problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
8. There are 128 ones written on the blackboard. In each step, you can erase any two numbers $a$ and $b$ on the blackboard, and write the number $a \cdot b + 1$. After doing this 127 times, only one number remains. Denote the maximum possible value of this remaining number as $A$. Find the last digit of $A$.
| 2 |
3. Given quadrilateral $A B C D$ is an isosceles trapezoid, $A B / / C D, A B=6, C D=16 . \triangle A C E$ is a right triangle, $\angle A E C=90^{\circ}, C E=B C=A D$. Then the length of $A E$ is . $\qquad$ | 4 \sqrt{6} |
41. On one of the three cards, the number 18 was written, on another - the number 75, and on the third - some two-digit number. If you add up all the different six-digit numbers that can be formed by arranging these cards in a row, the result is 2606058. What number is written on the third card? | 36 |
In the Cartesian coordinate plane $(xOy)$, the terminal side of angle $\theta$ passes through point $P(-2, t)$, and $\sin \theta + \cos \theta = \frac{\sqrt{5}}{5}$. Determine the value of the real number $t$. | 4 |
2. As shown in Figure 1, in the right trapezoid $A B C D$, $A B=B C=4$, $M$ is a point on the leg $B C$, and $\triangle A D M$ is an equilateral triangle. Then $S_{\triangle C D M}$ : $S_{\triangle A B M}=$ $\qquad$ . | 2 |
3. Let the set
$$
A=\left\{y \left\lvert\, y=\sin \frac{k \pi}{4}(k=0, \pm 1, \pm 2, \cdots)\right.\right\} \text {. }
$$
Then the number of proper subsets of set $A$ is $\qquad$ (2009, Tongji University Independent Admission Examination) | 31 |
For each [positive integer](https://artofproblemsolving.com/wiki/index.php/Positive_integer) $k$, let $S_k$ denote the [ increasing](https://artofproblemsolving.com/wiki/index.php/Increasing_sequence) [arithmetic sequence](https://artofproblemsolving.com/wiki/index.php/Arithmetic_sequence) of [integers](https://artofpr... | 12 |
Given the curve C: $$\frac {x^{2}}{4} + \frac {y^{2}}{9} = 1$$, and the line l: $$\begin{cases} x=2+t \\ y=2-2t \end{cases}$$ (where $t$ is a parameter)
(Ⅰ) Write the parametric equations for curve C and the standard equation for line l.
(Ⅱ) For any point P on curve C, construct a line that forms a 30° angle with l... | \frac {2 \sqrt {5}}{5} |
4. Three people, A, B, and C, are training in a broadcast tournament format. In each round, 2 people play a singles match, and the other 1 person is the referee. The loser of each round becomes the referee for the next round, and the previous referee challenges the winner. At the end of the half-day training, it was fo... | A |
3. As shown in Figure 1, Angel used 18 toothpicks to build a three-story staircase. Following this pattern, to build a five-story staircase, she would need ( ) more toothpicks.
(A) 9
(B) 18
(C) 20
(D) 22
(E) 24 | 22 |
Given the circle $x^{2}+y^{2}=4$, find the number of points on the circle that have a distance of $1$ to the line $3x-4y+5=0$. | 3 |
4. Given $a, b, c, d \in \mathbf{Z}$,
$$
f(x)=a x^{3}+b x^{2}+c x+d \text {. }
$$
Then among $A(1,1)$, $B(2009,1)$, $C(-6020,2)$, $D(2,-6020)$, at most $\qquad$ of them can be on the curve $y=f(x)$. | 3 |
5. Let the area of $\triangle A B C$ be $1, D$ be a point on side $A B$, and $\frac{A D}{A B}=\frac{1}{3}$. If a point $E$ is taken on side $A C$ such that the area of quadrilateral $D E C B$ is $\frac{3}{4}$, then the value of $\frac{C E}{E A}$ is ( ).
(A) $\frac{1}{2}$
(B) $\frac{1}{3}$
(C) $\frac{1}{4}$
(D) $\frac{1... | B |
11. (5 points) If the six-digit number $\overline{201 a b 7}$ is divisible by 11 and 13, then the two-digit number $\overline{a b}=$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 48 |
I am twice as old as you were when I was as old as you are now. But when you are as old as I am now, the sum of our ages will be 140. How old are you, and how old am I? | 62\frac{2}{9},46\frac{2}{3} |
What is the area of the region defined by the [inequality](https://artofproblemsolving.com/wiki/index.php/Inequality) $|3x-18|+|2y+7|\le3$?
$\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ \frac{7}{2}\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ \frac{9}{2}\qquad\mathrm{(E)}\ 5$ | 3 |
6.1. In the numerical example АБВ $+9=$ ГДЕ, the letters А, Б, В, Г, Д, and Е represent six different digits. What digit is represented by the letter Д? | 0 |
## Task 4
From class 1a, 3 Pioneers receive certificates for good performance. An equal number of Pioneers from class 1b also receive certificates.
How many Pioneers will be awarded? | 6 |
When $x^5, x+\frac{1}{x}$ and $1+\frac{2}{x} + \frac{3}{x^2}$ are multiplied, the product is a polynomial of degree.
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$ | 6 |
6. For all real numbers $x, y$ satisfying
$$
|x|+|y| \leqslant 1
$$
the inequality
$$
\left|2 x-3 y+\frac{3}{2}\right|+|y-1|+|2 y-x-3| \leqslant a
$$
always holds. Then the minimum value of the real number $a$ is . $\qquad$ | \frac{23}{2} |
The solution set of the quadratic inequality $ax^2 + bx + 1 > 0$ is $\{x \mid -1 < x < \frac{1}{3}\}$. Find the value of $ab$. | 6 |
Given that the sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n$, and $a\_2 = 12$, $S\_n = kn^2 - 1 (n \in \mathbb{N}^*)$. Find the sum of the first $n$ terms of the sequence $\{\frac{1}{S\_n}\}$. | \frac{n}{2n + 1} |
13. (Shanghai · Senior High School) Using the digits $2,4,6$: to construct a 6-digit number, but it is not allowed to have two consecutive 2s in the 6-digit number (for example, 626442 is allowed, 224626 is not allowed) - then the number of such 6-digit numbers is $\qquad$. | 448 |
One afternoon, taxi driver Xiao Wang's operation was all conducted on the north-south oriented Jianshe Street. If heading south is considered positive and heading north is considered negative, the distances he drove that afternoon (in kilometers) were: +3, +10, -21, +7, -6, -8, +12, +2, -3, -4, +6, -5. Please answer th... | 26.1 \, \text{liters} |
6.1. A cube lies on a plane. Each face is marked with 1 to 6 points such that the sum of points on opposite faces is always 7. The cube is rolled over the plane as shown in the picture. How many points will be on the top face of the cube when it lands on the last cell?
 For the function $f(x)$, if $f(x)=x$, then $x$ is called a "fixed point" of $f(x)$; if $f(f(x))=x$, then $x$ is called a "stable point" of $f(x)$. The sets of "fixed points" and "stable points" of the function $f(x)$ are denoted as $A$ and $B$, respectively, i.e., $A=\{x \mid f(x)=x\}, B=\{x \mid f(f(x)... | \left[-\frac{1}{4}, \frac{3}{4}\right] |
2. Let $f(x)=\frac{a^{x}}{1+a^{x}}(a>0, a \neq 1), [m]$ denote the greatest integer not exceeding the real number $m$, then the range of the function $\left[f(x) -\frac{1}{2}\right]+\left[f(-x)-\frac{1}{2}\right]$ is $\qquad$ . | \{-1,0\} |
Let's calculate the following expressions to three decimal places:
1.
$$
\frac{2 \sqrt{3}}{\sqrt{3}-\sqrt{2}}
$$
2.
$$
\frac{(3+\sqrt{3})(1+\sqrt{5})}{(5+\sqrt{5})(1+\sqrt{3})}
$$ | 0.775 |
7. (10 points) There is an octagon, each of whose sides has a length that is an integer number of centimeters. If any three sides taken from this octagon cannot form a triangle. Then the shortest perimeter of an octagon that meets these conditions is $\qquad$ cm. | 54 |
A student mistakenly copied $4 \times (\square - 3)$ as $4 \times \square - 3$. If the incorrect answer is $y$ and the correct answer is $x$, then calculate $x - y$. | -9 |
Let the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ be $S_n$. If $a_3=2a_4=2$, then $S_6=$ ______. | \frac{63}{4} |
## Task 4 - 200934
Determine all pairs $(a ; b)$ of natural numbers $a, b$ with $a > b$, for which the following statements (1), (2), (3) are true!
(1) The number $a$ is (in decimal notation) two-digit, the number $b$ as well.
(2) If the digits of $a$ are swapped, one obtains $b$.
(3) Subtracting $b^2$ from $a^2$ r... | (65,56) |
1. The arithmetic progression $a_{n}$ is such that its difference is not equal to zero, and $a_{10}, a_{13}$, and $a_{19}$ are consecutive terms of some geometric progression. Find the ratio $a_{12}: a_{18}$. | 5:11 |
6. Given $x=1+\sqrt{3}$. Then, $\frac{1}{x+2}+\frac{1}{x^{2}-4}-\frac{1}{x-2}$ equals $\qquad$ | -\frac{\sqrt{3}}{2} |
9. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=2, a_{n}=2\left(n+a_{n-1}\right), n=2,3, \cdots$, find the general term formula of the sequence $\left\{a_{n}\right\}$. | a_{n}=2^{n+2}-2n-4 |
Given the set $M=\{x|1 < x \leqslant 3\}$, if $N=\{x|2 < x \leqslant 5\}$, then $M\cup N=$ ()
A: $\{x|1 < x \leqslant 5\}$
B: $\{x|2 < x \leqslant 3\}$
C: $\{x|1 \leqslant x < 2\}$ or $\{3 \leqslant x \leqslant 5\}$
D: $\{x|1 \leqslant x \leqslant 5\}$ | A |
5. Tanya was sequentially writing down numbers of the form $n^{7}-1$ for natural numbers $n=2,3, \ldots$ and noticed that for $n=8$ the resulting number is divisible by 337. For what smallest $n>1$ will she get a number divisible by $2022?$ | 79 |
Given a complex number $z$ satisfying $$\frac {1+i}{1-i} \cdot z = 3+4i,$$ the conjugate of $z$ is ( )
A: 4+3i
B: -4+3i
C: -4-3i
D: 4-3i | \text{A} |
6. Given that $n$ is an integer greater than 1, and $2^{2^{n}}+1$ is a prime number. Then $2^{2^{n}}-1$ must be ( ).
(A) a prime number
(B) a composite number with a factor of 3
(C) a composite number with a factor of 5
(D) a composite number with a factor of 15 | D |
4. The sequence $2,3,5,6,7,8,10,11, \ldots$ is an enumeration of the positive integers which are not perfect squares. What is the 150th term of this sequence? | 162 |
Task 5. (20 points) In the center of a circular field stands a geologists' cabin. From it, 6 straight roads extend, dividing the field into 6 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... | 0.5 |
Example 5 Find the smallest real number $m$, such that for any positive numbers $a, b, c$ satisfying $a+b+c=1$, we have
$$
m\left(a^{3}+b^{3}+c^{3}\right) \geqslant 6\left(a^{2}+b^{2}+c^{2}\right)+1 \text {. }
$$
(3rd China Southeast Mathematical Olympiad) | 27 |
Simplify first, then evaluate: $({\frac{{2m+1}}{m}-1})÷\frac{{{m^2}-1}}{m}$, where $m=2$. | 1 |
Find the smallest positive value taken by $a^3 + b^3 + c^3 - 3abc$ for positive integers $a$, $b$, $c$ .
Find all $a$, $b$, $c$ which give the smallest value | 4 |
A store purchases goods at a cost of 10 yuan per item and sells them for 18 yuan each. At this price, they can sell 60 items per day. After conducting market research, the store manager discovers that for each 1-yuan increase in price above 18 yuan, daily sales decrease by 5 units, and for each 1-yuan decrease in price... | 500 |
Given the function $f(x)=\sin 2x+2\cos ^{2}x-1$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ When $x∈[0,\frac{π}{2}]$, find the minimum value of $f(x)$ and the corresponding value of the independent variable $x$. | \frac{\pi}{2} |
In which base number systems is it true that the rule for divisibility by four is the same as the rule for divisibility by nine in the decimal system, and the rule for divisibility by nine is the same as the rule for divisibility by four in the decimal system? | 12n+9 |
Calculate
(1) $\log_{2}25 \cdot \log_{3}4 \cdot \log_{5}9$
(2) $\frac{1}{2}\lg\frac{32}{49} - \frac{4}{3}\lg\sqrt{8} + \lg\sqrt{245}$. | \frac{1}{2} |
## Task Condition
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{15 ;-20 ;-1\}$
$p=\{0 ; 2 ; 1\}$
$q=\{0 ; 1 ;-1\}$
$r=\{5 ;-3 ; 2\}$ | -6p+q+3r |
(1) Let $f(x)=2^{x}, g(x)=4^{x}$, if $g[g(x)]>g[f(x)]>f[g(x)]$, find the range of $x$;
(2) If the range of the function $y=4^{x}-3 \cdot 2^{x}+3$ is $[1,7]$, find the maximum range of $x$. | 0<x<1 |
8.1. (12 points) In triangle $ABC$, the bisector $BL$ is drawn. Find the area of the triangle if it is known that $AL=2, BL=3\sqrt{10}$, and $CL=3$. | \frac{15\sqrt{15}}{4} |
15. (15 points) Definition: $[a]$ represents the greatest natural number not exceeding $a$, for example $[0.6]=0,[1.25]=1$. If $[5 a-0.9]$ $=3 a+0.7$, then the value of $a$ is. | 1.1 |
1st Putnam 1938 Problem B2 Find all solutions of the differential equation zz" - 2z'z' = 0 which pass through the point x=1, z=1. Solution | \frac{1}{A(x-1)+1} |
Given $x∈[{0,\frac{1}{3}}]$, the range of the function $g(x)=x+\sqrt{1-3x}$ is ____. | [\frac{1}{3},1] |
1. Solve the inequality $2^{\log _{2}^{2} x}-12 \cdot x^{\log _{0.5} x}<3-\log _{3-x}\left(x^{2}-6 x+9\right)$. | x\in(2^{-\sqrt{2}},2)\cup(2,2^{\sqrt{2}}) |
B3. In the final of the Just Dance dance competition, an evaluation system was used to determine the winning pair. First, the expert committee awarded points and recorded them in a table:
| Dance Pair No. | 4 | 27 | 15 | 8 | 11 |
| :--- | ---: | ---: | ---: | ---: | ---: |
| Footwork Technique (T) | 3 | 2 | 3 | 1 | 3 ... | 18,choreography,11,2T+1K+1F+3Ior3T+1K+1F+3I |
Find all positive integer $n$ and prime number $p$ such that $p^2+7^n$ is a perfect square | (n, p) = (1, 3) |
A three-digit number has a digit sum of 16. If in this number the digits in the hundreds and tens places are swapped, the number decreases by 360. If in the original number the digits in the tens and units places are swapped, the number increases by 54.
Find that three-digit number.
(L. Hozová) | 628 |
Calculate: $\sqrt{3}\tan 30^{\circ}+2\cos 60^{\circ}-\sqrt{2}\sin 45^{\circ}$. | 1 |
If $(1+mx)^6 = a + a_1x + \ldots + a_6x^6$, and $a + a_1 + a_2 + \ldots + a_6 = 64$, then the value of the real number $m$ is. | m = 1 \text{ or } m = -3 |
Two lines with slopes $\frac{1}{4}$ and $\frac{5}{4}$ intersect at $(1,1)$. What is the area of the triangle formed by these two lines and the vertical line $x=5$ ?
(A) 5
(B) 10
(C) 8
(D) 12
(E) 15 | 8 |
1. [5 points] Point $D$ lies on side $A C$ of triangle $A B C$. The circle with diameter $B D$ intersects sides $A B$ and $B C$ at points $P$ and $T$ respectively. Points $M$ and $N$ are the midpoints of segments $A D$ and $C D$ respectively. It is known that $P M \| T N$.
a) Find the angle $A B C$.
b) Suppose additi... | \frac{5\sqrt{13}}{3\sqrt{2}} |
Given complex numbers $z_1 = \cos\theta - i$ and $z_2 = \sin\theta + i$, the maximum value of the real part of $z_1 \cdot z_2$ is \_\_\_\_\_\_, and the maximum value of the imaginary part is \_\_\_\_\_\_. | \sqrt{2} |
4・117 Solve the system of equations
$$\left\{\begin{array}{l}
\frac{x-y \sqrt{x^{2}-y^{2}}}{\sqrt{1-x^{2}+y^{2}}}=a \\
\frac{y-x \sqrt{x^{2}-y^{2}}}{\sqrt{1-x^{2}+y^{2}}}=b
\end{array}\right.$$
where \(a, b\) are known numbers. | \left\{\begin{array}{l}
x=\frac{a+b \sqrt{a^{2}-b^{2}}}{\sqrt{1-a^{2}+b^{2}}} \\
y=\frac{b+a \sqrt{a^{2}-b^{2}}}{\sqrt{1-a^{2}+b^{2}}}
\end{array}\right.
} |
Problem 5. Merlin decided to weigh King Arthur on enchanted scales that always err by the same weight in the same direction. When Merlin weighed Arthur, they showed a weight of 19 stones. Then Merlin weighed the royal horse and got a weight of 101 stones. Finally, Merlin weighed Arthur on the horse, and the scales show... | 13 |
## Task 2 - 341212
Determine the number of all natural numbers $n$ that satisfy the following conditions (1), (2), (3):
(1) The number $n$ is divisible by 5.
(2) The number $n$ and its digit sum both contain no digit zero in their decimal representation.
(3) The digit sum of the digit sum of $n$ is 2. | 32 |
2. The City Mathematics Invitational Competition sets up three types of medals: gold, silver, and bronze. The organizing committee puts these medals in five boxes, with each box containing only one type of medal. The number of medals in each box is 3, 6, 9, 14, and 18, respectively. It is known that the silver medals a... | 12, 14, 24 |
It is a custom of five good friends that they wear a different tie each day, each one the one they wore the longest ago. Each of them has at least two ties, but none of them has a dozen. None of them has two ties of the same color, and there are no two of them who have the same number of ties. We have the following dat... | green |
Let's denote by $s(n)$ the sum of the digits of the number $n$. For example, $s(2345) = 2 + 3 + 4 + 5 = 14$. Observe that:
$40 - s(40) = 36 = 9 \times 4; 500 - s(500) = 495 = 9 \times 55; 2345 - s(2345) = 2331 = 9 \times 259$.
(a) What can we say about the number $n - s(n)$?
(b) Using the previous item, calculate $s... | 5 |
Let $a$, $b$, $c$ be positive numbers. Prove that: $\dfrac{bc}{a}+\dfrac{ca}{b}+\dfrac{ab}{c}\geqslant a+b+c$. | \dfrac{bc}{a}+ \dfrac{ac}{b}+ \dfrac{ab}{c}\geqslant a+b+c |
When boiling water, the temperature reaches 100℃ and starts to boil. The table below shows the data of two variables, time \( t \) (min) and temperature \( T \) (℃), recorded by a student during an experiment on "observing the boiling of water":
| \( t \) (min) | 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | ... |
|-------------... | \text{A} |
Test $3 P$ is a point inside $\triangle A B C$, line $A C$ and $B P$ intersect at $Q$, line $A B$ and $C P$ intersect at $R$, it is known that $A R=R B=C P, C Q=P Q$. Find $\angle B R C$. | 120 |
During the coming-of-age ceremony held in a class, one of the three students, A, B, and C, received a gift.
A said, "The gift is not with me";
B said, "The gift is with me";
C said, "The gift is not with B."
If only one of the three told the truth, who received the gift? \_\_\_\_\_\_ (Fill in "A", "B", or "C"). | \text{A} |
Example 7. Solve the integral equation
$$
\varphi(x)=x e^{-x}+\lambda \int_{0}^{\infty} J_{0}(2 \sqrt{x t}) \varphi(t) d t \quad(|\lambda| \neq 1)
$$ | \varphi(x)=e^{-x}(\frac{x}{1+\lambda}+\frac{\lambda}{1-\lambda^{2}}) |
How many lines are there that are tangent to both the circle $x^2+y^2+4x-4y+7=0$ and the circle $x^2+y^2-4x-10y+13=0$? | 3 |
The minimum positive period and maximum value of the function $f\left(x\right)=\sin \frac{x}{3}+\cos \frac{x}{3}$ are respectively ( ).
A: $3\pi$ and $\sqrt{2}$
B: $3\pi$ and $2$
C: $6\pi$ and $\sqrt{2}$
D: $6\pi$ and $2$ | C |
Students A and B are walking home from school. Student A walks $$\frac {1}{5}$$ more distance than student B, while student B takes $$\frac {1}{11}$$ less time than student A. What is the ratio of their speeds?
A: 11：12
B: 12：11
C: 6：5
D: 5：6 | \text{B} |
Given the function $f(x) = x^2 + ax + b$ ($a, b \in \mathbb{R}$) has a range of $[0, +\infty)$, if the solution set of the inequality $f(x) < c$ is $(m, m+8)$, then the value of the real number $c$ is \_\_\_\_\_\_. | 16 |
If set $M=\{x\,|\,|x|\leq2\}$, and $N=\{x\,|\,x^2-3x=0\}$, then $M\cap N=$ ? | \{0\} |
Given the function $f(x) = \frac{x^2}{\cos x}$, find the derivative $f'(π)$. | -2π |
4.4.9 ** Let $x_{1}, x_{2}, \cdots, x_{n}$ and $a_{1}, a_{2}, \cdots, a_{n}(n>2)$ be two sets of arbitrary real numbers satisfying the conditions:
(1) $x_{1}+x_{2}+\cdots+x_{n}=0$;
(2) $\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|=1$;
(3) $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n}$. To m... | \frac{1}{2} |
【Question 12】
It is known that $2^{29}$ is a nine-digit number consisting of nine different digits. Which digit is missing? | 4 |
Given an arithmetic sequence $\{a_n\}$, if $a_3 + a_{11} = 22$, then $a_7 = \ $ | 11 |
11) What is the negation of "all perfect numbers are even"? (It is not necessary to know what a perfect number is.)
(A) All perfect numbers are odd
(B) there is at least one odd perfect number
(C) there is at least one even number that is not perfect
(D) no odd number is perfect
(E) no even number is perfect. | B |
There are 25 red balls, which is 3 more than twice the number of blue balls. How many blue balls are there? | 11 |
7. Find the area of the region in the $x y$-plane satisfying $x^{6}-x^{2}+y^{2} \leq 0$. | \pi/2 |
Given that the moving circle $P$ passes through the fixed point $A(-3,0)$ and is internally tangent to the fixed circle $B:{(x-3)}^{2}+{{y}^{2}}=100$ (as shown in the figure on the right), find the trajectory equation of the moving circle's center $P$. | \frac{{x}^{2}}{25}+\frac{{y}^{2}}{16}=1 |
1. Given that the function $f(x)$ is a monotonically increasing function defined on $[-4,+\infty)$, to ensure that for all real numbers $x$,
$$
f\left(\cos x-b^{2}\right) \geqslant f\left(\sin ^{2} x-b-3\right)
$$
holds true. Find the range of real numbers $b$. | b \in\left[\frac{1}{2}-\sqrt{2}, 1\right] |
Define the symbol "$\oplus$" as an operation, that is, $a\otimes b=ab-a-b(a,b$ are positive real numbers). If positive numbers $x$ and $y$ satisfy $x\otimes y=3$, then the minimum value of $xy$ is ____. | 9 |
A line passing through any two vertices of a cube has a total of 28 lines. Among them, the number of pairs of skew lines is ( ).
A: 32
B: 72
C: 174
D: 189 | C |
## Task 4
Solve the equations:
$$
x+47=547 \quad y-60=380 \quad 570+z=710
$$ | 500;440;140 |
(4) For the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{m}=1$, the circle with the eccentricity as the radius and the right focus as the center is tangent to the asymptotes of the hyperbola, then $m=(\quad$.
(A) $\frac{3}{2}$
(B) $\frac{4}{3}$
(C) $\frac{5}{4}$
(D) $\frac{6}{5}$ | \frac{4}{3} |
A1. The equation $0, \overline{3}-\frac{1}{\frac{1}{x}-\frac{1}{3}}=3$ has the solution:
(A) $x=-24$
(B) $x=3$
(C) $x=-3$
(D) $x=24$
(E) $x=0$ | -24 |
7. Car A and Car B start from locations $\mathrm{A}$ and $\mathrm{B}$ respectively at the same time, heading towards each other, and meet at point $\mathrm{C}$, which is 1600 meters from $\mathrm{A}$. After meeting, Car B maintains its speed, while Car A's speed is halved. Car A continues to drive to $\mathrm{B}$ and i... | 2400 |
Given the equations of two circles, $(x^{2}+y^{2}=1)$ and $(x^{2}+y^{2}-6x-8y+9=0)$, determine the number of common tangent lines. The options are:
A: $4$
B: $3$
C: $2$
D: $1$ | B |
29. If the value of the fraction $\frac{|x|-2}{3 x-2}$ is negative, then the range of $x$ is ( ).
A. $\frac{2}{3}<x<2$.
B. $x>\frac{2}{3}$ or $x<-2$.
C. $-2<x<2$ and $x \neq \frac{2}{3}$.
D. $\frac{2}{3}<x<2$ or $x<-2$. | D |
51st Putnam 1990 Problem B6 C is a non-empty, closed, bounded, convex subset of the plane. Given a support line L of C and a real number 0 ≤ α ≤ 1, let B α be the band parallel to L, situated midway between L and the parallel support line on the other side of C, and of width α times the distance between the two support... | \frac{1}{3} |
11. The diagram shows a circle with centre $O$ and a triangle $O P Q$. Side $P Q$ is a tangent to the circle. The area of the circle is equal to the area of the triangle.
What is the ratio of the length of $P Q$ to the circumference of the circle?
A $1: 1$
B $2: 3$
C $2: \pi$
D $3: 2$
E $\pi: 2$ | 1:1 |
Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ that satisfy:
$$
f^{a b c-a}(a b c)+f^{a b c-b}(a b c)+f^{a b c-c}(a b c)=a+b+c
$$
For all $a, b, c \geq 2$ where $f^{k}(n)$ denotes the $k$-th iterate of $f$. | f(n)=n-1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.