problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given that the line $x= \frac{\pi}{6}$ is an axis of symmetry for the graph of the function $f(x)=\sin (\omega x+ \frac{\pi}{3})$ (where $|ω| < 6$), determine the value of $ω$. | ω = 1 \text{ or } ω = -5 |
Among the following functions, the one that is both an even function and monotonically decreasing on $\left(0,+\infty\right)$ is:
A: $y=-x$
B: $y=\cos x$
C: $y=x^{2/3}$
D: $y=-x^{2}$ | D |
Given three lines $l_1$: $4x+y-4=0$, $l_2$: $mx+y=0$, $l_3$: $x-my-4=0$ that do not intersect at the same point:
(1) When these three lines cannot form a triangle, find the value of the real number $m$.
(2) When $l_3$ is perpendicular to both $l_1$ and $l_2$, find the distance between the two foot points. | \frac{4\sqrt{17}}{17} |
If $a-b=-3$, $c+d=2$, then $(b+c)-(a-d)=$ ? | 5 |
In $\triangle ABC$, if $b^2+c^2-a^2=\sqrt {3}bc$, then $A=$ ( )
A: 30°
B: 60°
C: 120°
D: 150° | A |
Continuously toss a fair coin 3 times. Given that there is at least one head facing up, the probability of exactly one tail facing up is ______. | \frac{3}{7} |
170. During spring, Zhenya lost $20 \%$ of his weight, then gained $30 \%$ during summer, lost $20 \%$ again during autumn, and gained $10 \%$ during winter. Did Zhenya gain or lose weight over the year? | 91.52 |
Factorization Analysis
$x^3y - xy =$ . | xy(x - 1)(x + 1) |
Given $3^{a}=2$, $b=\ln 2$, $c=2^{0.3}$, the relationship between $a$, $b$, and $c$ is ( ).
A: $a \gt b \gt c$
B: $c \gt b \gt a$
C: $b \gt c \gt a$
D: $c \gt a \gt b$ | B |
Given that the distance from the focus of the hyperbola $x^{2}+ \frac{y^{2}}{b^{2}-4}=1$ to its asymptote is $2$, determine the equation of the hyperbola's asymptote(s).
A: $y=± \frac{1}{2}x$
B: $y=± \sqrt {3}x$
C: $y=±2x$
D: $y=± \frac{ {\sqrt {3}}}{3}x$ | C |
From a group of 10 university graduates, 3 are to be chosen to serve as village chief assistants. In how many different ways can this be done such that at least one of either candidate A or B is selected, and candidate C is not selected? | 49 |
2. If a straight line bisects both the area and the perimeter of a triangle, then the line must pass through the ( ) of the triangle.
(A) incenter
(B) circumcenter
(C) centroid
(D) orthocenter | A |
8. (3 points) A store has two sizes of gift boxes for the same type of pastry, with the larger box priced at 85.6 yuan per box and containing 32 pastries, and the smaller box priced at 46.8 yuan per box and containing 15 pastries. If Wang Lei spent 654 yuan to buy 9 boxes of pastries, then he can get $\qquad$ pastries. | 237 |
Given the proposition $p$: $∀x∈R$, $cosx≦1$, then $¬p($ $)$
A: $∃x₀∈R$, $cosx₀ > 1$
B: $∀x∈R$, $cosx > 1$
C: $∃x₀∈R$, $cosx₀ ≧ 1$
D: $∀ x∈R$, $cosx ≧ 1$ | A |
In a chess tournament, each pair of players competes in exactly one game. The winner of each game scores 2 points, the loser scores 0 points, and in the event of a draw, each player scores 1 point. There are 4 people who have tallied the total scores from this tournament, but due to carelessness, their data are all dif... | 45 |
Problem 11.8. Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$. A point $X$ is chosen on the edge $A_{1} D_{1}$, and a point $Y$ is chosen on the edge $B C$. It is known that $A_{1} X=5, B Y=3, B_{1} C_{1}=14$. The plane $C_{1} X Y$ intersects the ray $D A$ at point $Z$. Find $D Z$.
 have signed up for the school sports meet. Apart from the tug-of-war competition where the entire class participates, each student must participate in at least one of the other three events. It is known that 39 students in the class have signed up for the kick shuttlecock competition, ... | 22 |
1.033. $\frac{\left(\frac{5}{8}+2 \frac{17}{24}\right): 2.5}{\left(1.3+\frac{23}{30}+\frac{4}{11}\right) \cdot \frac{110}{401}} \cdot 0.5$. | 1 |
For complex numbers $z_{1}$ and $z_{2}$ corresponding to points $A$ and $B$ in the complex plane, respectively, where $z_{1}=3+4i$, point $A$ is rotated $90^{\circ}$ counterclockwise around the origin $O$ to obtain point $B$. Then, the conjugate of $z_{2}$ is
( )
A: $3-4i$
B: $-4-3i$
C: $-4+3i$
D: $-3-4i$ | \text{B} |
Problem 10.7. The graph of the function $f(x)=\frac{1}{12} x^{2}+a x+b$ intersects the $O x$ axis at points $A$ and $C$, and the $O y$ axis at point $B$, as shown in the figure. It turned out that for the point $T$ with coordinates $(3 ; 3)$, the condition $T A=T B=T C$ is satisfied. Find $b$.
$, and $\cos 2x = a$, find the value of $\sin x = (\ \ \ )$.
A: $\sqrt{\frac{1 - a}{2}}$
B: $-\sqrt{\frac{1 - a}{2}}$
C: $\sqrt{\frac{1 + a}{2}}$
D: $-\sqrt{\frac{1 + a}{2}}$ | B |
Given the proposition $p$: $\exists x_{0} \in \mathbb{R}$, $ax_{0}^2 + x_{0} + \frac{1}{2} \leqslant 0$. If proposition $p$ is false, then the range of values for the real number $a$ is ________. | \left( \frac{1}{2}, +\infty \right) |
If $a$, $b$, $c$ are real numbers, then statement P: $ac < 0$, is for statement Q: The equation $ax^2 + bx + c = 0$ has one positive root and one negative root, a ()
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither sufficient nor necess... | \text{C} |
The variance of the data 3, 1, 0, -1, -3 is . | 4 |
Among the following propositions, the correct one is\\((\\) \\()\\)
A: If two vectors are equal, then their starting points and endpoints coincide respectively.
B: Two parallel vectors with equal magnitudes are equal vectors.
C: If _\\(a\\)_ and _\\(b\\)_ are both unit vectors, then _\\(a\\)_\\(=\\) _\\(b\\)_.
D: Two ... | D |
The function $f$ satisfies the condition $$f (x + 1) = \frac{1 + f (x)}{1 - f (x)}$$ for all real $x$, for which the function is defined. Determine $f(2012)$, if we known that $f(1000)=2012$. | 2012 |
5. Yesterday at the market, with one hundred tugriks you could buy 9 gingerbreads and 7 pastries (and even get some change), but today this amount is no longer enough. However, with the same one hundred tugriks today, you can buy two gingerbreads and 11 pastries (also with some change), but yesterday this amount would ... | 5 |
For the one-variable quadratic equation $x^{2}-2x+m=0$ with real roots, the value of $m$ cannot be ( )
A: $-1$
B: $0$
C: $1$
D: $1.2$ | D |
7. In 1981, the United States developed the world's first space shuttle, the “$\qquad$”号, which successfully completed its first manned space flight test.
A. Houston
B. New York
C. Columbia
D. Apollo
Note: The Chinese character "号" in the blank should be replaced with the corresponding English name, which is "Columbia... | C |
3. (48th Slovenian Mathematical Olympiad) Mathieu first wrote down all the numbers from 1 to 10000 in order, then erased those numbers that are neither divisible by 5 nor by 11. Among the remaining numbers, what is the number at the 2004th position? | 7348 |
6. For the equation in terms of $x$
$$
\frac{1}{x^{2}-x}+\frac{k-5}{x^{2}+x}=\frac{k-1}{x^{2}-1}
$$
if it has an extraneous root $x=1$. Then the value of $k$ is ( ).
(A) 2
(B) 3
(C) 6
(D) 10 | B |
5.2.4. $B L$ is the bisector of triangle $A B C$. Find its area, given that $|A L|=4$, $|B L|=2 \sqrt{15},|C L|=5$. | \frac{9\sqrt{231}}{4} |
A surveillance service is to be installed in a park in the form of a network of stations. The stations must be connected by telephone lines so that any of the stations can communicate with all the others, either by a direct connection or through at most one other station. Each station can be directly connected by a cab... | 10 |
11. As shown in the figure, the graph of the linear function $y=k x+b$ passes through the point $P(1,3)$, and intersects the positive halves of the $x$-axis and $y$-axis at points $A, B$, respectively. Point $O$ is the origin. The minimum value of the area of $\triangle A O B$ is $\qquad$ | 6 |
Moving the decimal point of 76.08 one place to the right results in ( )
A: 76.308
B: 763.8
C: 7638
D: 760.8 | \text{D} |
If only the binomial coefficient of the third term in the expansion of $({2x-\frac{1}{{\sqrt{x}}}})^n$ ($n\in{N^*}$) is the largest, then the coefficient of the term ${x^{\frac{5}{2}}}$ in the expansion is ( ).
A: $32$
B: $-32$
C: $16$
D: $-16$ | B |
How does the number of solutions of the equation
$$
|x+2|-|2 x+8|=a
$$
depend on the value of the real parameter $a$? | >2 |
11.069. The lateral edge of a regular triangular prism is equal to the height of the base, and the area of the section made through this lateral edge and the height of the base is $Q$. Determine the volume of the prism. | Q\sqrt{Q/3} |
B4. A square with a side length of $\sqrt{2} \mathrm{~cm}$ is circumscribed by a circle. Over each side of the square, a semicircle is drawn that lies outside the square. What is the total area of the figures bounded by the drawn semicircles and the circumscribed circle?
\\) be a differentiable function, satisfying the condition \\( \lim\limits_{x \to 0} \frac{f(x+1)-f(1)}{2x}=3 \\). Then, the slope of the tangent line to the curve \\(y=f(x)\\) at the point \\((1,f(1))\\) is \\((\quad)\).
A: \\( \frac{3}{2} \\)
B: \\(3\\)
C: \\(6\\)
D: Cannot be determined | C |
(11) The smallest positive period of the function $f(x)=2 \sin \frac{x}{2}-\sqrt{3} \cos x$ is | 4\pi |
3. The solution set of the inequality $\sqrt{\log _{2} x-1}+\frac{1}{2} \log _{\frac{1}{2}} x^{3}+2>0$ is
(A) $[2,3)$
(B) $(2,3]$
(C) $[2,4)$
(D) $(2,4]$ | [2,4) |
Among 100 products, there are 95 genuine products and 5 defective ones. If 6 products are drawn from these, the events are as follows: at least 1 genuine product; at least 3 are defective; all 6 are defective; there are 2 defective and 4 genuine products. Among these four events, the number of random events is ( )
A: 3... | C |
Given that the function $f(x)={\log_a}({\sqrt{{x^2}+3a}-x})$ is an odd function, find the value of $a$. | \frac{1}{3} |
Given a sequence $\{a_n\}$ where the sum of the first $n$ terms is $S_n$ and the product of the first $n$ terms is $T_n$, satisfying $T_n = 2^{n(1-n)}$.
1. Find $a_1$;
2. Prove that the sequence $\{a_n\}$ is a geometric sequence;
3. Is there a constant $a$ such that $(S_{n+1} - a)^2 = (S_{n+2} - a)(S_n - a)$ holds for ... | a = \frac{4}{3} |
Which of the following phenomena is impossible? ( )
A: A conductor heats up when conducting electricity
B: Three non-collinear points determine a plane
C: Seeds germinate without water
D: Someone wins the lottery for two consecutive weeks | \text{C} |
Problem 7.8. Given an isosceles triangle $ABC (AB = BC)$. On the ray $BA$ beyond point $A$, point $E$ is marked, and on side $BC$, point $D$ is marked. It is known that
$$
\angle ADC = \angle AEC = 60^{\circ}, AD = CE = 13.
$$
Find the length of segment $AE$, if $DC = 9$.
=3 \cos \left(\omega x+\frac{\pi}{6}\right)-\sin \left(\omega x-\frac{\pi}{3}\right)(\omega>0)$ has the smallest positive period of $\pi$, then the maximum value of $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$ is $\qquad$ . | 2\sqrt{3} |
Which of the following functions is both odd and increasing within its domain?
A: $y=2^x$
B: $y=x^3+x$
C: $y=- \frac {1}{x}$
D: $y=-\log_{2}x$ | B |
5. Tic-Tac-Toe (from 6th grade, 1 point). A tic-tac-toe board (three by three cells) is randomly filled with six X's and three O's. Find the probability that all three O's will end up in a "winning position," that is, in a row either vertically, horizontally, or diagonally. | \frac{2}{21} |
Condition of the problem
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{4}}(\operatorname{tg} x)^{\operatorname{ctg} x}$ | 1 |
3. Given $a^{2}+b^{2}+c^{2}=1$. Then the range of $a b+b c+a c$ is . $\qquad$ | \left[-\frac{1}{2}, 1\right] |
Given the sets $M=\{y|y=2^{x}, x\in \mathbb{R}\}$ and $N=\{y|y=x^{2}, x\in \mathbb{R}\}$, then $M\cap N$ equals to ( )
A: 4, 2
B: (4, 2)
C: N
D: M | D |
Given the function $f(x)= \begin{cases} x(x-b), & x\geqslant 0 \\ ax(x+2), & x < 0 \end{cases} (a,b\in\mathbb{R})$ is an odd function, then the value of $f(a+b)$ is \_\_\_\_\_\_. | -1 |
6. 84 For a finite set $A$, there exists a function $f: N \rightarrow A$ with the following property: if $|i-j|$ is a prime number, then $f(i) \neq f(j), N=\{1,2, \cdots\}$, find the minimum number of elements in the finite set $A$.
| 4 |
The maximum value of the function $y=-4x^{2}+8x+3$ for $x\in (0,3]$ is ______. | 7 |
If $(2x+3)^{8}=a_{0}+a_{1}(x+1)+a_{2}(x+1)^{2}+\ldots +a_{8}(x+1)^{8}$, then $a_{0}+a_{2}+a_{4}+a_{6}+a_{8}=\left(\ \ \right)$
A: $6562$
B: $3281$
C: $3280$
D: $6560$ | B |
$17 \cdot 80$ A women's barracks is located 300 meters from a certain straight road, and a men's barracks is located on this straight road but 500 meters from the women's barracks. Now it is desired to build a restaurant on this straight road, equidistant from each barracks. Then this distance should be
(A) 400 meters.... | 312.5 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $(2\overrightarrow{a}-\overrightarrow{b})\cdot(\overrightarrow{a}+\overrightarrow{b})=6$ and have magnitudes $|\overrightarrow{a}|=2$ and $|\overrightarrow{b}|=1$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} |
Determine all positive integers $n$ such that $4k^2 +n$ is a prime number for all non-negative integer $k$ smaller than $n$. | n = 3, 7 |
Problem 1. Let $\left(a_{n}\right)_{n \geq 1}$ be a sequence of strictly positive real numbers, with the property that the sequence $\left(a_{n+1}-a_{n}\right)_{n \geq 1}$ is convergent, with a non-zero limit. Calculate the limit
$$
\lim _{n \rightarrow \infty}\left(\frac{a_{n+1}}{a_{n}}\right)^{n}
$$
Mathematical Ga... | e |
Given real numbers $x$ and $y$ satisfying $x + 2y = 2$, the minimum value of $3^x + 9^y$ is ______. | 6 |
8. Given that the number of elements in sets $A$, $B$, and $A \cup B$ are $12$, $13$, and $15$ respectively, set $C$ satisfies the following three conditions: (1) $C \subseteq A \cup B$; (2) The number of elements in $C \cap A \cap B$ is 1; (3) $C \nsubseteq B$. How many such sets $C$ are there? $\qquad$ | 240 |
6. Let $a_{1}, a_{2}, \cdots, a_{2014}$ be a permutation of the positive integers $1,2, \cdots$, 2014. Denote
$$
S_{k}=a_{1}+a_{2}+\cdots+a_{k}(k=1,2, \cdots, 2014) \text {. }
$$
Then the maximum number of odd numbers in $S_{1}, S_{2}, \cdots, S_{2014}$ is $\qquad$ | 1511 |
From the 20 natural numbers 1, 2, 3, ..., 20, if three numbers are randomly selected and their sum is an even number greater than 10, then there are $\boxed{\text{answer}}$ such sets of numbers. | 563 |
$2 \log_{5}{10} + \log_{5}{0.25} = \_\_\_\_\_\_\_\_\_$. | 2 |
[ Irrational Equations $]$ [ Monotonicity and Boundedness
Solve the equation $2017 x^{2017}-2017+x=\sqrt[2017]{2018-2017 x}$.
# | 1 |
Given a hyperbola with the equation $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\ (a > 0, b > 0)$, let $F\_1$ and $F\_2$ represent the foci, and $A\_1$ and $A\_2$ the vertices. A line parallel to one of the asymptotes of the hyperbola passes through focus $F\_2$ and intersects the other asymptote at point $M$. If poi... | (1, \sqrt{2}) |
4. The function $y=a \cos x+b(a, b$ are constants $)$ has a maximum value of 1 and a minimum value of $-7, y=$ $3+a b \sin x$ has a maximum value of $\qquad$. | 15 |
Let set $A=\{1,2,3,4,5\}$, and set $B=\{3,4,5,6,7\}$. The number of sets $S$ that satisfy $S \subseteq A$ and $S \cap B \neq \emptyset$ is ( )
A: 32
B: 28
C: 24
D: 8 | \text{B} |
8. Real numbers $x, y, z$ satisfy $x+y+z=0$, and $x^{2}+y^{2}+z^{2}=1$. Let $m$ be the maximum of $x^{2}, y^{2}, z^{2}$, then the minimum value of $m$ is $\qquad$ . | \frac{1}{2} |
11. (10 points) If 2 tennis rackets and 7 tennis balls cost a total of 220 yuan, and 1 tennis racket is 83 yuan more expensive than 1 tennis ball. Find the unit price of the tennis ball. | 6 |
Given a sequence $\{a_n\}$ is an arithmetic sequence with the first term $a$ and common difference $b$, and the sequence $\{b_n\}$ is a geometric sequence with the first term $b$ and common ratio $a$, and it is known that $a_1 < b_1 < a_2 < b_2 < a_3$, where $a$, $b$, $m$, $n\in\mathbb{N}^*$.
(I) Find the value of $a... | 3\cdot 2^{n-1} |
Binkov A, A:
In a football championship, 16 teams participated. Each team played against each of the others once, with 3 points awarded for a win, 1 point for a draw, and 0 points for a loss. We will call a team successful if it scored at least half of the maximum possible number of points. What is the maximum number ... | 15 |
Alex launches his boat into a river and heads upstream at a constant speed. At the same time at a point $8$ miles upstream from Alex, Alice launches her boat and heads downstream at a constant speed. Both boats move at $6$ miles per hour in still water, but the river is owing downstream at $2\frac{3}{10}$ miles per ho... | 52 |
Example 8 Find all real-coefficient polynomials $f(x)$ that satisfy the following conditions:
(1) $f(x)=a_{0} x^{2 n}+a_{1} x^{2 n-2}+\cdots+a_{2 n-2} x^{2}+a_{2 n}, a_{0}>0$;
(2) $\sum_{j=0}^{n} a_{2 j} a_{2 n-2 j} \leqslant c_{2 n}^{n} a_{0} a_{2 n}$;
(3) $f(x)$ has $2 n$ roots that are all purely imaginary. | f(x)=a_{0}(x^{2}+r^{2})^{n}(a_{0}>0,r>0) |
(4) $l$ and $l^{\prime}$ are mutually perpendicular skew lines, $l$ is parallel to plane $\pi$, and $l^{\prime}$ lies in plane $\pi$. The locus of points in plane $\pi$ that are equidistant from $l$ and $l^{\prime}$ is ( ).
(A) Line
(B) Ellipse
(C) Parabola
(D) Hyperbola | D |
12. As shown in the figure, $ABCD$ is a tetrahedron, $AB=41$, $AC=7$, $AD=18$, $BC=36$, $BD=27$, $CD=13$. Let $d$ be the distance between the midpoints of $AB$ and $CD$. Find the value of $d^{2}$. | 137 |
Convex pentagon $ABCDE$ has side lengths $AB=5$, $BC=CD=DE=6$, and $EA=7$. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$. | 60 |
Let X and Y be the following sums of arithmetic sequences:
\begin{eqnarray*}X &=& 10+12+14+\cdots+100,\\ Y &=& 12+14+16+\cdots+102.\end{eqnarray*}
What is the value of $Y - X?$
$\textbf{(A)}\ 92\qquad\textbf{(B)}\ 98\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 102\qquad\textbf{(E)}\ 112$ | 92 |
1. The graph of a quadratic function passes through $(1,0),(5, 0)$, the axis of symmetry is parallel to the $y$-axis, but does not pass through points above the line $y=2x$. Then the product of the maximum and minimum values of the coordinates of its vertex is $\qquad$ . | 4 |
906. Compute surface integrals of the first kind (over the area of the surface):
1) $I=\iint_{\sigma}(6 x+4 y+3 z) d s$, where $\sigma$ is the part of the plane $x+2 y+$ $+3 z=6$ located in the first octant.
2) $K=\iint_{W}\left(y+z+V \sqrt{a^{2}-x^{2}}\right) d s$, where $W-$ is the surface of the cylinder $x^{2}+y^{2... | 54\sqrt{14} |
Example 2. How much work is required to stretch a spring by $0.06 m$, if a force of 1 N stretches it by 0.01 m? | 0.18 |
137 Given that $x, y, z$ are all real numbers, and $x^{2}+y^{2}+z^{2}=x+y+z$, then the sum of the maximum and minimum values of $x+y+z$ is
A. 2
B. $2 \sqrt{2}$
C. 3
D. $2 \sqrt{3}$ | 3 |
During four tests, the number of sit-ups per minute completed by Xiao Li were 50, 45, 48, and 47, respectively. The median of this data set is _______. | 47.5 |
16. Given the function $f(x)=2\sin x+\sin 2x$, the minimum value of $f(x)$ is __________. | -\frac{3\sqrt{3}}{2} |
It is known that one focus of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ coincides with the focus of the parabola $y^2 = 4\sqrt{10}x$, and the eccentricity of the hyperbola is $\frac{\sqrt{10}}{3}$. The equation of the hyperbola is:
A: $x^2 - \frac{y^2}{9} = 1$
B: $x^2 - y^2 = 15$
C: $\frac{x^2}{9} - y^2 = 1... | C |
Problem 6.1. Five consecutive natural numbers are written in a row. The sum of the three smallest of them is 60. What is the sum of the three largest? | 66 |
When $x=-1$, the value of the function $y=-2x$ is equal to:
A: $2$
B: $-2$
C: $\frac{1}{2}$
D: $-\frac{1}{2}$ | \text{A} |
8. Find all values of the parameter $a$, for each of which the solution set of the inequality $\frac{x^{2}+(a+1) x+a}{x^{2}+5 x+4} \geq 0$ is the union of three non-overlapping intervals. In your answer, specify the sum of the three smallest integer values of $a$ from the obtained interval. | 9 |
[Text problems] [Case enumeration]
In a box, there are blue, red, and green pencils. In total, there are 20 pieces. The number of blue pencils is 6 times the number of green pencils, and the number of red pencils is less than the number of blue pencils. How many red pencils are in the box? | 6 |
52. One fountain fills the pool in 2.5 hours, the other in 3.75 hours. How long will it take for both fountains to fill the pool? | 1.5 |
4. Let $O$ be the center of the base $\triangle ABC$ of the regular tetrahedron $P-ABC$. The dihedral angles between each pair of lateral edges are $\alpha$, and the angle between $PC$ and the plane $PAB$ is $\beta$. Denote the distance from point $O$ to each face as $d$. A moving plane through point $O$ intersects $PC... | \frac{\sin \beta}{d} |
3. The base of a right quadrilateral prism is a rhombus with side length 2 and an angle of $30^{\circ}$, and the height is 1. A section is made through the base edge at an angle of $60^{\circ}$ to the base. The area of this section is $\qquad$ | \frac{4}{\sqrt{3}} |
Let \\(i\\) be the imaginary unit, and for the complex number \\((2-i)z=1+i\\), find the quadrant in which the conjugate of \\(z\\), denoted as \\(\overline{z}\\), is located in the complex plane.
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | D |
The two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a$, $b$ on the board. How many odd numbers will be on the board after $10$ such operations?
[i]Proposed by Michael Kural[/i] | 683 |
Given that $α$ is an acute angle, and $cos(π+α)=-\frac{\sqrt{3}}{2}$, find $tanα=\_\_\_\_\_\_.$ | \frac{\sqrt{3}}{3} |
There are $n$ positive integers on the board. We can add only positive integers $c=\frac{a+b}{a-b}$, where $a$ and $b$ are numbers already writted on the board.
$a)$ Find minimal value of $n$, such that with adding numbers with described method, we can get any positive integer number written on the board
$b)$ For such... | \{1, 2\} |
Given circle $C$: $(x-3)^{2}+(y-4)^{2}=4$ and points $A(-1,0)$, $B(1,0)$, $P(x,y)$ is any point on circle $C$, find the minimum value of $|AP|^{2}+|BP|^{2}$. | 20 |
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