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A room is shaped like an 'L'. One part is a rectangle that is 23 feet long and 15 feet wide, and a square with a side of 8 feet is attached at one end of the rectangle, extending its width. Calculate the ratio of the total length of the room to its perimeter.
1:4
0
30,401
Solve the equations: ① $3(x-1)^3 = 24$; ② $(x-3)^2 = 64$.
-5
21.09375
30,402
Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. At least how many problems are needed?
13
5.46875
30,403
Given complex numbers $u$ and $v$ such that $|u+v| = 3$ and $|u^2 + v^2| = 10,$ determine the largest possible value of $|u^3 + v^3|.$
31.5
0.78125
30,404
Let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, \pi]$ and $[-\frac{\pi}{2}, \frac{\pi}{2}]$, respectively. Define $P(\alpha)$ as the probability that \[\cos^2{x} + \cos^2{y} < \alpha\] where $\alpha$ is a constant with $1 < \alpha \leq 2$. Find the maximum value of $P(\alpha)$. A) $\frac{\pi}{4}$ B) $\frac{3\pi}{4}$ C) $\pi$ D) $\frac{\pi}{2}$ E) 1
\frac{\pi}{2}
10.9375
30,405
$O$ is the origin, and $F$ is the focus of the parabola $C:y^{2}=4x$. A line passing through $F$ intersects $C$ at points $A$ and $B$, and $\overrightarrow{FA}=2\overrightarrow{BF}$. Find the area of $\triangle OAB$.
\dfrac{3\sqrt{2}}{2}
32.03125
30,406
A "progressive number" refers to a positive integer in which, except for the highest digit, each digit is greater than the digit to its left (for example, 13456 and 35678 are both five-digit "progressive numbers"). (I) There are _______ five-digit "progressive numbers" (answer in digits); (II) If all the five-digit "progressive numbers" are arranged in ascending order, the 110th five-digit "progressive number" is _______.
34579
3.125
30,407
A spinner has four sections labeled 1, 2, 3, and 4, each section being equally likely to be selected. If you spin the spinner three times to form a three-digit number, with the first outcome as the hundreds digit, the second as the tens digit, and the third as the unit digit, what is the probability that the formed number is divisible by 8? Express your answer as a common fraction.
\frac{1}{8}
70.3125
30,408
In a rectangular coordinate system, a point whose coordinates are both integers is called a lattice point. How many lattice points \((x, y)\) satisfy the inequality \((|x|-1)^{2} + (|y|-1)^{2} < 2\)?
16
80.46875
30,409
Find the number of real solutions to the equation \[\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{120}{x - 120} = x.\]
121
68.75
30,410
Consider the infinite series defined by the following progression: \[2 + \frac{1}{3} + \frac{1}{9} + \frac{1}{3^2} + \frac{1}{9^2} + \frac{1}{3^3} + \frac{1}{9^3} + \cdots\] Determine the limit of this series as it extends to infinity. A) $\frac{1}{3}$ B) $3$ C) $\frac{21}{8}$ D) $2\frac{5}{8}$ E) $2.5$
\frac{21}{8}
26.5625
30,411
Given point $P$ is a moving point on the ellipse $\frac{x^2}{8} + \frac{y^2}{4} = 1$ in the first quadrant, through point $P$, two tangents $PA$ and $PB$ are drawn to the circle $x^2 + y^2 = 4$, with the points of tangency being $A$ and $B$ respectively. The line $AB$ intersects the $x$-axis and $y$-axis at points $M$ and $N$ respectively. Find the minimum value of the area of $\triangle OMN$.
\sqrt{2}
1.5625
30,412
Add the square of the smallest area from squares of size $1 \times 1, 2 \times 2,$ and $3 \times 3,$ such that the number of squares of each size is the same.
14
28.125
30,413
Given that the four vertices of the quadrilateral $MNPQ$ are on the graph of the function $f(x)=\log_{\frac{1}{2}} \frac{ax+1}{x+b}$, and it satisfies $\overrightarrow{MN}= \overrightarrow{QP}$, where $M(3,-1)$, $N\left( \frac{5}{3},-2\right)$, then the area of the quadrilateral $MNPQ$ is \_\_\_\_\_\_.
\frac{26}{3}
0.78125
30,414
A circle has an area of $16\pi$ square units. What are the lengths of the circle's diameter and circumference, in units?
8\pi
18.75
30,415
How many distinct sequences of five letters can be made from the letters in FREQUENCY if each sequence must begin with F, end with Y, and no letter can appear in a sequence more than once? Further, the second letter must be a vowel.
60
14.84375
30,416
Given the parabola \( C: x^{2} = 2py \) with \( p > 0 \), two tangents \( RA \) and \( RB \) are drawn from the point \( R(1, -1) \) to the parabola \( C \). The points of tangency are \( A \) and \( B \). Find the minimum area of the triangle \( \triangle RAB \) as \( p \) varies.
3 \sqrt{3}
0.78125
30,417
Among four people, A, B, C, and D, they pass a ball to each other. The first pass is from A to either B, C, or D, and the second pass is from the receiver to any of the other three. This process continues for several passes. Calculate the number of ways the ball can be passed such that it returns to A on the fourth pass.
21
18.75
30,418
The base of the quadrilateral pyramid \( S A B C D \) is a rhombus \( A B C D \) with an acute angle at vertex \( A \). The height of the rhombus is 4, and the point of intersection of its diagonals is the orthogonal projection of vertex \( S \) onto the plane of the base. A sphere with radius 2 touches the planes of all the faces of the pyramid. Find the volume of the pyramid, given that the distance from the center of the sphere to the line \( A C \) is \( \frac{2 \sqrt{2}}{3} A B \).
8\sqrt{2}
6.25
30,419
Let $b_0 = \sin^2 \left( \frac{\pi}{30} \right)$ and for $n \geq 0$, \[ b_{n + 1} = 4b_n (1 - b_n). \] Find the smallest positive integer $n$ such that $b_n = b_0$.
15
38.28125
30,420
Given the function $f(x)=\ln (1+x)- \frac {x(1+λx)}{1+x}$, if $f(x)\leqslant 0$ when $x\geqslant 0$, calculate the minimum value of $λ$.
\frac {1}{2}
64.84375
30,421
Given that the last initial of Mr. and Mrs. Alpha's baby's monogram is 'A', determine the number of possible monograms in alphabetical order with no letter repeated.
300
71.09375
30,422
Let the set $U=\{1, 3a+5, a^2+1\}$, $A=\{1, a+1\}$, and $\mathcal{C}_U A=\{5\}$. Find the value of $a$.
-2
45.3125
30,423
Evaluate the product $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{6561} \cdot \frac{19683}{1}$.
729
13.28125
30,424
In a bike shed, there are bicycles (two wheels), tricycles, and cars (four wheels). The number of bicycles is four times the number of cars. Several students counted the total number of wheels in the shed, but each of them obtained a different count: $235, 236, 237, 238, 239$. Among these, one count is correct. Smart kid, please calculate the number of different combinations of the three types of vehicles that satisfy the given conditions. (For example, if there are 1 bicycle, 2 tricycles, and 3 cars or 3 bicycles, 2 tricycles, and 1 car, it counts as two different combinations).
19
6.25
30,425
Given the ellipse $\frac{x^2}{16} + \frac{y^2}{b^2} = 1$, a line passing through its left focus intersects the ellipse at points $A$ and $B$, and the maximum value of $|AF_{2}| + |BF_{2}|$ is $10$. Find the eccentricity of the ellipse.
\frac{1}{2}
5.46875
30,426
Three of the four vertices of a square are $(2, 8)$, $(13, 8)$, and $(13, -3)$. What is the area of the intersection of this square region and the region inside the graph of the equation $(x - 2)^2 + (y + 3)^2 = 16$?
4\pi
69.53125
30,427
From the 1000 natural numbers ranging from 1 to 1000, a certain number of them are drawn. If the sum of any two numbers drawn is not a multiple of 5, then the maximum number of numbers that can be drawn from these 1000 natural numbers is     .
401
65.625
30,428
Determine the number of times and the positions in which it appears $\frac12$ in the following sequence of fractions: $$ \frac11, \frac21, \frac12 , \frac31 , \frac22 , \frac13 , \frac41,\frac32,\frac23,\frac14,..., \frac{1}{1992} $$
664
3.125
30,429
In the arithmetic sequence $\{a_{n}\}$, given that $a_{3}=-2, a_{n}=\frac{3}{2}, S_{n}=-\frac{15}{2}$, find the value of $a_{1}$.
-\frac{19}{6}
2.34375
30,430
Calculate:<br/>$(1)(\sqrt{\frac{1}{3}})^{2}+\sqrt{0.{3}^{2}}-\sqrt{\frac{1}{9}}$;<br/>$(2)(\sqrt{6}-\sqrt{\frac{1}{2}})-(\sqrt{24}+2\sqrt{\frac{2}{3}})$;<br/>$(3)(\frac{\sqrt{32}}{3}-4\sqrt{\frac{1}{2}}+3\sqrt{27})÷2\sqrt{2}$;<br/>$(4)(\sqrt{3}+\sqrt{2}-1)(\sqrt{3}-\sqrt{2}+1)$.
2\sqrt{2}
78.125
30,431
A 5-digit natural number \(abcde\) is called a "\(\pi_1\)" number if and only if it satisfies \(a < b < c\) and \(c > d > e\). Determine the total number of "\(\pi_1\)" numbers among all 5-digit numbers.
2142
1.5625
30,432
The postal department stipulates that for letters weighing up to $100$ grams (including $100$ grams), each $20$ grams requires a postage stamp of $0.8$ yuan. If the weight is less than $20$ grams, it is rounded up to $20$ grams. For weights exceeding $100$ grams, the initial postage is $4$ yuan. For each additional $100$ grams beyond $100$ grams, an extra postage of $2$ yuan is required. In Class 8 (9), there are $11$ students participating in a project to learn chemistry knowledge. If each answer sheet weighs $12$ grams and each envelope weighs $4$ grams, and these $11$ answer sheets are divided into two envelopes for mailing, the minimum total amount of postage required is ____ yuan.
5.6
12.5
30,433
How many four-digit numbers, without repeating digits, that can be formed using the digits 0, 1, 2, 3, 4, 5, are divisible by 25?
21
1.5625
30,434
A set \( \mathcal{T} \) of distinct positive integers has the following property: for every integer \( y \) in \( \mathcal{T}, \) the arithmetic mean of the set of values obtained by deleting \( y \) from \( \mathcal{T} \) is an integer. Given that 2 belongs to \( \mathcal{T} \) and that 1024 is the largest element of \( \mathcal{T}, \) what is the greatest number of elements that \( \mathcal{T} \) can have?
15
3.125
30,435
Given the universal set $U=\{2,3,5\}$, and $A=\{x|x^2+bx+c=0\}$. If $\complement_U A=\{2\}$, then $b=$ ____, $c=$ ____.
15
20.3125
30,436
Find the smallest number \( n > 1980 \) such that the number $$ \frac{x_{1} + x_{2} + x_{3} + \ldots + x_{n}}{5} $$ is an integer for any given integer values \( x_{1}, x_{2}, x_{3}, \ldots, x_{n} \), none of which is divisible by 5.
1985
76.5625
30,437
Given that the point F(0,1) is the focus of the parabola $x^2=2py$, (1) Find the equation of the parabola C; (2) Points A, B, and C are three points on the parabola such that $\overrightarrow{FA} + \overrightarrow{FB} + \overrightarrow{FC} = \overrightarrow{0}$, find the maximum value of the area of triangle ABC.
\frac{3\sqrt{6}}{2}
8.59375
30,438
Calculate the remainder when $1 + 11 + 11^2 + \cdots + 11^{1024}$ is divided by $500$.
25
0
30,439
Let $\triangle ABC$ be a triangle with $AB=5, BC=6, CA=7$ . Suppose $P$ is a point inside $\triangle ABC$ such that $\triangle BPA\sim \triangle APC$ . If $AP$ intersects $BC$ at $X$ , find $\frac{BX}{CX}$ . [i]Proposed by Nathan Ramesh
25/49
7.03125
30,440
What is the smallest positive integer $n$ such that $\sqrt{n}-\sqrt{n-1}<0.02$? A) 624 B) 625 C) 626 D) 627 E) 628
626
90.625
30,441
Given the function $f\left(x\right)=ax^{2}-bx-1$, sets $P=\{1,2,3,4\}$, $Q=\{2,4,6,8\}$, if a number $a$ and a number $b$ are randomly selected from sets $P$ and $Q$ respectively to form a pair $\left(a,b\right)$.<br/>$(1)$ Let event $A$ be "the monotonically increasing interval of the function $f\left(x\right)$ is $\left[1,+\infty \right)$", find the probability of event $A$;<br/>$(2)$ Let event $B$ be "the equation $|f\left(x\right)|=2$ has $4$ roots", find the probability of event $B$.
\frac{11}{16}
17.1875
30,442
Given that 20% of the birds are geese, 40% are swans, 10% are herons, and 20% are ducks, and the remaining are pigeons, calculate the percentage of the birds that are not herons and are ducks.
22.22\%
10.9375
30,443
A factory received a task to process 6000 pieces of part P and 2000 pieces of part Q. The factory has 214 workers. Each worker spends the same amount of time processing 5 pieces of part P as they do processing 3 pieces of part Q. The workers are divided into two groups to work simultaneously on different parts. In order to complete this batch of tasks in the shortest time, the number of people processing part P is \_\_\_\_\_\_.
137
14.0625
30,444
In the polar coordinate system, the equation of circle C is given by ρ = 2sinθ, and the equation of line l is given by $ρsin(θ+ \frac {π}{3})=a$. If line l is tangent to circle C, find the value of the real number a.
- \frac {1}{2}
29.6875
30,445
Find the maximum and minimum values of the function $f(x)=1+x-x^{2}$ in the interval $[-2,4]$.
-11
52.34375
30,446
The maximum value of the function $y=4^x+2^{x+1}+5$, where $x\in[1,2]$, is to be found.
29
78.90625
30,447
Given that in $\triangle ABC$, $\sin A + 2 \sin B \cos C = 0$, find the maximum value of $\tan A$.
\frac{\sqrt{3}}{3}
59.375
30,448
Find the area of the triangle formed by the axis of the parabola $y^{2}=8x$ and the two asymptotes of the hyperbola $(C)$: $\frac{x^{2}}{8}-\frac{y^{2}}{4}=1$.
2\sqrt{2}
2.34375
30,449
A metal bar at a temperature of $20^{\circ} \mathrm{C}$ is placed in water at a temperature of $100^{\circ} \mathrm{C}$. After thermal equilibrium is established, the temperature becomes $80^{\circ} \mathrm{C}$. Then, without removing the first bar, another identical metal bar also at $20^{\circ} \mathrm{C}$ is placed in the water. What will be the temperature of the water after thermal equilibrium is established?
68
3.90625
30,450
The area of an equilateral triangle ABC is 36. Points P, Q, R are located on BC, AB, and CA respectively, such that BP = 1/3 BC, AQ = QB, and PR is perpendicular to AC. Find the area of triangle PQR.
10
12.5
30,451
Sixty points, of which thirty are coloured red, twenty are coloured blue and ten are coloured green, are marked on a circle. These points divide the circle into sixty arcs. Each of these arcs is assigned a number according to the colours of its endpoints: an arc between a red and a green point is assigned a number $1$ , an arc between a red and a blue point is assigned a number $2$ , and an arc between a blue and a green point is assigned a number $3$ . The arcs between two points of the same colour are assigned a number $0$ . What is the greatest possible sum of all the numbers assigned to the arcs?
180
2.34375
30,452
The product \( 29 \cdot 11 \), and the numbers 1059, 1417, and 2312, are each divided by \( d \). If the remainder is always \( r \), where \( d \) is an integer greater than 1, what is \( d - r \) equal to?
15
62.5
30,453
Alice celebrated her birthday on Friday, March 15 in the year 2012. Determine the next year when her birthday will next fall on a Monday. A) 2021 B) 2022 C) 2023 D) 2024 E) 2025
2025
0.78125
30,454
A certain teacher received $10$, $6$, $8$, $5$, $6$ letters from Monday to Friday, then the variance of this data set is $s^{2}=$____.
3.2
93.75
30,455
What is the largest number, with its digits all different and none of them being zero, whose digits add up to 20?
9821
0.78125
30,456
In square $ABCD$, a point $P$ is chosen at random. The probability that $\angle APB < 90^{\circ}$ is ______.
1 - \frac{\pi}{8}
4.6875
30,457
The rules of table tennis competition stipulate: In a game, before the opponent's score reaches 10-all, one side serves twice consecutively, then the other side serves twice consecutively, and so on. Each serve, the winning side scores 1 point, and the losing side scores 0 points. In a game between player A and player B, the probability of the server scoring 1 point on each serve is 0.6, and the outcomes of each serve are independent of each other. Player A serves first in a game. (1) Find the probability that the score is 1:2 in favor of player B at the start of the fourth serve; (2) Find the probability that player A is leading in score at the start of the fifth serve.
0.3072
1.5625
30,458
An ellipse whose axes are parallel to the coordinate axes is tangent to the $x$-axis at $(6, 0)$ and tangent to the $y$-axis at $(0, 2)$. Find the distance between the foci of the ellipse.
4\sqrt{2}
7.03125
30,459
Suppose that \( ABCDEF \) is a regular hexagon with sides of length 6. Each interior angle of \( ABCDEF \) is equal to \( 120^{\circ} \). (a) A circular arc with center \( D \) and radius 6 is drawn from \( C \) to \( E \). Determine the area of the shaded sector. (b) A circular arc with center \( D \) and radius 6 is drawn from \( C \) to \( E \), and a second arc with center \( A \) and radius 6 is drawn from \( B \) to \( F \). These arcs are tangent (touch) at the center of the hexagon. Line segments \( BF \) and \( CE \) are also drawn. Determine the total area of the shaded regions. (c) Along each edge of the hexagon, a semi-circle with diameter 6 is drawn. Determine the total area of the shaded regions; that is, determine the total area of the regions that lie inside exactly two of the semi-circles.
18\pi - 27\sqrt{3}
1.5625
30,460
Calculate the value of $x$ when the arithmetic mean of the following five expressions is 30: $$x + 10 \hspace{.5cm} 3x - 5 \hspace{.5cm} 2x \hspace{.5cm} 18 \hspace{.5cm} 2x + 6$$
15.125
0
30,461
In the Cartesian coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system using the same unit of length. The parametric equations of line $l$ are given by $\begin{cases}x=2+t \\ y=1+t\end{cases} (t \text{ is a parameter})$, and the polar coordinate equation of circle $C$ is given by $\rho=4 \sqrt{2}\sin (θ+ \dfrac{π}{4})$. (1) Find the standard equation of line $l$ and the Cartesian coordinate equation of circle $C$; (2) Let points $A$ and $B$ be the intersections of curve $C$ and line $l$, and point $P$ has Cartesian coordinates $(2,1)$. Find the value of $||PA|-|PB||$.
\sqrt{2}
72.65625
30,462
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b\cos C=3a\cos B-c\cos B$, $\overrightarrow{BA}\cdot \overrightarrow{BC}=2$, find the area of $\triangle ABC$.
2\sqrt{2}
68.75
30,463
The equilateral triangle has sides of \(2x\) and \(x+15\) as shown. Find the perimeter of the triangle in terms of \(x\).
90
93.75
30,464
Given Mindy made four purchases for $2.96, 6.57, 8.49, and 12.38. Each amount needs to be rounded up to the nearest dollar except the amount closest to a whole number, which should be rounded down. Calculate the total rounded amount.
31
14.0625
30,465
Given a line $y=-x+1$ and an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), they intersect at points A and B. (1) If the eccentricity of the ellipse is $\frac{\sqrt{3}}{3}$ and the focal distance is 2, find the length of the segment AB. (2) (For Liberal Arts students) If segment OA is perpendicular to segment OB (where O is the origin), find the value of $\frac{1}{a^2} + \frac{1}{b^2}$. (3) (For Science students) If segment OA is perpendicular to segment OB (where O is the origin), and when the eccentricity of the ellipse $e$ lies in the interval $\left[ \frac{1}{2}, \frac{\sqrt{2}}{2} \right]$, find the maximum length of the major axis of the ellipse.
\sqrt{6}
7.03125
30,466
Triangle $ABC$ has $\angle BAC=90^\circ$ . A semicircle with diameter $XY$ is inscribed inside $\triangle ABC$ such that it is tangent to a point $D$ on side $BC$ , with $X$ on $AB$ and $Y$ on $AC$ . Let $O$ be the midpoint of $XY$ . Given that $AB=3$ , $AC=4$ , and $AX=\tfrac{9}{4}$ , compute the length of $AO$ .
39/32
0.78125
30,467
In triangle $ABC$, it is known that $\sqrt{3}\sin{2B} = 1 - \cos{2B}$. (1) Find the value of angle $B$; (2) If $BC = 2$ and $A = \frac{\pi}{4}$, find the area of triangle $ABC$.
\frac{3 + \sqrt{3}}{2}
73.4375
30,468
Given any point $P$ on the line $l: x-y+4=0$, two tangent lines $AB$ are drawn to the circle $O: x^{2}+y^{2}=4$ with tangent points $A$ and $B$. The line $AB$ passes through a fixed point ______; let the midpoint of segment $AB$ be $Q$. The minimum distance from point $Q$ to the line $l$ is ______.
\sqrt{2}
43.75
30,469
What is the sum of the different prime factors of $210630$?
93
0
30,470
Two circles \( C_{1} \) and \( C_{2} \) touch each other externally and the line \( l \) is a common tangent. The line \( m \) is parallel to \( l \) and touches the two circles \( C_{1} \) and \( C_{3} \). The three circles are mutually tangent. If the radius of \( C_{2} \) is 9 and the radius of \( C_{3} \) is 4, what is the radius of \( C_{1} \)?
12
5.46875
30,471
Given that \( x \) and \( y \) are non-zero real numbers and they satisfy \(\frac{x \sin \frac{\pi}{5} + y \cos \frac{\pi}{5}}{x \cos \frac{\pi}{5} - y \sin \frac{\pi}{5}} = \tan \frac{9 \pi}{20}\), 1. Find the value of \(\frac{y}{x}\). 2. In \(\triangle ABC\), if \(\tan C = \frac{y}{x}\), find the maximum value of \(\sin 2A + 2 \cos B\).
\frac{3}{2}
0.78125
30,472
Originally, there were 5 books on the bookshelf. If 2 more books are added, but the relative order of the original books must remain unchanged, then there are $\boxed{\text{different ways}}$ to place the books.
42
2.34375
30,473
Let $\triangle XYZ$ be a triangle in the plane, and let $W$ be a point outside the plane of $\triangle XYZ$, so that $WXYZ$ is a pyramid whose faces are all triangles. Suppose that the edges of $WXYZ$ have lengths of either $24$ or $49$, and no face of $WXYZ$ is equilateral. Determine the surface area of the pyramid $WXYZ$.
48 \sqrt{2257}
16.40625
30,474
The total number of toothpicks used to build a rectangular grid 15 toothpicks high and 12 toothpicks wide, with internal diagonal toothpicks, is calculated by finding the sum of the toothpicks.
567
0.78125
30,475
Rectangle $ABCD$ has $AB = CD = 3$ and $BC = DA = 5$. The rectangle is first rotated $90^\circ$ clockwise around vertex $D$, then it is rotated $90^\circ$ clockwise around the new position of vertex $C$ (after the first rotation). What is the length of the path traveled by point $A$? A) $\frac{3\pi(\sqrt{17} + 6)}{2}$ B) $\frac{\pi(\sqrt{34} + 5)}{2}$ C) $\frac{\pi(\sqrt{30} + 5)}{2}$ D) $\frac{\pi(\sqrt{40} + 5)}{2}$
\frac{\pi(\sqrt{34} + 5)}{2}
45.3125
30,476
Given that $a > 0$, $b > 0$, $c > 1$, and $a + b = 1$, find the minimum value of $( \frac{a^{2}+1}{ab} - 2) \cdot c + \frac{\sqrt{2}}{c - 1}$.
4 + 2\sqrt{2}
0
30,477
A real number $a$ is chosen randomly and uniformly from the interval $[-10, 15]$. Find the probability that the roots of the polynomial \[ x^4 + 3ax^3 + (3a - 3)x^2 + (-5a + 4)x - 3 \] are all real.
\frac{23}{25}
4.6875
30,478
Juan rolls a fair regular dodecahedral die marked with the numbers 1 through 12. Then Amal rolls a fair eight-sided die. What is the probability that the product of the two rolls is a multiple of 4?
\frac{7}{16}
23.4375
30,479
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 $XR \perp PY$. Given $XR = 6$, $PQ = 12$, and $XY = 7$, find the length of $PY$.
14
28.90625
30,480
Find $x$ such that $\lceil x \rceil \cdot x = 210$. Express $x$ as a decimal.
14
1.5625
30,481
If $x=\sqrt2+\sqrt3+\sqrt6$ is a root of $x^4+ax^3+bx^2+cx+d=0$ where $a,b,c,d$ are integers, what is the value of $|a+b+c+d|$ ?
93
46.875
30,482
If Sarah is leading a class of 35 students, and each time Sarah waves her hands a prime number of students sit down, determine the greatest possible number of students that could have been standing before her third wave.
31
35.9375
30,483
Let $p, q, r,$ and $s$ be the roots of the polynomial $3x^4 - 8x^3 - 15x^2 + 10x - 2 = 0$. Find $pqrs$.
\frac{2}{3}
8.59375
30,484
Given a parabola $C:y^2=2px (p > 0)$, the sum of the distances from any point $Q$ on the parabola to a point inside it, $P(3,1)$, and the focus $F$, has a minimum value of $4$. (I) Find the equation of the parabola; (II) Through the focus $F$, draw a line $l$ that intersects the parabola $C$ at points $A$ and $B$. Find the value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$.
-3
66.40625
30,485
Among all the five-digit numbers formed without repeating digits using 0, 1, 2, 3, and 4, if arranged in ascending order, what position would the number 12340 occupy?
10
17.96875
30,486
Find a four-digit number such that the square of the sum of the two-digit number formed by its first two digits and the two-digit number formed by its last two digits is exactly equal to the four-digit number itself.
2025
96.875
30,487
Given that \( O \) is the circumcenter of \( \triangle ABC \), and the equation \[ \overrightarrow{A O} \cdot \overrightarrow{B C} + 2 \overrightarrow{B O} \cdot \overrightarrow{C A} + 3 \overrightarrow{C O} \cdot \overrightarrow{A B} = 0, \] find the minimum value of \( \frac{1}{\tan A} + \frac{1}{\tan C} \).
\frac{2\sqrt{3}}{3}
33.59375
30,488
The rules for a race require that all runners start at $A$, touch any part of the 1500-meter wall, touch any part of the opposite 1500-meter wall, and stop at $B$. What is the minimum distance a participant must run? Assume that $A$ is 400 meters directly south of the first wall, and that $B$ is 600 meters directly north of the second wall. The two walls are parallel and are 1500 meters apart. Express your answer to the nearest meter.
2915
6.25
30,489
Point $A$ , $B$ , $C$ , and $D$ form a rectangle in that order. Point $X$ lies on $CD$ , and segments $\overline{BX}$ and $\overline{AC}$ intersect at $P$ . If the area of triangle $BCP$ is 3 and the area of triangle $PXC$ is 2, what is the area of the entire rectangle?
15
28.125
30,490
Evaluate $|5 - e|$ where $e$ is the base of the natural logarithm.
2.28172
3.90625
30,491
The imaginary part of the complex number $z = -4i + 3$ is $-4i$.
-4
67.1875
30,492
Determine the monotonicity of the function $f(x) = \frac{x}{x^2 + 1}$ on the interval $(1, +\infty)$, and find the maximum and minimum values of the function when $x \in [2, 3]$.
\frac{3}{10}
50.78125
30,493
A shooter, in a shooting training session, has the probabilities of hitting the 10, 9, 8, and 7 rings as follows: 0.21, 0.23, 0.25, 0.28, respectively. Calculate the probability that the shooter in a single shot: (1) Hits either the 10 or 9 ring; (2) Scores less than 7 rings.
0.03
89.84375
30,494
We call a pair of natural numbers \((a, p)\) good if the number \(a^3 + p^3\) is divisible by \(a^2 - p^2\), with \(a > p\). (a) (1 point) Specify any possible value of \(a\) for which the pair \((a, 13)\) is good. (b) (3 points) Find the number of good pairs for which \(p\) is a prime number less than 20.
24
7.03125
30,495
Let point $P$ be a point on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$. Let $F_1$ and $F_2$ respectively be the left and right foci of the ellipse, and let $I$ be the incenter of $\triangle PF_1F_2$. If $S_{\triangle IPF_1} + S_{\triangle IPF_2} = 2S_{\triangle IF_1F_2}$, then the eccentricity of the ellipse is ______.
\frac{1}{2}
60.9375
30,496
Given a set of data $x_1, x_2, x_3, x_4, x_5$ with a mean of 8 and variance of 2, find the mean and variance of a new set of data: $4x_1+1, 4x_2+1, 4x_3+1, 4x_4+1, 4x_5+1$.
32
77.34375
30,497
In how many ways can 9 distinct items be distributed into three boxes so that one box contains 3 items, another contains 2 items, and the third contains 4 items?
7560
83.59375
30,498
Form a three-digit number using the digits 0, 1, 2, 3. Repeating digits is not allowed. ① How many three-digit numbers can be formed? ② If the three-digit numbers from ① are sorted in ascending order, what position does 230 occupy? ③ If repeating digits is allowed, how many of the formed three-digit numbers are divisible by 3?
16
4.6875
30,499
A square has vertices \( P, Q, R, S \) labelled clockwise. An equilateral triangle is constructed with vertices \( P, T, R \) labelled clockwise. What is the size of angle \( \angle RQT \) in degrees?
135
0