problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Jiřka constructed two congruent equilateral triangles as shown in the figure. She further wants to construct all circles that will have their center at one of the vertices and will pass through another vertex of one of the triangles.
Construct and calculate all circles that meet Jiřka's requirements.
(K. Pazourek)
!... | 6 |
G9.3 If the lines $2 y+x+3=0$ and $3 y+c x+2=0$ are perpendicular, find the value of $c$. | -6 |
$\square$ Example 7 Let $x, y, z$ be positive numbers, and $x^{2}+y^{2}+z^{2}=1$, find the minimum value of $\frac{x^{5}}{y^{2}+z^{2}-y z}+$ $\frac{y^{5}}{z^{2}+x^{2}-z x}+\frac{z^{5}}{x^{2}+y^{2}-x y}$. (2006 Turkish Mathematical Olympiad Problem) | \frac{\sqrt{3}}{3} |
Given two points A(1, 2) and B(4, -2), the distances from these points to line l are 1 and 4 respectively. The number of lines l that satisfy these conditions is ( ).
A: 1 line
B: 2 lines
C: 3 lines
D: 4 lines | \text{C} |
7. (10 points) On the way from home to school, Xiaolong passes a store and a playground. The distance from home to the store is 500 meters, which took 7 minutes; from the store to the playground, he walks at a speed of 80 meters/minute for 8 minutes; the distance from the playground to school is 300 meters, and he walk... | 72 |
In the plane, $n$ lines have been drawn in such a way that no two lines are parallel and no three lines are concurrent. How many regions are delimited by these lines? | u_{n}=\frac{n(n+1)}{2}+1 |
Find the value of: $\tan 55^{\circ}+\tan 65^{\circ}-\sqrt{3}\tan 55^{\circ}\tan 65^{\circ}=$____. | -\sqrt{3} |
75. Archaeologists have determined using remote sensing and geophysical exploration methods that the underground palace of the Mausoleum of the First Qin Emperor is 35 meters deep, 170 meters long from east to west, and 145 meters wide from north to south, forming a rectangular shape. If the earth and stone from above ... | 2747.61 |
Example 8. Arrange $n^2$ distinct real numbers into the following table:
$$
\begin{array}{l}
a_{11}, a_{12}, \cdots, a_{1 n}, \\
a_{21}, a_{22}, \cdots, a_{2 n}, \\
\ldots \ldots \ldots \ldots \ldots \ldots \ldots \\
a_{1_{1}}, a_{a_{2}}, \cdots, a_{n} .
\end{array}
$$
Take the maximum number from each row to get $n$ ... | x \geqslant y |
In the geometric sequence $\{a_n\}$, all terms are positive, and $a_1a_5=4$, $a_4=1$. Then, the common ratio $q$ of $\{a_n\}$ is ( ).
A: $2$
B: $\dfrac{1}{2}$
C: $±\dfrac{1}{2}$
D: $±2$ | B |
$31 \cdot 23$ If $x=9, y=-4$ is a set of integer solutions to the binary quadratic equation $2 x^{2}+5 x y+3 y^{2}=30$, the number of different integer solutions to this equation is
(A) 2 sets.
(B) 6 sets.
(C) 12 sets.
(D) 16 sets.
(11th "Hope Cup" National Mathematics Invitational Competition, 1990) | 16 |
1. Six natural numbers are written on the board, such that for any two $a$ and $b$ among them (where $b>a$), $\log _{a} b$ is an integer. What is the smallest value that the maximum of these numbers can take? The answer can be written in the form of a power of a number: $m^{n}$ is denoted as $\mathrm{m}^{\wedge} \mathr... | 4294967296 |
4. Given that line segment $A D / /$ plane $\alpha$, and the distance to plane $\alpha$ is 8, point $B$ is a moving point on plane $\alpha$, and satisfies $A B=10$. If $A D=21$, then the minimum distance between point $D$ and point $B$ is $\qquad$ . | 17 |
The imaginary part of the complex number $z=1-2i$ (where $i$ is the imaginary unit) is \_\_\_\_\_\_. | -2 |
Given that vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are both non-zero, and $(\overrightarrow{a} - 2\overrightarrow{b}) \perp \overrightarrow{a}, (\overrightarrow{b} - 2\overrightarrow{a}) \perp \overrightarrow{b}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} |
8.4. In the country, there are 15 cities, some of which are connected by roads. Each city is assigned a number equal to the number of roads leading out of it. It turned out that there are no roads between cities with the same number. What is the maximum number of roads that can be in the country? | 85 |
If "$x^{2}-2x-3 > 0$" is a necessary but not sufficient condition for "$x < a$", then the maximum value of $a$ is ______. | -1 |
If $a < 0$ and $-1 < b < 0$, which of the following inequalities is correct? $(\quad)$
A: $a < ab^{2} < ab$
B: $ab^{2} < a < ab$
C: $a < ab < ab^{2}$
D: $ab^{2} < ab < a$ | \text{A} |
2 Given that $\sqrt{2 x+y}+\sqrt{x^{2}-9}=0$, find the value(s) of $y-x$.
(A) -9
(B) -6
(C) -9 or 9
(D) -3 or 3
(E) None of the above | -9or9 |
If $P = \{x | x \leq 3\}$, then ( )
A: $-1 \subseteq P$
B: $\{-1\} \in P$
C: $\emptyset \in P$
D: $\{-1\} \subseteq P$ | \text{D} |
Read the material and answer the following questions: When $m$ and $n$ are real numbers and satisfy $m-n=6$, the point $E(m-1,3n+1)$ is called a "friendly point". For example, the point $E(3,1)$, let $\left\{\begin{array}{l}m-1=3\\ 3n+1=1\end{array}\right.$, we get $\left\{\begin{array}{l}m=4\\ n=0\end{array}\right.$, ... | 10 |
In the book "Nine Chapters on the Mathematical Art," there is a mathematical problem stated as follows: During the Qingming Festival outing to the park, a total of eight boats were taken. Each large boat holds six people, each small boat holds four people. There were thirty-eight students in total, and all the boats we... | A |
In the Cartesian coordinate system, there is a circle \\(C\\) with its center on the line \\(x-2y=0\\) and passing through point \\(A(4,0)\\), but not through the origin. The chord formed by the intersection of the line \\(4x-3y=0\\) and circle \\(C\\) has a length of \\(4\\).
\\((1)\\) Find the general equation of cir... | 2x-5y+3=0 |
If the three interior angles A, B, and C of a triangle satisfy $A > 3B$ and $C < 2B$, then this triangle is ( )
A: Obtuse triangle
B: Right triangle
C: Acute triangle but not equilateral
D: Equilateral triangle | A |
Given that $\alpha$ is an acute angle, $\overrightarrow{a} = \left(\frac{3}{4}, \sin\alpha\right)$, $\overrightarrow{b} = \left(\cos\alpha, \frac{1}{\sqrt{3}}\right)$, and $\overrightarrow{a} \parallel \overrightarrow{b}$, then $\alpha$ is ( ).
A: $15^{\circ}$
B: $30^{\circ}$
C: $30^{\circ}$ or $60^{\circ}$
D: $15^... | C |
2. To get to the Solovyov family's cottage, you first need to walk 3 km along the highway and then 2 km along a path. Upon arriving at the station, Mom called her son Vasya at the cottage and asked him to meet her on a bicycle. They started moving towards each other at the same time. Mom walks at a constant speed of 4 ... | 800 |
Given that vector $\overrightarrow{a}$ and vector $\overrightarrow{b}$ are unit vectors, and $ \langle \overrightarrow{a},\overrightarrow{b} \rangle = \frac{\pi}{3}$, $\overrightarrow{c}=\overrightarrow{a}-\overrightarrow{b}$, then the projection vector of vector $\overrightarrow{c}$ onto vector $\overrightarrow{a}$ is... | A |
Write down an irrational number that is greater than $2$. | \sqrt{5} |
4. There are 20 cards each inscribed with the numbers $1,2, \cdots, 19,20$. They are placed in a box, and 4 people each draw one card. The two people who draw the smaller numbers are in one group, and the two people who draw the larger numbers are in another group. If two of them draw 5 and 14, the probability that the... | D |
5. Find all pairs of integers $(x ; y)$ that satisfy the equation
$$
x^{2}+x y+y^{2}=x+20
$$ | (1;-5),(5;-5),(-4;0),(5;0),(-4;4),(1;4) |
24 (1152). Sequences $\left(x_{n}\right)$ and ( $y_{n}$ ) are defined by the formulas $x_{n}=2 n-1$ and $y_{n}=n^{2}$. If all common terms of these sequences are written in ascending order, the resulting sequence is $\left(c_{n}\right)$. Write the formula for the $n$-th term of the sequence $\left(c_{n}\right)$. | (2n-1)^{2} |
Given the function $f(x)=|x-4|-|x+2|$.$(1)$ If $f(x)-a^{2}+5a\geqslant 0$ always holds, find the range of values for $a$;$(2)$ If the maximum value of $f(x)$ is $M$, and positive real numbers $a$, $b$, $c$ satisfy $a+b+c=M$, find the maximum value of $\sqrt{a+1}+\sqrt{b+2}+\sqrt{c+3}$. | 6 |
1. Calculate:
$$
\frac{1}{\sqrt{2}-1}-\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}-1}-\frac{1}{\sqrt{3}+1}
$$ | 3 |
Find all non-zero natural numbers $a$ and $b$ such that
$$
a^{b}=b^{a^{2}}
$$ | (1,1),(2,16),(3,27) |
In an isosceles triangle, the length of the base is $6$, and the length of the legs is one of the roots of the quadratic equation $x^{2}-7x+12=0$. What is the perimeter of this triangle?
A: $12$
B: $13$
C: $14$
D: $12$ or $14$ | C |
3. Two baby squirrels are located at points $A$ and $B$, and start jumping simultaneously along lines $A O$ and $B O$ towards point $O$ (after passing point $O$, each continues moving along its own line). The distance $A O=120$ meters, $B O=80$ meters, and the angle $A O B=60^{\circ}$. The baby squirrels have a constan... | 20\sqrt{3} |
The number of integer values of $x$ for which $\frac{3}{x+1}$ is an integer is
(A) 4
(B) 3
(C) 5
(D) 1
(E) 6 | 4 |
Alice and Bob play the following game. They alternate selecting distinct nonzero digits (from $1$ to $9$) until they have chosen seven such digits, and then consider the resulting seven-digit number by concatenating the digits in the order selected, with the seventh digit appearing last (i.e. $\overline{A_1B_2A_3B_4A_6... | \text{Alice} |
Given a sample of size $8$ with a mean of $5$ and a variance of $2$, a new data point $5$ is added to the sample, increasing the sample size to $9$. Let $\overline {x}$ and $s^{2}$ be the new mean and variance, respectively. Then,
A: $\overline {x}=5$, $s^{2} 2$
C: $\overline {x} > 5$, $s^{2} 5$, $s^{2} > 2$ | A |
For how many positive integers $m$ does there exist at least one positive integer n such that $m \cdot n \le m + n$?
$\textbf{(A) } 4\qquad \textbf{(B) } 6\qquad \textbf{(C) } 9\qquad \textbf{(D) } 12\qquad \textbf{(E) } \text{infinitely many}$ | \textbf{(E)} |
5. Let $x$ be a natural number. Solve the equation
$$
\frac{x-1}{x}+\frac{x-2}{x}+\frac{x-3}{x} \cdots+\frac{1}{x}=3
$$ | 7 |
1. At a concert, there are 20 singers who will perform. For each singer, there is a set of other singers (possibly an empty set) such that he wishes to perform later than all the singers in this set. Question: Is there a way to have exactly 2,010 ways to order the singers so that all their wishes are satisfied? | 2010 |
8. Place 10 numbers on a given circle such that their total sum is 200, and the sum of any three consecutive numbers is not less than 58. Then the maximum value of the largest number among all sets of 10 numbers that satisfy the above requirements is $\qquad$ | 26 |
16. Team A and Team B each send out 7 players to compete in a Go broadcast tournament, following a predetermined order. The competition starts with the No. 1 players from both teams. The loser is eliminated, and the winner then faces the No. 2 player from the losing team, $\cdots$, until all players from one team are e... | 210 |
2. Solve the system of equations $\left\{\begin{array}{l}2 y-x-2 x y=-1, \\ 4 x^{2} y^{2}+x^{2}+4 y^{2}-4 x y=61 \text {. }\end{array}\right.$ | (-6;-\frac{1}{2}),(1;3),(1;-\frac{5}{2}),(5;-\frac{1}{2}) |
Example 1. Given $\angle x o y=$ $60^{\circ}, M$ is a point inside $\angle x o y$, the distances from $M$ to the two sides are 2 and 11, respectively. Find the length of $O M$. (1978, Shanxuan Mathematics Competition) | 14 |
Alex has $\$ 2.65$. He has only dimes (worth $\$ 0.10$ each) and quarters (worth $\$ 0.25$ each). He has more dimes than quarters. What is the smallest number of coins that Alex could have?
(A) 25
(B) 16
(C) 13
(D) 19
(E) 22 | 16 |
If real numbers $x$ and $y$ satisfy $2^{x}+2^{y+1}=1$, $m=x+y$, $n={(\frac{1}{2})^x}+{(\frac{1}{2})^{y-1}}$, then which of the following is true?
A: $x \lt 0$ and $y \lt -1$
B: The maximum value of $m$ is $-3$
C: The minimum value of $n$ is $7$
D: $n\cdot 2^{m} \lt 2$ | ABD |
2. $\frac{1}{2} n(n+1)$ distinct numbers are randomly arranged in a triangle:
Let $M_{\mathrm{k}}$ be the maximum number in the $k$-th row (counting from the top), find the probability that $M_{1}<M_{2}<M_{3}<\cdots<M_{\mathrm{n}}$ holds. | \frac{2^{n}}{(n+1)!} |
How many four-digit numbers, without repeating digits, that can be formed using the digits 0, 1, 2, 3, 4, 5, are divisible by 25? | 21 |
2. $A B$ is a chord of $\odot O$, $P$ is a point outside $\odot O$, $P B$ is tangent to $\odot O$ at $B$, $P A$ intersects $\odot O$ at $C$, and $A C = B C$, $P D \perp A B$ at $D$, $E$ is the midpoint of $A B$, $D E = 1000$. Then $P B=$ | 2000 |
Example 2. Investigate the function for maximum and minimum
$$
f(x)=\frac{1}{3} x^{3}-\frac{1}{2} x^{2}-6 x+2 \frac{2}{3}
$$ | -10\frac{5}{6} |
2. $\frac{\sqrt{2} \cos 55^{\circ}-\sin 20^{\circ}}{\sqrt{2} \cos 5^{\circ}+\sin 20^{\circ}}=$ | 2-\sqrt{3} |
Example 2 Let real numbers $s, t$ satisfy
$$
\begin{array}{l}
19 s^{2}+99 s+1=0, \\
t^{2}+99 t+19=0(s t \neq 1) . \\
\text { Find the value of } \frac{s t+4 s+1}{t} \text { . }
\end{array}
$$
(1999, National Junior High School Mathematics Competition) | -5 |
On International Labor Day, the school committee organized a speech competition with the theme "I work, I am happy." Five students, A, B, C, etc., from a certain class participated and the order of appearance was determined by drawing lots. Under the premise that student A must appear before student B, the probability ... | B |
Find all triplets $ (x,y,z) $ of real numbers such that
\[ 2x^3 + 1 = 3zx \]\[ 2y^3 + 1 = 3xy \]\[ 2z^3 + 1 = 3yz \] | (1, 1, 1) |
Given a geometric sequence $\{a_n\}$ with the first term being 2 and the common ratio being 3, and the sum of the first $n$ terms being $S_n$. If $\log_{3}\left[\frac{1}{2}a_n\cdot(S_{4m}+1)\right]=9$, then the minimum value of $\frac{1}{n}+ \frac{4}{m}$ is \_\_\_\_\_. | 2.5 |
In $\triangle ABC$, it is known that $\sin (A+B)=2\sin A\cos B$. Then, $\triangle ABC$ must be
A: An isosceles right triangle
B: A right triangle
C: An isosceles triangle
D: An equilateral triangle | C |
[ Parallelism of lines and planes $]$ [ Systems of segments, lines, and circles ]
In space, there are $n$ segments, no three of which are parallel to the same plane. For any two segments, the line connecting their midpoints is perpendicular to both segments. What is the largest $n$ for which this is possible? | 2 |
Auxiliary similar triangles Pythagorean theorem (direct and inverse).
In triangle $ABC$, altitudes $AA_1$ and $BB_1$ are drawn. Find $AC$, if
a) $AA_1=4, BB_1=5, BC=6$
b) $A_1C=8, B_1C=5, BB_1=12$. | 4.8 |
Let's determine all natural numbers for which $1!+2!+\ldots+n!$ is a perfect square (where $n!$ denotes the product of the first $n$ natural numbers). | n=1n=3 |
Given that point $P(x, y)$ moves within a region defined by inequalities on the plane, the range of values for $z = x - y$ is
A: $[-2, -1]$
B: $[-2, 1]$
C: $[-1, 2]$
D: $[1, 2]$ | \text{C} |
If we walk along the fence from north to south, the distances between its posts are initially the same. From a certain post, the distance decreases to 2.9 meters and remains so until the southern end of the fence. Between the 1st and 16th posts (counted from the north), the distances do not change, and the distance bet... | 20 |
7. Highway (from 7th grade, 3 points). A highway running from west to east intersects with $n$ equal roads, numbered from 1 to $n$ in order. Cars travel on these roads from south to north and from north to south. The probability that a car will approach the highway from each of these roads is $\frac{1}{n}$. Similarly, ... | \frac{2kn-2k^{2}+2k-1}{n^{2}} |
In the rectangular coordinate system $O-xyz$, a normal vector of the plane $OAB$ is $\overrightarrow{n}=(2,-2,1)$. Given point $P(-1,3,2)$, the distance $d$ from point $P$ to the plane $OAB$ is equal to _____. | 2 |
922. Find the fraction with the smallest possible denominator that lies between $\frac{7}{25}$ and $\frac{8}{25}$. List all solutions. | \frac{2}{7} |
Given $f(x) = x$, if the graph of $f(x)$ is symmetric about the line $x = 1$ and the corresponding function is $g(x)$, then the expression for $g(x)$ is $\boxed{3x - 2}$. | 3x - 2 |
Problem 11.1. The sequence $\left\{x_{n}\right\}_{n=1}^{\infty}$ is defined as:
$$
x_{1}=3, \quad x_{n+1}=x_{n}^{2}-3 x_{n}+4, \quad n=1,2,3, \ldots
$$
(a) Prove that $\left\{x_{n}\right\}_{n=1}^{\infty}$ is monotone increasing and unbounded.
(b) Prove that the sequence $\left\{y_{n}\right\}_{n=1}^{\infty}$ defined... | 1 |
A school arranges for 5 classes to go to 4 factories for social practice, with each class going to one factory and each factory hosting at least one class. The number of different arrangements is
A: 180
B: 720
C: 240
D: 60 | C |
2 - 49 Find the smallest natural number $n$ that satisfies the following properties:
(1) The unit digit of $n$ is 6;
(2) If the unit digit 6 of $n$ is moved to the front of the other digits, the resulting new number is 4 times $n$. | 153846 |
5・146 Find integer coefficient polynomials $P(x)$ and $Q(x)$, such that
$$\frac{P(\sqrt{2}+\sqrt{3}+\sqrt{5})}{Q(\sqrt{2}+\sqrt{3}+\sqrt{5})}=\sqrt{3}+\sqrt{2}$$ | P(x)=3 x^{4}-20 x^{2}+24, Q(x)=4 x^{3} |
77. Simplify the expression
$$
\frac{3 a^{2}+3 a b+3 b^{2}}{4 a+4 b} \cdot \frac{2 a^{2}-2 b^{2}}{9 a^{3}-9 b^{3}}
$$ | \frac{1}{6} |
3. [5] What is the largest $x$ such that $x^{2}$ divides $24 \cdot 35 \cdot 46 \cdot 57$ ? | 12 |
Let \\(f(x) = a - \frac{3}{2^x + 1} (x \in \mathbb{R})\\) be an odd function. Then
A: \\(a = \frac{3}{2}\\), and \\(f(x)\\) is an increasing function
B: \\(a = -1\\), and \\(f(x)\\) is an increasing function
C: \\(a = \frac{3}{2}\\), and \\(f(x)\\) is a decreasing function
D: \\(a = -1\\), and \\(f(x)\\) is a de... | \text{A} |
10. One hundred rectangles are arranged edge-to-edge in a continuation of the pattern shown. Each rectangle measures $3 \mathrm{~cm}$ by $1 \mathrm{~cm}$. What is the perimeter, in $\mathrm{cm}$, of the completed shape?
A 800
B 700
С 602
D 600
E 502 | 602 |
Given a sequence \\(\{a_n\}\) where \\(a_1=1\\) and \\(a_{n+1}=2a_n+4(n\in \mathbb{N}^*)\\), find the general formula for \\(a_n=\\) . | 5\times2^{n-1}-4 |
6.200. $\left\{\begin{array}{l}(x+y)^{2}+2 x=35-2 y, \\ (x-y)^{2}-2 y=3-2 x\end{array}\right.$ | (-5,-2),(3,2),(-3,-4),(1,4) |
\section*{Problem 2}
Find the smallest positive integer which can be represented as \(36^{\mathrm{m}}-5^{\mathrm{n}}\).
\section*{Answer}
11
| 11 |
13. A rectangular cuboid (including a cube), all of its edges are integer centimeters, the total length of all edges is 36 centimeters. There are $\qquad$ types of such cuboids. | 7 |
In triangle $\triangle ABC$, where the sides opposite to the angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, given that $\frac{a}{b} = \frac{b + \sqrt{3}c}{a}$ and $\sin C = 2\sqrt{3}\sin B$, find the value of $\tan A$. | \tan A = \frac{\sqrt{3}}{3} |
5. [4] Call a positive integer $n$ weird if $n$ does not divide $(n-2)$ !. Determine the number of weird numbers between 2 and 100 inclusive. | 26 |
7. Let the set $M=\left\{(x, y) \left\lvert\, \frac{1}{\sqrt{x}}-\frac{1}{\sqrt{y}}=\frac{1}{\sqrt{45}}\right., x, y \in \mathbf{N}^{*}\right\}$, then the number of elements in the set $M$ is
A. 0
B. 1
C. 2
D. 3 | 1 |
7. Let $[x]$ denote the greatest integer not exceeding the real number $x$, and $\{x\}=x-[x]$. Then
$$
\left\{\frac{2015!}{2017}\right\}=
$$
$\qquad$ | \frac{1}{2017} |
If $10 \leq x \leq 20$ and $40 \leq y \leq 60$, the largest value of $\frac{x^{2}}{2 y}$ is
(A) 5
(B) $\frac{5}{6}$
(C) $\frac{10}{3}$
(D) $\frac{5}{4}$
(E) 10 | 5 |
7.157. It is known that $\log _{a} x=\alpha, \log _{b} x=p, \log _{c} x=\gamma, \log _{d} x=\delta$ and $x \neq 1$. Find $\log _{a b c d} x$. | \frac{\alpha\beta\gamma\delta}{\beta\gamma\delta+\alpha\gamma\delta+\alpha\beta\delta+\alpha\beta\gamma} |
Four, As shown in the figure, points $M, N$ are on the sides $B C$, $C D$ of square $A B C D$, respectively. It is known that the perimeter of $\triangle M C N$ is equal to half the perimeter of square $A B C D$. Find $\angle M A N$ | 45^{\circ} |
If $b>1$, $\sin x>0$, $\cos x>0$, and $\log_b \sin x = a$, then $\log_b \cos x$ equals
$\text{(A)} \ 2\log_b(1-b^{a/2}) ~~\text{(B)} \ \sqrt{1-a^2} ~~\text{(C)} \ b^{a^2} ~~\text{(D)} \ \frac 12 \log_b(1-b^{2a}) ~~\text{(E)} \ \text{none of these}$ | (D)\\frac{1}{2}\log_b(1-b^{2a}) |
Trailer $\underline{\text { P. }}$.
The numbers from 1 to 50 are written on the board. It is allowed to erase any two numbers and write down one number, which is the absolute value of their difference. After repeating this procedure 49 times, one number will remain on the board. What can this number be? | 1,3,5,\ldots,49 |
85. There is a magical five-digit number, which can be divided by $1, 3, 5, 7, 9, 11, 13, 15$ at the same time, but cannot be divided by any of the numbers 2, 4, 6, 8, 10, 12, 14, 16. Then, this five-digit number is . $\qquad$ | 45045 |
Example 1. As shown in the figure, in $\triangle A B C$, $A C=B C, \angle C$ $=20^{\circ}$. Also, $M$ is on $A C$, $N$ is on $B C$, and it satisfies $\angle B A N=50^{\circ}, \angle A B M=60^{\circ}$. Then $\angle N M B=$ $\qquad$ . (1991, Qinfen Cup Mathematics Invitational Competition) | 30^{\circ} |
Is there a 6-digit number that becomes 6 times larger when its last three digits are moved to the beginning of the number, keeping the order of the digits? | 142857 |
In space, "line $a \not\subset$ plane $\alpha$" is a ______ condition for "line $a \parallel$ plane $\alpha$". (Fill in the blank with "sufficient but not necessary", "sufficient and necessary", or "necessary but not sufficient") | \text{necessary but not sufficient} |
Use the Horner's method to find the value of $v_2$ when the polynomial $f(x) = x^6 + 6x^4 + 9x^2 + 208$ is evaluated at $x = -4$.
A: $-4$
B: $1$
C: $17$
D: $22$ | D |
Solve the following equation:
$$
(x+y)^{2}=(x+1)(y-1)
$$ | -1,1 |
28. [14] Johnny the grad student is typing all the integers from 1 to $\infty$, in order. The 2 on his computer is broken however, so he just skips any number with a 2. What's the 2008th number he types? | 3781 |
Example 18 (1993 National High School League Question) Real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}=5$. If $s=$ $x^{2}+y^{2}$, then $\frac{1}{s_{\max }}+\frac{1}{s_{\text {min }}}=$ | \frac{8}{5} |
If the fractional equation about $x$ $\dfrac{x}{x+1}-\dfrac{m+1}{x^{2}+x}=\dfrac{x+1}{x}$ has increasing roots, then the value of $m$ is ( ).
A: $-1$ or $-2$
B: $-1$ or $2$
C: $1$ or $2$
D: $0$ or $-2$ | \text{D} |
Define a function $f(x)$ on $\mathbb{R}$ that is an odd function and also a periodic function with a period of 2. Then, calculate $f(1) + f(2) + f(3)$. | 0 |
The equation of the function obtained by shifting the graph of the function $y=5^x$ 3 units to the right and then 2 units downward is ( )
A: $y=5^{x+3}-2$
B: $y=5^{x-3}+2$
C: $y=5^{x-3}-2$
D: $y=5^{x+3}+2$ | C |
Given that the function $y=f\left(x\right)$ is differentiable, and $\lim_{{Δx\to0}}\frac{{f({1+Δx})-f(1)}}{{Δx}}=1$, then the slope of the tangent line at the point $\left(1,f\left(1\right)\right)$ on the curve $y=f\left(x\right)$ is ( ).
A: $45^{\circ}$
B: $60^{\circ}$
C: $120^{\circ}$
D: $135^{\circ}$ | A |
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