problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
8. $\sum_{0 \leqslant i<j \leqslant 50} \mathrm{C}_{50}^{i} \mathrm{C}_{50}^{j}$ modulo 31 is | 16 |
7. Let positive integers $a, b, c, d, m, n$ satisfy: $a+b+c$ $+d=m^{2}, a^{2}+b^{2}+c^{2}+d^{2}=1989$, and the largest of $a, b$, $c, d$ is $n^{2}$. Find the values of $m, n$. | (,n)=(9,6) |
$a_1=-1$, $a_2=2$, and $a_n=\frac {a_{n-1}}{a_{n-2}}$ for $n\geq 3$. What is $a_{2006}$?
$
\textbf{(A)}\ -2
\qquad\textbf{(B)}\ -1
\qquad\textbf{(C)}\ -\frac 12
\qquad\textbf{(D)}\ \frac 12
\qquad\textbf{(E)}\ 2
$ | 2 |
2.060. $\left(\frac{3}{2 x-y}-\frac{2}{2 x+y}-\frac{1}{2 x-5 y}\right): \frac{y^{2}}{4 x^{2}-y^{2}}$. | \frac{24}{5y-2x} |
11.4. Find all pairs $(m, n)$ such that any pair $(x, y)$ satisfying the equation $\frac{x}{y}=m$, satisfies the equation $(x+y)^{2}=n$. | =-1,n=0 |
3. Given that $\triangle A B C$ is an equilateral triangle, the ellipse $\Gamma$ has one focus at $A$, and the other focus $F$ lies on the line segment $B C$. If the ellipse $\Gamma$ passes exactly through points $B$ and $C$, then its eccentricity is $\qquad$. | \frac{\sqrt{3}}{3} |
There is a "syllogism" reasoning for a differentiable function $f(x)$, with the major premise: If $f'(x_0) = 0$, then $x = x_0$ is an extremum point of the function $f(x)$; minor premise: Since the derivative of the function $f(x) = x^3$ at $x = 0$ is $f'(0) = 0$, conclusion: Therefore, $x = 0$ is an extremum point of ... | \text{major premise} |
A basketball player scored $12$, $a$, $8$, $15$, and $23$ points in the recent $5$ games, where $a > 0$. If the median of the points scored in these $5$ games is $12$, then the average score cannot be ($\quad\quad$).
A: $\frac{68}{5}$
B: $\frac{69}{5}$
C: $\frac{71}{5}$
D: $14$ | \text{C} |
A six-digit number is formed by repeating a three-digit number, such as 256256, or 678678, etc. This type of number cannot be divided by ( )
A: 11
B: 101
C: 13
D: 1001 | \text{B} |
Find the natural number $A$ such that there are $A$ integer solutions to $x+y\geq A$ where $0\leq x \leq 6$ and $0\leq y \leq 7$.
[i]Proposed by David Tang[/i] | 10 |
The organizing committee of an art festival arranges for five volunteers, Xiao Zhang, Xiao Zhao, Xiao Li, Xiao Luo, and Xiao Wang, to participate in service activities. Each person is engaged in one of four jobs: translator, tour guide, etiquette, or driver, and each job must have at least one person participating. If ... | \text{D} |
"x2 - 5x + 4 < 0" is a condition for "|x - 2| < 1" to be ( )
A: A sufficient but not necessary condition
B: A necessary but not sufficient condition
C: A necessary and sufficient condition
D: Neither sufficient nor necessary condition | \text{B} |
6. In the triangle $A B C, A C=2 B C, \angle C=90^{\circ}$ and $D$ is the foot of the altitude from $C$ onto $A B$. A circle with diameter $A D$ intersects the segment $A C$ at $E$. Find $A E: E C$. | 4 |
To promote mathematical culture and stimulate students' interest in learning, a certain school launched a mathematics subject month activity. The seventh grade conducted four projects: $A$. Reading mathematical classics; $B$. Telling mathematical stories; $C$. Making mathematical models; $D$. Challenging mathematical g... | \frac{1}{2} |
Find the equation of the line that passes through the point (-2, 1) and is parallel to the line $2x - 3y + 5 = 0$. | 2x - 3y + 7 = 0 |
Which is the positive integer that is a three-digit number in both the decimal and octal number systems, and the sum of its digits is fourteen in both cases? | 455 |
Given the complex number $$z= \frac {1+2i}{i}$$, where $i$ is the imaginary unit. The imaginary part of $z$ is ( )
A: i
B: -i
C: 1
D: -1 | \text{D} |
3. $A B C D E F$ - a regular hexagon, point O - its center. How many different isosceles triangles with vertices at the specified seven points can be constructed? Triangles that differ only in the order of vertices are considered the same (for example, AOB and BOA).
 = tan(2x), which of the following statements is incorrect?
A: The minimum positive period of y = f(x) is π
B: The minimum positive period of y = f(x) is π
C: y = f(x) is an odd function
D: The symmetry center of y = f(x) is $( \frac {kπ}{4}，0)(k∈Z)$ | \text{A} |
20.3. The collective farm placed a certain amount of money in the savings bank. If the number of hundreds is added to the number formed by the last two digits, then the result is the annual income from this amount, calculated at $2 \%$ per annum. What is the amount of the deposit placed in the savings bank?
$(7-8$ gra... | 4950 |
4. Bag $A$ contains two 10-yuan banknotes and three 1-yuan banknotes, and bag $B$ contains four 5-yuan banknotes and three 1-yuan banknotes. Now, two banknotes are randomly drawn from each of the two bags. Then the probability that the sum of the remaining banknotes in $A$ is greater than the sum of the remaining bankn... | \frac{9}{35} |
The complex number $\dfrac {3-i}{i}=$ ()
A: $1+3i$
B: $-1-3i$
C: $-1+3i$
D: $1-3i$ | B |
If the function $f(x) = x^2 + (m-1)x - 3$ is an even function, then $m =$ ( )
A: -2
B: -1
C: 1
D: 2 | C |
Given two propositions: \\
$p$: "The necessary and sufficient condition for event $A$ to be opposite to event $B$ is that event $A$ and event $B$ are mutually exclusive"; \\
$q$: "The graph of an even function is necessarily symmetric about the $y$-axis". Among the following propositions, the false one is $(\quad)$.
A... | B |
For which value of $m$, there is no triple of integer $(x,y,z)$ such that $3x^2+4y^2-5z^2=m$?
$ \textbf{(A)}\ 16
\qquad\textbf{(B)}\ 14
\qquad\textbf{(C)}\ 12
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ 8
$ | 10 |
2. Plot the graph of the function:
\[
y=\frac{x^{3}-x^{2}-x+1}{x^{2}-1}
\] | x-1 |
8. Given that the number of integer points (points with integer coordinates) on the closed region (including the boundary) enclosed by the circle $x^{2}+y^{2}=8$ is one-fifth of the number of integer points on the closed region (including the boundary) enclosed by the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{4}=1$, th... | 22\leq23 |
In Lhotě, they elected a mayor. Two citizens ran for the position: Ing. Schopný and his wife Dr. Schopná. There were three polling stations in the village. In the first and second polling stations, Dr. Schopná received more votes. In the first station, the ratio of votes was $7: 5$, and in the second station, it was $5... | 24:24:25 |
Problem 1. a) Determine the real numbers a, b, c knowing that $a^{2}+b^{2}+c^{2}=14$ and $2a+4b+6c=28$.
b) Let $t=(\sqrt{2}+\sqrt{3}) \cdot(a \sqrt{2}+b \sqrt{3})$, where a and b are integers. If $t$ is a rational number, calculate $x=a|b|+b|a|$. | =1,b=2,=3 |
The number of sets $A$ that satisfy $\{1\} \subseteq A \subseteq \{1,2,3,4,5\}$ is ( ).
A: $15$
B: $16$
C: $18$
D: $31$ | B |
Given $x^2 + xy = 3$, and $xy + y^2 = -2$, find the value of $2x^2 - xy - 3y^2$. | 12 |
The roots of the equation $x^{2}-2x = 0$ can be obtained graphically by finding the abscissas of the points of intersection of each of the following pairs of equations except the pair:
$\textbf{(A)}\ y = x^{2}, y = 2x\qquad\textbf{(B)}\ y = x^{2}-2x, y = 0\qquad\textbf{(C)}\ y = x, y = x-2\qquad\textbf{(D)}\ y = x^{2}... | \textbf{(C)} |
4. Let $n$ be a natural number, and $a_{n}=\sqrt[3]{n^{2}+2 n+1}$ $+\sqrt[3]{n^{2}-1}+\sqrt[3]{n^{2}-2 n+1}$. Then the value of $\frac{1}{a_{1}}+\frac{1}{a_{3}}+\frac{1}{a_{5}}$ $+\cdots+\frac{1}{a_{997}}+\frac{1}{a_{999}}$ is ( ).
(A) 3
(B) 4
(C) 5
(D) 6 | 5 |
For a positive integer $n$, let $\omega(n)$ denote the number of positive prime divisors of $n$. Find the smallest positive tinteger $k$ such that $2^{\omega(n)}\leq k\sqrt[4]{n}\forall n\in\mathbb{N}$. | 5 |
Task 5. (20 points) In the center of a circular field stands a geologists' cabin. From it, 8 straight roads extend, dividing the field into 8 equal sectors. Two geologists set out on a journey from their cabin at a speed of 4 km/h along a road each arbitrarily chooses. Determine the probability that the distance betwee... | 0.375 |
4. As shown in the figure, two national defense highways, each 40 m wide, intersect at an angle of $\alpha$. Then, the area of the common part (the shaded area in the figure) of the road surface at the intersection is ( ).
(A) $\frac{1600}{\sin \alpha}\left(\mathrm{m}^{2}\right)$
(B) $\frac{1600}{\cos \alpha}\left(\mat... | A |
14. (20 points) Given sets $A$ and $B$ are both sets of positive integers, and $|A|=20,|B|=16$. Set $A$ satisfies the following condition: if $a, b, m, n \in A$, and $a+b=$ $m+n$, then $\{a, b\}=\{m, n\}$. Define
$$
A+B=\{a+b \mid a \in A, b \in B\} \text {. }
$$
Determine the minimum value of $|A+B|$. | 200 |
18. (15 points) Let $\left\{a_{n}\right\}$ be a sequence of positive numbers, and its first $n$ terms sum $S_{n}$ satisfies $S_{n}=\frac{1}{4}\left(a_{n}-1\right)\left(a_{n}+3\right)$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) Let $b_{n}=\frac{1}{S_{n}}$, try to find the first $n$ te... | \frac{3}{4}-\frac{2 n+3}{2(n+1)(n+2)} |
If $(x+i)i = -1 + 2i$ where $x \in \mathbb{R}$, then $x = \boxed{2}$. | 2 |
16. Let $\alpha, \beta$ be real numbers. If for any real numbers $x, y, z$, we have $\alpha(x y+y z+z x) \leqslant M \leqslant \beta\left(x^{2}+y^{2}+z^{2}\right)$ always holds, where $M=\sqrt{x^{2}+x y+y^{2}} \cdot \sqrt{y^{2}+y z+z^{2}}+\sqrt{y^{2}+y z+z^{2}} \cdot \sqrt{z^{2}+z x+x^{2}}+\sqrt{z^{2}+z x+x^{2}} \cdot ... | 3 |
10. According to the pattern of the following equations, find the sum of the 2018th equation.
$$
2+3,3+7,4+11,5+15,6+19 \ldots
$$ | 10090 |
Given in $\triangle ABC$, the sides opposite to $A$, $B$, $C$ are respectively $a$, $b$, $c$. If $a= \sqrt{10}$, $c=3$, $\cos A= \frac{1}{4}$, then $b$ equals to ( )
A: $ \sqrt{2}$
B: $ \sqrt{3}$
C: $2$
D: $3$ | C |
In the polar coordinate system, the curve $C\_1$: $ρ=2\cos θ$, and the curve $C\_2$: $ρ\sin ^{2}θ=4\cos θ$. Establish a rectangular coordinate system $(xOy)$ with the pole as the coordinate origin and the polar axis as the positive semi-axis $x$. The parametric equation of the curve $C$ is $\begin{cases} x=2+ \frac {1}... | \frac {11}{3} |
$$
\begin{array}{l}
\text { 6. Given } x, y \in \mathbf{R}, \\
z_{1}=x+(y+2) \mathrm{i}, \\
z_{2}=(x-2)+y \mathrm{i}, \\
\left|z_{1}\right|+\left|z_{2}\right|=4 .
\end{array}
$$
Then the maximum value of $|x+y|$ is | 2\sqrt{2} |
## Task Condition
Approximately calculate using the differential.
$y=\sqrt[3]{x}, x=7.76$ | 1.98 |
1003. Find the total differentials of the functions:
$$
\text { 1) } z=5 x^{2} y^{3} ; \quad \text { 2) } z=\operatorname{arctg}\left(x^{2}+3 y\right)
$$ | =5xy^{2}(2y+3x)=\frac{1}{1+(x^{2}+3y)^{2}}(2x+3) |
Given a positive integer $n$, let real numbers $a_{1}$, $a_{2}, \cdots, a_{n}$ and positive real numbers $b_{1}, b_{2}, \cdots, b_{n}$ satisfy
$$
\begin{array}{l}
\sum_{i=1}^{n} a_{i} \geqslant 8 \text { and } \sum_{i=1}^{n} b_{i} \leqslant 4 . \\
\text { Find the minimum value of } \prod_{i=1}^{n} \mathrm{e}^{\frac{\l... | \mathrm{e}^{16} |
Let $n$ be a fixed positive integer. Find the maximum possible value of
$$ \sum_{1 \leqslant r<s \leqslant 2 n}(s-r-n) x_{r} x_{s}, $$
where $-1 \leqslant x_{i} \leqslant 1$ for all $i=1,2, \ldots, 2 n$. (Austria) Answer. $n(n-1)$. | n(n-1) |
4. Let $S=\left\{r_{1}, r_{2}, \cdots, r_{n}\right\} \subseteq\{1,2,3, \cdots, 50\}$, and any two numbers in $S$ cannot have a sum that is divisible by 7, then the maximum value of $n$ is $\qquad$ | 23 |
In triangle $\triangle ABC$, $\angle C=90^{\circ}$, $BC=3$, point $D$ is the trisection point of segment $AB$ closer to point $B$. Then $\overrightarrow{CB} \cdot \overrightarrow{CD}=$
A: $\frac{9}{4}$
B: $6$
C: $\frac{27}{4}$
D: $9$ | B |
$8 \cdot 15$ Let $a_{1}=3, b_{1}=100$, for $n \geqslant 1$
$$a_{n+1}=3^{a_{n}}, b_{n+1}=100^{b}{ }_{n}$$
Find the smallest positive integer $m$ such that $b_{m}>a_{100}$. | 99 |
Given a function $f(x) = dx^3 + cx^2 + bx + a$, satisfying $f'(x) = -3x^2 + 3$.
(1) Find the values of the real numbers $d$, $c$, and $b$;
(2) Find the extreme values of the function $f(x)$;
(3) For what value of the real number $a$ does the function $f(x)$ have only two intersections with the x-axis. | a = 2 \text{ or } a = -2 |
Example 4 Given a positive integer $n>1$. Let $a_{1}, a_{2}, \cdots, a_{n}$ be a permutation of $1,2, \cdots, n$. If $i<j$ and $a_{i}<a_{j}$, then $(a_{i}, a_{j})$ is called an ordered pair. $X$ is the number of ordered pairs in $a_{1}, a_{2}, \cdots, a_{n}$. Try to find $E(X)$.
(2020, University of Science and Technol... | \frac{n(n-1)}{4} |
On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?
$\textbf{(A) }50\qquad\textbf{(B) }70\qquad\textbf{(C) }... | 80 |
In the Cartesian coordinate system, the coordinates of point $P(-2, \sqrt{5})$ with respect to the $x$-axis are ______. | (-2, -\sqrt{5}) |
Simplify and then evaluate: $\left(1+\dfrac{1}{a-1}\right)\div \dfrac{2a}{a^{2}-1}$, where $a=-2$. | -\frac{1}{2} |
1. In the confectionery, there were either people with excess weight or sweet lovers (possibly both). It turned out that $80 \%$ of people with excess weight love sweets, and $70 \%$ of sweet lovers have excess weight. What fraction of those present love sweets but do not suffer from excess weight? | \frac{12}{47} |
Example 4 If real numbers $a, b, c$ satisfy $a+b+c=0$, $abc=2, c>0$, then ( ).
(A) $ab<0$
(B) $|a|+|b| \geqslant 2$
(C) $|a|+|b| \geqslant 4$
(D) $0<|a|+|b| \leqslant 1$ | B |
6) Let $\left\{a_{n}\right\}$ be a sequence with the sum of the first $n$ terms denoted as $S_{n}$. Let $T_{n}=\frac{S_{1}+S_{2}+\cdots+S_{n}}{n}$, which is called the "average" of the sequence $a_{1}, a_{2}, \cdots, a_{n}$. Given that the "average" of the sequence $a_{1}, a_{2}, \cdots, a_{1005}$ is 2012, what is the ... | 2009 |
For any real numbers $a$, $b$, $c$, consider the following propositions:
$(1)$ "$a=b$" is a necessary and sufficient condition for "$ac=bc$";
$(2)$ "$a+5$ is irrational" is a necessary and sufficient condition for "$a$ is irrational";
$(3)$ "$a > b$" is a sufficient condition for "$a^{2} > b^{2}$";
$(4)$ "$a < ... | B |
Example 13 Determine the largest real number $z$, such that $x+y+z=5, xy+yz+zx=3$, and $x, y$ are also real numbers.
(7th Canadian Intermediate Mathematics Competition) | \frac{13}{3} |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x \in [0, +\infty)$, $f(x) = x(2^{-x} + 1)$, determine the form of $f(x)$ when $x \in (-\infty, 0)$. | f(x) = x(2^x + 1) |
[Mathematical logic (miscellaneous).] [ Arithmetic. Mental arithmetic, etc. ]
Rabbits are sawing a log. They made 10 cuts. How many chunks did they get?
# | 11 |
Four people are sitting around a round table, with identical coins placed in front of each person. Everyone flips their coin simultaneously. If the coin lands heads up, the person stands up; if it lands tails up, the person remains seated. What is the probability that no two adjacent people stand up?
A: $\dfrac{1}{4}$... | \text{B} |
Problem 4. Solve the inequality
$$
\frac{7}{|x-1|-3} \geqslant|x+2|
$$ | [-\sqrt{7}-2;-2),(4;5] |
A student was using a calculator to find the approximate solution (accurate to 0.1) of the equation $\lg x = 2 - x$. They let $f(x) = \lg x + x - 2$, and found that $f(1) 0$. In the process, they used the "bisection method" to choose 4 more values of $x$, calculated the signs of their function values, and concluded th... | 1.5, 1.75, 1.875, 1.8125 |
Given the function $f(x)=|2x-1|-|x-a|$, where $a\leqslant 0$.
$(1)$ When $a=0$, find the solution set for the inequality $f(x) < 1$;
$(2)$ If the area of the triangle formed by the graph of $f(x)$ and the x-axis is greater than $\dfrac{3}{2}$, find the range of values for $a$. | (-\infty, -1) |
Given that the coordinates of the two foci of an ellipse are $(-2,0)$ and $(2,0)$, and it passes through the point $(2\sqrt{3},\sqrt{3})$, what is the standard equation of the ellipse? | \frac{x^{2}}{16}+\frac{y^{2}}{12}=1 |
Solve the following system of equations:
$$
\begin{aligned}
\left(x^{3}+y^{3}\right)\left(x^{2}+y^{2}\right) & =64 \\
x+y & =2
\end{aligned}
$$ | {\begin{pmatrix}x_{1}=1+\sqrt{\frac{5}{3}},\\y_{1}=1-\sqrt{\frac{5}{3}};\end{pmatrix}\quad |
Similar to a problem found in the ancient Chinese mathematical text "算法统宗", there is a question: "A journey of 441 li, the first day's brisk walking is not difficult, the distance walked decreases by half due to foot pain from the second day onwards, reaching the destination in six days. To find the distance walked on ... | B |
Rishi got the following marks on four math tests: $71,77,80$, and 87 . He will write one more math test. Each test is worth the same amount and all marks are between 0 and 100. Which of the following is a possible average for his five math tests?
(A) 88
(B) 62
(C) 82
(D) 84
(E) 86 | 82 |
G6.4 If $M=\left(10^{2}+10 \times 1+1^{2}\right)\left(10^{2}-1^{2}\right)\left(10^{2}-10 \times 1+1^{2}\right)$, find $M$. | 999999 |
Anna and Kati are celebrating their birthdays today. Three years from now, Anna will be four times as old as Kati was when Anna was two years older than Kati is now. How old is Anna if Kati is currently in high school? | 25 |
Given that $\sin(\pi-\alpha) = \frac{1}{3}$, find the value of $\cos\left( \frac{\pi}{2}+\alpha\right)$. | -\frac{1}{3} |
A right circular cone has for its base a circle having the same radius as a given sphere.
The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is:
$\textbf{(A)}\ \frac{1}{1}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{2}{3}\qquad\textbf{(D)}... | 2 |
[ Inscribed and Circumscribed Circles] Law of Sines $\quad]$
The radius of the circle circumscribed around triangle $K L M$ is $R$. Through vertex $L$, a line perpendicular to side $K M$ is drawn. This line is intersected at points $A$ and $B$ by the perpendicular bisectors of sides $K L$ and $L M$. It is known that $... | \frac{R^{2}}{} |
If $a-3b=-3$, then the value of the algebraic expression $5-a+3b$ is ( )
A: 0
B: 2
C: 5
D: 8 | D |
If $A=\{x|x^{2}-1 \lt 0\}$, $B=\{x|y=\ln x\}$, then $A\cup B=$
A: $\{x|-1 \lt x \lt 1\}$
B: $\{x|0 \lt x \lt 1\}$
C: $\{x|x \gt -1\}$
D: $\{x|x \gt 0\}$ | C |
Given that there exists a point symmetric about the $y$-axis on the graph of the function $f(x) = \ln x + 3x^{2}$ and the function $g(x) = 4x^{2} - ax$ where $a \neq 0$, the range of real number $a$ is ______. | (-\infty, -1] |
The function $f(x)$ is an even function, and it is increasing on $[0, +\infty)$. If $f(ax+1) \leq f(x-2)$ always holds on $[\frac{1}{2}, 1]$, then the range of the real number $a$ is \_\_\_\_\_\_. | [-2, 0] |
Given the universal set $U=\mathbb{R}$, set $A=\{x|x^2-3x-18\geq0\}$, and $B=\left\{x\middle| \frac{x+5}{x-14}\leq0\right\}$.
(1) Find $(\complement_{U}B) \cap A$.
(2) If set $C=\{x|2a<x<a+1\}$, and $B\cap C=C$, find the range of the real number $a$. | a \geq -\frac{5}{2} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $a=2$, $3b\sin C-5c\sin B\cos A=0$, find the maximum area of $\triangle ABC$. | 2 |
In the Mathematical Competition of HMS (Hellenic Mathematical Society) take part boys and girls who are divided into two groups : [i]Juniors[/i] and [i]seniors.[/i]The number of the boys taking part of this year competition is 55% of the number of all participants. The ratio of the number of... | \frac{11}{9} |
For which positive integer values of $n$ will $2^{n}+1$ be a power of a positive integer greater than one? | 3 |
4. As shown in Figure 1, in the cube $A B C D-A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, points $E$, $F$, and $G$ are moving points on segments $B B^{\prime}$, $A B$, and $A^{\prime} C$, respectively. The following four propositions are given:
(1) For any point $E$, there exists a point $F$ such that $D^{\prime} F \... | \frac{2}{3} |
Two positive integers $a_{1}, a_{2}, \cdots, a_{2006}$ (which can be the same) are such that $\frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \cdots, \frac{a_{2005}}{a_{2006}}$ are all distinct. How many different numbers are there at least among $a_{1}$, $a_{2}, \cdots, a_{2006}$?
（Chen Yonggao, problem contributor） | 46 |
If $y=x+\frac{1}{x}$, then $x^4+x^3-4x^2+x+1=0$ becomes:
$\textbf{(A)}\ x^2(y^2+y-2)=0 \qquad \textbf{(B)}\ x^2(y^2+y-3)=0\\ \textbf{(C)}\ x^2(y^2+y-4)=0 \qquad \textbf{(D)}\ x^2(y^2+y-6)=0\\ \textbf{(E)}\ \text{none of these}$ | D |
Given that no matter what positive real number $a$ is, the graph of $y=a^{x+2}-3$ always passes through a fixed point, then the coordinates of this fixed point are | (-2, -2) |
6. (30 points) The bases $AB$ and $CD$ of trapezoid $ABCD$ are 15 and 19, respectively. $AH$ and $BG$ are heights to the line $DC$, and $CF$ is a height to the line $AB$. Points $K, L, M$, and $N$ are the midpoints of segments $AB, CF, CD$, and $AH$ respectively. Find the ratio of the area of trapezoid $ABCD$ to the ar... | 2 |
Given a sequence $\{a_n\}$ with each term being a positive number, and satisfying $a_2=5$, $$a_{n+1}=a_{n}^2-2na_n+2 \quad (n \in \mathbb{N}^*).$$
(1) Conjecture the general formula for $\{a_n\}$.
(2) Let $b_n=2^{n-1}$ and $c_n=a_n+b_n$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_n$. | 2^n+n^2+2n-1 |
The smallest number in the set $\left\{\frac{1}{2}, \frac{2}{3}, \frac{1}{4}, \frac{5}{6}, \frac{7}{12}\right\}$ is
(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) $\frac{1}{4}$
(D) $\frac{5}{6}$
(E) $\frac{7}{12}$ | \frac{1}{4} |
Given the function $f(x) = \begin{cases} \sin \frac{\pi x}{6}, & 0\leq x\leq 2 \\ 2f(x-2), & x>2 \end{cases}$, then $f(2017)$ equals to ( )
A: 0
B: $\frac{1}{2}$
C: $2^{1007}$
D: $2^{1008}$ | C |
6. Given that the three sides of $\triangle A B C$ are $8, 12, 18$, and that $\triangle A_{1} B_{1} C_{1}$ also has one side of length 12, and is similar but not congruent to $\triangle A B C$. Then the number of such $\triangle A_{1} B_{1} C_{1}$ is ( ).
(A) 0
(B) 1
(C) 2
(D) 3 | C |
Let the function be $f(x)=x^{2}+ax-\ln x$ $(a\in\mathbb{R})$.
$(1)$ If $a=1$, find the intervals of monotonicity for the function $f(x)$;
$(2)$ Draw a tangent line from the origin $O$ to the curve $y=f(x)$, and prove that the abscissa of the tangent point is $1$. | 1 |
2. In a right-angled triangle, the lengths of the leg $a$ and the hypotenuse $c$ are expressed as two-digit integers, and both numbers use the same set of digits. The length of the leg $b$ is a positive rational number. What can the length of the hypotenuse be? (Provide all answers and prove that there are no others.) | 65 |
12.75 Find the integer solution $m$ of the equation $\left(1+\frac{1}{m}\right)^{m+1}=\left(1+\frac{1}{1988}\right)^{1988}$.
(22nd All-Soviet Union Mathematical Olympiad, 1988) | =1988or=-1989 |
Definition: If the derivative of the function $f(x)$ is $f''(x)$, and there exist $x_1$ and $x_2$ ($a < x_1 < x_2 < b$) in the interval $[a,b]$ such that $f''(x_1) = \frac{f(b) - f(a)}{b-a}$ and $f''(x_2) = \frac{f(b) - f(a)}{b-a}$, then $f(x)$ is called a "double mean value function" on the interval $[a,b]$. Given tha... | \text{D} |
Given vectors $$\overrightarrow {a}=(2x-1,x+3)$$, $$\overrightarrow {b}=(x,2x+1)$$, and $$\overrightarrow {c}=(1,2)$$, and $$( \overrightarrow {a}- \overrightarrow {b})\perp \overrightarrow {c}$$, find the value of the real number $x$. | 3 |
2. The sequence $a_{n}$ is defined recursively as
$$
\begin{aligned}
& a_{0}=-1 \\
& a_{n}=\frac{2 a_{n-1}-3}{3 a_{n-1}-4}, \quad n \in \mathbf{N}
\end{aligned}
$$
Determine the formula for $a_{n}$. | a_{n}=\frac{6n-1}{6n+1} |
Example 16 Han Xin Counts the Soldiers: There is a troop of soldiers. If they form a column of five, there is one person left over at the end. If they form a column of six, there are five people left over at the end. If they form a column of seven, there are four people left over at the end. If they form a column of el... | 2111 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.