problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
The radius of circle $O_1$ is 3 cm, and the radius of circle $O_2$ is 2 cm. The distance between their centers, $O_1O_2$, is 4 cm. The positional relationship between these two circles is ( )
A: Contained within each other
B: Tangent internally
C: Intersecting
D: Tangent externally | C |
Four athletes participate in a table tennis competition, with every two athletes playing one match and determining the winner and loser. Let the number of wins and losses for the i-th athlete be xi and yi respectively (i=1,2,3,4). The incorrect conclusion is ( ):
A: x1+x2+x3+x4\=y1+y2+y3+y4
B: x1+x2+x3+x4\=y1+y2+y3+y4
... | D |
If the sequence $\{ \frac{a_n}{n} + 1 \}$ is a geometric sequence with a common ratio of $2$, and $a_1 = 1$, then the sum of the first $n$ terms of the sequence $\{ a_n \}$ is $\_\_\_\_\_\_\_\_\_.$ | \frac{(n - 1) \cdot 2^{n+2} - (n^2 + n - 4)}{2} |
In the Cartesian coordinate system $(xOy)$, given points $A(-1,0)$ and $B(0,1)$, the number of points $P$ that satisfy $PA^{2}-PB^{2}=4$ and lie on the circle $x^{2}+y^{2}=4$ is $\_\_\_\_$. | 2 |
A square with a side length of 12 meters, its area is square meters. | 144 |
815. Find the differentials of the functions:
1) $y=x^{3}$ at the point $x=0$, if $\left.\Delta x=0.3 ; 2\right) y=x^{3}-x^{2}+$ $+3 x-1$; 3) $r=\varphi^{4}+2^{\sin 3 \varphi}$; 4) $y=\ln \left(x^{2}+1\right)+\operatorname{arctg} \sqrt{x}$ at the point $x=1$, if $\Delta x=0.1$. | 0.125 |
Given that $x=a$ is a solution to the equation $x^{2}-5x-1=0$, find $3a^{2}-15a$. | 3 |
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the equality
$$ f([x] y)=f(x)[f(y)] . $$
holds for all $x, y \in \mathbb{R}$. Here, by $[x]$ we denote the greatest integer not exceeding $x$. (France) Answer. $f(x)=$ const $=C$, where $C=0$ or $1 \leq C<2$. | f(x) = C, \text{ where } C = 0 \text{ or } 1 \leq C < 2 |
Example 1. If $a>1, b$ is a positive rational number, $a^{b}+a^{-0}$ $=2 \sqrt{2}$, find the value of $a^{b}-a^{-b}$. | 2 |
Evaluate or simplify:
1. $\frac{\sqrt{1 - 2\sin 15^{\circ}\cos 15^{\circ}}}{\cos 15^{\circ} - \sqrt{1 - \cos^{2} 165^{\circ}}}$
2. Given $|\vec{a}| = 4$, $|\vec{b}| = 2$, and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{2\pi}{3}$, find the value of $|\vec{a} + \vec{b}|$. | 2\sqrt{3} |
Which of the following transformations, from left to right, represents factorization?
A: $(a-1)(a+1) = a^2 - 1$
B: $m^2 - 4 = (m+2)(m-2)$
C: $a^2 - b^2 + 1 = (a+b)(a-b) + 1$
D: $2\pi R + 2\pi r = 2\pi r + 2\pi R$ | B |
Which natural numbers $m$ and $n$ satisfy the equation $2^{n}+1=m^{2}$? | =n=3 |
52. Solve the equation $2^{x}=-x^{2}+2 x+8$. | x_{1}\approx-1.9,x_{2}\approx2.7 |
A speaker talked for sixty minutes to a full auditorium. Twenty percent of the audience heard the entire talk and ten percent slept through the entire talk. Half of the remainder heard one third of the talk and the other half heard two thirds of the talk. What was the average number of minutes of the talk heard by memb... | 33 |
11. Let the three interior angles $A, B, C$ of $\triangle ABC$ form an arithmetic sequence, and the reciprocals of the three side lengths $a, b, c$ also form an arithmetic sequence. Try to find $A, B, C$. | A=B=C=60 |
For each positive number $x$, let
$f(x)=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2} {\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}$.
The minimum value of $f(x)$ is
$\textbf{(A) }1\qquad \textbf{(B) }2\qquad \textbf{(C) }3\qquad \textbf{(D) }4\qquad \textbf{(E) }6$ | 6 |
6. Given positive numbers $a, b, c$ satisfying $a+b+c=13$. Then the minimum value of $\frac{a^{2}+b^{3}+c^{4}+2019}{10 b+123 c+26}$ is $\qquad$ . | 4 |
Birgit has a list of four numbers. Luciano adds these numbers together, three at a time, and gets the sums 415, 442, 396, and 325. What is the sum of Birgit's four numbers? | 526 |
Problem 4. Find the number of all natural numbers $n, 4 \leq n \leq$ 1023, such that their binary representations do not contain three consecutive equal digits.
Emil Kolev
| 228 |
9. As shown in Figure 2, on the right branches of the hyperbolas \(x^{2}-y^{2}=1\) and \(\frac{x^{2}}{4}-\frac{y^{2}}{9}=1\), there are points \(A, B\) and \(C, D\) respectively, such that quadrilateral \(A B C D\) is a square with sides parallel to the coordinate axes. Find the area of square \(A B C D\) (accurate to ... | 0.8506 |
Given sets $A=\{2^a, 3\}$, $B=\{2, 3\}$. If $A \cup B = \{1, 2, 3\}$, then the value of the real number $a$ is ____. | 0 |
40. Segment $\overline{A B}$ with endpoints at points $A(-3 ; 2)$ and $B(4 ;-5)$ is divided by point $C$ in the ratio $\lambda=-3$. Find the coordinates of the point of division.
Fig. 8 | C(7.5,-8.5) |
Point $M(\sqrt{2},1)$ is on the ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a > b > 0)$, and the sum of the distances from point $M$ to the two foci of the ellipse is $2\sqrt{5}$.
$(1)$ Find the equation of the ellipse $C$;
$(2)$ Given a moving line $y=k(x+1)$ intersects the ellipse $C$ at points $A$ and $B$. If $P\... | \frac{4}{9} |
An increasing arithmetic sequence with infinitely many terms is determined as follows. A single die is thrown and the number that appears is taken as the first term. The die is thrown again and the second number that appears is taken as the common difference between each pair of consecutive terms. Determine with proo... | 27 |
The y-intercept of the line $y = -\sqrt{3}x + 4$ on the $y$-axis is ________. | 4 |
11. a) How many integers less than 1000 are not divisible by 5 or 7?
b) How many of these numbers are not divisible by 3, 5, or 7? | 457 |
Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i,j)$ and $(j,i)$ do not both appear for any $i$ and $... | 761 |
$$
\begin{array}{l}
\left(\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2020}\right)\left(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2019}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2020}\right)\left(\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2019}\right) \\
=
\end{array}
$$ | \frac{1}{2020} |
Given that the function $f(x)$ is an even function defined on $(-\infty, +\infty)$. When $x \in (-\infty, 0)$, $f(x) = x - x^4$. Find the expression for $f(x)$ when $x \in (0, +\infty)$. | -x^4 - x |
Problem 1. In two tubs, there are $4 \frac{1}{6}$ and $3 \frac{1}{2}$ liters of water respectively. How much water needs to be transferred from the first tub to the second tub so that after the transfer, both tubs have the same amount of water? Explain your answer! | \frac{1}{3} |
## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{1+2+3+\ldots+n}{\sqrt{9 n^{4}+1}}
$$ | \frac{1}{6} |
1. The Mad Hatter's clock is fast by 15 minutes per hour, while the March Hare's clock is slow by 10 minutes per hour. One day, they set their clocks by the Dormouse's clock (which is stopped and always shows 12:00) and agreed to meet at 5 o'clock in the evening for their traditional
five-o'clock tea. How long will the... | 2 |
Three, (12 points) Let the equation about $x$ be $x^{2}+\left(\frac{a}{x}\right)^{2}-7 x$ $-\frac{7 a}{x}+2 a+12=0$ and suppose it has equal roots. Find the value of $a$.
untranslated portion:
设关于 $x$ 的方程 $x^{2}+\left(\frac{a}{x}\right)^{2}-7 x$ $-\frac{7 a}{x}+2 a+12=0$ 有相等鳫根. 求 $a$ 的值.
translated portion:
Let t... | 4 \text{ or } 2 \frac{1}{4} |
2. If $3 \sin ^{2} \alpha+2 \sin ^{2} \beta-2 \sin \alpha=0$, then the range of $\cos ^{2} \alpha+\cos ^{2} \beta$ is
A. $\left[\frac{3}{2}, \frac{7}{2}\right]$
B. $\left[\frac{3}{2}, 2\right]$
C. $\left[\frac{14}{9}, 2\right]$
D. $[0,2]$ | [\frac{14}{9},2] |
If the slope of the tangent line to the curve $y=f(x)$ at a point $(x_{0}, f(x_{0}))$ is $2$, then the curve cannot be ( )
A: $y=\sin 2x$
B: $y=x^{3}+2x$
C: $y=\frac{1}{\ln x}$
D: $y=xe^{x}$ | \text{C} |
If the function $f(x)= \sqrt {8-ax-2x^{2}}$ is an even function, then the domain of this function is \_\_\_\_\_. | [-2,2] |
Let $m$ and $n$ be any two odd numbers, with $n$ less than $m$.
The largest integer which divides all possible numbers of the form $m^2-n^2$ is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$ | \textbf{(D)} |
10. Given a sequence $\left\{a_{n}\right\}$ with 100 terms, satisfying $a_{1}=0, a_{100}=475$, and $\left|a_{k+1}-a_{k}\right|=5(k=1,2, \cdots, 99)$, then the number of different sequences that meet the conditions is $\qquad$. | 4851 |
1. (8 points) The calculation result of the expression $(11 \times 24-23 \times 9) \div 3+3$ is | 22 |
The negation of the proposition "There exists an $x \in \mathbb{R}$, such that $x^2+x+1<0$" is ( )
A: "For any $x \in \mathbb{R}$, $x^2+x+1<0$"
B: "For any $x \in \mathbb{R}$, $x^2+x+1\geq0$"
C: "There exists an $x \in \mathbb{R}$, such that $x^2+x+1\geq0$"
D: "There does not exist an $x \in \mathbb{R}$, such that $x^2... | \text{B} |
If $x \gt y$, which of the following expressions is incorrect?
A: $x-3 \gt y-3$
B: $\frac{x}{3} \gt \frac{y}{3}$
C: $x+3 \gt y+3$
D: $-3x \gt -3y$ | D |
5. The solution set of the system of equations $\left\{\begin{array}{l}\sqrt{x+\frac{1}{y}}-\sqrt{x+y-3}=\sqrt{3}, \\ 2 x+y+\frac{1}{y}=6\end{array}\right.$
(1998, Shandong Province Junior High School Mathematics Competition) | (x, y)=(2,1),(4,-1) |
Example 8 When $m$ satisfies what conditions, the equation
$$
(m+3) x^{2}-4 m x+2 m-1=0
$$
has roots of opposite signs and the absolute value of the negative root is greater than the positive root? | -3<m<0 |
SG. 1 Given $\tan \theta=\frac{5}{12}$, where $180^{\circ} \leq \theta \leq 270^{\circ}$. If $A=\cos \theta+\sin \theta$, find the value of $A$. | -\frac{17}{13} |
It's not difficult to verify that the power of $3^{m}$, where $m=3^{2}$, starts with the digit sequence 1968. - On the other hand, someone claims that in the sequence of powers of $2^{n}$ - where $n$ is a natural number - the first one that starts with the digit sequence 1968 has more than $2^{k}$ digits, where $k=2^{3... | 868 |
4. On the plane, there are four points $A, B, C, D$ such that $A B=B C=C D, B D=D A=A C$. Find the angles of the quadrilateral with vertices at these points. | 72,72,108,108 |
In acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, satisfying $\frac{{bcosC}}{{ccosB}}=\frac{{1+cos2C}}{{1+cos2B}}$.$(1)$ If $cosA=\frac{1}{3}$, find $\cos B$;$(2)$ If $b=5$ and $cosA=\frac{3}{4}$, find $a$. | a = \frac{5\sqrt{2}}{2} |
Let $A B C D$ and $E F G H$ be squares with side lengths 33 and 12, respectively, with $E F$ lying on side $D C$ (as shown in the figure below). Let $X$ be the intersection point of segments $H B$ and $D C$. Suppose that $\overline{D E}=18$.
,(18;2) |
282. Solve the equation:
$$
x^{4}-5 x^{3}+8 x^{2}-5 x+1=0
$$ | 1,1,\frac{3\\sqrt{5}}{2} |
Example 4 Let $S=\frac{1}{1^{3}}+\frac{1}{2^{3}}+\cdots+\frac{1}{2011^{3}}$.
Then the integer part of $4 S$ is ( ). ${ }^{[2]}$
(A) 4
(B) 5
(C) 6
(D) 7
(2011, "Mathematics Weekly" Cup National Junior High School Mathematics Competition) | 4 |
4. Connect the right focus $F$ of the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ with a moving point $A$ on the ellipse, and construct a square $F A B C (F, A, B, C$ in clockwise order). When point $A$ moves around the ellipse once, the trajectory equation of the moving point $C$ is | \frac{(x-\sqrt{5})^{2}}{4}+\frac{(y-\sqrt{5})^{2}}{9}=1 |
The angles of a triangle are in the ratio $2: 3: 4$. The largest angle in the triangle is
(A) $100^{\circ}$
(B) $60^{\circ}$
(C) $80^{\circ}$
(D) $90^{\circ}$
(E) $160^{\circ}$ | 80 |
Let $T$ be a triangle with area 1 . We let $T_{1}$ be the medial triangle of $T$, i.e. the triangle whose vertices are the midpoints of sides of $T$. We then let $T_{2}$ be the medial triangle of $T_{1}, T_{3}$ the medial triangle of $T_{2}$, and so on. What is the sum of the areas of $T_{1}, T_{2}, T_{3}, T_{4}, \ldot... | \frac{1}{3} |
a) Find all solutions of the system of equations (1)-(3). - Furthermore, provide a system of equations that differs from the given one by exactly one coefficient and
b) has infinitely many solutions,
c) which is not satisfied by any triple of real numbers $x, y, z$.
$$
\begin{gathered}
2 x+3 y+z=1 \\
4 x-y+2 z=2 \\
... | x=\frac{1}{2},y=0,z=0 |
207. Zaimu is promised a reward in the form of the larger of two parts, which add up to 20, and the product of these parts is 96. How great is the reward? | 12 |
Example 1.8 Let the number of 4-digit even numbers with distinct digits formed by $1,2,3,4,5,6$ be $N$, and their sum be denoted as $M$. Find $N$ and $M$. | N=180,M=680040 |
The zero point of the function $f(x)=2^{x}- \frac {2}{x}-a$ is within the interval $(1,2)$, then the range of the real number $a$ is \_\_\_\_\_\_. | (0,3) |
One, (20 points) Find the integer part of $\left(\frac{1+\sqrt{5}}{2}\right)^{19}$. | 9349 |
## Task A-4.3. (4 points)
Determine the real number $A$, given that in the expansion of the polynomial
$$
\left(1+x^{4}\right)^{12}+A\left(x\left(1-x^{2}\right)^{2}\right)^{12}
$$
the coefficient of $x^{12}$ is 100. | -120 |
Given the center of a circle is at (1, 2), find the standard equation of the circle that is tangent to the x-axis. | (x - 1)^2 + (y - 2)^2 = 4 |
Sure, here is the translated text:
```
Solve the equation:
\[
\left|\begin{array}{ccc}
-4 & 2 & -4+3 i \\
3 & -5 & 2-5 i \\
-4+3 i & 2-5 i & x
\end{array}\right|=0
\]
``` | x=1+5i |
Among $7$ integers from $2$ to $8$, randomly select $3$ different numbers. The number of different ways to select $3$ numbers whose product is a multiple of $3$ is ______. | 25 |
If the distance from one focus of the hyperbola \\(x^{2}- \frac {y^{2}}{b^{2}}=1\\) to one of its asymptotes is \\(2 \sqrt {2}\\), then the focal length of this hyperbola is \\((\quad)\)
A: \\(3\\)
B: \\(6\\)
C: \\(2 \sqrt {2}\\)
D: \\(4 \sqrt {2}\\) | B |
Given that there are countless lines in a plane that are parallel to another plane, what is the relationship between these two planes?
A: They are definitely parallel
B: They definitely intersect
C: They are parallel or intersect
D: They definitely coincide | \text{C} |
A game has the following rules: There are 5 red balls and 5 yellow balls in a pocket. You draw 5 balls at a time. If all 5 balls are the same color, you get 100 points. If 4 of the balls are the same color and 1 is different, you get 50 points. Otherwise, you get no points. What is the expected score when Zhang draws b... | \frac{75}{7} |
The radii of the top and bottom bases of a frustum are 2 and 3, respectively. The area of the circle formed by the midsection is \_\_\_\_\_\_. | \frac{25\pi}{4} |
If the complex number $z$ satisfies $(2+i)z=1+3i$, then the point corresponding to $z$ in the complex plane is in
A: the first quadrant
B: the second quadrant
C: the third quadrant
D: the fourth quadrant | \text{A} |
In a convex quadrilateral $ABCD$, the angles in $B$ and $C$ are greater than $120^{\circ}$. Show that
$$|AC| + |BD| > |AB|+|BC|+|CD|.$$ | |AC| + |BD| > |AB| + |BC| + |CD| |
A hexagon is called equiangular when it has six equal internal angles. Consider the equiangular hexagon $A B C D E F$ with sides $3, y, 5, 4, 1$, and $x$, as shown in the figure below. Determine the unknown lengths $x$ and $y$.
^{\log _{2}\left(\frac{9 x}{4}\right)}=\frac{729}{x^{4}}$. | \frac{243}{4},\frac{1}{9} |
1. Nine positive real numbers $a_{1}, a_{2}, \cdots, a_{9}$ form a geometric sequence, and
$$
a_{1}+a_{2}=\frac{3}{4}, a_{3}+a_{4}+a_{5}+a_{6}=15 .
$$
Then $a_{7}+a_{8}+a_{9}=$ . $\qquad$ | 112 |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. If $m > 1$, and $a_{m-1} + a_{m+1} - a_m^2 - 1 = 0$, $S_{2m-1} = 39$, then $m = \underline{\quad}$. | 20 |
Given the function $f(x)=\frac{1}{9}x^3+x^2-2ax+1$, if the function $f(x)$ has an extremum on the interval $(1,2)$, then the range of real number $a$ is ______. | \left(\frac{7}{6}, \frac{8}{3}\right) |
5.1. (14 points) In an acute-angled triangle $A B C$, angle $A$ is equal to $35^{\circ}$, segments $B B_{1}$ and $C C_{1}$ are altitudes, points $B_{2}$ and $C_{2}$ are the midpoints of sides $A C$ and $A B$ respectively. Lines $B_{1} C_{2}$ and $C_{1} B_{2}$ intersect at point $K$. Find the measure (in degrees) of ang... | 75 |
21.5. (Austria, 83). Find all values of $a$ for which the roots $x_{1}, x_{2}, x_{3}$ of the polynomial $x^{3}-6 x^{2}+a x+a$ satisfy the equation
$$
\left(x_{1}-3\right)^{3}+\left(x_{2}-3\right)^{3}+\left(x_{3}-3\right)^{3}=0
$$ | -9 |
Given the function $f(x)=x^{3}-2x^{2}+x+t$ with a domain of $\mathbb{R}$
(1) Find the intervals where $f(x)$ is monotonically increasing or decreasing.
(2) If $f(x)$ has three real roots in $\mathbb{R}$, find the range of values for the real number $t$. | -\frac{4}{27} < t < 0 |
4. Let $s=\frac{1}{(1+x)^{2}}+\frac{1}{(1-x)^{2}}$. Then, the relationship between $s$ and 2 is ( ).
(A) $s=2$
(B) $s<2$
(C) $s>2$
(D) The relationship between $s$ and 2 depends on the value of $x$ | D |
36. [24] (Mathematicians A to Z) Below are the names of 26 mathematicians, one for each letter of the alphabet. Your answer to this question should be a subset of $\{A, B, \cdots, Z\}$, where each letter represents the corresponding mathematician. If two mathematicians in your subset have birthdates that are within 20 ... | 24 |
1. In the complex plane, the vertices $A, B, C$ of the right triangle $\triangle ABC$ correspond to the complex numbers $z+1, 2z+1, (z+1)^2$, respectively, with $A$ being the right-angle vertex, and $|z|=2$. Let the set $M=\left\{m \mid z^{m} \in \mathbf{R}, m \in \mathbf{N}_{+}\right\}, P=\left\{x \left\lvert\, x=\fra... | \frac{1}{7} |
10. Let $m$ be the product of all positive integer divisors of 360,000 . Suppose the prime factors of $m$ are $p_{1}, p_{2}, \ldots, p_{k}$, for some positive integer $k$, and $m=p_{1}^{e_{1}} p_{2}^{e_{2}} \cdot \ldots p_{k}^{e_{k}}$, for some positive integers $e_{1}, e_{2}, \ldots, e_{k}$. Find $e_{1}+e_{2}+\ldots+e... | 630 |
In an extracurricular basketball shooting competition, it is stipulated that each student shoots from a fixed point 10 times. It was measured that the average number of successful shots for the students in a class was 4.0. Among them, the average number of successful shots by male students was 25% higher than that of f... | 3.6 |
1. Given the sets
$$
A=\left\{1,2, \frac{1}{2}\right\}, B=\left\{y \mid y=x^{2}, x \in A\right\} \text {. }
$$
Then $A \cap B=(\quad)$.
(A) $\left\{\frac{1}{2}\right\}$
(B) $\{2\}$
(C) $\{1\}$
(D) $\varnothing$ | C |
10. For any positive integer $n \geqslant 1, 1+2+\cdots+n$ when expressed as a decimal, the last digit of the number can take which of the following values? | 0,1,3,5,6,8 |
The function $y=2\sin^2x$ is ( )
A: An even function with a period of $2\pi$
B: An even function with a period of $\pi$
C: An odd function with a period of $2\pi$
D: An odd function with a period of $\pi$ | \text{B} |
## Problem Statement
Calculate the definite integral:
$$
\int_{-3}^{0}\left(x^{2}+6 x+9\right) \sin 2 x \, d x
$$ | -\frac{17+\cos6}{4} |
4. In the theater, there are $k$ rows of seats. 770 spectators came to the theater and sat down (possibly not occupying all the seats). After the intermission, all the spectators forgot which seats they had and sat down differently. For what largest $k$ will there definitely be 4 spectators who sat in the same row both... | 16 |
Given $a= \int_{-1}^{1} 5x^{\frac{2}{3}}dx$, find the constant term in the binomial expansion of $(\sqrt{t}-\frac{a}{6t})^{a}$ $\quad$ (Fill in the blank with the numerical value) | 15 |
Given the function $f(x) = (x+1)\ln x - a(x-1)$.
(I) When $a=4$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$.
(II) If $f(x) > 0$ when $x \in (1, +\infty)$, find the range of values for $a$. | (-\infty, 2] |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{b}|=3$, and the projection of $\overrightarrow{a}$ on the direction of $\overrightarrow{b}$ is $\frac{3}{2}$, then $\overrightarrow{a} \cdot \overrightarrow{b}=$ ______. | \frac{9}{2} |
Given the following pseudocode:
```
x ← 2
i ← 1
s ← 0
While i ≤ 4
s ← s × x + 1
i ← i + 1
End While
Print s
```
What is the output of the program after it is run? | 15 |
$32 \cdot 30$ is greater than 10, less than 100, when written in decimal, the number of integers whose digits, when reversed, are 9 more than the original number is
(A) 0.
(B) 1.
(C) 8.
(D) 9.
(E) 10.
(27th American High School Mathematics Examination, 1976) | 8 |
B1. John walks from home to school with a constant speed, and his sister Joan bikes twice as fast. The distance between their home and school is $3 \mathrm{~km}$. If Joan leaves home 15 minutes after John then they arrive to school at the same time. What is the walking speed (in $\mathrm{km} / \mathrm{h}$ ) of John? | 6 |
The value of $999+999$ is
(A) 2999
(B) 181818
(C) 1998
(D) 999999
(E) 198 | 1998 |
B4. We define the number $A$ as the digit 4 followed by the digit combination 84 repeated 2024 times, and the number $B$ as the digit combination 84 repeated 2024 times followed by a 7. To illustrate, we have
$$
A=4 \underbrace{84 \ldots 84}_{4048 \text { digits }} \text { and } B=\underbrace{84 \ldots 84}_{4048 \text... | \frac{4}{7} |
Let $\alpha$ and $\beta$ be two non-overlapping planes, and $l$ and $m$ be two non-overlapping lines in space. Which of the following statements is incorrect?
A: If $l \perp \alpha$ and $l \perp \beta$, then $\alpha \parallel \beta$.
B: If $l \perp \alpha$ and $m \perp \alpha$, then $l \parallel m$.
C: If $l \perp \... | D |
5. Find the largest positive integer $x$ such that $x$ is divisible by all the positive integers $\leq \sqrt[3]{x}$. | 420 |
5. A football is sewn from 256 pieces of leather: white ones in the shape of hexagons and black ones in the shape of pentagons. Black pentagons only border white hexagons, and any white hexagon borders three black pentagons and three white hexagons. Find the number of white hexagons on the football. | 160 |
According to the following descriptions, which one can determine the position of a point?
A: In front of the school library
B: Row 3, Seat 6 at Phoenix Cinema
C: Carriage No. 2 of Harmony Train
D: North-northeast direction at $40^{\circ}$ | B |
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