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30,300 | Arrange the positive integers whose digits sum to 4 in ascending order. Which position does the number 2020 occupy in this sequence? | 28 | 1.5625 |
30,301 | Calculate the distance between the points $(5, -3)$ and $(-7, 4)$. Additionally, determine if the points $(5, -3)$, $(-7, 4)$, and $(5, 4)$ form a right triangle. | \sqrt{193} | 23.4375 |
30,302 | A right triangle has one angle measuring $30^\circ$. This triangle shares its hypotenuse with a second triangle that is also right-angled. The two triangles together form a quadrilateral. If the other acute angle in the second triangle is $45^\circ$, find the area of the quadrilateral given that the hypotenuse common to both triangles is 10 units long. | \frac{25\sqrt{3} + 50}{2} | 21.875 |
30,303 | Assuming that $x$ is a multiple of $18711$, determine the greatest common divisor of $g(x) = (4x+5)(5x+3)(6x+7)(3x+11)$ and $x$. | 1155 | 0.78125 |
30,304 | (1) Solve the inequality $|x+1| + 2|x-1| < 3x + 5$.
(2) Given $a, b \in [0, 1]$, find the maximum value of $ab + (1-a-b)(a+b)$. | \frac{1}{3} | 46.09375 |
30,305 | Find the maximum real number \( k \) such that for any simple graph \( G \) with \( n \) vertices (\( n \geq 3 \)), the inequality \( x^3 \geq k y^2 \) holds, where \( x \) is the number of edges of \( G \) and \( y \) is the number of triangles in \( G \). | \frac{9}{2} | 79.6875 |
30,306 | Calculate $8 \cdot 9\frac{2}{5}$. | 75\frac{1}{5} | 78.90625 |
30,307 | In a rectangle that is sub-divided into 3 identical squares of side length 1, if \(\alpha^{\circ} = \angle ABD + \angle ACD\), find the value of \(\alpha\). | 45 | 10.9375 |
30,308 | A right cone has a base with a circumference of $20\pi$ inches and a height of 40 inches. The height of the cone is reduced while the circumference stays the same. After reduction, the volume of the cone is $400\pi$ cubic inches. What is the ratio of the new height to the original height, and what is the new volume? | 400\pi | 35.9375 |
30,309 | Regular hexagon $GHJKLMN$ is the base of a right pyramid $QGHJKLMN$. If $QGM$ is an equilateral triangle with side length 10, then what is the volume of the pyramid? | 187.5 | 5.46875 |
30,310 | Four people, A, B, C, and D, stand on a staircase with 7 steps. If each step can accommodate up to 3 people, and the positions of people on the same step are not distinguished, then the number of different ways they can stand is (answer in digits). | 2394 | 0 |
30,311 | How many integers from 1 to 16500
a) are not divisible by 5;
b) are not divisible by either 5 or 3;
c) are not divisible by either 5, 3, or 11? | 8000 | 48.4375 |
30,312 | Given the ellipse $$C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a>b>0)$$ with its left and right foci being F<sub>1</sub> and F<sub>2</sub>, and its top vertex being B. If the perimeter of $\triangle BF_{1}F_{2}$ is 6, and the distance from point F<sub>1</sub> to the line BF<sub>2</sub> is $b$.
(1) Find the equation of ellipse C;
(2) Let A<sub>1</sub> and A<sub>2</sub> be the two endpoints of the major axis of ellipse C, and point P is any point on ellipse C different from A<sub>1</sub> and A<sub>2</sub>. The line A<sub>1</sub>P intersects the line $x=m$ at point M. If the circle with MP as its diameter passes through point A<sub>2</sub>, find the value of the real number $m$. | 14 | 11.71875 |
30,313 | The positive integers \(a\), \(b\) are such that \(15a + 16b\) and \(16a - 15b\) are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | 481 | 0 |
30,314 | Given right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+2$, where $b<150$, determine the number of possible triangles. | 12 | 18.75 |
30,315 | A modified deck of cards contains 60 cards consisting of 15 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Prince, and Princess) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, $\clubsuit$). Hence, each suit now has 15 cards. Four suits are still divided into two colors with $\spadesuit$ and $\clubsuit$ being black and $\heartsuit$ and $\diamondsuit$ being red. The deck is shuffled. What is the probability that the top card is either a $\diamondsuit$ or one of the new ranks (Prince or Princess)? | \frac{7}{20} | 16.40625 |
30,316 | A rectangle has a perimeter of 120 inches and each side has an even integer length. How many non-congruent rectangles meet these criteria? | 15 | 49.21875 |
30,317 | Calculate the area of an isosceles triangle if the length of its altitude, drawn to one of the equal sides, is $12 \mathrm{~cm}$, and the length of the base is $15 \mathrm{~cm}$. | 90 | 97.65625 |
30,318 | Given points O(0,0) and A(1,1), the slope angle of line OA is $\boxed{\text{answer}}$. | \frac{\pi}{4} | 60.9375 |
30,319 | A valuable right-angled triangular metal plate $A B O$ is placed in a plane rectangular coordinate system (as shown in the diagram), with $A B = B O = 1$ (meter) and $A B \perp O B$. Due to damage in the shaded part of the triangular plate, a line $M N$ passing through the point $P\left(\frac{1}{2}, \frac{1}{4}\right)$ is needed to cut off the damaged part. How should the slope of the line $M N$ be determined such that the area of the resulting triangular plate $\triangle A M N$ is maximized? | -\frac{1}{2} | 12.5 |
30,320 | A train passenger knows that the speed of their train is 40 km/h. As soon as a passing train started to go by the window, the passenger started a stopwatch and noted that the passing train took 3 seconds to pass completely. Determine the speed of the passing train, given that its length is 75 meters. | 50 | 65.625 |
30,321 | Let $P$ and $Q$ be points on the circle $(x-0)^2+(y-6)^2=2$ and the ellipse $\frac{x^2}{10}+y^2=1$, respectively. What is the maximum distance between $P$ and $Q$?
A) $5\sqrt{2}$
B) $\sqrt{46}+\sqrt{2}$
C) $7+\sqrt{2}$
D) $6\sqrt{2}$ | 6\sqrt{2} | 9.375 |
30,322 | Given the function $f(x)=\sin (2x+ \frac {π}{3})- \sqrt {3}\sin (2x- \frac {π}{6})$
(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;
(2) When $x\in\[- \frac {π}{6}, \frac {π}{3}\]$, find the maximum and minimum values of $f(x)$, and write out the values of the independent variable $x$ when the maximum and minimum values are obtained. | -\sqrt {3} | 0 |
30,323 | In a bag, there are three balls of different colors: red, yellow, and blue, each color having one ball. Each time a ball is drawn from the bag, its color is recorded and then the ball is put back. The drawing stops when all three colors of balls have been drawn, what is the probability of stopping after exactly 5 draws? | \frac{14}{81} | 18.75 |
30,324 | Determine if there exists a positive integer \( m \) such that the equation
\[
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}=\frac{m}{a+b+c}
\]
has infinitely many solutions in positive integers \( (a, b, c) \). | 12 | 10.15625 |
30,325 | Express $0.6\overline{03}$ as a common fraction. | \frac{104}{165} | 0 |
30,326 | Arrange the numbers 3, 4, 5, 6 into two natural numbers A and B, so that the product A×B is maximized. Find the value of A×B. | \left( 3402 \right) | 0 |
30,327 | Determine the number of digits in the product of $84,123,457,789,321,005$ and $56,789,234,567,891$. | 32 | 3.90625 |
30,328 | Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2020$, and $a^2 - b^2 + c^2 - d^2 = 2024$. Find the number of possible values of $a$. | 503 | 0.78125 |
30,329 | A food factory processes mushrooms, with a cost of 20 yuan per kilogram of mushrooms, and a processing fee of t yuan per kilogram (t is a constant, and $2 \leq t \leq 5$). Assume the factory price per kilogram of mushrooms is x yuan ($25 \leq x \leq 40$). According to market research, the sales volume q is inversely proportional to $e^x$, and when the factory price per kilogram of mushrooms is 30 yuan, the daily sales volume is 100 kilograms.
(1) Find the relationship between the factory's daily profit y yuan and the factory price per kilogram of mushrooms x yuan;
(2) If $t = 5$, what is the factory price per kilogram of mushrooms x for the factory's profit y to be maximized, and find the maximum value. | 100e^4 | 76.5625 |
30,330 | Given the function \( f(x) = ax^2 + 8x + 3 \) where \( a < 0 \). For a given negative \( a \), there exists a largest positive real number \( L(a) \) such that the inequality \( |f(x)| \leq 5 \) holds for the entire interval \([0, L(a)]\). Find the value of \( a \) that maximizes \( L(a) \) and determine this maximum \( L(a) \). | \frac{\sqrt{5} + 1}{2} | 0 |
30,331 | If the tangent line of the curve $y=\ln x$ at point $P(x_{1}, y_{1})$ is tangent to the curve $y=e^{x}$ at point $Q(x_{2}, y_{2})$, then $\frac{2}{{x_1}-1}+x_{2}=$____. | -1 | 19.53125 |
30,332 | A wishing well is located at the point $(11,11)$ in the $xy$ -plane. Rachelle randomly selects an integer $y$ from the set $\left\{ 0, 1, \dots, 10 \right\}$ . Then she randomly selects, with replacement, two integers $a,b$ from the set $\left\{ 1,2,\dots,10 \right\}$ . The probability the line through $(0,y)$ and $(a,b)$ passes through the well can be expressed as $\frac mn$ , where $m$ and $n$ are relatively prime positive integers. Compute $m+n$ .
*Proposed by Evan Chen* | 111 | 7.8125 |
30,333 | Given that point $M(x\_0, y\_0)$ moves on the circle $x^{2}+y^{2}=4$, $N(4,0)$, and point $P(x,y)$ is the midpoint of segment $MN$.
(1) Find the trajectory equation of point $P(x,y)$;
(2) Find the maximum and minimum distances from point $P(x,y)$ to the line $3x+4y-86=0$. | 15 | 80.46875 |
30,334 | Let $ABCD$ be a convex quadrilateral with $AB=AD, m\angle A = 40^{\circ}, m\angle C = 130^{\circ},$ and $m\angle ADC - m\angle ABC = 20^{\circ}.$ Find the measure of the non-reflex angle $\angle CDB$ in degrees. | 35 | 69.53125 |
30,335 | Let $ABCD$ be a square of side length $4$ . Points $E$ and $F$ are chosen on sides $BC$ and $DA$ , respectively, such that $EF = 5$ . Find the sum of the minimum and maximum possible areas of trapezoid $BEDF$ .
*Proposed by Andrew Wu* | 16 | 28.90625 |
30,336 | For all \( m \) and \( n \) satisfying \( 1 \leq n \leq m \leq 5 \), the polar equation
$$
\rho = \frac{1}{1 - C_{m}^{n} \cos \theta}
$$
represents how many different hyperbolas? | 10 | 18.75 |
30,337 | If $x$ is a real number and $k$ is a nonnegative integer, compute the value of
\[
\frac{\binom{1/2}{2015} \cdot 4^{2015}}{\binom{4030}{2015}} \, .
\] | -\frac{1}{4030 \cdot 4029 \cdot 4028} | 0 |
30,338 | Given points $A(-3, -4)$ and $B(6, 3)$ in the xy-plane; point $C(1, m)$ is taken so that $AC + CB$ is a minimum. Find the value of $m$. | -\frac{8}{9} | 3.125 |
30,339 | Divide 2 teachers and 6 students into 2 groups, each group consisting of 1 teacher and 3 students. Calculate the total number of different arrangements to assign them to locations A and B to participate in social practice activities. | 40 | 2.34375 |
30,340 | Compute $\arccos (\cos 3).$ All functions are in radians. | 3 - 2\pi | 0.78125 |
30,341 | In the rectangular coordinate system on a plane, the parametric equations of curve $C$ are given by $\begin{cases} x=5\cos \alpha \\ y=\sin \alpha \end{cases}$ where $\alpha$ is a parameter, and point $P$ has coordinates $(3 \sqrt {2},0)$.
(1) Determine the shape of curve $C$;
(2) Given that line $l$ passes through point $P$ and intersects curve $C$ at points $A$ and $B$, and the slope angle of line $l$ is $45^{\circ}$, find the value of $|PA|\cdot|PB|$. | \frac{7}{13} | 33.59375 |
30,342 | Let the function $f(x)= \begin{cases} \log_{2}(1-x) & (x < 0) \\ g(x)+1 & (x > 0) \end{cases}$. If $f(x)$ is an odd function, determine the value of $g(3)$. | -3 | 100 |
30,343 | Find the value of $a_0 + a_1 + a_2 + \cdots + a_6$ given that $(2-x)^7 = a_0 + a_1(1+x)^2 + \cdots + a_7(1+x)^7$. | 129 | 50 |
30,344 | Xiaofan checked the step count on the smartwatch app before going out and found that the step count was a two-digit number. After walking downstairs, he found that the tens digit and the units digit had swapped. When he reached the entrance of the residential area, he found that there was an extra $1$ between the two digits he saw after walking downstairs. He walked a total of $586$ steps from leaving home to the entrance of the residential area. What was the step count he saw when he left home? | 26 | 0 |
30,345 | The polynomial $-400x^5+2660x^4-3602x^3+1510x^2+18x-90$ has five rational roots. Suppose you guess a rational number which could possibly be a root (according to the rational root theorem). What is the probability that it actually is a root? | 1/36 | 3.90625 |
30,346 | Calculate the winning rate per game;
List all possible outcomes;
Calculate the probability of satisfying the condition "$a+b+c+d \leqslant 2$". | \dfrac{11}{16} | 10.15625 |
30,347 | Let $r(x)$ have a domain $\{0,1,2,3\}$ and a range $\{1,3,5,7\}$. Let $s(x)$ be defined on the domain $\{1,2,3,4,5,6\}$ with the function rule $s(x) = 2x + 1$. Determine the sum of all possible values of $s(r(x))$ where $r(x)$ outputs only odd numbers. | 21 | 5.46875 |
30,348 | On a busy afternoon, David decides to drink a cup of water every 20 minutes to stay hydrated. Assuming he maintains this pace, how many cups of water does David drink in 3 hours and 45 minutes? | 11.25 | 0 |
30,349 | Given that the domain of the function $f(x)$ is $\mathbb{R}$, if $f(x+1)$ and $f(x-1)$ are both odd functions, then the function $y=f(x)$ has at least \_\_\_\_\_ zeros in the interval $[0,100]$. | 50 | 20.3125 |
30,350 | Given that the length of the arc of a sector is $\pi$ and the radius is 3, the radian measure of the central angle of the sector is ______, and the area of the sector is ______. | \frac{3\pi}{2} | 92.96875 |
30,351 | Vasya wrote a set of distinct natural numbers on the board, each of which does not exceed 2023. It turned out that for any two written numbers \(a\) and \(b\), the number \(a + b\) is not divisible by the number \(a - b\). What is the maximum number of numbers Vasya might have written? | 675 | 6.25 |
30,352 | In triangle $XYZ$, we have $\angle X = 90^\circ$, $YZ = 20$, and $\tan Z = 3 \sin Y$. Calculate $XY$.
--- | \frac{40\sqrt{2}}{3} | 3.90625 |
30,353 | Two right triangles share a side as follows: Triangle ABC and triangle ABD have AB as their common side. AB = 8 units, AC = 12 units, and BD = 8 units. There is a rectangle BCEF where point E is on line segment BD and point F is directly above E such that CF is parallel to AB. What is the area of triangle ACF? | 24 | 4.6875 |
30,354 | A number is considered a visible factor number if it is divisible by each of its non-zero digits. For example, 204 is divisible by 2 and 4 and is therefore a visible factor number. Determine how many visible factor numbers exist from 200 to 250, inclusive. | 16 | 6.25 |
30,355 | An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, find the ratio of the shorter side to the longer side of this frame. | 2:3 | 0 |
30,356 | In trapezoid $PQRS$ with $PQ$ parallel to $RS$, the diagonals $PR$ and $QS$ intersect at $T$. If the area of triangle $PQT$ is 75 square units, and the area of triangle $PST$ is 45 square units, what is the area of trapezoid $PQRS$? | 192 | 15.625 |
30,357 | Convex quadrilateral $ABCD$ has $AB = 10$ and $CD = 15$. Diagonals $AC$ and $BD$ intersect at $E$, where $AC = 18$, and $\triangle AED$ and $\triangle BEC$ have equal perimeters. Calculate the length of $AE$.
**A)** $6$
**B)** $7.2$
**C)** $7.5$
**D)** $9$
**E)** $10$ | 7.2 | 39.84375 |
30,358 | Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | 24 | 70.3125 |
30,359 | For each of the possible outcomes of rolling three coins (TTT, THT, TTH, HHT, HTH, HHH), a fair die is rolled for each two heads that appear together. What is the probability that the sum of the die rolls is odd? | \frac{1}{4} | 43.75 |
30,360 | Given the set of vectors \(\mathbf{v}\) such that
\[
\mathbf{v} \cdot \mathbf{v} = \mathbf{v} \cdot \begin{pmatrix} 4 \\ -16 \\ 32 \end{pmatrix}
\]
determine the volume of the solid formed in space. | 7776\pi | 92.96875 |
30,361 | Yan is between his home and the library. To get to the library, he can either walk directly to the library or walk home and then ride his bicycle to the library. He rides 5 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the library? | \frac{2}{3} | 74.21875 |
30,362 | Given a set of data: $2$, $x$, $4$, $6$, $10$, with an average value of $5$, find the standard deviation of this data set. | 2\sqrt{2} | 10.9375 |
30,363 | In the polar coordinate system, the equation of curve $C$ is $\rho^{2}\cos 2\theta=9$. Point $P(2 \sqrt {3}, \frac {\pi}{6})$ is given. Establish a Cartesian coordinate system with the pole $O$ as the origin and the positive half-axis of the $x$-axis as the polar axis.
$(1)$ Find the parametric equation of line $OP$ and the Cartesian equation of curve $C$;
$(2)$ If line $OP$ intersects curve $C$ at points $A$ and $B$, find the value of $\frac {1}{|PA|}+ \frac {1}{|PB|}$. | \sqrt {2} | 0 |
30,364 | In right triangle $XYZ$ with $\angle Z = 90^\circ$, the lengths of sides $XY$ and $XZ$ are 8 and 3 respectively. Find $\cos Y$. | \frac{3}{8} | 50 |
30,365 | Seventy-five percent of a ship's passengers are women, and fifteen percent of those women are in first class. What is the number of women in first class if the ship is carrying 300 passengers? | 34 | 95.3125 |
30,366 | Given $f(α) = \frac{\sin(2π-α)\cos(π+α)\cos\left(\frac{π}{2}+α\right)\cos\left(\frac{11π}{2}-α\right)}{2\sin(3π + α)\sin(-π - α)\sin\left(\frac{9π}{2} + α\right)}$.
(1) Simplify $f(α)$;
(2) If $α = -\frac{25}{4}π$, find the value of $f(α)$. | -\frac{\sqrt{2}}{4} | 48.4375 |
30,367 | Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length? | 1236 | 0 |
30,368 | Given the random variable $ξ∼B\left(5,0.5\right)$, and $η=5ξ$, find the respective values of $Eη$ and $Dη$. | \frac{125}{4} | 0 |
30,369 | A gambler starts with $100. During a series of 4 rounds, they bet each time one-third of the amount they have. They win twice and lose twice, but now the wins are twice the staked amount, and losses mean losing the staked amount. Determine the final amount of money the gambler has, assuming the wins and losses happen in an arbitrary fixed order.
A) $\frac{8000}{81}$
B) $\frac{5000}{81}$
C) $\frac{7500}{81}$
D) $\frac{7200}{81}$ | \frac{8000}{81} | 23.4375 |
30,370 | Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters A, B, C, and D — some of these letters may not appear in the sequence — where A is never immediately followed by B or D, B is never immediately followed by C, C is never immediately followed by A, and D is never immediately followed by A. How many eight-letter good words are there? | 512 | 0 |
30,371 | Define the function \(f(n)\) on the positive integers such that \(f(f(n)) = 3n\) and \(f(3n + 1) = 3n + 2\) for all positive integers \(n\). Find \(f(729)\). | 729 | 0.78125 |
30,372 | What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? | 16.67\% | 89.0625 |
30,373 | Positive integers \( d, e, \) and \( f \) are chosen such that \( d < e < f \), and the system of equations
\[ 2x + y = 2010 \quad \text{and} \quad y = |x-d| + |x-e| + |x-f| \]
has exactly one solution. What is the minimum value of \( f \)? | 1006 | 10.9375 |
30,374 | Find the number of diagonals of a polygon with 150 sides and determine if 9900 represents half of this diagonal count. | 11025 | 31.25 |
30,375 | The updated stem-and-leaf plot shows the duration of rides for each of the $21$ top-rated roller coasters. Each entry in the plot represents ride time, where, for example, $3 \ 05$ means $3$ minutes, $5$ seconds. Convert this time to seconds to find the median of the data set.
\begin{tabular}{c|cccccc}
0&28&28&50&55&&\\
1&00&02&10&&&\\
2&20&25&35&43&45&50\\
3&00&00&00&05&30&36\\
4&00&10&15&20&&\\
\end{tabular} | 163 | 26.5625 |
30,376 | The states of Sunshine and Prairie have adopted new license plate configurations. Sunshine license plates are formatted "LDDLDL" where L denotes a letter and D denotes a digit. Prairie license plates are formatted "LDDLDD". Assuming all 10 digits and 26 letters are equally likely to appear in their respective positions, how many more license plates can Sunshine issue compared to Prairie? | 10816000 | 7.8125 |
30,377 | The volume of a regular octagonal prism is 8 cubic meters, and its height is 2.2 meters. Find the lateral surface area of the prism. | 16 \sqrt{2.2 (\sqrt{2} - 1)} | 0 |
30,378 | Two types of shapes composed of unit squares, each with an area of 3, are placed in an $8 \times 14$ rectangular grid. It is required that there are no common points between any two shapes. What is the maximum number of these two types of shapes that can be placed in the $8 \times 14$ rectangular grid? | 16 | 0.78125 |
30,379 | Given an equilateral triangle $ABC$ with a circle of radius $3$ tangent to line $AB$ at $B$ and to line $AC$ at $C$, find the area of the circle that passes through vertices $A$, $B$, and $C$. | 36\pi | 50.78125 |
30,380 | A river flows at a constant speed. Piers A and B are located upstream and downstream respectively, with a distance of 200 kilometers between them. Two boats, A and B, depart simultaneously from piers A and B, traveling towards each other. After meeting, they continue to their respective destinations, immediately return, and meet again for the second time. If the time interval between the two meetings is 4 hours, and the still water speeds of boats A and B are 36 km/h and 64 km/h respectively, what is the speed of the current in km/h? | 14 | 1.5625 |
30,381 | In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_1$ is $\begin{cases} x=t^{2} \\ y=2t \end{cases}$ (where $t$ is the parameter), and in the polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C_2$ is $\rho=5\cos \theta$.
$(1)$ Write the polar equation of curve $C_1$ and the Cartesian coordinate equation of curve $C_2$;
$(2)$ Let the intersection of curves $C_1$ and $C_2$ in the first quadrant be point $A$, and point $B$ is on curve $C_1$ with $\angle AOB= \frac {\pi}{2}$, find the area of $\triangle AOB$. | 20 | 14.0625 |
30,382 | Let $p_1,p_2,p_3,p_4$ be four distinct primes, and let $1=d_1<d_2<\ldots<d_{16}=n$ be the divisors of $n=p_1p_2p_3p_4$ . Determine all $n<2001$ with the property that $d_9-d_8=22$ . | 1995 | 1.5625 |
30,383 | The midpoints of the sides of a parallelogram with area $P$ are joined to form a smaller parallelogram inside it. What is the ratio of the area of the smaller parallelogram to the area of the original parallelogram? Express your answer as a common fraction. | \frac{1}{4} | 54.6875 |
30,384 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a} \cdot \overrightarrow{b} = -\sqrt{3}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{5\pi}{6} | 92.1875 |
30,385 | Given that $\frac{3\pi}{4} < \alpha < \pi$, find the values for the following:
1. $\tan\alpha$ given $\tan\alpha + \frac{1}{\tan\alpha} = -\frac{10}{3}$.
2. The value of $$\frac{5\sin^2\left(\frac{\alpha}{2}\right) + 8\sin\left(\frac{\alpha}{2}\right)\cos\left(\frac{\alpha}{2}\right) + 11\cos^2\left(\frac{\alpha}{2}\right) - 8}{\sqrt{2}\sin\left(\alpha - \frac{\pi}{4}\right)}.$$ | -\frac{5}{4} | 5.46875 |
30,386 | The solutions to the equation $x(5x+2) = 6(5x+2)$ are ___. | -\frac{2}{5} | 0 |
30,387 | Given that $α \in ( \frac{5}{4}π, \frac{3}{2}π)$, and it satisfies $\tan α + \frac{1}{\tan α} = 8$, find the value of $\sin α \cos α$ and $\sin α - \cos α$. | - \frac{\sqrt{3}}{2} | 13.28125 |
30,388 | Given that vertex $E$ of right $\triangle ABE$ (with $\angle ABE = 90^\circ$) is inside square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and hypotenuse $AE$, find the area of $\triangle ABF$ where the length of $AB$ is $\sqrt{2}$. | \frac{1}{2} | 57.8125 |
30,389 | Given the sequence $\{a_n\}$ satisfies $a_1=\frac{1}{2}$, $a_{n+1}=1-\frac{1}{a_n} (n\in N^*)$, find the maximum positive integer $k$ such that $a_1+a_2+\cdots +a_k < 100$. | 199 | 44.53125 |
30,390 | Find the number of pairs $(n,C)$ of positive integers such that $C\leq 100$ and $n^2+n+C$ is a perfect square. | 180 | 100 |
30,391 | A projectile is launched with an initial velocity $u$ at an angle $\phi$ from the ground. The trajectory can be described by the parametric equations:
\[
x = ut \cos \phi, \quad y = ut \sin \phi - \frac{1}{2} g t^2,
\]
where $t$ is the time, $g$ is the acceleration due to gravity, and the angle $\phi$ varies over $0^\circ \le \phi \le 90^\circ$. The highest points of each trajectory are plotted as $\phi$ changes. Determine the area of the curve traced by these highest points, expressed as $d \cdot \frac{u^4}{g^2}$. | \frac{\pi}{8} | 1.5625 |
30,392 | When $1 + 9 + 9^2 + \cdots + 9^{2023}$ is divided by $500$, a remainder of $M$ is obtained. Determine the value of $M$. | 45 | 5.46875 |
30,393 | Compute $\arccos (\cos 3).$ All functions are in radians. | 3 - 2\pi | 0.78125 |
30,394 | Barbara, Edward, Abhinav, and Alex took turns writing this test. Working alone, they could finish it in $10$ , $9$ , $11$ , and $12$ days, respectively. If only one person works on the test per day, and nobody works on it unless everyone else has spent at least as many days working on it, how many days (an integer) did it take to write this test? | 12 | 21.09375 |
30,395 | Let $w$ and $z$ be complex numbers such that $|w+z|=2$ and $|w^2+z^2|=18$. Find the smallest possible value of $|w^3+z^3|$. | 50 | 58.59375 |
30,396 | 1. Given that the terminal side of angle $\alpha$ passes through point $P(4, -3)$, find the value of $2\sin\alpha + \cos\alpha$.
2. Given that the terminal side of angle $\alpha$ passes through point $P(4a, -3a)$ ($a \neq 0$), find the value of $2\sin\alpha + \cos\alpha$.
3. Given that the ratio of the distance from a point $P$ on the terminal side of angle $\alpha$ to the x-axis and to the y-axis is 3:4, find the value of $2\sin\alpha + \cos\alpha$. | -\frac{2}{5} | 27.34375 |
30,397 | Given that the product of two positive integers $a$ and $b$ is $161$ after reversing the digits of the two-digit number $a$, find the correct value of the product of $a$ and $b$. | 224 | 35.15625 |
30,398 | The solutions to the equations $z^2 = 9 + 9\sqrt{7}i$ and $z^2 = 5 + 5\sqrt{2}i$, where $i = \sqrt{-1}$, form the vertices of a parallelogram in the complex plane. Determine the area of this parallelogram, which can be written in the form $p\sqrt{q} - r\sqrt{s}$, where $p, q, r, s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number.
A) $\frac{2}{3}\sqrt{35} - \frac{2}{3}\sqrt{7}$
B) $\frac{3}{4}\sqrt{35} - \frac{3}{4}\sqrt{15}$
C) $\frac{1}{2}\sqrt{21} - 3\sqrt{3}$
D) $\frac{3}{8}\sqrt{15} - \frac{3}{8}\sqrt{7}$ | \frac{3}{4}\sqrt{35} - \frac{3}{4}\sqrt{15} | 14.0625 |
30,399 | The random variable $X$ follows a normal distribution $N(1, 4)$, where the mean $\mu = 1$ and the variance $\sigma^2 = 4$. Given that $P(X \geq 2) = 0.2$, calculate the probability $P(0 \leq X \leq 1)$. | 0.3 | 26.5625 |
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