problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Among the following propositions in space, the correct one is
A: The parallel projection of parallel lines coincide
B: Two planes parallel to the same line are parallel
C: Two planes perpendicular to the same plane are parallel
D: Two lines perpendicular to the same plane are parallel | \text{D} |
Quadrilateral $A B C D$ has $\angle B C D=\angle D A B=90^{\circ}$. The perimeter of $A B C D$ is 224 and its area is 2205. One side of $A B C D$ has length 7. The remaining three sides have integer lengths. The sum of the squares of the side lengths of $A B C D$ is $S$. What is the integer formed by the rightmost two ... | 60 |
## 29. Leonie and Cats
When old lady Leonie is asked how many cats she has, she melancholically replies: “Four fifths of my cats plus four fifths of a cat.” How many cats does she have?
 | 4 |
In June, also known as graduation month, the senior students in high school need to take a group photo with their classmates whom they have been with for three years. There are 4 male students and 2 female students taking the photo. If the male student named "甲" cannot stand at either end and the female students must s... | \text{B} |
8. In the expansion of $\left(4 x^{2}-2 x-5\right)\left(1+\frac{1}{x^{2}}\right)^{5}$, the constant term is $\qquad$ | 15 |
For the function $f(x)$, if $f\left(x_{0}\right)=$ $x_{0}$, then $x_{0}$ is called a fixed point of $f(x)$. Given the function
$$
f(x)=\frac{x}{a x+b}(b>0), f(2)=1,
$$
and it has a unique fixed point in its domain.
(1) Find the expression for $f(x)$;
(2) If the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=\pi, a... | \frac{2011(2+1005 \pi)}{2 \pi} |
$n$ balls are placed independently uniformly at random into $n$ boxes. One box is selected at random, and is found to contain $b$ balls. Let $e_n$ be the expected value of $b^4$. Find $$\lim_{n \to
\infty}e_n.$$ | 15 |
5. A bag contains five white balls, four red balls, and three yellow balls. If four balls are drawn at random, the probability that balls of all colors are included is $\qquad$ . | \frac{6}{11} |
Given 6 people $A$, $B$, $C$, $D$, $E$, $F$, they are randomly arranged in a line. The probability of the event "$A$ is adjacent to $B$ and $A$ is not adjacent to $C$" is $\_\_\_\_\_\_$. | \frac{4}{15} |
$$
\begin{array}{c}
\text { 3. Given } x+y+z=3 a \quad(a \neq 0) \text {, then } \\
\frac{(x-a)(y-a)+(y-a)(z-a)+(z-a)(x-a)}{(x-a)^{2}+(y-a)^{2}+(z-a)^{2}}
\end{array}
$$
The value is $\qquad$ | -\frac{1}{2} |
3. Let there be a non-empty set $A \subseteq\{1,2, \cdots, 7\}$, and when $a \in A$, it must also be that $8-a \in A$. Then the number of such sets $A$ is $\qquad$ . | 15 |
9.1. The bathtub fills up from the hot water tap in 17 minutes, and from the cold water tap in 11 minutes. After how many minutes from opening the hot water tap should the cold water tap be opened so that by the time the bathtub is full, there is one third more hot water than cold water? | 5 |
Triangle $ABC$ has right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\angle APB = \angle BPC = \angle CPA$. Find $PC$.
[asy]
pair A=(0,5), B=origin, C=(12,0), D=rotate(-60)*C, F=rotate(60)*A, P=intersectionpoint(A--D, C--F);
draw(A--P--B--A--C--B^^C--P);
dot(A^^B^^C^^P);
pair point=P;
l... | 33 |
3. Given $0<\alpha \leqslant \beta \leqslant \gamma, \alpha+\beta+\gamma=\pi$. Then the range of $\min \left\{\frac{\sin \beta}{\sin \alpha}, \frac{\sin \gamma}{\sin \beta}\right\}$ is . $\qquad$ | \left[1, \frac{\sqrt{5}+1}{2}\right) |
Given a function \\(f(x)\\) defined on \\(\mathbb{R}\\) that satisfies \\(f(x-2)=f(-2-x)\\), and when \\(x \geqslant -2\\), \\(f(x)=2^{x}-3.\\) If the function \\(f(x)\\) has a zero point in the interval \\((k,k+1) (k \in \mathbb{Z})\\), then the value of \\(k\\) is \\((\quad)\).
A: \\(1\\) or \\(-6\\)
B: \\(0\\) or \\... | \text{A} |
17. A courier company undertakes courier services between 13 cities in a certain area. If each courier can take on the courier services for at most four cities, to ensure that there is at least one courier between every two cities, the courier company needs at least $\qquad$ couriers. | 13 |
Let $a, b \in \mathbb{R}$, and $i$ is the imaginary unit. Then "ab=0" is the condition for "the complex number $a+ \frac{b}{i}$ is purely imaginary" is
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither sufficient nor necessary condition | \text{B} |
$$
\begin{array}{l}
\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2005}\right)\left(1+\frac{1}{2}+\cdots+\frac{1}{2004}\right)- \\
\left(1+\frac{1}{2}+\cdots+\frac{1}{2005}\right)\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2004}\right)
\end{array}
$$ | \frac{1}{2005} |
There are 12 students in Mr. DoBa's math class. On the final exam, the average score of the top 3 students was 8 more than the average score of the other students, and the average score of the entire class was 85. Compute the average score of the top $3$ students.
[i]Proposed by Yifan Kang[/i] | 91 |
Example 2 Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=2, a_{n}=2^{2 n} a_{n-1}+2^{n^{2}} n \quad (n=2,3, \cdots) \text {. }
$$
Find the general term $a_{n} \ (n=1,2, \cdots)$.
(Sixth Northern Mathematical Olympiad Invitational Competition) | a_{n}=2^{n^{2}}\left(2^{n+1}-n-2\right) |
2. For an arithmetic sequence $\left\{a_{n}\right\}$ with a common difference of $d$ and all terms being positive integers, if $a_{1}=1919, a_{m}=1949, a_{n}=2019$, then the minimum value of the positive integer $m+n$ is . $\qquad$ | 15 |
If $2^{x}=5$ and $4^{y}=3$, then the value of $2^{x+2y}$ is ____. | 15 |
$-7\times 45+\left(-7\right)\times 55$. | -700 |
Given the function $f(x) = e^{mx} + x^2 - mx$.
(1) Discuss the monotonicity of $f(x)$;
(2) If for any $x_1, x_2 \in [-1, 1]$, we have $f(x_1) - f(x_2) \leq e - 1$, find the range of values for $m$. | [-1, 1] |
4. Andrija lent 54 kuna to his friends. He lent Mark ten kuna more than Ivan, and he lent Goran twice as much as Mark. How much money did he lend to each of his friends? | Ivan:6\,kn,Mark:16\,kn,Goran:32\,kn |
Given vectors $\overrightarrow{a}=(2,2,0)$ and $\overrightarrow{b}=(-2,0,2)$, if there exists a unit vector $\overrightarrow{n}$ such that $\overrightarrow{n} \perp \overrightarrow{a}$ and $\overrightarrow{n} \perp \overrightarrow{b}$, find $\overrightarrow{n}$. | \left(\frac{\sqrt{3}}{3}, -\frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}\right) |
A train line is divided into 10 segments by stations $A, B, C, D, E, F, G, H, I, J$ and $K$. The distance from $A$ to $K$ is equal to $56 \mathrm{~km}$. The route of two consecutive segments is always less than or equal to $12 \mathrm{~km}$ and the route of three consecutive segments is always greater than or equal to ... | )5,b)22,)29 |
6. Find the value of $\frac{\sin 80^{\circ}}{\sin 20^{\circ}}-\frac{\sqrt{3}}{2 \sin 80^{\circ}}$ | 2 |
Show that $5 x^{2}+y^{2}+1 \geqslant 4 x y+2 x$. Find the cases of equality. | (x,y)=(1,2) |
8.36 If $|x-\lg y|=x+\lg y$, where $x$ and $\lg y$ are real numbers, then
(A) $x=0$.
(B) $y=1$.
(C) $x=0$ and $y=1$.
(D) $x(y-1)=0$.
(E) None of these.
(23rd American High School Mathematics Examination, 1972) | D |
7. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=a_{2}=1, a_{n+2} a_{n}-a_{n+1}^{2}-a_{n+1} a_{n}=0 \text {. }
$$
Then $a_{2008}=$ $\qquad$ | 2007! |
4. In the acute triangle $\triangle A B C$, $2 \angle B=\angle C, A B: A C$ the range of values ( ).
A. $(0,2)$
B. $(0, \sqrt{2})$
C. $(\sqrt{2}, 2)$
D. $(\sqrt{2}, \sqrt{3})$ | (\sqrt{2},\sqrt{3}) |
2. Let $D$ be any point on the side $A B$ of $\triangle A B C$ (excluding points $A$ and $B$). $D$ moves parallel to $B C$ to point $E$ on side $A C$, then from $E$ moves parallel to $A B$ to point $F$ on side $B C$; then from $F$ moves parallel to $C A$ to point $G$ on side $A B$; and so on. A movement along a line pa... | C |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{2 \pi} \sin ^{2} x \cos ^{6} x d x
$$ | \frac{5\pi}{2^{6}} |
Given $\overrightarrow{a}=(2,4,5)$ and $\overrightarrow{b}=(3,x,y)$. If $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, then find the values of $x$ and $y$.
A: $x=6$, $y=15$
B: $x=3$, $y=\frac{15}{2}$
C: $x=3$, $y=15$
D: $x=6$, $y=\frac{15}{2}$ | D |
Example 19 Let $f(x)=|1-2 x|, x \in[0,1]$. Then, the number of solutions to the equation $f\{f[f(x)]\}=\frac{1}{2} x$ is $\qquad$ . | 8 |
2. Given that the altitude to the hypotenuse of right $\triangle A B C$ is 4. Then the minimum value of the area of $\triangle A B C$ is $\qquad$ . | 16 |
8. In a square $A B C D$ with side length 8, $M$ is the midpoint of $B C$, and $N$ is a point on side $D A$ such that $D N=3 N A$. If for a constant $m$, there are exactly 6 different points $P$ on the perimeter of square $A B C D$ such that $\overrightarrow{P M} \cdot \overrightarrow{P N}=m$, then the range of the rea... | \in(-1,8) |
2. For integers $x, y, z$ it holds that $x^{2}+y-z=10, x^{2}-y+z=22$. Find the smallest possible value of the expression $x^{2}+y^{2}+z^{2}$. | 34 |
7. To color 8 small squares on a $4 \times 4$ chessboard black, such that each row and each column has exactly two black squares, there are $\qquad$ different ways (answer with a number). | 90 |
1. [3] Joe B. first places the black king in one corner of the board. In how many of the 35 remaining squares can he place a white bishop so that it does not check the black king? | 30 |
3.1. A transgalactic ship has encountered an amazing meteorite stream. Some of the meteorites are flying along a straight line, one after another, at equal speeds and at equal distances from each other. Another part is flying the same way but along another straight line parallel to the first, at the same speeds but in ... | 9.1 |
## Task A-4.5.
A cube is formed from 27 identical white cubes, and all its outer faces are painted black.
(a) One of these cubes is randomly selected and placed on a table on a randomly chosen face. What is the probability that all five visible faces of the cube will be white?
(b) A cube is on the table with all fiv... | \frac{1}{2} |
If $a$, $b$, and $c$ are non-zero rational numbers, and $a+b+c=0$, then all possible values of $\frac{a}{{|a|}}+\frac{b}{{|b|}}+\frac{c}{{|c|}}-\frac{{abc}}{{|abc|}}$ are ( ).
A: $0$
B: $1$ or $-1$
C: $0$ or $-2$
D: $2$ or $-2$ | D |
1. A triangle has three sides of integer lengths, with the longest side being 11. The number of such triangles is ( ) .
A. 32
B. 34
C. 36
D. 40 | C |
11. As shown in Figure 4, the ellipse $\frac{x^{2}}{4}+y^{2}=1$, draw two parallel lines through points $P(1,0)$ and $Q(-1,0)$, intersecting the ellipse at points $A$, $B$, $C$, and $D$. Find the maximum value of the area of quadrilateral $A B C D$. | 2\sqrt{3} |
Determine the value of the expression $( \frac {64}{27})\,^{ \frac {1}{2}}+\log _{3} \frac {10}{9}+\log _{3} \frac {9}{10}$. | \frac {8 \sqrt {3}}{9} |
6. In the sequence $\left\{a_{n}\right\}$, $a_{1}=1$, the sum of the first $n$ terms is $S_{n}$, for any $n \geqslant 2\left(n \in \mathbf{N}_{+}\right), S_{n-1}-8, a_{n}$, $8 S_{n}-4$ are always in arithmetic progression. Then $\lim _{n \rightarrow \infty}\left(a_{n}{ }^{-}+S_{n}\right)=$ $\qquad$ | \frac{4}{3} |
The complex number $z=\frac{{3+i}}{{4-i}}$ corresponds to a point in the complex plane located in which quadrant?
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | \text{A} |
Given a complex number $z$ satisfies $z\left(1+i\right)=1-i$ (where $i$ is the imaginary unit), then $|z|$ is
A: $\frac{1}{2}$
B: $\frac{\sqrt{2}}{2}$
C: $\sqrt{2}$
D: $1$ | D |
2-110 Let the sequence of positive real numbers $x_{0}, x_{1}, \cdots, x_{1995}$ satisfy the following two conditions:
(1) $x_{0}=x_{1995}$;
(2) $x_{i-1}+\frac{2}{x_{i-1}}=2 x_{i}+\frac{1}{x_{i}}, i=1,2, \cdots, 1995$.
Find the maximum value of $x_{0}$ for all sequences satisfying the above conditions. | 2^{997} |
Example 2 Let $P(x)$ be a polynomial with real coefficients $P(x)=a x^{3}+b x^{2}+c x+d$. Prove: If for any $|x|<1$, we have $|P(x)| \leqslant 1$, then $|a|+|b|+|c|+|d| \leqslant 7$. (37th IMO) | 7 |
If positive numbers $a$ and $b$ satisfy $4a+3b-1=0$, then the minimum value of $\dfrac{1}{2a+b}+\dfrac{1}{a+b}$ is:
A: $3+2\sqrt{2}$
B: $1+2\sqrt{2}$
C: $2+3\sqrt{2}$
D: $2\sqrt{2}$ | \text{A} |
1. (10 points) We will call a date diverse if in its representation in the form DD/MM/YY (day-month-year) all digits from 0 to 5 are present. How many diverse dates are there in the year 2013? | 2 |
6. As shown in Figure 2, in rectangle $A B C D$, it is known that the diagonal length is 2, and $\angle 1=\angle 2=\angle 3=\angle 4$. Then the perimeter of quadrilateral $E F G H$ is ( ).
(A) $2 \sqrt{2}$
(B) 4
(C) $4 \sqrt{2}$
(D) 6 | B |
Through the use of theorems on logarithms
\[\log{\frac{a}{b}} + \log{\frac{b}{c}} + \log{\frac{c}{d}} - \log{\frac{ay}{dx}}\]
can be reduced to:
$\textbf{(A)}\ \log{\frac{y}{x}}\qquad \textbf{(B)}\ \log{\frac{x}{y}}\qquad \textbf{(C)}\ 1\qquad \\ \textbf{(D)}\ 140x-24x^2+x^3\qquad \textbf{(E)}\ \text{none of these}... | \log\frac{x}{y} |
4. Given that $m$ and $n$ are rational numbers, and the equation
$$
x^{2}+m x+n=0
$$
has a root $\sqrt{5}-2$. Then $m+n=$ $\qquad$ .
(2001, National Junior High School Mathematics Competition, Tianjin Preliminary Round) | 3 |
Which of the following is equivalent to $\sqrt{\frac{x}{1-\frac{x-1}{x}}}$ when $x < 0$?
$\mathrm{(A) \ } -x\qquad \mathrm{(B) \ } x\qquad \mathrm{(C) \ } 1\qquad \mathrm{(D) \ } \sqrt{\frac{x}{2}}\qquad \mathrm{(E) \ } x\sqrt{-1}$ | -x |
Given the function $f(\log_{2}x)$ has a domain of $[1,4]$, then the domain of $f(x)$ is ( )
A: $[2,16]$
B: $[1,2]$
C: $[0,8]$
D: $[0,2]$ | D |
8. (10 points) As shown in the figure, a large cube with an edge length of 4 is formed by 64 smaller cubes with an edge length of 1. Then, 4 smaller cubes (the shaded part in the figure) are removed from top to bottom. If the inner and outer surfaces of the remaining solid are painted red, how many smaller cubes have e... | 28 |
9. Four black $1 \times 1 \times 1$ cubes and four white $1 \times 1 \times 1$ cubes can form $\qquad$ different $2 \times 2 \times 2$ cubes (cubes that are the same after rotation are considered the same type). | 7 |
3-3. Side $AD$ of parallelogram $ABCD$ is divided into $n$ equal parts. The first division point $P$ is connected to vertex $B$. Prove that line $BP$ cuts off a segment $AQ$ on diagonal $AC$ which is $\frac{1}{n+1}$ of the diagonal $\left(AQ=\frac{AC}{n+1}\right)$. | \frac{AC}{n+1} |
There are six cards, each marked with the numbers 0, 1, 2, 4, 6, and 9, respectively. The cards with 6 and 9 can be used interchangeably (6 can be flipped to represent 9). By selecting any three cards to form a three-digit number, the number of three-digit numbers whose digits sum to a multiple of 3 is $\boxed{40}$. | 40 |
4. Let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$. There exist two points $A$ and $B$ on the ellipse such that $\overrightarrow{F_{1} A}=3 \overrightarrow{F_{2} B}$. Then the range of the eccentricity of the ellipse is ( ).
(A) $\left(0, \frac{1}{2}\... | (\frac{1}{2},1) |
2. If the complex number $z$ satisfies $|z|<1$ and $\left|\bar{z}+\frac{1}{z}\right|=\frac{5}{2}$, then $|z|=$ ( ).
(A) $\frac{4}{5}$
(B) $\frac{3}{4}$
(C) $\frac{1}{2}$
(D) $\frac{2}{3}$ | \frac{1}{2} |
What is the value of cos(-300°)?
A: $- \frac { \sqrt {3}}{2}$
B: $- \frac {1}{2}$
C: $\frac {1}{2}$
D: $\frac { \sqrt {3}}{2}$ | \text{C} |
6.251. For what positive $p$ are the roots of the equation
$5 x^{2}-4(p+3) x+4=p^{2}$
of opposite signs? Find these roots. | p>2;x_{1}=\frac{-p+2}{5},x_{2}=p+2 |
If positive numbers $x$ and $y$ satisfy $x+3y=5xy$, when $3x+4y$ reaches its minimum value, the value of $x+4y$ is:
A: $2$
B: $3$
C: $4$
D: $5$ | B |
To enhance students' physical fitness, our school has set up sports interest classes for seventh graders. Among them, the basketball interest class has $x$ students, the number of students in the soccer interest class is $2$ less than twice the number of students in the basketball interest class, and the number of stud... | 19 |
9. Let the line $y=x+\sqrt{2}$ intersect the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ at points $M, N$, and $O M \perp O N$ (where $O$ is the origin). If $|M N|=\sqrt{6}$, find the equation of the ellipse. | \frac{x^{2}}{4+2\sqrt{2}}+\frac{y^{2}}{4-2\sqrt{2}}=1 |
Given that the sequence $\{a_n\}$ is an arithmetic sequence with a non-zero common difference, $a_1=2$, and $a_1$, $a_3$, and $a_{11}$ form a geometric sequence.
1. Find the general formula for the sequence $\{a_n\}$.
2. Let $b_n = a_n - 2^n - \frac {1}{2}$, find the sum of the first $n$ terms $T_n$ for the sequence $\... | \frac {3}{2}n^2 - 2^{n+1} + 2 |
Example 4 If $x \neq 0$, find the maximum value of
$$
\frac{\sqrt{1+x^{2}+x^{4}}-\sqrt{1+x^{4}}}{x}
$$
(1992, National Junior High School Mathematics League) | \sqrt{3}-\sqrt{2} |
2. Let $\log _{x}\left(2 x^{2}+x-1\right)>\log _{x} 2-1$. Then the range of $x$ is $(\quad)$.
(A) $\frac{1}{2}\frac{1}{2}, x \neq 1$
(C) $x>1$
(D) $0<x<1$ | B |
(12) The line $y=b$ intersects the graph of the function $y=|x^2-4x+3|$ at least at three points, then the range of the real number $b$ is | 0<b\leqslant1 |
Given the parametric curve $\begin{cases} x=2t \\ y=1+4t \end{cases}$ (where $y=f(x)$) and the circle $\rho=2 \sqrt {2}\sin \theta$, determine their relative positions.
A: Separated
B: Tangent
C: Intersecting
D: Indeterminate | C |
For arithmetic sequences $\{a_n\}$ and $\{b_n\}$, the sums of the first $n$ terms are denoted as $S_n$ and $T_n$, respectively. If $\frac{S_n}{T_n} = \frac{2n}{3n+1}$, then $\frac{a_n}{b_n}$ equals ( ).
A: $\frac{2}{3}$
B: $\frac{2n-1}{3n-1}$
C: $\frac{2n+1}{3n+1}$
D: $\frac{2n-1}{3n+4}$ | \text{B} |
The equation of circle A is $x^2+y^2-2x-2y-7=0$, and the equation of circle B is $x^2+y^2+2x+2y-2=0$. Determine whether circle A and circle B intersect. If they intersect, find the equation of the line passing through the two intersection points and the distance between these two points; if they do not intersect, expla... | \frac{\sqrt{238}}{4} |
1. Given positive integers $a, b$ satisfy
$$
|b-2|+b-2=0,|a-b|+a-b=0,
$$
and $a \neq b$. Then the value of $a b$ is $\qquad$ . | 2 |
The coordinates of the point $M(5,2)$ with respect to the $x$-axis are ( ).
A: $(-5,2)$
B: $(-5,-2)$
C: $(5,-2)$
D: $(2,-5)$ | C |
In $\triangle ABC$, $A= \dfrac {\pi}{3}$, $AB=4$, and the area of $\triangle ABC$ is $2 \sqrt {3}$. Find the length of side $a$. | 2 \sqrt {3} |
Given that $a, b \in \mathbb{R}$ and $a + 3b - 2 = 0$, find the minimum value of $2^a + 8^b$. | 4 |
Given \\(x > 0\\), \\(y > 0\\), and \\(2x+3y=6\\), find the maximum value of \\(xy\\). | \dfrac {3}{2} |
9. There are 50 consecutive even numbers: $2,4,6,8, \cdots \cdots, 98,100$. From these 50 numbers, the minimum number of numbers to be taken such that their sum is 2022 (each number can be taken at most once) is | 28 |
Oldjuk meg az
$$
x^{4}-14 x^{3}+71 x^{2}-154 x+120=0
$$
egyenletet, ha tudjuk azt, hogy két gyökének összege 5, a másik két gyökének szorzata 20.
| x_{1}=2,x_{2}=3,x_{3}=4,x_{4}=5 |
C6. I have four identical black beads and four identical white beads. Carefully explain how many different bracelets I can make using all the beads. | 8 |
If three planes intersect pairwise, and the three lines of intersection are mutually parallel, then these three planes divide space into (\boxed{C}) parts.
A 5 B 6 C 7 D 8 | C |
4. Given that the average of the data $x_{1}, x_{2}, \ldots, x_{10}$ is 6, and the standard deviation is $\sqrt{2}$, then the range of the average of the data $x_{1}, x_{2}, \ldots, x_{5}$ is $\qquad$ | [6-\sqrt{2},6+\sqrt{2}] |
A function $y = 6\cos x$ is defined on the interval $(0, \frac{\pi}{2})$. Its graph intersects the graph of the function $y = 5\tan x$ at point $P$. A line is drawn perpendicular to the $x$-axis passing through point $P$, with the foot of the perpendicular being $P\_1$. The line segment $PP\_1$ intersects the graph of ... | \frac{1}{3} |
11.2. On the sides $A B$ and $A C$ of an equilateral triangle $A B C$ with side length 10, points $P$ and $Q$ are taken such that the segment $P Q$ is tangent to the inscribed circle of the triangle and its length is 4. Find the area of triangle $A P Q$. | \frac{5}{\sqrt{3}} |
Liu Hui was one of the most outstanding mathematicians in ancient China. His most significant contribution to the history of Chinese arithmetic was his annotation of "The Nine Chapters on the Mathematical Art". In the chapter on circle measurement, Liu Hui proposed the idea of "repeatedly dividing until it cannot be di... | 3 |
The fraction $\frac{\sqrt{a^2+x^2}-\frac{x^2-a^2}{\sqrt{a^2+x^2}}}{a^2+x^2}$ reduces to:
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac{2a^2}{a^2+x^2} \qquad \textbf{(C)}\ \frac{2x^2}{(a^2+x^2)^{\frac{3}{2}}}\qquad \textbf{(D)}\ \frac{2a^2}{(a^2+x^2)^{\frac{3}{2}}}\qquad \textbf{(E)}\ \frac{2x^2}{a^2+x^2}$ | \frac{2a^2}{(^2+x^2)^{\frac{3}{2}}} |
## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(1 ; 2 ;-3)$
$M_{2}(1 ; 0 ; 1)$
$M_{3}(-2 ;-1 ; 6)$
$M_{0}(3 ;-2 ;-9)$ | 2\sqrt{6} |
Given positive real numbers $x$ and $y$ satisfying $x+2y=1$, the maximum value of $x \cdot y$ is \_\_\_\_\_\_. | \frac{1}{8} |
Jamal has a drawer containing $6$ green socks, $18$ purple socks, and $12$ orange socks. After adding more purple socks, Jamal noticed that there is now a $60\%$ chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
$\textbf{(A)}\ 6\qquad~~\textbf{(B)}\ 9\qquad~~\textbf{(... | 9 |
Calculate: $(\sqrt{3}-1)^{0}+2^{-1}=\_\_\_\_\_\_$. | \frac{3}{2} |
1. 51 Let $n$ be an even number, and select four different numbers $a, b, c, d$ from the integers $1, 2, \cdots, n$ such that $a+c=b+d$. Prove: The number of ways to select (without considering the order of $a, b, c, d$) is $\frac{n(n-2)(2 n-5)}{24}$. | \frac{n(n-2)(2n-5)}{24} |
Each of a group of $50$ girls is blonde or brunette and is blue eyed of brown eyed. If $14$ are blue-eyed blondes,
$31$ are brunettes, and $18$ are brown-eyed, then the number of brown-eyed brunettes is
$\textbf{(A) }5\qquad \textbf{(B) }7\qquad \textbf{(C) }9\qquad \textbf{(D) }11\qquad \textbf{(E) }13$ | 13 |
Given that $i$ is the imaginary unit, and the complex number $z$ satisfies $(z-3)(2-i)=5$.
(I) Find $z$ and $|z-2+3i|$;
(II) If $z\cdot(a+i)$ is a pure imaginary number, find the value of the real number $a$. | \frac {1}{5} |
Given $2\left(x-1\right)^{2}=8$, find the value of $x$. | x = 3 \text{ or } x = -1 |
Problem 4. Determine the smallest consecutive natural numbers $a, b, c, d$ knowing that $a$ is divisible by $8$, $b$ by $7$, $c$ by $6$, and $d$ by $5$. | 832,833,834,835 |
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