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40.3k
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stringlengths 10
5.15k
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stringlengths 1
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100
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|---|---|---|---|
29,500 | Given an isosceles right triangle \(ABC\) with hypotenuse \(AB\). Point \(M\) is the midpoint of side \(BC\). A point \(K\) is chosen on the smaller arc \(AC\) of the circumcircle of triangle \(ABC\). Point \(H\) is the foot of the perpendicular dropped from \(K\) to line \(AB\). Find the angle \(\angle CAK\), given that \(KH = BM\) and lines \(MH\) and \(CK\) are parallel. | 22.5 | 30.46875 |
29,501 | Given that $(a + 1)x^2 + (a^2 + 1) + 8x = 9$ is a quadratic equation in terms of $x$, find the value of $a$. | 2\sqrt{2} | 0 |
29,502 | Given that the probability of Team A winning a single game is $\frac{2}{3}$, calculate the probability that Team A will win in a "best of three" format, where the first team to win two games wins the match and ends the competition. | \frac{16}{27} | 0.78125 |
29,503 | On the section of the river from $A$ to $B$, the current is so small that it can be ignored; on the section from $B$ to $C$, the current affects the movement of the boat. The boat covers the distance downstream from $A$ to $C$ in 6 hours, and upstream from $C$ to $A$ in 7 hours. If the current on the section from $A$ to $B$ were the same as on the section from $B$ to $C$, the entire journey from $A$ to $C$ would take 5.5 hours. How much time would the boat take to travel upstream from $C$ to $A$ under these conditions? The boat's own speed remains unchanged in all cases. | 7.7 | 0.78125 |
29,504 | There are \( N \geq 5 \) natural numbers written on the board. It is known that the sum of all the numbers is 80, and the sum of any five of them is not more than 19. What is the smallest possible value of \( N \)? | 26 | 2.34375 |
29,505 | Rectangle \(ABCD\) is divided into four parts by \(CE\) and \(DF\). It is known that the areas of three of these parts are \(5\), \(16\), and \(20\) square centimeters, respectively. What is the area of quadrilateral \(ADOE\) in square centimeters? | 19 | 0.78125 |
29,506 | What is the smallest prime factor of 1739? | 1739 | 0 |
29,507 | For how many even positive integers $n$ less than or equal to 800 is $$(\sin t - i\cos t)^n = \sin nt - i\cos nt$$ true for all real $t$? | 200 | 67.96875 |
29,508 | Let $[x]$ represent the greatest integer less than or equal to the real number $x$. How many positive integers $n \leq 1000$ satisfy the condition that $\left[\frac{998}{n}\right]+\left[\frac{999}{n}\right]+\left[\frac{1000}{n}\right]$ is not divisible by 3? | 22 | 82.8125 |
29,509 | In triangle $PQR,$ $PQ = 12,$ $QR = 13,$ $PR = 15,$ and point $H$ is the intersection of the medians. Points $P',$ $Q',$ and $R',$ are the images of $P,$ $Q,$ and $R,$ respectively, after a $180^\circ$ rotation about $H.$ What is the area of the union of the two regions enclosed by the triangles $PQR$ and $P'Q'R'?$ | 20\sqrt{14} | 15.625 |
29,510 | If $\lceil{\sqrt{x}}\rceil=12$, how many possible integer values of $x$ are there? | 25 | 4.6875 |
29,511 | During what year after 1994 did sales increase the most number of dollars? | 2000 | 8.59375 |
29,512 | What percent of square $EFGH$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.7));
xaxis(0,6,Ticks(1.0,NoZero));
yaxis(0,6,Ticks(1.0,NoZero));
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle);
fill((3,0)--(4,0)--(4,4)--(0,4)--(0,3)--(3,3)--cycle);
fill((5,0)--(6,0)--(6,6)--(0,6)--(0,5)--(5,5)--cycle);
label("$E$",(0,0),SW);
label("$F$",(0,6),N);
label("$G$",(6,6),NE);
label("$H$",(6,0),E);[/asy] | 61.11\% | 14.84375 |
29,513 | If
\[
\sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq
\]
for relatively prime positive integers $p,q$ , find $p+q$ .
*Proposed by Michael Kural* | 9901 | 0 |
29,514 | Let \( A = (2,0) \) and \( B = (8,6) \). Let \( P \) be a point on the parabola \( y^2 = 8x \). Find the smallest possible value of \( AP + BP \). | 10 | 60.9375 |
29,515 | Given an ellipse $\frac{x^{2}}{8} + \frac{y^{2}}{2} = 1$, and a point $A(2,1)$ on the ellipse. The slopes of the lines $AB$ and $AC$ connecting point $A$ to two moving points $B$ and $C$ on the ellipse are $k_{1}$ and $k_{2}$, respectively, with $k_{1} + k_{2} = 0$. Determine the slope $k$ of line $BC$. | \frac{1}{2} | 18.75 |
29,516 | Given that $x > 0$, $y > 0$, and $x + 2y = 2$, find the minimum value of $xy$. | \frac{1}{2} | 93.75 |
29,517 | Let the odd function $f(x)$ defined on $\mathbb{R}$ satisfy $f(2-x) = f(x)$, and it is monotonically decreasing on the interval $[0, 1)$. If the equation $f(x) = -1$ has real roots in the interval $[0, 1)$, then the sum of all real roots of the equation $f(x) = 1$ in the interval $[-1, 7]$ is $\_\_\_\_\_\_$. | 12 | 41.40625 |
29,518 | Find $7463_{8} - 3254_{8}$. Express your answer first in base $8$, then convert it to base $10$. | 2183_{10} | 22.65625 |
29,519 | Let $\phi$ be the smallest acute angle for which $\cos \phi,$ $\cos 2 \phi,$ $\cos 3 \phi$ form an arithmetic progression, in some order. Find $\sin \phi.$ | \frac{\sqrt{3}}{2} | 21.875 |
29,520 | Given that the domains of functions f(x) and g(x) are both $\mathbb{R}$, and $f(x) + g(2-x) = 5$, $g(x) - f(x-4) = 7$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, $g(2) = 4$, calculate the value of $\sum _{k=1}^{22}f(k)$. | -24 | 2.34375 |
29,521 | A school organizes a social practice activity during the summer vacation and needs to allocate 8 grade 10 students to company A and company B evenly. Among these students, two students with excellent English scores cannot be allocated to the same company, and neither can the three students with strong computer skills. How many different allocation schemes are there? (Answer with a number) | 36 | 3.90625 |
29,522 | Charlie and Daisy each arrive at a cafe at a random time between 1:00 PM and 3:00 PM. Each stays for 20 minutes. What is the probability that Charlie and Daisy are at the cafe at the same time? | \frac{4}{9} | 0 |
29,523 | The triangle $\triangle ABC$ is an isosceles triangle where $AC = 6$ and $\angle A$ is a right angle. If $I$ is the incenter of $\triangle ABC,$ then what is $BI$? | 6\sqrt{2} - 6 | 7.8125 |
29,524 | In the fourth grade, there are 20 boys and 26 girls. The percentage of the number of boys to the number of girls is %. | 76.9 | 0 |
29,525 | In triangle $ABC$, $AC = \sqrt{7}$, $BC = 2$, and $\angle B = 60^\circ$. What is the area of $\triangle ABC$? | \frac{3\sqrt{3}}{2} | 16.40625 |
29,526 | Given that the polynomial \(x^2 - kx + 24\) has only positive integer roots, find the average of all distinct possibilities for \(k\). | 15 | 100 |
29,527 | Given \( f(x) = \sum_{k=1}^{2017} \frac{\cos k x}{\cos^k x} \), find \( f\left(\frac{\pi}{2018}\right) \). | -1 | 25.78125 |
29,528 | Compute
\[
\frac{(1 + 23) \left( 1 + \dfrac{23}{2} \right) \left( 1 + \dfrac{23}{3} \right) \dotsm \left( 1 + \dfrac{23}{25} \right)}{(1 + 25) \left( 1 + \dfrac{25}{2} \right) \left( 1 + \dfrac{25}{3} \right) \dotsm \left( 1 + \dfrac{25}{23} \right)}.
\] | 600 | 0 |
29,529 | Let $f : [0, 1] \rightarrow \mathbb{R}$ be a monotonically increasing function such that $$ f\left(\frac{x}{3}\right) = \frac{f(x)}{2} $$ $$ f(1 0 x) = 2018 - f(x). $$ If $f(1) = 2018$ , find $f\left(\dfrac{12}{13}\right)$ .
| 2018 | 5.46875 |
29,530 | Given a moving point P in the plane, the difference between the distance from point P to point F(1, 0) and the distance from point P to the y-axis is equal to 1.
(Ⅰ) Find the equation of the trajectory C of the moving point P;
(Ⅱ) Draw two lines $l_1$ and $l_2$ through point F, both with defined slopes and perpendicular to each other. Suppose $l_1$ intersects trajectory C at points A and B, and $l_2$ intersects trajectory C at points D and E. Find the minimum value of $\overrightarrow {AD} \cdot \overrightarrow {EB}$. | 16 | 21.875 |
29,531 | In the triangle \( \triangle ABC \), \( \angle C = 90^{\circ} \), and \( CB > CA \). Point \( D \) is on \( BC \) such that \( \angle CAD = 2 \angle DAB \). If \( \frac{AC}{AD} = \frac{2}{3} \) and \( \frac{CD}{BD} = \frac{m}{n} \) where \( m \) and \( n \) are coprime positive integers, then what is \( m + n \)?
(49th US High School Math Competition, 1998) | 14 | 0.78125 |
29,532 | Five cards have the numbers 101, 102, 103, 104, and 105 on their fronts. On the reverse, each card has one of five different positive integers: \(a, b, c, d,\) and \(e\) respectively. We know that \(a + 2 = b - 2 = 2c = \frac{d}{2} = e^2\).
Gina picks up the card which has the largest integer on its reverse. What number is on the front of Gina's card? | 105 | 10.15625 |
29,533 | Let the function \( f(x) \) defined on \( (0, +\infty) \) satisfy \( f(x) > -\frac{3}{x} \) for any \( x \in (0, +\infty) \) and \( f\left(f(x) + \frac{3}{x}\right) = 2 \). Find \( f(5) \). | \frac{7}{5} | 3.125 |
29,534 | Find the sum of the distinct prime factors of $7^7 - 7^4$. | 24 | 0.78125 |
29,535 | Let points $A = (0,0)$, $B = (2,3)$, $C = (5,4)$, and $D = (6,0)$. Quadrilateral $ABCD$ is divided into two equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\left (\frac{p}{q}, \frac{r}{s} \right )$, where these fractions are in lowest terms. Determine $p + q + r + s$.
A) 50
B) 58
C) 62
D) 66
E) 70 | 58 | 12.5 |
29,536 | Find all real numbers $x$ such that the product $(x + 2i)((x + 1) + 2i)((x + 2) + 2i)((x + 3) + 2i)$ is purely imaginary. | -2 | 0 |
29,537 | Let \( b = \frac{\pi}{2010} \). Find the smallest positive integer \( m \) such that
\[
2\left[\cos(b) \sin(b) + \cos(4b) \sin(2b) + \cos(9b) \sin(3b) + \cdots + \cos(m^2b) \sin(mb)\right]
\]
is an integer. | 67 | 9.375 |
29,538 | 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 | 45.3125 |
29,539 | Define an ordered quadruple of integers $(a, b, c, d)$ as captivating if $1 \le a < b < c < d \le 15$, and $a+d > 2(b+c)$. How many captivating ordered quadruples are there? | 200 | 0 |
29,540 | A hexagon has its vertices alternately on two concentric circles with radii 3 units and 5 units and centered at the origin on a Cartesian plane. Each alternate vertex starting from the origin extends radially outward to the larger circle. Calculate the area of this hexagon. | \frac{51\sqrt{3}}{2} | 2.34375 |
29,541 | A set containing three real numbers can be represented as $\{a, \frac{b}{a}, 1\}$, and also as $\{a^2, a+b, 0\}$. Find the value of $a^{2003} + b^{2004}$. | -1 | 53.125 |
29,542 | If $b$ is an even multiple of $7786$, find the greatest common divisor of $8b^2 + 85b + 200$ and $2b + 10$. | 10 | 35.9375 |
29,543 | From the set of three-digit numbers that do not contain the digits $0,1,2,3,4,5$, several numbers were written down in such a way that no two numbers could be obtained from each other by swapping two adjacent digits. What is the maximum number of such numbers that could have been written? | 40 | 0 |
29,544 | How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1 \le x \le 5$ and $1 \le y \le 3$? | 416 | 13.28125 |
29,545 | Find the maximum number of elements in a set $S$ that satisfies the following conditions:
(1) Every element in $S$ is a positive integer not exceeding 100.
(2) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the greatest common divisor (gcd) of $a + b$ and $c$ is 1.
(3) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the gcd of $a + b$ and $c$ is greater than 1. | 50 | 71.875 |
29,546 | Let $\triangle XYZ$ have side lengths $XY=15$, $XZ=20$, and $YZ=25$. Inside $\angle XYZ$, there are two circles: one is tangent to the rays $\overline{XY}$, $\overline{XZ}$, and the segment $\overline{YZ}$, while the other is tangent to the extension of $\overline{XY}$ beyond $Y$, $\overline{XZ}$, and $\overline{YZ}$. Compute the distance between the centers of these two circles. | 25 | 0.78125 |
29,547 | Geoff and Trevor each roll a fair eight-sided die (the sides are labeled 1 through 8). What is the probability that the product of the numbers they roll is even or a prime number? | \frac{7}{8} | 3.90625 |
29,548 | Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded region). What is the maximum number of resulting regions? | 22 | 0 |
29,549 | Define $\operatorname{gcd}(a, b)$ as the greatest common divisor of integers $a$ and $b$. Given that $n$ is the smallest positive integer greater than 1000 that satisfies:
$$
\begin{array}{l}
\operatorname{gcd}(63, n+120) = 21, \\
\operatorname{gcd}(n+63, 120) = 60
\end{array}
$$
Then the sum of the digits of $n$ is ( ). | 18 | 86.71875 |
29,550 | A $1 \times n$ rectangle ( $n \geq 1 $ ) is divided into $n$ unit ( $1 \times 1$ ) squares. Each square of this rectangle is colored red, blue or green. Let $f(n)$ be the number of colourings of the rectangle in which there are an even number of red squares. What is the largest prime factor of $f(9)/f(3)$ ? (The number of red squares can be zero.) | 37 | 59.375 |
29,551 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $|\overrightarrow{a}-\overrightarrow{b}|=\sqrt{7}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 96.09375 |
29,552 | Two cars, $A$ and $B$, depart from one city to another. In the first 5 minutes, they traveled the same distance. Then, due to an engine failure, $B$ had to reduce its speed to 2/5 of its original speed, and thus arrived at the destination 15 minutes after car $A$, which continued at a constant speed. If the failure had occurred 4 km farther from the starting point, $B$ would have arrived only 10 minutes after $A$. How far apart are the two cities? | 18 | 7.8125 |
29,553 | Given the original spherical dome has a height of $55$ meters and can be represented as holding $250,000$ liters of air, and Emily's scale model can hold only $0.2$ liters of air, determine the height, in meters, of the spherical dome in Emily's model. | 0.5 | 0.78125 |
29,554 | Two positive integers that only have 1 as a common factor are called coprime numbers. For example, 2 and 7 are coprime, as are 3 and 4. In any permutation of 2, 3, 4, 5, 6, 7, where each pair of adjacent numbers are coprime, there are a total of \_\_\_\_\_\_\_\_ different permutations (answer with a number). | 72 | 100 |
29,555 | What is the minimum number of straight cuts needed to cut a cake in 100 pieces? The pieces do not need to be the same size or shape but cannot be rearranged between cuts. You may assume that the cake is a large cube and may be cut from any direction. | 11 | 0 |
29,556 | Observe the table below:
1,
2,3
4,5,6,7
8,9,10,11,12,13,14,15,
...
Question: (1) What is the last number of the $n$th row in this table?
(2) What is the sum of all numbers in the $n$th row?
(3) Which row and position is the number 2008? | 985 | 21.09375 |
29,557 | Find the smallest positive integer $n\neq 2004$ for which there exists a polynomial $f\in\mathbb{Z}[x]$ such that the equation $f(x)=2004$ has at least one, and the equation $f(x)=n$ has at least $2004$ different integer solutions. | (1002!)^2 + 2004 | 0 |
29,558 | Let \( g_{1}(x) = \sqrt{2-x} \), and for integers \( n \geq 2 \), define
\[ g_{n}(x) = g_{n-1}\left(\sqrt{(n+1)^2 - x}\right) \].
Let \( M \) be the largest value of \( n \) for which the domain of \( g_n \) is non-empty. For this value of \( M \), the domain of \( g_M \) consists of a single point \(\{d\}\). Compute \( d \). | 25 | 2.34375 |
29,559 | Calculate the area of the crescent moon enclosed by the portion of the circle of radius 4 centered at (0,0) that lies in the first quadrant, the portion of the circle with radius 2 centered at (0,1) that lies in the first quadrant, and the line segment from (0,0) to (4,0). | 2\pi | 0 |
29,560 | Juan takes a number, adds 3 to it, squares the result, then multiplies the answer by 2, subtracts 3 from the result, and finally divides that number by 2. If his final answer is 49, what was the original number? | \sqrt{\frac{101}{2}} - 3 | 0 |
29,561 | How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle). | 12 | 36.71875 |
29,562 | Twelve tiles numbered $1$ through $12$ are turned up at random, and an 8-sided die (sides numbered from 1 to 8) is rolled. Calculate the probability that the product of the numbers on the tile and the die will be a square. | \frac{7}{48} | 24.21875 |
29,563 | In the plane rectangular coordinate system $xOy$, it is known that $MN$ is a chord of the circle $C: (x-2)^{2} + (y-4)^{2} = 2$, and satisfies $CM\perp CN$. Point $P$ is the midpoint of $MN$. As the chord $MN$ moves on the circle $C$, there exist two points $A$ and $B$ on the line $2x-y-3=0$, such that $\angle APB \geq \frac{\pi}{2}$ always holds. Find the minimum value of the length of segment $AB$. | \frac{6\sqrt{5}}{5} + 2 | 0.78125 |
29,564 | Given $0 < \beta < \frac{\pi}{2} < \alpha < \pi$ and $\cos\left(\alpha - \frac{\beta}{2}\right) = -\frac{1}{9}$, $\sin\left(\frac{\alpha}{2} - \beta\right) = \frac{2}{3}$, find the value of $\cos(\alpha + \beta)$. | -\frac{239}{729} | 53.90625 |
29,565 | If vectors $\mathbf{a} = (1,0)$, $\mathbf{b} = (0,1)$, $\mathbf{c} = k\mathbf{a} + \mathbf{b}$ ($k \in \mathbb{R}$), $\mathbf{d} = \mathbf{a} - \mathbf{b}$, and $\mathbf{c} \parallel \mathbf{d}$, calculate the value of $k$ and the direction of vector $\mathbf{c}$. | -1 | 1.5625 |
29,566 | Given the coordinates of the foci of an ellipse are $F_{1}(-1,0)$, $F_{2}(1,0)$, and a line perpendicular to the major axis through $F_{2}$ intersects the ellipse at points $P$ and $Q$, with $|PQ|=3$.
$(1)$ Find the equation of the ellipse;
$(2)$ A line $l$ through $F_{2}$ intersects the ellipse at two distinct points $M$ and $N$. Does the area of the incircle of $\triangle F_{1}MN$ have a maximum value? If it exists, find this maximum value and the equation of the line at this time; if not, explain why. | \frac {9}{16}\pi | 0 |
29,567 | There are 100 points located on a line. Mark the midpoints of all possible segments with endpoints at these points. What is the minimum number of marked points that can result? | 197 | 7.8125 |
29,568 | The sum of the digits of the integer equal to \( 777777777777777^2 - 222222222222223^2 \) can be found by evaluating the expression. | 74 | 1.5625 |
29,569 | In the third year of high school, the class organized a fun sports competition. After multiple rounds of competition, Class A and Class B entered the finals. There were three events in the finals, with the winner of each event receiving 2 points and the loser receiving -1 point. There were no draws. The class with the highest total score after the three events would be the champion. It is known that the probabilities of Class A winning in the three events are 0.4, 0.5, and 0.8 respectively, and the results of each event are independent of each other.
$(1)$ Find the probability of Class A winning the championship.
$(2)$ Let $X$ represent the total score of Class B. Find the distribution table and expectation of $X$. | 0.9 | 0 |
29,570 | How many 6-digit numbers have at least two zeros? | 73,314 | 0 |
29,571 | (1) Given $$x^{ \frac {1}{2}}+x^{- \frac {1}{2}}=3$$, find the value of $x+x^{-1}$;
(2) Calculate $$( \frac {1}{8})^{- \frac {1}{3}}-3^{\log_{3}2}(\log_{3}4)\cdot (\log_{8}27)+2\log_{ \frac {1}{6}} \sqrt {3}-\log_{6}2$$. | -3 | 90.625 |
29,572 | Through the vertices \( A, C, D \) of the parallelogram \( ABCD \) with sides \( AB = 7 \) and \( AD = 4 \), a circle is drawn that intersects the line \( BD \) at point \( E \), and \( DE = 13 \). Find the length of diagonal \( BD \). | 15 | 2.34375 |
29,573 | In trapezoid \(ABCD\), the bases \(AD\) and \(BC\) are equal to 8 and 18, respectively. It is known that the circumcircle of triangle \(ABD\) is tangent to lines \(BC\) and \(CD\). Find the perimeter of the trapezoid. | 56 | 3.125 |
29,574 | Given $|\vec{a}|=1$, $|\vec{b}|=6$, and $\vec{a}\cdot(\vec{b}-\vec{a})=2$, calculate the angle between $\vec{a}$ and $\vec{b}$. | \dfrac{\pi}{3} | 96.09375 |
29,575 | Find the smallest possible side of a square in which five circles of radius $1$ can be placed, so that no two of them have a common interior point. | 2 + 2\sqrt{2} | 27.34375 |
29,576 | The amplitude, period, frequency, phase, and initial phase of the function $y=3\sin \left( \frac {1}{2}x- \frac {\pi}{6}\right)$ are ______, ______, ______, ______, ______, respectively. | - \frac {\pi}{6} | 8.59375 |
29,577 | In how many ways can you rearrange the letters of ‘Alejandro’ such that it contains one of the words ‘ned’ or ‘den’? | 40320 | 0 |
29,578 | In the expression $x \cdot y^z - w$, the values of $x$, $y$, $z$, and $w$ are 1, 2, 3, and 4, although not necessarily in that order. What is the maximum possible value of the expression? | 161 | 0 |
29,579 | Given that \(9^{-1} \equiv 90 \pmod{101}\), find \(81^{-1} \pmod{101}\), as a residue modulo 101. (Give an answer between 0 and 100, inclusive.) | 20 | 56.25 |
29,580 | Given that $F$ is the focus of the parabola $C_{1}$: $y^{2}=2ρx (ρ > 0)$, and point $A$ is a common point of one of the asymptotes of the hyperbola $C_{2}$: $\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1 (a > 0, b > 0)$ and $AF \perp x$-axis, find the eccentricity of the hyperbola. | \sqrt{5} | 2.34375 |
29,581 | The positive integer divisors of 294, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 21? | 49 | 3.90625 |
29,582 | Compute
\[\frac{\lfloor \sqrt{1} \rfloor \cdot \lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{3} \rfloor \cdot \lfloor \sqrt{5} \rfloor \dotsm \lfloor \sqrt{15} \rfloor}{\lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{4} \rfloor \cdot \lfloor \sqrt{6} \rfloor \dotsm \lfloor \sqrt{16} \rfloor}.\] | \frac{3}{8} | 3.90625 |
29,583 | What is the least integer whose square is 36 more than three times its value? | -6 | 4.6875 |
29,584 | Calculate the sum of all four-digit numbers that can be formed using the digits 0, 1, 2, 3, and 4, with no repeated digits. | 259980 | 23.4375 |
29,585 | The cards in a stack are numbered consecutively from 1 to $2n$ from top to bottom. The top $n$ cards are removed to form pile $A$ and the remaining cards form pile $B$. The cards are restacked by alternating cards from pile $B$ and $A$, starting with a card from $B$. Given this process, find the total number of cards ($2n$) in the stack if card number 201 retains its original position. | 402 | 32.8125 |
29,586 | A sequence of positive integers \(a_{1}, a_{2}, \ldots\) is such that for each \(m\) and \(n\) the following holds: if \(m\) is a divisor of \(n\) and \(m < n\), then \(a_{m}\) is a divisor of \(a_{n}\) and \(a_{m} < a_{n}\). Find the least possible value of \(a_{2000}\). | 128 | 0 |
29,587 | Design a computer operation program:
1. Initial values \( x = 3 \), \( y = 0 \).
2. \( x = x + 2 \).
3. \( y = y + x \).
4. If \( y \geqslant 10000 \), proceed to (5); otherwise, go back to (2).
5. Print \( x \).
6. Stop running. What will be the printed result when this program is executed? | 201 | 97.65625 |
29,588 | Find the $2019$ th strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$ . | 37805 | 9.375 |
29,589 | On a circle, points $A,B,C,D$ lie counterclockwise in this order. Let the orthocenters of $ABC,BCD,CDA,DAB$ be $H,I,J,K$ respectively. Let $HI=2$ , $IJ=3$ , $JK=4$ , $KH=5$ . Find the value of $13(BD)^2$ . | 169 | 10.9375 |
29,590 | Determine the number of $8$ -tuples $(\epsilon_1, \epsilon_2,...,\epsilon_8)$ such that $\epsilon_1, \epsilon_2, ..., 8 \in \{1,-1\}$ and $\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 +...+ 8\epsilon_8$ is a multiple of $3$ . | 88 | 77.34375 |
29,591 | An electronic clock displays time from 00:00:00 to 23:59:59. How much time throughout the day does the clock show a number that reads the same forward and backward? | 96 | 46.875 |
29,592 | Given that point $P$ is a moving point on the parabola $y^{2}=2x$, find the minimum value of the sum of the distance from point $P$ to point $(0,2)$ and the distance from $P$ to the directrix of the parabola. | \dfrac { \sqrt {17}}{2} | 0 |
29,593 | Given that \( P Q R S \) is a square, and point \( O \) is on the line \( R Q \) such that the distance from \( O \) to point \( P \) is 1, what is the maximum possible distance from \( O \) to point \( S \)? | \frac{1 + \sqrt{5}}{2} | 2.34375 |
29,594 | Monica is renovating the floor of her 15-foot by 20-foot dining room. She plans to place two-foot by two-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. Additionally, there is a 1-foot by 1-foot column at the center of the room that needs to be tiled as well. Determine the total number of tiles she will use. | 78 | 3.125 |
29,595 | Two students are preparing to register for the independent admission tests of Zhejiang University, Fudan University, and Shanghai Jiao Tong University, with the requirement that each person can choose at most two schools. Calculate the number of different registration results. | 36 | 24.21875 |
29,596 | The number obtained from the last two nonzero digits of $80!$ is equal to $n$. Find the value of $n$. | 76 | 0 |
29,597 | In triangle \(ABC\), the perpendicular bisectors of sides \(AB\) and \(AC\) are drawn, intersecting lines \(AC\) and \(AB\) at points \(N\) and \(M\) respectively. The length of segment \(NM\) is equal to the length of side \(BC\) of the triangle. The angle at vertex \(C\) of the triangle is \(40^\circ\). Find the angle at vertex \(B\) of the triangle. | 50 | 0.78125 |
29,598 | How many four-digit numbers contain one even digit and three odd digits, with no repeated digits? | 1140 | 10.9375 |
29,599 | On a circle, points \(B\) and \(D\) are located on opposite sides of the diameter \(AC\). It is known that \(AB = \sqrt{6}\), \(CD = 1\), and the area of triangle \(ABC\) is three times the area of triangle \(BCD\). Find the radius of the circle. | 1.5 | 0 |
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