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40.3k
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stringlengths 10
5.15k
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100
|
|---|---|---|---|
19,400 |
On the extension of side $AD$ of rhombus $ABCD$, point $K$ is taken beyond point $D$. The lines $AC$ and $BK$ intersect at point $Q$. It is known that $AK=14$ and that points $A$, $B$, and $Q$ lie on a circle with a radius of 6, the center of which belongs to segment $AA$. Find $BK$.
|
20
| 6.25 |
19,401 |
Below is the graph of \( y = a \sin(bx + c) \) for some constants \( a > 0 \), \( b > 0 \), and \( c \). The graph reaches its maximum value at \( 3 \) and completes one full cycle by \( 2\pi \). There is a phase shift where the maximum first occurs at \( \pi/6 \). Find the values of \( a \), \( b \), and \( c \).
|
\frac{\pi}{3}
| 0 |
19,402 |
Which of the followings gives the product of the real roots of the equation $x^4+3x^3+5x^2 + 21x -14=0$ ?
|
-2
| 61.71875 |
19,403 |
Mrs. Crabapple now teaches two different classes of British Literature. Her first class has 12 students and meets three times a week, while her second class has 9 students and meets twice a week. How many different sequences of crabapple recipients are possible in a week for both classes combined?
|
139,968
| 0 |
19,404 |
Two isosceles triangles are given with equal perimeters. The base of the second triangle is 15% larger than the base of the first, and the leg of the second triangle is 5% smaller than the leg of the first triangle. Find the ratio of the sides of the first triangle.
|
\frac{2}{3}
| 25 |
19,405 |
Given that the terms of the geometric sequence $\{a_n\}$ are positive, and the common ratio is $q$, if $q^2 = 4$, then $$\frac {a_{3}+a_{4}}{a_{4}+a_{5}}$$ equals \_\_\_\_\_\_.
|
\frac {1}{2}
| 97.65625 |
19,406 |
Let $c$ be a real number randomly selected from the interval $[-20,20]$. Then, $p$ and $q$ are two relatively prime positive integers such that $\frac{p}{q}$ is the probability that the equation $x^4 + 36c^2 = (9c^2 - 15c)x^2$ has at least two distinct real solutions. Find the value of $p + q$.
|
29
| 2.34375 |
19,407 |
Let \\(x,y\\) satisfy the constraint conditions \\(\begin{cases} & x+y-2\geqslant 0 \\\\ & x-y+1\geqslant 0 \\\\ & x\leqslant 3 \end{cases}\\). If the minimum value of \\(z=mx+y\\) is \\(-3\\), then the value of \\(m\\) is .
|
- \dfrac{2}{3}
| 75.78125 |
19,408 |
Given an arithmetic sequence $\{a_n\}$ with common difference $d \neq 0$, and its first term $a_1 = d$. The sum of the first $n$ terms of the sequence $\{a_n^2\}$ is denoted as $S_n$. Additionally, there is a geometric sequence $\{b_n\}$ with a common ratio $q$ that is a positive rational number less than $1$. The first term of this geometric sequence is $b_1 = d^2$, and the sum of its first $n$ terms is $T_n$. Find the possible value(s) of $q$ such that $\frac{S_3}{T_3}$ is a positive integer.
|
\frac{1}{2}
| 35.15625 |
19,409 |
Let $a, b \in \mathbb{R}$. If the line $l: ax+y-7=0$ is transformed by the matrix $A= \begin{bmatrix} 3 & 0 \\ -1 & b\end{bmatrix}$, and the resulting line is $l′: 9x+y-91=0$. Find the values of the real numbers $a$ and $b$.
|
13
| 1.5625 |
19,410 |
The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules:
- $D(1) = 0$ ;
- $D(p)=1$ for all primes $p$ ;
- $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$ .
Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$ .
|
31
| 62.5 |
19,411 |
Let the equations $x^2+ax+b=0$ and $x^2+bx+a=0$ ($a<0$, $b<0$, $a \neq b$) have one common root. Let the other two roots be $x_1$ and $x_2$.
(1) Find the value of $x_1+x_2$.
(2) Find the maximum value of $x_1x_2$.
|
\frac{1}{4}
| 43.75 |
19,412 |
If a die is rolled, event \( A = \{1, 2, 3\} \) consists of rolling one of the faces 1, 2, or 3. Similarly, event \( B = \{1, 2, 4\} \) consists of rolling one of the faces 1, 2, or 4.
The die is rolled 10 times. It is known that event \( A \) occurred exactly 6 times.
a) Find the probability that under this condition, event \( B \) did not occur at all.
b) Find the expected value of the random variable \( X \), which represents the number of occurrences of event \( B \).
|
\frac{16}{3}
| 0 |
19,413 |
A fair dice is rolled twice, and the scores obtained are denoted as $m$ and $n$ respectively. Let the angle between vector $a=(m,n)$ and vector $b=(1,-1)$ be $\theta$. The probability that $\theta$ is an acute angle is $\_\_\_\_\_\_\_\_\_\_\_\_\_.$
|
\frac{5}{12}
| 99.21875 |
19,414 |
Given that $\cos \alpha = \dfrac{1}{7}$, $\cos (\alpha - \beta) = \dfrac{13}{14}$, and $0 < \beta < \alpha < \dfrac{\pi}{2}$.
1. Find the value of $\tan 2\alpha$.
2. Find the value of $\cos \beta$.
|
\dfrac{1}{2}
| 70.3125 |
19,415 |
A curve C is established in the polar coordinate system with the coordinate origin O as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of the curve C is given by $$ρ^{2}= \frac {12}{4-cos^{2}\theta }$$
1. Find the rectangular coordinate equation of the curve C.
2. Suppose a line l passes through the point P(1, 0) with a slope angle of 45° and intersects the curve C at two points A and B. Find the value of $$\frac {1}{|PA|}+ \frac {1}{|PB|}$$.
|
\frac{4}{3}
| 39.84375 |
19,416 |
Two lines with slopes $-\frac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$.
|
22.5
| 19.53125 |
19,417 |
Let $a$, $b$, $c$, $a+b-c$, $a+c-b$, $b+c-a$, $a+b+c$ be seven distinct prime numbers, and among $a$, $b$, $c$, the sum of two numbers is 800. Let $d$ be the difference between the largest and the smallest of these seven prime numbers. Find the maximum possible value of $d$.
|
1594
| 81.25 |
19,418 |
Each of the symbols $\diamond$ and $\circ$ represents an operation in the set $\{+,-,\times,\div\}$, and $\frac{15 \diamond 3}{8 \circ 2} = 3$. What is the value of $\frac{9 \diamond 4}{14 \circ 7}$? Express your answer as a common fraction.
|
\frac{13}{7}
| 29.6875 |
19,419 |
From a set of integers $\{1, 2, 3, \ldots, 12\}$, eight distinct integers are chosen at random. What is the probability that, among those selected, the third smallest number is $4$?
|
\frac{56}{165}
| 19.53125 |
19,420 |
From the set of integers $\{1,2,3,\dots,3009\}$, choose $k$ pairs $\{a_i,b_i\}$ with $a_i<b_i$ so that no two pairs share a common element. Each sum $a_i+b_i$ must be distinct and less than or equal to $3009$. Determine the maximum possible value of $k$.
|
1504
| 76.5625 |
19,421 |
Given that the lines $ax+2y+6=0$ and $x+\left(a-1\right)y+a^{2}-1=0$ are parallel to each other, find the value(s) of the real number $a$.
|
-1
| 7.03125 |
19,422 |
In the arithmetic sequence $\{a_n\}$, we have $a_2=4$, and $a_4+a_7=15$.
(Ⅰ) Find the general term formula for the sequence $\{a_n\}$.
(Ⅱ) Let $b_n= \frac{1}{a_n a_{n+1}}$, calculate the value of $b_1+b_2+b_3+\dots+b_{10}$.
|
\frac{10}{39}
| 86.71875 |
19,423 |
In the plane rectangular coordinate system $xOy$, the graph of the quadratic function $y=a\left(x-h\right)^{2}+k\left(a \lt 0\right)$ passes through the points $A\left(-3,m\right)$ and $B\left(-1,n\right)$.<br/>$(1)$ When $m=n$, find the length of the line segment $AB$ and the value of $h$;<br/>$(2)$ If the point $C\left(1,0\right)$ also lies on the graph of the quadratic function $y=a\left(x-h\right)^{2}+k\left(a \lt 0\right)$, and $m \lt 0 \lt n$,<br/>① find the abscissa of the other intersection point of the graph of the quadratic function $y=a\left(x-h\right)^{2}+k\left(a \lt 0\right)$ with the $x$-axis (expressed in terms of $h$) and the range of values for $h$;<br/>② if $a=-1$, find the area of $\triangle ABC$;<br/>③ a line passing through point $D(0$,$h^{2})$ perpendicular to the $y$-axis intersects the parabola at points $P(x_{1}$,$y_{1})$ and $(x_{2}$,$y_{2})$ (where $P$ and $Q$ are not coincident), and intersects the line $BC$ at point $(x_{3}$,$y_{3})$. Is there a value of $a$ such that $x_{1}+x_{2}-x_{3}$ is always a constant? If so, find the value of $a$; if not, explain why.
|
-\frac{1}{4}
| 3.125 |
19,424 |
Tim continues the prank into the next week after a successful first week. This time, he starts on Monday with two people willing to do the prank, on Tuesday there are three options, on Wednesday everyone from Monday and Tuesday refuses but there are six new people, on Thursday four of Wednesday's people can't participate but two additional new ones can, and on Friday two people from Monday are again willing to help along with one new person. How many different combinations of people could Tim involve in this prank across the week?
|
432
| 28.125 |
19,425 |
Let $k$ be a positive real number. In the $X-Y$ coordinate plane, let $S$ be the set of all points of the form $(x,x^2+k)$ where $x\in\mathbb{R}$ . Let $C$ be the set of all circles whose center lies in $S$ , and which are tangent to $X$ -axis. Find the minimum value of $k$ such that any two circles in $C$ have at least one point of intersection.
|
\frac{1}{4}
| 15.625 |
19,426 |
A function $g$ is defined by $g(z) = (3 - 2i) z^2 + \beta z + \delta$ for all complex numbers $z$, where $\beta$ and $\delta$ are complex numbers and $i^2 = -1$. Suppose that $g(1)$ and $g(-i)$ are both real. What is the smallest possible value of $|\beta| + |\delta|$?
|
\sqrt{13}
| 2.34375 |
19,427 |
According to the notice from the Ministry of Industry and Information Technology on the comprehensive promotion of China's characteristic enterprise new apprenticeship system and the strengthening of skills training, our region clearly promotes the new apprenticeship system training for all types of enterprises, deepens the integration of production and education, school-enterprise cooperation, and the apprenticeship training goal is to cultivate intermediate and senior technical workers that meet the needs of business positions. In the year 2020, a certain enterprise needs to train 200 apprentices. After the training, an assessment is conducted, and the statistics of obtaining corresponding job certificates are as follows:
| Job Certificate | Junior Worker | Intermediate Worker | Senior Worker | Technician | Senior Technician |
|-----------------|---------------|---------------------|--------------|-----------|------------------|
| Number of People | 20 | 60 | 60 | 40 | 20 |
$(1)$ Now, using stratified sampling, 10 people are selected from these 200 people to form a group for exchanging skills and experiences. Find the number of people in the exchange group who have obtained job certificates in the technician category (including technicians and senior technicians).
$(2)$ From the 10 people selected in (1) for the exchange group, 3 people are randomly chosen as representatives to speak. Let the number of technicians among these 3 people be $X$. Find the probability distribution and the mathematical expectation of the random variable $X$.
|
\frac{9}{10}
| 53.90625 |
19,428 |
A point $Q$ is chosen within $\triangle DEF$ such that lines drawn through $Q$, parallel to the sides of $\triangle DEF$, divide it into three smaller triangles with areas $9$, $16$, and $25$ respectively. Determine the area of $\triangle DEF$.
|
144
| 9.375 |
19,429 |
What is the sum of the greatest common divisor of $45$ and $4410$ and the least common multiple of $45$ and $4410$?
|
4455
| 58.59375 |
19,430 |
If for any real numbers $u,v$, the inequality ${{(u+5-2v)}^{2}}+{{(u-{{v}^{2}})}^{2}}\geqslant {{t}^{2}}(t > 0)$ always holds, then the maximum value of $t$ is
|
2 \sqrt{2}
| 42.96875 |
19,431 |
In \\(\triangle ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively. It is given that \\(a+b=5\\), \\(c=\sqrt{7}\\), and \\(4{{\left( \sin \frac{A+B}{2} \right)}^{2}}-\cos 2C=\frac{7}{2}\\).
\\((1)\\) Find the magnitude of angle \\(C\\);
\\((2)\\) Find the area of \\(\triangle ABC\\).
|
\frac {3 \sqrt {3}}{2}
| 0 |
19,432 |
Inside the ellipse $4x^2+9y^2=144$, there exists a point $P(3, 2)$. Find the slope of the line that contains the chord with point P as its midpoint.
|
-\frac{2}{3}
| 64.0625 |
19,433 |
Use \((a, b)\) to represent the greatest common divisor of \(a\) and \(b\). Let \(n\) be an integer greater than 2021, and \((63, n+120) = 21\) and \((n+63, 120) = 60\). What is the sum of the digits of the smallest \(n\) that satisfies the above conditions?
|
15
| 18.75 |
19,434 |
Observe the following set of equations:
\\(S_{1}=1\\),
\\(S_{2}=2+3=5\\),
\\(S_{3}=4+5+6=15\\),
\\(S_{4}=7+8+9+10=34\\),
\\(S_{5}=11+12+13+14+15=65\\),
\\(\ldots\\)
Based on the equations above, guess that \\(S_{2n-1}=(2n-1)(an^{2}+bn+c)\\), then \\(a\cdot b\cdot c=\\) \_\_\_\_\_\_.
|
-4
| 53.90625 |
19,435 |
The maximum area of a right-angled triangle with a hypotenuse of length 8 is
|
16
| 98.4375 |
19,436 |
Given real numbers $x$ and $y$ that satisfy the equation $x^{2}+y^{2}-4x+6y+12=0$, find the minimum value of $|2x-y-2|$.
|
5-\sqrt{5}
| 64.0625 |
19,437 |
If \( x \) and \( y \) are real numbers such that \( x + y = 4 \) and \( xy = -2 \), then the value of \( x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y \) is:
|
440
| 67.1875 |
19,438 |
Given that $0 < β < \dfrac{π}{2} < α < π$, and $\cos (α- \dfrac{β}{2} )= \dfrac{5}{13} $, $\sin ( \dfrac{α}{2}-β)= \dfrac{3}{5} $. Find the values of:
$(1) \tan (α- \dfrac{β}{2} )$
$(2) \cos ( \dfrac{α+β}{2} )$
|
\dfrac{56}{65}
| 51.5625 |
19,439 |
Given an equilateral triangle $PQR$ with a side length of 8 units, a process similar to the previous one is applied, but here each time, the triangle is divided into three smaller equilateral triangles by joining the midpoints of its sides, and the middle triangle is shaded each time. If this procedure is repeated 100 times, what is the total area of the shaded triangles?
A) $6\sqrt{3}$
B) $8\sqrt{3}$
C) $10\sqrt{3}$
D) $12\sqrt{3}$
E) $14\sqrt{3}$
|
8\sqrt{3}
| 42.96875 |
19,440 |
If A and B can only undertake the first three tasks, while the other three can undertake all four tasks, calculate the total number of different selection schemes for the team leader group to select four people from five volunteers to undertake four different tasks.
|
72
| 6.25 |
19,441 |
Given the function $f(x)$ defined on the interval $[-2011, 2011]$ and satisfying $f(x_1+x_2) = f(x_1) + f(x_2) - 2011$ for any $x_1, x_2 \in [-2011, 2011]$, and $f(x) > 2011$ when $x > 0$, determine the value of $M+N$.
|
4022
| 84.375 |
19,442 |
Given real numbers $x$, $y$ satisfying $x > y > 0$, and $x + y \leqslant 2$, the minimum value of $\dfrac{2}{x+3y}+\dfrac{1}{x-y}$ is
|
\dfrac {3+2 \sqrt {2}}{4}
| 0 |
19,443 |
A square is inscribed in a circle. The number of inches in the perimeter of the square equals the number of square inches in the area of the circumscribed circle. What is the radius, in inches, of the circle? Express your answer in terms of pi.
|
\frac{4\sqrt{2}}{\pi}
| 38.28125 |
19,444 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c = a \cos B + 2b \sin^2 \frac{A}{2}$.
(1) Find angle $A$.
(2) If $b=4$ and the length of median drawn to side $AC$ is $\sqrt{7}$, find $a$.
|
\sqrt{13}
| 2.34375 |
19,445 |
Given the distribution list of the random variable $X$, $P(X=\frac{k}{5})=ak$, where $k=1,2,3,4,5$.
1. Find the value of the constant $a$.
2. Find $P(X\geqslant\frac{3}{5})$.
3. Find $P(\frac{1}{10}<X<\frac{7}{10})$.
|
\frac{2}{5}
| 9.375 |
19,446 |
Out of the digits 0 through 9, three digits are randomly chosen to form a three-digit number without repeating any digits. What is the probability that this number is not divisible by 3?
|
2/3
| 78.90625 |
19,447 |
From the four numbers $0,1,2,3$, we want to select $3$ digits to form a three-digit number with no repeating digits. What is the probability that this three-digit number is divisible by $3$?
|
\dfrac{5}{9}
| 64.84375 |
19,448 |
Given an arithmetic sequence $\{a_{n}\}$ with a non-zero common difference, $S_{n}$ represents the sum of its first $n$ terms. If $S_{5}=0$, then the number of distinct values in $S_{i}$ for $i=1,2,\ldots,100$ is ______.
|
98
| 23.4375 |
19,449 |
For any real number $a$, let $\left[a\right]$ denote the largest integer not exceeding $a$. For example, $\left[4\right]=4$, $[\sqrt{3}]=1$. Now, for the number $72$, the following operations are performed: $72\stackrel{1st}{→}[\sqrt{72}]=8\stackrel{2nd}{→}[\sqrt{8}]=2\stackrel{3rd}{→}[\sqrt{2}]=1$. In this way, the number $72$ becomes $1$ after $3$ operations. Similarly, among all positive integers that become $2$ after $3$ operations, the largest one is ____.
|
6560
| 43.75 |
19,450 |
Given that a triangle with integral sides is isosceles and has a perimeter of 12, find the area of the triangle.
|
4\sqrt{3}
| 60.15625 |
19,451 |
If the line \( x = \frac{\pi}{4} \) intercepts the curve \( C: (x - \arcsin a)(x - \arccos a) + (y - \arcsin a)(y + \arccos a) = 0 \) at a chord of length \( d \), find the minimum value of \( d \) as \( a \) varies.
|
\frac{\pi}{2}
| 38.28125 |
19,452 |
What is the modular inverse of $13$, modulo $2000$?
Express your answer as an integer from $0$ to $1999$, inclusive.
|
1077
| 2.34375 |
19,453 |
If $∀x∈(0,+\infty)$, $ln2x-\frac{ae^{x}}{2}≤lna$, then find the minimum value of $a$.
|
\frac{2}{e}
| 28.125 |
19,454 |
There are 20 rooms, with some lights on and some lights off. The people in these rooms want to have their lights in the same state as the majority of the other rooms. Starting with the first room, if the majority of the remaining 19 rooms have their lights on, the person will turn their light on; otherwise, they will turn their light off. Initially, there are 10 rooms with lights on and 10 rooms with lights off, and the light in the first room is on. After everyone has had their turn, how many rooms will have their lights off?
|
20
| 0 |
19,455 |
For a real number $x$ , let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$ .
(1) Find the minimum value of $f(x)$ .
(2) Evaluate $\int_0^1 f(x)\ dx$ .
*2011 Tokyo Institute of Technology entrance exam, Problem 2*
|
\frac{1}{4} + \frac{1}{2} \ln 2
| 0.78125 |
19,456 |
Find the smallest positive integer $b$ for which $x^2 + bx + 1760$ factors into a product of two polynomials, each having integer coefficients.
|
108
| 0.78125 |
19,457 |
Given that the coefficients of the first three terms of the expansion of $(x+ \frac {1}{2})^{n}$ form an arithmetic sequence. Let $(x+ \frac {1}{2})^{n} = a_{0} + a_{1}x + a_{2}x^{2} + \ldots + a_{n}x^{n}$. Find:
(1) The value of $n$;
(2) The value of $a_{5}$;
(3) The value of $a_{0} - a_{1} + a_{2} - a_{3} + \ldots + (-1)^{n}a_{n}$.
|
\frac {1}{256}
| 37.5 |
19,458 |
How many integers are between $(11.2)^3$ and $(11.3)^3$?
|
38
| 40.625 |
19,459 |
Find the smallest, positive five-digit multiple of $18$.
|
10008
| 88.28125 |
19,460 |
The term containing \(x^7\) in the expansion of \((1 + 2x - x^2)^4\) arises when \(x\) is raised to the power of 3 in three factors and \(-x^2\) is raised to the power of 1 in one factor.
|
-8
| 81.25 |
19,461 |
In a right triangle ABC with sides 9, 12, and 15, a small circle with center Q and radius 2 rolls around the inside of the triangle, always remaining tangent to at least one side of the triangle. When Q first returns to its original position, through what distance has Q traveled?
|
24
| 35.9375 |
19,462 |
Each of the integers 334 and 419 has digits whose product is 36. How many 3-digit positive integers have digits whose product is 36?
|
21
| 32.03125 |
19,463 |
Given the random variables $\xi + \eta = 8$, if $\xi \sim B(10, 0.6)$, then calculate $E\eta$ and $D\eta$.
|
2.4
| 71.09375 |
19,464 |
Simplify $\frac{1}{1+\sqrt{3}} \cdot \frac{1}{1-\sqrt{5}}$.
|
\frac{1}{1 - \sqrt{5} + \sqrt{3} - \sqrt{15}}
| 2.34375 |
19,465 |
An ice cream shop offers 8 different flavors of ice cream. What is the greatest number of sundaes that can be made if each sundae can consist of 1, 2, or 3 scoops, with each scoop possibly being a different type of ice cream and no two sundaes having the same combination of flavors?
|
92
| 7.8125 |
19,466 |
In the rectangular coordinate system on the plane, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar coordinate equation of the curve C₁ is ρ²-6ρcosθ+5=0, and the parametric equation of the curve C₂ is $$\begin{cases} x=tcos \frac {π}{6} \\ y=tsin \frac {π}{6}\end{cases}$$ (t is the parameter).
(1) Find the rectangular coordinate equation of the curve C₁ and explain what type of curve it is.
(2) If the curves C₁ and C₂ intersect at points A and B, find the value of |AB|.
|
\sqrt {7}
| 0 |
19,467 |
The sides opposite to the internal angles $A$, $B$, and $C$ of $\triangle ABC$ are $a$, $b$, and $c$ respectively. It is given that $b\sin C + c\sin B = 4a\sin B\sin C$ and $b^2 + c^2 - a^2 = 8$. The area of $\triangle ABC$ is __________.
|
\frac{2\sqrt{3}}{3}
| 35.9375 |
19,468 |
In the arithmetic sequence $\{a\_n\}$, it is given that $a\_3 + a\_4 = 12$ and $S\_7 = 49$.
(I) Find the general term formula for the sequence $\{a\_n\}$.
(II) Let $[x]$ denote the greatest integer not exceeding $x$, for example, $[0.9] = 0$ and $[2.6] = 2$. Define a new sequence $\{b\_n\}$ where $b\_n = [\log_{10} a\_n]$. Find the sum of the first 2000 terms of the sequence $\{b\_n\}$.
|
5445
| 16.40625 |
19,469 |
In rectangle $EFGH$, we have $E=(1,1)$, $F=(101,21)$, and $H=(3,y)$ for some integer $y$. What is the area of rectangle $EFGH$?
A) 520
B) 1040
C) 2080
D) 2600
|
1040
| 11.71875 |
19,470 |
In a certain grade with 1000 students, 100 students are selected as a sample using systematic sampling. All students are numbered from 1 to 1000, and are evenly divided into 100 groups (1-10, 11-20, ..., 991-1000 in order). If the number drawn from the first group is 6, then the number that should be drawn from the tenth group is __________.
|
96
| 99.21875 |
19,471 |
Given that the function $f(x) = \sqrt{3}\sin\omega x - 2\sin^2\left(\frac{\omega x}{2}\right)$ ($\omega > 0$) has a minimum positive period of $3\pi$,
(I) Find the maximum and minimum values of the function $f(x)$ on the interval $[-\pi, \frac{3\pi}{4}]$;
(II) In $\triangle ABC$, where $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and $a < b < c$, with $\sqrt{3}a = 2c\sin A$, find the measure of angle $C$;
(III) Under the conditions of (II), if $f\left(\frac{3}{2}A + \frac{\pi}{2}\right) = \frac{11}{13}$, find the value of $\cos B$.
|
\frac{12 + 5\sqrt{3}}{26}
| 15.625 |
19,472 |
No math tournament exam is complete without a self referencing question. What is the product of
the smallest prime factor of the number of words in this problem times the largest prime factor of the
number of words in this problem
|
1681
| 1.5625 |
19,473 |
On $5\times 5$ squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.
|
24
| 25.78125 |
19,474 |
Let \( M \) be a subset of the set \(\{1, 2, 3, \cdots, 15\}\), and suppose that the product of any three different elements in \( M \) is not a perfect square. Let \( |M| \) denote the number of elements in the set \( M \). Find the maximum value of \( |M| \).
|
10
| 85.15625 |
19,475 |
For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\sin x = \sin (n^2 x)$ on the interval $[0, 2\pi]$. What is $\sum_{n=2}^{100} G(n)$?
|
676797
| 24.21875 |
19,476 |
Given the function $f(x)=\sin(2x+\varphi)$ where $(0 < \varphi < \pi)$ satisfies $f(x) \leq |f(\frac{\pi}{6})|$, and $f(x_{1}) = f(x_{2}) = -\frac{3}{5}$, calculate the value of $\sin(x_{2}-x_{1})$.
|
\frac{4}{5}
| 68.75 |
19,477 |
Given the function $f(x)=\cos x\cdot\sin \left(x+ \frac {\pi}{3}\right)- \sqrt {3}\cos ^{2}x+ \frac { \sqrt {3}}{4}$, $x\in\mathbb{R}$.
(I) Find the smallest positive period of $f(x)$.
(II) Find the maximum and minimum values of $f(x)$ on the closed interval $\left[- \frac {\pi}{4}, \frac {\pi}{4}\right]$.
|
- \frac {1}{2}
| 70.3125 |
19,478 |
Given $\tan \alpha =2$, find the values of the following expressions:
$(1) \frac{\sin \alpha - 3\cos \alpha}{\sin \alpha + \cos \alpha}$
$(2) 2\sin ^{2} \alpha - \sin \alpha \cos \alpha + \cos ^{2} \alpha$
|
\frac{7}{5}
| 93.75 |
19,479 |
Julio cuts off the four corners, or vertices, of a regular tetrahedron. How many vertices does the remaining shape have?
|
12
| 11.71875 |
19,480 |
Let $x_1,x_2,y_1,y_2$ be real numbers satisfying the equations $x^2_1+5x^2_2=10$ , $x_2y_1-x_1y_2=5$ , and $x_1y_1+5x_2y_2=\sqrt{105}$ . Find the value of $y_1^2+5y_2^2$
|
23
| 34.375 |
19,481 |
Assume every 7-digit whole number is a possible telephone number except those that begin with a digit less than 3. What fraction of telephone numbers begin with $9$ and have $3$ as their middle digit (i.e., fourth digit)?
A) $\frac{1}{60}$
B) $\frac{1}{70}$
C) $\frac{1}{80}$
D) $\frac{1}{90}$
E) $\frac{1}{100}$
|
\frac{1}{70}
| 99.21875 |
19,482 |
Let $Q$ be the product of the first $50$ positive even integers. Find the largest integer $j$ such that $Q$ is divisible by $2^j$.
|
97
| 89.0625 |
19,483 |
Q. A light source at the point $(0, 16)$ in the co-ordinate plane casts light in all directions. A disc(circle along ith it's interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\sqrt{n}$ where $m, n$ are positive integers and $n$ is squarefree. Find $m + n$ .
|
21
| 10.9375 |
19,484 |
In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{\begin{array}{l}{x=2+3\cos\alpha,}\\{y=3\sin\alpha}\end{array}\right.$ ($\alpha$ is the parameter). Taking the coordinate origin $O$ as the pole and the non-negative $x$-axis as the polar axis to establish a polar coordinate system, the polar coordinate equation of the line $l$ is $2\rho \cos \theta -\rho \sin \theta -1=0$.
$(1)$ Find the Cartesian equation of curve $C$ and the rectangular coordinate equation of line $l$;
$(2)$ If line $l$ intersects curve $C$ at points $A$ and $B$, and point $P(0,-1)$, find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$.
|
\frac{3\sqrt{5}}{5}
| 96.875 |
19,485 |
Given triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, and $b \sin A = \sqrt{3} a \cos B$.
1. Find the measure of angle $B$.
2. If $b = 3$ and $\sin C = 2 \sin A$, find the values of $a$ and $c$.
|
2\sqrt{3}
| 60.15625 |
19,486 |
For how many non-negative real values of $x$ is $\sqrt{169-\sqrt[4]{x}}$ an integer?
|
14
| 89.84375 |
19,487 |
Five rays $\overrightarrow{OA}$ , $\overrightarrow{OB}$ , $\overrightarrow{OC}$ , $\overrightarrow{OD}$ , and $\overrightarrow{OE}$ radiate in a clockwise order from $O$ forming four non-overlapping angles such that $\angle EOD = 2\angle COB$ , $\angle COB = 2\angle BOA$ , while $\angle DOC = 3\angle BOA$ . If $E$ , $O$ , $A$ are collinear with $O$ between $A$ and $E$ , what is the degree measure of $\angle DOB?$
|
90
| 94.53125 |
19,488 |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0$, $b > 0$), a line with an angle of $60^\circ$ passes through one of the foci and intersects the y-axis and the right branch of the hyperbola. Find the eccentricity of the hyperbola if the point where the line intersects the y-axis bisects the line segment between one of the foci and the point of intersection with the right branch of the hyperbola.
|
2 + \sqrt{3}
| 20.3125 |
19,489 |
Find $ [\sqrt{19992000}]$ where $ [x]$ is the greatest integer less than or equal to $ x$ .
|
4471
| 100 |
19,490 |
Find the largest positive integer $n$ such that $\sigma(n) = 28$ , where $\sigma(n)$ is the sum of the divisors of $n$ , including $n$ .
|
12
| 80.46875 |
19,491 |
Find the product of the values of $x$ that satisfy the equation $|5x| + 7 = 47$.
|
-64
| 97.65625 |
19,492 |
Given eleven books consisting of three Arabic, two English, four Spanish, and two French, calculate the number of ways to arrange the books on the shelf keeping the Arabic books together, the Spanish books together, and the English books together.
|
34560
| 35.9375 |
19,493 |
The average of the numbers $1, 2, 3, \dots, 149,$ and $x$ is $150x$. What is $x$?
|
\frac{11175}{22499}
| 79.6875 |
19,494 |
If $\alpha$ and $\beta$ are acute angles, and $\sin \alpha = \frac{\sqrt{5}}{5}$, $\cos \beta = \frac{3\sqrt{10}}{10}$, then $\sin (\alpha + \beta) =$____, $\alpha + \beta =$____.
|
\frac{\pi}{4}
| 82.03125 |
19,495 |
A wizard is crafting a magical elixir. For this, he requires one of four magical herbs and one of six enchanted gems. However, one of the gems cannot be used with three of the herbs. Additionally, another gem can only be used if it is paired with one specific herb. How many valid combinations can the wizard use to prepare his elixir?
|
18
| 14.0625 |
19,496 |
How many ways are there to put 7 balls in 4 boxes if the balls are not distinguishable and neither are the boxes?
|
11
| 16.40625 |
19,497 |
Given the sequences $\{a_{n}\}$ and $\{b_{n}\}$ satisfying $2a_{n+1}+a_{n}=3$ for $n\geqslant 1$, $a_{1}=10$, and $b_{n}=a_{n}-1$. Find the smallest integer $n$ that satisfies the inequality $|{{S_n}-6}|＜\frac{1}{{170}}$.
|
10
| 25 |
19,498 |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $$\begin{cases} x=3\cos \theta \\ y=2\sin \theta \end{cases} (\theta \text{ is the parameter}),$$ and the parametric equation of the line $l$ is $$\begin{cases} x=t-1 \\ y=2t-a-1 \end{cases} (t \text{ is the parameter}).$$
(Ⅰ) If $a=1$, find the length of the line segment cut off by line $l$ from curve $C$.
(Ⅱ) If $a=11$, find a point $M$ on curve $C$ such that the distance from $M$ to line $l$ is minimal, and calculate the minimum distance.
|
2\sqrt{5}-2\sqrt{2}
| 12.5 |
19,499 |
In the diagram, $\triangle PQR$ is isosceles with $PQ = PR = 39$ and $\triangle SQR$ is equilateral with side length 30. The area of $\triangle PQS$ is closest to:
|
75
| 20.3125 |
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