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40.3k
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|---|---|---|---|
19,600 |
Let $X,$ $Y,$ and $Z$ be points on the line such that $\frac{XZ}{ZY} = 3$. If $Y = (2, 6)$ and $Z = (-4, 8)$, determine the sum of the coordinates of point $X$.
|
-8
| 6.25 |
19,601 |
We know that every natural number has factors. For a natural number $a$, we call the positive factors less than $a$ the proper factors of $a$. For example, the positive factors of $10$ are $1$, $2$, $5$, $10$, where $1$, $2$, and $5$ are the proper factors of $10$. The quotient obtained by dividing the sum of all proper factors of a natural number $a$ by $a$ is called the "perfect index" of $a$. For example, the perfect index of $10$ is $\left(1+2+5\right)\div 10=\frac{4}{5}$. The closer the "perfect index" of a natural number is to $1$, the more "perfect" we say the number is. If the "perfect index" of $21$ is _______, then among the natural numbers greater than $20$ and less than $30$, the most "perfect" number is _______.
|
28
| 52.34375 |
19,602 |
A rubber tire has an outer diameter of 25 inches. Calculate the approximate percentage increase in the number of rotations in one mile when the radius of the tire decreases by \(\frac{1}{4}\) inch.
|
2\%
| 27.34375 |
19,603 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that the area of $\triangle ABC$ is $3\sin A$, the perimeter is $4(\sqrt{2}+1)$, and $\sin B + \sin C = \sqrt{2}\sin A$.
1. Find the values of $a$ and $\cos A$.
2. Find the value of $\cos (2A - \frac{\pi}{3})$.
|
\frac{4\sqrt{6} - 7}{18}
| 37.5 |
19,604 |
Choose three digits from the odd numbers 1, 3, 5, 7, 9 and two digits from the even numbers 2, 4, 6, 8 to form a five-digit number with no repeating digits, such that the odd and even digits alternate. How many such five-digit numbers can be formed?
|
720
| 14.0625 |
19,605 |
Fill in the blanks with unique digits in the following equation:
\[ \square \times(\square+\square \square) \times(\square+\square+\square+\square \square) = 2014 \]
The maximum sum of the five one-digit numbers among the choices is:
|
35
| 3.125 |
19,606 |
A person has $440.55$ in their wallet. They purchase goods costing $122.25$. Calculate the remaining money in the wallet. After this, calculate the amount this person would have if they received interest annually at a rate of 3% on their remaining money over a period of 1 year.
|
327.85
| 13.28125 |
19,607 |
The terms of the sequence $(b_i)$ defined by $b_{n + 2} = \frac {b_n + 2021} {1 + b_{n + 1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $b_1 + b_2$.
|
90
| 64.0625 |
19,608 |
Find the root $x$ of the equation $\log x = 4 - x$ where $x \in (k, k+1)$, and $k \in \mathbb{Z}$. What is the value of $k$?
|
k = 3
| 35.15625 |
19,609 |
Triangle $ABC$ has $AB = 15, BC = 16$, and $AC = 17$. The points $D, E$, and $F$ are the midpoints of $\overline{AB}, \overline{BC}$, and $\overline{AC}$ respectively. Let $X \neq E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. Determine $XA + XB + XC$.
A) $\frac{480 \sqrt{39}}{7}$
B) $\frac{960 \sqrt{39}}{14}$
C) $\frac{1200 \sqrt{39}}{17}$
D) $\frac{1020 \sqrt{39}}{15}$
|
\frac{960 \sqrt{39}}{14}
| 3.90625 |
19,610 |
Given the hyperbola $C:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a > 0,b > 0)$, the line $l$ passing through point $P(3,6)$ intersects $C$ at points $A$ and $B$, and the midpoint of $AB$ is $N(12,15)$. Determine the eccentricity of the hyperbola $C$.
|
\frac{3}{2}
| 59.375 |
19,611 |
Given a complex number $z$ satisfying the equation $|z-1|=|z+2i|$ (where $i$ is the imaginary unit), find the minimum value of $|z-1-i|$.
|
\frac{9\sqrt{5}}{10}
| 61.71875 |
19,612 |
The minimum value of the function $f(x) = \cos^2 x + \sin x$ is given by $\frac{-1 + \sqrt{2}}{2}$.
|
-1
| 57.03125 |
19,613 |
Three identical square sheets of paper each with side length \(8\) are stacked on top of each other. The middle sheet is rotated clockwise \(20^\circ\) about its center and the top sheet is rotated clockwise \(50^\circ\) about its center. Determine the area of the resulting polygon.
A) 178
B) 192
C) 204
D) 216
|
192
| 55.46875 |
19,614 |
A fair six-sided die with uniform quality is rolled twice in succession. Let $a$ and $b$ denote the respective outcomes. Find the probability that the function $f(x) = \frac{1}{3}x^3 + \frac{1}{2}ax^2 + bx$ has an extreme value.
|
\frac{17}{36}
| 6.25 |
19,615 |
Mark has 75% more pencils than John, and Luke has 50% more pencils than John. Find the percentage relationship between the number of pencils that Mark and Luke have.
|
16.67\%
| 7.8125 |
19,616 |
Given a point on the parabola $y^2=6x$ whose distance to the focus is twice the distance to the y-axis, find the x-coordinate of this point.
|
\frac{3}{2}
| 91.40625 |
19,617 |
Compute \((1+i^{-100}) + (2+i^{-99}) + (3+i^{-98}) + \cdots + (101+i^0) + (102+i^1) + \cdots + (201+i^{100})\).
|
20302
| 2.34375 |
19,618 |
Given that the sum of the first $n$ terms of the positive arithmetic geometric sequence ${a_n}$ is $S_n$, if $S_2=3$, $S_4=15$, find the common ratio $q$ and $S_6$.
|
63
| 39.84375 |
19,619 |
(Experimental Class Question) Given that $\cos \alpha = \frac{1}{7}$ and $\cos (\alpha - \beta) = \frac{13}{14}$, with $0 < \beta < \alpha < \pi$.
1. Find the value of $\sin (2\alpha - \frac{\pi}{6})$;
2. Find the value of $\beta$.
|
\frac{\pi}{3}
| 47.65625 |
19,620 |
In a school there are 1200 students. Each student must join exactly $k$ clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of $k$ .
|
23
| 46.875 |
19,621 |
Given a geometric sequence $\{a_n\}$, where the sum of the first $n$ terms is denoted as $S_n$, and $S_n = a\left(\frac{1}{4}\right)^{n-1} + 6$, find the value of $a$.
|
-\frac{3}{2}
| 62.5 |
19,622 |
A chord of length √3 divides a circle of radius 1 into two arcs. R is the region bounded by the chord and the shorter arc. What is the largest area of a rectangle that can be drawn in R?
|
\frac{\sqrt{3}}{2}
| 11.71875 |
19,623 |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $|\overrightarrow {a}| = 4$ and the projection of $\overrightarrow {b}$ on $\overrightarrow {a}$ is $-2$, find the minimum value of $|\overrightarrow {a} - 3\overrightarrow {b}|$.
|
10
| 35.9375 |
19,624 |
A worker first receives a 25% cut in wages, then undergoes a 10% increase on the reduced wage. Determine the percent raise on his latest wage that the worker needs to regain his original pay.
|
21.21\%
| 39.0625 |
19,625 |
Given that the final mathematics scores of high school seniors in a certain city follow a normal distribution $X\sim N(85,\sigma ^{2})$, and $P(80 < X < 90)=0.3$, calculate the probability that a randomly selected high school senior's score is not less than $90$ points.
|
0.35
| 75.78125 |
19,626 |
Given a circle of radius $3$, there are multiple line segments of length $6$ that are tangent to the circle at their midpoints. Calculate the area of the region occupied by all such line segments.
|
9\pi
| 22.65625 |
19,627 |
Let $p, q, r, s, t, u$ be positive real numbers such that $p+q+r+s+t+u = 11$. Find the minimum value of
\[
\frac{1}{p} + \frac{9}{q} + \frac{25}{r} + \frac{49}{s} + \frac{81}{t} + \frac{121}{u}.
\]
|
\frac{1296}{11}
| 75 |
19,628 |
A rare genetic disorder is present in one out of every 1000 people in a certain community, and it typically shows no outward symptoms. There is a specialized blood test to detect this disorder. For individuals who have this disorder, the test consistently yields a positive result. For those without the disorder, the test has a $5\%$ false positive rate. If a randomly selected individual from this population tests positive, let $p$ be the probability that they actually have the disorder. Calculate $p$ and select the closest answer from the provided choices.
A) $0.001$
B) $0.019$
C) $0.050$
D) $0.100$
E) $0.190$
|
0.019
| 68.75 |
19,629 |
Let $P(x) = 3\sqrt{x}$, and $Q(x) = x^2 + 1$. Calculate $P(Q(P(Q(P(Q(4))))))$.
|
3\sqrt{1387}
| 67.1875 |
19,630 |
Given that the sum of all coefficients in the expansion of $(x+ \frac {1}{x})^{2n}$ is greater than the sum of all coefficients in the expansion of $(3 3x -x)^{n}$ by $240$.
$(1)$ Find the constant term in the expansion of $(x+ \frac {1}{x})^{2n}$ (answer with a number);
$(2)$ Find the sum of the binomial coefficients in the expansion of $(2x- \frac {1}{x})^{n}$ (answer with a number)
|
16
| 0.78125 |
19,631 |
The lengths of the edges of a rectangular parallelepiped extending from one vertex are 8, 8, and 27. Divide the parallelepiped into four parts that can be assembled into a cube.
|
12
| 79.6875 |
19,632 |
A self-employed person plans to distribute two types of goods, A and B. According to a survey, when the investment amount is $x(x\geqslant 0)$ ten thousand yuan, the profits obtained from distributing goods A and B are $f(x)$ ten thousand yuan and $g(x)$ ten thousand yuan, respectively, where $f(x)=a(x-1)+2(a > 0)$; $g(x)=6\ln (x+b)$, $(b > 0)$. It is known that when the investment amount is zero, the profit is zero.
$(1)$ Try to find the values of $a$ and $b$;
$(2)$ If the self-employed person is ready to invest 5 ten thousand yuan in these two types of goods, please help him develop a capital investment plan to maximize his profit, and calculate the maximum value of his income (accurate to $0.1$, reference data: $\ln 3\approx1.10$).
|
12.6
| 81.25 |
19,633 |
Given $(b_1, b_2, ..., b_{12})$ is a list of the first 12 positive integers, where for each $2 \leq i \leq 12$, either $b_i + 1$, $b_i - 1$, or both appear somewhere in the list before $b_i$, and all even integers precede any of their immediate consecutive odd integers, find the number of such lists.
|
2048
| 1.5625 |
19,634 |
Given that b is an even number between 1 and 11 (inclusive) and c is any natural number, determine the number of quadratic equations x^{2} + bx + c = 0 that have two distinct real roots.
|
50
| 60.15625 |
19,635 |
Given three numbers $1$, $3$, $4$, find the value of x such that the set $\{1, 3, 4, x\}$ forms a proportion.
|
12
| 28.90625 |
19,636 |
There are 54 students in a class, and there are 4 tickets for the Shanghai World Expo. Now, according to the students' ID numbers, the tickets are distributed to 4 students through systematic sampling. If it is known that students with ID numbers 3, 29, and 42 have been selected, then the ID number of another student who has been selected is ▲.
|
16
| 81.25 |
19,637 |
A rectangle is divided into 40 identical squares. The rectangle contains more than one row of squares. Andrew coloured all the squares in the middle row. How many squares did he not colour?
|
32
| 37.5 |
19,638 |
The sequence $\{a_n\}$ satisfies $a_1=1$, $a_2=1$, $a_{n+2}=(1+\sin^2 \frac{n\pi}{2})a_n+2\cos^2 \frac{n\pi}{2}$. Find the sum of the first $20$ terms of this sequence.
|
1123
| 17.96875 |
19,639 |
Dr. Green gives bonus points on a test for students who score above the class average. In a class of 150 students, what is the maximum number of students who can score above the average if their scores are integers?
|
149
| 11.71875 |
19,640 |
For how many integer values of $n$ between 1 and 2000 inclusive does the decimal representation of $\frac{n}{2940}$ terminate?
|
13
| 93.75 |
19,641 |
There are six identical red balls and three identical green balls in a pail. Four of these balls are selected at random and then these four balls are arranged in a line in some order. Find the number of different-looking arrangements of the selected balls.
|
15
| 59.375 |
19,642 |
A rectangular garden measures $12$ meters in width and $20$ meters in length. It is paved with tiles that are $2$ meters by $2$ meters each. A cat runs from one corner of the rectangular garden to the opposite corner but must leap over a small pond that exactly covers one tile in the middle of the path. How many tiles does the cat touch, including the first and the last tile?
|
13
| 54.6875 |
19,643 |
How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 7, 7, 7\}$ if no digit may be used more times than it appears in the given set of available digits?
|
43
| 74.21875 |
19,644 |
Imagine you own 8 shirts, 5 pairs of pants, 4 ties, and 3 different jackets. If an outfit consists of a shirt, a pair of pants, and optionally a tie and/or a jacket, how many different outfits can you create?
|
800
| 67.1875 |
19,645 |
Let $f(x) = 4x - 9$ and $g(f(x)) = 3x^2 + 4x - 2.$ Find $g(-10).$
|
\frac{-45}{16}
| 0 |
19,646 |
Consider a regular hexagon where each of its $6$ sides and the $9$ diagonals are colored randomly and independently either red or blue, each color with the same probability. What is the probability that there exists at least one triangle, formed by three of the hexagon’s vertices, in which all sides are of the same color?
A) $\frac{253}{256}$
B) $\frac{1001}{1024}$
C) $\frac{815}{819}$
D) $\frac{1048575}{1048576}$
E) 1
|
\frac{1048575}{1048576}
| 46.09375 |
19,647 |
In the Cartesian coordinate plane $(xOy)$, given vectors $\overrightarrow{AB}=(6,1)$, $\overrightarrow{BC}=(x,y)$, $\overrightarrow{CD}=(-2,-3)$, and $\overrightarrow{AD}$ is parallel to $\overrightarrow{BC}$.
(1) Find the relationship between $x$ and $y$;
(2) If $\overrightarrow{AC}$ is perpendicular to $\overrightarrow{BD}$, find the area of the quadrilateral $ABCD$.
|
16
| 27.34375 |
19,648 |
Let $\overrightarrow{a}=(\sin x, \frac{3}{4})$, $\overrightarrow{b}=( \frac{1}{3}, \frac{1}{2}\cos x )$, and $\overrightarrow{a} \parallel \overrightarrow{b}$. Find the acute angle $x$.
|
\frac{\pi}{4}
| 91.40625 |
19,649 |
A set \( A \) consists of 40 elements chosen from \(\{1, 2, \ldots, 50\}\), and \( S \) is the sum of all elements in the set \( A \). How many distinct values can \( S \) take?
|
401
| 97.65625 |
19,650 |
The point is chosen at random within the rectangle in the coordinate plane whose vertices are $(0, 0), (3030, 0), (3030, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{3}$. Find $d$ to the nearest tenth.
|
0.3
| 43.75 |
19,651 |
Two rectangles, one measuring $2 \times 4$ and another measuring $3 \times 5$, along with a circle of diameter 3, are to be contained within a square. The sides of the square are parallel to the sides of the rectangles and the circle must not overlap any rectangle at any point internally. What is the smallest possible area of the square?
|
49
| 14.84375 |
19,652 |
In the expansion of $(1-\frac{y}{x})(x+y)^{8}$, the coefficient of $x^{2}y^{6}$ is ____ (provide your answer as a number).
|
-28
| 83.59375 |
19,653 |
Given the fraction $\frac{987654321}{2^{30}\cdot 5^6}$, determine the minimum number of digits to the right of the decimal point required to express this fraction as a decimal.
|
30
| 46.875 |
19,654 |
Given the ellipse $\frac{{x}^{2}}{3{{m}^{2}}}+\frac{{{y}^{2}}}{5{{n}^{2}}}=1$ and the hyperbola $\frac{{{x}^{2}}}{2{{m}^{2}}}-\frac{{{y}^{2}}}{3{{n}^{2}}}=1$ share a common focus, find the eccentricity of the hyperbola ( ).
|
\frac{\sqrt{19}}{4}
| 21.875 |
19,655 |
Given the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with left and right foci $F\_1$, $F\_2$, $b = 4$, and an eccentricity of $\frac{3}{5}$. A line passing through $F\_1$ intersects the ellipse at points $A$ and $B$. Find the perimeter of $\triangle ABF\_2$.
|
20
| 88.28125 |
19,656 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b= \sqrt {2}a$, $\sqrt {3}\cos B= \sqrt {2}\cos A$, $c= \sqrt {3}+1$. Find the area of $\triangle ABC$.
|
\frac { \sqrt {3}+1}{2}
| 0 |
19,657 |
The function $g$ is defined on positive integers as follows:
\[g(n) = \left\{
\begin{array}{cl}
n + 12 & \text{if $n < 12$}, \\
g(n - 7) & \text{if $n \ge 12$}.
\end{array}
\right.\]
Find the maximum value of the function.
|
23
| 85.9375 |
19,658 |
Five identical white pieces and ten identical black pieces are arranged in a row. It is required that the right neighbor of each white piece must be a black piece. The number of different arrangements is .
|
252
| 72.65625 |
19,659 |
Three couples sit for a photograph in $2$ rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many such arrangements are possible?
|
96
| 96.875 |
19,660 |
Given $\tan\left( \frac{\pi}{4} + \alpha \right) = \frac{1}{7}$, with $\alpha \in \left( \frac{\pi}{2}, \pi \right)$, find the value of $\tan\alpha$ and $\cos\alpha$.
|
-\frac{4}{5}
| 53.90625 |
19,661 |
Find the sum of distances from a point on the ellipse $7x^{2}+3y^{2}=21$ to its two foci.
|
2\sqrt{7}
| 100 |
19,662 |
Square $EFGH$ has a side length of $40$. Point $Q$ lies inside the square such that $EQ = 16$ and $FQ = 34$. The centroids of $\triangle{EFQ}$, $\triangle{FGQ}$, $\triangle{GHQ}$, and $\triangle{HEQ}$ are the vertices of a convex quadrilateral. Calculate the area of this quadrilateral.
|
\frac{3200}{9}
| 1.5625 |
19,663 |
A national team needs to select 4 out of 6 sprinters to participate in the 4×100m relay at the Asian Games. If one of them, A, cannot run the first leg, and another, B, cannot run the fourth leg, how many different methods are there to select the team?
|
252
| 64.84375 |
19,664 |
Let $m$ and $n$ be integers such that $m + n$ and $m - n$ are prime numbers less than $100$ . Find the maximal possible value of $mn$ .
|
2350
| 92.1875 |
19,665 |
In the Cartesian coordinate system $xOy$, it is known that $P$ is a moving point on the graph of the function $f(x)=\ln x$ ($x > 0$). The tangent line $l$ at point $P$ intersects the $x$-axis at point $E$. A perpendicular line to $l$ through point $P$ intersects the $x$-axis at point $F$. If the midpoint of the line segment $EF$ is $T$ with the $x$-coordinate $t$, then the maximum value of $t$ is \_\_\_\_\_\_.
|
\dfrac {1}{2}(e+ \dfrac {1}{e})
| 0 |
19,666 |
Express 826,000,000 in scientific notation.
|
8.26 \times 10^{8}
| 0 |
19,667 |
If the polynomial $x^3+x^{10}=a_0+a_1(x+1)+\ldots+a_9(x+1)^9+a_{10}(x+1)^{10}$, then $a_2=$ ______.
|
42
| 42.1875 |
19,668 |
A bicycle costs 389 yuan, and an electric fan costs 189 yuan. Dad wants to buy a bicycle and an electric fan. He will need approximately \_\_\_\_\_\_ yuan.
|
600
| 0 |
19,669 |
Given points F₁(-1, 0), F₂(1, 0), line l: y = x + 2. If the ellipse C, with foci at F₁ and F₂, intersects with line l, calculate the maximum eccentricity of ellipse C.
|
\frac {\sqrt {10}}{5}
| 0 |
19,670 |
Let $\overline{AB}$ have a length of 8 units and $\overline{A'B'}$ have a length of 6 units. $D$ is located 3 units away from $A$ on $\overline{AB}$ and $D'$ is located 1 unit away from $A'$ on $\overline{A'B'}$. If $P$ is a point on $\overline{AB}$ such that $x$ (the distance from $P$ to $D$) equals $2$ units, find the sum $x + y$, given that the ratio of $x$ to $y$ (the distance from the associated point $P'$ on $\overline{A'B'}$ to $D'$) is 3:2.
A) $\frac{8}{3}$ units
B) $\frac{9}{3}$ units
C) $\frac{10}{3}$ units
D) $\frac{11}{3}$ units
E) $\frac{12}{3}$ units
|
\frac{10}{3}
| 96.09375 |
19,671 |
A boss schedules a meeting at a cafe with two of his staff, planning to arrive randomly between 1:00 PM and 4:00 PM. Each staff member also arrives randomly within the same timeframe. If the boss arrives and any staff member isn't there, he leaves immediately. Each staff member will wait for up to 90 minutes for the other to arrive before leaving. What is the probability that the meeting successfully takes place?
|
\frac{1}{4}
| 2.34375 |
19,672 |
Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci denoted as $F_{1}$ and $F_{2}$, respectively. Draw a line $l$ passing through the right focus that intersects the ellipse at points $P$ and $Q$. What is the maximum area of the inscribed circle of triangle $\triangle F_{1} P Q$?
|
\frac{9\pi}{16}
| 7.03125 |
19,673 |
The integer \( n \) is the smallest positive number that satisfies the following conditions:
1. \( n \) is a multiple of 75.
2. \( n \) has exactly 75 positive divisors (including 1 and itself).
Find the value of \( \frac{n}{75} \).
|
432
| 73.4375 |
19,674 |
In $\triangle ABC$, $a=1$, $B=45^{\circ}$, $S_{\triangle ABC}=2$, calculate the diameter of the circumcircle of $\triangle ABC$.
|
5\sqrt{2}
| 41.40625 |
19,675 |
Calculate the product: $100 \times 29.98 \times 2.998 \times 1000 = $
|
2998^2
| 0 |
19,676 |
What is the value of $49^3 + 3(49^2) + 3(49) + 1$?
|
125000
| 96.09375 |
19,677 |
Given 8 volunteer positions to be allocated to 3 schools, with each school receiving at least one position and the allocations being unequal, find the number of ways to distribute the positions.
|
12
| 26.5625 |
19,678 |
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$?
|
60
| 77.34375 |
19,679 |
Let \( P \) be a regular 2006-sided polygon. A diagonal of \( P \) is called a "good edge" if its endpoints divide the polygon into two parts, each containing an odd number of the polygon's sides. Each side of \( P \) is also considered a "good edge".
Given that 2003 non-intersecting diagonals divide \( P \) into several triangles, determine the maximum number of isosceles triangles, under this division, that have two "good edges".
|
1003
| 84.375 |
19,680 |
A builder has two identical bricks. She places them side by side in three different ways, resulting in shapes with surface areas of 72, 96, and 102. What is the surface area of one original brick?
|
54
| 1.5625 |
19,681 |
Given that the math scores of a certain high school approximately follow a normal distribution N(100, 100), calculate the percentage of students scoring between 80 and 120 points.
|
95.44\%
| 0 |
19,682 |
In the Cartesian coordinate system $xOy$, suppose the terminal side of the obtuse angle $\alpha$ intersects the circle $O: x^{2}+y^{2}=4$ at point $P(x_{1},y_{1})$. If point $P$ moves clockwise along the circle for a unit arc length of $\frac{2\pi}{3}$ to reach point $Q(x_{2},y_{2})$, then the range of values for $y_{1}+y_{2}$ is \_\_\_\_\_\_; if $x_{2}= \frac{1}{2}$, then $x_{1}=$\_\_\_\_\_\_.
|
\frac{1-3\sqrt{5}}{4}
| 3.125 |
19,683 |
Find the smallest three-digit number such that the following holds:
If the order of digits of this number is reversed and the number obtained by this is added to the original number, the resulting number consists of only odd digits.
|
209
| 94.53125 |
19,684 |
Find the number of different arrangements for a class to select 6 people to participate in two volunteer activities, with each activity accommodating no more than 4 people.
|
50
| 14.84375 |
19,685 |
Given the geometric sequence $\{a_n\}$, $a_3$ and $a_7$ are the extreme points of the function $f(x) = \frac{1}{3}x^3 + 4x^2 + 9x - 1$. Calculate the value of $a_5$.
|
-3
| 32.8125 |
19,686 |
Given line segments $OA$, $OB$, $OC$ are pairwise perpendicular, with $OA=1$, $OB=1$, $OC=2$. If the projections of line segments $OA$, $OB$, $OC$ on line $OP$ have equal lengths, then the length of these projections is $\_\_\_\_\_\_.$
|
\frac{2}{3}
| 87.5 |
19,687 |
Find the area of the triangle with vertices $(2, -3),$ $(1, 4),$ and $(-3, -2).$
|
17
| 58.59375 |
19,688 |
In $\triangle ABC$, given $BC=2$, $AC=\sqrt{7}$, $B=\dfrac{2\pi}{3}$, find the area of $\triangle ABC$.
|
\dfrac{\sqrt{3}}{2}
| 46.09375 |
19,689 |
Given that Jeff, Maria, and Lee paid $90, $150, and $210 respectively, find j - m where Jeff gave Lee $j dollars and Maria gave Lee $m dollars to settle the debts such that everyone paid equally.
|
60
| 71.09375 |
19,690 |
If the lengths of the sides of a triangle are positive integers not greater than 5, how many such distinct triangles exist?
|
22
| 28.90625 |
19,691 |
Given the function $f(x)=x^{3}- \frac {3}{2}x^{2}+ \frac {3}{4}x+ \frac {1}{8}$, find the value of $\sum\limits_{k=1}^{2016}f( \frac {k}{2017})$.
|
504
| 23.4375 |
19,692 |
Given the sequence $\{a\_n\}$, the sum of its first $n$ terms is $S\_n=1-5+9-13+17-21+…+(-1)^{n+1}(4n-3)$. Find the value of $S\_{15}+S\_{22}-S\_{31}$.
|
-76
| 73.4375 |
19,693 |
Five volunteers participate in community service for two days, Saturday and Sunday. Each day, two people are selected from the group to serve. Determine the number of ways to select exactly one person to serve for both days.
|
60
| 72.65625 |
19,694 |
Given the function $f(x)=a^{2}\sin 2x+(a-2)\cos 2x$, if its graph is symmetric about the line $x=-\frac{\pi}{8}$, determine the maximum value of $f(x)$.
|
4\sqrt{2}
| 47.65625 |
19,695 |
We glue together $990$ one inch cubes into a $9$ by $10$ by $11$ inch rectangular solid. Then we paint the outside of the solid. How many of the original $990$ cubes have just one of their sides painted?
|
382
| 83.59375 |
19,696 |
Let \(O\) be the origin. There exists a scalar \(k'\) so that for any points \(A\), \(B\), \(C\), and \(D\) if
\[4 \overrightarrow{OA} - 3 \overrightarrow{OB} + 6 \overrightarrow{OC} + k' \overrightarrow{OD} = \mathbf{0},\]
then the four points \(A\), \(B\), \(C\), and \(D\) are coplanar. Find \(k'\).
|
-7
| 73.4375 |
19,697 |
Given the polynomial $f(x)=3x^{9}+3x^{6}+5x^{4}+x^{3}+7x^{2}+3x+1$, calculate the value of $v_{5}$ when $x=3$ using Horner's method.
|
761
| 5.46875 |
19,698 |
The function $y=|x-1|+|x-2|+\ldots+|x-10|$, when $x$ takes values in the real number range, the minimum value of $y$ is.
|
25
| 97.65625 |
19,699 |
Calculate the probability that athlete A cannot run the first leg and athlete B cannot run the last leg in a 4x100 meter relay race selection from 6 short-distance runners, including athletes A and B, to form a team of 4 runners.
|
\frac{7}{10}
| 21.875 |
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