problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given that the terminal side of angle \\(\varphi\\) lies on the ray \\(y= \sqrt {3}x(x\leqslant 0)\\), and the distance between two adjacent axes of symmetry of the graph of the function \\(f(x)=\cos (\omega x+\varphi)(\omega > 0)\\) is equal to \\( \dfrac {\pi}{3}\\), then \\(f( \dfrac {\pi}{6})=\\)
A: \\(- \dfrac {... | D |
(10 From a material site beside a road, electric poles are to be transported along a straight road in the same direction to be planted at the roadside 500 m away. At the 500 m point, one pole is planted, and then one pole is planted every 50 m along the roadside. Knowing that the transport vehicle can carry a maximum o... | 14000 |
## Task B-1.5.
Determine all integer solutions of the equation
$$
(m n-3)^{2}=m^{2}+n^{2}-4 m n
$$ | (,n)\in{(1,4),(4,1),(-1,-4),(-4,-1),(0,-3),(-3,0),(0,3),(3,0)} |
7. The real solution $(x, y, z)=$ $\qquad$ of the equation $2 \sqrt{x-4}+3 \sqrt{y-9}+4 \sqrt{z-16}=\frac{1}{2}(x+y+z)$ | (8,18,32) |
In triangle \( \triangle ABC \), \( \angle B = 120^{\circ} \), \( AC = 7 \), \( AB = 5 \), then the area of \( \triangle ABC \) is _______. | \frac{15 \sqrt{3}}{4} |
2. If $3 \leqslant x+y \leqslant 5, -1 \leqslant x-y \leqslant 1$, then the maximum value of $3 x+y$ is | 11 |
1. Given $f(x)=\frac{10}{x+1}-\frac{\sqrt{x}}{3}$. Then the set
$$
M=\left\{n \in \mathbf{Z} \mid f\left(n^{2}-1\right) \geqslant 0\right\}
$$
has $\qquad$ elements. | 6 |
$(1)$ If $a$ and $b$ represent the tens digit and ones digit of a two-digit number, then this two-digit number can be represented as: ______;
$(2)$ If we exchange the tens digit and ones digit of this two-digit number, the resulting number can be represented as: ______;
$(3)$ Can $9$ divide the difference between t... | \text{Yes} |
3.4. Solve the system of equations:
$$
\left\{\begin{array}{l}
3^{x+y}-3^{x-y}=18 \\
3^{x+y}-27 \cdot 3^{y-x}=24
\end{array}\right.
$$ | \frac{5}{2},\frac{1}{2} |
1. If $a, b$ are integers, and $x^{2}-x-1$ is a factor of $a x^{17}$ $+b x^{16}+1$, then $a=$ $\qquad$ | 987 |
# 2. Solution:
$\sin ^{4} 2025 x+\cos ^{2019} 2016 x \cdot \cos ^{2018} 2025 x=1 \Leftrightarrow\left(1-\cos ^{2} 2025 x\right)^{2}+\cos ^{2019} 2016 x \cdot \cos ^{2018} 2025 x=1 \Leftrightarrow$ $\cos ^{4} 2025 x-2 \cos ^{2} 2025 x+\cos ^{2019} 2016 x \cdot \cos ^{2018} 2025 x=0 \Leftrightarrow$ $\cos ^{2} 2025 x\le... | \frac{\pi}{4050}+\frac{\pin}{2025},n\inZ,\frac{\pik}{9},k\inZ |
8. The solution set of the inequality $\log _{5}\left(5 x^{2}+2 x\right) \cdot \log _{5}\left(5+\frac{2}{x}\right)>\log _{5} 5 x^{2}$ is $\qquad$ . | (\frac{-1-\sqrt{2}}{5},-\frac{2}{5})\cup(\frac{\sqrt{2}-1}{5},+\infty) |
Given $a=\sin\frac{1}{3}$, $b=\left(\frac{1}{3}\right)^{0.9}$, $c=\frac{1}{2}\log_{27}9$, then which of the following is true?
A: $a \lt c \lt b$
B: $a \lt b \lt c$
C: $b \lt a \lt c$
D: $c \lt a \lt b$ | A |
9. Let the inequality $\left|2^{x}-a\right|<\left|5-2^{x}\right|$ hold for all $x \in[1,2]$, find the range of real number $a$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | (3,5) |
Compute the expression: $4\sin 60^{\circ}-|-2|-\sqrt {12}+(-1)^{2016}$. | -1 |
13. Given the polynomial
$$
\begin{array}{l}
(1+x)+(1+x)^{2}+\cdots+(1+x)^{n} \\
=b_{0}+b_{1} x+\cdots+b_{n} x^{n},
\end{array}
$$
and $b_{1}+b_{2}+\cdots+b_{n}=1013$.
Then a possible value of the positive integer $n$ is | 9 |
36. Calculate: $\frac{1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots \cdots+\frac{1}{2019}-\frac{1}{2020}}{\frac{1}{1011 \times 2020}+\frac{1}{1012 \times 2019}+\frac{1}{1013 \times 2018}+\cdots \cdots+\frac{1}{1515 \times 1516}}=$ | 3031 |
From the set $\{1,2,3,…,10\}$, select three different numbers such that at least two of them are adjacent. The number of different ways to do this is ( )
A: $72$
B: $70$
C: $66$
D: $64$ | D |
SUBIECTUL I . If the real numbers $a^{2}+b^{2}-2 a \sqrt{2}-2 b \sqrt{3}+5=0$, prove that $\left(\frac{2}{a}+\frac{3}{b}\right) \cdot(b-a)=1$ | 1 |
Example 2 Solve the system of simultaneous equations
$$
\left\{\begin{array}{l}
x+y+z=3, \\
x^{2}+y^{2}+z^{2}=3, \\
x^{5}+y^{5}+z^{5}=3 .
\end{array}\right.
$$
Find all real or complex roots.
(2nd USA Mathematical Olympiad) | 1 |
16. Let $D$ be a point inside $\triangle A B C$ such that $\angle B A D=\angle B C D$ and $\angle B D C=90^{\circ}$. If $A B=$ $5, B C=6$ and $M$ is the midpoint of $A C$, find the length of $D M$.
(2 marks)
設 $D$ 為 $\triangle A B C$ 內的一點, 使得 $\angle B A D=\angle B C D$ 及 $\angle B D C=90^{\circ}$ 。若 $A B=5$, $B C=6$, ... | \frac{\sqrt{11}}{2} |
For the quadratic equation $ax^{2}+x-2=0$, if one of its roots is $1$, then $a=\_\_\_\_\_\_.$ | 1 |
1. Given the function $f(x)=x^{3}+a x^{2}+b x+a^{2}$ has an extremum of 10 at $x=1$. Then the real number pair $(a, b)$ is $\qquad$ . | (4,-11) |
2. Let the 20 vertices of a regular 20-sided polygon inscribed in the unit circle in the complex plane correspond to the complex numbers $z_{1}, z_{2}, \cdots, z_{20}$. Then the number of different points corresponding to the complex numbers $z_{1}^{1995}, z_{2}^{1995}, \cdots, z_{20}^{1995}$ is
A. 4
B. 5
C. 10
D. 20 | 4 |
Four, (18 points) The real numbers $x_{1}, x_{2}, \cdots, x_{2001}$ satisfy
$$
\begin{array}{l}
\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001 . \\
\text { Let } y_{k}=\frac{1}{k}\left(x_{1}+x_{2}+\cdots+x_{k}\right), k=1,2 .
\end{array}
$$
$\cdots, 2$ 001. Find the maximum possible value of $\sum_{k=1}^{2000}\left|y... | 2000 |
Given $a$, $b$, $c$ are the lengths of the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$, respectively, and $a\sin C- \sqrt {3}c\cos A=0$.
(1) Find the angle $A$;
(2) If $a=2$ and the area of $\triangle ABC$ is $\sqrt {3}$, find $b$ and $c$. | 2 |
3. As shown in Figure 6, given the quadratic function $\boldsymbol{y}=$ $(x+m)^{2}+k-m^{2}$, the graph intersects the $x$-axis at two distinct points $A\left(x_{1}, 0\right)$ and $B\left(x_{2}, 0\right)$, and intersects the $y$-axis at point $C$. Then the other intersection point $D$ of the circumcircle of $\triangle A... | (0,1) |
Task 5. (20 points) In the center of a circular field stands a geologists' cabin. From it, 8 straight roads extend, dividing the field into 8 equal sectors. Two geologists set out on a journey from their cabin at a speed of 5 km/h along a road each arbitrarily chooses. Determine the probability that the distance betwee... | 0.375 |
I4.2 Consider fractions of the form $\frac{n}{n+1}$, where $n$ is a positive integer. If 1 is subtracted from both the numerator and the denominator, and the resultant fraction remains positive and is strictly less than $\frac{\alpha}{7}$, determine, $\beta$, the number of these fractions. | 5 |
Given that $(\overrightarrow{a}+\overrightarrow{b})\cdot \overrightarrow{b}=7$, and $|\overrightarrow{a}|=\sqrt{3}$, $|\overrightarrow{b}|=2$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{6} |
(treatise)
For what condition on $y \in \mathbb{N}$ is $y^{2}+5 y+12$ a square? | 3 |
The vertex of the parabola is at the origin, and its focus is on the y-axis. If a point P(m, 1) on the parabola is 5 units away from the focus, then the standard equation of the parabola is \_\_\_\_\_\_. | x^2 = 16y |
If line $a$ is parallel to plane $\alpha$, and line $b$ is contained within plane $\alpha$, then the relationship between $a$ and $b$ is ( )
A: Parallel
B: Intersecting
C: Parallel or Skew
D: None of the above | \text{C} |
Given an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity $e = \frac{\sqrt{3}}{3}$. The left and right foci are $F_1$ and $F_2$, respectively, with $F_2$ coinciding with the focus of the parabola $y^2 = 4x$.
(I) Find the standard equation of the ellipse;
(II) If a line passing th... | \frac{16\sqrt{3}}{5} |
## Task 3
$1 \mathrm{~m}=\ldots \mathrm{cm} ; \quad 40 \mathrm{~mm}=\ldots \mathrm{cm} ; \quad 10 \mathrm{~cm}=\ldots \mathrm{dm}$ | 1 |
7.6 On a horizontal line, points $A$ and $B$ are marked, the distance between which is 4. Above the line, two semicircles with a radius of 2 are constructed, centered at points
A and B. Additionally, one circle, also with a radius of 2, is constructed, for which the point of intersection of these semicircles is the lo... | 8 |
In how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?
$\textbf{(A)} ~10\qquad\textbf{(B)} ~18\qquad\textbf{(C)} ~24 \qquad\textbf{(D)} ~32 \qquad\textbf{(E)} ~44$ | 32 |
6. Let $a, b, c$ be the lengths of the three sides of $\triangle ABC$, and $t_{a}, t_{b}, t_{c}$ be the lengths of the corresponding angle bisectors of $\triangle ABC$. Given the following conclusions:
(1) $t_{a} t_{b} t_{c} \geqslant a b c$,
(2) $t_{a} t_{b} t_{c}<\frac{\sqrt{3}}{2} a b c$,
(3) $t_{a}^{2}+t_{b}^{2}+t_... | D |
11. Given real numbers $x_{1}, x_{2}, x_{3}$ satisfy
$$
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{1} x_{2}+x_{2} x_{3}=2 \text {. }
$$
Then the maximum value of $\left|x_{2}\right|$ is $\qquad$ | 2 |
Given an ellipse $E$ with its center at the origin of the coordinate system, its foci on the coordinate axes, and passing through points $A(-2, 0)$, $B(2, 0)$, $C(1, \frac{3}{2})$:
(1) Find the equation of the ellipse.
(2) If the left and right foci of this ellipse are $F_1$ and $F_2$, respectively, and a line $L$ pas... | 8 |
119. What is the greatest possible value of the ratio of a three-digit number to the sum of its digits? | 100 |
If the sum of the coefficients of all terms in the expansion of $(1-2x)^5$ is $m$, and the coefficient of $x^5$ in the expansion of $(1+x^3)(1-2x)^6$ is $n$, then find the value of $m \cdot n$. | 132 |
11. Every year, a large number of students participate in the International Mathematical Olympiad. One twelfth of them win a gold medal, another sixth win a silver medal, a further quarter win a bronze medal, and the remaining half receive a handshake. If we encounter a group of six participants chosen at random, what ... | 0.4 |
Example 4 Try to find the unit digit of the integer part of $(\sqrt{2}+\sqrt{3})^{2012}$.
[2] | 7 |
7. The equation of the hyperbola $C$ is $x^{2}-\frac{y^{2}}{3}=1$, with the left and right foci being $F_{1}$ and $F_{2}$, respectively. A line is drawn through point $F_{2}$ intersecting the right branch of the hyperbola $C$ at points $P$ and $Q$, such that $\angle F_{1} P Q=90^{\circ}$. Then, the radius of the incirc... | \sqrt{7}-1 |
5. Solve the inequality
$$
\frac{\left(2 \cdot 3^{-\log _{x} 2}-6\right) \sqrt{2-\sqrt{2 \log _{x} 2+3}}}{2+\sqrt{\log _{x} 2+2}}>\frac{\left(3^{-\log _{x} 2}-3\right) \sqrt{2-\sqrt{2 \log _{x} 2+3}}}{\sqrt{\log _{x} 2+2}-1}
$$ | x\in(0;1/2)\cup(1/2;1/\sqrt[3]{4}]\cup(4;+\infty) |
## Problem Statement
Calculate the lengths of the arcs of the curves given by equations in a rectangular coordinate system.
$$
y=1-\ln \left(x^{2}-1\right), 3 \leq x \leq 4
$$ | 1+\ln\frac{6}{5} |
If the square root $\sqrt{\frac{1}{{x-1}}}$ is meaningful, then the range of values for $x$ is ______. | x > 1 |
Given $P=\{x|a-4 < x < a+4\}$, $Q=\{x|x^2-4x+3 < 0\}$, and $x \in P$ is a necessary condition for $x \in Q$, find the range of the real number $a$. | -1 \leqslant a \leqslant 5 |
3. On a very long road, a race was organized. 21 runners started at different times, each running at a constant speed. The race continued until each runner had overtaken all the slower ones. The speeds of the runners who started first and last were the same, and the speeds of the others were different from theirs and d... | 190;19 |
10. The system of equations
$$
\left\{\begin{array}{l}
a+b+c+d=-2, \\
a b+a c+a d+b c+b d+c d=-3, \\
b c d+a c d+a b d+a b c=4, \\
a b c d=3
\end{array}\right.
$$
has a set of real solutions $(a, b, c, d)=$ . $\qquad$ | \left(\frac{-1+\sqrt{5}}{2}, \frac{-1-\sqrt{5}}{2}, \frac{-1+\sqrt{13}}{2}, \frac{-1-\sqrt{13}}{2}\right) |
If $\frac{a}{b}=\frac{2}{3}$, then $\frac{a}{a-b}=$____. | -2 |
The graph of the linear function $y=-x-2$ does not pass through which quadrant?
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | A |
71. Solve the system of equations
$$
\left\{\begin{array}{l}
5 x+3 y=12 \\
2 x-y=7
\end{array}\right.
$$ | (3,-1) |
Given the function $f(x) = x^2 - 1$, let $M = \{(x, y) | f(x) + f(y) \leq 0\}$ and $N = \{(x, y) | f(x) - f(y) \geq 0\}$. Find the area of the planar region represented by the intersection of sets $M$ and $N$. | \pi |
9. If $\vec{a}=\{8,9,-12\}$, vector $\overrightarrow{A B}$ is in the same direction as $\vec{a}$, point $A(2,-1,7),|B A|=34$, then the coordinates of point $B$ are | (18,17,-17) |
G3.2 Suppose the equation $x^{2} y-x^{2}-3 y-14=0$ has only one positive integral solution $\left(x_{0}, y_{0}\right)$. If $x_{0}+y_{0}=b$, find the value of $b$. | 20 |
Define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. Define $\{x \} = x - \lfloor x \rfloor$. For example, $\{ 3 \} = 3-3 = 0$, $\{ \pi \} = \pi - 3$, and $\{ - \pi \} = 4-\pi$. If $\{n\} + \{ 3n\} = 1.4$, then find the sum of all possible values of $100\{n\}$.
[i]Proposed by Isabella Grabski
[... | 145 |
To obtain the graph of $y=\sin\frac{x}{2}$, it is only necessary to shift the graph of the function $y=\cos({\frac{x}{2}-\frac{π}{4}})$ by ( ).
A: Shift left by $\frac{π}{4}$ units
B: Shift right by $\frac{π}{4}$ units
C: Shift left by $\frac{π}{2}$ units
D: Shift right by $\frac{π}{2}$ units | D |
If the curve $f(x)=\dfrac{1}{2e}{{x}^{2}}$ and the curve $g(x)=a\ln x$ ($a > 0$) have a common tangent line at their common point $P(s,t)$, then the real number $a$ equals
A: $-2$
B: $\dfrac{1}{2}$
C: $1$
D: $2$ | C |
Given vectors $\overrightarrow{a}=(\sin x,\cos x)$, $\overrightarrow{b}=(\sin x+\cos x,\sin x-\cos x)(x\in R)$, if $\overrightarrow{a} \perp \overrightarrow{b}$, find the set of all possible values for $x$, denoted as $(\ \ \ \ )$.
A: $\{x|x= \frac {kπ}{2}+ \frac {π}{8},k\in Z\}$
B: $\{x|x=kπ+ \frac {π}{8},k\in Z\}$
... | A |
# Problem 4. (3 points)
In how many ways can the number 1500 be represented as the product of three natural numbers (variants where the multipliers are the same but differ in order are considered the same)?
# | 32 |
The integers $ 1,2,...,n$ are placed in order so that each value is either strictly bigger than all the preceding values or is strictly smaller than all preceding values. In how many ways can this be done? | 2^{n-1} |
Example 11 The equation about $x$
$$
x^{4}+(k-1) x^{3}+k x^{2}+(k-1) x+1=0
$$
has no real roots. Then the range of real number $k$ is $\qquad$ | 0<k<4 |
Let the midpoint of segment $AB$ be $F$, and let $Z$ be any point on segment $AF$. We erect a perpendicular to $AB$ at $F$ and measure the distance $FX = FA$. We also erect a perpendicular to $AB$ at $B$ and measure the distance $BY = AZ$ such that $X$ and $Y$ are on the same side of line $AB$. What could be the measur... | 45 |
How many strings of ones and zeroes of length 10 are there such that there is an even number of ones, and no zero follows another zero? | 72 |
18. There are four boxes of apples. By weighing any three of them, four different weights can be obtained: $70 \mathrm{~kg}$, $80 \mathrm{~kg}$, $73 \mathrm{~kg}$, $77 \mathrm{~kg}$. The weights of these four boxes of apples are $\qquad$ kg, $\qquad$ kg, $\qquad$ kg, $\qquad$ $\mathrm{kg}$. | 30,20,27,23 |
Given that $k$ is a positive integer, and $1 \leqslant k \leqslant 2017$, then the number of $k$ that satisfies the equation $\sin 1^{\circ}+\sin 2^{\circ}+\ldots+\sin k^{\circ}=\sin 1^{\circ}\cdot\sin 2^{\circ}\ldots\sin k^{\circ}$ is \_\_\_\_\_\_. | 11 |
When negating the statement "Among the natural numbers \(a\), \(b\), \(c\), exactly one is even", the correct assumption for the negation is ( )
A: \(a\), \(b\), \(c\) are all odd
B: \(a\), \(b\), \(c\) are all even
C: There are at least two even numbers among \(a\), \(b\), \(c\)
D: \(a\), \(b\), \(c\) are all odd o... | \text{D} |
The 15 animators go on a training session together. Each day, 3 animators supervise the pool. It is noted that each pair of animators has supervised the pool together exactly once. How many days will the training session last? | 35 |
Given functions $f(x)=\sqrt{|x-1|}+\sqrt{|x+1|}-3$ and $g(x)=\ln(\sqrt{{x}^{2}+1}-|x|)$, the number of intersection points between the graphs of $f\left(x\right)$ and $g\left(x\right)$ is:
A: $6$
B: $4$
C: $3$
D: $2$ | \text{D} |
3. As shown in Figure 2, in rectangle $A B C D$, $E$ is a point on side $A B$, $A B=14, C E=13, D E$ $=15, C F \perp D E$ at point $F$, connect $A F$ and $B F$. Find the area of $\triangle A B F$. | 36.96 |
22. A chicken, a duck, a goose, and a rabbit are standing in a row. The chicken is 5 meters to the left of the goose, the rabbit is 3 meters to the left of the duck, and the goose is 2 meters to the right of the rabbit. The distance between the two animals on the far left and the far right is __ meters. | 6 |
6) A rectangular room has walls facing the four cardinal directions and there are four access doors. Three people are in the room and make the following statements. First person: "There are no doors on the South wall". Second person: "There are doors only on the North wall". Third person: "There is at most one door on ... | D |
Given a circle passes through the origin and point P(1, 1), and the center of the circle lies on the line $2x+3y+1=0$, find the equation of the circle. | (x-4)^2+(y+3)^2=25 |
Given positive integers $a$, $b$, $c$ satisfying: $a < b < c$, and $ab + bc + ca = abc$. Find all sets of $a$, $b$, $c$ that meet these conditions. | a = 2, b = 3, c = 6 |
Example 1. (IMO-29-5) In the right triangle $\triangle ABC$, $AD$ is the altitude on the hypotenuse $BC$. The line connecting the incenter of $\triangle ABD$ and the incenter of $\triangle ACD$ intersects sides $AB$ and $AC$ at $K$ and $L$, respectively. Let $S_{\triangle B C}=S, S_{\triangle K L}=T$. Prove: $S \geqsla... | S \geqslant 2 T |
Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$ | 8 |
4. In the tetrahedron $P-ABC$,
$$
\begin{array}{l}
\angle APB=\angle BPC=\angle CPA=90^{\circ}, \\
PA=4, PB=PC=3 .
\end{array}
$$
Then the minimum value of the sum of the squares of the distances from any point on the base $ABC$ to the three side faces is | \frac{144}{41} |
2. [2] Let $A B C$ be an equilateral triangle. Let $\Omega$ be a circle inscribed in $A B C$ 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 |
Given a sequence $\{a_n\}$, where $a_1=1$, and for $n\geqslant 2$, the sum of its first $n$ terms $S_n$ satisfies $S_n^2=a_n(S_n- \frac {1}{2})$.
(I) Find the expression for $S_n$;
(II) Let $b_n= \frac {S_n}{2n+1}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$. | \frac {n}{2n+1} |
When $x=2$, the value of the algebraic expression $mx^{2}-2x+n$ is $2$. Then when $x=-2$, the value of this algebraic expression is ____. | 10 |
The value of $sin\frac{2023π}{3}$ is:
A: $\frac{1}{2}$
B: $-\frac{1}{2}$
C: $\frac{\sqrt{3}}{2}$
D: $-\frac{\sqrt{3}}{2}$ | C |
1. Nine positive real numbers $a_{1}, a_{2}, \cdots, a_{9}$ form a geometric sequence, and
$$
a_{1}+a_{2}=\frac{3}{4}, a_{3}+a_{4}+a_{5}+a_{6}=15 .
$$
Then $a_{7}+a_{8}+a_{9}=$ . $\qquad$ | 112 |
70. How many members will result after expanding all the brackets in the expression: $(a+1)(b+1)(c+1)(d+1)(e+1)(f+1) \times$ $\times(g+1)$ ? | 128 |
# 2. Solution:
According to the condition, the center of the second blot is at a distance of no less than 2 cm from the edge of the sheet, i.e., inside a rectangle of 11 cm by 16 cm. Consider the event A "The blots intersect." For this to happen, the center of the second blot must fall within a circle of radius 4 cm w... | \frac{\pi}{11} |
## Task Condition
Find the derivative.
$$
y=\frac{6^{x}(\sin 4 x \cdot \ln 6-4 \cos 4 x)}{16+\ln ^{2} 6}
$$ | 6^{x}\sin4x |
8.3. Find the perimeter of a parallelogram if the bisector of one of its angles divides a side of the parallelogram into segments of 7 and 14. | 56or70 |
Given the universal set $U = \{1,2,3\}$, set $B = \{1,2\}$, and $A \cap B = \{1\}$, the number of sets $A$ that satisfy the condition is $(\ \ \ \ )$
A: $0$
B: $1$
C: $2$
D: $3$ | C |
Given sets $M=\{0, 1, 2, 3\}$, $N=\{-1, 1\}$, which of the following statements is true? ( )
A: $N \subseteq M$
B: $M \cup N = M$
C: $M \cap N = N$
D: $M \cap N = \{1\}$ | D |
A specific type of computer virus that invades memory space starts by occupying 2KB of memory when the computer is turned on. Then, every 3 minutes, it replicates itself, and the memory it occupies doubles. If the virus occupies 64MB of memory (1MB = $2^{10}$KB), then after the computer is turned on, it will take ( ) ... | A |
Calculate the sum of $C_{10}^{9} + C_{10}^{8}$. | 55 |
A sugar solution with $ag$ sugar contains $bg$ sugar. If an additional $mg$ sugar is added (where $a > b > 0$, $m > 0$), common sense tells us that the sugar solution will be sweeter after the added sugar completely dissolves. Based on this common sense, can you derive an inequality? Try to provide a proof. | \frac{b+m}{a+m} > \frac{b}{a} |
[ Decimal numeral system ]
What is the smallest integer of the form 111...11 that is divisible by 333...33 (100 threes)? | 1\ldots1 |
## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{2 x^{3}+6 x^{2}+7 x+4}{(x+2)(x+1)^{3}} d x
$$ | 2\cdot\ln|x+2|-\frac{1}{2(x+1)^{2}}+C |
Randomly select a number $a$ from the set $\{2,3,4,5\}$, and a number $b$ from the set $\{1,3,5\}$. Then, the probability that the vector $m=(a,b)$ is perpendicular to the vector $n=(1,-1)$ is ______. | \frac{1}{6} |
When $x=1$, the function $f\left(x\right)=a\ln x+\frac{b}{x}$ reaches a maximum value of $-2$. Then, ${f'}\left(2\right)=\left(\ \ \right)$
A: $-1$
B: $-\frac{1}{2}$
C: $\frac{1}{2}$
D: $1$ | B |
The vertices of an equilateral triangle lie on the hyperbola $xy=1$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?
$\textbf{(A)}\ 48\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 108\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 169$ | 108 |
4. As shown in Figure $2, D, E$ are points on the sides $BC, AD$ of the equilateral $\triangle ABC$ respectively, $\angle BEC$ $=90^{\circ}, \angle CED=30^{\circ}$. Then the value of $BD: CD$ is ( ).
(A) $2: 1$
(B) 3: 1
(C) $5: 2$
(D) $20: 9$ | A |
Is there a number in which every non-zero digit is used exactly once, and for which the number formed by the first $k$ digits is divisible by $k$ for every $k$ between 1 and 9? | 381654729 |
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