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30,700 | Two positive integers $m$ and $n$ are both less than $500$ and $\text{lcm}(m,n) = (m-n)^2$ . What is the maximum possible value of $m+n$ ? | 840 | 83.59375 |
30,701 | On the display of a certain instrument, each indicator light can show different signals by lighting up in red, yellow, or blue. It is known that there are 6 indicator lights in a row, and each time 3 of them light up, with only 2 being adjacent, then the total number of different signals that can be displayed is ______. | 324 | 15.625 |
30,702 | The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence plus the square of the first term. What is the value of the first term which exceeds 10000? | 16384 | 64.84375 |
30,703 | Determine the total number of real solutions for the equation
\[
\frac{x}{50} = \sin x.
\] | 32 | 43.75 |
30,704 | $(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits. | 1944 | 0 |
30,705 | How many distinct lines pass through the point $(0, 2016)$ and intersect the parabola $y = x^2$ at two lattice points? (A lattice point is a point whose coordinates are integers.) | 36 | 8.59375 |
30,706 | Acute-angled $\triangle ABC$ is inscribed in a circle with center at $O$. The measures of arcs are $\stackrel \frown {AB} = 80^\circ$ and $\stackrel \frown {BC} = 100^\circ$. A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. Find the ratio of the magnitudes of $\angle OBE$ and $\angle BAC$. | 10 | 0 |
30,707 | Given a point P on the parabola $y^2=4x$, let the distance from point P to the directrix of this parabola be $d_1$, and the distance from point P to the line $x+2y-12=0$ be $d_2$. | \frac{11 \sqrt{5}}{5} | 0 |
30,708 | Let $a$ and $b$ be positive integers such that $2a - 9b + 18ab = 2018$ . Find $b - a$ . | 223 | 38.28125 |
30,709 | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5 \times 5$ square array of dots? | 100 | 18.75 |
30,710 | From the 20 natural numbers 1, 2, 3, ..., 20, if three numbers are randomly selected and their sum is an even number greater than 10, then there are $\boxed{\text{answer}}$ such sets of numbers. | 563 | 60.9375 |
30,711 | There is a rectangular field that measures $20\text{m}$ by $15\text{m}$ . Xiaoyu the butterfly is sitting at the perimeter of the field on one of the $20\text{m}$ sides such that he is $6\text{m}$ from a corner. He flies in a straight line to another point on the perimeter. His flying path splits the field into two parts with equal area. How far in meters did Xiaoyu fly? | 17 | 42.96875 |
30,712 | Given that three fair coins are tossed once and for each head, one fair die is rolled, with a second roll occurring if the first roll is 6, determine the probability that the sum of all die rolls is odd. | \frac{7}{16} | 38.28125 |
30,713 | Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x, y, \) and \( z \) such that \( x \mid y^{3} \), \( y \mid z^{3} \), and \( z \mid x^{3} \), it always holds that \( x y z \mid (x+y+z)^{n} \). | 13 | 15.625 |
30,714 | Binbin's height is 1.46 meters, his father is 0.32 meters taller than Binbin, and his mother's height is 1.5 meters.
(1) How tall is Binbin's father?
(2) How much shorter is Binbin's mother than his father? | 0.28 | 94.53125 |
30,715 | Given that $\frac{{\cos 2\alpha}}{{\sin(\alpha+\frac{\pi}{4})}}=\frac{4}{7}$, find the value of $\sin 2\alpha$. | \frac{41}{49} | 33.59375 |
30,716 | When $10^{95} - 95 - 2$ is expressed as a single whole number, calculate the sum of the digits. | 840 | 7.03125 |
30,717 | In how many ways every unit square of a $2018$ x $2018$ board can be colored in red or white such that number of red unit squares in any two rows are distinct and number of red squares in any two columns are distinct. | 2 * (2018!)^2 | 0 |
30,718 | Let \( m \) and \( n \) be positive integers such that \( m > n \). If the last three digits of \( 2012^m \) and \( 2012^n \) are identical, find the smallest possible value of \( m+n \). | 104 | 1.5625 |
30,719 | Given the function $f(x)=e^{ax}$, a line parallel to the $y$-axis is drawn through $A(a,0)$ and intersects the function $f(x)$ at point $P$. A tangent line to $f(x)$ at $P$ intersects the $x$-axis at point $B$. Find the minimum value of the area of $\triangle APB$. | \dfrac { \sqrt {2e}}{2} | 0 |
30,720 | Consider a hyperbola with the equation $x^2 - y^2 = 9$. A line passing through the left focus $F_1$ of the hyperbola intersects the left branch of the hyperbola at points $P$ and $Q$. Let $F_2$ be the right focus of the hyperbola. If the length of segment $PQ$ is 7, then calculate the perimeter of $\triangle F_2PQ$. | 26 | 40.625 |
30,721 | Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $(x^{2}- \dfrac {y^{2}}{24}=1)$, and $P$ is a point on the hyperbola, find the area of $\triangle F\_1PF\_2$. | 24 | 52.34375 |
30,722 | An investor placed $\$15,000$ in a three-month term deposit that offered a simple annual interest rate of $8\%$. After three months, the total value of the deposit was reinvested in another three-month term deposit. After these three months, the investment amounted to $\$15,735$. Determine the annual interest rate, $s\%$, of the second term deposit. | 11.36 | 0 |
30,723 | Given that ${(1-2x)^{2016}}=a_{0}+a_{1}(x-2)+a_{2}(x-2)^{2}+\cdots+a_{2015}(x-2)^{2015}+a_{2016}(x-2)^{2016}$ $(x\in\mathbb{R})$, find the value of $a_{1}-2a_{2}+3a_{3}-4a_{4}+\cdots+2015a_{2015}-2016a_{2016}$. | 4032 | 46.875 |
30,724 | Given points P and Q are on a circle of radius 7 and PQ = 8, find the length of the line segment PR, where R is the midpoint of the minor arc PQ. | \sqrt{98 - 14\sqrt{33}} | 16.40625 |
30,725 | The inclination angle of the line $\sqrt{3}x + y - 1 = 0$ is ______. | \frac{2\pi}{3} | 54.6875 |
30,726 | There are 3 female and 2 male volunteers, a total of 5 volunteers, who need to be distributed among 3 communities to participate in volunteer services. Each community can have 1 to 2 people. Female volunteers A and B must be in the same community, and male volunteers must be in different communities. The number of different distribution methods is \_\_\_\_\_\_. | 12 | 21.875 |
30,727 | Alice and Bob each arrive at a meeting at a random time between 8:00 and 9:00 AM. If Alice arrives after Bob, what is the probability that Bob arrived before 8:45 AM? | \frac{9}{16} | 40.625 |
30,728 | Find the smallest integer starting with the digit 7 that becomes three times smaller when this digit is moved to the end. Find all such numbers. | 7241379310344827586206896551 | 50.78125 |
30,729 | Freshmen go for a long walk in the suburbs after the start of school. They arrive at point \( A \) 6 minutes later than the originally planned time of 10:10, and they arrive at point \( C \) 6 minutes earlier than the originally planned time of 13:10. There is exactly one point \( B \) between \( A \) and \( C \) that is reached according to the original planned time. What is the time of arrival at point \( B \)? | 11:40 | 71.875 |
30,730 | Let \( f(n) \) be the sum of the squares of the digits of positive integer \( n \) (in decimal). For example, \( f(123) = 1^{2} + 2^{2} + 3^{2} = 14 \). Define \( f_{1}(n) = f(n) \), and \( f_{k+1}(n) = f\left(f_{k}(n)\right) \) for \( k = 1, 2, 3, \ldots \). What is the value of \( f_{2005}(2006) \)? | 145 | 7.03125 |
30,731 | For a positive number $x$, define $f(x)=\frac{x}{x+1}$. For example, $f(1)=\frac{1}{1+1}=\frac{1}{2}$, $f(2)=\frac{2}{2+1}=\frac{2}{3}$, $f(\frac{1}{2})=\frac{\frac{1}{2}}{\frac{1}{2}+1}=\frac{1}{3}$.
$(1)$ Find the value of: $f(3)+f(\frac{1}{3})=$______; $f(4)+f(\frac{1}{4})=$______.
$(2)$ Conjecture: $f(x)+f(\frac{1}{x})=$______.
$(3)$ Application: Based on the conclusion of $(2)$, calculate the value of the following expression: $f(2023)+f(2022)+f(2021)+\ldots +f(2)+f(1)+f(\frac{1}{2})+⋯+f(\frac{1}{2021})+f(\frac{1}{2022})+f(\frac{1}{2023})$. | 2022.5 | 24.21875 |
30,732 | Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), a line passing through points A($-a, 0$) and B($0, b$) has an inclination angle of $\frac{\pi}{6}$, and the distance from origin to this line is $\frac{\sqrt{3}}{2}$.
(1) Find the equation of the ellipse.
(2) Suppose a line with a positive slope passes through point D($-1, 0$) and intersects the ellipse at points E and F. If $\overrightarrow{ED} = 2\overrightarrow{DF}$, find the equation of line EF.
(3) Is there a real number $k$ such that the line $y = kx + 2$ intersects the ellipse at points P and Q, and the circle with diameter PQ passes through point D($-1, 0$)? If it exists, find the value of $k$; if not, explain why. | k = \frac{7}{6} | 0 |
30,733 | Given that the general term of the sequence $\{a_n\}$ is $a_n=2^{n-1}$, and the general term of the sequence $\{b_n\}$ is $b_n=3n$, let set $A=\{a_1,a_2,\ldots,a_n,\ldots\}$, $B=\{b_1,b_2,\ldots,b_n,\ldots\}$, $n\in\mathbb{N}^*$. The sequence $\{c_n\}$ is formed by arranging the elements of set $A\cup B$ in ascending order. Find the sum of the first 28 terms of the sequence $\{c_n\}$, denoted as $S_{28}$. | 820 | 0 |
30,734 | Given the curve $C$ represented by the equation $\sqrt {x^{2}+2 \sqrt {7}x+y^{2}+7}+ \sqrt {x^{2}-2 \sqrt {7}x+y^{2}+7}=8$, find the distance from the origin to the line determined by two distinct points on the curve $C$. | \dfrac {12}{5} | 23.4375 |
30,735 | In a hypothetical scenario, a small country is planning an international event in 2023. Let \( A \), \( B \), and \( C \) be distinct positive integers such that their product \( A \cdot B \cdot C = 2023 \). Determine the largest possible value of the sum \( A + B + C \). | 297 | 17.1875 |
30,736 | Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction. | \frac{1}{8} | 4.6875 |
30,737 | Let $f(x) = 2a^{x} - 2a^{-x}$ where $a > 0$ and $a \neq 1$. <br/> $(1)$ Discuss the monotonicity of the function $f(x)$; <br/> $(2)$ If $f(1) = 3$, and $g(x) = a^{2x} + a^{-2x} - 2f(x)$, $x \in [0,3]$, find the minimum value of $g(x)$. | -2 | 17.96875 |
30,738 | Given the function $f(x)=\tan(\omega x + \phi)$ $(\omega \neq 0, \left|\phi\right| < \frac{\pi}{2})$, points $\left(\frac{2\pi}{3}, 0\right)$ and $\left(\frac{7\pi}{6}, 0\right)$ are two adjacent centers of symmetry for $f(x)$, and the function is monotonically decreasing in the interval $\left(\frac{2\pi}{3}, \frac{4\pi}{3}\right)$, find the value of $\phi$. | -\frac{\pi}{6} | 17.96875 |
30,739 | For a finite sequence $B = (b_1, b_2, \dots, b_m)$ of numbers, the Cesaro sum of $B$ is defined to be
\[\frac{T_1 + T_2 + \dots + T_m}{m},\]
where $T_k = b_1 + b_2 + \dots + b_k$ for $1 \leq k \leq m$.
If the Cesaro sum of the 50-term sequence $(b_1, b_2, \dots, b_{50})$ is 500, what is the Cesaro sum of the 51-term sequence $(2, b_1, b_2, \dots, b_{50})$? | 492 | 7.03125 |
30,740 | Find both the sum and the product of the coordinates of the midpoint of the segment with endpoints $(8, 15)$ and $(-2, -3)$. | 18 | 13.28125 |
30,741 | Yura has a calculator that allows multiplying a number by 3, adding 3 to a number, or (if the number is divisible by 3) dividing by 3. How can you obtain the number 11 from the number 1 using this calculator? | 11 | 3.125 |
30,742 | In the $xy$-plane, a triangle has vertices with coordinates $(x, y)$, where $x$ and $y$ are integers satisfying $1 \leqslant x \leqslant 4$ and $1 \leqslant y \leqslant 4$. How many such triangles are there?
(Source: 44th American High School Mathematics Exam, 1993) | 516 | 73.4375 |
30,743 | Given that the heights of the first ten students formed a geometric sequence, with the fourth student 1.5 meters tall and the tenth student 1.62 meters tall, determine the height of the seventh student. | \sqrt{2.43} | 0 |
30,744 | The area of the triangle formed by the tangent line at point $(1,1)$ on the curve $y=x^3$, the x-axis, and the line $x=2$ is $\frac{4}{3}$. | \frac{8}{3} | 35.9375 |
30,745 | You are given the digits $0$, $1$, $2$, $3$, $4$, $5$. Form a four-digit number with no repeating digits.
(I) How many different four-digit numbers can be formed?
(II) How many of these four-digit numbers have a tens digit that is larger than both the units digit and the hundreds digit? | 100 | 68.75 |
30,746 | A rectangle has side lengths $6$ and $8$ . There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$ . | 11 | 3.90625 |
30,747 | The cars Niva and Toyota are driving on a ring track of a test site, a quarter of which is on a dirt road and the remaining part is on asphalt. The speed of the Niva on the dirt road is 80 km/h, and on asphalt it is 90 km/h. The speed of the Toyota on the dirt road is 40 km/h, and on asphalt it is 120 km/h. The cars start simultaneously at the beginning of the dirt road section and first drive through this dirt part. On which lap will one of the cars overtake the other for the first time? | 11 | 2.34375 |
30,748 | Three dice are thrown, and the sums of the points that appear on them are counted. In how many ways can you get a total of 5 points and 6 points? | 10 | 32.03125 |
30,749 | Given the circle with radius $6\sqrt{2}$, diameter $\overline{AB}$, and chord $\overline{CD}$ intersecting $\overline{AB}$ at point $E$, where $BE = 3\sqrt{2}$ and $\angle AEC = 60^{\circ}$, calculate $CE^2+DE^2$. | 216 | 3.90625 |
30,750 | 27 people went to a mall to buy water to drink. There was a promotion in the mall where three empty bottles could be exchanged for one bottle of water. The question is: For 27 people, the minimum number of bottles of water that need to be purchased so that each person can have one bottle of water to drink is $\boxed{18}$ bottles. | 18 | 83.59375 |
30,751 | Given a geometric sequence $\{a_n\}$ where each term is positive, and $a_1a_3=4$, $a_7a_9=25$, find $a_5$. | \sqrt{10} | 60.9375 |
30,752 |
A group of one hundred friends, including Petya and Vasya, live in several cities. Petya found the distance from his city to the city of each of the other 99 friends and summed these 99 distances, obtaining a total of 1000 km. What is the maximum possible total distance that Vasya could obtain using the same method? Assume cities are points on a plane and if two friends live in the same city, the distance between their cities is considered to be zero. | 99000 | 5.46875 |
30,753 | Given the function $f(x)=2\cos^2\frac{x}{2}+\sin x-1$. Find:
- $(Ⅰ)$ The minimum positive period, monotonic decreasing interval, and symmetry center of $f(x)$.
- $(Ⅱ)$ When $x\in \left[-\pi ,0\right]$, find the minimum value of $f(x)$ and the corresponding value of $x$. | -\frac{3\pi}{4} | 71.875 |
30,754 | Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2023,0),(2023,2024),$ and $(0,2024)$. What is the probability that $x > 9y$? Express your answer as a common fraction. | \frac{2023}{36432} | 30.46875 |
30,755 | Five points, no three of which are collinear, are given. What is the least possible value of the numbers of convex polygons whose some corners are from these five points? | 16 | 49.21875 |
30,756 | Given an arithmetic sequence $\{a_n\}$ with the first term being a positive number, and the sum of the first $n$ terms is $S_n$. If $a_{1006}$ and $a_{1007}$ are the two roots of the equation $x^2 - 2012x - 2011 = 0$, then the maximum value of the positive integer $n$ for which $S_n > 0$ holds is ______. | 2011 | 10.9375 |
30,757 | Inside a circle \(\omega\) there is a circle \(\omega_{1}\) touching \(\omega\) at point \(K\). Circle \(\omega_{2}\) touches circle \(\omega_{1}\) at point \(L\) and intersects circle \(\omega\) at points \(M\) and \(N\). It turns out that the points \(K, L,\) and \(M\) are collinear. Find the radius of circle \(\omega\), given that the radii of circles \(\omega_{1}\) and \(\omega_{2}\) are 4 and 7, respectively. | 11 | 11.71875 |
30,758 | Alice has 10 green marbles and 5 purple marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 8 marbles. What is the probability that exactly four of the marbles that she removes are green? Express your answer as a decimal rounded to the nearest thousandth. | 0.171 | 80.46875 |
30,759 | After a track and field event, each athlete shook hands once with every athlete from every other team, but not with their own team members. Afterwards, two coaches arrived, each only shaking hands with each athlete from their respective teams. If there were a total of 300 handshakes at the event, what is the fewest number of handshakes the coaches could have participated in? | 20 | 18.75 |
30,760 | A sphere is inscribed in a right cone with base radius \(15\) cm and height \(30\) cm. Determine the radius of the sphere if it can be expressed as \(b\sqrt{d} - b\) cm. What is the value of \(b + d\)? | 12.5 | 10.15625 |
30,761 | For a real number $a$ and an integer $n(\geq 2)$ , define $$ S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}} $$ Find every value of $a$ s.t. sequence $\{S_n(a)\}_{n\geq 2}$ converges to a positive real. | 2019 | 14.84375 |
30,762 | A marine biologist interested in monitoring a specific fish species population in a coastal area. On January 15, he captures and tags 80 fish, then releases them back into the water. On June 15, he captures another sample of 100 fish, finding that 6 of them are tagged. He assumes that 20% of the tagged fish have died or migrated out of the area by June 15, and also that 50% of the fish in the June sample are recent additions due to birth or migration. How many fish were in the coastal area on January 15, based on his assumptions? | 533 | 60.15625 |
30,763 | In the diagram, \(ABCD\) is a rectangle with \(AD = 13\), \(DE = 5\), and \(EA = 12\). The area of \(ABCD\) is | 60 | 1.5625 |
30,764 | A rectangular flag is divided into four triangles, labelled Left, Right, Top, and Bottom. Each triangle is to be colored one of red, white, blue, green, and purple such that no two triangles that share an edge are the same color. Determine the total number of different flags that can be made. | 260 | 20.3125 |
30,765 | Given that in $\triangle ABC$, $C = 2A$, $\cos A = \frac{3}{4}$, and $2 \overrightarrow{BA} \cdot \overrightarrow{CB} = -27$.
(I) Find the value of $\cos B$;
(II) Find the perimeter of $\triangle ABC$. | 15 | 45.3125 |
30,766 | Let points $P$ and $Q$ be moving points on the curve $f(x)=x^{2}-\ln x$ and the line $x-y-2=0$, respectively. The minimum value of the distance between points $P$ and $Q$ is _______. | \sqrt{2} | 43.75 |
30,767 | When the expression $3(x^2 - 3x + 3) - 8(x^3 - 2x^2 + 4x - 1)$ is fully simplified, what is the sum of the squares of the coefficients of the terms? | 2395 | 22.65625 |
30,768 | Eight congruent disks are arranged around a square with side length 2 in such a manner that each disk is tangent to two others, with four disks centered on the midpoints of the square’s sides and four at the corners. The disks completely cover the boundary of the square. Calculate the total area of the eight disks. | 4\pi | 1.5625 |
30,769 | In $\triangle ABC$, the ratio $AC:CB$ is $2:3$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). Determine the ratio $PA:AB$. | 2:1 | 4.6875 |
30,770 | Given the sets $A=\{x|x=2n-1,n\in\mathbb{N}^*\}$ and $B=\{x|x=2^n,n\in\mathbb{N}^*\}$. Arrange all elements of $A\cup B$ in ascending order to form a sequence $\{a_n\}$. Let $S_n$ denote the sum of the first $n$ terms of the sequence $\{a_n\}$. Find the smallest value of $n$ such that $S_n > 12a_{n+1}$. | 27 | 69.53125 |
30,771 | In a particular right triangle, the lengths of the two legs are 30 inches and 24 inches. Calculate both the area of the triangle and the length of its hypotenuse. | \sqrt{1476} | 0 |
30,772 | Given the functions $f(x)=2\sin \left(\pi x+\frac{\pi }{3}\right)$ and $g(x)=2\cos \left(\pi x+\frac{\pi }{3}\right)$, find the area of the triangle formed by the intersection points of the two functions within the interval $\left[-\frac{4}{3},\frac{7}{6}\right]$. | 2\sqrt{2} | 36.71875 |
30,773 | Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers $1$ to $16$ ? | 11 | 62.5 |
30,774 | Imagine a dodecahedron (a polyhedron with 12 pentagonal faces) and an ant starts at one of the top vertices. The ant will randomly walk to one of three adjacent vertices, denoted as vertex A. From vertex A, the ant then walks to one of another three randomly selected adjacent vertices, signified as vertex B. Calculate the probability that vertex B is one of the five bottom vertices of the dodecahedron. Express your answer as a common fraction. | \frac{1}{3} | 17.96875 |
30,775 | For every non-empty subset of the natural number set $N^*$, we define the "alternating sum" as follows: arrange the elements of the subset in descending order, then start with the largest number and alternately add and subtract each number. For example, the alternating sum of the subset $\{1, 2, 4, 6, 9\}$ is $9 - 6 + 4 - 2 + 1 = 6$. Then, the total sum of the alternating sums of all non-empty subsets of the set $\{1, 2, 3, 4, 5, 6, 7\}$ is | 448 | 42.96875 |
30,776 | In a store where all items are priced in whole numbers of rubles, there are two special offers:
1) A customer who buys at least three items at once can select one item for free, provided that its price does not exceed the minimum of the paid items' prices.
2) A customer who buys exactly one item priced at no less than \(N\) rubles receives a 20% discount on their next purchase (which can include any number of items).
A customer, visiting this store for the first time, wants to buy exactly four items with a total price of 1000 rubles, where the cheapest item is priced at no less than 99 rubles. Determine the maximum \(N\) such that the second offer is more beneficial for the customer than the first one. | 504 | 2.34375 |
30,777 | Out of a batch of 5000 electric lamps, a self-random non-repeating sample of 300 was selected. The average burning time of the lamps in the sample was found to be 1450 hours, and the variance was 40,000. What is the probability that the average burning time of the lamps in the entire batch falls within the range from 1410 to 1490 hours? | 0.99964 | 0 |
30,778 | For a nonnegative integer $n$, let $r_9(n)$ be the remainder when $n$ is divided by $9$. Consider all nonnegative integers $n$ for which $r_9(7n) \leq 5$. Find the $15^{\text{th}}$ entry in an ordered list of all such $n$. | 21 | 18.75 |
30,779 | $A$ and $B$ plays a game, with $A$ choosing a positive integer $n \in \{1, 2, \dots, 1001\} = S$ . $B$ must guess the value of $n$ by choosing several subsets of $S$ , then $A$ will tell $B$ how many subsets $n$ is in. $B$ will do this three times selecting $k_1, k_2$ then $k_3$ subsets of $S$ each.
What is the least value of $k_1 + k_2 + k_3$ such that $B$ has a strategy to correctly guess the value of $n$ no matter what $A$ chooses? | 28 | 0 |
30,780 | In the sequence $\{a_n\}$, two adjacent terms $a_n$ and $a_{n+1}$ are the roots of the equation $x^2 + 3nx + b_n = 0$. Given that $a_{10} = -17$, calculate the value of $b_{51}$. | 5840 | 7.8125 |
30,781 | Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions:
a) No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$ , and
b) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$ . | 10 | 16.40625 |
30,782 | Martha can make 24 cookies with 3 cups of flour. How many cookies can she make with 5 cups of flour, and how many cups of flour are needed to make 60 cookies? | 7.5 | 97.65625 |
30,783 | Compute $\sqrt[3]{466560000}$. | 360 | 3.90625 |
30,784 | Given vectors $\overrightarrow {a}$ = (4, -7) and $\overrightarrow {b}$ = (3, -4), find the projection of $\overrightarrow {a}$ - $2\overrightarrow {b}$ in the direction of $\overrightarrow {b}$. | -2 | 0 |
30,785 | Given the function $f(x) = e^{\sin x + \cos x} - \frac{1}{2}\sin 2x$ ($x \in \mathbb{R}$), find the difference between the maximum and minimum values of the function $f(x)$. | e^{\sqrt{2}} - e^{-\sqrt{2}} | 0 |
30,786 | Given two unit vectors $a$ and $b$, and $|a-2b| = \sqrt{7}$, then the angle between $a$ and $b$ is ______. | \frac{2\pi}{3} | 93.75 |
30,787 | Calculate the harmonic mean of the numbers $1999$ and $2001$. | 2000 | 34.375 |
30,788 | Find the number of solutions to the equation
\[\tan (3 \pi \cos \theta) = \cot (3 \pi \sin \theta)\]
where $\theta \in (0, 3\pi)$. | 16 | 46.875 |
30,789 | Given the polynomial $f(x) = 3x^6 + 5x^5 + 6x^4 + 20x^3 - 8x^2 + 35x + 12$ and $x = -2$, apply Horner's method to calculate the value of $v_4$. | -16 | 42.1875 |
30,790 | Through a simulation experiment, 20 groups of random numbers were generated: 830, 3013, 7055, 7430, 7740, 4422, 7884, 2604, 3346, 0952, 6807, 9706, 5774, 5725, 6576, 5929, 9768, 6071, 9138, 6754. If exactly three numbers are among 1, 2, 3, 4, 5, 6, it indicates that the target was hit exactly three times. Then, the probability of hitting the target exactly three times in four shots is approximately \_\_\_\_\_\_\_\_. | 25\% | 0 |
30,791 | Given that all three vertices of \(\triangle ABC\) lie on the parabola defined by \(y = 4x^2\), with \(A\) at the origin and \(\overline{BC}\) parallel to the \(x\)-axis, calculate the length of \(BC\), given that the area of the triangle is 128. | 4\sqrt[3]{4} | 85.9375 |
30,792 | For how many integers $n$ between 1 and 500 inclusive is $$(\sin t + i\cos t)^n = \sin(2nt) + i\cos(2nt)$$ true for all real $t$? | 125 | 57.03125 |
30,793 | Two symmetrical coins are flipped. What is the probability that both coins show numbers on their upper sides? | 0.25 | 10.15625 |
30,794 | What is the largest integer $n$ that satisfies $(100^2-99^2)(99^2-98^2)\dots(3^2-2^2)(2^2-1^2)$ is divisible by $3^n$ ? | 49 | 62.5 |
30,795 | Let \(f_{0}(x)=\frac{1}{c-x}\) and \(f_{n}(x)=f_{0}\left(f_{n-1}(x)\right)\), \(n=1,2,3, \ldots\). If \(f_{2000}(2000)=d\), find the value of \(d\). | 2000 | 14.0625 |
30,796 | Given that the sequence $\left\{a_n\right\}$ is an arithmetic sequence, if $\dfrac{a_{11}}{a_{10}} < -1$, and the sum of its first $n$ terms, $S_n$, has a maximum value, calculate the maximum value of $n$ for which $S_n > 0$. | 19 | 64.84375 |
30,797 | Given that the first character can be chosen from 5 digits (3, 5, 6, 8, 9), and the third character from the left can be chosen from 4 letters (B, C, D), and the other 3 characters can be chosen from 3 digits (1, 3, 6, 9), and the last character from the left can be chosen from the remaining 3 digits (1, 3, 6, 9), find the total number of possible license plate numbers available for this car owner. | 960 | 7.8125 |
30,798 | Let $T$ be the set of all positive real numbers. Let $g : T \to T$ be a function such that
\[ g(x) + g(y) = x \cdot y \cdot g(g(x) + g(y)) \]
for all $x, y \in T$ where $x + y \neq 1$.
Let $m$ be the number of possible values of $g(3),$ and let $t$ be the sum of all possible values of $g(3).$ Find $m \times t.$ | \frac{1}{3} | 42.96875 |
30,799 | An object is moving towards a converging lens with a focal length of \( f = 10 \ \mathrm{cm} \) along the line defined by the two focal points at a speed of \( 2 \ \mathrm{m/s} \). What is the relative speed between the object and its image when the object distance is \( t = 30 \ \mathrm{cm} \)? | 1.5 | 0 |
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