathbb{R}$.
(Ⅰ) Find the minimal positive period of $f(x)$;
(Ⅱ) If the graph of the function $y = f(x)$ is translated by $\overrightarrow{b}=\left( \frac{\pi}{4}, \frac{\sqrt{3}}{2} \right)$ to obtain the graph of the function $y = g(x)$, find the maximum value of $y=g(x)$ on the interval $\left[0, \frac{\pi}{4}\right]$. | \frac{3\sqrt{3}}{2} | 61.71875 |
22,047 | Given that $0 < a < \pi, \tan a=-2$,
(1) Find the value of $\cos a$;
(2) Find the value of $2\sin^{2}a - \sin a \cos a + \cos^{2}a$. | \frac{11}{5} | 96.09375 |
22,048 | The number halfway between $\dfrac{1}{8}$ and $\dfrac{1}{3}$ is
A) $\dfrac{11}{48}$
B) $\dfrac{11}{24}$
C) $\dfrac{5}{24}$
D) $\dfrac{1}{4}$
E) $\dfrac{1}{5}$ | \dfrac{11}{48} | 100 |
22,049 | A new dump truck delivered sand to a construction site, forming a conical pile with a diameter of $12$ feet. The height of the cone was $50\%$ of its diameter. However, the pile was too large, causing some sand to spill, forming a cylindrical layer directly around the base of the cone. The height of this cylindrical layer was $2$ feet and the thickness was $1$ foot. Calculate the total volume of sand delivered, expressing your answer in terms of $\pi$. | 98\pi | 39.84375 |
22,050 | In the Cartesian coordinate system $(xOy)$, a pole is established at the origin $O$ with the non-negative semi-axis of the $x$-axis as the polar axis, forming a polar coordinate system. Given that the equation of line $l$ is $4ρ\cos θ-ρ\sin θ-25=0$, and the curve $W$ is defined by the parametric equations $x=2t, y=t^{2}-1$.
1. Find the Cartesian equation of line $l$ and the general equation of curve $W$.
2. If point $P$ is on line $l$, and point $Q$ is on curve $W$, find the minimum value of $|PQ|$. | \frac{8\sqrt{17}}{17} | 25.78125 |
22,051 | Given the function $f(x)=\sin (x+ \frac{7\pi}{4})+\cos (x- \frac{3\pi}{4})$, where $x\in R$.
(1) Find the smallest positive period and the minimum value of $f(x)$;
(2) Given that $f(\alpha)= \frac{6}{5}$, where $0 < \alpha < \frac{3\pi}{4}$, find the value of $f(2\alpha)$. | \frac{31\sqrt{2}}{25} | 34.375 |
22,052 | Given that triangle $PQR$ is a right triangle, each side being the diameter of a semicircle, the area of the semicircle on $\overline{PQ}$ is $18\pi$, and the arc of the semicircle on $\overline{PR}$ has length $10\pi$, calculate the radius of the semicircle on $\overline{QR}$. | \sqrt{136} | 0 |
22,053 | Acute-angled $\triangle ABC$ is inscribed in a circle with center at $O$; $\stackrel \frown {AB} = 100^\circ$ and $\stackrel \frown {BC} = 80^\circ$.
A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. The task is to determine the ratio of the magnitudes of $\angle OBE$ and $\angle BAC$. | \frac{1}{2} | 8.59375 |
22,054 | Given that $x_{0}$ is a zero of the function $f(x)=2a\sqrt{x}+b-{e}^{\frac{x}{2}}$, and $x_{0}\in [\frac{1}{4}$,$e]$, find the minimum value of $a^{2}+b^{2}$. | \frac{{e}^{\frac{3}{4}}}{4} | 0.78125 |
22,055 | What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal? | 50 | 1.5625 |
22,056 | Compute $9 \cdot (-5) - (7 \cdot -2) + (8 \cdot -6)$. | -79 | 18.75 |
22,057 | $A$,$B$,$C$,$D$,$E$,$F$ are 6 students standing in a row to participate in a literary performance. If $A$ does not stand at either end, and $B$ and $C$ must be adjacent, then the total number of different arrangements is ____. | 144 | 46.09375 |
22,058 | "Modulo $m$ graph paper" consists of a grid of $13^2$ points, representing all pairs of integer residues $(x,y)$ where $0\le x, y <13$. To graph a congruence on modulo $13$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. Consider the graph of $$4x \equiv 3y + 1 \pmod{13}.$$ Find the sum of the $x$-intercept and the $y$-intercept, where the intercepts are represented as $(x_0,0)$ and $(0,y_0)$ with $0\le x_0,y_0<13$. | 14 | 17.1875 |
22,059 | Simplify the expression $\dfrac {\cos 40 ^{\circ} }{\cos 25 ^{\circ} \sqrt {1-\sin 40 ^{\circ} }}$. | \sqrt{2} | 40.625 |
22,060 | Given a $6 \times 6$ square of $36$ square blocks, find the number of different combinations of $4$ blocks that can be selected so that no two are in the same row or column. | 5400 | 92.1875 |
22,061 | Given the complex number $z= \frac {(1+i)^{2}+2(5-i)}{3+i}$.
$(1)$ Find $|z|$;
$(2)$ If $z(z+a)=b+i$, find the values of the real numbers $a$ and $b$. | -13 | 23.4375 |
22,062 | Given that the sum of the first $n$ terms of the geometric sequence $\{a\_n\}$ is $S\_n$, and it satisfies $S\_n=(\frac{1}{2})^{n}-1$, find the limit as $n$ approaches infinity of $(a\_1+a\_3+...+a\_2n-1)$ . | -\frac{2}{3} | 93.75 |
22,063 | Given that in the expansion of $\left(1+x\right)^{n}$, the coefficient of $x^{3}$ is the largest, then the sum of the coefficients of $\left(1+x\right)^{n}$ is ____. | 64 | 77.34375 |
22,064 | How many integers between $200$ and $300$ have three different digits in increasing order? | 21 | 24.21875 |
22,065 | The minimum positive period of the function $f(x)=\sin x$ is $\pi$. | 2\pi | 47.65625 |
22,066 | Given $$\frac {\pi}{2} < \alpha < \pi$$, $$0 < \beta < \frac {\pi}{2}$$, $$\tan\alpha = -\frac {3}{4}$$, and $$\cos(\beta-\alpha) = \frac {5}{13}$$, find the value of $\sin\beta$. | \frac {63}{65} | 18.75 |
22,067 | Given the function $f(x) = 2\sin(\frac{1}{3}x - \frac{π}{6})$, where $x \in \mathbb{R}$.
(1) Find the value of $f(\frac{5π}{4})$;
(2) Let $\alpha, \beta \in [0, \frac{π}{2}], f(3\alpha + \frac{π}{2}) = \frac{10}{13}, f(3\beta + 2π) = \frac{6}{5}$, find the value of $\cos(\alpha + \beta)$. | \frac{16}{65} | 80.46875 |
22,068 | A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction. | \frac{9}{2} | 60.15625 |
22,069 | Given that Connie adds $3$ to a number and gets $45$ as her answer, but she should have subtracted $3$ from the number to get the correct answer, determine the correct number. | 39 | 94.53125 |
22,070 | Regular hexagon $PQRSTU$ has vertices $P$ and $R$ at $(0,0)$ and $(8,2)$, respectively. What is its area? | 102\sqrt{3} | 16.40625 |
22,071 | From milk with a fat content of $5\%$, cottage cheese with a fat content of $15.5\%$ is produced, while there remains whey with a fat content of $0.5\%$. How much cottage cheese is obtained from 1 ton of milk? | 0.3 | 18.75 |
22,072 | Find the positive value of $k$ such that the equation $4x^3 + 9x^2 + kx + 4 = 0$ has exactly one real solution in $x$. | 6.75 | 0 |
22,073 | Rationalize the denominator of $\frac{2+\sqrt{5}}{3-\sqrt{5}}$. When you write your answer in the form $A+B\sqrt{C}$, where $A$, $B$, and $C$ are integers, what is $ABC$? | 275 | 56.25 |
22,074 | Let \( ABC \) be a triangle with \( AB = 5 \), \( AC = 4 \), \( BC = 6 \). The angle bisector of \( \angle C \) intersects side \( AB \) at \( X \). Points \( M \) and \( N \) are drawn on sides \( BC \) and \( AC \), respectively, such that \( \overline{XM} \parallel \overline{AC} \) and \( \overline{XN} \parallel \overline{BC} \). Compute the length \( MN \). | \frac{3 \sqrt{14}}{5} | 0 |
22,075 | Given that $α$ and $β ∈ ( \frac{π}{2},π)$, and $sinα + cosα = a$, $cos(β - α) = \frac{3}{5}$.
(1) If $a = \frac{1}{3}$, find the value of $sinαcosα + tanα - \frac{1}{3cosα}$;
(2) If $a = \frac{7}{13}$, find the value of $sinβ$. | \frac{16}{65} | 0.78125 |
22,076 | Given points $M(4,0)$ and $N(1,0)$, any point $P$ on curve $C$ satisfies: $\overset{→}{MN} \cdot \overset{→}{MP} = 6|\overset{→}{PN}|$.
(I) Find the trajectory equation of point $P$;
(II) A line passing through point $N(1,0)$ intersects curve $C$ at points $A$ and $B$, and intersects the $y$-axis at point $H$. If $\overset{→}{HA} = λ_1\overset{→}{AN}$ and $\overset{→}{HB} = λ_2\overset{→}{BN}$, determine whether $λ_1 + λ_2$ is a constant value. If it is, find this value; if not, explain the reason. | -\frac{8}{3} | 71.09375 |
22,077 | Given two points $A(-2, 0)$ and $B(0, 2)$, point $C$ is any point on the circle $x^2 + y^2 - 2x = 0$, the minimum value of the area of $\triangle ABC$ is \_\_\_\_\_\_. | 3 - \sqrt{2} | 60.15625 |
22,078 | How many even divisors does \(10!\) have? | 240 | 80.46875 |
22,079 | A circle is inscribed in trapezoid \( PQRS \).
If \( PS = QR = 25 \) cm, \( PQ = 18 \) cm, and \( SR = 32 \) cm, what is the length of the diameter of the circle? | 24 | 68.75 |
22,080 | Find the minimum value of
\[\sqrt{x^2 + (2 - x)^2} + \sqrt{(2 - x)^2 + (2 + x)^2}\]over all real numbers $x.$ | 2\sqrt{5} | 64.0625 |
22,081 | Let $ABCDE$ be a regular pentagon. The star $ACEBD$ has area 1. $AC$ and $BE$ meet at $P$ , while $BD$ and $CE$ meet at $Q$ . Find the area of $APQD$ . | 1/2 | 5.46875 |
22,082 | Given vectors $\overrightarrow{m}=(\sin x, -1)$ and $\overrightarrow{n}=(\sqrt{3}\cos x, -\frac{1}{2})$, let $f(x)=(\overrightarrow{m}+\overrightarrow{n})\cdot \overrightarrow{m}$.
(1) Find the analytic expression for $f(x)$ and its intervals of monotonic increase;
(2) Given that $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$ in triangle $\triangle ABC$, respectively, and $A$ is an acute angle with $a=2\sqrt{3}$ and $c=4$. If $f(A)$ is the maximum value of $f(x)$ on the interval $[0, \frac{\pi}{2}]$, find $A$, $b$, and the area $S$ of $\triangle ABC$. | 2\sqrt{3} | 25.78125 |
22,083 | Given the function $f(x)= \frac {2}{x+1}$, point $O$ is the coordinate origin, point $A_{n}(n,f(n))(n∈N^{})$, vector $ \overrightarrow{j}=(0,1)$, and $θ_{n}$ is the angle between vector $ \overrightarrow{OA_{n}}$ and $ \overrightarrow{j}$, determine the value of $\frac {cos θ_{1}}{sin θ_{1}}+ \frac {cos θ_{2}}{sin θ_{2}}+ \frac {cos θ_{1}}{sin θ_{1}}+…+ \frac {cos θ_{2016}}{sin θ_{2016}}$. | \frac{4032}{2017} | 83.59375 |
22,084 | Find the number of positive integers less than or equal to $1200$ that are neither $5$-nice nor $6$-nice. | 800 | 13.28125 |
22,085 | Given $|x+2|+|1-x|=9-|y-5|-|1+y|$, find the maximum and minimum values of $x+y$. | -3 | 57.8125 |
22,086 | What is the smallest positive integer that has eight positive odd integer divisors and sixteen positive even integer divisors? | 3000 | 0 |
22,087 | Petya and Vasya are playing the following game. Petya thinks of a natural number \( x \) with a digit sum of 2012. On each turn, Vasya chooses any natural number \( a \) and finds out the digit sum of the number \( |x-a| \) from Petya. What is the minimum number of turns Vasya needs to determine \( x \) with certainty? | 2012 | 21.09375 |
22,088 | Calculate the area of the parallelogram formed by the vectors $\begin{pmatrix} 4 \\ 2 \\ -3 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ -4 \\ 5 \end{pmatrix}$. | 6\sqrt{30} | 60.9375 |
22,089 | Consider the function $g(x) = \frac{x^2}{2} + 2x - 1$. Determine the sum of all distinct numbers $x$ such that $g(g(g(x))) = 1$. | -4 | 1.5625 |
22,090 | What is the first year after 2000 for which the sum of the digits is 15? | 2049 | 78.125 |
22,091 | In the Cartesian coordinate system $xOy$, the parametric equations of curve $C$ are $\begin{cases}x=2\cos \alpha \\ y=\sin \alpha \\ \end{cases}$ (where $\alpha$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis.
1. Find the polar coordinate equation of curve $C$.
2. If two points $M$ and $N$ on curve $C$ have $OM\bot ON$, find the minimum value of the area of triangle $OMN$. | \frac{4}{5} | 22.65625 |
22,092 | Given the function $f(x)=2\sin (\pi-x)\cos x$.
- (I) Find the smallest positive period of $f(x)$;
- (II) Find the maximum and minimum values of $f(x)$ in the interval $\left[- \frac {\pi}{6}, \frac {\pi}{2}\right]$. | - \frac{ \sqrt{3}}{2} | 85.15625 |
22,093 | In the interval $[0,\pi]$, a number $x$ is randomly selected. The probability that $\sin x$ falls between $0$ and $\frac{1}{2}$ is ______. | \frac{1}{3} | 76.5625 |
22,094 | Given the function $f(x)=(2-a)(x-1)-2\ln x$ $(a\in \mathbb{R})$.
(Ⅰ) If the tangent line at the point $(1,g(1))$ on the curve $g(x)=f(x)+x$ passes through the point $(0,2)$, find the interval where the function $g(x)$ is decreasing;
(Ⅱ) If the function $y=f(x)$ has no zeros in the interval $\left(0, \frac{1}{2}\right)$, find the minimum value of $a$. | 2-4\ln 2 | 66.40625 |
22,095 | Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$? | 60 | 81.25 |
22,096 | Let \(a\), \(b\), \(c\), and \(d\) be distinct positive integers such that \(a+b\), \(a+c\), and \(a+d\) are all odd and are all squares. Let \(L\) be the least possible value of \(a + b + c + d\). What is the value of \(10L\)? | 670 | 8.59375 |
22,097 | 28 apples weigh 3 kilograms. If they are evenly divided into 7 portions, each portion accounts for $\boxed{\frac{1}{7}}$ of all the apples, and each portion weighs $\boxed{\frac{3}{7}}$ kilograms. | \frac{3}{7} | 97.65625 |
22,098 | Perform the calculations.
$(54+38) \times 15$
$1500-32 \times 45$
$157 \times (70 \div 35)$ | 314 | 51.5625 |
22,099 | A rectangle has dimensions $4$ and $2\sqrt{3}$. Two equilateral triangles are contained within this rectangle, each with one side coinciding with the longer side of the rectangle. The triangles intersect, forming another polygon. What is the area of this polygon?
A) $2\sqrt{3}$
B) $4\sqrt{3}$
C) $6$
D) $8\sqrt{3}$ | 4\sqrt{3} | 19.53125 |
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