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29,600 | There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \dots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \); circle \( C_{6} \) passes through exactly 6 points in \( M \); ..., circle \( C_{1} \) passes through exactly 1 point in \( M \). Determine the minimum number of points in set \( M \). | 12 | 8.59375 |
29,601 | In the $xy$-plane, the segment with endpoints $(-3,0)$ and $(27,0)$ is the diameter of a circle. A vertical line $x=k$ intersects the circle at two points, and one of the points has a $y$-coordinate of $12$. Find the value of $k$. | 21 | 44.53125 |
29,602 | The diagram shows a smaller rectangle made from three squares, each of area \(25 \ \mathrm{cm}^{2}\), inside a larger rectangle. Two of the vertices of the smaller rectangle lie on the mid-points of the shorter sides of the larger rectangle. The other two vertices of the smaller rectangle lie on the other two sides of the larger rectangle. What is the area, in \(\mathrm{cm}^{2}\), of the larger rectangle? | 150 | 75 |
29,603 | When evaluated, the sum of the digits of the integer equal to \(10^{2021} - 2021\) is: | 18185 | 15.625 |
29,604 | How many four-digit positive integers are multiples of 7? | 1286 | 99.21875 |
29,605 | In the diagram, the rectangular wire grid contains 15 identical squares. The length of the rectangular grid is 10. What is the length of wire needed to construct the grid? | 76 | 5.46875 |
29,606 | A rectangle has a perimeter of 80 inches and an area greater than 240 square inches. How many non-congruent rectangles meet these criteria? | 13 | 24.21875 |
29,607 | If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$ . If $n$ is a positive integer, let $S(n)$ be defined by
\[
S(n)
= \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
+ 10 \left( n - 10^{\lfloor \log n \rfloor}
\cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
\right) \, .
\]
(All the logarithms are base 10.) How many integers $n$ from 1 to 2011 (inclusive) satisfy $S(S(n)) = n$ ? | 108 | 89.84375 |
29,608 | A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing 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. | 8.8\% | 0 |
29,609 | How many lattice points lie on the hyperbola \( x^2 - y^2 = 1800^2 \)? | 250 | 8.59375 |
29,610 | Let $\theta$ be the angle between the line
\[\frac{x - 2}{4} = \frac{y + 1}{5} = \frac{z - 4}{7}\]
and the plane $3x + 4y - 7z = 5.$ Find $\sin \theta.$ | \frac{17}{\sqrt{6660}} | 0 |
29,611 | Druv has a $33 \times 33$ grid of unit squares, and he wants to color each unit square with exactly one of three distinct colors such that he uses all three colors and the number of unit squares with each color is the same. However, he realizes that there are internal sides, or unit line segments that have exactly one unit square on each side, with these two unit squares having different colors. What is the minimum possible number of such internal sides? | 66 | 21.875 |
29,612 | Find the largest positive integer $n$ such that there exist $n$ distinct positive integers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying
$$
x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=2017.
$$ | 16 | 0.78125 |
29,613 | Given that there are $12$ different cards, with $3$ cards each of red, yellow, green, and blue, select $3$ cards such that they cannot all be of the same color and there can be at most $1$ blue card. | 189 | 14.84375 |
29,614 | A pair of natural numbers is called "good" if one of the numbers is divisible by the other. The numbers from 1 to 30 are divided into 15 pairs. What is the maximum number of good pairs that could be formed? | 13 | 0 |
29,615 | If $a$, $b$, and $c$ are positive numbers such that $ab = 24\sqrt{3}$, $ac = 30\sqrt{3}$, and $bc = 40\sqrt{3}$, find the value of $abc$. | 120\sqrt{6} | 6.25 |
29,616 | Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $(1, \frac{2\sqrt{3}}{3})$, with its left and right foci being $F_1$ and $F_2$ respectively. The chord formed by the intersection of the circle $x^2 + y^2 = 2$ and the line $x + y + b = 0$ has a length of $2$.
(Ⅰ) Find the standard equation of the ellipse $C$;
(Ⅱ) Let $Q$ be a moving point on the ellipse $C$ that is not on the $x$-axis, with $Q$ being the origin. A line parallel to $OQ$ passing through $F_2$ intersects the ellipse $C$ at two distinct points $M$ and $N$.
(1) Investigate whether $\frac{|MN|}{|OQ|^2}$ is a constant. If it is, find the constant; if not, explain why.
(2) Let the area of $\triangle QF_2M$ be $S_1$ and the area of $\triangle OF_2N$ be $S_2$, and let $S = S_1 + S_2$. Find the maximum value of $S$. | \frac{2\sqrt{3}}{3} | 1.5625 |
29,617 | Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, find the maximum area of $\triangle PAB$. | 20\sqrt{5} | 0.78125 |
29,618 | Let \( x, y, z \) be positive numbers that satisfy the following system of equations:
\[
\begin{cases}
x^2 + xy + y^2 = 108 \\
y^2 + yz + z^2 = 9 \\
z^2 + xz + x^2 = 117
\end{cases}
\]
Find the value of the expression \( xy + yz + xz \). | 36 | 94.53125 |
29,619 | Let $b_n$ be the number obtained by writing the integers $1$ to $n$ from left to right and then subtracting $n$ from the resulting number. For example, $b_4 = 1234 - 4 = 1230$ and $b_{12} = 123456789101112 - 12 = 123456789101100$. For $1 \le k \le 100$, how many $b_k$ are divisible by 9? | 22 | 85.9375 |
29,620 | What is the least positive integer with exactly $12$ positive factors? | 108 | 0 |
29,621 | For the function $f(x)=4\sin \left(2x+\frac{\pi }{3}\right)$, the following propositions are given:
$(1)$ From $f(x_1)=f(x_2)$, it can be concluded that $x_1-x_2$ is an integer multiple of $\pi$; $(2)$ The expression for $f(x)$ can be rewritten as $f(x)=4\cos \left(2x-\frac{\pi }{6}\right)$; $(3)$ The graph of $f(x)$ is symmetric about the point $\left(-\frac{\pi }{6},0\right)$; $(4)$ The graph of $f(x)$ is symmetric about the line $x=-\frac{\pi }{6}$; Among these, the correct propositions are ______________. | (2)(3) | 0 |
29,622 | Let $T$ be a subset of $\{1,2,3,...,100\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$? | 41 | 82.03125 |
29,623 | At a conference, there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\frac{3}{7}$. What fraction of the people in the conference are married men? | \frac{4}{11} | 90.625 |
29,624 | Hexagon $ABCDEF$ is divided into five rhombuses, $P, Q, R, S,$ and $T$ , as shown. Rhombuses $P, Q, R,$ and $S$ are congruent, and each has area $\sqrt{2006}.$ Let $K$ be the area of rhombus $T$ . Given that $K$ is a positive integer, find the number of possible values for $K.$ [asy] // TheMathGuyd size(8cm); pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6); draw(A--B--C--D--EE--F--cycle); draw(F--G--(2.1,0)); draw(C--H--(2.1,0)); draw(G--(2.1,-3.2)); draw(H--(2.1,-3.2)); label("$\mathcal{T}$",(2.1,-1.6)); label("$\mathcal{P}$",(0,-1),NE); label("$\mathcal{Q}$",(4.2,-1),NW); label("$\mathcal{R}$",(0,-2.2),SE); label("$\mathcal{S}$",(4.2,-2.2),SW); [/asy] | 89 | 0 |
29,625 | Given the function $$f(x)= \begin{cases} a^{x}, x<0 \\ ( \frac {1}{4}-a)x+2a, x\geq0\end{cases}$$ such that for any $x\_1 \neq x\_2$, the inequality $$\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}}<0$$ holds true. Determine the range of values for the real number $a$. | \frac{1}{2} | 12.5 |
29,626 | Two lines with slopes $\dfrac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$. | 8.625 | 0 |
29,627 | How many positive four-digit integers of the form $\_\_50$ are divisible by 50? | 90 | 91.40625 |
29,628 | There are 5 balls of the same shape and size in a bag, including 3 red balls and 2 yellow balls. Now, balls are randomly drawn from the bag one at a time until two different colors of balls are drawn. Let the random variable $\xi$ be the number of balls drawn at this time. Find $E(\xi)=$____. | \frac{5}{2} | 22.65625 |
29,629 | Given an ellipse $(C)$: $\frac{x^{2}}{3m} + \frac{y^{2}}{m} = 1 (m > 0)$ with the length of its major axis being $2\sqrt{6}$, and $O$ is the coordinate origin.
(I) Find the equation and eccentricity of the ellipse $(C)$;
(II) Let moving line $(l)$ intersect with the $y$-axis at point $B$, and the symmetric point $P(3, 0)$ about line $(l)$ lies on the ellipse $(C)$. Find the minimum value of $|OB|$. | \sqrt{6} | 0.78125 |
29,630 | Let the function $f(x)=2x-\cos x$, and $\{a_n\}$ be an arithmetic sequence with a common difference of $\dfrac{\pi}{8}$. If $f(a_1)+f(a_2)+\ldots+f(a_5)=5\pi$, calculate $\left[f(a_3)\right]^2-a_2a_3$. | \dfrac{13}{16}\pi^2 | 0 |
29,631 | Andy is attempting to solve the quadratic equation $$64x^2 - 96x - 48 = 0$$ by completing the square. He aims to rewrite the equation in the form $$(ax + b)^2 = c,$$ where \(a\), \(b\), and \(c\) are integers and \(a > 0\). Determine the value of \(a + b + c\). | 86 | 15.625 |
29,632 | Let \( S \) be a subset of \(\{1,2,3, \ldots, 199,200\}\). We say that \( S \) is pretty if, for every pair of elements \( a \) and \( b \) in \( S \), the number \( a - b \) is not a prime number. What is the maximum number of elements in a pretty subset of \(\{1,2,3, \ldots, 199,200\}\)? | 50 | 20.3125 |
29,633 | In a subject test, the average score of Xiaofang's four subjects: Chinese, Mathematics, English, and Science, is 88. The average score of the first two subjects is 93, and the average score of the last three subjects is 87. What is Xiaofang's English test score? | 95 | 11.71875 |
29,634 | Points $K$, $L$, $M$, and $N$ lie in the plane of the square $ABCD$ such that $AKB$, $BLC$, $CMD$, and $DNA$ are isosceles right triangles. If the area of square $ABCD$ is 25, find the area of $KLMN$. | 25 | 1.5625 |
29,635 | What is the least positive integer with exactly $12$ positive factors? | 72 | 0 |
29,636 | Given that $A$, $B$, and $P$ are three distinct points on the hyperbola ${x^2}-\frac{{y^2}}{4}=1$, and they satisfy $\overrightarrow{PA}+\overrightarrow{PB}=2\overrightarrow{PO}$ (where $O$ is the origin), the slopes of lines $PA$ and $PB$ are denoted as $m$ and $n$ respectively. Find the minimum value of ${m^2}+\frac{{n^2}}{9}$. | \frac{8}{3} | 1.5625 |
29,637 | We have an equilateral triangle with circumradius $1$ . We extend its sides. Determine the point $P$ inside the triangle such that the total lengths of the sides (extended), which lies inside the circle with center $P$ and radius $1$ , is maximum.
(The total distance of the point P from the sides of an equilateral triangle is fixed )
*Proposed by Erfan Salavati* | 3\sqrt{3} | 26.5625 |
29,638 | Calculate the corrected average score and variance of a class of 50 students, given that the original average was 70 and the original variance was 102, after two students' scores were corrected from 50 to 80 and from 90 to 60. | 90 | 27.34375 |
29,639 | Find the remainder when the value of $m$ is divided by 1000 in the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_6 \le 1500$ such that $a_i-i$ is odd for $1\le i \le 6$. The total number of sequences can be expressed as ${m \choose n}$ for some integers $m>n$. | 752 | 0.78125 |
29,640 | Given points P(-2,-3) and Q(5,3) in the xy-plane; point R(x,m) is such that x=2 and PR+RQ is a minimum. Find m. | \frac{3}{7} | 2.34375 |
29,641 | In isosceles triangle $\triangle ABC$, $CA=CB=6$, $\angle ACB=120^{\circ}$, and point $M$ satisfies $\overrightarrow{BM}=2 \overrightarrow{MA}$. Determine the value of $\overrightarrow{CM} \cdot \overrightarrow{CB}$. | 12 | 0.78125 |
29,642 | For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\frac{n}{1260}$ terminate? | 47 | 1.5625 |
29,643 | The largest three-digit number divided by an integer, with the quotient rounded to one decimal place being 2.5, will have the smallest divisor as: | 392 | 9.375 |
29,644 | Given that the magnitude of the star Altair is $0.75$ and the magnitude of the star Vega is $0$, determine the ratio of the luminosity of Altair to Vega. | 10^{-\frac{3}{10}} | 0 |
29,645 | Given that in triangle $\triangle ABC$, $a=2$, $\angle A=\frac{π}{6}$, $b=2\sqrt{3}$, find the measure of angle $\angle C$. | \frac{\pi}{2} | 85.15625 |
29,646 | Flights are arranged between 13 countries. For $ k\ge 2$ , the sequence $ A_{1} ,A_{2} ,\ldots A_{k}$ is said to a cycle if there exist a flight from $ A_{1}$ to $ A_{2}$ , from $ A_{2}$ to $ A_{3}$ , $ \ldots$ , from $ A_{k \minus{} 1}$ to $ A_{k}$ , and from $ A_{k}$ to $ A_{1}$ . What is the smallest possible number of flights such that how the flights are arranged, there exist a cycle? | 79 | 4.6875 |
29,647 | Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, find the maximum area of $\triangle PAB$. | 20\sqrt{5} | 0 |
29,648 | Arrange 5 people to be on duty from Monday to Friday, with each person on duty for one day and one person arranged for each day. The conditions are: A and B are not on duty on adjacent days, while B and C are on duty on adjacent days. The number of different arrangements is $\boxed{\text{answer}}$. | 36 | 22.65625 |
29,649 | Given the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. Point $A(4,2\sqrt{2})$ lies on the ellipse, and $AF\_2$ is perpendicular to the $x$-axis.
1. Find the equation of the ellipse.
2. A line passing through point $F\_2$ intersects the ellipse at points $B$ and $C$. Find the maximum area of triangle $COB$. | 8\sqrt{2} | 17.96875 |
29,650 | Given the function \( f(x)=\frac{\sin (\pi x)-\cos (\pi x)+2}{\sqrt{x}} \) for \( \frac{1}{4} \leqslant x \leqslant \frac{5}{4} \), find the minimum value of \( f(x) \). | \frac{4\sqrt{5}}{5} - \frac{2\sqrt{10}}{5} | 0 |
29,651 | In a tetrahedron V-ABC with edge length 10, point O is the center of the base ABC. Segment MN has a length of 2, with one endpoint M on segment VO and the other endpoint N inside face ABC. If point T is the midpoint of segment MN, then the area of the trajectory formed by point T is __________. | 2\pi | 2.34375 |
29,652 | Three distinct diameters are drawn on a unit circle such that chords are drawn as shown. If the length of one chord is \(\sqrt{2}\) units and the other two chords are of equal lengths, what is the common length of these chords? | \sqrt{2-\sqrt{2}} | 3.90625 |
29,653 | The Lions are competing against the Eagles in a seven-game championship series. The Lions have a probability of $\dfrac{2}{3}$ of winning a game whenever it rains and a probability of $\dfrac{1}{2}$ of winning when it does not rain. Assume it's forecasted to rain for the first three games and the remaining will have no rain. What is the probability that the Lions will win the championship series? Express your answer as a percent, rounded to the nearest whole percent. | 76\% | 6.25 |
29,654 | Assuming that the clock hands move without jumps, determine how many minutes after the clock shows 8:00 will the minute hand catch up with the hour hand. | 43 \frac{7}{11} | 12.5 |
29,655 | Determine the number of relatively prime dates in the month with the second fewest relatively prime dates. | 11 | 1.5625 |
29,656 | Given that $x \sim N(-1,36)$ and $P(-3 \leqslant \xi \leqslant -1) = 0.4$, calculate $P(\xi \geqslant 1)$. | 0.1 | 48.4375 |
29,657 | A man and a dog are walking. The dog waits for the man to start walking along a path and then runs to the end of the path and back to the man a total of four times, always moving at a constant speed. The last time the dog runs back to the man, it covers the remaining distance of 81 meters. The distance from the door to the end of the path is 625 meters. Find the speed of the dog if the man is walking at a speed of 4 km/h. | 16 | 3.90625 |
29,658 | Let the set \( S = \{1, 2, 3, \cdots, 50\} \). Find the smallest positive integer \( n \) such that every subset of \( S \) with \( n \) elements contains three numbers that can be the side lengths of a right triangle. | 42 | 9.375 |
29,659 | Find four consecutive odd numbers, none of which are divisible by 3, such that their sum is divisible by 5. What is the smallest possible value of this sum? | 40 | 30.46875 |
29,660 | Given the function $f(x)= \begin{cases} \sqrt {x}+3,x\geqslant 0 \\ ax+b,x < 0 \end{cases}$ that satisfies the condition: for all $x_{1}∈R$ and $x_{1}≠ 0$, there exists a unique $x_{2}∈R$ and $x_{1}≠ x_{2}$ such that $f(x_{1})=f(x_{2})$, determine the value of the real number $a+b$ when $f(2a)=f(3b)$ holds true. | -\dfrac{\sqrt{6}}{2}+3 | 0 |
29,661 | Given the ratio of women to men is $7$ to $5$, and the average age of women is $30$ years and the average age of men is $35$ years, determine the average age of the community. | 32\frac{1}{12} | 0.78125 |
29,662 | Given that in a mathematics test, $20\%$ of the students scored $60$ points, $25\%$ scored $75$ points, $20\%$ scored $85$ points, $25\%$ scored $95$ points, and the rest scored $100$ points, calculate the difference between the mean and the median score of the students' scores on this test. | 6.5 | 0.78125 |
29,663 | In a triangle configuration, each row consists of increasing multiples of 3 unit rods. The number of connectors in a triangle always forms an additional row than the rods, with connectors enclosing each by doubling the requirements of connecting joints from the previous triangle. How many total pieces are required to build a four-row triangle? | 60 | 9.375 |
29,664 | A circle passes through the vertices $K$ and $P$ of triangle $KPM$ and intersects its sides $KM$ and $PM$ at points $F$ and $B$, respectively. Given that $K F : F M = 3 : 1$ and $P B : B M = 6 : 5$, find $K P$ given that $B F = \sqrt{15}$. | 2 \sqrt{33} | 0 |
29,665 | If an irrational number $a$ multiplied by $\sqrt{8}$ is a rational number, write down one possible value of $a$ as ____. | \sqrt{2} | 42.96875 |
29,666 | A mischievous child mounted the hour hand on the minute hand's axle and the minute hand on the hour hand's axle of a correctly functioning clock. The question is, how many times within a day does this clock display the correct time? | 22 | 4.6875 |
29,667 | In a dark room, a drawer contains 120 red socks, 100 green socks, 70 blue socks, and 50 black socks. A person selects socks one by one from the drawer without being able to see their color. What is the minimum number of socks that must be selected to ensure that at least 15 pairs of socks are selected, with no sock being counted in more than one pair? | 33 | 7.8125 |
29,668 | Determine how many integer values of $n$ between 1 and 180 inclusive ensure that the decimal representation of $\frac{n}{180}$ terminates. | 60 | 40.625 |
29,669 | Given three sequences $\{F_n\}$, $\{k_n\}$, $\{r_n\}$ satisfying: $F_1=F_2=1$, $F_{n+2}=F_{n+1}+F_n$ ($n\in\mathbb{N}^*$), $r_n=F_n-3k_n$, $k_n\in\mathbb{N}$, $0\leq r_n<3$, calculate the sum $r_1+r_3+r_5+\ldots+r_{2011}$. | 1509 | 3.125 |
29,670 | Tetrahedron $PQRS$ is such that $PQ=6$, $PR=5$, $PS=4\sqrt{2}$, $QR=3\sqrt{2}$, $QS=5$, and $RS=4$. Calculate the volume of tetrahedron $PQRS$.
**A)** $\frac{130}{9}$
**B)** $\frac{135}{9}$
**C)** $\frac{140}{9}$
**D)** $\frac{145}{9}$ | \frac{140}{9} | 32.8125 |
29,671 | How many points does one have to place on a unit square to guarantee that two of them are strictly less than 1/2 unit apart? | 10 | 3.90625 |
29,672 | Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | 2028 | 47.65625 |
29,673 | Given that the polynomial \(x^2 - kx + 24\) has only positive integer roots, find the average of all distinct possibilities for \(k\). | 15 | 100 |
29,674 | Given any point $P$ on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\; \; (a > b > 0)$ with foci $F\_{1}$ and $F\_{2}$, if $\angle PF\_1F\_2=\alpha$, $\angle PF\_2F\_1=\beta$, $\cos \alpha= \frac{ \sqrt{5}}{5}$, and $\sin (\alpha+\beta)= \frac{3}{5}$, find the eccentricity of this ellipse. | \frac{\sqrt{5}}{7} | 0 |
29,675 | Given the expansion of $(1+\frac{a}{x}){{(2x-\frac{1}{x})}^{5}}$, find the constant term. | 80 | 3.90625 |
29,676 | Xiao Ming arranges chess pieces in a two-layer hollow square array (the diagram shows the top left part of the array). After arranging the inner layer, 60 chess pieces are left. After arranging the outer layer, 32 chess pieces are left. How many chess pieces does Xiao Ming have in total? | 80 | 5.46875 |
29,677 | Let $\{a_{n}\}$ be an integer sequence such that for any $n \in \mathbf{N}^{*}$, the condition \((n-1) a_{n+1} = (n+1) a_{n} - 2 (n-1)\) holds. Additionally, \(2008 \mid a_{2007}\). Find the smallest positive integer \(n \geqslant 2\) such that \(2008 \mid a_{n}\). | 501 | 0 |
29,678 | For which $x$ and $y$ is the number $x x y y$ a square of a natural number? | 7744 | 6.25 |
29,679 | Given that $P$ is any point on the hyperbola $\frac{x^2}{3} - y^2 = 1$, a line perpendicular to each asymptote of the hyperbola is drawn through point $P$, with the feet of these perpendiculars being $A$ and $B$. Determine the value of $\overrightarrow{PA} \cdot \overrightarrow{PB}$. | -\frac{3}{8} | 14.0625 |
29,680 | You have a whole cake in your pantry. On your first trip to the pantry, you eat one-third of the cake. On each successive trip, you eat one-third of the remaining cake. After four trips to the pantry, what fractional part of the cake have you eaten? | \frac{40}{81} | 0 |
29,681 | The line joining $(2,3)$ and $(5,1)$ divides the square shown into two parts. What fraction of the area of the square is above this line? The square has vertices at $(2,1)$, $(5,1)$, $(5,4)$, and $(2,4)$. | \frac{2}{3} | 17.1875 |
29,682 | $n$ balls are placed independently uniformly at random into $n$ boxes. One box is selected at random, and is found to contain $b$ balls. Let $e_n$ be the expected value of $b^4$ . Find $$ \lim_{n \to
\infty}e_n. $$ | 15 | 64.84375 |
29,683 | What is the least possible sum of two positive integers $a$ and $b$ where $a \cdot b = 10! ?$ | 3960 | 0 |
29,684 | Given an increasing sequence $\{a_{n}\}$ where all terms are positive integers, the sum of the first $n$ terms is $S_{n}$. If $a_{1}=3$ and $S_{n}=2023$, calculate the value of $a_{n}$ when $n$ takes its maximum value. | 73 | 3.125 |
29,685 |
Determine the volume of the released gas:
\[ \omega\left(\mathrm{SO}_{2}\right) = n\left(\mathrm{SO}_{2}\right) \cdot V_{m} = 0.1122 \cdot 22.4 = 2.52 \text{ liters} \] | 2.52 | 66.40625 |
29,686 | A pentagon is formed by placing an equilateral triangle on top of a rectangle. The side length of the equilateral triangle is equal to the width of the rectangle, and the height of the rectangle is twice the side length of the triangle. What percent of the area of the pentagon is the area of the equilateral triangle? | \frac{\sqrt{3}}{\sqrt{3} + 8} \times 100\% | 0 |
29,687 | Given a class with 21 students, such that at least two of any three students are friends, determine the largest possible value of k. | 10 | 12.5 |
29,688 | In triangle $ABC$, $\angle C = 4\angle A$, $a = 36$, and $c = 60$. Determine the length of side $b$. | 45 | 4.6875 |
29,689 | Evaluate $\sqrt[3]{1+27} \cdot \sqrt[3]{1+\sqrt[3]{27}}$. | \sqrt[3]{112} | 44.53125 |
29,690 | Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and $S_{n}=n^{2}+n+1$. In the positive geometric sequence ${b_n}$, $b_3=a_2$, $b_4=a_4$. Find:
1. The general term formulas for ${a_n}$ and ${b_n}$;
2. If $c_n$ is defined as $c_n=\begin{cases} a_{n},(n\text{ is odd}) \\ b_{n},(n\text{ is even}) \end{cases}$, and $T_n=c_1+c_2+…+c_n$, find $T_{10}$. | 733 | 37.5 |
29,691 | There are 7 cards, each with a number written on it: 1, 2, 2, 3, 4, 5, 6. Three cards are randomly drawn from these 7 cards. Let the minimum value of the numbers on the drawn cards be $\xi$. Find $P\left(\xi =2\right)=$____ and $E\left(\xi \right)=$____. | \frac{12}{7} | 4.6875 |
29,692 | A rectangular piece of paper with dimensions 8 cm by 6 cm is folded in half horizontally. After folding, the paper is cut vertically at 3 cm and 5 cm from one edge, forming three distinct rectangles. Calculate the ratio of the perimeter of the smallest rectangle to the perimeter of the largest rectangle. | \frac{5}{6} | 3.125 |
29,693 | Arrange the letters a, a, b, b, c, c into three rows and two columns, such that in each row and each column, the letters are different. How many different arrangements are there? | 12 | 21.875 |
29,694 | An equilateral triangle with side length $12$ is completely filled with non-overlapping equilateral triangles of side length $2$. Calculate the number of small triangles required. | 36 | 93.75 |
29,695 | In the parallelepiped $ABCD-A_{1}B_{1}C_{1}D_{1}$, where $AB=4$, $AD=3$, $AA_{1}=3$, $\angle BAD=90^{\circ}$, $\angle BAA_{1}=60^{\circ}$, $\angle DAA_{1}=60^{\circ}$, find the length of $AC_{1}$. | \sqrt{55} | 0 |
29,696 | Let $n$ be an odd integer with exactly 12 positive divisors. Find the number of positive divisors of $27n^3$. | 256 | 17.1875 |
29,697 | Let's say you have three numbers A, B, and C, which satisfy the equations $2002C + 4004A = 8008$, and $3003B - 5005A = 7007$. What is the average of A, B, and C?
**A)** $\frac{7}{3}$
**B)** $7$
**C)** $\frac{26}{9}$
**D)** $\frac{22}{9}$
**E)** $0$ | \frac{22}{9} | 4.6875 |
29,698 | On the board, the number 0 is written. Two players take turns appending to the expression on the board: the first player appends a + or - sign, and the second player appends one of the natural numbers from 1 to 1993. The players make 1993 moves each, and the second player uses each of the numbers from 1 to 1993 exactly once. At the end of the game, the second player receives a reward equal to the absolute value of the algebraic sum written on the board. What is the maximum reward the second player can guarantee for themselves? | 1993 | 14.84375 |
29,699 | A club has 20 members and needs to choose 2 members to be co-presidents and 1 member to be a treasurer. In how many ways can the club select its co-presidents and treasurer? | 3420 | 96.09375 |
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