problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given $a = \int_{1}^{e} (x + \frac{1}{x}) dx$, calculate the value of $a$. | \frac{1}{2}e^2 + \frac{1}{2} |
The coefficient of $x^7$ in the expansion of $\left(\frac{x^2}{2}-\frac{2}{x}\right)^8$ is:
$\textbf{(A)}\ 56\qquad \textbf{(B)}\ -56\qquad \textbf{(C)}\ 14\qquad \textbf{(D)}\ -14\qquad \textbf{(E)}\ 0$ | -14 |
How many different positive integer solutions does the equation $x y=10^{n}$ have? | (n+1)^2 |
9. In the five-digit number abcde, $a, b, c, d, e$ represent its ten-thousands, thousands, hundreds, tens, and units place values, respectively. If $\mathrm{d}>\mathrm{e}, \mathrm{c}>\mathrm{d}+\mathrm{e}, \mathrm{b}>\mathrm{c}+\mathrm{d}+\mathrm{e}, \mathrm{a}>\mathrm{b}+\mathrm{c}+\mathrm{d}+\mathrm{e}$. Then, the la... | 95210 |
9. A gardener is preparing to plant a row of 20 trees, with two types of trees available: maple trees or sycamore trees. The number of trees between any two maple trees (not including these two maple trees) cannot be equal to 3. Therefore, the maximum number of maple trees among the 20 trees is $\qquad$ trees. | 12 |
Given that $f'(x)$ is the derivative of the function $f(x)$, if the equation of the tangent line to the graph of the function $f(x)$ at the point $x=5$ is $x+y-5=0$, then $f(5)+f'(5)=$ （ ）
A: 1
B: -1
C: -2
D: 0 | \text{B} |
1. Let $A B C D$ be a unit square. Draw a quadrant of a circle with $A$ as centre and $B, D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A, C$ as end points of the arc. Inscribe a circle $\Gamma$ touching the arc $A C$ externally, the arc $B D$ internally and also touching ... | \frac{1}{6} |
Let the sequence $\{a_n\}$ have the $n$th partial sum $S_n$, $a_1=1$, and $na_{n+1}=(n+2)S_{n}$ for $n \in \mathbb{N}^*$.
(1) Prove that the sequence $\left\{\frac{S_n}{n}\right\}$ is a geometric sequence.
(2) Find the sum of the first $n$ terms of the sequence $\{S_n\}$, denoted $T_n$. | T_n = (n-1)2^n + 1 |
$16 \cdot 19$ On a plane, given three non-collinear points $A, B, C$, construct a line $l$ such that the distances from points $A, B, C$ to line $l$ are in the ratio $1: 1: 2$ or $1: 2: 1$ or $2: 1: 1$. The number of such lines is
(A) 3.
(B) 6.
(C) 9.
(D) 12.
(E) 15.
(China Junior High School Mathematics Correspondence... | 12 |
The imaginary part of the complex number $z=i(1-i)$ is. | 1 |
30 Given a natural number $a \geqslant 2$, the set $A=\left\{y \mid y=a^{x}, x \in \mathbf{N}\right\}$. Does there exist $b$ in the interval $[1, a]$ such that $C=A \cap B \neq \emptyset$? Here $B=\{y \mid y=(a+1) x+b, x \in \mathbf{N}\}$. If it exists, find all possible values of $b$ and the corresponding set $C$; if ... | 1 |
For real numbers $a$ and $b$, define a new operation "★" as follows: $a★b= \begin{cases} a^{2}b+a, & \text{when } a\geq b \\ ab^{2}+b, & \text{when } a<b \end{cases}$. If $2★m=36$, then the real number $m$ equals ( )
A: 8.5
B: 4
C: 4 or -4.5
D: 4 or -4.5 or 8.5 | B |
9. Solve the system of equations in the set of real numbers $\left\{\begin{array}{l}\sqrt{x+1}+\sqrt{y-1}=5, \\ x+y=13 .\end{array}\right.$ | (8,5),(3,10) |
17 (12 points) Given the ellipse $T: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ and the hyperbola $S$ : $\frac{x^{2}}{m^{2}}-\frac{y^{2}}{n^{2}}=1(m>0, n>0)$ have the same focus $F(2,0)$. Let the asymptote of the hyperbola $S$ in the first quadrant be $l$. If the focus $F$ and the upper vertex $B$ of the ellipse... | \frac{11x^{2}}{60}+\frac{11y^{2}}{16}=1\frac{5x^{2}}{4}-\frac{5y^{2}}{16}=1 |
Example 1.11.2. Let $a, b$, c be non-negative real numbers. For each $k \geq 0$, find the minimum of the expression
$$\frac{a+k b}{c+k b}+\frac{b+k c}{a+k c}+\frac{c+k a}{b+k a}$$ | 3 |
Given the parabola $y^{2}=2px(p > 0)$, a line passing through its focus and with a slope of $2$ intersects the parabola at points $A$ and $B$. If the x-coordinate of the midpoint of segment $AB$ is $3$, then the equation of the directrix of this parabola is \_\_\_\_\_\_. | x=-2 |
If the slope of the tangent line to the curve $y=2x^2+1$ at point $M$ is $-4$, then the coordinates of point $M$ are _____. | (-1, 3) |
## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$$
f(x)=\left\{\begin{array}{c}
\sqrt{1+\ln \left(1+3 x^{2} \cos \frac{2}{x}\right)}-1, x \neq 0 \\
0, x=0
\end{array}\right.
$$ | 0 |
Marek is playing with a calculator. He wrote down one number on a piece of paper. He entered it into the calculator and then pressed the buttons in sequence: plus, four, divide, four, minus, four, times, four. He wrote down the result on the paper. Then he repeated the same process with this number, again: plus, four, ... | 38;26;14;2 |
18. In the diagram below, the lengths of the three sides of the triangle are $a \mathrm{~cm}, b \mathrm{~cm}$ and $c \mathrm{~cm}$. It is given that $\frac{a^{2}+b^{2}}{c^{2}}=2011$. Find the value of $\frac{\cot C}{\cot A+\cot B}$. | 1005 |
11. Let the function $f_{n}(x)=1+x+\frac{1}{2!} x^{2}+\cdots+\frac{1}{n!} x^{n}$.
(1) Prove that when $x \in(0,+\infty), n \in \mathbf{N}_{+}$, $\mathrm{e}^{x}>f_{n}(x)$;
(2) Let $x>0, n \in \mathbf{N}_{+}$, if there exists $y \in \mathbf{R}$ such that $\mathrm{e}^{x}=f_{n}(x)+\frac{\mathrm{e}^{y}}{(n+1)!} x^{n+1}$, pr... | 0<y<x |
In a math activity class, Xiaoming wants to use three wooden sticks to make a triangle model by connecting them end to end. He has two sticks with lengths of $2cm$ and $5cm$ respectively. The length of the third stick can be ( )
A: $1cm$
B: $2cm$
C: $3cm$
D: $4cm$ | D |
1. Find the equation of the line that is parallel to the line $3x + 4y - 12 = 0$ and has a distance of $7$ from it.
2. Find the equation of the line that is perpendicular to the line $x + 3y - 5 = 0$ and has a distance of $\frac{3}{5} \sqrt{10}$ from point $P(-1, 0)$. | 3x - y - 3 = 0 |
Determine the value of $a$ in the equation $\int_{0}^{a}{2dx=2}$. | 1 |
Problem 5.6. On the board, there is one three-digit number and two two-digit numbers. The sum of the numbers that have a seven in their notation is 208. The sum of the numbers that have a three in their notation is 76. Find the sum of all three numbers. | 247 |
Given that $y$ is directly proportional to $x-1$, and when $x=3$, $y=4$.
$(1)$ Find the analytical expression for the relationship between $y$ and $x$;
$(2)$ If the point $(-1,m)$ lies on the graph of this function, find the value of $m$. | m = -4 |
2. Given positive numbers $a, b, c, d$. Find the minimum value of the expression
$$
A=\left(a^{2}+\frac{1}{b c}\right)^{3}+\left(b^{2}+\frac{1}{c d}\right)^{3}+\left(c^{2}+\frac{1}{d a}\right)^{3}+\left(d^{2}+\frac{1}{a b}\right)^{3}
$$ | 32 |
Given that the sum of the first $n$ terms of the sequence ${a_n}$, denoted as $S_n$, satisfies $S_n + S_m = S_{n+m}$ ($n$, $m\in\mathbb{N}^*$) and $a_1 = 5$, find the value of $a_8$ ( options: A: 40, B: 35, C: 12, D: 5 ). | D |
A line passing through the right focus $F_2$ of the hyperbola $\frac {x^{2}}{3}- \frac {y^{2}}{6}=1$ with an inclination angle of $30^{\circ}$ intersects the hyperbola at points $A$ and $B$. Find the length of $|AB|$. | \frac {16}{5} \sqrt {3} |
B2. Without using a pocket calculator, calculate the value of the expression
$$
8 \cdot \sqrt[20]{32}-9 \cdot \sqrt[5]{\sqrt[6]{9}}-4 \cdot \sqrt[16]{16}+4 \cdot \sqrt[5]{\sqrt[9]{27}}+5 \cdot \sqrt[5]{\sqrt[12]{81}}
$$ | 4\cdot\sqrt[4]{2} |
Given triangle ABC, vectors $\overrightarrow {m}$ = ($\sqrt {3}$sinA, sinB) and $\overrightarrow {n}$ = (cosB, $\sqrt {3}$cosA). If the dot product of $\overrightarrow {m}$ and $\overrightarrow {n}$ is equal to 1 + cos(A + B), find the value of angle C. | \frac {2π}{3} |
5. In the number $2 * 0 * 1 * 6 * 02 *$, each of the 5 asterisks needs to be replaced with any of the digits $0,2,4,7,8,9$ (digits can repeat) so that the resulting 11-digit number is divisible by 12. In how many ways can this be done? | 1296 |
\section*{Problem 9}
Every city in a certain state is directly connected by air with at most three other cities in the state, but one can get from any city to any other city with at most one change of plane. What is the maximum possible number of cities?
| 10 |
If the terminal side of angle $\alpha$ passes through point $P(a, 2a) (a < 0)$, then $\cos \alpha =$ _____. | -\frac{\sqrt{5}}{5} |
3. A football invitational tournament involves sixteen cities, each sending two teams, Team A and Team B. According to the competition rules, after several days of matches, it was found that except for Team A of City A, all other teams had played a different number of matches. How many matches has Team B of City A play... | 15 |
8. If the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{n}=\left[(2+\sqrt{3})^{2^{n}}\right]\left(n \in \mathbf{Z}_{+}\right),
$$
then the last two digits of $a_{2020}$ are $\qquad$ | 53 |
3. In the rectangular coordinate plane, given $A(-\sqrt{3}, 0)$, $B(\sqrt{3}, 0)$, point $P$ moves on the line $y=\frac{\sqrt{3}}{3}(x+4)+1$. When $\angle A P B$ is maximized, the value of $\frac{P A}{P B}$ is $\qquad$. | \sqrt{3} - 1 |
Problem 3. Solve in positive integers the equation:
$$
1+5^{x}=2^{y}+2^{z} \cdot 5^{t}
$$
| (x,y,z,)=(2,4,1,1) |
\section*{Problem 6 - 081046}
The figure shows a cube \(W = A B C D E F G H\) with edge length \(a\).
In the side faces \(A B C D, A B F E, A D H E, B C G F, D C G H, E F G H\) of \(W\), ... | \frac{1}{2} |
The solution set of the inequality $|x^{2}-2| < 2$ is ( ).
A: $(-1,1)$
B: $(-2,2)$
C: $(-1,0)\cup(0,1)$
D: $(-2,0)\cup(0,2)$ | D |
The complex number $z = m(m-1) + (m-1)i$ is a pure imaginary number. Find the value of the real number $m$. | 0 |
(1) Simplify $\frac{\cos(180^\circ + \alpha) \cdot \sin(\alpha + 360^\circ)}{\sin(-\alpha - 180^\circ) \cdot \cos(-180^\circ - \alpha)}$.
(2) Given that $\tan \alpha = -\frac{3}{4}$, find the value of $\frac{\cos(\frac{\pi}{2} + \alpha) \cdot \sin(-\pi - \alpha)}{\cos(\frac{11\pi}{2} - \alpha) \cdot \sin(\frac{11\pi}... | \frac{3}{4} |
## Task A-4.1.
Nikola has thought of five numbers. The first number he thought of is -2, and the fifth is 6. The first four numbers are consecutive terms of an arithmetic sequence, and the last three are consecutive terms of a geometric sequence. What numbers did Nikola think of? | -2,-\frac{2}{3},\frac{2}{3},2,6 |
9. (16 points) Given $a, b \in [1,3], a+b=4$.
Find the maximum value of $f(a, b)=\left|\sqrt{a+\frac{1}{b}}-\sqrt{b+\frac{1}{a}}\right|$. | 2-\frac{2}{\sqrt{3}} |
What is the largest possible value of the expression $$gcd \,\,\, (n^2 + 3, (n + 1)^2 + 3 )$$ for naturals $n$?
[hide]original wording]Kāda ir izteiksmes LKD (n2 + 3, (n + 1)2 + 3) lielākā iespējamā vērtība naturāliem n? [/hide] | 13 |
For which value of the parameter $k$ can the expression
$$
(3 k-2) x(x+k)+k^{2}(k-1)
$$
be written in the form $(a x+b)^{2}$? | 2 |
14. Let $\left\{a_{n}\right\}$ be a sequence of positive numbers, with the sum of its first $n$ terms being $S_{n}$, and for all natural numbers $n$, we have
$$
S_{n}=\frac{3 n+1}{2}-\frac{n}{2} a_{n} \text {. }
$$
(1) Write down the first three terms of the sequence $\left\{a_{n}\right\}$;
(2) Find the general term fo... | S_{n}^{\prime}=\frac{n(n+1)(n+2)(n+3)}{8} |
Given a set of data arranged in ascending order: -1, 0, 4, x, 7, 14, if the median is 5, then the mean of this set of data is \_\_\_\_\_\_, and the variance is \_\_\_\_\_\_. | \frac{74}{3} |
If $b \neq 0$, we define $a \oslash b=a b-\frac{a}{b}$. For example, $3 \oslash 1=3(1)-\frac{3}{1}=0$.
If $x=4 \triangleright 2$ and $y=2 \oslash 2$, what is the value of $x^{y}$ ? | 216 |
8. We consider all possible sets consisting of 2017 different natural numbers, where in each set no number can be represented as the sum of two other numbers in the set. What is the smallest value that the largest number in such a set can take?
Answer: 4032. | 4032 |
3. The value of the positive integer $n$ that makes $2^{n}+256$ a perfect square is $\qquad$ | 11 |
In an isosceles triangle, the lengths of two sides are $3$ and $6$ respectively. The perimeter of this isosceles triangle is ( ).
A: $12$
B: $15$
C: $12$ or $15$
D: $18$ | B |
## problem statement
Are the vectors $c_{1 \text { and }} c_{2}$, constructed from vectors $a_{\text {and }} b$, collinear?
$$
\begin{aligned}
& a=\{-2 ;-3 ;-2\} \\
& b=\{1 ; 0 ; 5\} \\
& c_{1}=3 a+9 b \\
& c_{2}=-a-3 b
\end{aligned}
$$ | c_{1}=-3\cdotc_{2} |
Solve the following system of equations:
$$
\frac{x+y}{x^{3}+y^{3}}=\frac{2}{5}, \quad \frac{x^{2}+y^{2}}{x^{4}+y^{4}}=\frac{8}{23}
$$ | \begin{gathered}x_{1}=\frac{1}{8}(\sqrt{70}+\sqrt{30}),\quady_{1}=\frac{1}{8}(\sqrt{70}-\sqrt{30})\\x_{2}=\frac{1}{8}(\sqrt{70}-\sqrt{30}),\quady_{2}=\frac{1}{8}(\sqrt{70} |
$ABCD$ is a convex quadrilateral such that $AB<AD$. The diagonal $\overline{AC}$ bisects $\angle BAD$, and $m\angle ABD=130^\circ$. Let $E$ be a point on the interior of $\overline{AD}$, and $m\angle BAD=40^\circ$. Given that $BC=CD=DE$, determine $m\angle ACE$ in degrees. | 55^\circ |
The standard equation of the circle that passes through points A(-1, 3) and B(3, -1) and whose center lies on the line x - 2y - 1 = 0 is ( )
A: (x+1)2+(y-1)2=4
B: (x+1)2+(y+1)2=16
C: (x-1)2+y2=13
D: (x-1)2+y2=5 | \text{B} |
From 10 college graduates, 3 individuals are to be selected to serve as village assistants. The number of different ways to select them such that at least one of A or B is chosen, and C is not chosen, is ______. | 49 |
10. In the geometric sequence $\left\{a_{n}\right\}$, $a_{1}>0, S_{n}=80$, the largest term among the first $n$ terms is $54, S_{2 n}=$ 6560, the general term formula of this sequence $a_{n}=$ $\qquad$, and $\lim _{n \rightarrow \infty} \frac{a_{n}}{S_{n}}=$ $\qquad$ | a_{n}=2\cdot3^{n-1},\frac{2}{3} |
In parallelogram $ABCD$, $\overrightarrow{AC}-\overrightarrow{BC}=$
A: $\overrightarrow{DA}$
B: $\overrightarrow{BD}$
C: $\overrightarrow{BA}$
D: $\overrightarrow{DC}$ | D |
Given a quadratic function $f\left(x\right)=-x^{2}+2ax-3$.
$(1)$ When $a=-2$, the inequality $f\left(x\right) \lt 0$ has solutions.
$(2)$ If $x\in \left[1,5\right]$, the inequality $f\left(x\right) \lt 3ax$ always holds. Find the range of real numbers. | (-2\sqrt{3}, +\infty) |
Exercise 5. Let $x, y, z$ be strictly positive real numbers such that
$$
x+\frac{y}{z}=y+\frac{z}{x}=z+\frac{x}{y}=2
$$
Determine all possible values that the number $x+y+z$ can take. | 3 |
Given the proposition p: ∀x≤0, $\sqrt {x^{2}}$ = -x, then the negation of p (¬p) is ( )
A: ∀x≤0, $\sqrt {x^{2}}$ ≠ -x
B: ∀x≤0, $\sqrt {x^{2}}$ ≠ -x
C: ∀x>0, $\sqrt {x^{2}}$ = -x
D: ∃x0≤0, $\sqrt {x_{0}^{2}}$ ≠ -x0 | \text{D} |
8.310. $\sin ^{3} x(1-\operatorname{ctg} x)+\cos ^{3} x(1-\operatorname{tg} x)=1.5 \cos 2 x$. | \frac{\pi}{4}(2k+1),k\inZ |
The central angle of a sector is $\frac{\pi}{3}$, and the chord length opposite to it is 3 cm. Then, the area of this sector is \_\_\_\_\_\_ cm2. | \frac{3\pi}{2} |
If $\frac {1}{x} - \frac {1}{y} = \frac {1}{z}$, then $z$ equals:
$\textbf{(A)}\ y - x\qquad \textbf{(B)}\ x - y\qquad \textbf{(C)}\ \frac {y - x}{xy}\qquad \textbf{(D)}\ \frac {xy}{y - x}\qquad \textbf{(E)}\ \frac {xy}{x - y}$ | \frac{xy}{y-x} |
Given vectors $\vec{a}=(m,3)$ and $\vec{b}=(\sqrt{3},1)$, if the angle between vectors $\vec{a}$ and $\vec{b}$ is $30^\circ$, then the real number $m=$ ______. | \sqrt{3} |
There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries,... | 20 |
4. For the equation $x^{2}+(2 m-1) x+(m-6)=0$, one root is no greater than -1, and the other root is no less than 1. Try to find:
(1) the range of values for the parameter $m$;
(2) the maximum and minimum values of the sum of the squares of the roots.
(9th Junior High School Mathematics Competition, Jiangsu Province) | -4 \leqslant m \leqslant 2, \text{ Maximum value } 101, \text{ Minimum value } \frac{43}{4} |
Which of the following functions is both an odd function and a decreasing function within its domain?
A: $y=\log_{\frac{1}{2}}x(x>0)$
B: $y=-x^{2}(x\in R)$
C: $y=-x^{3}-x(x\in R)$
D: $y=-\frac{1}{x}(x\in R$ and $x\neq 0)$ | C |
9 The function $f(x)$ defined on $\mathbf{R}$ satisfies $f\left(x+\frac{3}{2}\right)+f(x)=0$, and the function $y=$ $f\left(x-\frac{3}{4}\right)$ is an odd function. Given the following propositions: (1) The smallest positive period of the function $f(x)$ is $\frac{3}{2}$; (2) The graph of the function $y=f(x)$ is symm... | B |
362. Calculate the integral $\int_{1}^{4} x^{2} d x$ using Simpson's formula. | 21 |
Let \[\mathcal{S} = \sum_{i=1}^{\infty}\left(\prod_{j=1}^i \dfrac{3j - 2}{12j}\right).\] Then $(\mathcal{S} + 1)^3 = \tfrac mn$ with $m$ and $n$ coprime positive integers. Find $10m + n$.
[i]Proposed by Justin Lee and Evan Chang[/i] | 43 |
[ Properties and characteristics of an isosceles triangle. ]
Through the vertices $A$ and $C$ of triangle $A B C$, lines are drawn perpendicular to the bisector of angle $A B C$ and intersect the lines $C B$ and $B A$ at points $K$ and $M$ respectively. Find $A B$, if $B M=8, K C=1$. | 7or9 |
1. a) Solve in $Z$ the equation: $5 \cdot(2 \cdot|3 x-4|+4)-30=10$ | 2 |
1. [4] The distance from a certain point inside a regular hexagon to three of its consecutive vertices is 1, 1, and 2, respectively. What is the side length of this hexagon?
M. Evdokimov | \sqrt{3} |
14. Given the line $y=k x-1$ intersects the left branch of the hyperbola $x^{2}-$ $y^{2}=1$ at points $A$ and $B$. If another line $l$ passes through point $P(-2,0)$ and the midpoint $Q$ of segment $A B$, find the range of the y-intercept $b$ of line $l$. | (-\infty,-2) \cup(2+\sqrt{2},+\infty) |
II. (25 points) As shown in Figure 3, in the right triangle $\triangle ABC$, $\angle C=90^{\circ}, CD \perp AB$ at $D$, the incenter of $\triangle ADC$ and $\triangle CDB$ are $I_{1}$ and $I_{2}$ respectively, and $I_{1} I_{2}$ intersects $CD$ at $K$. Prove that: $\frac{1}{AC}+\frac{1}{BC}=\frac{1}{CK}$. | \frac{1}{AC} + \frac{1}{BC} = \frac{1}{CK} |
Example 12 (1991 Japan Mathematical Olympiad) Find the positive integer solutions of the indeterminate equation $\frac{1}{x+1}+\frac{1}{y}+\frac{1}{(x+1) y}=$ $\frac{1}{1991}$. | 64 |
Given a set $M \subseteq \{1, 2, 3, 4\}$, and $M \cap \{1, 2\} = \{1, 2\}$, the number of sets $M$ is ( )
A: 1
B: 2
C: 3
D: 4 | D |
14) In the following arithmetic expression, Chinese characters represent the numbers 1 to 9, with different characters representing different numbers. If the character “祝” represents the number “4” and the character “贺” represents the number “8”, find the integer represented by “华杯赛”.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果... | 7632 |
13. A confectionery company produces two types of nougat, using the same white paste and hazelnuts, but in different proportions. In type A nougat, hazelnuts represent $30 \%$ of the weight and $40 \%$ of the volume; in type B nougat, hazelnuts represent $60 \%$ of the weight. What percentage of the volume do the hazel... | 70 |
Let $a$ be a positive constant. Evaluate the following definite integrals $A,\ B$.
\[A=\int_0^{\pi} e^{-ax}\sin ^ 2 x\ dx,\ B=\int_0^{\pi} e^{-ax}\cos ^ 2 x\ dx\].
[i]1998 Shinsyu University entrance exam/Textile Science[/i] | B = \frac{1}{2} \left( \frac{1}{a} + \frac{a}{a^2 + 4} \right) (1 - e^{-a\pi}) |
Problem 10.8. Rectangle $ABCD$ is such that $AD = 2AB$. Point $M$ is the midpoint of side $AD$. Inside the rectangle, there is a point $K$ such that $\angle AMK = 80^{\circ}$ and ray $KD$ is the bisector of angle $MKC$. How many degrees does angle $KDA$ measure?
=x\sin x$, then the relationship among $f\left(\frac{\pi}{11}\right)$, $f(-1)$, and $f\left(-\frac{\pi}{3}\right)$ is ( )
A: $f\left(-\frac{\pi}{3}\right) > f(-1) > f\left(\frac{\pi}{11}\right)$
B: $f(-1) > f\left(-\frac{\pi}{3}\right) > f\left(\frac{\pi}{11}\right)$
C: $f\left(\frac{\pi}{11}... | A |
1. If all the diagonals of a convex $n$-sided polygon $F (n \geqslant 4)$ are equal, then
A. $F \in\{$ quadrilateral $\}$
B. $F \in\{$ pentagon $\}$
C. $F \in\{$ quadrilateral $\} \bigcup\{$ pentagon $\}$
D. $F \in\{$ equilateral polygon $\} \bigcup\{$ equiangular polygon $\}$ | C |
Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?
$\textbf{(A) } \frac14 \qquad\textbf{(B) } \frac13 \qquad\textbf{(C) } \frac12 \qquad\textbf{(D) } \frac23 \qquad\textbf{(E) } \frac34$ | \frac{1}{3} |
Given an geometric sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. If $S\_3$, $S\_9$, and $S\_6$ form an arithmetic sequence and $a\_2 + a\_5 = 4$, find the value of $a\_8$. | 2 |
Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_2=3$, if $|a_{n+1}-a_n|=2^n$ $(n\in\mathbb{N}^*)$, and the sequence $\{a_{2n-1}\}$ is increasing while $\{a_{2n}\}$ is decreasing, then $\lim\limits_{n\to\infty} \frac{a_{2n-1}}{a_{2n}}=$ ______. | -\frac{1}{2} |
Given the function $f(x)=\ln x$, with $0 < a < b < c < 1$, determine the relationship in magnitude among $\frac{f(a)}{a}$, $\frac{f(b)}{b}$, and $\frac{f(c)}{c}$. | \frac{f(a)}{a} < \frac{f(b)}{b} < \frac{f(c)}{c} |
Given that the ground temperature at a certain location is $20^{\circ}\mathrm{C}$, if the temperature decreases by $6^{\circ}\mathrm{C}$ for every $1000m$ increase in altitude, then the functional relationship between temperature $t\left(^{\circ}\mathrm{C}\right)$ and altitude $h\left(m\right)$ is ____. | t = -0.006h + 20 |
2. Let $a$ and $b$ be real numbers that satisfy the equations
$$
\frac{a}{b}+\frac{b}{a}=\frac{5}{2} \quad \text { and } \quad a-b=\frac{3}{2} \text {. }
$$
Find all possible values of $a^{2}+2 a b+b^{2}+2 a^{2} b+2 a b^{2}+a^{2} b^{2}$. | 081 |
5. A ladybird is free to fly between the $1 \times 1$ cells of a $10 \times 10$ square grid. She may begin in any $1 \times 1$ cell of the grid. Every second she flies to a different $1 \times 1$ cell that she has not visited before.
Find the smallest number of cells the ladybird must visit, including her starting cell... | 76 |
Given that the sequence $\{a_{n}\}$ is a geometric sequence, if $a_{4}=2$ and $a_{8}=32$, then $a_{6}=\left(\ \ \right)$
A: $\pm 8$
B: $8$
C: $16$
D: $\pm 16$ | B |
11. (6 points) In an exam, the average score of $A, B, C, D, E$ is 90 points. If the average score of $A, B, C$ is 86 points, and the average score of $B, D, E$ is 95 points, then the score of $B$ is $\qquad$ points. | 93 |
Given the sequence $\{a_n\}$ with the sum of the first $n$ terms $S_n=n^2-n$, and a positive geometric sequence $\{b_n\}$ where $b_2=a_3$, and $b_{n+3}b_{n-1}=4b_n^2$ $(n\geqslant 2,n\in\mathbb{N}_+)$, then $\log_2b_n=$ ()
A: $n$
B: $2n-1$
C: $n-2$
D: $n-1$ | A |
Extend 1 Let non-negative real numbers $a, b, c, x, y, z$ satisfy $a+b+c=x+y+z=1$.
Find the minimum value of $\left(a-x^{2}\right)\left(b-y^{2}\right)\left(c-z^{2}\right)$. | -\frac{1}{4} |
Problem 8.2. In a volleyball tournament for the Euro- African cup the European teams are 9 more than the African teams. Every two teams met exactly once and the European teams gained 9 times more points than the African teams (the winner takes 1 point and the loser takes 0 point). What are the maximum possible points g... | 11 |
1. Let set $A=\left\{x \mid x^{2}-3 x-10 \leq 0\right\}, B=\{x \mid m+1 \leq x \leq 2 m-1\}$, if $A \cap B=B$, then the range of real number $m$ is $\qquad$. | \leq3 |
21.12. Find a four-digit number that is a perfect square and has the first two digits the same and the last two digits also the same. | 7744 |
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