problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
7・3 If $x$ is a real number satisfying $\sec x-\operatorname{tg} x=2$, then the value of $\sec x+\operatorname{tg} x$ is
(A)0.1.
(B) 0.2.
(C) 0.3.
(D) 0.4.
(E)0.5.
(50th American High School Mathematics Examination, 1999) | 0.5 |
Let \\(\alpha \in \{1,2,3, \frac{1}{2},-1\}\\), then the value of \\(\alpha\\) that makes \\(y=x^{\alpha}\\) an odd function and its graph above the line \\(y=x\\) in the interval \\((0,1)\\) is \_\_\_\_\_\_. | -1 |
Given the function $f(x)= \sqrt {3}\sin \dfrac {ωx+φ}{2}\cos \dfrac {ωx+φ}{2}+ \sin ^{2} \dfrac {ωx+φ}{2}(ω > 0,0 < φ < \dfrac {π}{2}).$ The distance between two adjacent symmetry centers of the graph is $\dfrac {π}{2}$, and the graph passes through the point $(\dfrac {π}{3},1)$.
(I) Find the analytical expression of ... | \dfrac{5}{12}π |
Given that there are 6 white balls and 4 yellow balls in an opaque bag (these balls are identical except for color). Person A draws one ball from the bag. If a white ball is drawn, the drawn white ball is returned to the bag, and then one yellow ball is taken out and replaced with a white ball. If a yellow ball is draw... | \frac{19}{125} |
[ Algebraic problems on the triangle inequality ]
In a triangle, two sides are equal to 3.14 and 0.67. Find the third side, given that its length is an integer.
# | 3 |
Huai Luo Geng was a great mathematician in China in the last century. Mathematical research achievements named after him include "Huai's Theorem," "Huai's Inequality," and "Huai's Method." In addition to theoretical mathematics research, he also vigorously promoted the "Optimization Method" and "Overall Planning Method... | B |
Given the complex number $z= \frac {2i}{1+i}$, the imaginary part of the conjugate of $z$ is ( )
A: -1
B: -i
C: 1
D: i | A |
If points $A(-1, y_{1})$, $B(2, y_{2})$, and $C(3, y_{3})$ lie on the graph of the inverse proportion function $y=-\frac{6}{x}$, then the relationship between $y_{1}$, $y_{2}$, and $y_{3}$ is ______. | y_{1} > y_{3} > y_{2} |
234*. Once in a room, there were several inhabitants of an island where only truth-tellers and liars live. Three of them said the following.
- There are no more than three of us here. All of us are liars.
- There are no more than four of us here. Not all of us are liars.
- There are five of us. Three of us are liars.
... | 4 |
5. Given the equation of $\odot O$ is $x^{2}+y^{2}=4$, and the equation of $\odot M$ is
$$
(x-5 \cos \theta)^{2}+(y-5 \sin \theta)^{2}=1(\theta \in \mathbf{R}),
$$
a line is drawn from any point $P$ on $\odot M$ to $\odot O$, forming two tangents $P E$ and $P F$, with the points of tangency being $E$ and $F$. Then the... | 6 |
11. Given that vectors $\alpha, \beta$ are two mutually perpendicular unit vectors in a plane, and
$$
(3 \boldsymbol{\alpha}-\boldsymbol{\gamma}) \cdot(4 \boldsymbol{\beta}-\boldsymbol{\gamma})=0 .
$$
Then the maximum value of $|\boldsymbol{\gamma}|$ is $\qquad$ | 5 |
Let's find all right-angled triangles whose sides are integers, and when 6 is added to the hypotenuse, we get the sum of the legs. | (7,24,25),(8,15,17),(9,12,15) |
During the torch relay event of the National Youth Games in 2015, there were 5 torchbearers with numbers 1, 2, 3, 4, and 5. If 2 torchbearers are randomly selected, the probability that the selected torchbearers have consecutive numbers is ( )
A: $ \dfrac{3}{10} $
B: $ \dfrac{5}{8} $
C: $ \dfrac{7}{10} $
D: ... | D |
37. There are two-digit numbers where the sum of the tens digit and the units digit is 8. There are $\qquad$ such two-digit numbers. | 8 |
Given that the even function $f(x)$ is monotonically decreasing on $[0,+\infty)$, determine the relationship between $f(1)$ and $f(-6)$. The options are:
A: $f(1) f(-6)$
C: $f(1)=f(-6)$
D: The relationship between $f(1)$ and $f(-6)$ cannot be determined | \text{B} |
18. Given that $a, b, c$ are non-zero rational numbers, and satisfy $a b^{2}=\frac{c}{a}-b$, then $\left(\frac{a^{2} b^{2}}{c^{2}}-\frac{2}{c}+\frac{1}{a^{2} b^{2}}+\frac{2 a b}{c^{2}}-\frac{2}{a b c}\right) \div\left(\frac{2}{a b}-\frac{2 a b}{c}\right) \div \frac{101}{c}=$ $\qquad$ | -\frac{1}{202} |
A [i]strip[/i] is the region between two parallel lines. Let $A$ and $B$ be two strips in a plane. The intersection of strips $A$ and $B$ is a parallelogram $P$. Let $A'$ be a rotation of $A$ in the plane by $60^\circ$. The intersection of strips $A'$ and $B$ is a parallelogram with the same area as $P$. Let $x^\c... | 150^\circ |
Given the functions $f(x)=a^{3x^{2}-3}$ and $g(x)=( \frac {1}{a})^{5x+5}$, where $a > 0$ and $a\neq 1$.
1. If $0 < a < 1$, find the set of values for $x$ that satisfy the inequality $f(x) < 1$.
2. Find the set of solutions for the inequality $f(x)\geqslant g(x)$ in terms of $x$. | [-1,- \frac {2}{3}] |
3.4. The angle at the base of an acute isosceles triangle \(ABC (AB = BC)\) is \(\alpha\). In what ratio, counting from vertex \(A\), does the height \(BD\) divide the height \(AE\) ? | -\frac{1}{\cos2\alpha} |
8. Reviews (from 7th grade. 1 point). Angry reviews about the work of an online store are left by $80 \%$ of dissatisfied customers (those who were poorly served in the store). Only $15 \%$ of satisfied customers leave positive reviews. A certain online store has received 60 angry and 20 positive reviews. Using this st... | 0.64 |
Given: For all \\(x \in \mathbb{R}^{+}\\), \\(a < x+ \frac {1}{x}\\) always holds, then the range of the real number \\(a\\) is \_\_\_\_\_\_. | a < 2 |
7. $n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns
$$
\begin{array}{l}
a_{11} a_{12} a_{13} a_{14} \cdots a_{1 n} \\
a_{21} a_{22} a_{23} a_{24} \cdots a_{2 n} \\
a_{31} a_{32} a_{33} a_{34} \cdots a_{3 n} \\
\cdots \cdots \cdots \cdots \cdots \\
a_{n 1} a_{n 2} a_{n 3} a_{n 4} \cdots a... | 2-\frac{1}{2^{n-1}}-\frac{n}{2^{n}} |
$\begin{array}{l} \text { Given } a>0, b>0, a+b=1, \\ \sqrt{2}<\sqrt{a+\frac{1}{2}}+\sqrt{b+\frac{1}{2}} \leqslant 2 .\end{array}$ | \sqrt{a+\frac{1}{2}}+\sqrt{b+\frac{1}{2}}>\frac{\sqrt{2}}{2}+\frac{\sqrt{6}}{2} |
2. A bag contains $x$ green and $y$ red sweets. A sweet is selected at random from the bag and its colour noted. It is then replaced into the bag together with 10 additional sweets of the same colour. A second sweet is next randomly drawn. Find the probability that the second sweet is red.
(A) $\frac{y+10}{x+y+10}$
(B)... | \frac{y}{x+y} |
Convert the parametric equation $\begin{cases} x= \frac{a}{2}\left( t+ \frac{1}{t} \right), \\ y= \frac{b}{2}\left( t- \frac{1}{t} \right) \end{cases}$ (where $t$ is the parameter) into a standard equation. | \frac{{x}^{2}}{{a}^{2}}- \frac{{y}^{2}}{{b}^{2}}=1 |
Given that point $A(2,-1)$ lies on the terminal side of angle $\theta$, then $\frac{sin\theta-cos\theta}{sin\theta+cos\theta}=$____. | -3 |
Given a point P on the curve $y=e^x$ and a point Q on the curve $y=\ln x$, the minimum value of $|PQ|$ is ( )
A: $\ln 2$
B: $\sqrt {2}$
C: $1+ \sqrt {2}$
D: $\sqrt {2} - 1$ | B |
Given $f(x)=\sin\left(2x-\frac{\pi}{6}\right)-1$, and a triangle $ABC$ with sides $a$, $b$, and $c=\sqrt{3}$ opposite to angles $A$, $B$, and $C$ respectively, and $f(C)=0$;
(I) If $\sin(B)=2\sin(A)$, find the values of $a$ and $b$;
(II) Find the maximum area $S$ of $\triangle ABC$. | S_{max}=\frac{3\sqrt{3}}{4} |
28 Three sides $O A B, O A C$ and $O B C$ of a tetrahedron $O A B C$ are right-angled triangles, i.e. $\angle A O B=\angle A O C=\angle B O C=90^{\circ}$. Given that $O A=7, O B=2$ and $O C=6$, find the value of
(Area of $\triangle O A B)^{2}+(\text { Area of } \triangle O A C)^{2}+(\text { Area of } \triangle O B C)^{... | 1052 |
Let P be a point inside (including the boundary) the diagonal face BDD1B1 of the cube ABCD-A1B1C1D1. If the distance from point P to the planes ABC, ABA1, and ADA1 are equal, then the point P that meets the condition is ( )
A: Only one
B: A finite number
C: An infinite number
D: Does not exist | \text{A} |
Given $|a|=2$, $b=-7$, and the opposite number of $c$ is $-5$, try to find the value of $a^2 + (-b) + (-c)$. | 6 |
In triangle $\triangle ABC$, point $M$ is a point on $AC$ such that $\overrightarrow{AM}=3\overrightarrow{MC}$. If $\overrightarrow{BM}=λ\overrightarrow{BA}+μ\overrightarrow{BC}$ ($λ,μ∈{R}$), then $\mu -\lambda =\left(\ \ \right)$
A: $-\frac{1}{3}$
B: $-\frac{1}{2}$
C: $\frac{1}{3}$
D: $\frac{1}{2}$ | D |
Given the function $f(x)=\ln x-mx+m$, where $(m∈R)$.
(1) Discuss the monotonicity of the function $f(x)$;
(2) If the function $f(x)\leqslant 0$ holds for any $x∈(0,+∞)$, find the range of values for the real number $m$. | m=1 |
4. Calculate $\lim _{x \rightarrow 1} \frac{\sqrt{\operatorname{arctg} x}-\sqrt{\arccos \left(\frac{\sqrt{2}}{2} x\right)}}{\sqrt[3]{1+\ln [x(2 x-1)]}-\sqrt[3]{1+\ln \left(x^{3}-4 x+4\right)}}$.
Iulian Danielescu, Brăila | \frac{9}{8\sqrt{\pi}} |
Example 9. Solve the inequality
$$
\frac{\lg ^{2} x-3 \lg x+3}{\lg x-1}<1
$$ | 0<x<10 |
\section*{Task 1 - V11021}
The industrial production of the Soviet Union increased by 10\% in 1960. In the following years, production is also expected to grow annually by approximately this magnitude.
Calculate how many times the industrial production of the USSR will increase:
a) in the next 10 years,
b) in the n... | 2.6,6.8,45.3 |
Given that $(1 + 2ai) \cdot i = 1 - bi$, where $a, b \in \mathbb{R}$, find the magnitude $|a + bi|$. | \frac{\sqrt{5}}{2} |
The following propositions:
\\(①\\) "If \\(x=1\\)," then it is a sufficient but not necessary condition for "\\({{x}^{2}}-3x+2=0\\)."
\\(②\\) The contrapositive of the proposition "If \\(x-3x+2=0\\), then \\(x=1\\)" is "If \\(x \neq 1\\), then \\(x-3x+2 \neq 0\\)."
\\(③\\) For the proposition \\(p: \exists x > 0 \\)... | C |
Given the function $f(x)=a(x^{2}-x)-\ln x$ $(a\in \mathbb{R})$.
- $(1)$ If $f(x)$ has an extremum at $x=1$, find the value of $a$;
- $(2)$ If $f(x)\geqslant 0$ holds for all $x$ in $[1,+\infty)$, find the range of values for $a$. | a\geqslant 1 |
2. Arrange the positive even numbers in 5 columns as shown in Table 1.
Table 1
\begin{tabular}{|l|c|c|c|c|c|}
\hline & Column 1 & Column 2 & Column 3 & Column 4 & Column 5 \\
\hline Row 1 & & 2 & 4 & 6 & 8 \\
\hline Row 2 & 16 & 14 & 12 & 10 & \\
\hline Row 3 & & 18 & 20 & 22 & 24 \\
\hline Row 4 & 32 & 30 & 28 & 26 & ... | D |
Example 2 $x$ is a positive real number, find the minimum value of the function $y=x^{2}+x+\frac{3}{x}$. | 5 |
A company has sales outlets in four regions: A, B, C, and D, with 150, 120, 180, and 150 outlets respectively. To investigate the sales situation of its products, the company needs to draw a sample of 100 outlets from these 600 outlets, which is referred to as investigation ①. In region C, there are 10 large sales outl... | \text{B} |
Given a sequence $\{a_n\}$ that satisfies: $a_1 = m$, where $m$ is a positive integer, and $a_{n+1} = \begin{cases} \frac{a_n}{2}, & \text{when } a_n \text{ is even} \\ 3a_n + 1, & \text{when } a_n \text{ is odd} \end{cases}$, if $a_6 = 1$, then all possible values of $m$ are ( )
A: $\{4，5\}$
B: $\{4，32\}$
C: $\{4，... | \text{C} |
In how many ways can $7$ red, $7$ white balls be distributed into $7$ boxes such that every box contains exactly $2$ balls?
$ \textbf{(A)}\ 163
\qquad\textbf{(B)}\ 393
\qquad\textbf{(C)}\ 858
\qquad\textbf{(D)}\ 1716
\qquad\textbf{(E)}\ \text{None}
$ | 393 |
Given two planar vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $\overrightarrow{a}=(1,2)$ and $|\overrightarrow{b}|=\sqrt{10}$, with $|\overrightarrow{a}+\overrightarrow{b}|=5$, then the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is ( )
A: $\dfrac{\pi}{4}$
B: $\dfrac{\pi}... | A |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $\dfrac{S_3}{S_6} = \dfrac{1}{3}$, then $\dfrac{S_6}{S_{12}}=$ ()
A: $\dfrac{3}{10}$
B: $\dfrac{1}{3}$
C: $\dfrac{1}{8}$
D: $\dfrac{1}{9}$ | A |
Given a positive integer $m \in \mathbb{N}^*$, let $x$ be a positive number defined as $x = \frac{a_1}{2} + \frac{a_2}{2^2} + \dots + \frac{a_m}{2^m}$, where $|a_i| = 1$ for $1 \leq i \leq m$, $i \in \mathbb{N}^*$. Let $X_m$ denote the sum of all such numbers $x$.
1. Find $X_3$ and $X_4$.
2. Find $X_m$. | 2^{m-2} |
The solution to the equation $4^x - 2^x - 6 = 0$ is. | \log_2{3} |
1. (5 points) Find the value of the function $f(x)$ at the point $x_{0}=6000$, if $f(0)=1$ and for any $x$ the equality $f(x+3)=f(x)+2 x+3$ holds. | 12000001 |
In the arithmetic sequence $\{a_{n}\}$, if $a_{6}$ and $a_{7}$ are two roots of the equation $x^{2}+2x-3=0$, then the sum of the first $12$ terms of $\{a_{n}\}$ is ( )
A: $12$
B: $18$
C: $-18$
D: $-12$ | \text{D} |
## Task A-3.3.
Determine all real numbers $x$ for which there exists a real number $y$ such that
$$
\frac{\sin (2 x)}{\sin (2 y)}=4 \sin (x+y) \cos (x-y)
$$ | x\in\bigcup_{k\in\mathbb{Z}}\langlek\pi,\frac{(2k+1)\pi}{2}\rangle |
334. A machine manufactures balls. A ball is considered suitable if the deviation $X$ of the ball's diameter from the design size in absolute value is less than 0.7 mm. Assuming that the random variable $X$ is normally distributed with a standard deviation $\sigma=0.4$ mm, find how many suitable balls on average will b... | 92 |
(8 points) Use the Horner's method to calculate the value of $f(x) = 2x^4 + 3x^3 + 5x - 4$ at $x = 2$. | 62 |
Example 4 Find all pairs of positive integers $(m, n)$ such that $m^{6}=n^{n+1}+n-1$. [6]
(2013, Turkey National Team Selection Exam) | =n=1 |
1. Given two groups of numerical sequences, each consisting of 15 arithmetic progressions containing 10 terms each. The first terms of the progressions in the first group are $1,2,3, \ldots, 15$, and their differences are respectively $2,4,6, \ldots, 30$. The second group of progressions has the same first terms $1,2,3... | \frac{160}{151} |
1. Find the sum of all positive integers whose largest proper divisor is 55 . (A proper divisor of $n$ is divisor that is strictly less than $n$.)
Proposed by: Michael Tang | 550 |
Find all integers $n\ge1$ such that $2^n-1$ has exactly $n$ positive integer divisors.
[i]Proposed by Ankan Bhattacharya [/i] | n = 1, 2, 4, 6, 8, 16, 32 |
Given a complex number $z=\dfrac{5+3i}{1-i}$, which of the following statements is correct?
A: The imaginary part of $z$ is $4i$
B: The conjugate of $z$ is $1-4i$
C: $|z|=5$
D: The point corresponding to $z$ in the complex plane is in the second quadrant | B |
Galperin G.A.
Solve the equation in natural numbers: $x+\frac{1}{y+\frac{1}{z}}=\frac{10}{7}$. | 1,2,3 |
1. A natural number has a digit sum of 2013. The next number has a smaller digit sum and is not divisible by 4. What is the digit sum of the next natural number. | 2005 |
If the simplest quadratic radicals $\sqrt{3a-8}$ and $\sqrt{17-2a}$ can be combined, find the range of values for $x$ that makes $\sqrt{4a-2x}$ meaningful. | x \leq 10 |
4. Use $1,2,3,4,5,6$ to form a six-digit number without repeating digits. By arbitrarily extracting any two adjacent digits, 5 different two-digit numbers can be obtained. The maximum sum of these 5 two-digit numbers is $\qquad$ . | 219 |
8. (6 points) Given that $\frac{x}{9}, \frac{y}{15}, \frac{z}{14}$ are all simplest proper fractions, and their product is $\frac{1}{6}$, then $x+y+z=$ $\qquad$ | 21 |
For any real numbers $a$, $b$, $c$, and $d$, define the operation "$\triangle$" between the ordered pairs of real numbers $(a, b)$ and $(c, d)$ as: $(a, b) \triangle (c, d) = (ac + bd, ad + bc)$. If for any real numbers $u$, $v$, it holds that $(u, v) \triangle (x, y) = (u, v)$, then the pair $(x, y)$ is
A: $(0, 1)$ ... | \text{B} |
Given the set $\{1, a, \frac{b}{2}\} = \{0, a^2, a+b\}$, find the value of $a^{2013} + b^{2014}$. | -1 |
3B. Determine all pairs of integers $(x, y)$ that satisfy the equation
$$
x y + 3 y = x^2 + 6 x + 12
$$ | (-4,-4),(-2,4),(0,4),(-6,-4) |
Ten birds land on a $10$-meter-long wire, each at a random point chosen uniformly along the wire. (That is, if we pick out any $x$-meter portion of the wire, there is an $\tfrac{x}{10}$ probability that a given bird will land there.) What is the probability that every bird sits more than one meter away from its closest... | \frac{1}{10^{10}} |
1. For what smallest natural $k$ does the number 2016 $\cdot$ 20162016 $\cdot$ 201620162016 $\cdot$... 20162016...2016( $k$ factors) divide without remainder by $3^{67}$? | 27 |
In how many ways can two different squares be chosen from an $8 \times 8$ chessboard such that the midpoint of the segment connecting their centers also coincides with the center of a square? | 480 |
$\mathbf{5 . 2} 020$ balls are placed into 2020 different boxes, with each ball having an equal probability of landing in any box, and the probabilities of different balls landing in the boxes are independent of each other. The expected number of empty boxes is $\qquad$ | \frac{2019^{2020}}{2020^{2019}} |
Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if nece... | \frac{1}{2013} |
19.1.9 * Find all integers $n$ such that $n^{4}+6 n^{3}+11 n^{2}+3 n+31$ is a perfect square. | 10 |
26. As shown in the figure, the side length ratio of squares $E F G H$ and $A B C D$ is 1:3, and $E F / / A B, M, N$ are the midpoints of $C F, D H$ respectively. If $M N=\sqrt{17}$, then the area of square $A B C D$ is | 36 |
In the future, each country in the world produces its Olympic athletes via cloning and strict training
programs. Therefore, in the fi nals of the 200 m free, there are two indistinguishable athletes from each
of the four countries. How many ways are there to arrange them into eight lanes? | 2520 |
99. There are 6 identical-looking coins, but 4 are genuine, of the same weight, while 2 are counterfeit, lighter, and also weigh the same. What is the minimum number of weighings on a balance scale without weights that are needed to find both counterfeit coins? | 3 |
9. Arrange the positive integers (excluding 0) that are neither perfect squares nor perfect cubes in ascending order, resulting in 2, $3,5,6,7,10, \cdots \cdots$, what is the 1000th number in this sequence? | 1039 |
Calculate: $({2+\sqrt{3}})^0+3\tan30°-|\sqrt{3}-2|+(\frac{1}{2})^{-1}$. | 1+2\sqrt{3} |
$36 \cdot 15$ In a certain office, six people $A, B, C, D, E, F$ take turns working the night shift. The order of the shifts is $A \rightarrow B \rightarrow C \rightarrow D \rightarrow E \rightarrow F \rightarrow A \rightarrow B \cdots$. In the first week, Sunday is covered by $A$, Monday by $B$, $\cdots$, Friday by $F... | B |
[Trigonometric Identities.]
From the condition $\operatorname{tg} \phi=1 / \cos \alpha \cos \beta+\operatorname{tg} \alpha \operatorname{tg} \beta$ derive that $\cos 2 \phi \leq 0$.
# | \cos2\phi\leq0 |
11. (16 points) As shown in Figure 1, the area of $\triangle ABC$ is 1. Through a point $O$ inside $\triangle ABC$, draw three lines parallel to the sides of the triangle: $DE$, $FG$, and $HI$. Points $D$, $E$, $F$, $G$, $H$, and $I$ are all on the sides of $\triangle ABC$. Find the minimum area of the hexagon $DGHFEI$... | \frac{2}{3} |
6. The general term formula of the sequence $\left\{a_{n}\right\}$ is $a_{n}=\tan n \cdot \tan (n-1)$, and for any positive integer $n$, there is $a_{1}+a_{2}+\cdots+a_{n}=A \tan n+B n$ holds, then $A=$ $\qquad$ ,$B=$ $\qquad$ | A=\frac{1}{\tan1},B=-1 |
8. Given $\theta=\arctan \frac{5}{12}$. Then, the principal value of the argument of the complex number $z=\frac{\cos 2 \theta+\mathrm{i} \sin 2 \theta}{239+\mathrm{i}}$ is $\qquad$ | \frac{\pi}{4} |
3. In triangle $\triangle A B C$, the base $A B$ of which lies on the x-axis, altitudes $A M, B N$ and $C K$ are drawn. Find the length of the base $A B$, if the coordinates of points $M(2,2)$ and $N(4,4)$ are known. | 4\sqrt{5} |
23. (5 points)
There are some natural numbers, like $121$ and $2442$, which read the same from left to right as from right to left. We call such numbers palindromic numbers. The difference between the smallest four-digit palindromic number and the smallest three-digit palindromic number is | 900 |
6. (8 points) By expanding the expression $(1+\sqrt{7})^{211}$ using the binomial theorem, we obtain terms of the form $C_{211}^{k}(\sqrt{7})^{k}$. Find the value of $k$ for which such a term attains its maximum value. | 153 |
The article in the fifth issue of our journal in 1983, titled "A Simple Method for Compiling Radical Equations," states that the equation
$$
\sqrt{5 \mathrm{x}-1}+\sqrt{2 \mathrm{x}}=3 \mathrm{x}-1
$$
"will produce a quartic equation after squaring twice, which may be quite troublesome to solve." In fact, this equation... | x=2 |
07.1. Find one solution in positive integers to the equation
$$
x^{2}-2 x-2007 y^{2}=0
$$
| (225,5) |
39. Find all three-digit numbers $x$, in the notation of which the digits do not repeat, such that the difference between this number and the number written with the same digits but in reverse order is also a three-digit number consisting of the same digits as the number $x$. | 954459 |
Given the sequence {an}, the sum of its first n terms is Sn, and an + an+2 = 2an+1. Also, a1 = 13 and a2 = 11. When Sn reaches its maximum value, n equals ( ).
A: 5
B: 6
C: 7
D: 8 | \text{C} |
Compute: $-1^{2018} - 6\tan30^\circ + \left(\frac{1}{2}\right)^{-2} + |1 - \sqrt{3}|$. | 2 - \sqrt{3} |
1. If $M=\left\{(x, y)|| \operatorname{tg} \pi y \mid+\sin ^{2} \pi x=0\right\}$,
$$
N=\left\{(x, y)\left|x^{2}+y^{2}\right| \leqslant 2\right\},
$$
then the number of elements in $M \cap N$ is
(A) 4 ;
(B) 5 ;
(C) 8 ;
(D) 9 . | 9 |
A car travels from place A to place B, first driving on a regular road and then on a highway. It is known that the car travels at a speed of 60km/h on the regular road and 100km/h on the highway. The distance traveled on the highway is twice that of the regular road, and the total travel time from A to B is 2.2 hours. ... | 1 |
Example 3. Find $\lim _{x \rightarrow 4} \frac{x^{2}-6 x+8}{x-4}$. | 2 |
8. Evaluate $\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\frac{1}{99}+\frac{1}{143}+\frac{1}{195}$. | \frac{7}{15} |
If $x > 0$, $y > 0$, and $\frac{2}{x} + \frac{8}{y} = 1$, find the product $xy$ and the minimum value of $x+y$, and when are these minimum values attained? | 18 |
The price of a taxi ride is $\mathrm{R} \$ 2.50$ fixed (the "flag-down fare") plus $R \$ 0.10$ per 100 meters traveled. I have only $R \$ 10.00$ in my pocket. Therefore, I have enough money for a ride of, at most, how many kilometers?
(a) 2.5
(b) 5.0
(c) 7.5
(d) 10.0
(e) 12.5 | 7.5 |
## Task 1 - V10721
All countries of the socialist camp together produced:
| Product | Pre-war year | 1959 |
| :--- | :--- | :--- |
| Electric power | 84.7 billion kWh | 418.2 billion kWh |
| Steel | 25.4 million t | 92.7 million t |
| Cement | 14.5 million t | 73.3 million t |
By how many percent did production incr... | 393.7,265,405.5 |
Given the polar equation of curve $C_1$ is $\rho^2=\frac {2}{3+\cos2\theta}$, establish a rectangular coordinate system with the pole O as the origin and the polar axis as the positive direction of the x-axis. After stretching all the x-coordinates of points on curve $C_1$ to twice their original values and shortening ... | \frac{13\sqrt{2}}{4} |
Given a quadratic function $f(x)$ with a minimum value of $1$, and $f(0)=f(2)=3$,
$(1)$ Find the explicit expression for $f(x)$;
$(2)$ If $f(x)$ is not monotonic on the interval $[2a, a+1]$, determine the range of values for the real number $a$;
$(3)$ On the interval $[-1, 1]$, the graph of $y=f(x)$ is always above tha... | m < -1 |
11. (3 points) Among the 100 natural numbers from 1 to 100, the sum of all numbers that cannot be divided by 6 is | 4234 |
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