problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
The result of $-1÷(-5)÷(-\frac{1}{5})$ is:
A: $-1$
B: $-\frac{1}{25}$
C: $\frac{1}{25}$
D: $1$ | \text{A} |
Given the line 5x + 3y - 15 = 0, let 'a' be the x-intercept and 'b' be the y-intercept. Then, ( )
A: a=5, b=3
B: a=3, b=5
C: a=-3, b=5
D: a=-3, b=-5 | \text{B} |
[ Weighing ]
There are 2018 weights with masses of 1 g, 2 g, ..., 2018 g. The Hare placed two weights on one pan of the scales. The Wolf tried to balance them with two other weights on the other pan, but failed. Which weights could the Hare have taken?
# | (1\;2\);(1\;3\);(2016\;2018\);(2017\;2018\) |
20. (3 marks) A rectangular piece of paper has integer side lengths. The paper is folded so that a pair of diagonally opposite vertices coincide, and it is found that the crease is of length 65. Find a possible value of the perimeter of the paper.
(3 分) A rectangular piece of paper has integer side lengths. The paper i... | 408 |
Given $(a-2i)i=b-i$, where $a$, $b \in \mathbb{R}$ and $i$ is the imaginary unit, find the complex number $a+bi$. | -1 + 2i |
Given the function $f(x) = \cos \left( 2x - \frac{4\pi}{3} \right) + 2 \cos^2 x$.
(1) Find the maximum value of $f(x)$, and write out the set of $x$ values for which $f(x)$ attains its maximum;
(2) In $\triangle ABC$, let $a$, $b$, and $c$ denote the lengths of the sides opposite to angles $A$, $B$, and $C$, respecti... | 1 |
Example. Find the maximum and minimum values of the function
$$
y=\frac{10 x+10}{x^{2}+2 x+2}
$$
on the interval $[-1,2]$. | =0,M=5 |
12. The size order of $\sin 1, \sin 2, \sin 3, \sin 4, \sin 6$ is $\qquad$ | \sin2>\sin1>\sin3>\sin6>\sin4 |
A and B are playing chess. The probability of A winning is 60%, and the probability of A not losing is 80%. The probability of A and B drawing is \_\_\_\_\_\_. | 20\% |
Given the ellipse $C: \frac{x^2}{2} + y^2 = 1$ with its right focus at $F$ and the right directrix as $l$, point $A$ lies on $l$, and the line segment $AF$ intersects $C$ at point $B$. If $\overrightarrow{FA} = 3 \overrightarrow{FB}$, then the magnitude of $\overrightarrow{|AF|} =$ ( )
A: $\sqrt{2}$
B: $2$
C: $\sqrt... | A |
Example 9 The decimal representation of a natural number $x$ does not
contain the digit 0, and it satisfies $x \cdot \bar{x}=1000+\Pi(x)$, where $\bar{x}$ is the natural number obtained by writing the digits of $x$ in reverse order, and $\Pi(x)$ is the product of the digits of $x$. Find all such natural numbers $x$.
(1... | 24 \text{ or } 42 |
Given that $y=f\left(x\right)$ is an odd function defined on $R$, when $x \gt 0$, $f\left(x\right)=x^{2}+x+1$, then when $x \lt 0$, $f\left(x\right)=$____. | -x^2 + x - 1 |
Given that $α$ is an angle in the fourth quadrant and $\tan α=-\frac{3}{4}$, find the value of $\sin α=(\ \ \ )$.
A: $-\frac{3}{5}$
B: $\frac{3}{5}$
C: $\frac{4}{5}$
D: $-\frac{4}{5}$ | A |
If the function $f(x) = (a^2 - 2a - 3)x^2 + (a - 3)x + 1$ has both its domain and range as $\mathbb{R}$, then the range of $a$ is ( )
A: $a = -1$ or $3$
B: $a = -1$
C: $a > 3$ or $a < -1$
D: $-1 < a < 3$ | \text{B} |
A sixth of the students in a class wear glasses. Among those who wear glasses, a third are girls; in addition, four boys wear glasses. How many students are there in this class? | 36 |
2A. Let the complex number $i$ be given. Determine all complex numbers $z$ such that the number $a=\frac{u-\bar{u} z}{1-z}$ is a real number. | z\neq1 |
Given vectors $\overrightarrow{a} = (1, k)$ and $\overrightarrow{b} = (2, 2)$, and that $\overrightarrow{a} + \overrightarrow{b}$ is collinear with $\overrightarrow{a}$, find the cosine of the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | 1 |
The modulus of the complex number $z= \frac {-3+i}{2+i}$ is $\boxed{\text{answer}}$. | \sqrt {2} |
306. Lightning Rod. A strong gust of wind broke the lightning rod's pole, causing its top to hit the ground 20 m from the base of the pole. The pole was repaired, but it broke again under a gust of wind, 5 m lower than before, and its top hit the ground 30 m from the base.
What is the height of the pole? In both cases... | 50\mathrm{} |
XXIV OM - I - Problem 12
In a class of n students, a Secret Santa event was organized. Each student draws the name of the person for whom they are to buy a gift, so student $ A_1 $ buys a gift for student $ A_2 $, $ A_2 $ buys a gift for $ A_3 $, ..., $ A_k $ buys a gift for $ A_1 $, where $ 1 \leq k \leq n $. Assumin... | \frac{1}{n} |
4. For which values of the real parameter $m$ does the equation
$$
x^{2}-(m+1) x+2 m-4=0
$$
have real solutions, and at the same time, the sum of their squares is the smallest possible? | 1 |
Let $x<y<z$. Solve the following equation in the set of natural numbers:
$$
3^{x}+3^{y}+3^{z}=179415
$$ | 4,7,11 |
Calculate: $(2\sqrt{2}-1)^{2}+(1+\sqrt{3})(1-\sqrt{3})$. | 7-4\sqrt{2} |
7-5. In a row, there are 1000 toy bears. The bears can be of three colors: white, brown, and black. Among any three consecutive bears, there is a toy of each color. Iskander is trying to guess the colors of the bears. He made five guesses:
- The 2nd bear from the left is white;
- The 20th bear from the left is brown;
... | 20 |
Source: 2017 Canadian Open Math Challenge, Problem A2
-----
An equilateral triangle has sides of length $4$cm. At each vertex, a circle with radius $2$cm is drawn, as shown in the figure below. The total area of the shaded regions of the three circles is $a\cdot \pi \text{cm}^2$. Determine $a$.
[center][asy]
size(2.5c... | 2 |
Three. (This question is worth $\mathbf{50}$ points) Let the set $A=\{1,2, \cdots, n\}, X, Y$ be non-empty subsets of $A$ (allowing $X=Y$). The maximum element of $X$ and the minimum element of $Y$ are denoted as $\max X, \min Y$, respectively. Find the number of ordered pairs of sets $(X, Y)$ that satisfy $\max X>\min... | 2^{2n}-2^{n}(n+1) |
Given a line with a slope angle of 60° that passes through the origin and intersects the circle $x^2 + y^2 - 4x = 0$ at points A and B, find the length $|AB|$. | 2 |
3. Given are the sides $a = BC$ and $b = CA$ of triangle $ABC$. Determine the length of the third side if it is equal to the length of the corresponding height. For which values of $a$ and $b$ does the problem have a solution? | ^{2}=\frac{1}{5}(^{2}+b^{2}\\sqrt{D}) |
If the lateral surface area of a cylinder with a square cross-section is $4\pi$, then the volume of the cylinder is ( )
A: $\pi$
B: $2\pi$
C: $4\pi$
D: $8\pi$ | B |
4. (5th "Hope Cup" Senior High School Competition) The functions $f_{1}(x)=\left|\sin \frac{x}{2}\right| \cdot\left|\cos \frac{x}{2}\right|, f_{2}(x)=\sin \frac{2 x}{3} +\cos \frac{2 x}{3}, f_{3}(x)=\arccos (\sin x)$ have the smallest positive periods $T_{1}, T_{2}, T_{3}$, respectively, then
A. $T_{1}<T_{2}<T_{3}$
B. ... | C |
Given that the function $f(x+1)$ is an odd function defined on $\mathbb{R}$, if for any given unequal real numbers $x_{1}, x_{2}$, the inequality $x_{1}f(x_{1})+x_{2}f(x_{2}) > x_{1}f(x_{2})+x_{2}f(x_{1})$ always holds, then the solution set of the inequality $f(2-x) < 0$ is ( )
A: $(-\infty ,0)$
B: $(2,+\infty) $
... | D |
Reduce $$\frac{2}{\sqrt{4-3\sqrt[4]{5} + 2\sqrt[4]{25}-\sqrt[4]{125}}}$$ to its lowest form.
Then generalize this result and show that it holds for any positive $n$. | 1 + \sqrt[4]{5} |
4.13. The lateral edges of a triangular pyramid are pairwise perpendicular and have lengths $a, b$ and $c$. Find the volume of the pyramid. | \frac{}{6} |
5. As shown in Figure 1, given $\triangle A B C$, a tangent line to the circumcircle through point $A$ intersects the extension of $B C$ at point $P, \frac{P C}{P A}=\frac{\sqrt{2}}{2}$, point $D$ is on $A C$ such that $\frac{A D}{C D}=\frac{1}{2}$, and the extension of $P D$ intersects $A B$ at point $E$. Then the val... | A |
The standard equation of a parabola with the directrix $x=3$ is ( )
A: $y^2=-6x$
B: $y^2=-12x$
C: $y^2=6x$
D: $y^2=12x$ | \text{B} |
The minimum positive period of the function $y=\tan \left( \frac{\pi}{2}x- \frac{\pi}{3}\right)$ is ______. | 2 |
Given that $(a_n)$ is an arithmetic sequence, and $a_7-2a_4=-1$, $a_3=0$, then the common difference $d=$ ()
A: $- \frac{1}{2}$
B: $-2$
C: $ \frac{1}{2}$
D: $2$ | \text{A} |
3. To make the equation
$$
x^{4}+(m-4) x^{2}+2(1-m)=0
$$
have exactly one real root not less than 2, the range of values for $m$ is $\qquad$ | m \leqslant -1 |
The minimum value of $|x-4|+|x+2|$ is ______; the minimum value of $|\frac{5}{6}x-1|+|\frac{1}{2}x-1|+|\frac{2}{3}x-1|$ is ______. | \frac{1}{2} |
Let $m$ and $n$ be positive integers such that $x=m+\sqrt{n}$ is a solution to the equation $x^2-10x+1=\sqrt{x}(x+1)$. Find $m+n$. | 55 |
Given that the eccentricity of the ellipse $C$ with foci on the $x$-axis is $\frac{\sqrt{3}}{2}$, and it passes through the point $A(-2,1)$.
(1) Find the standard equation of the ellipse $C$;
(2) Line $l:x-y+2=0$ intersects the ellipse $C$ at points $P$ and $Q$, with $O$ being the origin, find the area of $\Delta OPQ... | \frac{4\sqrt{6}}{5} |
10.4. Find all functions $f(x)$ defined on the entire real line that satisfy the equation $f(y-f(x))=1-x-y$ for any $x$ and $y$. | f(x)=\frac{1}{2}-x |
4. Vasya thought of two numbers. Their sum equals their product and equals their quotient. What numbers did Vasya think of? | \frac{1}{2},-1 |
Evaluate the following expression:
$$\frac { \sqrt {3}tan12 ° -3}{sin12 ° (4cos ^{2}12 ° -2)}$$ | -4 \sqrt {3} |
36th BMO 2000 Problem 3 Find positive integers m, n such that (m 1/3 + n 1/3 - 1) 2 = 49 + 20 (6 1/3 ). | =48,n=288 |
Given the function $f(x) = x(x - c)^2$, the value of $c$ when the function has a minimum value at $x = 3$ is $(\ \ \ \ )$.
A: $3$ or $9$
B: $9$
C: $3$
D: $6$ | C |
Given the quadratic function $y=a(a+1)x^{2}-(2a+1)x+1$, when $a$ takes the values of $1, 2, 3, \dots, n$, the lengths of the line segments intercepted by the parabola on the $x$-axis are $d_{1}, d_{2}, d_{3}, \dots, d_{n}$ respectively. Find the limit $\lim\limits_{n→∞}(d_{1}+d_{2}+⋯+d_{n})$. | 1 |
The condition "$mn < 0$" is the ( ) for the equation "$mx^{2}+ny^{2}=1$" to represent a hyperbola with foci on the $y$-axis.
A: Necessary and sufficient condition
B: Neither necessary nor sufficient condition
C: Sufficient but not necessary condition
D: Necessary but not sufficient condition | D |
18.40 In parallelogram $ABCD$, the area is $10$, $AB=3$, and $BC=5$. Points $E$, $F$, and $G$ are taken on $AB$, $BC$, and $AD$ respectively, such that $AE=BF=AG=2$. A line through $G$ parallel to $EF$ intersects $CD$ at $H$. Then the area of quadrilateral $EFHG$ is
(A) 4.
(B) 4.5.
(C) 5.
(D) 5.5.
(E) 6.
(43rd America... | 5 |
Given non-zero vectors $\overrightarrow{a}$, $\overrightarrow{b}$, with $|\overrightarrow{a}|=m$, $|\overrightarrow{b}|=n$, if vector $\overrightarrow{c} = \lambda_1 \overrightarrow{a} + \lambda_2 \overrightarrow{b}$, then the maximum value of $|\overrightarrow{c}|$ is ( )
A: $\lambda_1 m + \lambda_2 n$
B: $|\lamb... | \text{B} |
Let the function $f(x) = (x-3)^3 + x - 1$, and let $\{a_n\}$ be an arithmetic sequence with a non-zero common difference. If $f(a_1) + f(a_2) + \ldots + f(a_7) = 14$, then the value of $a_1 + a_2 + \ldots + a_7$ is ( ).
A: 0
B: 7
C: 14
D: 21 | \text{D} |
11. (20 points) Let the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ satisfy:
$$
a_{1}=3, b_{1}=1
$$
and for any $n \in \mathbf{Z}_{+}$, we have
$$
\left\{\begin{array}{l}
a_{n+1}=a_{n}+b_{n}+\sqrt{a_{n}^{2}-a_{n} b_{n}+b_{n}^{2}}, \\
b_{n+1}=a_{n}+b_{n}-\sqrt{a_{n}^{2}-a_{n} b_{n}+b_{n}^{2}} .
\end{arr... | 9 |
The arithmetic mean of a set of $50$ numbers is $38$. If two numbers of the set, namely $45$ and $55$, are discarded,
the arithmetic mean of the remaining set of numbers is:
$\textbf{(A)}\ 38.5 \qquad \textbf{(B)}\ 37.5 \qquad \textbf{(C)}\ 37 \qquad \textbf{(D)}\ 36.5 \qquad \textbf{(E)}\ 36$ | 37.5 |
[ [Geometric progression]
It is known that the sum of the first n terms of a geometric progression consisting of positive numbers is $\mathrm{S}$, and the sum of the reciprocals of the first n terms of this progression is R. Find the product of the first n terms of this progression.
# | (S/R)^{n/2} |
A $\pm 1$-sequence is a sequence of 2022 numbers $a_{1}, \ldots, a_{2022}$, each equal to either +1 or -1. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \leqslant t_{1}<\ldots<t_{k} \leqslant 2022$ so that $t_{i+1}-t_{i} \leqslant 2$ for all $i$, and
$$ \left|\... | 506 |
3. (8 points) In the figure, the radius of the large circle is 20 cm, and the radii of the 7 small circles are all 10 cm. The area of the shaded region is $\qquad$ square centimeters ( $\pi$ is taken as 3.14). | 942 |
Which of the following equations is a quadratic equation?
A: $-6x+2=0$
B: $2x^{2}-y+1=0$
C: $x^{2}+2x=0$
D: $\dfrac{1}{x^{2}}+x=2$ | \text{C} |
Find the largest $K$ satisfying the following:
Given any closed intervals $A_1,\ldots, A_N$ of length $1$ where $N$ is an arbitrary positive integer. If their union is $[0,2021]$, then we can always find $K$ intervals from $A_1,\ldots, A_N$ such that the intersection of any two of them is empty. | K = 1011 |
4. From the town "U ${ }^{\prime}$ " to the town " $A x$ ", Ivan set off on his bicycle at $11^{00}$ AM, having traveled two fifths of the distance, he passed the town " $O x$ ", from which at that moment Peter set off on foot towards the town "Ux". At the moment when Ivan arrived in the town " $A x$ ", from there in t... | 5 |
If $f(x) = \ln(e^{3x} + 1) + ax$ is an even function, then $a = \_\_\_\_\_\_$. | -\frac{3}{2} |
Calculate the limit of the following sequence:
$$\underset{n\to \infty }{\lim }\ \,\frac{1+3+{{3}^{2}}+\cdots +{{3}^{n}}}{{{3}^{n}}+{{2}^{n}}}=\ \_\_\_\_\_\_\_\_\_\_\_\_.$$ | \frac{3}{2} |
Xiao Ming participated in the themed speech contest "Celebrating the 20th Birthday, Pursuing Dreams in Youth". He scored $9$ points in speech image, $8$ points in content, and $8$ points in effectiveness. If the scores of the three items are weighted by $30\%$, $40\%$, and $30\%$ respectively to determine the final sco... | 8.3 |
Given that $a$ and $b$ are two non-coincident lines, and $\alpha$ and $\beta$ are two different planes, then the correct proposition is ( )
A: If $a \perp \alpha$, $\beta \perp \alpha$, then $a \parallel \beta$
B: If $a, b \subset \alpha$, $a \parallel \beta$, $b \parallel \beta$, then $\alpha \parallel \beta$
C: If $a... | D |
In the polar coordinate system, the equation of line $l$ is $\rho\sin \left( \frac {\pi}{6}-\theta\right)=2$, and the equation of curve $C$ is $\rho=4\cos \theta$. Find the length of the chord cut by line $l$ on curve $C$. | 2 \sqrt {3} |
Let $f(x) = |\lg x|$, and $a$, $b$ are real numbers satisfying $f(a) = f(b) = 2f\left(\frac{a+b}{2}\right)$, where $0 < a < b$.
$(1)$ Prove that: $a < 1 < b$;
$(2)$ Prove that: $2 < 4b - b^{2} < 3$. | 2 < 4b - b^{2} < 3 |
6. Given the origin $O$ and point $A(2,-2)$, $B$ is a point on the coordinate axis. If $\triangle A O B$ is an isosceles triangle, then the number of such points $B$ is ( ) .
(A) 4
(B) 5
(C) 6
(D) 8 | D |
15. A triangle with all sides of different lengths has one altitude of 10 and another altitude of 15. The third altitude cannot be ( ).
A. 5
B. 11
C. 17
D. 20
E. 25 | A |
Consider the situation such that $ n$ circles with radius $ r$ are contained between 2 concentric circles with radius $ 1,\ 1\minus{}2r$ without intersecting one anoyher.Let $ r$ vary with being fixed $ n\geq 2.$
(1) Show that $ r$ should be satisfied $ 0<r\leq \frac{\sin \frac{\pi}{n}}{1\plus{}\sin \frac{\pi}{n}}.$... | r = \frac{2}{n + 4} |
11.6*. The perimeter of triangle $A B C$ is $2 p$. Points $M$ and $N$ are taken on sides $A B$ and $A C$ such that $M N \| B C$ and $M N$ is tangent to the inscribed circle of triangle $A B C$. Find the maximum value of the length of segment $M N$. | \frac{p}{4} |
2. Find the minimum value of $x^{2}+4 y^{2}-2 x$, where $x$ and $y$ are real numbers that satisfy $2 x+8 y=3$. | -\frac{19}{20} |
Kevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write? | 63 |
The complex number $z=(x^2-1)+(x-1)i$ is a pure imaginary number, then the real number $x=$. | -1 |
Find the primes $p$ such that there exist $x, y \in \mathbb{N}^{*}$ such that
$$
x\left(y^{2}-p\right)+y\left(x^{2}-p\right)=5 p
$$ | (1,4,2),(2,3,3),(3,4,7),(4,3,7) |
In a geometric sequence, the first term is $\frac{9}{8}$, the last term is $\frac{1}{3}$, and the common ratio is $\frac{2}{3}$. The number of terms $n$ equals \_\_\_\_\_\_. | 4 |
19. As shown in Figure 4, in rectangle $A B C D$, $A B=6, B C$ $=3$, point $E$ is on side $B C$, point $F$ is on side $C E$, $B E=$ $E F=F C$. Connecting $A E$ and $A F$, they intersect $B D$ at points $P$ and $Q$ respectively. If $B P: P Q: Q D=r: s: t\left(r, s, t \in \mathbf{Z}_{+},(r, s, t)=1\right)$, then $r+s+t=(... | E |
In the 2012 Hubei college entrance examination, a palindrome is defined as a positive integer that reads the same from left to right and from right to left, such as 22, 121, 3443, 94249, etc. It is evident that there are 9 two-digit palindromes: 11, 22, 33, ..., 99. There are 90 three-digit palindromes: 101, 111, 121, ... | 9 \times 10^n |
Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the ... | 93 |
Given points A(2,0), B(0,4), and O as the coordinate origin, the standard equation of the circumcircle of △AOB is _____. | (x-1)^2+(y-2)^2=5 |
8,9 [ Properties and characteristics of an isosceles triangle. ]
[ The ratio in which the bisector divides the side ]
In triangle $A B C$, $A C=8, B C=5$. A line parallel to the bisector of the external angle $C$ passes through the midpoint of side $A B$ and point $E$ on side $A C$. Find $A E$.
# | 1.5 |
In triangle $ABC$, which of the following judgments is correct?
A: $a=7,b=18,A=30^\circ$ has two solutions
B: $a=9,c=10,A=60^\circ$ has no solution
C: $b=6,c=9,B=45^\circ$ has two solutions
D: $a=28,b=24,A=150^\circ$ has one solution | D |
Three, (This question 20 points) Given a triangle with one side length of 2, the median on this side is 1, and the sum of the other two sides is $1+\sqrt{3}$. Find the area of the triangle.
| \frac{\sqrt{3}}{2} |
3. Given that $100^{2}+1^{2}=65^{2}+76^{2}=p q$ for some primes $p$ and $q$. Find $p+q$. | 210 |
To obtain the graph of the function $y=\sin (2x+\frac{\pi}{3})$, one only needs to shift the graph of the function $y=\sin 2x$ ( )
A: to the left by $\frac{\pi}{6}$ units
B: to the right by $\frac{\pi}{3}$ units
C: to the left by $\frac{\pi}{3}$ units
D: to the right by $\frac{\pi}{6}$ units | \text{A} |
2. The number $A$ in decimal form is represented as $A=\overline{7 a 631 b}$, where $a, b$ are non-zero digits. The number $B$ is obtained by summing all distinct six-digit numbers, including $A$, that are formed by cyclic permutations of the digits of $A$ (the first digit moves to the second position, the second to th... | 796317 |
7. (USS 5) For which digits $a$ do there exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a$? | a \in\{5,6\} |
1. Find natural numbers $a, b, c$, such that for any $n \in \mathbf{N}, n>2$, we have
$$
b-\frac{c}{(n-2)!}<\frac{2^{3}-a}{2!}+\frac{3^{3}-a}{3!}+\cdots+\frac{n^{3}-a}{n!}<b .
$$ | =5,b=9,c_{\text{}}=4 |
39. Calculate: $2379 \cdot 23782378-2378 \cdot 23792379$. | 0 |
8. Right triangles $M D C$ and $A D K$ have a common right angle $D$. Point $K$ lies on $C D$ and divides it in the ratio $2: 3$, counting from point $C$. Point $M$ is the midpoint of side $A D$. Find the sum of the degree measures of angles $A K D$ and $M C D$, if $A D: C D=2: 5$. | 45 |
The smallest positive period of the function $ y = \tan(2x + \frac{\pi}{4})$ is __________.
A: $ \frac{\pi}{4} $
B: $ \frac{\pi}{2} $
C: $ \pi $
D: $ 2\pi $ | B |
Let $\left\{a_{n}\right\}$ be an arithmetic sequence with the sum of its first $n$ terms denoted as $S_{n}$. If $S_{17}=255$ and $a_{10}=20$, then the common difference of the sequence $\left\{a_{n}\right\}$ is:
A: $3$
B: $4$
C: $5$
D: $6$ | C |
18.1. $[10.2$ (15 points)] A wooden parallelepiped, all sides of which are expressed in whole centimeters, was painted red, and then sawn parallel to the faces into cubes with a side of 1 cm. It turned out that one third of the resulting cubes have at least one red face, while the remaining two thirds have all faces un... | 18 |
Which of the following functions cannot be considered as a "Tai Chi function" of circle O: $x^2+y^2=25$, which divides both the circumference and the area of the circle O into two equal parts? ( )
A: $f(x)=4x^3+x$
B: $f(x)=\ln \frac{6-x}{6+x}$
C: $f(x)=\tan \frac{x}{2}$
D: $f(x)=e^x+e^{-x}$ | \text{D} |
9. In the example $(* *)^{3}=* * * 9$, replace the asterisks with digits to obtain a correct equation. | 19^{3}=6859 |
Example 7. Determine the number of roots of the equation
$$
z^{6}-6 z+10=0
$$
inside the circle $|z|<1$. | 0 |
3. Two numbers x and y satisfy the equation $280 x^{2}-61 x y+3 y^{2}-13=0$ and are the fourth and ninth terms, respectively, of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression. | -5 |
4. In the tetrahedron $O A B C$, $O A \perp$ plane $A B C, A B \perp$ $A C$, point $P$ satisfies $O P=l O A+m O B+n O C$, where $l, m, n$ are positive numbers and $l+m+n=1$. If the line $O P$ is formed by points that are equidistant from the planes $O B C$, $O C A$, and $O A B$, then the cosine value of the dihedral an... | \frac{n}{l} |
245. A discrete random variable $X$ is given by the distribution law
$$
\begin{array}{lll}
X & 0.3 & 0.6 \\
p & 0.2 & 0.8
\end{array}
$$
Using Chebyshev's inequality, estimate the probability that $|X-M(X)|<0.2$. | 0.64 |
# 4.1. Condition:
In front of the elevator stand people weighing 50, 51, 55, 57, 58, 59, 60, 63, 75, and 140 kg. The elevator's load capacity is 180 kg. What is the minimum number of trips needed to get everyone up? | 4 |
8. (5 points) As shown in the figure, in square $A B C D$, point $E$ is on side $A D$, and point $F$ is on side $D C$, with $A E=2 E D$ and $D F=3 F C$. The ratio of the area of $\triangle B E F$ to the area of square $A B C D$ is $\qquad$ . | \frac{5}{12} |
5. Let the function $f(x)$ be defined on $\mathbf{R}$, and consider the following three statements:
(1) The graph of a function satisfying $f(x+2)+f(2-x)=4$ is symmetric about the point $(2,2)$;
(2) The graph of a function satisfying $f(x+2)=f(2-x)$ is symmetric about the line $x=2$;
(3) The graphs of the functions $f(... | D |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.