problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
22. Three friends make the following statements.
Ben says, "Exactly one of Dan and Cam is telling the truth."
Dan says, "Exactly one of Ben and Cam is telling the truth."
Cam says, "Neither Ben nor Dan is telling the truth."
Which of the three friends is lying?
A Just Ben
B Just Dan
C Just Cam
D Each of Ben and Cam
E E... | C |
4. Let the angle between vectors $\boldsymbol{a}, \boldsymbol{b}$ be $60^{\circ}$, and $|\boldsymbol{a}|=1$, $|\boldsymbol{a}-2 \boldsymbol{b}|=\sqrt{13}$. Then $\boldsymbol{b}=(\quad)$.
(A) $\sqrt{2}$
(B) $\frac{3}{2}$
(C) $2 \sqrt{2}$
(D) 2 | 2 |
7. In $\triangle A B C$, the internal angles $A, B, C$ are opposite the sides $a, b, c$ respectively, and $\sin C \cos \frac{A}{2}=(2-\cos C) \sin \frac{A}{2}$, $\cos A=\frac{3}{5}, a=4$, then the area of $\triangle A B C$ is $\qquad$ . | 6 |
(7) From $\{1,2,3, \cdots, 100\}$, if 5 numbers (repetition allowed) are randomly selected, then the expected number of composite numbers is $\qquad$ . | \frac{37}{10} |
3. Ten consecutive natural numbers are written on the board. What is the maximum number of them that can have a digit sum equal to a perfect square? | 4 |
Given a complex number $z = \frac {a}{2+i}+ \frac {2+i}{5}$, the sum of its real part and imaginary part is $1$, find the value of the real number $a$. The options are:
A: $0$
B: $1$
C: $2$
D: $3$ | C |
$17 \cdot 99$ In triangle $ABC$, point $F$ divides $AC$ in the ratio 1:2. Let $E$ be the intersection of side $BC$ and line $AG$, where $G$ is the midpoint of $BF$. Then the ratio in which $E$ divides side $BC$ is
(A) $1: 4$
(B) $1: 3$.
(C) $2: 5$.
(D) $4: 11$.
(E) $3: 8$. | 1:3 |
The range of the independent variable $x$ in the function $y=\sqrt{2x-3}$ is ______. | x \geqslant \frac{3}{2} |
64. In the reading room, there are six textbooks on probability theory, three of which are in hardcover. The librarian randomly took two textbooks. Find the probability that both textbooks will be in hardcover. | 0.2 |
Let the function $f(x)=\begin{cases} 2\cos \frac{\pi }{2}x, & |x|\leqslant 1, \\ x^{2}-1, & |x| > 1 \end{cases}$, if $|f(x)+f(x+l)-2|+|f(x)-f(x+l)|\geqslant 2$ ($l > 0$) holds for any real number $x$, then the minimum value of $l$ is. | 2\sqrt{3} |
In the sequence $\{a_n\}$, $a_1 = -2$, $a_{n+1} = 2a_n + n$, then $a_3 =$ （ ）
A: $-6$
B: $-5$
C: $-4$
D: $-3$ | \text{C} |
## Problem Statement
Calculate the areas of figures bounded by the graphs of the functions:
$$
y=x^{2} \cdot \sqrt{16-x^{2}}, y=0 ;(0 \leq x \leq 4)
$$ | 16\pi |
Given the function $f(x)=\cos ( \frac {π}{2}+x)+\sin ^{2}( \frac {π}{2}+x)$, $x\in R$, find the maximum value of $f(x)$. | \frac {5}{4} |
Given the equation in terms of $x$: $\left(k-2\right)x^{2}-2x+1=0$ has two real roots.$(1)$ Find the range of values for $k$;$(2)$ When $k$ takes the maximum integer value, find the roots of the equation. | x_1 = x_2 = 1 |
Given that $\sqrt{2+\frac{2}{3}}=2\sqrt{\frac{2}{3}}$, $\sqrt{3+\frac{3}{8}}=3\sqrt{\frac{3}{8}}$, $\sqrt{4+\frac{4}{15}}=4\sqrt{\frac{4}{15}}$, ..., if $\sqrt{6+\frac{a}{b}}=6\sqrt{\frac{a}{b}}$ (where $a,b$ are real numbers), then the values of $a$ and $b$ can be inferred to be _______. | 6, 35 |
Given the function $f(x)$, for $x$, $y \in \mathbb{R}$, it always holds that $f(x+y)=f(x)+f(y)$. When $x > 0$, $f(x) > 0$
$(1)$ Prove that $f(x)$ is an odd function;
$(2)$ If $f(1)= \frac {1}{2}$, find the maximum and minimum values of $f(x)$ in the interval $[-2,6]$;
$(3)$ Does there exist an $m$ such that $f(2(... | m > \frac {9}{8} |
Task B-3.3. The number $n$ is the smallest positive multiple of 15 whose digits are all 0 or 8. Determine the largest prime number by which the number $n$ is divisible. | 37 |
Given functions $f(x)=\frac{{a+1+\ln x}}{x}$ and $g(x)=3e^{1-x}+1$, where $a$ is a real number.
$(1)$ If $f(x)\leq e$ always holds, find the range of real number $a$.
$(2)$ If the equation $f(x)=g(x)$ has exactly $3$ distinct real roots, find the value of real number $a$. | a = 3 |
The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$. What is the average of $a$ and $b$?
$\textbf{(A) } 0 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 30 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 60$ | 30 |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $a_3 + a_{11} = 12$, then $S_{13} =$ （ ）
A: 60
B: 78
C: 156
D: Indeterminate | \text{B} |
Given a moving circle $C$ that passes through points $A(4,0)$ and $B(0,-2)$, and intersects with the line passing through point $M(1,-2)$ at points $E$ and $F$. Find the minimum value of $|EF|$ when the area of circle $C$ is at its minimum. | 2\sqrt{3} |
Solve the equations:
1. $A_{2x}^4 = 60A_x^3$
2. $C_{n+3}^{n+1} = C_{n+1}^{n-1} + C_{n+1}^{n} + C_{n}^{n-2}$. | n = 4 |
6. In $\triangle A B C$, if $\sin A=\frac{4}{5}, \operatorname{tg} B=\frac{12}{5}$, then $\angle A$ ).
(A) is an acute angle
(B) is an obtuse angle
(C) is an acute or obtuse angle
(D) could be an obtuse angle | A |
(7) Let $a, b, c$ be non-negative real numbers, then the minimum value of $\frac{c}{a}+\frac{a}{b+c}+\frac{b}{c}$ is | 2 |
Solve the following quadratic equations:
(1) $x^{2}-8x+12=0$;
(2) $(x-3)^{2}=2x(x-3)$. | x_{1}=3, \; x_{2}=-3 |
Given that $\sin \alpha \cos (\alpha - \beta) - \cos \alpha \sin (\alpha - \beta) = \frac{4}{5}$, find the value of $\sin \beta =$ _____. | \frac{4}{5} |
In the Cartesian coordinate system, the parametric equation of curve $C_{1}$ is
$$
\begin{cases}
x=3+ \sqrt {2}\cos \alpha \\
y=1+ \sqrt {2}\sin \alpha
\end{cases}
(\alpha \text{ is the parameter})
$$
With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is es... | 2 |
7.079. $2 \log _{x} 27-3 \log _{27} x=1$. | \frac{1}{27};9 |
Three. (50 points) Let $S=\{1,2, \cdots, 100\}$. Find the largest integer $k$, such that the set $S$ has $k$ distinct non-empty subsets with the property: for any two different subsets among these $k$ subsets, if their intersection is non-empty, then the smallest element in their intersection is different from the larg... | 2^{99}-1 |
Given that $f(x)$ is an odd function on $\mathbb{R}$, $f(1)=1$, and $f(x+6)=f(x)+f(3)$ holds for any $x∈\mathbb{R}$, find the value of $f(2015)+f(2016)$. | -1 |
16. Stephen's calculator displays only one digit, as shown in the diagram. Unfortunately, the calculator is broken. Each time he switches it on, each of the seven bars will either illuminate (show up) or not, with probability 0.5 . The resultant display correctly shows one of the ten digits 0 - 9 with probability $\fra... | 173 |
Propositions:
① A plane parallel to two sides of a triangle is parallel to the third side of the triangle;
② A line perpendicular to two sides of a triangle is perpendicular to the third side;
③ A plane equidistant from the three vertices of a triangle is parallel to the plane containing the triangle.
Among the... | \text{B} |
Let $n$ be a positive integer, set $S_n = \{ (a_1,a_2,\cdots,a_{2^n}) \mid a_i=0 \ \text{or} \ 1, 1 \leq i \leq 2^n\}$. For any two elements $a=(a_1,a_2,\cdots,a_{2^n})$ and $b=(b_1,b_2,\cdots,b_{2^n})$ of $S_n$, define
\[ d(a,b)= \sum_{i=1}^{2^n} |a_i - b_i| \]
We call $A \subseteq S_n$ a $\textsl{Good Subset}$ if ... | 2^{n+1} |
Three, (25 points) Find the smallest positive integer $k$, such that there exists a positive integer $n$, satisfying $10^{n}=29 k+2$.
---
The translation maintains the original text's format and line breaks. | 3448275862 |
1. [2] Suppose $x$ is a rational number such that $x \sqrt{2}$ is also rational. Find $x$. | 0 |
$8 \quad$ Given that $x$ and $y$ are real numbers satisfying the following equations:
$$
x+x y+y=2+3 \sqrt{2} \text { and } x^{2}+y^{2}=6 \text {, }
$$
find the value of $|x+y+1|$.
(A) $1+\sqrt{3}$
(B) $2-\sqrt{3}$
(C) $2+\sqrt{3}$
(D) $3-\sqrt{2}$
(E) $3+\sqrt{2}$ | 3+\sqrt{2} |
NT4. Find all the three digit numbers $\overline{a b c}$ such that
$$
\overline{a b c}=a b c(a+b+c)
$$
| 135144 |
Let $x_{1}$ and $x_{2}$ be the two roots of the equation $x^{2}-2x-5=0$. Find the value of $\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}$. | -\frac{2}{5} |
5. Why are the parameters $a$ and $b$ of a uniformly distributed random variable $X$ estimated by the formulas
$$
a^{*}=\bar{x}_{\mathrm{B}}-\sqrt{3} \sigma_{\mathrm{B}}, \quad b^{*}=\bar{x}_{\mathrm{B}}+\sqrt{3} \sigma_{\mathrm{B}} ?
$$ | ^{*}=\bar{x}_{\mathrm{B}}-\sqrt{3}\sigma_{\mathrm{B}},\quadb^{*}=\bar{x}_{\mathrm{B}}+\sqrt{3}\sigma_{\mathrm{B}} |
Find all polynomials $P(x)$ of degree $1$ such that
$\underset {a\le x\le b}{max} P(x) - \underset {a\le x\le b}{min} P(x) =b-a$ , $\forall a,b\in R$ where $a < b$ | P(x) = x + c \text{ and } P(x) = -x + c |
In the triangle $ABC$, the area $S$ and the angle $C$ are known. Find the values of the sides $a$ and $b$ so that the side $c$ is as short as possible.
## SOLUTION:
The above text has been translated into English, preserving the original text's line breaks and format. | =b=\sqrt{\frac{2S}{\sinC}} |
The chord intercepted on the x-axis by a circle centered at point A(4, 4) on the parabola $y^2 = 4x$ and tangent to the directrix of the parabola has a length of __________. | 6 |
6. Let $a$, $b$, $c$, $a+b-c$, $a+c-b$, $b+c-a$, $a+b+c$ be 7 distinct prime numbers, and among $a$, $b$, $c$, the sum of two of them is 800. Let $d$ be the difference between the largest and smallest of these 7 prime numbers. Find the maximum possible value of $d$.
(2001, China Mathematical Olympiad) | 1594 |
A certain parking lot charges temporary parking fees based on the duration of parking. The charging standard is that each car is free for up to half an hour of parking, and any part exceeding half an hour is charged $3$ dollars per hour (rounded up to the nearest hour). Now, two individuals, A and B, are parking tempor... | \frac{49}{72} |
There are five people, named A, B, C, D, and E, standing in a row. It is required that both A and B are not adjacent to C. How many different arrangements are possible? | 36 |
Given that $\{a\_n\}$ is an arithmetic sequence, $a\_1=-26$, $a\_8+a\_13=5$, when the sum of the first $n$ terms $S\_n$ takes the minimum value, $n$ is equal to ( )
A: $8$
B: $9$
C: $10$
D: $11$ | B |
Given an ellipse $(C)$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{6}}{3}$, and it passes through the point $(0, 1)$. A quadrilateral $MNPQ$ has all four vertices on the ellipse $(C)$. The diagonal $MP$ has a slope of $(-1)$, and $MN = MQ$, $PN = PQ$.
(I) Find the ... | 3 |
11. As shown in the figure, the teacher wrote 9 numbers on the blackboard, and asked to select 3 numbers to form a three-digit number, with any two selected numbers not coming from the same row or the same column. A total of $\qquad$ three-digit numbers that can be divided by 4 can be formed. | 8 |
A and B are shooting free throws with the probabilities of scoring being $$\frac {1}{2}$$ and $$\frac {2}{5}$$, respectively. Both A and B shoot once from the free-throw line.
(1) Find the probability that both shots are successful.
(2) Find the probability that at least one shot is successful. | \frac {7}{10} |
3. From cities $A$ and $B$, which are $250 \mathrm{~km}$ apart, two motorcyclists set off towards each other at the same time. The speed of one of them is $10 \mathrm{~km} / \mathrm{h}$ greater than the speed of the other. After two hours of travel, they still had $30 \mathrm{~km}$ left to meet. Determine the speed of ... | 60\mathrm{~}/\mathrm{} |
5. As shown in Figure $3, A B$ is the diameter of the semicircle $\odot O$, $A C$ and $A D$ are both chords, and $\angle B A C = \angle C A D$. Which of the following equations is true? ( ).
(A) $A C^{2} = A B \cdot A D$
(B) $A B \cdot A D > A C^{2}$
(C) $A B + A D = 2 A C$
(D) $A B + A D < 2 A C$ | D |
## Task A-2.6.
Let $a$ and $b$ be natural numbers such that $a < b$ and let $S = \{a^2, a^2 + 1, a^2 + 2, \ldots, b^2\}$. Determine all pairs of numbers $a$ and $b$ for which exactly $1\%$ of the elements in the set $S$ are squares of natural numbers. | (1,100),(35,68),(49,60),(51,60),(65,68),(99,100) |
Among the following sets of numbers, the equal one is:
A: $-3^{2}$ and $-2^{3}$
B: $\left(-3\times 2\right)^{2}$ and $-3\times 2^{2}$
C: $-3^{2}$ and $\left(-3\right)^{2}$
D: $-2^{3}$ and $\left(-2\right)^{3}$ | \text{D} |
One, (20 points) If \( x, y \in [0,1] \), find the maximum value of
\[
x \sqrt{1-y} + y \sqrt{1-x}
\] | 1 |
4. The teacher asked the students: "How many of you reviewed math yesterday?"
Zhang: "No one."
Li: "One person."
Wang: "Two people."
Zhao: "Three people."
Liu: "Four people."
The teacher knew that some of them reviewed yesterday afternoon, and some did not. Those who reviewed spoke the truth, and those who did not revi... | 1 |
Given a function $f(x) = 2^x - \frac{1}{2^{|x|}}$ defined on $\mathbb{R}$.
(1) If $f(x) = \frac{3}{2}$, find the value of $x$;
(2) If $2^t f(2t) + mf(t) \geq 0$ holds for all $t \in [1, 2]$, find the range of the real number $m$. | [-5, +\infty) |
6. Let $x$ be an acute angle. Then the maximum value of the function $y=\sin x \cdot \sin 2 x$ is $\qquad$ | \frac{4 \sqrt{3}}{9} |
Given $n=5\int_{0}^{\pi}\sin x dx$, find the coefficient of $a^2bc^{n-3}$ in the expansion of the binomial $(2a-3b+c)^n$. | -4320 |
5. A music lover, having numbered five newly listened to compact discs with the digits $1,2,3,4,5$, placed them in the cassette player in a random order. What is the probability that discs №1 and №2 will be placed next to each other in the cassette player and in ascending order of their numbers? | \frac{1}{5} |
12) Let 4 positive integers $a, b, c, d$ be fixed such that $\frac{a}{b} \leq \frac{c}{d} \leq 1$. Which of the following inequalities is certainly true?
(A) $\frac{a+c}{b+d} \geq \frac{a}{b}+\frac{c}{d}$
(B) $\frac{a+c}{b+d} > 1$
(C) $\frac{a+c}{b+d} \leq \frac{c}{d}$
(D) $\frac{a+c}{b+d} < 1$
(E) none of the above | C |
Let $0 < a < 1$, and the function $f(x) = \log_a(a^{2x} - 2a^x - 2)$. Find the range of $x$ for which $f(x) < 0$.
A: $(-\infty, 0)$
B: $(0, +\infty)$
C: $(-\infty, \log_a 3)$
D: $(\log_a 3, +\infty)$ | \text{C} |
On the 13th, at a working meeting of the National Railway Administration, Fei Dongbin, the director of the National Railway Administration, reviewed the brilliant achievements of railway development in the new era over the past 10 years. From 2012 to 2022, the main framework of the "Four Vertical and Four Horizontal" h... | C |
The radian measure of the angle of inclination of the tangent line to the curve $y=x^{3}-2x+4$ at the point $(1,3)$ is ______. | \frac{\pi}{4} |
7.102. $2 \lg \sqrt{4-x}+\lg (6-x)=1$. | 5-\sqrt{11} |
Shapovalov A.V.
Among the angles of each lateral face of a pentagonal prism, there is an angle $\varphi$. Find all possible values of $\varphi$. | 90 |
Five identical rectangles are arranged to form a larger rectangle $P Q R S$, as shown. The area of $P Q R S$ is 4000 . The length, $x$, of each of the identical rectangles is closest to
(A) 35
(B) 39
(C) 41
(D) 37
(E) 33
(x^{4}-2x^{3}+2x^{2}-2x+2) |
Problem 1. (Option 1)
Calculate $x^{3}+3 x$, where $x=\sqrt[3]{2+\sqrt{5}}-\frac{1}{\sqrt[3]{2+\sqrt{5}}}$. | 4 |
Let $V$ be the region in the Cartesian plane consisting of all points $(x,y)$ satisfying the simultaneous conditions
$$|x|\le y\le|x|+3\text{ and }y\le4.$$Find the centroid of $V$. | (0, 2.31) |
Given that a railway bridge is $600$ meters long, if a train passes through the bridge, it takes a total of $30$ seconds from the beginning of the bridge to the end. The train is completely on the bridge for $20$ seconds. Find the length of the train. | 120 |
9. Find the maximum value of the function $f(x)=3 \sin x+4 \cos x$. | 5 |
Mr. Green receives a $10\%$ raise every year. His salary after four such raises has gone up by what percent?
$\text{(A)}\ \text{less than }40\% \qquad \text{(B)}\ 40\% \qquad \text{(C)}\ 44\% \qquad \text{(D)}\ 45\% \qquad \text{(E)}\ \text{more than }45\%$ | E |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $A=60^\circ$, $b=4$, and $S_{\triangle ABC}=2\sqrt{3}$, find the value of $a$. | a=2\sqrt{3} |
* *Proof: There exist infinitely many positive integers $n$ such that $n \mid 2^{n}+1$. Find all prime numbers $n$ that satisfy the above condition. | 3 |
The number of positive integer solutions to the inequality $x-1 \lt \sqrt{5}$ is ( ).
A: $3$
B: $4$
C: $5$
D: $6$ | A |
## 51. Mixed-Up Hands.
- Yesterday, between two and three o'clock, - said Colonel Crackham, - I looked at my watch and, mistaking the hour hand for the minute hand, I was wrong in my estimate of the time. The erroneous time was 55 minutes less than the actual time. What was the actual time? | 2 |
10. Let $M=\{1,2,3, \cdots, 1995\}, A \subseteq M$, and when $x \in A$, $19 x \notin A$. Find the maximum value of $|A|$.
untranslated portion:
设 $M=\{1,2,3, \cdots, 1995\}, A \subseteq M$, 且当 $x \in A$ 时, $19 x \notin A$, 求 $|A|$的最大值.
translated portion:
Let $M=\{1,2,3, \cdots, 1995\}, A \subseteq M$, and when $x ... | 1895 |
The opposite of $2$ is
A: $\frac{1}{2}$
B: $-2$
C: $2$
D: $-\frac{1}{2}$ | B |
In the diagram, the ratio of the number of shaded triangles to the number of unshaded triangles is
(A) $5: 2$
(B) $5: 3$
(C) $8: 5$
(D) $5: 8$
(E) $2: 5$
 | 5:3 |
$\begin{array}{l}\text { Given } \frac{x}{y+z}=\frac{y}{z+x}=\frac{z}{x+y} \\ =k \text {. Find the value of } k \text { . }\end{array}$ | k=\frac{1}{2} \text{ or } k=-1 |
The expression$\frac{x^2-3x+2}{x^2-5x+6}\div \frac{x^2-5x+4}{x^2-7x+12},$ when simplified is:
$\textbf{(A)}\ \frac{(x-1)(x-6)}{(x-3)(x-4)} \qquad\textbf{(B)}\ \frac{x+3}{x-3}\qquad\textbf{(C)}\ \frac{x+1}{x-1}\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2$ | 1 |
## 1. Six-digit number
Each digit of the six-digit number, starting from the thousands place, is equal to the sum of the two preceding digits (which are to its left). What is the three-digit ending of the largest number with this property?
## Result: $\quad 369$ | 369 |
17. During rush hour, Jerry's father needs 20 minutes to drive him to school. One day, they did not encounter rush hour traffic, and the car was 18 km/h faster, so it only took 12 minutes to get Jerry to school. The distance from home to school is ( ) km.
(A) 4
(B) 6
(C) 8
(D) 9
(E) 12 | 9 |
Given $\cos \alpha =\frac{1}{7}$, $\cos (\alpha -\beta )=\frac{13}{14}$, and $0 < \beta < \alpha < \frac{\pi }{2}$, find the value of $\beta$.
A: $\frac{\pi }{4}$
B: $\frac{\pi }{6}$
C: $\frac{\pi }{3}$
D: $\frac{5}{12}\pi $ | C |
7. Variant 1. Roma decided to create his own multiplication table. The rows correspond to the numbers 12, $13,14, \ldots, 60$, and the columns - to the numbers $15,16,17, \ldots, 40$. In the cells of the table, he wrote the products of the pairs of numbers from the row and column. How many of these products will be eve... | 962 |
The medians of a triangle are individually: $s_{1}=\frac{1}{2} \sqrt{673}, s_{2}=\frac{1}{2} \sqrt{592}$ and $s_{3}=\frac{1}{2} \sqrt{505}$; solve the triangle using trigonometric methods. | 13;14;15 |
The simplest common denominator of the fractions $\frac{1}{2x^{3}}$, $\frac{1}{6x^{2}y}$ is . | 6x^{3}y |
Given the plane vectors $\overrightarrow{a}=(2,1), \overrightarrow{b}=(2,x)$, and $(\overrightarrow{a}-2\overrightarrow{b}) \ / \!(2\overrightarrow{a}+3\overrightarrow{b})$, find the value of the real number $x$. | 1 |
Externally tangent circles with centers at points $A$ and $B$ have radii of lengths $5$ and $3$, respectively. A line externally tangent to both circles intersects ray $AB$ at point $C$. What is $BC$?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 4.8\qquad\textbf{(C)}\ 10.2\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 14.4$ | 12 |
5. Given $a_{1}=1, a_{2}=3$,
$$
a_{n+2}=(n+3) a_{n+1}-(n+2) a_{n} \text {, }
$$
when $m \geqslant n$, $a_{m}$ is divisible by 9. Then the minimum value of $n$ is $\qquad$ . | 5 |
When solving the equation $x^{2}-4x-11=0$ using the formula method, $\Delta =\left(\ \ \right)$
A: $-43$
B: $-28$
C: $45$
D: $60$ | D |
$9 \cdot 8$ Given three numbers $x, y=x^{x}, \quad z=x^{x^{x}}$, where $0.9<x<1.0$, the increasing order of these numbers is
(A) $x<z<y$.
(B) $x<y<z$.
(C) $y<x<z$.
(D) $y<z<x$.
(E) $z<x<y$.
(19th American High School Mathematics Examination, 1968) | A |
11. The car license plates in a certain town consist of four characters, the first of which must be a vowel $(A, E, I, O, U)$, the second and third must be different non-vowel letters, and the fourth must be a digit from $0$ to $9$. If a license plate is chosen at random from those that meet these conditions, the proba... | B |
59. During the winter vacation, Xue Mei read a storybook, planning to read 7 pages every day, but actually read 10 pages every day, and as a result, finished reading 12 days earlier than planned. This storybook has $\qquad$ pages. | 280 |
Given the complex number $z$ that satisfies $(1+i) z=(1-i)^{2}$, find $\bar{z}$ = ( )
A: $-1+i$
B: $1+i$
C: $1-i$
D: $-1-i$ | \text{A} |
Given that line $l$ is tangent to circle $O: x^2+y^2=1$ at point $C$ in the first quadrant, and intersects the $x$ and $y$ axes at points $A$ and $B$ respectively, the minimum value of $|AB|$ is \_\_\_\_\_\_. | 2 |
The traffic on a certain east-west highway moves at a constant speed of 60 miles per hour in both directions. An eastbound driver passes 20 west-bound vehicles in a five-minute interval. Assume vehicles in the westbound lane are equally spaced. Which of the following is closest to the number of westbound vehicles prese... | 200 |
Can a circle be circumscribed around quadrilateral $A B C D$, if $\angle A D C=30^{\circ}, A B=3, B C=4, A C=6$? | No |
Let $\tau (n)$ denote the number of positive integer divisors of $n$. Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$. | 540 |
Given that $a$ and $b$ are real numbers, then "$|a+b|=|a|+|b|$" is a ( ) condition for "$ab>0$".
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Sufficient and necessary condition
D: Neither sufficient nor necessary condition | \text{B} |
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