problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
4. The sum of all roots of the equation $[3 x+1]=2 x-\frac{1}{2}$ is $(\quad)$
A. -1
B. -2
C. $-\frac{5}{4}$
D. 0 | -2 |
2. On the written math test, the first problem was not solved by $12 \%$ of the students, $32 \%$ solved it partially, and the remaining 14 students solved it correctly. How many students were in this class? | 25 |
5. A5 (NET 2) ${ }^{\mathrm{IMO}}$ Let $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$ be a regular hexagon. Each of its diagonals $A_{i-1} A_{i+1}$ is divided into the same ratio $\frac{\lambda}{1-\lambda}$, where $0<\lambda<1$, by a point $B_{i}$ in such a way that $A_{i}, B_{i}$, and $B_{i+2}$ are collinear ( $i \equiv$ $1, \... | \frac{1}{\sqrt{3}} |
$31 \cdot 53$ satisfies the system of equations
$$
\left\{\begin{array}{l}
x^{2}-3 x y+2 y^{2}-z^{2}=31, \\
-x^{2}+6 y z+2 z^{2}=44, \\
x^{2}+x y+8 z^{2}=100 .
\end{array}\right.
$$
The number of ordered triples $(x, y, z)$ is
(A) 0.
(B) 1.
(C) 2.
(D) A finite number greater than 2.
(E) Infinitely many.
(31st American... | A |
Find the value of $\cos(\alpha - \beta)$ given that $\cos(\alpha + \beta) = \frac{1}{5}$ and $\tan\alpha \tan\beta = \frac{1}{2}$. | \frac{3}{5} |
Order the numbers $2^{300}, 10^{100}$, and $3^{200}$ from least to greatest, and prove that your ordering is correct. | 2^{300}<3^{200}<10^{100} |
1. A rhombus with a side length of 1 and an acute angle of $60^{\circ}$ is covered by three equal circles. Find the smallest possible radius.
The rhombus with a side length of 1 and an acute angle of $60^{\circ}$ is covered by three equal circles. Find the smallest possible radius. | \frac{1}{2} |
Given vectors $\overrightarrow{a}=(\sin \theta, -2)$ and $\overrightarrow{b}=(\cos \theta, 1)$, if $\overrightarrow{a} \parallel \overrightarrow{b}$, then $\tan 2\theta=$ ______. | \dfrac {4}{3} |
In the rectangular coordinate system on the plane, with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the polar coordinates of two points $M$ and $N$ on the line $l$ are $(3,π)$ and $(√3, \frac{π}{2})$, respectively.
(I) Let $P$ be a moving point on the line segment $MN$. Fi... | 3 |
\%In the Cartesian coordinate system $x O y$, the coordinates of points $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ are both one-digit positive integers, the angle between $O A$ and the positive direction of the $x$-axis is greater than $45^{\circ}$, the angle between $O B$ and the positive direction ... | 1985 |
Example 3 Given that the longest diagonal of a regular octagon is equal to $a$, and the shortest diagonal is equal to $b$. Then the area of this regular octagon is ( ). ${ }^{[3]}$
(A) $a^{2}+b^{2}$
(B) $a^{2}-b^{2}$
(C) $a+b$
(D) $a b$
$(2009$, Beijing Middle School Mathematics Competition (Grade 8)) | ab |
Observe the following equations: $a + b = 1$, $a^2 + b^2 = 3$, $a^3 + b^3 = 4$, $a^4 + b^4 = 7$, $a^5 + b^5 = 11$, ..., then $a^{10} + b^{10} = \ $( )
A: 28
B: 76
C: 123
D: 199 | \text{C} |
8. The solution set of the inequality
$$
\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9
$$
is $\qquad$ . | [-\frac{1}{2},0)\cup(0,\frac{45}{8}) |
The eccentricity of the ellipse $\dfrac{x^{2}}{a+8} + \dfrac{y^{2}}{9} = 1$ is $e = \dfrac{1}{2}$. Find the value of $a$.
A: $10$ or $-\dfrac{7}{2}$
B: $4$ or $-\dfrac{5}{4}$
C: $4$ or $-\dfrac{7}{2}$
D: $10$ or $-\dfrac{5}{4}$ | \text{B} |
Given that $P$ is a point on the circle $(x-3)^{2}+(y-4)^{2}=4$, then the maximum distance from point $Q(\cos \alpha ,\sin \alpha )$ to point $P$ is ______. | 8 |
50. 24 cows can finish all the grass on a 4-hectare pasture in 42 days. 36 cows can finish all the grass on an 8-hectare pasture in 84 days. 10 hectares of pasture can support $\qquad$ cows for 63 days. | 50 |
In rectangle $ABCD, AB=5$ and $BC=3$. Points $F$ and $G$ are on $\overline{CD}$ so that $DF=1$ and $GC=2$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$.
$\textbf{(A) } 10 \qquad\textbf{(B) } \frac{21}{2} \qquad\textbf{(C) } 12 \qquad\textbf{(D) } \frac{25}{2} \qquad\textbf{(E) } 15$ | \frac{25}{2} |
Determine the interval(s) where the function $f(x) = \left(\frac{1}{2}\right)^{x^2 - x - 1}$ is strictly increasing.
A) $\left(-\infty, \frac{1 - \sqrt{5}}{2}\right]$
B) $\left[\frac{1 + \sqrt{5}}{2}, +\infty\right)$
C) $(-\infty, \frac{1}{2})$
D) $(\frac{1}{2}, +\infty)$ | \text{C} |
Among the following statements, the number of incorrect propositions is ( )
① The negation of the proposition "For all $x \in \mathbb{R}, x^2 - x + 1 \leq 0$" is "$\exists x_{0} \in \mathbb{R}, x_{0}^{2} - x_{0} + 1 > 0$";
② If "p ∨ q" is a false proposition, then both p and q are false propositions;
③ "mn > 0" ... | B |
The function $f(x) = \begin{cases} a^{x}, & \text{for } x<1 \\ (a-3)x+4a, & \text{for } x\geq 1 \end{cases}$ satisfies that for any $x_1 \neq x_2$, the inequality $\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}} < 0$ holds. Then, the range of values for $a$ is \_\_\_\_\_\_. | 0 < a \leq \frac {3}{4} |
Example 2-62 A binary number of $2n$ bits composed of $n$ ones and $n$ zeros, requires that the cumulative number of 1s is greater than the cumulative number of 0s when scanning the first $2n-1$ bits from left to right. Find the number of such numbers. This problem can be reduced to finding the number of paths from poi... | \frac{1}{n}\binom{2n-2}{n-1}=h_{n+1} |
Four (18 points) As shown in Figure 3, $ABCD$ is a square with a side length of 1, $U$ and $V$ are points on $AB$ and $CD$ respectively, $AV$ intersects with $DU$ at point $P$, and $BV$ intersects with $CU$ at point $Q$. Find the maximum area of quadrilateral $PUQV$. | \frac{1}{4} |
The relationship in magnitude among $\sin \frac{1}{2}$, $\cos \frac{1}{2}$, and $\tan \frac{1}{2}$ is ( )
A: $\sin \frac{1}{2} > \cos \frac{1}{2} > \tan \frac{1}{2}$
B: $\cos \frac{1}{2} > \tan \frac{1}{2} > \sin \frac{1}{2}$
C: $\tan \frac{1}{2} > \sin \frac{1}{2} > \cos \frac{1}{2}$
D: $\tan \frac{1}{2} > \cos \... | \text{B} |
During the second Wuzhen Internet Conference, in order to enhance the security level while also facilitating accommodation arrangements, it was decided to arrange hotel accommodations for personnel from five participating countries. These five countries had to choose one among three hotels, labeled a, b, and c, with th... | D |
Among the following functions, which one is an odd function and is an increasing function within its domain?\\((\\) \\()\\)
A: $y=e^{x}$
B: $y=\tan x$
C: $y=x^{3}-x$
D: $y=\ln \frac {2+x}{2-x}$ | D |
Let $n$ be an integer greater than 1, and its all positive divisors are $d_{1}, d_{2}, \cdots, d_{k}$, where $1=d_{1}<d_{2}<\cdots<d_{k}=n$. It is known that $d_{2}-d_{1}, d_{3}-d_{2}, \cdots, d_{k}-d_{k-1}$ are all the positive divisors of another positive integer (all distinct but not necessarily in increasing order)... | 2^{}(\in{N}^{*}) |
Let $ABCD$ be a square of side length $4$. Points $E$ and $F$ are chosen on sides $BC$ and $DA$, respectively, such that $EF = 5$. Find the sum of the minimum and maximum possible areas of trapezoid $BEDF$.
[i]Proposed by Andrew Wu[/i] | 16 |
942. A wire of length $20 m$ is required to fence a flower bed, which should have the shape of a circular sector. What radius of the circle should be taken so that the area of the flower bed is maximized? | 5\mathrm{~} |
$i^{2010} =$ （ ）
A: $2i$
B: $-i$
C: $-1$
D: $1$ | \text{C} |
A right-angled triangle has a hypotenuse $a$, while the sum of three times one leg and four times the other is as large as possible. What are the lengths of the legs in these cases? | \frac{3}{5},\frac{4}{5} |
9.3. Add a digit to the left and right of the number 10 so that the resulting number is divisible by 72.
$$
(5-6 \text { grade) }
$$ | 4104 |
Before a total solar eclipse - according to calculations - at 9:13, the distance between the centers of the Sun and Moon disks was $4 \frac{7}{10}$ lunar diameters. Both disks had the same apparent size and both were moving in the same apparent direction, from west to east. The Moon traveled $1 \frac{1}{16}$ lunar diam... | )14:01,b)12:59\frac{34}{47} |
Given the sets $A=\{x \,|\, |x|<1\}$ and $B=\{x \,|\, -2<x<0\}$, then $A \cap B=$ ? | (-1, 0) |
Find the value of $\cos^{2} \frac{\pi}{12} - \sin^{2} \frac{\pi}{12} = \_\_\_\_\_\_\_\_.$ | \frac{\sqrt{3}}{2} |
Factorization: $x^{2}-25y^{2}=\_\_\_\_\_\_.$ | (x-5y)(x+5y) |
Task B-4.1. (20 points) Solve the system of equations (in the set of natural numbers)
$$
\left\{\begin{array}{l}
\binom{x}{y}=\binom{x}{y+2} \\
\binom{x}{2}=153
\end{array}\right.
$$ | 8 |
It is known about the triangle $ABC$ that $3 BC = CA + AB$. Let the $A$-symmedian of triangle $ABC$ intersect the circumcircle of triangle $ABC$ at point $D$. Prove that $\frac{1}{BD}+ \frac{1}{CD}= \frac{6}{AD}$.
(Ercole Suppa, Italy)
| \frac{1}{BD} + \frac{1}{CD} = \frac{6}{AD} |
4. In $\triangle A B C$, $A B=15, A C=13$, and the altitude $A D=12$. Let the radius of the circle that can completely cover $\triangle A B C$ be $R$. Then the minimum value of $R$ is $\qquad$ | \frac{65}{8} \text{ or } \frac{15}{2} |
14. Let $z=\cos \theta+i\sin \theta(0<\theta<\pi)$, given that $\omega=\frac{1-(\bar{z})^{4}}{1+z^{4}},|\omega|=\frac{\sqrt{3}}{3}, \arg \omega<\frac{\pi}{2}$, find $\theta$. | \theta=\frac{\pi}{12} |
55. Roller Calibration. One of the components of a gasoline engine has the shape of a roller. To measure the thickness of the roller, a steel plate is used, in which 15 holes are drilled in a row with precisely established dimensions. The first hole has a diameter of 10 mm, and each subsequent hole has a diameter 0.04 ... | 4 |
3.4. A natural number $n$ is written with different digits, the sum of which is 44. What can the sum of the digits of the number $n-1$ be? Find all possible options.
Answer, option 1. 36 and 45 (two answers).
Answer, option 2. 38 and 47 (two answers).
Answer, option 3. 41 and 50 (two answers).
Answer, option 4. 43 ... | 3645 |
Example 8 When $s$ and $t$ take all real numbers, then the minimum value that $(s+5-3|\cos t|)^{2}+(s-2|\sin t|)^{2}$ can achieve is $\qquad$
(1989, National High School Mathematics Competition) | 2 |
【Question 5】
Two line segments are parallel, forming a pair of parallel line segments. As shown in the figure, among the 12 edges of a rectangular prism, there are $\qquad$ pairs of parallel line segments. | 18 |
An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have
$$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$
Find the largest constant $K = K(n)$ such... | K(n) = \frac{1-n}{2} |
$24 . N$ is the smallest number among those which leave a remainder of 2 when divided by 5, a remainder of 3 when divided by 7, and a remainder of 4 when divided by 9. Then the sum of the digits of $N$ is ( ).
(A) 4
(B) 8
(C) 13
(D) 22
(E) 40 | C |
【Question 3】
Several students stand in a row, Pippi counted the people on his left and right, there are 6 people wearing glasses, and 7 people not wearing glasses. There are $\qquad$ people in this row. | 14 |
From the seven digits $l$, $2$, $3$, $4$, $5$, $6$, $7$, the number of seven-digit positive integers with exactly two even numbers adjacent to each other is $\_\_\_\_\_\_$. | 2880 |
3. Problem: Let $\left\{x_{1}, x_{2}, x_{3}, \ldots, x_{n}\right\}$ be a set of $n$ distinct positive integers, such that the sum of any 3 of them is a prime number. What is the maximum value of $n$ ? | 4 |
9. The function $f(x)=$ $\frac{x+1}{x}+\frac{x+2}{x+1}+\cdots+\frac{x+2023}{x+2022}$ has a center of symmetry at $(a, b)$, then $2 a+b=$ $\qquad$ _. | 1 |
Example 1 (2003 Russian Mathematical Olympiad) Given in $\triangle A B C$, $\angle C$ is a right angle, $D$ is a point on side $A C$, $K$ is a point on side $B D$, and $\angle A B C=\angle K A D=\angle A K D$. Prove: $B K=2 D C$.
---
The translation maintains the original text's formatting and line breaks. | BK=2CD |
Fomin C.B.
Kolya and Vasya received 20 grades each in January, and Kolya received as many fives as Vasya received fours, as many fours as Vasya received threes, as many threes as Vasya received twos, and as many twos as Vasya received fives. Moreover, their average grade for January is the same. How many twos did Koly... | 5 |
Eva wrote down consecutive natural numbers: 1234567891011. . What digit did she write on the 2009th place?
(M. Volfová) | 0 |
In triangle $\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to the internal angles $A$, $B$, $C$ respectively, and satisfy $\left(b-a\right)\left(\sin B+\sin A\right)=c(\sqrt{3}\sin B-\sin C)$. Find the value of $A$. | \frac{\pi}{6} |
1.2. [4] On the sheet, the numbers 1 and $x$, which is not a natural number, are written. In one move, the reciprocal of any of the already written numbers or the sum or difference of two of the written numbers can be written on the sheet. Is it possible for the number $x^{2}$ to appear on the sheet after several moves... | x^2 |
Given that the sum of the first $n$ terms of a sequence of positive numbers $\{a\_n\}$ is $S\_n$, and it satisfies $S\_n= \frac {1}{4}(a\_n+1)^{2}$,
(1) Find the general term formula for the sequence $\{a\_n\}$;
(2) If $b\_n= \frac {1}{a\_n a_{n+1}}$, let $T\_n$ denote the sum of the first $n$ terms of the sequence $\{... | \frac {n}{2n+1} |
For any real number $x$, there exists an eigenvector for the matrix $$\begin{bmatrix} x & 2+m \\ 3-m & 3\end{bmatrix}.$$ What is the range of values for $m$? | [-2, 3] |
## Task 3 - 130833
In a square $A B C D$ with side length $a$, the side $A B$ is divided into 4 equal segments by the points $A_{1}, A_{2}, A_{3}$, the side $B C$ by the points $B_{1}, B_{2}, B_{3}$, the side $C D$ by $C_{1}, C_{2}, C_{3}$, and $D A$ by the points $D_{1}, D_{2}, D_{3}$. Furthermore, the segments $A_{1... | \frac{4}{15}^{2} |
At 7:00 a.m. yesterday, Sherry correctly determined what time it had been 100 hours before. What was her answer? (Be sure to include "a.m." or "p.m." in your answer.) | 3:000 |
Ribamko A.V.
In the lower left corner of a $n \times n$ chessboard, there is a knight. It is known that the minimum number of moves it takes for the knight to reach the upper right corner is equal to the minimum number of moves it takes to reach the lower right corner. Find $n$.
# | 7 |
From the numbers \\(1\\), \\(2\\), \\(3\\), and \\(4\\), two numbers are randomly selected without replacement. The probability that not both numbers are odd is ( ).
A: \\( \dfrac {1}{6}\\)
B: \\( \dfrac {5}{6}\\)
C: \\( \dfrac {1}{2}\\)
D: \\( \dfrac {1}{4}\\) | B |
Teacher gave Little Ming 10 math problems, requiring him to complete them in ascending order of their serial numbers. He must complete at least one problem per day. If time permits, he can do more, or even finish all of them in one day. The number of different ways Little Ming can complete the problems is ( ).
A: 55
B:... | D |
11. A picture, together with its frame, makes a square of side-length $80 \mathrm{~cm}$. The frame is $4 \mathrm{~cm}$ wide.
What percentage of the area of the square is covered by the frame?
A $15 \%$
B $18 \%$
C $19 \%$
D $20 \%$
E $24 \%$ | 19 |
Example 1.11 (Han Jingjun) Real numbers $a_{i}, b_{i}(i=1,2, \cdots, n)$ satisfy $\sum_{i=1}^{n} a_{i}=\sum_{i=1}^{n} b_{i}=1$, prove that
$$\sum_{1 \leqslant i<j \leqslant n} a_{i} a_{j}+\sum_{1 \leqslant i<j \leqslant n} b_{i} b_{j}+\sum_{i=1}^{n} a_{i} b_{i} \leqslant 1$$ | 1 |
Given triangle $ABC$, if $\sin A + 2\sin B\cos C = 0$, find the maximum value of $\tan A$. | \frac{1}{\sqrt{3}} |
147. Diophantine system. Solve the following system of equations in positive integers:
$$
\left\{\begin{aligned}
a^{3}-b^{3}-c^{3} & =3 a b c \\
a^{2} & =2(b+c)
\end{aligned}\right.
$$ | =2,b==1 |
What is the smallest possible perimeter of a triangle with integer coordinate vertices, area $\frac12$, and no side parallel to an axis? | \sqrt{5} + \sqrt{2} + \sqrt{13} |
5. If $a, b$ are nonzero real numbers such that $a^{2}+b^{2}=8 a b$, find the value of $\left|\frac{a+b}{a-b}\right|$. | \frac{\sqrt{15}}{3} |
8. In a $5 \times 5$ matrix, each element is either 0 or 1, and it satisfies: the sum of the elements in each of the five rows is equal, but the sum of the elements in the five columns are all different. The number of such matrices is $\qquad$ | 26400 |
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms $S_n$ satisfies $S_3=0$ and $S_5=-5$,
$(1)$ find the general formula for $\{a_n\}$;
$(2)$ find the sum of the first $n$ terms of the sequence $\left\{ \frac {1}{a_{2n-1}a_{2n+1}} \right\}$. | -\frac {n}{2n-1} |
Find the solution set of the inequality $2|x-1|-1 < 0$. | \frac{1}{2} < x < \frac{3}{2} |
8,9
On the plane, points $A(1 ; 2), B(2 ; 1), C(3 ;-3), D(0 ; 0)$ are given. They are the vertices of a convex quadrilateral $A B C D$. In what ratio does the point of intersection of its diagonals divide the diagonal $A C$? | 1:3 |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors, and the angle between them is $120^{\circ}$, find the value of $| \overrightarrow{a}-2 \overrightarrow{b}|$. | \sqrt{7} |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 1} \frac{\sqrt{x^{2}-x+1}-1}{\tan \pi x}$ | \frac{1}{2\pi} |
Guizhou Province belongs to the subtropical humid monsoon climate zone, with a warm and humid climate, and an average annual temperature of around $15^{\circ}\mathrm{C}$. The unique climate characteristics make Guizhou an ideal destination for leisure travel and summer vacations. A school research and practice group is... | C |
Problem 16. (5 points)
Maxim Sergeevich works as a financial manager in a large industrial company. He has long dreamed of acquiring a large plot of land with a house on it. Maxim Sergeevich saved money for a down payment on a mortgage for about two years, and finally, when a sufficient amount was in his account, he c... | MaximSergeevich |
6 . For the parabola $y^{2}=4 x$, a line is drawn through its focus intersecting the parabola at points $P, Q$. Then, the equation of the locus of the midpoint of segment $P Q$ is (
(A) $y=2 x-1$
(D) $y^{2}=2 x-2$
(C) $y'=-2 x+1$
(D) $y^{2}=-2 x+2$ | B |
## Task 1.
Let $a, b, c, d$ be real numbers such that
$$
\begin{gathered}
a+b+c+d=10 \\
ab+bc+cd+da=25 \\
abc+bcd+cda+dab=50
\end{gathered}
$$
Prove that then $a^{2}+b^{2}+c^{2}+d^{2}=30$. | 30 |
4. Solve the system of equations $\left\{\begin{array}{l}x+\sqrt{x+2 y}-2 y=\frac{7}{2}, \\ x^{2}+x+2 y-4 y^{2}=\frac{27}{2}\end{array}\right.$. | (\frac{19}{4};\frac{17}{8}) |
Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible... | 13 |
Find the range of real number $m$ such that the equation $x^{2}-2x+m+1=0$ has two positive roots for $x$. | -1 < m \leq 0 |
In rectangle $ABCD$, point $M$ is the midpoint of $AB$ and $P$ is a point on side $BC$. The perpendicular bisector of $MP$ intersects side $DA$ at point $X$. Given that $AB = 33$ and $BC = 56$, find the least possible value of $MX$.
[i]Proposed by Michael Tang[/i] | 33 |
1. First solution: Since the Rabbit ran at a speed twice that of Alice, by the time Alice arrived at the Duchess's, the Rabbit was again halfway. Since he was 10 minutes late, Alice spent 20 minutes on half the journey, and 40 minutes on the entire journey. | 40 |
Given a function $f(x)$ defined on $\mathbb{R}$, if $f(x)$ is an odd function and $f(x+1)$ is an even function, and when $0 \leq x \leq 1$, $f(x) = x^2$, then $f(2015) =$ （ ）
A: $-1$
B: $1$
C: $0$
D: $2015^2$ | A |
Palmer correctly computes the product of the first $1,001$ prime numbers. Which of the following is NOT a factor of Palmer's product?
$\text{(A) }2,002\qquad\text{(B) }3,003\qquad\text{(C) }5,005\qquad\text{(D) }6,006\qquad\text{(E) }7,007$ | 7,007 |
5. Given that the sum of 2008 consecutive positive integers is a perfect square. Then the minimum value of the largest number is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 2133 |
Given two non-coincident lines $l_1$ and $l_2$ with slopes $k_1$ and $k_2$ respectively, then "lines $l_1 \parallel l_2$" is a condition for "$k_1 = k_2$" to hold true. Choose the correct option:
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Neither sufficient nor necessary cond... | D |
Problem 12.5. A plane intersects a tetrahedron $A B C D$ and divides the medians of the triangles $D A B, D B C$ and $D C A$ through $D$ in ratios $1: 2,1: 3$ and $1: 4$ from $D$, respectively. Find the ratio of the volumes of the two parts of the tetrahedron cut by the plane.
Oleg Mushkarov | 1:47 |
Given vectors $\overrightarrow {AB}=(0,1)$, $\overrightarrow {BC}=(1,0)$, then vector $\overrightarrow {AC}=$ （ ）
A: (-1,1)
B: (-1,0)
C: (1,1)
D: (0,-1) | C |
Task B-2.2. Determine all complex numbers $z$ such that $z^{3}=\bar{z}$. | z_{1}=0,\quadz_{2}=i,\quadz_{3}=-i,\quadz_{4}=1,\quadz_{5}=-1 |
$\sqrt{2}\cos 30^{\circ}=\_\_\_\_\_\_.$ | \frac{\sqrt{6}}{2} |
20. There is a pentagon $A B C D E$. If the vertices $A$, $B$, $C$, $D$, $E$ are colored with one of the three colors: red, yellow, green, such that adjacent vertices are colored differently, then there are a total of different coloring methods. | 30 |
If proposition P: For all $x \in \mathbb{R}$, $\cos x \leq 1$, then ( )
A: $\neg P$: There exists an $x_0 \in \mathbb{R}$, $\cos x_0 > 1$
B: $\neg P$: For all $x \in \mathbb{R}$, $\cos x > 1$
C: $\neg P$: There exists an $x_0 \in \mathbb{R}$, $\cos x_0 \geq 1$
D: $\neg P$: For all $x \in \mathbb{R}$, $\cos x \geq ... | \text{A} |
3. The sum of the first $n$ terms of the sequence $\left\{a_{n}!\right.$ is $S_{n}$, given that $a_{1}=4, a_{n}=S_{n-1}+2 n+1$, then $a_{2003}=(\quad)$.
A. $3 \times 2^{2003}-2$
B. $6 \times 2^{2003}-2$
C. $11 \times 2^{21002}-2$
D. $11 \times 2^{2001}-2$ | 11\times2^{2001}-2 |
Yesterday, Alex, Beth, and Carl raked their lawn. First, Alex and Beth raked half of the lawn together in $30$ minutes. While they took a break, Carl raked a third of the remaining lawn in $60$ minutes. Finally, Beth joined Carl and together they finished raking the lawn in $24$ minutes. If they each rake at a constant... | 3 |
197. Test Series. Professor Tester conducts a series of tests, based on which he assigns the test subject an average score. After answering the last test, John realized that if he had scored 97 points on this last test, his average score would have been 90. On the other hand, if he had scored only 73 points on the last... | 8 |
A bag contains 6 black balls and 4 white balls. If 2 balls are drawn without replacement, find:$(1)$ The probability of drawing 2 black balls;$(2)$ The probability of drawing a black ball on the second draw given that a black ball was drawn on the first draw. | \frac{5}{9} |
Which of the following equations is a linear equation in two variables?
A: $xy=2$
B: $\frac{1}{x}+y=3$
C: $3x+y^{2}=1$
D: $2x+y=5$ | D |
The number of sets $A$ satisfying $\{0,1,2\} \subset A \subseteq \{0,1,2,3,4,5\}$ is $\boxed{7}$. | 7 |
For the function $f(x)=\sin(x+e^{x})$, the value of $f''(0)$ is
A: $1$
B: $2$
C: $3$
D: $0$ | B |
Given that $\overline{z}$ is the complex conjugate of the complex number $z = \frac{1}{2} - i$, find $(z - \overline{z})^{2016}(\ \ \ \ )$
A: $2^{2016}$
B: $-2^{2016}$
C: $2^{2016}i$
D: $-i$ | A |
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