problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Determine the number of zero points of the function f(x) = sin(πcosx) in the interval [0, 2π]. | 5 |
In a regression model with two variables, $y$ and $x$, four different models were chosen. Their coefficients of determination, $R^2$, are as follows: Model 1 has an $R^2$ of 0.98, Model 2 has an $R^2$ of 0.80, Model 3 has an $R^2$ of 0.50, and Model 4 has an $R^2$ of 0.25. Among these, the model with the best fitting e... | A |
14. (5 points)
The number of golden monkeys on the Monkey Mountain is $\frac{4}{7}$ of the number of long-tailed monkeys. After a few years, the number of golden monkeys becomes twice the original, and the number of long-tailed monkeys increases by 24. The number of golden monkeys is $\frac{4}{5}$ of the number of long... | 64 |
The graph of the function $f(x) = a^x - 3$ ($a > 0$ and $a \neq 1$) must pass through a fixed point $P$. The coordinates of point $P$ are. | (0, -2) |
G3.1 If $m$ and $n$ are positive integers with $m^{2}-n^{2}=43$, find the value of $m^{3}-n^{3}$. | 1387 |
If the complex number $z$ satisfies $(1+i)z=2-i$, then the point corresponding to the complex number $z$ in the complex plane is in ( ).
A: the first quadrant
B: the second quadrant
C: the third quadrant
D: the fourth quadrant | D |
4. For how many positive integer values of $n$ is $n^{2}+2 n$ prime?
A 0
B 1
C 2
D 3
E more than 3 | 1 |
Given the proposition $p$: There exists $x_{0} \in \mathbb{R}$, such that $\log_{2}(3^{x_{0}}+1) \leqslant 0$, then the negation of $p$, $\neg p$, is:
A: There exists $x_{0} \in \mathbb{R}$, such that $\log_{2}(3^{x_{0}}+1) > 0$
B: For all $x_{0} \in \mathbb{R}$, $\log_{2}(3^{x}+1) \leqslant 0$
C: For all $x \in \mathb... | C |
Problem 9.5. Positive numbers $a, b$, and $c$ satisfy the equations
$$
a^{2}+a b+b^{2}=1, \quad b^{2}+b c+c^{2}=3, \quad c^{2}+c a+a^{2}=4
$$
Find $a+b+c$. | \sqrt{7} |
If the tangent line to the curve $y = x^2 + ax + b$ at the point $(0, b)$ is given by the equation $x - y + 1 = 0$, then the values of $a$ and $b$ are respectively ________, ________. | 1 |
In a certain measurement, the measurement result $\xi$ follows a normal distribution $N^2 \sim \left(2,\delta^2\right) (\delta > 0)$, if the probability of $\xi$ taking values in $(0,2)$ is $0.4$, then the probability of $\xi$ taking values in $(0,+\infty)$ is ( )
A: $0.2$
B: $0.4$
C: $0.8$
D: $0.9$ | D |
5. Find the maximum value of the expression $(\sin x + \sin 2y + \sin 3z)(\cos x + \cos 2y + \cos 3z)$. (15 points) | 4.5 |
8. Let the function $y=3^{x}-3^{-x}$ be an odd function, $x \in\left(2^{2-a}-2,2^{a-2}\right)$, the real number $a=$ | 2 |
19. What angle do the minute and hour hands form exactly at 3:05? | 62.5 |
Consider \(n^2\) unit squares in the \(xy\) plane centered at point \((i,j)\) with integer coordinates, \(1 \leq i \leq n\), \(1 \leq j \leq n\). It is required to colour each unit square in such a way that whenever \(1 \leq i < j \leq n\) and \(1 \leq k < l \leq n\), the three squares with centres at \((i,k),(j,k),(j,... | 2n-1 |
The graph of $x^2-4y^2=0$ is:
$\textbf{(A)}\ \text{a parabola} \qquad \textbf{(B)}\ \text{an ellipse} \qquad \textbf{(C)}\ \text{a pair of straight lines}\qquad \\ \textbf{(D)}\ \text{a point}\qquad \textbf{(E)}\ \text{None of these}$ | \textbf{(C)} |
Bethany is told to create an expression from $2 \square 0 \square 1 \square 7$ by putting a + in one box, a in another, and $a x$ in the remaining box. There are 6 ways in which she can do this. She calculates the value of each expression and obtains a maximum value of $M$ and a minimum value of $m$. What is $M-m$ ? | 15 |
Given the function $f(x) = \log_a(2-ax)$ where $a>0$ and $a \neq 1$, and it is a decreasing function in the interval $[0,1]$, find the range of the real number $a$. | (1,2) |
[Coordinate method on the plane $]$ $[\underline{\text { Orthocenter and orthotriangle }]}$
Given points $A(5, -1), B(4, -8), C(-4, -4)$. Find the coordinates of the intersection point of the altitudes of triangle $ABC$.
# | (3,-5) |
8,9
Find the distance between the parallel lines $y=-3 x+5$ and $y=-3 x-4$.
# | \frac{9}{\sqrt{10}} |
Given that the domain of the function $f(x)$ is $(0,+∞)$, and $f(x)=2f\left( \frac{1}{x} \right)· \sqrt{x}-1$, find the analytical expression of $f(x)$. | f(x) = \frac{2}{3} \sqrt{x} + \frac{1}{3} |
Use the Euclidean algorithm to find the greatest common divisor (GCD) of 288 and 123. The answer is ( )
A: 42
B: 39
C: 13
D: 3 | D |
Among the four real numbers $1$, $\sqrt{3}$, $0$, $2$, the largest real number is:
A: $1$
B: $\sqrt{3}$
C: $0$
D: $2$ | D |
9.16. Find the sum
$$
1^{3}+3^{3}+5^{3}+\ldots+(2 n-1)^{3}
$$
## 9.4. Pairing
## | n^{2}(2n^{2}-1) |
8. Let $S=\{1,2, \cdots, 100\}$. If a three-element subset $A=\{a, b, c\}$ of $S$ satisfies $a+b=3 c$, then $A$ is said to have property $P$. The number of all three-element subsets of $S$ that have property $P$ is $\qquad$. | 1600 |
Example 2 Find the convergent fraction of $3+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}=[3,2,3, 4,5]$. | \frac{539}{157} |
If $f(x)$ is an even function and its derivative exists, the value of $f'(0)$ is ( )
A: 2
B: 1
C: 0
D: -1 | C |
Given points $M(-2, -1)$, $N(a, 3)$, and the distance $|MN| = 5$, find the real number $a$. | a = 1 \quad \text{or} \quad a = -5 |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{2 \sin (\pi(x+1))}{\ln (1+2 x)}$ | -\pi |
1. A trip to Nizhny Novgorod is being organized for 50 schoolchildren along with their parents, some of whom will be driving cars. Each car can accommodate 6 people, including the driver. What is the minimum number of parents that need to be invited on the trip?
ANSWER: 10 | 10 |
The equation of the tangent line to the curve $y=x^2+\frac{1}{x}$ at the point $(1,2)$ is __________. | x-y+1=0 |
Let $m$ and $n$ be two different lines, and $\alpha$, $\beta$, $\gamma$ be three different planes. Consider the following four propositions:
① If $\alpha \parallel \beta$ and $\alpha \parallel \gamma$, then $\beta \parallel \gamma$;
② If $\alpha \perp \beta$ and $m \parallel \alpha$, then $m \perp \beta$;
③ If $m... | A |
## Problem 3
Find all functions $f: \Re \rightarrow \Re$ such that
$$
x^{2} \cdot f(x)+f(1-x)=2 x-x^{4}
$$ | f(x)=1-x^{2} |
7. Let $M=\{(x, y) \mid | x y |=1, x>0\}, N=\{(x, y) \mid \arctan x+\operatorname{arccot} y=\pi\}$, then
A. $M \cup N=\{(x, y) \mid |x y|=1\}$
B. $M \cup N=M$
C. $M \cup N=N$
D. $M \cup N=\{(x, y) \mid | x y |=1$, and $x, y$ are not both negative $\}$ | B |
Given $f(x) = ax^3 + bx^9 + 2$ has a maximum value of 5 on the interval $(0, +\infty)$, find the minimum value of $f(x)$ on the interval $(-\infty, 0)$. | -1 |
Given the function $f(x)=\sin x$, $g(x)=e^{x}\cdot f'(x)$, where $e$ is the base of the natural logarithm.
(I) Find the equation of the tangent line to the curve $y=g(x)$ at the point $(0,g(0))$;
(II) If for any $x\in\left[- \frac {\pi}{2},0\right]$, the inequality $g(x)\geqslant x\cdot f(x)+m$ always holds, find t... | 2 |
Calculate:
1. $\sum_{k=0}^{n}\binom{n}{k}$
2. $\sum_{k=0}^{n} k\binom{n}{k}$ | n2^{n-1} |
Let $z$ be an integer $> 1$ and let $M$ be the set of all numbers of the form $z_k = 1+z + \cdots+ z^k, \ k = 0, 1,\ldots$. Determine the set $T$ of divisors of at least one of the numbers $z_k$ from $M.$ | T = \{ n \in \mathbb{Z}^+ \mid \gcd(n, z) = 1 \} |
Example 15 (Question from the 7th "Hope Cup" Invitational Competition) In the tetrahedron $P-ABC$, $\angle APC=\angle CPB=\angle BPA=\frac{\pi}{2}$, and $PA=PB=3, PC=4$. Also, $M$ is a point within the base $ABC$. The minimum value of the sum of the squares of the distances from $M$ to the three lateral faces of the te... | \frac{144}{41} |
11. (20 points) Let $x, y$ be complex numbers, find the maximum value of $\frac{|3 x+4 y|}{\sqrt{|x|^{2}+|y|^{2}+\left|x^{2}+y^{2}\right|}}$.
| \frac{5\sqrt{2}}{2} |
a) 2 white and 2 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 4, to the other black cat is 8. The sum of distances from the black cats to one white cat is 3, to the other white cat is 9. What cats are sitting on the edges?
b) 2 white and 3 black cats are sitting on t... | W_1 |
Given the complex number $z$ that satisfies the equation $\frac{1-i}{z} = -i$, where $i$ is the imaginary unit, find $\overset{ .}{z}$ = $\_\_\_\_\_\_$. | 1-i |
12. Given that $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ are all positive numbers, determine the largest real number $C$ such that the inequality $C\left(x_{1}^{2005}+x_{2}^{2005}+x_{3}^{2005}+x_{4}^{2005}+x_{5}^{2005}\right) \geqslant x_{1} x_{2} x_{3} x_{4} x_{5}\left(x_{1}^{125}+x_{2}^{125}+x_{3}^{125}+x_{4}^{125}+x_{5}^{... | 5^{15} |
Which of the following statements is correct? ( )
A: If $\dfrac{1}{a} > \dfrac{1}{b}$, then $a < b$
B: If proposition $p: \exists x \in (0,\pi), x + \dfrac{1}{\sin x} \leqslant 2$, then $\neg P$ is a true proposition
C: Given propositions $p$, $q$, "p is a true proposition" is a necessary and sufficient condition fo... | B |
NT 1. Find all the integers pairs $(x, y)$ which satisfy the equation
$$
x^{5}-y^{5}=16 x y
$$
| (x,y)=(0,0)or(-2,2) |
2. Find all functions $f(x)$ that simultaneously satisfy the following three conditions: 1) $f(x)>0$ for any $x>0$; 2) $f(1)=1$; 3) $f(a+b) \cdot(f(a)+f(b))=2 f(a) \cdot f(b) + a^{2} + b^{2}$ for any $a, b \in \mathbb{R}$. | f(x)=x |
12. A person bought a total of 12 packs of black and white sunflower seeds, spending 9 yuan and 9 jiao. Each pack of white sunflower seeds is 3 jiao more expensive than each pack of black sunflower seeds, and the number of packs of white sunflower seeds is more than that of black sunflower seeds. Question: How many pac... | x=3, y=9 |
Given the graph of the power function $y=f(x)$ passes through the point $\left(\frac{1}{3},\frac{\sqrt{3}}{3}\right)$, then the value of $\log_{2}f(2)$ is \_\_\_\_. | \frac{1}{2} |
## Problem Statement
Calculate the lengths of the arcs of the curves given by equations in a rectangular coordinate system.
$$
y=\frac{1-e^{x}-e^{-x}}{2}, 0 \leq x \leq 3
$$ | \frac{1}{2}(e^{3}-e^{-3}) |
Calculate: $(2\sqrt{2}-\pi )^{0}-4\cos 60^{\circ}+|\sqrt{2}-2|-\sqrt{18}$. | 1 - 4\sqrt{2} |
3. Given a right triangle $A B C$. On the extension of the hypotenuse $B C$, a point $D$ is chosen such that the line $A D$ is tangent to the circumcircle $\omega$ of triangle $A B C$. The line $A C$ intersects the circumcircle of triangle $A B D$ at point $E$. It turns out that the bisector of $\angle A D E$ is tangen... | AC:CE=1:2 |
Points $P$ and $Q$ are chosen on diagonal $AC$ of square $ABCD$ such that $AB=AP=CQ=1$. What is the measure of $\angle{PBQ}$ in degrees?
[i]2019 CCA Math Bonanza Lightning Round #1.3[/i] | 45^\circ |
1.21 A children's abacus has ten beads on each of the three rods (as shown in the figure). The beads on each rod are to be divided into two parts (no part can be empty). Now, we want the product of the three numbers of beads on the left side to equal the product of the three numbers of beads on the right side. How many... | 25 |
The number of dots on opposite faces of a regular die add to 7. Four regular dice are arranged as shown. Which of the following could be the sum of the number of dots hidden between the dice?
(A) 22
(B) 26
(C) 24
(D) 21
(E) 23
(x-3a-5) < 0\}$, the domain of the function $y=\lg (-x^{2}+5x+14)$ is set $B$.
$(1)$ If $a=4$, find the set $A\cap B$;
$(2)$ If "$x\in A$" is a sufficient condition for "$x\in B$", find the range of the real number $a$. | -\dfrac{7}{3}\leqslant a\leqslant \dfrac{2}{3} |
3.46. Solve the system of differential equations:
a) $\left\{\begin{array}{l}\frac{d y_{1}(t)}{d t}+y_{1}(t)-y_{2}(t)=e^{t}, \\ \frac{d y_{2}(t)}{d t}+3 y_{1}(t)-2 y_{2}(t)=2 e^{t}, \\ y_{1}(0)=1, \quad y_{2}(0)=1, t>0 .\end{array}\right.$
b) $\left\{\begin{array}{l}\frac{d x}{d t}=x+2 y, \\ \frac{d y}{d t}=2 x+y+1 ;\e... | y_{1}()=y_{2}()=e^{},\quadx()=-\frac{2}{3}-2e^{-}+\frac{8}{3}e^{3},\quady()=\frac{1}{3}+2e^{-}+\frac{8}{3}e^{3} |
Natural numbers $k, l,p$ and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$ then $a +\frac1b$ and $b + \frac1a$ are the roots of $x^2 -px + q = 0$. What is the sum of all possible values of $q$? | 4 |
11.007. Find the ratio of the volume of a cube to the volume of a regular tetrahedron, the edge of which is equal to the diagonal of the face of the cube. | 3 |
Determine all real values of $x$ for which
\[\frac{1}{\sqrt{x} + \sqrt{x - 2}} + \frac{1}{\sqrt{x} + \sqrt{x + 2}} = \frac{1}{4}.\] | \frac{257}{16} |
In the arithmetic sequence $\{a_n\}$:
(1) Given $a_4 + a_{14} = 2$, then $S_{17} = \_\_\_\_\_\_\_\_$;
(2) Given $a_{11} = 10$, then $S_{21} = \_\_\_\_\_\_\_\_$;
(3) Given $S_{11} = 55$, then $a_6 = \_\_\_\_\_\_\_\_$;
(4) Given $S_8 = 100$ and $S_{16} = 392$, then $S_{24} = \_\_\_\_\_\_\_\_$. | 876 |
8. Given $x, y \in [0,+\infty)$. Then the minimum value of $x^{3}+y^{3}-5 x y$ is $\qquad$ . | -\frac{125}{27} |
B2. In a department store, an escalator runs from the ground floor to the first floor. Dion goes up this escalator; while doing so, he also takes a few steps at a constant pace. Raymond walks on the same escalator, in the opposite direction, from top to bottom, and takes steps at the same pace as Dion. Both take one st... | 20 |
Task B-1.2. (20 points) Solve the inequality
$$
|||9-x|-x|+2 x| \leq 2009
$$ | x\in(-\infty,1000] |
Five. (20 points) As shown in Figure 1, given that the circumcenter of acute triangle $\triangle ABC$ is $O$, the midpoints of line segments $OA$ and $BC$ are points $M$ and $N$ respectively, $\angle ABC = 4 \angle OMN$, and $\angle ACB = 6 \angle OMN$. Find the degree measure of $\angle OMN$.
---
Translation:
Five.... | 12^{\circ} |
Given that the function $f(x)$ satisfies $f′(x) - f(x) = 2xe^{x}$, with $f(0) = 1$, where $f′(x)$ is the derivative of $f(x)$, find the maximum value of $\frac{f′(x)}{f(x)}$ when $x > 0$. The options are:
A: $\sqrt{2}$
B: $2$
C: $2\sqrt{2}$
D: $4$ | B |
1. Set $A=\{a \mid a=12 m+8 n+4 l, m, n, l \in \mathbf{Z}\}, B=\{b \mid b=20 p+$ $16 q+12 r, p, q, r \in \mathbf{Z}\}, A, B$ the relationship between is ( ).
A. $A=B$
B. $A \nsubseteq B, B \nsubseteq A$
C. $A \supset B$
D. $A \subset B$ | B |
Calculate: $-3^2+(\frac{1}{2})^{-2}+(π-3.14)^0-\sqrt[3]{-125}$. | 1 |
2. $[\mathbf{3}]$ Let $A B C$ be an equilateral triangle. Let $\Omega$ be its incircle (circle inscribed in the triangle) and let $\omega$ be a circle tangent externally to $\Omega$ as well as to sides $A B$ and $A C$. Determine the ratio of the radius of $\Omega$ to the radius of $\omega$. | 3 |
From a bag containing $n+1$ balls (of which $n$ are white and $1$ is black), $m$ balls are drawn ($0 < m \leq n$, $m, n \in \mathbb{N}$). There are a total of $C_{n+1}^{m}$ ways to draw the balls. These $C_{n+1}^{m}$ ways can be divided into two categories: one category is drawing $m$ balls without the black ball, whic... | C_{n+k}^{m} |
## Task Condition
Find the derivative.
$$
y=3 \cdot \sqrt[3]{\frac{x+1}{(x-1)^{2}}}
$$ | -\sqrt[3]{\frac{x-1}{(x+1)^{2}}}\cdot\frac{x+3}{(x-1)^{2}} |
30.10. Prove that
$$
1-\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}-\frac{1}{8}+\frac{1}{5}-\ldots=\frac{1}{2} \ln 2
$$ | \frac{1}{2}\ln2 |
Given $A=x^2+ax$ and $B=2bx^2-4x-1$, and the value of the polynomial $2A+B$ is independent of the value of $x$, find the values of $a$ and $b$. | a=2, b=-1 |
8. There are $n$ mathematicians attending a conference, and each mathematician has exactly 3 friends (friendship is mutual). Now, if we arrange some of the mathematicians to sit around a round table to discuss problems, it turns out that if everyone is seated next to their friends, then the number of people at the tabl... | 24 |
Given the universal set $U=\mathbb{R}$, and the sets $A=\{x\in\mathbb{Z}|-x^2+5x\leq0\}$, $B=\{x|x-4<0\}$, find the largest element in $(C_UA) \cap B$. | 3 |
# 8. Problem 8
A divisor of a natural number is called proper if it is different from 1 and the number itself. A number is called interesting if it has two proper divisors, one of which is prime, and the other is a perfect square, and the sum of which is also a perfect square (a perfect square is the square of an inte... | 70 |
An empty swimming pool was filled with water by two faucets $A$ and $B$, both with constant flow rates. For 4 hours, both faucets were open and filled $50 \%$ of the pool. Then, faucet B was turned off and for 2 hours faucet A filled $15 \%$ of the pool's volume. After this period, faucet A was turned off and faucet $\... | 7 |
In triangle $ABC$, the sides are known: $AB=6$, $BC=4$, $AC=8$. The bisector of angle $C$ intersects side $AB$ at point $D$. A circle is drawn through points $A$, $D$, and $C$, intersecting side $BC$ at point $E$. Find the area of triangle $ADE$.
# | \frac{3\sqrt{15}}{2} |
In the Cartesian coordinate system, the graph of the quadratic function $y=x^{2}+mx+m^{2}-m$ (where $m$ is a constant) passes through the point $\left(0,6\right)$, and its axis of symmetry is to the left of the $y$-axis. Then, this quadratic function has:
A: maximum value of $5$
B: maximum value of $\frac{15}{4}$
C:... | D |
3. If $x, y$ are real numbers, and $\log _{12}\left(x^{2}+3\right)-$ $\log _{12} x y+\log _{12}\left(y^{2}+3\right)=1$, then the ordered pair $(x, y)=$ | (\sqrt{3}, \sqrt{3}) \text{ or } (-\sqrt{3}, -\sqrt{3}) |
501. How many square centimeters are in a square meter | 10000 |
Example 2 Let non-negative real numbers $a, b, c$ satisfy $a+b+c=1$. Find the
$$
S=\sqrt{4 a+1}+\sqrt{4 b+1}+\sqrt{4 c+1}
$$
extreme values. | 2+\sqrt{5} \leqslant S \leqslant \sqrt{21} |
2. (8 points) Legend has it that those who can find a four-leaf clover in a field of three-leaf clovers are lucky people. One day, Jiajia was picking three-leaf clovers in a big forest. When she picked her first four-leaf clover, she found that the total number of leaves she had picked was exactly 1000. How many three-... | 332 |
8. Let $f(x)=\frac{4^{x}}{4^{x}+2}$, then $f\left(\frac{1}{1001}\right)+f\left(\frac{2}{1001}\right)+\cdots+f\left(\frac{1000}{1001}\right)=$ | 500 |
9. Given that $a$ and $b$ are the roots of the equation
$$
\log _{3 x} 3+\log _{27} 3 x=-\frac{4}{3}
$$
then $a+b=$ $\qquad$ | \frac{10}{81} |
If $m=x^2+2x+3$ ($x\in \mathbb{R}$), and $n=2$, then the relationship between $m$ and $n$ is \_\_\_\_\_\_. | m\geq n |
The coordinates of the point $M(1,2)$ with respect to the $x$-axis are ( ).
A: $(1,-2)$
B: $(-1,2)$
C: $(-1,-2)$
D: $(2,1)$ | A |
Let $f(x)=\sin (\omega x-\frac{π}{4})$ where $\omega > 0$. If $|f(x_{1})-f(x_{2})|=2$, the minimum value of $|x_{1}-x_{2}|$ is $\frac{π}{3}$, then which of the following statements is true?
A: The period of the function $f(x)$ is $\frac{π}{3}$.
B: Shifting the graph of the function $f(x)$ to the left by $\frac{π}{4}$... | \text{D} |
1. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that the equation
$$
f([x] y)=f(x)[f(y)]
$$
holds for all $x, y \in \mathbf{R}$ (where $[z]$ denotes the greatest integer less than or equal to the real number $z$).
(French contribution) | f(x)=c \text{ (constant), where, } c=0 \text{ or } 1 \leqslant c<2 |
16. Bridges. (From 9th grade, 3 points) Along the southern shore of an endless sea, there stretches an archipelago of an infinite number of islands. The islands are connected by an infinite chain of bridges, and each island is connected by a bridge to the shore. In the event of a strong earthquake, each bridge independ... | \frac{2}{3} |
The Hangzhou Asian Games have successfully concluded, thanks to the hard work of the Asian Games online car drivers. One morning, Lao Yao's operation was all on a straight road heading east. If it is stipulated that east is positive and west is negative, then his morning driving mileage (unit: kilometers) is recorded a... | 120\text{ yuan} |
The positional relationship between Circle $O_1$: $(x-1)^2+y^2=1$ and Circle $O_2$: $x^2+(y-3)^2=9$ is ( )
A: Intersecting
B: Tangent
C: Externally separate
D: Contained within each other | A |
The graph of the function $f(x)=\sin ωx$ (where $ω > 0$) is translated to the right by $\frac{π}{4}$ units. The resulting graph passes through the point $(\frac{3π}{4},0)$. Determine the minimum value of $ω$. | 2 |
3. Given an angle of $13^{0}$. How to obtain an angle of $11^{0}$? | 11 |
Given that the coordinates of the point corresponding to the complex number $z$ in the complex plane are $\left(-3,4\right)$, then $|\overline{z}|=\left(\ \ \right)$
A: $3$
B: $4$
C: $5$
D: $4\sqrt{2}$ | C |
If each exterior angle of a polygon is $45^{\circ}$, then the number of sides of this polygon is ______. | 8 |
Determine explicit formulas for the following recursively defined sequences:
(1) $u_{0}=2$ and $u_{n+1}=3 u_{n}^{2}-4 u_{n}+2$;
(2) $u_{0}=\frac{5}{2}$ and $u_{n+1}=u_{n}^{2}-2$. | u_{n}=\frac{4^{2^{n}}+2}{3} |
13. The mean of $p$ and $q$ is 13 ; the mean of $q$ and $r$ is 16 ; the mean of $r$ and $p$ is 7 . What is the mean of $p, q$ and $r$ ?
A 12
B 13
С 14
D 15
E 16 | 12 |
1.1. How many solutions in natural numbers does the equation $\operatorname{LCM}(n, k)=2^{3} 3^{5} 5^{7}$ have? | 1155 |
The square root of $\sqrt{81}$ is ____; the cube root of $-\frac{{64}}{{125}}$ is ____. | -\frac{4}{5} |
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