problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
6. Let $f(x)$ be a function defined on $\mathbf{R}$, satisfying
$$
\left|f(x)+\cos ^{2} x\right| \leqslant \frac{3}{4},\left|f(x)-\sin ^{2} x\right| \leqslant \frac{1}{4} .
$$
Then the analytical expression of the function $f(x)$ is . $\qquad$ | f(x)=\sin ^{2} x-\frac{1}{4} |
Given two vectors in the plane, $\overrightarrow{a} = (-1, 0)$ and $\overrightarrow{b} = (0, 2)$, then $2\overrightarrow{a} - 3\overrightarrow{b} =$ （ ）
A: $(6, 3)$
B: $(-2, -6)$
C: $(2, 1)$
D: $(7, 2)$ | \text{B} |
3. (7 points) On the radius $A O$ of a circle with center $O$, a point $M$ is chosen. On one side of $A O$ on the circle, points $B$ and $C$ are chosen such that $\angle A M B = \angle O M C = \alpha$. Find the length of $B C$ if the radius of the circle is $16$, and $\sin \alpha = \frac{\sqrt{55}}{8}$? | 12 |
A cycle of three conferences had constant success, that is, in each session there was the same number of participants. However, half of those who attended the first one did not return; one-third of those who attended the second conference attended only that one, and one-quarter of those who attended the third did not a... | 156,37 |
Represent the set $$\{(x, y) | \begin{cases} x+y=5 \\ 2x-y=1 \end{cases}$$ in the form of enumeration. The correct answer is ( )
A: {2, 3}
B: {(2, 3)}
C: {x=2, y=3}
D: (2, 3) | \text{B} |
3. Given an isosceles right triangle with legs equal to $\sqrt{2}$. Find the straight-line cut of the smallest length that divides the area of the triangle in half.
In the answer, specify the length of the cut, rounded to 2 decimal places after the decimal point. Answer: $0.91$. | 0.91 |
4. A positive integer, if added to 100 and 168 respectively, can result in two perfect squares. This positive integer is $\qquad$ | 156 |
Among 5 questions, there are 3 science questions and 2 liberal arts questions. If two questions are drawn sequentially without replacement, given that the first question drawn is a science question, the probability of drawing a science question on the second draw is ( )
A: $$\frac {3}{10}$$
B: $$\frac {3}{5}$$
C: $... | C |
5-141 Find all non-zero polynomials $f(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n}$ that satisfy the equation $f\left(x^{2}\right)=(f(x))^{2}$, where $x$ is a real number. | f(x)=x^{n} |
Start with a three-digit positive integer $A$. Obtain $B$ by interchanging the two leftmost digits of $A$. Obtain $C$ by doubling $B$. Obtain $D$ by subtracting $500$ from $C$. Given that $A + B + C + D = 2014$, fi nd $A$. | 344 |
The slope of the line $x-\sqrt{3}y+1=0$ is ( ).
A: $120^{\circ}$
B: $150^{\circ}$
C: $30^{\circ}$
D: $45^{\circ}$ | C |
If $\angle A$ is four times $\angle B$, and the complement of $\angle B$ is four times the complement of $\angle A$, then $\angle B=$
$\textbf{(A)}\ 10^{\circ} \qquad\textbf{(B)}\ 12^{\circ} \qquad\textbf{(C)}\ 15^{\circ} \qquad\textbf{(D)}\ 18^{\circ} \qquad\textbf{(E)}\ 22.5^{\circ}$ | 18 |
Convert the decimal number $123_{(10)}$ to binary. | 1111011_{(2)} |
Example 5 Find all natural numbers $a, b$, such that $\left[\frac{a^{2}}{b}\right]+$
$$
\left[\frac{b^{2}}{a}\right]=\left[\frac{a^{2}+b^{2}}{a b}\right]+a b
$$ | ^2+1,\in{N}^*orb^2+1,b\in{N}^* |
2. Let the function $f: \mathbf{R} \rightarrow \mathbf{R}$, for any real number $x$ we have $f\left(x^{2}+x\right)+2 f\left(x^{2}-3 x+2\right)=9 x^{2}-15 x$. Then $f(50)=(\quad$.
(A) 72
(B) 73
(C) 144
(D) 146 | D |
In triangle $ABC$, $AC=1, \angle ABC=30^{\circ}, \angle BAC=60^{\circ}$, and the foot of the altitude from $C$ is $D$. How far apart are the centers of the inscribed circles of triangles $ACD$ and $BCD$? | \sqrt{2-\sqrt{3}} |
Given $a \lt b$, which of the following expressions may not hold true?
A: $a-1 \lt b-1$
B: $ma \gt mb$
C: $\frac{1}{2}a+1 \lt \frac{1}{2}b+1$
D: $-2a \gt -2b$ | B |
There exists a function $f(x)$ such that for any $x \in \mathbb{R}$, we have ( )
A: $f(|x|) = x$
B: $f(|x|) = x^2 + 2x$
C: $f(|x+1|) = x$
D: $f(|x+1|) = x^2 + 2x$ | \text{D} |
Russian problems Problem 23 The triangle ABC satisfies 0 <= AB <= 1 <= BC <= 2 <= CA <= 3. What is the maximum area it can have? | 1 |
14. The radius of the clock face is 20 cm, and it is externally tangent to another smaller disk with a radius of 10 cm at the 12 o'clock position. The smaller disk has a fixed pointer, which initially points vertically upward. When the smaller disk rolls clockwise along the outer edge of the larger clock face, always r... | C |
Given that $ABCD$ is a square, point $M$ (different from points $B, C$) is on side $BC$, and the perpendicular bisector $l$ of line segment $AM$ intersects $AB$ and $CD$ at points $E$ and $F$ respectively.
(1) Which is longer, $BE$ or $DF$? Please explain your reasoning.
(2) If $AB=1$, find the range of $|BE - DF|$ as ... | \left(0, \frac{1}{4}\right] |
Let set $A=\{0, 1, 2, 4, 5, 7\}$, $B=\{1, 3, 6, 8, 9\}$, $C=\{3, 7, 8\}$. Find the set $(A \cap B) \cup C$. | \{1, 3, 7, 8\} |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_{1}$ is $\begin{cases} x=a\cos \phi \\ y=b\sin \phi \end{cases}$ ($a > b > 0$, $\phi$ is the parameter), and the point $M(2, \sqrt {3})$ on curve $C_{1}$ corresponds to the parameter $\phi= \dfrac {\pi}{3}$. Taking $O$ as the pole and the pos... | \dfrac {1}{\rho _{1}^{2}}+ \dfrac {1}{\rho _{2}^{2}}= \dfrac {1}{16}+ \dfrac {1}{4}= \dfrac {5}{16} |
For any real numbers $a$ and $b$, define $a*b = a(a+b) + b$. Given that $a*2.5 = 28.5$, the value of the real number $a$ is ____. | 4, -\frac{13}{2} |
Given the function $f(x)=2x^{2}-mx+5$, where $m\in\mathbb{R}$, it is monotonically decreasing in the interval $(-\infty,-2]$. Then, the range of $f(1)$ is $(\quad)$.
A: $f(1)=15$
B: $f(1) > 15$
C: $f(1)\leqslant 15$
D: $f(1)\geqslant 15$ | C |
Ivan and Mirka were sharing pears from a plate. Ivan always took two pears, and Mirka took half of what was left on the plate. They proceeded in this manner: Ivan, Mirka, Ivan, Mirka, and finally Ivan, who took the last two pears.
Determine who ended up with more pears and by how many.
(M. Dillingerová)
Hint. How ma... | 2 |
## Problem Statement
Calculate the volumes of solids formed by rotating the figures bounded by the graphs of the functions. The axis of rotation is $O y$.
$$
y=\arccos \frac{x}{3}, y=\arccos x, y=0
$$ | 2\pi^{2} |
2.5 A can run a circular track in 40 seconds, B runs in the opposite direction, and meets A every 15 seconds, how many seconds does it take for B to run the track? | 24 |
In a certain place, the slogan "Save water, protect the environment" is promoted. If saving $30cm^{3}$ of water is denoted as $+30cm^{3}$, then wasting $10cm^{3}$ of water is denoted as ( )
A: $+10cm^{3}$
B: $-10cm^{3}$
C: $+0cm^{3}$
D: $-20cm^{3}$ | B |
The coordinates of the vertex of the parabola $y=\left(x-2\right)^{2}-1$ are:
A: $(-2,1)$
B: $(-2,-1)$
C: $(2,1)$
D: $(2,-1)$ | D |
Task B-2.1. Solve the equation in the set of complex numbers
$$
z \cdot(4+i)+\bar{z} \cdot(2-3 i)=1-i^{11}
$$ | \frac{3}{2},2 |
Simplify first and then evaluate: $3x-2({x}^{2}-\frac{1}{2}{y}^{2})+(x-\frac{1}{2}{y}^{2})$, where $x$ and $y$ satisfy $|x-2|+\left(y+1\right)^{2}=0$. | \frac{1}{2} |
5. A race on an unpredictable distance is held as follows. On a circular running track, two points $A$ and $B$ are randomly selected (using a spinning arrow), after which the athletes run from $A$ to $B$ along the shorter arc. A spectator has bought a ticket to the stadium and wants the athletes to run past his seat (t... | \frac{1}{4} |
Last year, there were 30 more boys than girls in our scout troop. This year, the number of children in the troop increased by $10 \%$, while the number of boys increased by $5 \%$ and the number of girls increased by $20 \%$.
How many children do we have in the troop this year?
(L. Hozová) | 99 |
Calculate: $(\sqrt{5}+3)(3-\sqrt{5})-(\sqrt{3}-1)^2$. | 2\sqrt{3} |
10. Find the sum of the digits of all numbers in the sequence $1,2,3, \ldots, 99,100$. | 901 |
Given a plane $\alpha$, a plane $\beta$, and a line $m$, with the conditions: ① $m \parallel \alpha$; ② $m \perp \alpha$; ③ $m \subset \alpha$; ④ $\alpha \perp \beta$; ⑤ $\alpha \parallel \beta$. Among these five conditions, the combination that can deduce $m \parallel \beta$ is ( )
A: ①④
B: ①⑤
C: ②⑤
D: ③⑤ | D |
A servant confesses to his master that he stole one liter of wine from a 250-liter barrel every day for 50 days and always replaced the missing amount with water. What percentage of wine was still in the barrel? | 81.846 |
[ Classical combinatorics (other) $]$
In the cabinet of Anchuria, there are 100 ministers. Among them, there are both corrupt and honest ministers. It is known that among any ten ministers, at least one minister is a corrupt one. What is the smallest number of corrupt ministers that can be in the cabinet?
# | 91 |
Given the sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n = 2^n + a$ (where $a$ is a constant and $n \in \mathbb{N}^*$).
1. Find $a\_1$, $a\_2$, and $a\_3$.
2. If the sequence $\{a\_n\}$ is a geometric sequence, find the value of the constant $a$ and the expression for $a\_n$.
3. For $a\_n$ ob... | (-\frac{3}{4}, 0] |
What is the value of the sum $9^{20}+9^{20}+9^{20}$?
(a) $9^{20}$
(b) $3^{66}$
(c) $9^{23}$
(d) $3^{41}$
(e) $3^{23}$ | 3^{41} |
16th Irish 2003 Problem B3 Find all integer solutions to m 2 + 2m = n 4 + 20n 3 + 104n 2 + 40n + 2003. | =128,n=7or-17 |
Choose 3 out of 4 varieties of vegetables and plant them on 3 different plots of land. The number of different planting methods is ( )
A: 36
B: 64
C: 24
D: 81 | C |
5. In $\triangle A B C$, $A B<B C<C A$, and $A C-$ $A B=2, A D$ is the angle bisector of $\angle B A C$, $E$ is a point on side $A C$, and line $B E$ intersects $A D$ at point $G$, and $\frac{A C}{C D}=\frac{A E}{B D}=$ $2, \frac{A G}{G D}=2007$. Then the length of side $B C$ is $(\quad$.
(A) 2008
(B) 2007
(C.) 2006
(D... | B |
[ Problems on percentages and ratios ]
Little Man and Karlson together ate a jar of jam. In this case, Karlson ate $40 \%$ fewer spoons of jam than Little Man, but on the other hand, his spoon held $150 \%$ more jam than Little Man's spoon. What part of the jar of jam did Karlson eat? | \frac{3}{5} |
When randomly selecting 3 items from a batch of products, the probability of getting no defective items is 0.18, the probability of getting exactly one defective item is 0.53, the probability of getting exactly two defective items is 0.27, and the probability of all three items being defective is 0.02.
Find:
(1) Th... | 0.71 |
Given a line $m$ and planes $\alpha$ and $\beta$, which of the following four statements is correct?
A: If $\alpha \bot \beta$, $m\subset \beta$, then $m\bot \alpha$
B: If $m$ is parallel to $\alpha$ and $m$ is parallel to $\beta$, then $\alpha$ is parallel to $\beta$
C: If $\alpha$ is parallel to $\beta$, $m$ is pa... | \text{D} |
13. (6 points) A fraction, if the denominator is reduced by 1, simplifies to $\frac{1}{3}$; if the numerator is increased by 4, it simplifies to $\frac{1}{2}$. This fraction is $\qquad$ | \frac{7}{22} |
10. Write the number $a_{0}=9.3 \pm 0.5$ using a double inequality. | 8.8\leqslanta_{0}\leqslant9.8 |
If the temperature at noon was 2°C and it dropped by 3°C in the evening, what was the temperature in the evening? | -1 |
In the arithmetic sequence {an}, a2 = 8, and a3 + a5 = 4a2.
(I) Find the general term formula for the arithmetic sequence {an};
(II) Suppose that the geometric sequence {bn} with positive terms satisfies b4 = a1, b6 = a4, find the sum Sn of the first n terms of the sequence {bn - an}. | 2^{n-1} - \frac{1}{2} - 2n^{2} - 2n |
Define $\left\lVert A-B \right\rVert = (x_A-x_B)^2+(y_A-y_B)^2$ for every two points $A = (x_A, y_A)$ and $B = (x_B, y_B)$ in the plane.
Let $S$ be the set of points $(x,y)$ in the plane for which $x,y \in \left\{ 0,1,\dots,100 \right\}$.
Find the number of functions $f : S \to S$ such that $\left\lVert A-B \right\rVer... | 2040200 |
Given $t^{2}-3t+1=0$, find the value of $t+\frac{1}{t}$. | 3 |
The perimeter of $\triangle A B C$ is 32 . If $\angle A B C=\angle A C B$ and $B C=12$, the length of $A B$ is
(A) 20
(B) 12
(C) 10
(D) 8
(E) 16
 | 10 |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. Given that $a_3=12$ and $S_{12} > 0$, $S_{13} < 0$. Find the range of the common difference $d$. | - \frac{24}{7} < d < -3 |
Prove that $\sqrt {10}- \sqrt {5} < \sqrt {7}- \sqrt {2}$. | \sqrt {10}- \sqrt {5} < \sqrt {7}- \sqrt {2} |
Find the minimum value of the function $y=|x-2|+|3-x|$ on $\mathbb{R}$. | 1 |
Find the area of the triangle formed by the two tangent lines at the intersection points of the curve $y = \frac{1}{x}$ and $y=x^{2}$ with the $x$-axis. | \frac{3}{4} |
Example 7 (1994 National Training Team Question) Let $S=$ $\{1,2, \cdots, 10\}, A_{1}, A_{2}, \cdots, A_{k}$ be subsets of $S$, satisfying the conditions:
(1) $\left|A_{i}\right|=5(i=1,2, \cdots, k)$;
(2) $\left|A_{i} \cap A_{j}\right| \leqslant 2(1 \leqslant i<j \leqslant k)$.
Find the maximum value of $K$. | 6 |
The arithmetic square root of the real number $16$ is ____. | 4 |
Bogdanov I.I.
In a store, 21 white and 21 purple shirts are hanging in a row. Find the smallest $k$ such that for any initial order of the shirts, it is possible to remove $k$ white and $k$ purple shirts so that the remaining white shirts hang together and the remaining purple shirts also hang together. | 10 |
Given a set of sample data $x_{1}$, $x_{2}$, $\cdots$, $x_{6}$, where $x_{1}$ is the minimum value and $x_{6}$ is the maximum value, then ()
A: The average of $x_{2}$, $x_{3}$, $x_{4}$, $x_{5}$ is equal to the average of $x_{1}$, $x_{2}$, $\cdots$, $x_{6}$.
B: The median of $x_{2}$, $x_{3}$, $x_{4}$, $x_{5}$ is equal... | BD |
The radian measure of $-150^{\circ}$ is ( )
A: $- \dfrac {5\pi}{6}$
B: $ \dfrac {4\pi}{3}$
C: $- \dfrac {2\pi}{3}$
D: $- \dfrac {3\pi}{4}$ | A |
Given the function $f(x)=|x+1|-|x-a|$
(I) When $a=1$, find the solution set of the inequality $f(x) < 1$;
(II) If the maximum value of $f(x)$ is $6$, find the value of $a$. | a=5 |
Given that line $l$ passes through the intersection point $P$ of the lines $3x + 4y - 2 = 0$ and $2x + y + 2 = 0$.
(1) If line $l$ is perpendicular to the line $x - 2y - 1 = 0$, find the equation of line $l$;
(2) If line $l$ is parallel to the line $AB$ passing through points $A(8, -6)$ and $B(2, 2)$, find the equation... | 4x + 3y + 2 = 0 |
A student participates in a science and knowledge competition, where they must answer three questions. According to the competition rules, correctly answering the first, second, and third questions earns them 100 points, 100 points, and 200 points, respectively, while wrong answers earn zero points. Assuming that the p... | 0.564 |
Given the function $f(x)= \begin{cases}f(x-3),x > 0 \\ e^{x}+\log _{2}[8^{x+1}×( \frac {1}{4})^{-2}],x\leqslant 0\end{cases}$, find $f(2016)=$ _____ . | 8 |
2. In tetrahedron $ABCD$, $BD = DC = CB = \sqrt{2}$, $AC = \sqrt{3}$, $AB = AD = 1$. $M$ is the midpoint of $CD$. Then the cosine of the angle formed by the skew lines $BM$ and $AC$ is ( ).
(A) $\frac{\sqrt{2}}{3}$
(B) $\frac{\sqrt{2}}{2}$.
(C) $\frac{\sqrt{6}}{2}$.
(D) $\frac{\sqrt{3}}{9}$. | A |
Given the function $f(x) = ax^2 + 3x - 2$, the slope of the tangent line at the point $(2, f(2))$ is 7. The value of the real number $a$ is ( )
A: $-1$
B: $1$
C: $\pm 1$
D: $-2$ | \text{B} |
If $a \lt b \lt 0$, then which of the following inequalities must be true?
A: $\frac{1}{a} \gt \frac{1}{b}$
B: $\frac{1}{a-b} \gt \frac{1}{b}$
C: $\sqrt{-a} \gt \sqrt{-b}$
D: $|a| \gt -b$ | ACD |
5. Given $x \geq 1, y \geq 1$ and $\lg ^{2} x+\lg ^{2} y=\lg 10 x^{2}+\lg 10 y^{2}$, then the maximum value of $u=\lg x y$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 2+2\sqrt{2} |
Given the parametric equations of line \\(l\\) as: \\(\begin{cases}x=2t \\ y=1+4t\end{cases} \\) (where \\(t\\) is the parameter), and the polar equation of circle \\(C\\) as \\(\rho=2 \sqrt{2}\sin \left(\theta\right) \\), then the positional relationship between line \\(l\\) and circle \\(C\\) is
A: Tangent
B: Interse... | \text{B} |
Given $p:(x-m)^{2} \lt 9$, $q:\log _{4}(x+3) \lt 1$, if $\neg q$ is a necessary but not sufficient condition for $\neg p$, then the range of $m$ is ______. | [-2,0] |
11. (20 points) Let 2018 real numbers $a_{1}, a_{2}, \cdots, a_{2018}$ satisfy
$$
\sum_{i=1}^{2018} a_{i}=0, \sum_{i=1}^{2018} a_{i}^{2}=2018 .
$$
Find the maximum possible value of the minimum product of any two of these 2018 real numbers. | -1 |
Given $\lambda_{i} \in \mathbf{R}_{+}(i=1,2, \cdots, n)$. Try to find all positive real solutions $x_{1}, x_{2}, \cdots, x_{n}$ of the equation
$$
\sum_{i=1}^{n} \frac{\left(x_{i}+\lambda_{i}\right)^{2}}{x_{i+1}}=4 \sum_{i=1}^{n} \lambda_{i}
$$
(Here $x_{n+1}=x_{1}$ ).
(Xiao Zhengang) | x_{k}=\frac{1}{2^{n}-1}\sum_{i=1}^{n}2^{i-1}\lambda_{k-1+i},k=1,2,\cdots,n |
22. Two numbers in the $4 \times 4$ grid can be swapped to create a Magic Square (in which all rows, all columns and both main diagonals add to the same total).
What is the sum of these two numbers?
A 12
B 15
C 22
D 26
E 28
\begin{tabular}{|c|c|c|c|}
\hline 9 & 6 & 3 & 16 \\
\hline 4 & 13 & 10 & 5 \\
\hline 14 & 1 & 8 ... | 28 |
6. (10 points) A water pool contains one-eighteenth of its capacity. Two water pipes start filling the pool simultaneously. When the water level reaches two-ninths of the pool's capacity, the first pipe continues to fill the pool alone for 81 minutes, during which it injects an amount of water equal to what the second ... | 231 |
Find the area of inscribed convex octagon, if the length of four sides is $2$, and length of other four sides is $ 6\sqrt {2}$.
$\textbf{(A)}\ 120 \qquad\textbf{(B)}\ 24 \plus{} 68\sqrt {2} \qquad\textbf{(C)}\ 88\sqrt {2} \qquad\textbf{(D)}\ 124 \qquad\textbf{(E)}\ 72\sqrt {3}$ | 124 |
Problem 1. The faces of an orthogonal parallelepiped whose dimensions are natural numbers are painted green. The parallelepiped is partitioned into unit cubes by planes parallel to its faces. Find the dimensions of the parallelepiped if the number of cubes having no green face is one third of the total number of cubes... | (4,7,30),(4,8,18),(4,9,14),(4,10,12),(5,5,27),(5,6,12),(5,7,9),(6,6,8) |
Rectangle $A B C D$ has diagonal $B D$ with endpoints $B(4,2)$ and $D(12,8)$. Diagonal $A C$ lies on the line with equation $x+2 y-18=0$. Determine the area of $A B C D$. | 20\sqrt{5} |
1467. Calculate $\sin 10^{\circ}$ with an accuracy of 0.0001. | 0.1736 |
10. Find the area of the triangle if it is known that the radius of the inscribed circle is 1, and the lengths of all three altitudes are expressed as integers. | 3\sqrt{3} |
In the triangle $ABC$, point $D$ is given on the extension of side $CA$ beyond $A$, and point $E$ is given on the extension of side $CB$ beyond $B$, such that $AB = AD = BE$. The angle bisectors from $A$ and $B$ of triangle $ABC$ intersect the opposite sides at points $A_1$ and $B_1$, respectively. What is the area of ... | 6 |
Given the sets $P=\{x|1\leqslant x\leqslant 3\}$ and $Q=\{x|(x-1)^{2}\leqslant 4\}$, then $P\cap Q=$ .
A: $[-1,3]$
B: $[1,3]$
C: $[1,2]$
D: $(-\infty,3]$ | B |
The function $f(x) = \begin{cases} 2^{-x} & (x < 1) \\ \log_{81}x & (x \geq 1) \end{cases}$, then the value(s) of $x$ that satisfy $f(x) = \frac{1}{4}$ is ( )
A: 2
B: -2
C: 3
D: $\frac{1}{3}$ | C |
Simplify and compute:
(1) $3m^2 + 2mn - 5m^2 + 3mn$
(2) $(x^2 + 2x) - 2(x^2 - x)$ | -x^2 + 4x |
Given a positive sequence $\{a_n\}$ such that for any $p, q \in \mathbb{N}^*$, it holds that $a_{p+q} = a_p \cdot a_q$. If $a_2 = 4$, find the value of $a_9$. | 512 |
Given two different lines $m$ and $n$, and two different planes $α$ and $β$, consider the following four conclusions:
1. If $α \perp β$ and $m \parallel α$, then $m \perp β$.
2. If $m \perp α$, $n \perp β$, and $m \perp n$, then $α \perp β$.
3. If $m \perp α$ and $m \parallel β$, then $α \perp β$.
4. If $m \parallel α... | B |
Given a sequence $\{a_n\}$ that satisfies $a_1=1, a_{n+1}=\frac{1}{4}a_n^2+p$.
(I) If the sequence $\{a_n\}$ is a constant sequence, find the value of $p$.
(II) Prove that $a_n 1$.
(III) Find the maximum positive value of $p$ such that $a_n < 2$ holds for all integers $n$, and prove your conclusion. | 1 |
Example 1 As shown in Figure 5, in the complete quadrilateral $A B C D E F$, $G J \perp E F$ at point J. Then
$$
\begin{array}{l}
\angle B J A \\
=\angle D J C .
\end{array}
$$ | \angle B J A=\angle D J C |
In the pyramid $P-ABCD$, where the quadrilateral $ABCD$ is a square, $PA \perp$ plane $ABCD$, and $PA=6$, $AB=8$, the ratio of the surface area of the circumscribed sphere to the inscribed sphere of the pyramid $P-ABCD$ is ( ).
A: $\frac{{\sqrt{41}}}{2}$
B: $\frac{{41}}{4}$
C: $3$
D: $\frac{{11}}{2}$ | B |
Example 7 Divide $n$ stones into two piles at random, and record the product of the number of stones in the two piles; then divide one of the piles into two piles, and record the product of the number of stones in these two piles; then divide one of the three piles into two piles, and record the product of the number o... | S(n)=\frac{n(n-1)}{2} |
Given the sequence $$\frac {1}{2}, \frac {2}{3}, \frac {3}{4}, \frac {4}{5},$$ …, then among 0.98, 0.96, 0.94, the number of values that belong to this sequence is ( )
A: 0
B: 1
C: 2
D: 3 | C |
6. Let $A=(0,2 \pi), A$ be a mapping $f: t \rightarrow(\sin t, 2 \sin t \cos t)$ to the set of points on the coordinate plane. Also, let $B=$ $\{f(t) \mid t \in A\}, C(r)=\left\{(x, y) \mid x^{2}+y^{2} \leqslant r^{2}\right\}$ $(r>0)$. Find the minimum value of $r$ that satisfies $B \subseteq C(r)$. (US | \frac{5}{4} |
8.1. Points $K, L, M, N$ are the centers of the circles inscribed in the faces $S A B, S A C$, $S B C$ and $A B C$ of the tetrahedron $S A B C$. It is known that $A B=S C=5, A C=S B=7, B C=S A=8$. Find the volume of the tetrahedron $K L M N$. Round your answer to the nearest hundredth if necessary. | 0.66 |
If $P= \sqrt {a}+ \sqrt {a+7}$, $Q= \sqrt {a+3}+ \sqrt {a+4}$ $(a\geqslant 0)$, then the relationship between $P$ and $Q$ is: ______. | P < Q |
Sound is produced by the vibration of objects, which contains a sine function. The mathematical model of a pure tone is the function $y=A\sin \omega t$. The sound we hear is composed of pure tones, called composite tones. If the mathematical model of a composite tone is the function $f(x)=\sin\frac{1}{2}x-\frac{1}{2}\s... | \text{BC} |
Consider a charged capacitor made with two square plates of side length $L$, uniformly charged, and separated by a very small distance $d$. The EMF across the capacitor is $\xi$. One of the plates is now rotated by a very small angle $\theta$ to the original axis of the capacitor. Find an expression for the difference ... | \frac{\epsilon_0 L^3 \theta \xi}{d^2} |
7. If the quadratic function $f(x)=a x^{2}-2 x-a$ satisfies
$$
f(2)<f(1)<f(3)<f(0) \text {, }
$$
then the range of the real number $a$ is $\qquad$. | \left(\frac{1}{2}, \frac{2}{3}\right) |
Two identical bottles are filled with alcohol solutions. The ratio of alcohol to water by volume in one bottle is $m:1$, and in the other bottle, it is $n:1$. When the two solutions are mixed together, the ratio of alcohol to water in the mixture is ( )
A: $\boxed{\frac{m+n}{2}}$
B: $\boxed{\frac{2(m^2+mn+n^2)}{3(m+... | D |
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