Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
16,500 | Five years ago, Tim was three times as old as his sister Sarah, and three years before that, Tim was five times as old as Sarah. Determine the number of years it will take for the ratio of their ages to be 3 : 2. | 13 | 74.21875 |
16,501 | The minimum positive period of $y=\tan(4x+ \frac{\pi}{3})$ is $\pi$. | \frac{\pi}{4} | 99.21875 |
16,502 | Which one satisfies $n^{29} \equiv 7 \pmod {65}$? | 37 | 3.90625 |
16,503 | In triangle ABC, $a-b=4$, $a+c=2b$, and the largest angle is $120^\circ$. Find the perimeter of this triangle. | 30 | 71.09375 |
16,504 | In triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the interior angles $A$, $B$, and $C$, respectively, and $a\cos C+\left(2b+c\right)\cos A=0$.
$(1)$ Find the value of angle $A$.
$(2)$ If $D$ is the midpoint of segment $BC$ and $AD=\frac{7}{2}$, $AC=3$, find the area of triangle $ABC$... | 6\sqrt{3} | 59.375 |
16,505 | Regular hexagon $ABCDEF$ has an area of $n$. Let $m$ be the area of triangle $ACE$. What is $\tfrac{m}{n}?$
A) $\frac{1}{2}$
B) $\frac{2}{3}$
C) $\frac{3}{4}$
D) $\frac{1}{3}$
E) $\frac{3}{2}$ | \frac{2}{3} | 6.25 |
16,506 | Given the function $f(x)=\cos x\cdot \sin \left(x+\frac{\pi }{3}\right)-\sqrt{3}\cos ^{2}x+\frac{\sqrt{3}}{4}$, where $x\in R$.
(1) Find the smallest positive period of $f(x)$;
(2) Find the maximum and minimum values of $f(x)$ on the closed interval $\left[-\frac{\pi }{4},\frac{\pi }{4}\right]$. | -\frac{1}{2} | 64.84375 |
16,507 | From a batch of parts, 50 are drawn, and then 40 out of these 50 are inspected. It is found that there are 38 qualified products. Calculate the pass rate of this batch of products as a percentage. | 95\% | 17.1875 |
16,508 | For each integer \( n \geq 2 \), let \( A(n) \) be the area of the region in the coordinate plane defined by the inequalities \( 1 \leq x \leq n \) and \( 0 \leq y \leq x \left\lfloor \log_2{x} \right\rfloor \), where \( \left\lfloor \log_2{x} \right\rfloor \) is the greatest integer not exceeding \( \log_2{x} \). Find... | 99 | 73.4375 |
16,509 | Let \( x[n] \) denote \( x \) raised to the power of \( x \), repeated \( n \) times. What is the minimum value of \( n \) such that \( 9[9] < 3[n] \)?
(For example, \( 3[2] = 3^3 = 27 \); \( 2[3] = 2^{2^2} = 16 \).) | 10 | 11.71875 |
16,510 | Determine the largest constant $K\geq 0$ such that $$ \frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2 $$ holds for all positive real numbers $a,b,c$ such that $ab+bc+ca=abc$ .
*Proposed by Orif Ibrogimov (Czech Technical Uni... | 18 | 55.46875 |
16,511 | Suppose $x$ and $y$ are positive real numbers such that $x^2 - 2xy + 3y^2 = 9$. Find the maximum possible value of $x^2 + 2xy + 3y^2$. | 18 + 9\sqrt{3} | 5.46875 |
16,512 | Given that $\alpha$ is an acute angle and satisfies $\cos(\alpha+\frac{\pi}{4})=\frac{\sqrt{3}}{3}$.
$(1)$ Find the value of $\sin(\alpha+\frac{7\pi}{12})$.
$(2)$ Find the value of $\cos(2\alpha+\frac{\pi}{6})$. | \frac{2\sqrt{6}-1}{6} | 69.53125 |
16,513 | Evaluate $|\omega^2 + 7\omega + 40|$ if $\omega = 4 + 3i$. | 15\sqrt{34} | 84.375 |
16,514 | Given that the base edge length of a regular square pyramid is $2$, and the side edge length is $\sqrt{6}$, determine the volume of the pyramid. | \frac{8}{3} | 95.3125 |
16,515 | Given that z and w are complex numbers with a modulus of 1, and 1 ≤ |z + w| ≤ √2, find the minimum value of |z - w|. | \sqrt{2} | 60.15625 |
16,516 | Let be the set $ \mathcal{C} =\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f''\bigg|_{[0,1]} \right.\quad\exists x_1,x_2\in [0,1]\quad x_1\neq x_2\wedge \left( f\left(
x_1 \right) = f\left( x_2 \right) =0\vee f\left(
x_1 \right) = f'\left( x_1 \right) = 0\right) \wedge f''<1 \right\} , $ and $ f^*\in\ma... | 1/12 | 57.8125 |
16,517 | Given that there are a total of 25 students in an art class, including 10 boys and 15 girls. In the final art exhibition, the average number of artworks per boy is 25 with a variance of 1, and the average number of artworks per girl is 30 with a variance of 2. Find the variance of the number of artworks per student amo... | \frac{38}{5} | 0 |
16,518 | Among the non-empty subsets of the set \( A = \{1, 2, \cdots, 10\} \), how many subsets have the sum of their elements being a multiple of 10? | 103 | 75 |
16,519 | The arithmetic mean of eleven numbers is 32. If three numbers $x$, $y$, and $z$ are added to this list, the mean of the fourteen-member list becomes 45. What is the mean of $x$, $y$, and $z$? | \frac{278}{3} | 50 |
16,520 | A zookeeper distributes a pile of peaches among several monkeys. If each monkey gets 6 peaches, there are 57 peaches left. If each monkey gets 9 peaches, 5 monkeys get none, and one monkey gets only 3 peaches. How many peaches are there in total? | 273 | 49.21875 |
16,521 | Find $537_{8} - 261_{8}$. Verify by adding the result to $261_{8}$ and checking if it matches $537_{8}$. | 256_8 | 79.6875 |
16,522 | Find the least positive integer $ a$ such that $ 2001$ divides $ 55^n\plus{}a \cdot 32^n$ for some odd $ n$ . | 436 | 48.4375 |
16,523 | Aaron takes a square sheet of paper, with one corner labeled $A$ . Point $P$ is chosen at random inside of the square and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$ . Let $p$ be the probability that the remaining piece is... | 25 | 32.03125 |
16,524 | A lattice point is a point whose coordinates are integers. How many lattice points are on the boundary or inside the region bounded by \( y = |x| \) and \( y = -x^2 + 8 \)? | 33 | 2.34375 |
16,525 | In the Cartesian coordinate system, with the origin as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. The polar equation of curve C is $\rho - 2\cos\theta - 6\sin\theta + \frac{1}{\rho} = 0$, and the parametric equation of line l is $\begin{cases} x=3+ \fr... | 2\sqrt{6} | 71.875 |
16,526 | Given that real numbers $a$ and $b$ satisfy $\frac{1}{a} + \frac{2}{b} = \sqrt{ab}$, calculate the minimum value of $ab$. | 2\sqrt{2} | 56.25 |
16,527 | The Yellers are coached by Coach Loud. The Yellers have 15 players, but three of them, Max, Rex, and Tex, refuse to play together in any combination. How many starting lineups (of 5 players) can Coach Loud make, if the starting lineup can't contain any two of Max, Rex, and Tex together? | 2277 | 35.15625 |
16,528 | A natural number $n$ is said to be $good$ if $n$ is the sum or $r$ consecutive positive integers, for some $r \geq 2 $ . Find the number of good numbers in the set $\{1,2 \dots , 100\}$ . | 93 | 96.09375 |
16,529 | Given the function $f(x) = x^3 - 3x$,
(Ⅰ) Find the intervals of monotonicity for $f(x)$;
(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $[-3,2]$. | -18 | 76.5625 |
16,530 | Triangle $ABC$ has vertices $A(0, 10)$, $B(3, 0)$, $C(9, 0)$. A horizontal line with equation $y=s$ intersects line segment $\overline{AB}$ at $P$ and line segment $\overline{AC}$ at $Q$, forming $\triangle APQ$ with area 18. Compute $s$. | 10 - 2\sqrt{15} | 64.84375 |
16,531 | Given the function $f(x)= \begin{cases} 2x-10, & x\leqslant 7 \\ \frac {1}{f(x-2)}, & x > 7 \end{cases}$, and the sequence ${a_{n}}={f(n)}$ where $n\in\mathbb{N}^{*}$, find the sum of the first 50 terms of the sequence ${a_{n}}$. | \frac {225}{4} | 11.71875 |
16,532 | In $\triangle ABC$, the lengths of the sides opposite to angles A, B, and C are a, b, and c respectively. Given that a = 3, cosC = $- \frac{1}{15}$, and 5sin(B + C) = 3sin(A + C).
(1) Find the length of side c.
(2) Find the value of sin(B - $\frac{\pi}{3}$). | \frac{2\sqrt{14} - 5\sqrt{3}}{18} | 50.78125 |
16,533 | The diagram shows a polygon made by removing six $2\times 2$ squares from the sides of an $8\times 12$ rectangle. Find the perimeter of this polygon.
 | 52 | 13.28125 |
16,534 | A frustum of a cone has a lower base radius of 8 inches, an upper base radius of 4 inches, and a height of 5 inches. Calculate its lateral surface area and total surface area. | (80 + 12\sqrt{41})\pi | 0 |
16,535 | In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $b\sin C + c\sin B = 4a\sin B\sin C$ and $b^2 + c^2 - a^2 = 8$. Find the area of $\triangle ABC$. | \frac{2\sqrt{3}}{3} | 35.9375 |
16,536 | An equilateral triangle ABC has a side length of 4. A right isosceles triangle DBE, where $DB=EB=1$ and angle $D\hat{B}E = 90^\circ$, is cut from triangle ABC. Calculate the perimeter of the remaining quadrilateral. | 10 + \sqrt{2} | 32.8125 |
16,537 | In $\triangle ABC$, given $A(1,4)$, $B(4,1)$, $C(0,-4)$, find the minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB} + \overrightarrow{PB} \cdot \overrightarrow{PC} + \overrightarrow{PC} \cdot \overrightarrow{PA}$. | - \dfrac {62}{3} | 25.78125 |
16,538 | Around a circular table, there are 18 girls seated, 11 dressed in blue and 7 dressed in red. Each girl is asked if the girl to her right is dressed in blue, and each one responds with either yes or no. It is known that a girl tells the truth only when both of her neighbors, the one to her right and the one to her left,... | 11 | 39.84375 |
16,539 | A sequence \( b_1, b_2, b_3, \dots \) is defined recursively by \( b_1 = 2, b_2 = 2, \) and for \( k \ge 3, \)
\[
b_k = \frac{1}{2} b_{k - 1} + \frac{1}{3} b_{k - 2}.
\]
Evaluate \( b_1 + b_2 + b_3 + \dotsb. \) | 18 | 26.5625 |
16,540 | Let $x$, $y\in \mathbb{R}$, vectors $\overrightarrow{a}=(2,x)$, $\overrightarrow{b}=(y,1)$, $\overrightarrow{c}=(3,-3)$, and $\overrightarrow{a}⊥\overrightarrow{b}$, $\overrightarrow{b}∥\overrightarrow{c}$.
$(1)$ Find $|\overrightarrow{a}+\overrightarrow{b}|$;
$(2)$ Find the cosine value of the angle between vector... | \frac{3}{5} | 92.96875 |
16,541 | If each of the four numbers $3, 4, 6,$ and $7$ replaces a $\square$, what is the largest possible sum of the fractions shown? | $\frac{23}{6}$ | 0 |
16,542 | Calculate the angle $\theta$ for the sum
\[e^{3\pi i/60} + e^{11\pi i/60} + e^{19\pi i/60} + e^{27\pi i/60} + e^{35\pi i/60} + e^{43\pi i/60} + e^{51\pi i/60} + e^{59\pi i/60}\]
when expressed in the form of $r e^{i\theta}$, where $0 \leq \theta < 2\pi$. | \dfrac{31\pi}{60} | 0.78125 |
16,543 | Given that $a$, $b$, $c$ form an arithmetic sequence in triangle $ABC$, $\angle B=30^{\circ}$, and the area of $\triangle ABC$ is $\frac{1}{2}$, determine the value of $b$. | \frac{3+ \sqrt{3}}{3} | 0 |
16,544 | In triangle $ABC$, the sides opposite to angles A, B, and C are denoted by $a$, $b$, and $c$ respectively, with $A+C=\frac{2\pi}{3}$ and $b=1$.
(1) If we let angle A be $x$ and define $f(x)=a+c$, find the range of $f(x)$ when triangle $ABC$ is an acute triangle;
(2) Determine the maximum area of triangle $ABC$. | \frac{\sqrt{3}}{4} | 77.34375 |
16,545 | In an equilateral triangle $ABC$ with side length of 10, a similar process of division by midpoints and shading of one of these triangles occurs. If this dividing and shading process is repeated indefinitely, and the first triangle to be shaded is the triangle involving vertex $A$, the total shaded area will converge t... | \frac{25\sqrt{3}}{3} | 75.78125 |
16,546 | Given the variables $a$ and $b$ satisfying $b=-\frac{1}{2}a^2+3\ln(a)$ (with $a>0$), if point $Q(m,n)$ is on the line $y=2x+\frac{1}{2}$, calculate the minimum value of $(a-m)^2+(b-n)^2$. | \frac{9}{5} | 1.5625 |
16,547 | Let $(x,y)$ satisfy the constraints $\begin{cases} 8x - y - 4 \leqslant 0 \\ x + y + 1 \geqslant 0 \\ y - 4x \leqslant 0 \end{cases}$, and the maximum value of the objective function $z = ax + by (a > 0, b > 0)$ is $2$. Find the minimum value of $\frac{1}{a} + \frac{1}{b}$. | \frac{9}{2} | 42.96875 |
16,548 | There are 8 young people, among whom 5 are capable of doing English translation work, and 4 are capable of doing computer software design work (including one person who is capable of doing both tasks). Now, 5 young people are to be selected to undertake a task, with 3 people doing English translation work and 2 people ... | 42 | 14.84375 |
16,549 | The integer points $(x, y)$ in the first quadrant satisfy $x + y > 8$ and $x \leq y \leq 8$. Determine the number of such integer points $(x, y)$. | 20 | 12.5 |
16,550 | The sum of an infinite geometric series is 64 times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | \frac{1}{2} | 45.3125 |
16,551 | Given the function g defined on the set of positive rational numbers by g(x \cdot y) = g(x) + g(y) for all positive rational numbers x and y, and g(n) = n^2 for every prime number n, calculate g(x) for x = \frac{25}{21}. | -8 | 92.1875 |
16,552 | In a large library storage room, there are $1584$ boxes, each containing $45$ books. The library dean asks for these books to be repacked so that each new box contains $47$ books. How many books will be left over after repacking the books into as many full boxes of $47$ books each as possible? | 28 | 0 |
16,553 | In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle? | 50 | 74.21875 |
16,554 | Given real numbers $x$ and $y$ satisfying $x^2+4y^2=4$, find the maximum value of $\frac {xy}{x+2y-2}$. | \frac {1+ \sqrt {2}}{2} | 0 |
16,555 | Given the function $f(x)= \begin{cases} 2x,& x > 0 \\ f(x+1),& x\leqslant 0 \\ \end{cases}$, find $f(- \frac {4}{3})=$\_\_\_\_\_\_ and the maximum value of the real number $x_{0}$ that satisfies $f(f(x_{0}))=2$. | \frac{1}{2} | 75.78125 |
16,556 | The greatest common divisor of two integers is $(x+3)$ and their least common multiple is $x(x+3)$, where $x$ is a positive integer. If one of the integers is 30, what is the smallest possible value of the other one? | 162 | 3.125 |
16,557 | Jane Doe invested some amount of money into a savings account and mutual funds. The total amount she invested was \$320,000. If she invested 6 times as much in mutual funds as she did in the savings account, what was her total investment in mutual funds? | 274,285.74 | 0.78125 |
16,558 | On one particular Wednesday, Jack worked \( t-2 \) hours and earned \( 3t-2 \) dollars per hour. His coworker Bob worked 1.5 times more hours than Jack but earned \( 2t-7 \) dollars per hour less than Jack. After paying a fixed tax of $10 each, they both netted the same amount of earnings. Determine the value of \( t \... | \frac{19}{3} | 53.90625 |
16,559 | Define $n!!$ as in the original problem. Evaluate $\sum_{i=1}^{5} \frac{(2i-1)!!}{(2i)!!}$, and express the result as a fraction in lowest terms. | \frac{437}{256} | 44.53125 |
16,560 | Jessica enjoys vanilla ice cream, and she visits her favorite ice cream shop every day for a week. However, sometimes the shop serves only chocolate ice cream, so each day the shop has a 3/4 chance of serving vanilla ice cream. What is the probability that the shop serves vanilla ice cream exactly 3 out of the 7 days s... | \frac{945}{16384} | 16.40625 |
16,561 | The national security agency's wiretap recorded a conversation between two spies and found that on a 30-minute tape, starting from the 30-second mark, there was a 10-second segment of conversation containing information about the spies' criminal activities. Later, it was discovered that part of this conversation was er... | \frac{1}{45} | 16.40625 |
16,562 | Marsha now has two numbers, $a$ and $b$. When she divides $a$ by $60$ she gets a remainder of $58$. When she divides $b$ by $90$ she gets a remainder of $84$. What remainder does she get when she divides $a+b$ by $30$? | 22 | 85.9375 |
16,563 | Given the function $f(x) = |x^2 + bx|$ ($b \in \mathbb{R}$), when $x \in [0, 1]$, find the minimum value of the maximum value of $f(x)$. | 3-2\sqrt{2} | 15.625 |
16,564 | Two spinners are divided into fifths and sixths, respectively. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be an even number? The numbers on the first spinner are 3, 5, 6, 7, and 9. The numbers on the second spinner are 2, 4, 6, 8, 9, and 11. | \frac{11}{15} | 92.96875 |
16,565 | For a given list of three numbers, the operation "changesum" replaces each number in the list with the sum of the other two. For example, applying "changesum" to \(3,11,7\) gives \(18,10,14\). Arav starts with the list \(20,2,3\) and applies the operation "changesum" 2023 times. What is the largest difference between t... | 18 | 95.3125 |
16,566 | Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\heartsuit$ and the second card dealt is a $\clubsuit$? | \frac{13}{204} | 100 |
16,567 | If the price of a stamp is 50 cents, what is the maximum number of stamps that could be purchased with $50? Furthermore, if a customer buys more than 80 stamps, they receive a discount of 5 cents per stamp. How many stamps would then be purchased at maximum? | 111 | 86.71875 |
16,568 | Point P lies on the curve represented by the equation $$\sqrt {(x-5)^{2}+y^{2}}- \sqrt {(x+5)^{2}+y^{2}}=6$$. If the y-coordinate of point P is 4, then its x-coordinate is ______. | x = -3\sqrt{2} | 7.03125 |
16,569 | What is the smallest possible area, in square units, of a right triangle with two sides measuring $7$ units and $8$ units? | \frac{7\sqrt{15}}{2} | 74.21875 |
16,570 | What is the sum of all two-digit positive integers whose squares end with the digits 25? | 495 | 14.84375 |
16,571 | Given the function $f(x)=x^3+ax^2+bx+16$ has an extremum of $10$ at $x=1$;
(1) Find the values of $a$ and $b$;
(2) Find the maximum and minimum values of $f(x)$ on the interval $[0,2]$. | 10 | 49.21875 |
16,572 | Given that $f(x)$ is a function defined on $[1,+\infty)$, and
$$
f(x)=\begin{cases}
1-|2x-3|, & 1\leqslant x < 2, \\
\frac{1}{2}f\left(\frac{1}{2}x\right), & x\geqslant 2,
\end{cases}
$$
then the number of zeros of the function $y=2xf(x)-3$ in the interval $(1,2015)$ is ______. | 11 | 25.78125 |
16,573 | Given $a\in\{1,3,5\}$ and $b\in\{2,4,8\}$, find the probability that the function $y=\log_{\frac{b}{a}}{\frac{1}{x}}$ is an increasing function. | \frac{1}{3} | 27.34375 |
16,574 | A national team needs to select 4 out of 6 sprinters to participate in the 4×100 m relay at the Asian Games. If sprinter A cannot run the first leg and sprinter B cannot run the fourth leg, there are a total of ______ ways to participate. | 252 | 93.75 |
16,575 | One of the eight faces of a hexagonal prism will be transformed into the base of a new pyramid. Calculate the maximum sum of the number of exterior faces, vertices, and edges of the resultant composite shape (combining the prism and pyramid) when the pyramid is added to each type of face of the prism. | 50 | 26.5625 |
16,576 | A positive two-digit number is odd and is a multiple of 9. The product of its digits is a perfect square. What is this two-digit number? | 99 | 9.375 |
16,577 | Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$ | 1011 | 65.625 |
16,578 | In $\triangle ABC$, the sides opposite to angles A, B, and C are $a$, $b$, and $c$ respectively. Given that $$bsin(C- \frac {π}{3})-csinB=0$$
(I) Find the value of angle C;
(II) If $a=4$, $c=2 \sqrt {7}$, find the area of $\triangle ABC$. | 2 \sqrt {3} | 0 |
16,579 | Calculate:<br/>$(1)\left(-12\right)-5+\left(-14\right)-\left(-39\right)$;<br/>$(2)-2^{2}\times 5-\left(-12\right)\div 4-4$. | -21 | 86.71875 |
16,580 | A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% | 71.875 |
16,581 | A green point and a purple point are chosen at random on the number line between 0 and 1. What is the probability that the purple point is greater than the green point but less than three times the green point? | \frac{1}{3} | 12.5 |
16,582 | Let $P$ be a point not on line $XY$, and $Q$ a point on line $XY$ such that $PQ \perp XY.$ Meanwhile, $R$ is a point on line $PY$ such that $XR \perp PY.$ If $XR = 3$, $PQ = 6$, and $XY = 7$, then what is the length of $PY?$ | 14 | 30.46875 |
16,583 | The number $m$ is a prime number between 30 and 50. If you divide $m$ by 12, the remainder is 7. What is the value of $m$? | 43 | 29.6875 |
16,584 | In $\triangle ABC$, if $\sin B= \sqrt {3}\sin A$, $BC= \sqrt {2}$, and $C= \frac {\pi}{6}$, then the height to side $AC$ is ______. | \frac { \sqrt {2}}{2} | 0 |
16,585 | Set A has 30 elements, and set B has 25 elements. Set C has 10 elements. Calculate the smallest possible number of elements in the union A ∪ B ∪ C. | 30 | 59.375 |
16,586 | In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive $x$-axis as the polar axis, the parametric equation of curve $C_1$ is
$$
\begin{cases}
x=2+ \sqrt {3}\cos \theta \\
y= \sqrt {3}\sin \theta
\end{cases}
(\theta \text{ is the parameter}),
$$
and the polar equation of curve $C_2$ is... | 2 \sqrt {2} | 0 |
16,587 | Larry now only likes numbers that end with two digits making the whole number divisible by 4. He has some favorite numbers like 120, 1156, and 504. How many different pairs of the last two digits are possible in numbers that Larry likes? | 25 | 45.3125 |
16,588 | What is the value of \(a + b + c + d\) if
$$
\begin{gathered}
6a + 2b = 3848 \\
6c + 3d = 4410 \\
a + 3b + 2d = 3080
\end{gathered}
$$ | 1986 | 34.375 |
16,589 | The graph of the function $f(x)=\sin (\omega x+\frac{π}{3})$ ($\omega\ \ \gt 0$) is shifted to the left by $\frac{π}{2}$ units to obtain the curve $C$. If $C$ is symmetric about the $y$-axis, then find the minimum value of $\omega$. | \frac{1}{3} | 66.40625 |
16,590 | Let $\{a_n\}$ be an arithmetic sequence. If we select any 4 different numbers from $\{a_1, a_2, a_3, \ldots, a_{10}\}$ such that these 4 numbers still form an arithmetic sequence, then the maximum number of such arithmetic sequences is $\boxed{24}$. | 24 | 64.84375 |
16,591 | John has 8 green marbles and 7 purple marbles. He chooses a marble at random, records its color, and then does not put the marble back. He repeats this process 6 times. What is the probability that he chooses exactly three green marbles? | \frac{392}{1001} | 0 |
16,592 | The sides of rhombus \( EFGH \) are the hypotenuses of the isosceles right triangles \( EAF, FDG, GCH, \) and \( HBE \), and all these triangles have common interior points with the rhombus \( EFGH \). The sum of the areas of quadrilateral \( ABCD \) and rhombus \( EFGH \) is 12. Find \( GH \). | 2\sqrt{3} | 20.3125 |
16,593 | The given arithmetic sequences $\{a_{n}\}$ and $\{b_{n}\}$ have respective sums of the first $n$ terms, denoted by $S_{n}$ and $T_{n}$. The ratio $\frac{S_{n}}{T_{n}} = \frac{3n + 1}{n + 3}$. Determine the value of $\frac{a_{2} + a_{20}}{b_{7} + b_{15}}$. | \frac{8}{3} | 53.125 |
16,594 | How many distinct arrangements of the letters in the word "balloon" are there? | 1260 | 45.3125 |
16,595 | Given $\tan 2\alpha= \frac {3}{4}$, $\alpha\in(- \frac {\pi}{2}, \frac {\pi}{2})$, $f(x)=\sin (x+\alpha)+\sin (\alpha-x)-2\sin \alpha$, and for any $x\in\mathbb{R}$, it always holds that $f(x)\geqslant 0$, find the value of $\sin (\alpha- \frac {\pi}{4})$. | - \frac {2 \sqrt {5}}{5} | 0 |
16,596 | Let $T = (2+i)^{20} - (2-i)^{20}$, where $i = \sqrt{-1}$. Find $|T|$. | 19531250 | 26.5625 |
16,597 | Given $\sin\theta + \cos\theta = \frac{3}{4}$, where $\theta$ is an angle of a triangle, find the value of $\sin\theta - \cos\theta$. | \frac{\sqrt{23}}{4} | 41.40625 |
16,598 | Let $\mathbf{v}$ be a vector such that
\[\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.\]
Find the smallest possible value of $\|\mathbf{v}\|$. | 10 - 2 \sqrt{5} | 43.75 |
16,599 | The following is the process of simplifying fractions by Xiaobai. Please read carefully and complete the corresponding tasks.
Solution: $(\frac{3x+4}{x^2-1}-\frac{2}{x-1})÷\frac{x+2}{x^2-2x+1}$
$=[\frac{3x+4}{(x+1)(x-1)}-\frac{2}{x-1}]÷\frac{x+2}{(x-1)^2}\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ... | \frac{1}{3} | 89.0625 |
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