problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Find the number of lines that pass through the point $(0,1)$ and intersect the parabola $y^2=4x$ at only one point. | 3 |
Given that the pole of the polar coordinate system is at the origin of the rectangular coordinate system, the polar axis coincides with the non-negative x-axis and they share the same unit length. The curve $C\_1$ is defined by the equation $ρ\cos θ - 2ρ\sin θ - 7 = 0$, and the curve $C\_2$ is defined by the parametric... | \frac{8\sqrt{5}}{5} |
9. (40 points) The numbers $s_{1}, s_{2}, \ldots, s_{1008}$ are such that their sum is $2016^{2}$. It is known that
$$
\frac{s_{1}}{s_{1}+1}=\frac{s_{2}}{s_{2}+3}=\frac{s_{3}}{s_{3}+5}=\ldots=\frac{s_{1008}}{s_{1008}+2015}
$$
Find $s_{17}$. | 132 |
If the line $(a+1)x-y+2=0$ is parallel to the line $x+(a-1)y-1=0$, then the value of the real number $a$ is $\_\_\_\_\_\_$. | 0 |
## Task 4 - 311244
Let $P_{1} P_{2} P_{3}$ be a given arbitrary triangle; its area be $F$, and the incircle have the center $M$ and the radius $r$.
a) Prove that a pyramid $P_{1} P_{2} P_{3} S$ has the smallest possible surface area among all pyramids $P_{1} P_{2} P_{3} S$ with this base $P_{1} P_{2} P_{3}$ and a giv... | F+\sqrt{F^{2}+(\frac{3V}{r})^{2}} |
1. For the fraction $\frac{m}{n}$, where $m$ and $n$ are natural numbers, it holds that $\frac{1}{3}<\frac{m}{n}<1$. If a natural number is added to the numerator and the denominator is multiplied by this number, the value of the fraction does not change. Find all such fractions $\frac{m}{n}$. | \frac{2}{3},\frac{2}{4},\frac{2}{5} |
Every book has an International Standard Book Number (ISBN): A B C D E F G H I J, where A B C D E F G H I are arranged from nine digits, and J is the check digit. Let $S = 10A + 9B + 8C + 7D + 6E + 5F + 4G + 3H + 2I$, and $r$ is the remainder when $S$ is divided by 11. If $r$ is not equal to 0 or 1, then it is stipulat... | 7 |
There are $n$ football teams participating in a league, where each team plays a match against every other team exactly once. It is stipulated that a win earns 3 points, a draw earns 1 point, and a loss earns 0 points. After the league ends, some teams may be disqualified, and thus their match results will also be cance... | n(n-1) |
Given a function $f(x)=\left\{\begin{array}{l}{x^{\frac{1}{2}}}-2, x>0\\{e^x}, x\leq0\end{array}\right.$, then the value of $f\left(f\left(4\right)\right)$ is ( )
A: $-2$
B: $0$
C: $1$
D: $e$ | C |
Example 5 Calculate $\sec \frac{2 \pi}{9}+\sec \frac{4 \pi}{9}+\sec \frac{6 \pi}{9}+\sec \frac{8 \pi}{9}$. | 4 |
What is the smallest eight-digit positive integer that has exactly four digits which are 4 ? | 10004444 |
6・97 Let $x, y, z$ be positive numbers, and $x^{2}+y^{2}+z^{2}=1$, try to find the minimum value of the following expression
$$
S=\frac{x y}{z}+\frac{y z}{x}+\frac{z x}{y} .
$$ | \sqrt{3} |
Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly $ 5\over 4$ times greater than Vasya's result. What m... | 5 |
2. Five runners ran a relay. If the first runner ran twice as fast, they would have spent $5 \%$ less time. If the second runner ran twice as fast, they would have spent $10 \%$ less time. If the third runner ran twice as fast, they would have spent $12 \%$ less time. If the fourth runner ran twice as fast, they would ... | 8 |
From five people, A, B, C, etc., three people are randomly selected to do passing training. The training rule is to determine one person to pass the ball for the first time. Each time the ball is passed, the passer will pass the ball to either of the other two people with equal probability, and the ball must be passed ... | p_{n} = \frac{1}{3}\left[1 + \frac{(-1)^{n}}{{2}^{n-1}}\right] |
If the value of the fraction $\frac{{x}^{2}-4}{x}$ is $0$, then the value of $x$ is ( )
A: $2$ or $-2$
B: $2$
C: $-2$
D: $0$ | \text{A} |
Given two non-empty sets $A$ and $B$ satisfying $A \cup B = \{1, 2, 3\}$, the number of ordered pairs $(A, B)$ that meet the condition is ( )
A: 6
B: 8
C: 25
D: 27 | C |
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n=n(2n+1)$, find $a_2=$ ______. | 7 |
Given that the focus $F$ of the parabola $C\_1$: $y=ax^{2}(a > 0)$ is also a focus of the ellipse $C\_2$: $\frac{y^{2}}{4}+\frac{x^{2}}{b^{2}}=1 (b > 0)$, and the point $M$, $P(\frac{3}{2}, 1)$ are points on the curves $C\_1$ and $C\_2$ respectively. Find the minimum value of $(|MP|+|MF|)$. | 2 |
6. Let $P(x)$ be a quadratic polynomial with real coefficients such that $P(11)=181$ and $x^{2}-2 x+2 \leq P(x) \leq 2 x^{2}-4 x+3$ for any real number $x$. Find $P(21)$.
(1 mark)
設 $P(x)$ 為二次多項式, 其系數皆為實數。已知 $P(11)=181$, 且對任意實數 $x$ 皆有 $x^{2}-2 x+2 \leq P(x) \leq 2 x^{2}-4 x+3$, 求 $P(21)$ 。
(1 分) | 721 |
10. When less than 100 students perform a group dance, there are two combinations: one is a group of 5 in the middle, with the rest forming groups of 8 around the outer circle; the other is a group of 8 in the middle, with the rest forming groups of 5 around the outer circle. How many students are there at most? | 93 |
14. In a sequence, each term after the first two terms is the mean of all the terms which come before that term. The first term is 8 and the tenth term is 26 . What is the second term?
A 17
B 18
C 44
D 52
E 68 | 44 |
Vector $\overrightarrow{a}=(1,2)$, $\overrightarrow{b}=(2+m,3-m)$, $\overrightarrow{c}=(3m,1)$, $\overrightarrow{a}∥(\overrightarrow{c}-\overrightarrow{b})$, then $m=\left(\ \ \right)$
A: $2$
B: $-2$
C: $\frac{2}{3}$
D: $-\frac{2}{3}$ | C |
3. Given an arithmetic sequence $a_{1}, a_{2}, \cdots, a_{1000}$, the sum of the first 100 terms is 100, and the sum of the last 100 terms is 1000. Then $a_{1}=$ $\qquad$ | 0.505 |
Given sets $A=\{-1, 1, 2\}$ and $B=\{0, 1, 2, 7\}$, the number of elements in the set $A \cup B$ is ______. | 5 |
The price of a pencil is $a$ yuan, the price of an eraser is $b$ yuan. Buying $3$ pencils and $7$ erasers costs ______ yuan. | 3a+7b |
1. If points $A$ and $B$ are on the same side of the $x$-axis, you can first find the symmetric point $B^{\prime}\left(x_{2}, -y_{2}\right)$ (or $A^{\prime}\left(x_{1}, -y_{1}\right)$) of point $B$ (or $A$) with respect to the $x$-axis, then calculate $\left|A B^{\prime}\right|$. The value of $\left|A B^{\prime}\right|... | \sqrt{26} |
14.3 If $\frac{6}{b}<x<\frac{10}{b}$, find the value of $c=\sqrt{x^{2}-2 x+1}+\sqrt{x^{2}-6 x+9}$.
If $\frac{6}{b}<x<\frac{10}{b}$, determine the value of $c=\sqrt{x^{2}-2 x+1}+\sqrt{x^{2}-6 x+9}$. | 2 |
At the end of a day of activities, an amusement park collected 100 reais from the tickets of 100 people. We know that each adult had to pay 3 reais to enter, each young person 2 reais, and each child 30 cents. What is the minimum number of adults who entered the park on that day?
# | 2 |
In the 2022 Qatar World Cup, a total of 32 teams will compete in the final stage. The final stage is divided into group stage and knockout stage. The 32 teams are drawn into 8 groups, with 4 teams in each group, to play the group stage matches. The group stage matches are played in a round-robin format (each pair of te... | 64 |
The line $$l: y=k(x- \sqrt {2})$$ and the hyperbola $$x^2-y^2=1$$ have only one point in common. The value of the real number $k$ is ( )
A: 1
B: -1
C: 1 or -1
D: 1 or -1 or 0 | \text{C} |
3. (10 points) Connect the diagonals of square $A B C D$, and color the four vertices either red or yellow. Triangles with all vertices of the same color are called monochromatic triangles. The number of coloring methods that result in monochromatic triangles in the figure is ( )
A. 12
B. 17
C. 22
D. 10 | 10 |
The Richter scale $\left(M\right)$ is a scale used to indicate the magnitude of an earthquake. It is derived from the common logarithm calculation of the maximum amplitude of seismic waves recorded by a seismograph at an observation point $\left(A\right)$ and the amplitude that a standard seismic scale at the observati... | D |
1. For any given natural number $n$, if $n^{6}+3 a$ is a cube of a positive integer, where $a$ is a positive integer, then
(A) there are infinitely many such $a$,
(B) such $a$ exist, but only finitely many;
(C) such $a$ do not exist,
(D) none of the conclusions (A),
(B),
(C) are correct. | A |
Subject (3). Determine the natural numbers $m, n$ such that $85^{m}-n^{4}=4$. | =1,n=3 |
For what value of $m$ does the equation $x^2 - 4|x| + 5 - m = 0$ have four distinct real roots? | 1 < m < 5 |
6.3 .9 Find all integer pairs $(a, b)$ such that there exists an integer-coefficient polynomial $p(x)$, satisfying: the product $\left(x^{2}+a x+b\right) p(x)$ has the form: $x^{n}+c_{n-1} x^{n-1}+\cdots+c_{0}$, where each $c_{i}$ is either 1 or -1. | (\1,\1),(0,\1),(\2,1) |
Given $a:b:c=1:3:6$, $a+b+c=30$, find $c-b-a$. | 6 |
The slope of the line passing through points A(0, -1) and B(2, 4) is ( )
A: $$\frac {3}{2}$$
B: $$\frac {5}{2}$$
C: $$\frac {2}{5}$$
D: $$\frac {3}{2}$$ | \text{B} |
8,9
The side of a cell on a grid paper is 1. A rectangle with sides $m$ and $n$ is constructed along the grid lines.
Can a closed broken line be drawn within the rectangle along the grid lines that passes exactly once through each grid node located inside or on the boundary of the rectangle? If so, what is its length? | (n+1)(+1) |
13.024. A tractor team can plow 5/6 of a plot of land in 4 hours and 15 minutes. Before the lunch break, the team worked for 4.5 hours, after which 8 hectares remained unplowed. How large was the plot? | 68 |
Given the function $f(x)= \begin{cases} \log_{2}x,&x > 0 \\ 2^{x},&x\leqslant 0 \end{cases}.$ If $f(a)=2$, find the value of $a$. | 4 |
Given $a > 1$, then the point $P(1+a, 1-a)$ is in ( )
A: The first quadrant
B: The second quadrant
C: The third quadrant
D: The fourth quadrant | D |
## Task 1 - 230721
Uwe's way to school passes by the town hall and the train station. At the town hall, Uwe has completed a quarter of the journey; the town hall clock shows 7:30 AM. At the train station, Uwe has covered a third of the journey; the train station clock shows 7:32 AM.
At what time does Uwe arrive at sc... | 7:48 |
8. Let $a_{1}=1, a_{2}=2$, for $n \geqslant 2$ we have
$$
a_{n+1}=\frac{2 n}{n+1} a_{n}-\frac{n-1}{n+1} a_{n-1} .
$$
If for all positive integers $n \geqslant m$, we have $a_{n}>2+$ $\frac{2008}{2009}$, then the smallest positive integer $m$ is $\qquad$ . | 4019 |
Given a sequence ${a_n}$ where the sum of its first $n$ terms is $S_n$, $a_1=1$ and $a_{n+1}=2S_n+1$, $n\in\mathbb{N}^*$.
(1) Find the general term formula for the sequence ${a_n}$;
(2) Let $c_n=\log_3{a_{2n}}$, $b_n=\frac{1}{c_nc_{n+2}}$, and denote the sum of the first $n$ terms of the sequence ${b_n}$ as $T_n$. Find... | \frac{1}{3}-\frac{1}{4}\left(\frac{1}{2n+1}+\frac{1}{2n+3}\right) |
10. (20 points) Given the curve $C_{1}: \frac{x^{2}}{4}+y^{2}=1$, the curve $C_{2}: y=x^{2}-1$, and the point $M(0,-1)$. A line $l$ passing through the origin $O$ intersects $C_{2}$ at points $A, B$. The lines $M A, M B$ intersect $C_{1}$ at points $D, E$ $(D, E \neq M)$.
(1) Prove: $M D \perp M E$;
(2) Let the areas o... | \\frac{3}{2}x |
2. Consider the sequences $\left(x_{n}\right)_{n \geq 1}$ and $\left(y_{n}\right)_{n \geq 1}$ defined recursively by $x_{1}=4, y_{1}=1$,
$$
x_{n+1}=\frac{2 x_{n}+y_{n}}{3}, y_{n+1}=\frac{x_{n}+y_{n}}{2}, \forall n \in \mathbb{N}^{*}
$$
Prove that the given sequences are convergent and determine their limits. | \frac{14}{5} |
1. 10 runners start simultaneously: five in blue T-shirts from one end of the running track, and five in red T-shirts from the other. Their speeds are constant and different, with each runner's speed being greater than 9 km/h but less than 12 km/h. Upon reaching the end of the track, each runner immediately turns aroun... | 50 |
4. For any positive integers $n$ and $k(k \leqslant n)$, $f(n, k)$ represents the number of positive integers not exceeding $\left[\frac{n}{k}\right]$ (where $[x]$ denotes the greatest integer not exceeding the real number $x$) that are coprime with $n$. Then $f(100,3)=(\quad)$.
(A) 11
(B) 13
(C) 14
(D) 19 | C |
If the equation $x^{2}+ax-2=0$ has one root as $-1$, then the other root is ____. | 2 |
$\square$ Example 5 Given two positive integers $n \geqslant 2$ and $T \geqslant 2$, find all positive integers $a$, such that for any positive numbers $a_{1}, a_{2}, \cdots, a_{n}$, we have $\sum_{k=1}^{n} \frac{a k+\frac{a^{2}}{4}}{S_{k}}<T^{2} \sum_{k=1}^{n} \frac{1}{a_{k}}$, where $S_{k}$ $=a_{1}+a_{2}+\cdots+a_{k}... | 1,2,3, \cdots, 2(T-1) |
12.378. A regular truncated quadrilateral pyramid is inscribed in a sphere of radius $R$, with the larger base passing through the center of the sphere, and the lateral edge forming an angle $\beta$ with the plane of the base. Find the volume of the truncated pyramid. | \frac{2}{3}R^3\sin2\beta(1+\cos^22\beta-\cos2\beta) |
If vector $\overrightarrow{a}$ and vector $\overrightarrow{b}$ satisfy: $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=3$, and when $\lambda \in \mathbb{R}$, the minimum value of $|\overrightarrow{b}-\lambda \overrightarrow{a}|$ is $2\sqrt{2}$, then the projection of vector $\overrightarrow{a}+ \overrightarrow{b}$ in ... | C |
Dudeney, Amusements in Mathematics Problem 14 "This is queer, " said McCrank to his friend. "Twopence added to twopence is fourpence, and twopence multiplied by twopence is also fourpence." Of course, he was wrong in thinking you can multiply money by money. The multiplier must be regarded as an abstract number. It is ... | 1\frac{1}{2}3d |
Let the set $P=\{x|x^{2}- \sqrt {2}x\leqslant 0\}$, and $m=\sqrt{3}$, then among the following relations, the correct one is ( )
A: $m \not\subseteq P$
B: $m \notin P$
C: $m \in P$
D: $m \not\supseteq P$ | B |
Consider a hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$. Let $M$ be a point on the hyperbola $C$ and the center of a circle. This circle is tangent to the $x$-axis at one focus $F$ of the hyperbola and intersects the $y$-axis at points $P$ and $Q$. If $\triangle MPQ$ is an equilateral t... | B |
3. Regardless of the value of $m$, the vertex of the parabola
$$
y=x^{2}+2 m x+m^{2}+m-1
$$
always lies on a straight line. Then the function expression of this straight line is $\qquad$ . | y=-x-1 |
28, 10 identical oranges are placed into 3 different plates, allowing some plates to be empty. How many different ways are there to do this? $\qquad$ | 66 |
The sum of the coefficients of all terms without a constant term in the expansion of \\({{({{x}^{3}}-\dfrac{2}{\sqrt{x}})}^{7}}\\) is ______. | -449 |
Example 7. Find the angle $\Theta$ between the gradients of the functions
$$
u=\sqrt{x^{2}+y^{2}} \text { and } v=x+y+2 \sqrt{x y}
$$
at the point $M_{0}(1, 1)$. | \Theta=0 |
7. Let the ellipse $\frac{x^{2}}{m^{2}}+\frac{y^{2}}{n^{2}}=1$ pass through the fixed point $P(1,2)$. Then the minimum value of $m+n$ is . $\qquad$ | (1+2^{\frac{2}{3}})^{\frac{3}{2}} |
A die with uniform density has natural numbers $1$ to $6$ marked on its six faces. When thrown once, the probability that the number on the top face is even is ( ).
A: $\frac{1}{6}$
B: $\frac{1}{2}$
C: $\frac{1}{3}$
D: $\frac{1}{4}$ | B |
A space diagonal of a rectangular prism is $A A^{\prime}$. The plane through the endpoints $B, C, D$ of the edges starting from vertex $A$ divides the diagonal $A A^{\prime}$ in what ratio? | 1:2 |
3. Given $\tan \alpha+\tan \beta+\tan \gamma=\frac{17}{6}$, $\cot \alpha+\cot \beta+\cot \gamma=-\frac{4}{5}$, $\cot \alpha \cdot \cot \beta+\cot \beta \cdot \cot \gamma+\cot \gamma \cdot \cot \alpha=-\frac{17}{5}$. Then $\tan (\alpha+\beta+\gamma)=$ $\qquad$ | 11 |
Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\circ$, $60^\circ$, and $60.001^\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-... | \textbf{(E)}15 |
[ Transfer of side, diagonal, etc. ]
Find the height of the trapezoid, the lateral sides of which are 6 and 8, and the bases are 4 and 14.
# | 4.8 |
In the Cartesian coordinate system, if the terminal side of angle $\alpha$ passes through point P(1, -2), then $\sin 2\alpha =$ \_\_\_\_\_\_. | -\frac{4}{5} |
27. The number of five-digit numbers that are multiples of 4 and whose sum of digits is 40 is $\qquad$.
untranslated part: $\qwen$ (Note: This part is not translatable as it is a placeholder or variable in the original text.)
For the purpose of a complete translation, the final answer should look like this:
27. Th... | 20 |
5. (3 points) The unit digit of $2^{2012}$ is $\qquad$ . (where, $2^{n}$ represents $n$ twos multiplied together) | 6 |
Example 3 If $x, y$ are solutions to the system of equations
$$
\left\{\begin{array}{l}
1995 x+1997 y=5989 \\
1997 x+1995 y=5987
\end{array}\right. \text {, }
$$
then $\frac{x^{3} y^{2}}{x^{2}+y^{2}}=$ . $\qquad$ | \frac{4}{5} |
Given $$f(x)= \frac {x}{1+x}, x≥0$$, if $f_1(x)=f(x)$, $f_{n+1}(x)=f(f_n(x))$, where $n\in\mathbb{N}^+$, deduce the expression for $f_{2018}(x)$. | f_{2018}(x)= \frac {x}{1+2018x} |
The general term formula of a sequence is $a_n = 4n - 1$, then $a_6$ equals ( )
A: 21
B: 22
C: 23
D: 24 | C |
## Problem A3
Find all functions $\mathrm{f}$ on the positive integers with positive integer values such that (1) if $\mathrm{x}<\mathrm{y}$, then $f(x)<f(y)$, and $(2) f(y f(x))=x^{2} f(x y)$.
| f(x)=x^2 |
Given the complex number $z = -1 + i$, $\overline{z}$ is the conjugate of $z$. In the complex plane, the point corresponding to $\overline{z}$ is located in ( )
A. The first quadrant
B. The second quadrant
C. The third quadrant
D. The fourth quadrant | C |
If $p: x > 2$, and $q: x > 3$, then $p$ is a condition of $q$. (Choose the appropriate one from "sufficient but not necessary", "necessary but not sufficient", "necessary and sufficient", and "neither sufficient nor necessary" to fill in the blank) | \text{necessary but not sufficient} |
II. (Problem 16) The permutation $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$ of $1,2,3,4,5$ has the following property: for $1 \leqslant i \leqslant 4$, $a_{1}$, $a_{2} \cdots, a_{i}$ does not form a permutation of $1,2, \cdots, i$. Find the number of such permutations. | 71 |
Problem 1. Cut 2003 rectangles from an acute $\triangle A B C$ such that any of them has a side parallel to $A B$ and the sum of their areas is maximal. | \frac{2003}{2004}S |
Rotate the graph of the function $y= \sqrt{-x^{2}+4x}- \sqrt{3}$, where $x\in[1,3]$, counterclockwise around the origin by an angle $\theta$ (where $\theta$ is an acute angle). If the resulting curve is still the graph of a function, then the maximum value of $\theta$ is ____.
A: $\dfrac{\pi}{6}$
B: $\dfrac{\pi}{4}$... | C |
7. Let $P(x)=x^{4}+a x^{3}+b x^{2}+c x+d$, where
$a, b, c, d$ are real coefficients. Assume
$$
P(1)=7, P(2)=52, P(3)=97 \text {, }
$$
then $\frac{P(9)+P(-5)}{4}=$ $\qquad$ . (Vietnam) | 1202 |
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$
$\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 41 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\... | 50 |
For every $n$ the sum of n terms of an arithmetic progression is $2n + 3n^2$. The $r$th term is:
$\textbf{(A)}\ 3r^2 \qquad \textbf{(B) }\ 3r^2 + 2r \qquad \textbf{(C) }\ 6r - 1 \qquad \textbf{(D) }\ 5r + 5 \qquad \textbf{(E) }\ 6r+2\qquad$ | 6r-1 |
Calculate:$(1)\left(\pi -2\right)^{0}-|8|+(\frac{1}{3})^{-2}$;$(2)\left(2x\right)^{3}\cdot (-3xy^{2})\div (-2x^{2}y^{2})$;$(3)\left(4-x\right)^{2}-\left(x-2\right)\left(x+3)$;$(4)125^{2}-124\times 126$. (Please use simplified calculations) | 1 |
Find the value of $\frac{{\tan 20° + 4\sin 20°}}{{\tan 30°}}$. | 3 |
A certain company decided to recruit one advertising planner. The scores (in points) of a candidate in three qualities tests are as follows:
| Test Item | Innovation Ability | Comprehensive Knowledge | Language Expression |
|---------------|--------------------|-------------------------|---------------------|
| Te... | 83 |
“$x > 3$” is a condition for “$x^{2} > 9$” to be \\((\\) \\()\\)
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Both sufficient and necessary
D: Neither sufficient nor necessary | \text{A} |
Given that the real numbers x, y, z are in the interval [0, 4], and the sequence of x^2, y^2, z^2 is an arithmetic progression with a common difference of 2, find the minimum value of |x-y|+|y-z|. | 4 - 2\sqrt{3} |
75. $\int 2 \sin x d x$
Translation:
75. $\int 2 \sin x d x$ | -2\cosx+C |
2. The solution set of the equation about the real number $x$
$$
2^{1-2 \sin ^{2} x}=2+\log _{2}\left(1-\sin ^{2} x\right)
$$
is . $\qquad$ | \left\{x \mid x=k \pi\right. \text{ or } \left.\frac{k \pi}{2}+\frac{\pi}{4}, k \in \mathbf{Z}\right\} |
Given the function
$$
f(x)=\left\{\begin{array}{lc}
-x, & 0 \leqslant x \leqslant 1, \\
1+x, & -1 \leqslant x<0 .
\end{array}\right.
$$
Then its inverse function is $\qquad$ | f^{-1}(x)=\left\{\begin{array}{l} x-1, x \in[0,1), \\ -x, x \in[-1,0] \end{array}\right.} |
Let $a\_n (n=2,3,4,…)$ be the coefficient of the $x$ term in the expansion of $(3- \sqrt {x})^{n}$. Find the value of $\frac{\frac{a\_2}{3^{2}} + \frac{a\_3}{3^{3}} +…+ \frac{a\_{2015}}{3^{2015}}}{A\_{2016}^{3}}$. | \frac{1}{54} |
Evaluate the definite integral $\int_{-2}^{2}|x^2-2x|dx$. | 8 |
Given that the sequence $\{a_n\}$ is an arithmetic sequence, if $a_4 + 2a_6 + a_8 = 12$, then the sum of the first 11 terms of this sequence is. | 33 |
Among the following equations, the one that is a linear equation in one variable is ( )
A: $x-y+1=0$
B: $x^2-4x+4=0$
C: $\frac{1}{x} = 2$
D: $\pi x-2=0$ | D |
3. Let $a>b>c, a+b+c=1$, and $a^{2}+b^{2}+c^{2}=1$. Then ( )
(A) $a+b>1$
(B) $a+b=1$
(C) $a+b<1$
(D) Cannot be determined, depends on the specific values of $a, b$ | A |
Given that $x$ and $y$ are distinct nonzero real numbers such that $x+\tfrac{2}{x} = y + \tfrac{2}{y}$, what is $xy$?
$\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4\qquad$ | 2 |
2. Let the set $\left\{\left.\frac{3}{a}+b \right\rvert\, 1 \leqslant a \leqslant b \leqslant 2\right\}$ have the maximum element and the minimum element as $M, m$, respectively, then the value of $M-m$ is $\qquad$. | 5-2\sqrt{3} |
The eight points A, B,.., G and $H$ lie on five circles as shown in the figure. Each of these letters is replaced by one of the eight numbers 1, 2, ..., 7, and 8 such that the following conditions are met:
(i) Each of the eight numbers is used exactly once.
(ii) The sum of the numbers on each of the five circles is t... | 8 |
A company plans to invest in 3 different projects among 5 candidate cities around the Bohai Economic Rim, which are Dalian, Yingkou, Panjin, Jinzhou, and Huludao. The number of projects invested in the same city cannot exceed 2. How many different investment plans can the company have? (Answer with a number). | 120 |
4. We will call a number greater than 25 semi-prime if it is the sum of some two distinct prime numbers. What is the maximum number of consecutive natural numbers that can be semi-prime? | 5 |
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