problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
The maximum and minimum values of the function $f(x)=x- \sqrt {2}\sin x$ on the interval $[0,\pi]$ are respectively $(\quad)$
A: $\pi$，$0$
B: $ \dfrac {\pi}{2}- \sqrt {2}\;$,$0$
C: $\pi\;$, $ \dfrac {\pi}{4}-1$
D: $0\;$,$\; \dfrac {\pi}{4}-1$ | C |
The initial terms of the sequence $\{a_n\}$ are 1, 3, 5, 7, 9, 11, 13,... In the sequence $\{b_n\}$, $b_1=a_1$, $b_2=a_2$, $b_8=a_8$,... What is the value of $b_{20}$? | 39 |
Given the function f(x) = x² - 5x + 2lnx, determine the interval where the function is monotonically decreasing.
A: $(0, \frac{1}{2})$ and $(1, +\infty)$
B: $(0, 1)$ and $(2, +\infty)$
C: $(0, \frac{1}{2})$ and $(2, +\infty)$
D: $(\frac{1}{2}, 2)$ | \text{D} |
5. Andrej and Blaž drew a circle on the playground and marked diametrically opposite points on it. They each stood on one of these points and started walking in opposite directions at a uniform speed. By the time they first met, Andrej had walked 60 m. From the time they first met until they met again, Blaž had walked ... | 210\mathrm{~} |
1. The natural number $n$ when divided by 2 has a remainder $a$, when divided by 4 has a remainder $b$, and when divided by 6 has a remainder $c$. If $a+b+c=9$, determine the remainder when the number $n$ is divided by 12. | 11 |
7. Find the maximum value of the positive real number $M$ such that for all $x, y, z \in \mathbf{R}$, the inequality $x^{4}+y^{4}+z^{4}+$ $x y z(x+y+z) \geqslant M(x y+y z+z x)^{2}$ holds. (2004 Greek Mathematical Olympiad) | \frac{2}{3} |
The point corresponding to the complex number $z$ is in the second quadrant, its modulus is 3, and its real part is $-\sqrt{5}$. Then $\overline{z}$ is ( )
A: $-\sqrt{5}+2i$
B: $-\sqrt{5}-2i$
C: $\sqrt{5}+2i$
D: $\sqrt{5}-2i$ | \text{B} |
If $m \gt n$, which of the following inequalities is correct?
A: $m-2 \lt n-2$
B: $\frac{m}{4} \gt \frac{n}{4}$
C: $6m \lt 6n$
D: $-8m \gt -8n$ | B |
Among the following propositions, the incorrect one is\\((\\) \\()\\)
A: If \\(f(x)\\) is a continuous odd function, then \\( \int_{-α}^{α}f(x)dx=0\\)
B: If \\(f(x)\\) is a continuous even function, then \\( \int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx\\)
C: If \\(f(x)\\) is continuous and always positive on \\([a,b]\\), t... | D |
Given the line $x-y+2=0$ and the curve $y=\ln x+a$ are tangent, then the value of $a$ is __________. | 3 |
5. Let the complex number $\alpha=a+b i(a, b$ be real numbers, and $b>0) \frac{\alpha^{2}}{1+\alpha}$ and $\frac{\alpha}{1+\alpha^{2}}$ are both real numbers, then $\alpha=$ | -\frac{1}{2}+\frac{\sqrt{3}}{2} i |
4. When $x-y=1$, the value of $x^{4}-x y^{3}-x^{3} y-3 x^{2} y$ $+3 x y^{2}+y^{4}$ is ( ).
(A) -1
(B) 0
(C) 1
(D) 2 | C |
A regular tetrahedron $SABC$ of volume $V$ is given. The midpoints $D$ and $E$ are taken on $SA$ and $SB$ respectively and the point $F$ is taken on the edge $SC$ such that $SF: FC = 1: 3$. Find the volume of the pentahedron $FDEABC$. | \frac{15}{16}V |
Example 4 Given that $S$ is a subset of the set $\{1,2, \cdots, n\}$, $S$ contains no number that divides another, and no two numbers are coprime. Find the maximum value of $S$. | \left[\frac{n+2}{4}\right] |
Given that $\{a_n\}$ is an arithmetic sequence, and $a_2+a_4+a_{15}$ is a certain constant, then among the following options, which one is also a constant?
A: $a_5$
B: $a_6$
C: $a_7$
D: $a_8$ | \text{C} |
Let $x_1,x_2,y_1,y_2$ be real numbers satisfying the equations $x^2_1+5x^2_2=10$, $x_2y_1-x_1y_2=5$, and $x_1y_1+5x_2y_2=\sqrt{105}$. Find the value of $y_1^2+5y_2^2$ | 23 |
37. In $\angle ABC$, $AB=6, BC=5, CA=4$. If $\kappa$ is on $BC$ and $\frac{B K}{K C}=\frac{3}{2}$, then $A K=(\quad$.
(A) $2 \sqrt{3}$
(B) 4
(C) $3 \sqrt{2}$
(D) $2 \sqrt{\frac{21}{5}}$
(E) * | C |
Given $x+y=1$, $y>0$, $x>0$, then the minimum value of $\frac{1}{{2x}}+\frac{1}{y}$ is ( ).
A: $\sqrt{2}$
B: $2+\sqrt{2}$
C: $2\sqrt{2}$
D: $\frac{3}{2}+\sqrt{2}$ | D |
If $x < 0$ and $0 < y < 1$, then the order from smallest to largest of $\frac{y^2}{x}$, $\frac{y}{x}$, $\frac{1}{x}$ is __________. | \frac{1}{x} < \frac{y}{x} < \frac{y^2}{x} |
14. On the $x y$-plane, let $S$ denote the region consisting of all points $(x, y)$ for which
$$
\left|x+\frac{1}{2} y\right| \leq 10, \quad|x| \leq 10 \text { and } \quad|y| \leq 10 .
$$
The largest circle centred at $(0,0)$ that can be fitted in the region $S$ has area $k \pi$. Find the value of $k$. | 80 |
## Task 2 - 340512
Xaver and Yvette report: Each of us has thought of a natural number. We have shared these numbers with each other.
Xaver says: The successor of my number is divisible by the successor of Yvette's number.
Yvette says: The sum of the successor of my number and the successor of Xaver's number is an o... | 4\cdot9 |
The value of $\frac{1}{2}+\left(\frac{1}{2} \times \frac{1}{2}\right)$ is
(A) $\frac{3}{8}$
(B) 1
(C) $\frac{1}{6}$
(D) $\frac{1}{4}$
(E) $\frac{3}{4}$ | \frac{3}{4} |
## Problem Statement
Find the derivative.
$$
y=\ln \sqrt[3]{\frac{x-1}{x+1}}-\frac{1}{2}\left(\frac{1}{2}+\frac{1}{x^{2}-1}\right) \operatorname{arctan} x
$$ | \frac{5x^{2}+17}{12(x^{4}-1)}+\frac{x\cdot\operatorname{arctg}x}{(x^{2}-1)^{2}} |
Given the set $A=\{1, 2, a\}$ and the set $B=\{1, 7\}$. If $B \subseteq A$, then the value of the real number $a$ is ___. | 7 |
If $\sin \left( \frac {\pi}{2}+α \right)= \frac {3}{5}$, then $\cos 2α=$ ______. | - \frac {7}{25} |
Given a four-digit number where the digits are distinct non-negative integers and their sum is 12, find the number of such four-digit numbers. | 174 |
3. Let $f=uv\left(u, v \in \mathbf{R}_{+}\right)$. It is known that for all positive triples $(a, b, c)$ satisfying $a+b+c=1$, the inequality
$$
u(ab+bc+ca)+vabc \leqslant \frac{7}{3}
$$
always holds. Then the maximum value of $f$ is . $\qquad$ | \frac{441}{4} |
Example 1.7.7. Let $a, b, c$ be non-negative real numbers. Prove that
$$\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}+\sqrt{\frac{27(ab+bc+ca)}{a^2+b^2+c^2}} \geq \frac{7 \sqrt{2}}{2} .$$ | \frac{7 \sqrt{2}}{2} |
The eccentricity of the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ $(a > b > 0)$ is $e=\dfrac{\sqrt{5}-1}{2}$. If $A$ is the left vertex, $F$ is the right focus, and $B$ is one endpoint of the minor axis, then the angle $\angle ABF$ equals $(\quad)$.
A: $30^{\circ}$
B: $45^{\circ}$
C: $90^{\circ}$
D: $120^{\circ}$ | C |
49. A waterworks' reservoir contains 400 tons of water. The waterworks can inject 60 tons of water into the reservoir per hour, while the reservoir continuously supplies water to residential areas, with the total water supply in $t$ hours being $120 \sqrt{6 t}$ tons $(0 \leqslant t \leqslant 24)$.
(1) From the start of... | 8 |
Example 6 From the 100 positive integers $1,2,3, \cdots, 100$, if $n$ numbers are taken, among these $n$ numbers, there are always 4 numbers that are pairwise coprime. Find the minimum value of $n$.
| 75 |
The largest number by which the expression $n^3 - n$ is divisible for all possible integral values of $n$, is:
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$ | 6 |
13. (6 points) Mathematician Wiener is the founder of cybernetics. At the ceremony where he was granted his Ph.D. from Harvard University, someone, curious about his youthful appearance, asked him about his age. Wiener's response was quite interesting. He said, “The cube of my age is a four-digit number, and the fourth... | 18 |
1. (2 points) Anya and Kolya were collecting apples. It turned out that Anya collected as many apples as the percentage of the total apples collected by Kolya, and vice versa, Kolya collected as many apples as the percentage of the total apples collected by Anya. How many apples did Anya collect and how many did Kolya ... | 50 |
## SUBJECT $2(7 p=4 p+3 p)$
a) Determine the minimum value of the expression $E(x, y)=x^{4}+y^{4}+\frac{2}{x^{2} y^{2}}, x, y \in \mathbb{R}$. Specify the values of $x$ and $y$ for which this minimum is achieved.
b) Determine the real numbers $x, y, z$ knowing that: $x+y+z=\frac{3}{2}$ and $x^{2}+y^{2}+z^{2}=\frac{3}... | \frac{1}{2} |
9. Given the function
$$
f(x)=\sin \left(2 x-\frac{\pi}{6}\right)-m
$$
has two zeros in $\left[0, \frac{\pi}{2}\right]$. Then the range of $m$ is
(A) $\left(\frac{1}{2}, 1\right)$
(B) $\left[\frac{1}{2}, 1\right]$
(C) $\left[\frac{1}{2}, 1\right)$
(D) $\left(\frac{1}{2}, 1\right]$ | C |
Find the real root $x$ of the equation $\ln x = 8 - 2x$ that belongs to the interval $(k, k+1)$, where $k \in \mathbb{Z}$. Determine the value of $k$. | k = 3 |
To investigate the situation of food waste during lunch in the school cafeteria among 630 second-year high school students, it is planned to draw a sample of 45 students. Considering systematic sampling, the interval $k$ is ( )
A: 16
B: 14
C: 12
D: 22 | B |
# 3. Option 1.
In the village, 7 people live. Some of them are liars who always lie, while the rest are knights (who always tell the truth). Each resident said whether each of the others is a knight or a liar. A total of 42 answers were received, and in 24 cases, a resident called a fellow villager a liar. What is the... | 3 |
Given 463 Let $x_{i}, y_{i} \in [a, b] (0 < a < b, i = 1,2, \cdots, n), \sum_{i=1}^{n} x_{i}^{2} = \sum_{i=1}^{n} y_{i}^{2}$. Prove:
$$
\sum_{i=1}^{n} \frac{x_{i}^{3}}{y_{i}} + \sum_{i=1}^{n} \frac{y_{i}^{3}}{x_{i}} + 2 \sum_{i=1}^{n} x_{i} y_{i} < 4 n \cdot \frac{a^{4} b + b^{5}}{a^{3} + a b^{2}} \text{. }
$$ | 4 n \cdot \frac{a^{4} b+b^{5}}{a^{3}+a b^{2}} |
Example 1 Solve the congruence equation $589 x \equiv 1026(\bmod 817)$. | x \equiv 43 u + 17 \pmod{817}, \quad u = 0, 1, \cdots, 18 |
Form a six-digit number using the digits 1, 2, 3, 4, 5, 6 without repetition, requiring that any two adjacent digits have different parity, and 1 and 2 are adjacent. The number of such six-digit numbers is \_\_\_\_\_\_ (answer in digits). | 40 |
9.2. Ten numbers are written around a circle, the sum of which is 100. It is known that the sum of any three consecutive numbers is not less than 29. Indicate the smallest number $A$ such that in any set of numbers satisfying the condition, each number does not exceed $A$. | 13 |
Given an odd function $f(x)$ with domain $\mathbf{R}$, and $f(x+2)$ is an even function. If $f(1)=a$, then $f(1)+f(3)+f(5)+\cdots+f(2019)=$（ ）．
A: $0$
B: $a$
C: $2a$
D: $1010a$ | \text{C} |
Given that $f(x)$ is a differentiable function on $\mathbb{R}$ and $f'(x)=1+\cos x$, find an expression for the function $f(x)$ (only one valid function expression is needed). | f(x) = x + \sin x |
Given a positive sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and it satisfies: $2S_n=a_n^2+a_n$, $(n\in\mathbb{N}^+)$
$(1)$ Find the values of $a_1$, $a_2$, $a_3$
$(2)$ Find the general formula for the sequence $\{a_n\}$. | a_n=n |
A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$. The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity $v_2$. The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant... | 8 |
Find all pairs $ (m, n) $ of natural numbers such that $ n ^ 4 \ | \ 2m ^ 5 - 1 $ and $ m ^ 4 \ | \ 2n ^ 5 + 1 $. | (m, n) = (1, 1) |
3. In $\triangle A B C$, the lengths of the three sides are $a=3$, $b=4, c=6, h_{a}$ represents the altitude on side $B C$, and $h_{b} 、 h_{c}$ are defined similarly. Then
$$
\left(h_{a}+h_{b}+h_{c}\right)\left(\frac{1}{h_{a}}+\frac{1}{h_{b}}+\frac{1}{h_{c}}\right)=
$$
$\qquad$ | 9 \frac{3}{4} |
Among the following functions, the odd function is ( )
A: $f(x) = 2^x - 3^x$
B: $f(x) = x^3 + x^2$
C: $f(x) = \sin x \tan x$
D: $f(x) = \lg \frac{1-x}{1+x}$ | D |
1. Calculate the following sum:
$$
\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\cdots+\frac{10}{2^{10}} .
$$
(A) $\frac{503}{256}$;
(B) $\frac{505}{256}$;
(C) $\frac{507}{256}$;
(D) $\frac{509}{256}$;
(E) None of the above. | \frac{509}{256} |
9. Let $F$ be the left focus of the ellipse $E: \frac{x^{2}}{3}+y^{2}=1$. A line $l$ passing through point $F$ with a positive slope intersects $E$ at points $A$ and $B$. Lines $A M$ and $B N$ are drawn through points $A$ and $B$ respectively, such that $A M \perp l, B N \perp l$, and lines $A M, B N$ intersect the $x$... | \sqrt{6} |
28. A rectangular prism with integer centimeter edge lengths has all 6 of its faces painted red, and then it is completely cut into small cubes with edge lengths of 1 centimeter. Among these small cubes, there are 24 cubes with exactly 1 face painted red, and 28 cubes with exactly 2 faces painted red. The original volu... | 60 |
6. An aquarium in the shape of a rectangular parallelepiped has dimensions: length $2 m$, width - 600 mm, height - 60 cm. It is filled with water at a rate of 3 liters/minute. How many seconds after the start of filling will the aquarium be completely filled? (10 points) | 14400 |
If $9^{a}\div 81^{b}=9$, then the value of $a-2b$ is ____. | 1 |
## Task 1.
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$,
$$
x f(x) - y f(y) = (x - y)(f(x + y) - xy)
$$ | f(x)=x^{2}++ |
5, 50 different positive integers, their sum is 2012, how many of these numbers can be odd at most?
There are 50 different positive integers, their sum is 2012, how many of these numbers can be odd at most? | 44 |
Find the area of the plane figure bounded by the parabola $y=2x^2$ and the lines $x=2$, $y=0$. When the interval $[0, 2]$ is divided into $n$ equal parts, the $i$-th interval is ( )
A: $\left[ \frac{i-1}{n}, \frac{i}{n} \right]$
B: $\left[ \frac{i}{n}, \frac{i+1}{n} \right]$
C: $\left[ \frac{2(i-2)}{n}, \frac{2(i-1... | D |
Given $m= \frac {\tan 22.5^\circ }{1-\tan^{2}22.5^\circ }$, then the minimum value of the function $y=2m\cdot x+ \frac {3}{x-1}+1$ ($x>1$) is
A: 2
B: $2 \sqrt {3}$
C: $2+2 \sqrt {3}$
D: $2 \sqrt {3}-2$ | \text{C} |
Given that $\{a_n\}$ is an arithmetic sequence with the first term being 2 and a non-zero common difference, if $a_1$, $a_3$, $a_6$ form a geometric sequence, then the sum of the first 9 terms of $\{a_n\}$ is ( )
A: 26
B: 30
C: 36
D: 40 | C |
Task 1. Calculate:
$$
20130+2 \cdot(480 \cdot 4 \cdot 14+30 \cdot 44 \cdot 16)-(5 \cdot 80 \cdot 43+19 \cdot 400 \cdot 3) \cdot 2
$$ | 36130 |
3rd CaMO 1971 Problem 1 A diameter and a chord of a circle intersect at a point inside the circle. The two parts of the chord are length 3 and 5 and one part of the diameter is length 1. What is the radius the circle? | 8 |
3. For a regular triangle, a square, and a regular hexagon with the same perimeter, their areas are $S_{3}, S_{4}, S_{6}$ respectively, then
(A) $S_{3}>S_{4}>S_{6}$;
(B) $S_{6}>S_{4}>S_{3}$;
(C) $S_{6}>S_{3}>S_{4}$;
(D) $S_{3}>S_{6}>S_{4}$;
(E) $S_{4}>S_{6}>S_{3}$. | S_{6}>S_{4}>S_{3} |
02.2. In two bowls there are in total $N$ balls, numbered from 1 to $N$. One ball is moved from one of the bowls to the other. The average of the numbers in the bowls is increased in both of the bowls by the same amount, $x$. Determine the largest possible value of $x$.
| \frac{1}{2} |
Let the random variable $\xi$ follow the normal distribution $N(3, \sigma^2)$. If $P(\xi > m) = a$, then $P(\xi > 6-m)$ equals to ( )
A: $a$
B: $1-2a$
C: $2a$
D: $1-a$ | D |
8. A cube consists of eight smaller identical cubes. Three of the smaller cubes were replaced with ones of the same size but with a density three times greater. Determine the ratio of the final to the initial density of the large cube. (10 points) | 1.75 |
Task B-1.2. How many integer solutions does the inequality $\frac{x^{2}-2015 x+2014}{2013 x^{2}-2013 x} \leq 0$ have? | 2013 |
Find all positive integers $n$ such that $6^n+1$ it has all the same digits when it is writen in decimal representation. | n = 1 \text{ and } n = 5 |
If the base radius and height of a cylinder are both $a$, then its volume is ______ (expressed as an algebraic expression containing $a$). | \pi a^{3} |
Let \\(a = \left( \frac{5}{7} \right)^{-\frac{5}{7}}, b = \left( \frac{7}{5} \right)^{\frac{3}{5}}, c = \log_{3} \frac{14}{5}\\), then the order of \\(a\\), \\(b\\), and \\(c\\) is
A: \\(b 0\\)
D: \\(c < b < a \\) | D |
2. A tournament of the game "Rock, Paper, Scissors" ${ }^{3}$ is held according to the Olympic system, ${ }^{4}$ with 16 players registered. The organizers have determined who will play against whom in the first match, which winners of which matches will play against each other in the next, and so on. Each participant ... | 3\cdot2^{15} |
Given the expression "$\left(-1-8\right)\times 2-5$".$(1)$ Congcong copied the number "$5$" incorrectly and obtained a result of $-24$, then Congcong mistakenly wrote "$5$" as _______;$(2)$ Huihui accidentally mistook the operator "$\times $" as "$+$", find how much larger Huihui's calculated result is compared to the ... | 11 |
Example 9. Find the integral $\int \frac{d x}{\sin ^{2} 3 x}$. | -\frac{1}{3}\operatorname{ctg}^{2}3x+C |
Define "Equally Summed Sequence": In a sequence, if the sum of each term and its subsequent term is always a constant, then this sequence is called an Equally Summed Sequence. Given that $a\_1=2$ and the common sum is $5$, find the value of $a\_{18}$. | 3 |
10.1 A weirdo didn't mind and recorded 2022 numbers in a circle, and it turned out that each number coincides with the absolute difference of its two neighbors. Determine which numbers were recorded if their sum is 2022. | \frac{3}{2} |
Which of the following judgments is correct? ( )
A: If the function $y=f(x)$ is a differentiable function on $\mathbb{R}$, then $f'(x_0)=0$ is a necessary and sufficient condition for $x_0$ to be an extremum point of the function $f(x)$.
B: The negation of the proposition "There exists $x \in \mathbb{R}$, such that $x... | D |
The domain of the function $f(x)=\sqrt{2-x}+\lg(x+1)$ is ( )
A: $( -1,2 ]$
B: $[ -1,3 )$
C: $[ 2,+\infty )$
D: $( -\infty ,-1 )$ | A |
【Example 2】There are $m$ white balls and $n$ black balls $(m>n)$ in a jar. The balls are taken out one by one, without replacement. If at some point during the process, the number of white balls taken out equals the number of black balls taken out, it is said that a coincidence has occurred. Find the probability that a... | \frac{2n}{+n} |
3. Determine the three-digit number $\overline{a b c}$ if $\overline{a b} \cdot \overline{c c} \cdot \overline{a b c}=\overline{a b c a b c}$. | 137or911 |
10. Given the sequence $\left\{a_{n}\right\}, a_{1}=1, a_{n+1}$ $=\frac{\sqrt{3} a_{n}+1}{\sqrt{3}-a_{n}}$, then $\sum_{n=1}^{2022} a_{n}=$ $\qquad$ | 0 |
Given a complex number $Z = x + yi$ ($x, y \in \mathbb{R}$) such that $|Z - 4i| = |Z + 2|$, find the minimum value of $2^x + 4^y$. | 4\sqrt{2} |
From 6 boys and 4 girls, 3 representatives are to be selected, with the requirement that at least 1 girl must be included. How many different ways can this be done? | 100 |
3. On the sides of triangle $A B C$, points were marked: 10 - on side $A B$, 11 - on side $B C$, 12 - on side $A C$. At the same time, none of the vertices of the triangle were marked. How many triangles exist with vertices at the marked points? | 4951 |
Given $y=f(x)$ is an odd function. When $x>0$, $f(x)=2x(1-x)$. When $x<0$, $f(x)$ should be equal to ( ).
A: $-2x(1-x)$
B: $2x(1-x)$
C: $-2x(1+x)$
D: $2x(1+x)$ | \text{D} |
6.097. $\left\{\begin{array}{l}x+2 y+3 z=3, \\ 3 x+y+2 z=7, \\ 2 x+3 y+z=2 .\end{array}\right.$ | (2,-1,1) |
Given that the domain of the function $y=f(x+1)$ is $[-1,3]$, determine the domain of $y=f(x^{2})$.
A: $[0,4]$
B: $[0,16]$
C: $[-2,2]$
D: $[1,4]$ | \text{C} |
8.4. 23 ** As shown in the figure, given that $A B$ is the diameter of the semicircle $\odot C$, $\odot O, \odot O_{1}, \odot O_{2}$ are all tangent to $\odot C$ and $A B$, and $\odot O_{1}, \odot O_{2}$ are tangent to $\odot O$. Let $r, r_{1}, r_{2}$ be the radii of $\odot O, \odot O_{1}, \odot O_{2}$ respectively.
Pr... | \frac{1}{\sqrt{r_{1}}}+\frac{1}{\sqrt{r_{2}}}=\frac{2\sqrt{2}}{\sqrt{r}} |
10. Children are assembling robot toys. At first, every 2 children work together to assemble 1 small robot toy, then every 3 children work together to assemble 1 large robot toy, resulting in a total of 30 robots of various sizes. How many children are there $\qquad$? | 36 |
8・168 Let $A=\{1,2, \cdots, 17\}$. For the mapping $f: A \rightarrow A$, denote
$$
f^{(1)}(x)=f(x), f^{(k+1)}(x)=f\left(f^{(k)}(x)\right), k \in N \text {. }
$$
Consider a one-to-one mapping $f$ from $A$ to $A$ that satisfies the following conditions: there exists a natural number $M$ such that
(1) when $m<M, 1 \leqsl... | 8 |
The sum of the distances from points $A$ and $B$ on the parabola $y^{2}=4x$ to the focus is $7$. Then, the sum of the distances from $A$ and $B$ to the $y$-axis is __( ).
A: $8$
B: $7$
C: $6$
D: $5$ | D |
Given the function $f(x)=\sin (3x+ \frac {\pi}{3})+\cos (3x+ \frac {\pi}{6})+m\sin 3x$ ($m\in\mathbb{R}$), and $f( \frac {17\pi}{18})=-1$
$(1)$ Find the value of $m$;
$(2)$ In triangle $ABC$, with the sides opposite angles $A$, $B$, and $C$ being $a$, $b$, and $c$ respectively, if $f( \frac {B}{3})= \sqrt {3}$, and... | -3 \sqrt {3} |
Determine the value of $\sin 135^{\circ}\cos 15^{\circ}-\cos 45^{\circ}\sin (-15^{\circ})$. | \frac{\sqrt{3}}{2} |
9. Leilei went to buy instant noodles and handed the boss a 100 yuan note. After giving her the change, the boss said to her: "You only gave me 1 piece of money, but I gave you 16 pieces of money, plus instant noodles worth 5 yuan, you really got a great deal!" If the money the boss gave Leilei was either 10 yuan or 5 ... | 3 |
7, 10 consecutive four-digit numbers whose sum is a multiple of 37, the smallest sum of these 10 numbers is | 10175 |
The sum of all digits used in the numbers 1, 2, 3, ..., 999 is . | 13500 |
One of the symmetry centers of the function $y=\tan(2x- \frac {\pi}{4})$ is ( )
A: $(- \frac {\pi}{8}，0)$
B: $( \frac {\pi}{2}，0)$
C: $(0，0)$
D: $( \frac {\pi}{4}，0)$ | \text{A} |
Problem 10.1. Solve the system
$$
\left\lvert\, \begin{aligned}
& 3 \cdot 4^{x}+2^{x+1} \cdot 3^{y}-9^{y}=0 \\
& 2 \cdot 4^{x}-5 \cdot 2^{x} \cdot 3^{y}+9^{y}=-8
\end{aligned}\right.
$$
Ivan Landjev | \frac{1}{2},1+\frac{1}{2}\log_{3}2 |
Given the vertices of a triangle are A(2, -1, 4), B(3, 2, -6), and C(5, 0, 2), find the length of the median to side BC. | 2\sqrt{11} |
When $a > 0$ and $a \neq 1$, the graph of the function $y = a^{x-1} + 3$ must pass through the point ( ).
A: (4, 1)
B: (1, 4)
C: (1, 3)
D: (-1, 3) | \text{B} |
Given a convex pentagon $ABCDE$ with $AB=BC, CD=DE, \angle ABC=150^o, \angle CDE=30^o, BD=2$. Find the area of $ABCDE$.
(I.Voronovich) | 1 |
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