problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Two bodies move uniformly, on two intersecting straight lines at right angles, towards the point of intersection; the speed of one is $3 \mathrm{~m}$ per $\mathrm{mp}$, the other's is $4 \mathrm{~m}$. The distance between the two bodies is initially $20 \mathrm{~m}$, but after $2 \mathrm{mp}$ it is only $10 \mathrm{~m}... | 12\mathrm{~},16\mathrm{~} |
7. In 2021 , a first class postage stamp cost 85 p and a second class postage stamp cost 66 p. In order to spend an exact number of pounds and to buy at least one of each type, what is the smallest total number of stamps that should be purchased?
A 10
B 8
С 7
D 5
E 2 | 7 |
Example 1 Given a curve $C: x^{2}+y^{2}-2 x-2 y+1=0$ and a tangent line $l$ that intersects the $x$-axis and $y$-axis at points $A$ and $B$ respectively. $O$ is the origin, $|O A|=a,|O B|=b, a>2$, $b>2$.
(1) Prove: $(a-2)(b-2)=2$;
(2) Find the equation of the locus of the midpoint of segment $A B$;
(3) Find the minimum... | 2\sqrt{2}+3 |
## 1. task
On a shelf in a bookstore, there are six mathematics books whose prices are $62 \mathrm{kn}, 63 \mathrm{kn}, 66 \mathrm{kn}, 68 \mathrm{kn}, 69 \mathrm{kn}$, and 71 kn. Both Jure and Duje bought books from this shelf, and Duje spent four times as much as Jure, with only one book remaining on the shelf. Whic... | 69\mathrm{kn} |
9.5. A football team coach loves to experiment with the lineup. During training sessions, he divides 20 available field players into two teams of 10 players each, adds goalkeepers, and arranges a game between the teams. He wants any two field players to end up on different teams at some training session. What is the mi... | 5 |
Given $f(\alpha) = \frac{\sin\left(\alpha - \frac{5\pi}{2}\right)\cos\left(\frac{3\pi}{2} + \alpha\right)\tan\left(\pi - \alpha\right)}{\tan\left(-\alpha - \pi\right)\sin\left(\pi - \alpha\right)}$.
$(1)$ Simplify $f(\alpha)$.
$(2)$ If $\cos\left(\alpha + \frac{3\pi}{2}\right) = \frac{1}{5}$ and $\alpha$ is an angle in... | f(\alpha) = \frac{2\sqrt{6}}{5} |
1. Real numbers $x, y, z$ (non-zero) satisfy the equation: $x+y=z$. Find all possible values that the expression $\frac{z}{x}+\frac{y}{z}-\frac{x}{y}+\frac{x}{z}+\frac{z}{y}-\frac{y}{x}$ can take. | 3 |
7. (2004 National College Entrance Examination Chongqing Paper) Let the function $f(x)=x(x-1)(x-a)(a>1)$.
( I ) Find the derivative $f^{\prime}(x)$, and prove that $f(x)$ has two distinct extreme points $x_{1} 、 x_{2}$;
(II) If the inequality $f\left(x_{1}\right)+f\left(x_{2}\right) \leqslant 0$ holds, find the range o... | \geqslant2 |
6. (8 points) Let for positive numbers $x, y, z$ the following system of equations holds:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=108 \\
y^{2}+y z+z^{2}=9 \\
z^{2}+x z+x^{2}=117
\end{array}\right.
$$
Find the value of the expression $x y+y z+x z$. | 36 |
Given vectors $a = (3,1)$, $b = (1,3)$, $c = (k,2)$, if $(a-c) \perp b$, then $k = \boxed{\_\_\_\_\_\_}$ | 0 |
3. Let $a, b \in \mathbf{N}_{+}$, and satisfy
$$
56 \leqslant a+b \leqslant 59,0.9<\frac{a}{b}<0.91 \text {. }
$$
Then $b^{2}-a^{2}$ equals ( ).
(A) 171
(B) 177
(C) 180
(D) 182 | B |
In triangle $A B C A B=15, B C=12, A C=18$. In what ratio does the center $O$ of the inscribed circle of the triangle divide the bisector $C M$? | 2:1 |
Problem 1. Let $n$ be a natural number. Find the least natural number $k$ for which there exist $k$ sequences of 0 's and 1's of length $2 n+2$ with the following property: any sequence of 0 's and 1 's of length $2 n+2$ coincides in at least $n+2$ positions with some of these $k$ sequences.
| 4 |
The inhabitants of a planet speak a language only using the letters $A$ and $O$. To avoid mistakes, any two words of equal length differ at least on three positions. Show that there are not more than $\frac{2^n}{n+1}$ words with $n$ letters. | \frac{2^n}{n+1} |
The arithmetic square root of $(-4)^2$ is ( )
A: -4
B: 4
C: ±4
D: 16 | \text{B} |
315. Find the variance and standard deviation of a random variable $X$ that is uniformly distributed in the interval $(a, b)$. | D(X)=\frac{(b-)^2}{12},\sigma(X)=\frac{b-}{2\sqrt{3}} |
$10 \cdot 86$ Find all integers $n>3$, such that there exist $n$ points $A_{1}, A_{2}, \cdots$, $A_{n}$ and real numbers $r_{1}, r_{2}, \cdots, r_{n}$ in the plane, satisfying the following two conditions:
(1) No three points among $A_{1}, A_{2}, \cdots, A_{n}$ are collinear;
(2) For each triplet $\left\{A_{i}, A_{j}, ... | 4 |
6. (1995 National High School Mathematics Competition) Find all real numbers $p$ such that the cubic equation $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ has three roots that are all natural numbers.
| 76 |
Given the function $f\left(x\right)=\sin 2x-x$, $x\in \left(0,\pi \right)$, the point of maximum value of $f\left(x\right)$ is ______. | \frac{\pi}{6} |
The graph of the quadratic function $y=x^{2}+2x+c$ intersects the $x$-axis at two points $A(x_{1}$,$0)$ and $B(x_{2}$,$0)$, where $x_{1} \lt x_{2}$. Point $P\left(m,n\right)$ is a point on the graph. Which of the following statements is correct?
A: When $n \gt 0$, $m \lt x_{1}$
B: When $n \gt 0$, $m \gt x_{2}$
C: Wh... | D |
Solve for $x$: $2\log_{6}x=1-\log_{6}3$. | \sqrt{2} |
5. A passenger train always moves at a constant speed. When it passes a bridge that is $450 \mathrm{~m}$ long, it takes $33 \mathrm{~s}$ from the moment the front of the train enters the bridge until the rear of the train leaves the bridge; when it passes through a tunnel that is $760 \mathrm{~m}$ long, the entire trai... | \frac{600}{37}<t \leqslant \frac{300}{17} |
Consider an arithmetic sequence $\{a_n\}$ with the sum of its first $n$ terms given by $S_n$. Given that $a_2 + S_3 = -4$ and $a_4 = 3$, find the common difference of the sequence. | 2 |
Given a line $l$ with a slope of $1$ that intersects the parabola $y^{2}=2px (p > 0)$ at two distinct points $A$ and $B$ located above the $x$-axis, let $K_{1}$ and $K_{2}$ denote the slopes of the lines $OA$ and $OB$, respectively. Determine the range of values for $K_{1}+K_{2}$. | (4,+\infty) |
A $0,1,1,1,4,4,4,6$ digits can form how many 8-digit numbers that are divisible by 4? | 340 |
Two unit-radius spheres intersect in such a way that the resulting 3 parts have equal volume. How far apart are the centers of the spheres? | 4\cos\frac{4\pi}{9}\approx0.695 |
7. In the polar coordinate system, the distance from the point $\left(2,-\frac{\pi}{6}\right)$ to the line $\rho \sin \left(\theta-30^{\circ}\right)=1$ is | \sqrt{3}+1 |
161. $2^{x}+2^{x-1}-2^{x-3}=44$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
161. $2^{x}+2^{x-1}-2^{x-3}=44$. | 5 |
5. The sum of 2021 consecutive natural numbers is $S$, and $S$ is divisible by 9. Then the minimum value of $S \div 2021$ is
$\qquad$ | 1017 |
From the five numbers $0$, $1$, $2$, $3$, $4$, select four digits to form a four-digit number with no repeated digits. The number of even numbers among them is ( ).
A: $36$
B: $60$
C: $72$
D: $96$ | B |
11 Given that $x_{1}$ is a root of the equation $x+\lg x=3$, and $x_{2}$ is a root of the equation $x+10^{x}=3$, then the value of $x_{1}+x_{2}$ is ( ).
(A) 3
(B) 6
(C) 2
(D) 1 | 3 |
Given the universal set $U=\{0,1,2,3,4,5,6,7,8,9\}$, set $A=\{0,1,3,5,8\}$, and set $B=\{2,4,5,6,8\}$, then $(\complement_U A) \cap (\complement_U B)$ equals ( )
(A) $\{5,8\}$ (B) $\{7,9\}$
(C) $\{0,1,3\}$ (D) $\{2,4,6\}$ | \text{B} |
5. 3. 15 ** Let $D$ be a point inside acute $\triangle A B C$ such that $\angle A D B=\angle A C B+90^{\circ}$, and $A C \cdot B D=A D \cdot B C$. Find the value of $\frac{A B \cdot C D}{A C \cdot B D}$. | \sqrt{2} |
The positional relationship between the line $ax-y+2a=0$ and the circle $x^2+y^2=9$ is ( )
A: Separate
B: Intersect
C: Tangent
D: Uncertain | \text{B} |
Among the following numbers: $-1$, $3.14$, $0$, $\sqrt{2}$, $\pi$, $\frac{1}{3}$, the number of irrational numbers is ( ).
A: $1$
B: $2$
C: $3$
D: $4$ | B |
Given the equation $5x^{2}+kx-6=0$, one of its roots is $2$. Find its other root and the value of $k$. | -7 |
11.055. The center of the upper base of a cube is connected to the midpoints of the sides of the lower base. A quadrilateral angle is formed, each plane angle of which is equal to $\alpha$. Prove that $30^{\circ}<\alpha<45^{\circ}$. | 30<\alpha<45 |
The ratio of the radius of the circle inscribed in a triangle to the radius of the circle circumscribed around the triangle can vary between what values? When is it the largest? | \varrho/r\leqq1/2 |
In the Cartesian coordinate system xOy, "ab > 0" is the condition for "the curve of the equation $ax^2+by^2=1$ to be an ellipse" (Fill in one of the following: "sufficient but not necessary", "necessary but not sufficient", "both necessary and sufficient", and "neither sufficient nor necessary"). | \text{necessary but not sufficient} |
8. $x+y+z=1$, where $x \geqslant 0, y \geqslant 0, z \geqslant 0$, let $f(z)=(z-x)(z-y)$, then the range of $f(z)$ is $\qquad$ | [-\frac{1}{8},1] |
$2 \cdot 121$ In the equation $\frac{x(x-1)-(m+1)}{(x-1)(m-1)}=\frac{x}{m}$, if the roots are equal, then:
(A) $m=1$.
(B) $m=\frac{1}{2}$.
(C) $m=0$.
(D) $m=-1$.
(E) $m=-\frac{1}{2}$.
(2nd American High School Mathematics Examination, 1951) | -\frac{1}{2} |
2. In $\triangle A B C$, it is known that $\angle A=120^{\circ}, A B=5, B C$ =7. Then $\frac{\sin B}{\sin C}=(\quad)$.
(A) $\frac{8}{5}$
(B) $\frac{5}{8}$
(C) $\frac{5}{3}$
(D) $\frac{3}{5}$ | D |
Example 4: Arrange 5 white stones and 10 black stones in a horizontal row, such that the right neighbor of each white stone must be a black stone. How many arrangements are there?
(1996, Japan Mathematical Olympiad Preliminary) | 252 |
Define the set operation: $A \odot B = \{z | z = xy(x+y), x \in A, y \in B\}$. Given the sets $A = \{0, 1\}$ and $B = \{2, 3\}$, find the sum of all elements in the set $A \odot B$. | 18 |
1B. Determine all pairs of natural numbers $(a, b)$ such that $a \leq b$ and
$$
\left(a+\frac{6}{b}\right)\left(b+\frac{6}{a}\right)=25
$$ | (1,4),(2,2),(1,9),(3,3) |
Example 9 Given that $a, b$ are positive numbers, and the parabolas $y=x^{2}+a x+2 b$ and $y=x^{2}+2 b x+a$ both have common points with the $x$-axis. Then the minimum value of $a^{2}+b^{2}$ is $\qquad$
$(2000$, National Junior High School Mathematics League) | 20 |
15. Let the quadratic function $f(x)=a x^{2}+b x+c(a, b, c \in \mathbf{R}$, $a \neq 0$ ) satisfy the following conditions:
(1) For $x \in \mathbf{R}$, $f(x-4)=f(2-x)$, and $f(x) \geqslant x$;
(2) For $x \in(0,2)$, $f(x) \leqslant\left(\frac{x+1}{2}\right)^{2}$;
(3) The minimum value of $f(x)$ on $\mathbf{R}$ is 0.
Find... | 9 |
Observe the following equations: $(x-1)(x+1)=x^{2}-1$, $(x-1)(x^{2}+x+1)=x^{3}-1$, $(x-1)(x^{3}+x^{2}+x+1)=x^{4}-1$. Therefore, if $(x-1)(x^{4}+x^{3}+x^{2}+x+1)=-2$, then the value of $x^{2023}$ is ( ).
A: $0$
B: $1$
C: $-1$
D: $2^{2023}$ | \text{C} |
15. (16 points) In an activity class, the teacher asked each student to make a rectangular box with a lid as shown in Figure 6. The length, width, and height of the rectangular box are $x \mathrm{~cm}, y \mathrm{~cm}$, and $z \mathrm{~cm}$, respectively. When Xiao Yang presented his box, he told the class: “The length,... | 22 |
Let the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ be $S_n$. If $a_1 + 8a_4 = 0$, then $\frac{S_3}{S_4} =$
A: $\frac{6}{5}$
B: $\frac{14}{15}$
C: $\frac{7}{15}$
D: $-\frac{3}{5}$ | A |
3. The tetrahedron S-ABC is empty, with three pairs of edges being equal, sequentially $\sqrt{34}, \sqrt{41}, 5$. Then the volume of the tetrahedron is ( ).
(A) 20
(B) $10 \sqrt{7}$
(C) $20 \sqrt{3}$
(D) 30 | A |
Given that $x=1$ is a solution to the equation $2-3(a+x)=2x$ with respect to $x$, the value of $a$ is:
A: $1$
B: $-2$
C: $-1$
D: $2$ | C |
Xiao Wang and Xiao Zhang each completed 20 difficult math problems. If a problem is answered correctly, a two-digit number of points is added, and if a problem is answered incorrectly, another two-digit number of points is deducted. Now Xiao Wang scored 328 points, and Xiao Zhang scored 27 points. The number of points ... | 25 |
Observe the solutions of the equation $(x-1)(x+2)=0$. | x_1=1, x_2=-2 |
7. Given that the circumcenter of $\triangle A B C$ is $O$, and
$$
\overrightarrow{A O} \cdot \overrightarrow{B C}+2 \overrightarrow{B O} \cdot \overrightarrow{C A}+3 \overrightarrow{C O} \cdot \overrightarrow{A B}=0
$$
then the minimum value of $\cot A+\cot C$ is $\qquad$ | \frac{2 \sqrt{3}}{3} |
If $$\alpha \in (0, \frac{\pi}{2})$$ and $$\sin^2\alpha + \cos2\alpha = \frac{1}{4}$$, then the value of $\tan\alpha$ is ( )
A: $$\frac{\sqrt{2}}{2}$$
B: $$\frac{\sqrt{3}}{3}$$
C: 1
D: $$\sqrt{3}$$ | D |
1. Given that $α$ and $β$ are acute angles, and $\cos α= \frac{4}{5}$, $\cos (α+β)=- \frac{16}{65}$, find the value of $\cos β$.
2. Given that $0 < β < \frac{π}{4} < α < \frac{3}{4}π$, $\cos ( \frac{π}{4}-α)= \frac{3}{5}$, $\sin ( \frac{3π}{4}+β)= \frac{5}{13}$, find the value of $\sin (α+β)$. | \frac{56}{65} |
Given the equation of circle C is $(x-1)^2+(y-1)^2=4$, and a tangent line is drawn from a point M on the line $x-y-6=0$ to circle C at point N, find the minimum value of $|MN|$.
A: $2\sqrt{3}$
B: $\sqrt{14}$
C: 4
D: 3$\sqrt{2}$ | \text{B} |
Suppose that $f(x)=a(x-b)(x-c)$ is a quadratic function where $a, b$ and $c$ are distinct positive integers less than 10. For each choice of $a, b$ and $c$, the function $f(x)$ has a minimum value. What is the minimum of these possible minimum values? | -128 |
The digits 1, 2, 3, 4 can be arranged to form twenty-four different four digit numbers. If these twenty-four numbers are then listed from smallest to largest, in what position is 3142 ?
(A) 13 th
(B) 14th
(C) 15 th
(D) 16 th
(E) 17 th | 14th |
The graph of the function $f(x) = 2\sin(2x + \frac{\pi}{3} + \varphi)$ ($|\varphi| < \frac{\pi}{2}$) is shifted to the left by $\frac{\pi}{2}$ units and then becomes symmetric about the y-axis. Which of the following judgments is incorrect?
A: $f(x+ \frac{\pi}{4})$ is an odd function.
B: $(\frac{\pi}{4}, 0)$ is a cen... | \text{C} |
When $x=\sqrt{2}-1$, the value of $x^{2}+2x$ is:
A: $1$
B: $2$
C: $2\sqrt{2}-1$
D: $2\sqrt{2}+1$ | \text{A} |
Problem 6.8. There are exactly 120 ways to color five cells in a $5 \times 5$ table so that each column and each row contains exactly one colored cell.
There are exactly 96 ways to color five cells in a $5 \times 5$ table without a corner cell so that each column and each row contains exactly one colored cell.
How ma... | 78 |
Given that one of the internal angles of triangle ABC is $120^{\circ}$ and the lengths of its sides form an arithmetic sequence with the first term being 3, find the cosine value of the smallest angle in $\triangle ABC$. | \frac{13}{14} |
You are given a positive integer $n$. What is the largest possible number of numbers that can be chosen from the set
$\{1, 2, \ldots, 2n\}$ so that there are no two chosen numbers $x > y$ for which $x - y = (x, y)$?
Here $(x, y)$ denotes the greatest common divisor of $x, y$.
[i]Proposed by Anton Trygub[/i] | n |
Example $1^{\bullet}$. Consider the sequence $\left\{a_{n}\right\}$ defined inductively by $a_{1}=1, a_{n+1}=\frac{1}{2} a_{n}+1$ $(n=1,2,3, \cdots)$. Try to find the general term of the sequence $\left\{a_{n}\right\}$. | a_{n}=2-\left(\frac{1}{2}\right)^{n-1} |
Solve the following equation:
$$
(x-5)\left|\frac{x-2}{x-5}\right|+\left|\frac{1}{x-2}\right|=2
$$ | 1+\sqrt{2} |
Sally rolls an $8$-sided die with faces numbered $1$ through $8$. Compute the probability that she gets a power of $2$. | \frac{1}{2} |
Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products \[x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2\]
is divisible by $3$. | 80 |
6. [4] Circle $\omega$ has radius 5 and is centered at $O$. Point $A$ lies outside $\omega$ such that $O A=13$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $B C$ is tangent to $\omega$ and $\omega$ lies outside triangle $A B... | 17 |
Convert the number 19 into binary ( )
A: 10110(2)
B: 11010(2)
C: 10011(2)
D: 1011(2) | C |
Given a real number $a$ as a constant, and $U=\mathbb{R}$, let the set $A=\{x| \frac{x-3}{x+1} > 0\}$, $B=\{x|y= \sqrt{\log_{2}x-1}\}$, and $C=\{x|x^{2}-(4+a)x+4a \leq 0\}$.
(1) Find $A \cap B$;
(2) If $\complement_{U}A \subseteq C$, find the range of values for $a$. | a \leq -1 |
Two circles of radius 1 are to be constructed as follows. The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ and $(2,0)$. The center of circle $B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $(0,1)$ to $(2,1)$. Wha... | \frac{4\sqrt{3}-3}{4} |
$(1)$ Calculate: $(-\frac{1}{3})^{-2}+\sqrt[3]{-8}+|\sqrt{3}-2|-(\frac{1}{\sqrt{2023}-1})^{0}+2\sin 60^{\circ}$.$(2)$ Factorize: $(x^{2}+4x)^{2}-(x^{2}+4x)-20$. | (x+5)(x-1)(x+2)^{2} |
19 The sequence of positive integers $\left\{a_{n}\right\}$ satisfies: for any positive integers $m, n$, if $m \mid n, m<n$, then $a_{m} \mid a_{n}$, and $a_{m}<a_{n}$. Find the minimum possible value of $a_{2000}$. | 128 |
12. Let's number from 1 to 28 the squares placed along the edge of a common $8 \times 8$ chessboard (see the figure on the side). Initially, a cubic die is placed on square 1 so that its bottom face exactly overlaps the entire square. We move the die along the squares as follows: the first move consists of rotating the... | 84 |
38. If \(x+y+z=8, xy+yz+zx=17, xyz=10\), then the value of \((x+y)(y+z)(z+x)\) is | 126 |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, respectively. Line $l$ passes through $F\_2$ and intersects the ellipse $C$ at points $A$ and $B$. The angle of inclination of line $l$ is $60^{\circ}$, and the distance from $F\_1$ to li... | \frac{x^{2}}{9} + \frac{y^{2}}{5} = 1 |
4. Thirty students from five classes came up with 40 tasks, and students from one class came up with the same number of tasks, while students from different classes came up with a different number. How many students came up with one task? | 26 |
The function $f(x) = \cos{2x}$ is translated to the left by $\phi$ ($0 < \phi < \frac{\pi}{2}$) units to obtain the function $g(x)$. If the function $g(x)$ is monotonically decreasing in the interval $[-\frac{\pi}{6}, \frac{\pi}{6}]$, and the largest negative zero of the function $g(x)$ is in the interval $(-\frac{\pi}... | (\frac{\pi}{4}, \frac{\pi}{3}] |
Three regular solids: tetrahedron, hexahedron, and octahedron have equal surface areas. What is the ratio of their volumes? | 1:\sqrt[4]{3}:\sqrt[4]{4} |
$6 \cdot 9$ Suppose there are $n$ teams participating in a volleyball tournament, and the tournament is conducted in a knockout (elimination) format. In the tournament, every two teams that meet play one match, and each match must have a winner and a loser, with the losing team being eliminated from the tournament. How... | n-1 |
2. The function $f(x)$ is an even function with the domain $\mathbf{R}$, and it is decreasing on $(0,+\infty)$. Which of the following inequalities holds? $($
A. $f\left(-\frac{3}{4}\right)>f\left(a^{2}-a+1\right)$.
B. $f\left(-\frac{3}{4}\right) \geqslant f\left(a^{2}-a+1\right)$.
C. $f\left(-\frac{3}{4}\right)<f\left... | B |
Given that the function $f(x) = 2\sin{ωx}\cos{ωx} + 2\sqrt{3}\sin^2{ωx} - \sqrt{3} (ω > 0)$ has a minimum positive period of $π$.
1. Find the value of $ω$ and the interval where the function $f(x)$ is monotonically decreasing.
2. The graph of the function $f(x)$ is shifted to the left by $\frac{π}{6}$ units and then up... | \frac{59π}{12} |
Given that p: -2≤x≤2, q: x≤a, and p is a sufficient but not necessary condition for q, what is the range of values for a?
A: a≥2
B: a≤2
C: a≥-2
D: a≤-2 | \text{A} |
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ and two lines $l_1: y = \frac{1}{2}x$ and $l_2: y = -\frac{1}{2}x$, parallel lines to $l_1$ and $l_2$ pass through a point $P$ on the ellipse and meet $l_1$ and $l_2$ at points $M$ and $N$, respectively. If $|MN|$ is a constant value, then the value o... | 2 |
## Task B-4.4.
Points $A, B$, and $C$ are the vertices of a triangle. On side $\overline{A B}$, 6 points are marked, on side $\overline{B C}$, 7 points are marked, and on side $\overline{C A}$, 8 points are marked. The vertices of the triangle are not among the marked points. How many different quadrilaterals can we d... | 4347 |
Given two circles $A:(x+4)^2+y^2=25$ and $B:(x-4)^2+y^2=1$, a moving circle $M$ is externally tangent to both fixed circles. Let the locus of the center of moving circle $M$ be curve $C$.
(I) Find the equation of curve $C$;
(II) If line $l$ intersects curve $C$ at points $P$ and $Q$, and $OP \perp OQ$. Is $\frac{1}{|... | \frac{1}{6} |
1. There are 50 students attending the party, the first girl to arrive shook hands with all the boys, the second girl to arrive only missed shaking hands with 1 boy, the third girl to arrive only missed shaking hands with 2 boys, and so on, the last girl to arrive only shook hands with 7 boys, how many boys are there a... | 28 |
5. How many five-digit numbers of the form $\overline{a b 16 c}$ are divisible by 16? $(a, b, c-$ arbitrary digits, not necessarily different).
ANSWER: 90. | 90 |
Given the complex number $z= \frac{i+i^{2}+i^{3}+...+i^{2017}}{1+i}$, the point corresponding to the complex number $z$ in the complex plane is located in $\text{( } \ \text{ )}$.
A: the first quadrant
B: the second quadrant
C: the third quadrant
D: the fourth quadrant | A |
Given $a$, $b$, $c\in \left(1,+\infty \right)$, and ${e^a}=2a{e^{\frac{1}{2}}}$, ${e^b}=3b{e^{\frac{1}{3}}}$, ${e^c}=5c{e^{\frac{1}{5}}}$, then ( )
A: $bce^{a} \lt cae^{b} \lt abe^{c}$
B: $ace^{b} \lt bce^{a} \lt abe^{c}$
C: $bce^{a} \lt abe^{c} \lt ace^{b}$
D: $abe^{c} \lt ace^{b} \lt bce^{a}$ | \text{A} |
There are several groups of six integers whose product is 1 . Which of the following cannot be the sum of such a group of six integers?
(A) -6
(B) -2
(C) 0
(D) 2
(E) 6 | 0 |
The interval of monotonic increase for the function $y=4x^{2}+ \frac {1}{x}$ is __( )__.
A: $(0,+\infty)$
B: $(-\infty,1)$
C: $( \frac {1}{2},+\infty)$
D: $(1,+\infty)$ | C |
Given the function $f(x) = x^2 - 2e \ln x$ (where $e$ is the base of the natural logarithm),
1. Find the intervals where $f(x)$ is monotonically increasing or decreasing;
2. Find the equation of the tangent line to the graph of $f(x)$ at the point $(1, f(1))$. | (2e - 2)x + y + 1 - 2e = 0 |
Moskvitin N.A.
In triangle $A B C$, $A B = B C$. A perpendicular $E D$ is dropped from point $E$ on side $A B$ to $B C$. It turns out that $A E = E D$. Find the angle $D A C$. | 45 |
The mumbo-jumbo tribe's language consists of $k$ different words. The words only contain two letters, $A$ and $O$. No word can be obtained from another by removing some letters from the end. What is the minimum number of letters contained in the complete dictionary of the language? | kH+(2^{H}-k) |
Example 2 There are three piles of stones. Each time, A moves one stone from one pile to another, and A can receive a reward from B for each move, which is equal to the difference between the number of stones in the pile to which A moves the stone and the number of stones in the pile from which A moves the stone. If th... | 0 |
7. Given $\sin ^{2}\left(x+\frac{\pi}{8}\right)-\cos ^{2}\left(x-\frac{\pi}{8}\right)=$ $\frac{1}{4}$, and $x \in\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$. Then the value of $\tan x$ is $\qquad$ . | \frac{2 \sqrt{14}+\sqrt{7}}{7} |
A certain college has 18 senior teachers, 12 first-level teachers, and 6 top-level teachers. A sample of size $n$ needs to be drawn from them to serve as representatives. If both systematic sampling and stratified sampling methods are used for drawing the sample, no individuals need to be excluded; if the sample size i... | 6 |
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