problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
The average of sample data $x_{1}$, $x_{2}$, $\ldots$, $x_{6}$ is $x_{0}$, $x_{1} \lt x_{2} \lt \ldots \lt x_{6}$. Then ( )
A: The range of $x_{1}$, $x_{2}$, $\ldots$, $x_{6}$ is equal to the range of $x_{1}$, $x_{2}$, $\ldots$, $x_{6}$, $x_{0}$.
B: The average of $x_{1}$, $x_{2}$, $\ldots$, $x_{6}$ is equal to the a... | AB |
Given $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(-1,3)$.
$(1)$ When $k\overrightarrow{a}+\vec{b}$ is perpendicular to $\overrightarrow{a}-3\vec{b}$, find the value of $k$;
$(2)$ When $k\overrightarrow{a}+\overrightarrow{b}$ is parallel to $\overrightarrow{a}-3\overrightarrow{b}$, find the value of $k$. | -\frac{1}{3} |
5. Given a natural number $x=9^{n}-1$, where $n-$ is an odd natural number. It is known that $x$ has exactly three distinct prime divisors, one of which is 61. Find $x$. | 59048 |
Given the sets $A=\{0, 1, 2\}$ and $B=\{x|x^2-x\leq0\}$, then $A\cap B= \_\_\_\_\_\_$. | \{0,1\} |
Find the number of sets $\{a,b,c\}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$. | 728 |
6. Given three points $A, B, C$ on a plane satisfying $|\overrightarrow{A B}|=3,|\overrightarrow{B C}|=4,|\overrightarrow{C A}|=5$, then $\overrightarrow{A B} \cdot \overrightarrow{B C}+\overrightarrow{B C} \cdot \overrightarrow{C A}+\overrightarrow{C A} \cdot \overrightarrow{A B}$ is equal to | -25 |
$x^2+y^2=(x+y)^2+P=(x-y)^2+Q$, then $P=$____, $Q=$____. | 2xy |
Folklore
In a certain state, the airline system is arranged in such a way that each city is connected by air routes to no more than three other cities, and from any city, you can reach any other city with no more than one layover. What is the maximum number of cities that can be in this state? | 10 |
If $\frac{a}{b}=3$ and $\frac{b}{c}=2$, then the value of $\frac{a-b}{c-b}$ is
(A) -4
(B) $-\frac{1}{3}$
(C) $\frac{2}{3}$
(D) 2
(E) 6 | -4 |
Given a $3 \times 3$ table (like for playing tic-tac-toe). Four chips are placed randomly in four randomly chosen cells.
Find the probability that among these four chips, there will be three that are in a row either vertically, horizontally, or diagonally. | \frac{8}{21} |
4. A cone of height $H$ is circumscribed around a hemisphere of radius $r$, such that the bases of the hemisphere and the cone are concentric circles. Calculate the volume of the part of the cone that does not belong to the hemisphere, i.e., express that volume in terms of $r$ and $H$. | \frac{r^{2}\pi}{3}(\frac{H^{3}}{H^{2}-r^{2}}-2r) |
3. Felix chooses a positive integer as the starting number and writes it on the board. He then repeatedly performs the following step: he replaces the number $n$ on the board with $\frac{1}{2} n$ if $n$ is even and with $n^{2}+3$ if $n$ is odd.
(a) Prove that if Felix starts with an odd number, the next two numbers he... | 21 |
21. Let $A=\{1,2,3,4,5,6\}$. Find the number of distinct functions $f: A \rightarrow A$ such that $f(f(f(n)))=n$ for all $n \in A$. | 81 |
3. A square of unit area is inscribed in an isosceles triangle, with one side lying on the base of the triangle. Find the area of the triangle, given that the centroids of the triangle and the square coincide. | \frac{9}{4} |
2. Let $a \in \mathbf{R}$, if for all $x>0$, $\left(x^{2}+a x-5\right)(a x-1) \geq 0$ holds, then $a=$ | \frac{1}{2} |
2. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
\begin{array}{l}
a_{1}=2, a_{2}=1, \\
a_{n} a_{n+1} a_{n+2}=a_{n}+a_{n+1}+a_{n+2}\left(n \in \mathbf{N}_{+}\right) .
\end{array}
$$
Then $a_{1}+a_{2}+\cdots+a_{2011}=$ $\qquad$ . | 4022 |
Given that the sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n$, and it satisfies the equation $S\_{n-1} + S\_n = 2n^2 + 1$ for $n \geq 2, n \in \mathbb{N}^+$, and $a\_1 = x$, and $\{a\_n\}$ is monotonically increasing, determine the range of values for $x$. | (2, 3) |
3. In the country of Aviania, there are 40 cities, some of which are connected by two-way flights. Moreover, between any two cities, there is only one reasonable air route (i.e., a route where the same flight is not used in different directions).
For each city, the air distance to the capital was calculated. It is cal... | 780 |
Given $A=2x^{2}-3xy+y^{2}+2x+2y$, $B=4x^{2}-6xy+2y^{2}-3x-y$.
$(1)$ When $x=2$, $y=-\frac{1}{5}$, find the value of $B-2A$.
$(2)$ If $|x-2a|+\left(y-3\right)^{2}=0$, and $B-2A=a$, find the value of $a$. | -1 |
Example 2. Form the equations of the normal and the tangent plane to the sphere $x^{2}+y^{2}+z^{2}-2 x+4 y-6 z+5=0$ at the point $M_{0}(3,-1,5)$. | 2x+y+2z-15=0 |
Which of the following operations is correct?
A: $\sqrt{(-6)^2}=-6$
B: $(a-b)^{2}=a^{2}-ab+b^{2}$
C: $(-2x^{2})^{3}=-6x^{6}$
D: $x^{2}\cdot x^{3}=x^{5}$ | D |
If the function $f(x)= \frac{x^{2}-1}{x^{2}+1}$, then
$(1) \frac{f(2)}{f\left(\frac{1}{2}\right)}=$\_\_\_\_\_\_\_\_;
$(2) f(3)+f(4)+\ldots+f(2017)+f\left(\frac{1}{3}\right)+f\left(\frac{1}{4}\right)+\ldots+f\left(\frac{1}{2017}\right)=$\_\_\_\_\_\_\_\_. | 0 |
5. The sum of 10 positive integers is 24, the maximum value of the sum of their squares is $a$, and the minimum value is $b$, then $a +$ $b=$ . $\qquad$ | 294 |
Given a sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_1=-2$, $a_2=2$, and $a_{n+2}-a_n=1+(-1)^n$, then $S_{50}=$ ______. | 600 |
The squares of the natural numbers from 1 to 99 were written one after another, forming the number 14916253649... What is the digit that occupies the 100th position? (The positions are counted from left to right, so the $1^{\underline{a}}$ position is the 1, the $2^{\underline{a}}$ is the 4, and so on.) | 9 |
Given a positive integer $a$ satisfying $192 | a^3 + 191$ and $a < 2009$, find the sum of all possible positive integers $a$ that meet the condition. | 10571 |
Given sets $A=\{2,y\}$ and $B=\{x,3\}$, if $A=B$, then $x+y=$____. | 5 |
Given the function $f(x)= \frac{1}{2}{e}^{2x}-ax$, where $a\in R$ and $e$ is the base of the natural logarithm.
1. Discuss the monotonicity of the function $f(x)$.
2. If $a=1$, and the function $g(x)=(x-m)f(x)- \frac{1}{4}{e}^{2x}+{x}^{2}+x$ is increasing in the interval $(0,+\infty)$, find the maximum value of the in... | 1 |
20. Xiaoling, while calculating division, wrote the divisor 65 as 56, resulting in a quotient of 13 and a remainder of 52. The correct quotient should be $\qquad$. | 12 |
Let $f_n(x) = (1+2x)(1+2^2x)\ldots(1+2^nx)$ ($n \geq 2$, $n \in \mathbb{N}^*$).
(1) Let the coefficient of the $x$ term in the expansion of $f_n(x)$ be $a_n$. Find $a_n$.
(2) Let the coefficient of the $x^2$ term in the expansion of $f_n(x)$ be $b_n$. Prove that $b_{n+1} = b_n + 2^{n+1}a_n$.
(3) Does there exist con... | a=1, b=-1 |
Example 5 Find the real solution of the equation $\sqrt{1+\sqrt{1+x}}=\sqrt[3]{x}$.
$(2015$, Finnish High School Mathematics Competition) | 8 |
Given that $x^{m-2}+1=3$ is a linear equation in $x$, the value of the rational number $m$ is:
A: $3$
B: $2$
C: $1$
D: $0$ | A |
Find the equation of the line that passes through point A (2, 1) and is parallel to the line $2x - y + 3 = 0$. | 2x - y - 3 = 0 |
Given the function $f(x)=a\sin 3x+bx^{3}+4$ ($a\in \mathbb{R}, b\in \mathbb{R}$), where $f'(x)$ is the derivative of $f(x)$, then $f(2014)+f(-2014)+f'(2015)-f'(-2015)=$ ()
A: $8$
B: $2014$
C: $2015$
D: $0$ | A |
Which of the following operations is correct?
A: $a^2 \cdot a^3 = a^6$
B: $a^3 \div a = a^3$
C: $(a^2)^3 = a^5$
D: $(a^2b)^2 = a^4b^2$ | \text{D} |
The minute hand on a clock points at the 12. The minute hand then rotates $120^{\circ}$ clockwise. Which number will it be pointing at?
(A) 6
(B) 2
(C) 4
(D) 3
(E) 5
 | C |
7. (7 points) Use the digits $1,2,3,4,5,6,7,8$ to form two different four-digit numbers (each digit is used only once) so that their difference is the smallest. What is this difference? $\qquad$ . | 247 |
In triangle $\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to the internal angles $A$, $B$, $C$, respectively, and $2a\sin A = \left(2b+c\right)\sin B + \left(2c+b\right)\sin C$.
$(1)$ Find the measure of angle $A$;
$(2)$ If $a=7$ and the distance from vertex $A$ to side $BC$ is $\frac{15\sqrt... | b=5, c=3 |
Given a discrete random variable $X$ which follows a binomial distribution $X \sim B(n, p)$, and it is known that $E(X) = 3$ and $D(X) = 2$, find the values of $n$ and $p$. | n = 9, \quad p = \frac{1}{3} |
## Task B-3.4.
Four friends have decided to spend the weekend in a small family hotel that has five rooms, each decorated in a different style and color. They are the only guests in the hotel. They have left it to the hotel owner to assign them to rooms, but with the condition that no more than two friends are in one ... | 540 |
2. In $\triangle A B C$, it is known that $A B=a^{2}, B C=4 a$, $A C=b^{2}-4$, and $a, b$ are odd numbers greater than 3. Then the relationship between $a$ and $b$ is ( ).
(A) $a>b$
(B) $a<b$
(C) $a=b$
(D) cannot be determined | C |
If the proposition $\neg(p \vee q)$ is false, then ( )
A: At least one of $p$, $q$ is true.
B: At most one of $p$, $q$ is true.
C: Both $p$ and $q$ are true.
D: Both $p$ and $q$ are false. | \text{A} |
$\left[\begin{array}{l}\text { Angles between lines and planes } \\ {[\underline{\text { Linear dependence of vectors }}]}\end{array}\right]$
The lateral face of a regular quadrilateral pyramid forms an angle of $45^{\circ}$ with the plane of the base. Find the angle between the apothem of the pyramid and the plane of... | 30 |
$25 \cdot 18$ For some angle $\theta$, the number of triangles with vertices at $(-5,0),(5,0)$ and $(5 \cos \theta, 5 \sin \theta)$ that have an area of 10 is
(A) 0.
(B) 2.
(C) 4.
(D) 6.
(E) 8.
(49th American High School Mathematics Examination, 1998) | C |
3. Given $\triangle A B C$ with the three interior angles $\angle A, \angle B, \angle C$ and the corresponding side lengths $a, b, c$, where $a, b, c$ are rational numbers, and they satisfy
$$
(a+\sqrt{2})^{2}=(b+\sqrt{2})(c+\sqrt{2}) \text {. }
$$
Then $\angle A=(\quad)$.
(A) $60^{\circ}$
(B) $75^{\circ}$
(C) $90^{\c... | 60 |
## Problem A1
Find all real solutions to: $\mathrm{x}+\mathrm{y}-\mathrm{z}=-1 ; \mathrm{x}^{2}-\mathrm{y}^{2}+\mathrm{z}^{2}=1,-\mathrm{x}^{3}+\mathrm{y}^{3}+\mathrm{z}^{3}=-1$.
| (1,-1,1)or(-1,-1,-1) |
30th Putnam 1969 Problem A2 A is an n x n matrix with elements a ij = |i - j|. Show that the determinant |A| = (-1) n-1 (n - 1) 2 n-2 . | (-1)^{n-1}(n-1)2^{n-2} |
The real numbers $x, y$ and $z$ satisfy the three equations
$$
\begin{aligned}
x+y & =7 \\
x z & =-180 \\
(x+y+z)^{2} & =4
\end{aligned}
$$
If $S$ is the sum of the two possible values of $y$, then $-S$ equals
(A) 56
(B) 14
(C) 36
(D) 34
(E) 42 | 42 |
7. Let $M \Theta N$ denote the remainder of the division of the larger number by the smaller number among $M$ and $N$. For example, $3 \Theta 10=1$. For a non-zero natural number $A$ less than 40, given that $20 \Theta(A \bigodot 20)=7$, then $A=$ $\qquad$ | 33 |
The contrapositive statement of "If $x \neq 1$, then $x^2 - 1 \neq 0$" is a ________ statement. (Fill in "true" or "false") | \text{false} |
4. Let $a, b$ be any positive real numbers such that $a^{2}+b^{2}=1$. Find the smallest possible value of the expression
$$
\frac{a^{2}\left(a+b^{3}\right)}{b-b^{3}}+\frac{b^{2}\left(b+a^{3}\right)}{a-a^{3}}
$$
and determine for which pairs $(a, b)$ this value is achieved. (Tomáš Bárta) | 3 |
Given that points $A$, $B$, and $C$ lie on the straight line $l$, and the vectors $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\overrightarrow{OC}$ satisfy $\overrightarrow{OA} = [f(x) + 2f'(1)x]\overrightarrow{OB} - \ln x \cdot \overrightarrow{OC}$, determine the expression for the function $y = f(x)$. | f(x) = \ln x - \frac{2}{3}x + 1 |
Determine all monic polynomials $ p(x)$ of fifth degree having real coefficients and the following property: Whenever $ a$ is a (real or complex) root of $ p(x)$, then so are $ \frac{1}{a}$ and $ 1\minus{}a$. | P(x) = (x + 1)(x - 2)(x - \frac{1}{2})(x^2 - x + 1) |
(5 points) If $f(x)$ is an odd function and $f(3) = -2$, then compute the value of $f(-3) + f(0)$. | 2 |
A2. The point $E\left(4,-\frac{1}{2}\right)$ lies on the line
(A) $y=-4 x+\frac{1}{2}$
(B) $y=x-\frac{9}{2}$
(C) $y=4 x-\frac{1}{2}$
(D) $y=-x+\frac{9}{2}$
(E) $y=\frac{1}{2} x-4$ | x-\frac{9}{2} |
There are $\boxed{\text{answer}}$ thousands, $\boxed{\text{answer}}$ hundreds, and $\boxed{\text{answer}}$ tens in 9890. | 9, 8, 9 |
Given the universal set $U=\{x|x^{2}-2x-3\leqslant 0\}$ and the set $M=\{y|x^{2}+y^{2}=1\}$, find the complement of $M$ in $U$, denoted as $∁\_UM$.
A: $(-∞,-1)∪(1,+∞)$
B: $(1,3]$
C: $[-1,1]$
D: $[-1,3]$ | B |
Problem 4. Blagoya participated in a lottery where each ticket is marked with a three-digit number. He bought all the tickets marked with numbers where the product of the tens and units digits is equal to 12, and the sum of these two digits differs from the hundreds digit by 1. How many tickets and with which numbers d... | 8 |
Find the remainder of the division of the number 1000! by $10^{250}$.
# | 2\cdot10^{249} |
8. In a certain school, there are 5000 students. Each student is assigned an ID number from 0001 to 5000 . No two students can have the same ID number. If a student is selected uniformly at random, what is the probability that the ID number of the student does not contain any 2 s among its digits? | \frac{729}{1250} |
Which of the following cannot form a set?
A: All equilateral triangles
B: All students in the first year of high school
C: All overweight students in the first year of high school
D: All irrational numbers | \text{C} |
3. Let $S=\{1,2, \cdots, 2005\}$. If any set of $n$ pairwise coprime numbers in $S$ contains at least one prime number, find the minimum value of $n$.
(Tang Lihua) | 16 |
If the central angle of a regular polygon is 36°, then the number of sides of this regular polygon is _____. | 10 |
Let the function $f(x) = 2\ln x - x^2$, then ( )
A: $x=e$ is a point of local maximum
B: $x=1$ is a point of local maximum
C: $x=1$ is a point of local minimum
D: There are no points of local extremum | B |
4. Given $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ whose sum is 2,016. If these $n$ numbers can be divided into 32 groups with equal sums, and also into 63 groups with equal sums, find the minimum value of $n$.
| 94 |
7. (2nd "Friendship Cup" International Mathematics Competition Problem) Solve the system of equations
$$
\left\{\begin{array}{l}
y z=3 y+2 z-8, \\
z x=4 z+3 x-8, \\
x y=2 x+y-1 .
\end{array}\right.
$$ | (2,3,1),(3,\frac{5}{2},-1) |
Task B-2.1. Draw in the complex plane the set of points $(x, y)$, associated with complex numbers $z=x+yi$, for which
$$
\operatorname{Re} z \cdot \operatorname{Im} z<0 \quad \mathrm{and} \quad|z|^{2} \leq \operatorname{Im}\left(z^{2}\right)+1
$$
Calculate the area of this set of points. | 1 |
16. (CUB 2) Given $a, \theta \in \mathbb{R}, m \in \mathbb{N}$, and $P(x)=x^{2 m}-2|a|^{m} x^{m} \cos \theta+a^{2 m}$, factorize $P(x)$ as a product of $m$ real quadratic polynomials. | P(x)=\prod_{k=1}^{}(x^{2}-2||x\cos\frac{\theta+2(k-1)\pi}{}+^{2}) |
Let $f(x)=x+\frac{1}{2 x+\frac{1}{2 x+\frac{1}{2 x+} \ddots}}$ for $x>0$. Find $f(99) f^{\prime}(99)$. | 99 |
7. Given $z=\frac{(4-3 i)^{2}(-1-\sqrt{3} i)^{10}}{(1+i)^{12}}$, then $|z|=$ $\qquad$ | 400 |
4. In the sequence $\left\{a_{n}\right\}$, $a_{0}=2, a_{n}=a_{n-1}-n+3, n \geqslant 1$, then $a_{n}=$ | -\frac{n(n+1)}{2}+3n+2 |
The monthly wage $y$ (in yuan) of a worker changes according to the labor productivity $x$ (in thousand yuan) following the regression equation $\hat{y} = 50 + 60x$. Which of the following statements is correct?
A: When the labor productivity is 1,000 yuan, the wage is 110 yuan.
B: When the labor productivity increas... | \text{B} |
Among four polygons, the second has 4 more sides than the first, the third has 8 more sides than the second, while the fourth has 1 more side than the third. The sum of the number of all diagonals of the first and fourth polygons is equal to the sum of the number of all diagonals of the second and third polygons. How m... | 3,7,15,16 |
[
The ratio in which the bisector divides the side. [The ratio of the areas of triangles with a common base or common height]
In triangle $ABC$, the median $AD$ and the bisector $BE$ are perpendicular and intersect at point $F$. It is known that $S_{DEF}=5$. Find $S_{ABC}$.
# | 60 |
If an even function $f(x)$ defined on $\mathbb{R}$ is a decreasing function on $[0, +\infty)$, then we have ( )
A: $f(3) < f(-2) < f(1)$
B: $f(1) < f(-2) < f(3)$
C: $f(-2) < f(1) < f(3)$
D: $f(3) < f(1) < f(-2)$ | \text{A} |
Calculate \\( \left( \frac {1+i}{1-i} \right)^{2017} = \) \\()
A: \\(-1\\)
B: \\(i\\)
C: \\(-i\\)
D: \\(1\\) | B |
5. (20 points) Alexei came up with the following game. First, he chooses a number $x$ such that $2017 \leqslant x \leqslant 2117$. Then he checks if $x$ is divisible by 3, 5, 7, 9, and 11 without a remainder. If $x$ is divisible by 3, Alexei awards the number 3 points, if by 5 - then 5 points, ..., if by 11 - then 11 p... | 2079 |
The hosting of the Beijing Winter Olympics has not only attracted 300 million people to participate in ice and snow sports but also ignited the enthusiasm for national fitness. The development of ice and snow sports and the smooth advancement of national fitness have laid a solid foundation for building a sports powerh... | \frac{8}{15} |
Example 12. In a box, there are 15 red, 9 blue, and 6 green balls. Six balls are drawn at random. What is the probability that 1 green, 2 blue, and 3 red balls are drawn (event $A$)? | \frac{24}{145}\approx0.17 |
The function $f$ has the following properties:
(i) its domain is all real numbers,
(ii) it is an odd function (that is, $f(-x)=-f(x)$ for every real number $x$ ), and
(iii) $f(2 x-3)-2 f(3 x-10)+f(x-3)=28-6 x$ for every real number $x$.
Determine the value of $f(4)$. | 8 |
[b]p1.[/b] Consider a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$ such that the perimeter of the right triangle is numerically (ignoring units) equal to its area. Prove that there is only one possible value of $a + b - c$, and determine that value.
[b]p2.[/b] Last August, Jennifer McL... | 4 |
Prove the proposition using the method of contradiction: "For $a, b \in \mathbb{N}$, if $ab$ is divisible by $5$, then at least one of $a$ or $b$ is divisible by $5$." What is the assumption for the contradiction?
A: Both $a$ and $b$ are divisible by $5$
B: Neither $a$ nor $b$ is divisible by $5$
C: $a$ is not divisibl... | \text{B} |
1. The sum of all roots of the equation $|2 x-4|=5$ is ( ).
(A) -0.5
(B) 4.5
(C) 5
(D) 4 | D |
Problem 5.4. At the end-of-the-year school dance, there were twice as many boys as girls. Masha counted that there were 8 fewer girls, besides herself, than boys. How many boys came to the dance? | 14 |
Given the function $f(x) = a^x + x - b$ has a root $x_b \in (n, n+1)$ ($n \in \mathbb{Z}$), where the constants $a, b$ satisfy $2^a = 3$, $3^b = 2$, then the value of $n$ is ( )
A: -2
B: -1
C: 0
D: 1 | \text{B} |
Given in \\(\triangle ABC\\), the three interior angles \\(A\\), \\(B\\), and \\(C\\) form an arithmetic sequence, then \\(\sin B=\\) ( )
A: \\( \dfrac {1}{2}\\)
B: \\( \dfrac { \sqrt {3}}{2}\\)
C: \\( \dfrac { \sqrt {2}}{2}\\)
D: \\( \dfrac { \sqrt {3}}{3}\\) | \text{B} |
Let the two base angles of a triangle be $A$ and $B$, with $B$ larger than $A$.
The altitude to the base divides the vertex angle $C$ into two parts, $C_1$ and $C_2$, with $C_2$ adjacent to side $a$. Then:
$\textbf{(A)}\ C_1+C_2=A+B \qquad \textbf{(B)}\ C_1-C_2=B-A \qquad\\ \textbf{(C)}\ C_1-C_2=A-B \qquad \textbf{(D)... | B-A=C_1-C_2 |
5. Determine all pairs $(x, y)$ of integers with $x \leq y$ for which $\frac{1}{x}+\frac{1}{y}=\frac{1}{4}$. | (x,y)=(-4,2),(-12,3),(5,20),(6,12),(8,8) |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. It is given that $2c\cos B=2a-b$.
$(1)$ Find the magnitude of angle $C$,
$(2)$ Given that the sequence $\{a_n\}$ is a geometric sequence with common ratio $q=\cos (A+B)$, and $a_3=6\cos C$, find the general formu... | T_n=n^2+n |
The random variable $\xi$ takes the values $0$, $1$, $2$. If $P(\xi=0)=\frac{1}{4}$, $E(\xi)=1$, then $D(\xi)=$\_\_\_\_\_\_\_\_． | \frac{1}{2} |
3. Vicky starts with a list $(a, b, c, d)$ of four digits. Each digit is $0,1,2$, or 3 . Vicky enters the list into a machine to produce a new list $(w, x, y, z)$. In the new list, $w$ is the number of $0 \mathrm{~s}$ in the original list, while $x, y$ and $z$ are the numbers of $1 \mathrm{~s}, 2 \mathrm{~s}$ and $3 \m... | (6,2,1,0,0,0,1,0,0,0) |
If the calculation of data from a 2×2 contingency table gives a $\chi^2$ value of 4.073, then there is a ______ confidence level to believe that the two variables are related. [Given that $P(\chi^2 \geq 3.841) \approx 0.05$, $P(\chi^2 \geq 5.024) \approx 0.025$]. | 95\% |
Given right $\triangle ABC$ with legs $BC=3, AC=4$. Find the length of the shorter angle trisector from $C$ to the hypotenuse:
$\textbf{(A)}\ \frac{32\sqrt{3}-24}{13}\qquad\textbf{(B)}\ \frac{12\sqrt{3}-9}{13}\qquad\textbf{(C)}\ 6\sqrt{3}-8\qquad\textbf{(D)}\ \frac{5\sqrt{10}}{6}\qquad\textbf{(E)}\ \frac{25}{12}\qquad... | \textbf{(A)}\\frac{32\sqrt{3}-24}{13} |
If the solution set of the inequality $\left(1+k^{2}\right)x\leqslant k^{4}+4$ with respect to $x$ is $M$, then for any real constant $k$, it is always true that:
A: $2\in M$, $0\in M$
B: $2\notin M$, $0\notin M$
C: $2\in M$, $0\notin M$
D: $2\notin M$, $0\in M$ | \text{A} |
Define $n!!$ to be $n(n-2)(n-4)\cdots 3\cdot 1$ for $n$ odd and $n(n-2)(n-4)\cdots 4\cdot 2$ for $n$ even. When $\sum_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\dfrac{ab}{10}$. | 401 |
Given the function $f(x)=-x^{3}+ax^{2}-4$ reaches an extreme value at $x=2$, find the minimum value of $f(m)$ where $m\in\[-1,1\]$ . | -4 |
3. As shown in Figure 1, $C$ is any point on line segment $A B$, $\triangle A C D$ and $\triangle B C E$ are equilateral triangles, $A F$ and $B G$ are tangents to the circle passing through points $D$, $C$, and $E$, with $F$ and $G$ being the points of tangency. Then ( ).
(A) $A F>B G$
(B) $A F=B G$
(C) $A F<B G$
(D) ... | B |
3. Divide the sides of the equilateral $\triangle A B C$ into four equal parts, as shown in Figure 1. Then the number of pairs of congruent equilateral triangles in the figure is ( ).
(A) 100
(B) 121
(C) 144
(D) 169 | 144 |
5.2 Solve the equation $\sqrt{x^{2}-4}+2 \sqrt{x^{2}-1}=\sqrt{7} \cdot x$. | \frac{2}{3}\sqrt{21} |
[b]p17.[/b] Let the roots of the polynomial $f(x) = 3x^3 + 2x^2 + x + 8 = 0$ be $p, q$, and $r$. What is the sum $\frac{1}{p} +\frac{1}{q} +\frac{1}{r}$ ?
[b]p18.[/b] Two students are playing a game. They take a deck of five cards numbered $1$ through $5$, shuffle them, and then place them in a stack facedown, turnin... | -\frac{1}{8} |
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