problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
12. (10 points) Find the largest five-digit positive integer that is not a multiple of 11, and from which no number that can be divided by 11 can be obtained by deleting some of its digits. | 98765 |
The smallest micro-particles that can be filtered by the medical mask $N95$ are approximately $300$ nanometers. Given that $1$ nanometer $=0.00000001$ meters, the scientific notation for $300$ nanometers is ( ).
A: $3\times 10^{-6}$ meters
B: $3\times 10^{-7}$ meters
C: $3\times 10^{-8}$ meters
D: $3\times 10^{-9}$... | A |
Example 7.12 Find the number of second-kind circular permutations formed by selecting 6 elements with repetition from the 3-element set $A=\{a, b, c\}$. | 92 |
Find all polynomials with real coefficients such that $\forall a, b, c \in \mathbb{R}$,
$$
P(a+b-2 c)+P(b+c-2 a)+P(c+a-2 b)=3 P(a-b)+3 P(b-c)+3 P(c-a)
$$ | P(X)=aX^{2}+bX |
Given the complex numbers z1 = 1 + i, z2 = 3 - i, find the imaginary part of z1 • z2. | 2 |
Given the universal set $U=\{x|x>0\}$, and the set $M=\{x|2x-x^2>0\}$, then $\complement_U M=$ （ ）
A: $\{x|x\geq2\}$
B: $\{x|x>2\}$
C: $\{x|x\leq0 \text{ or } x\geq2\}$
D: $\{x|0<x<2\}$ | \text{A} |
If the function $f(x)$ is defined on $\mathbb{R}$ with a period of $2$ and is an odd function, and when $0 < x < 1$, $f(x)=4^{x}$, then $f\left(-\frac{5}{2}\right)+f(2)= \boxed{-2}$． | -2 |
12. Arrange all positive integers $m$ whose digits are no greater than 3 in ascending order to form a sequence $\left\{a_{n}\right\}$. Then $a_{2007}=$ $\qquad$ . | 133113 |
If $\sqrt{(2a-1)^{2}}=1-2a$, then the range of values for $a$ is ( )
A: $a \lt \frac{1}{2}$
B: $a≥\frac{1}{2}$
C: $a＞\frac{1}{2}$
D: $a≤\frac{1}{2}$ | D |
Given that a point $P$ is taken within a right circular cone $(M-ABC)$ such that the volume of the cone $(P-ABC)$ is not greater than one third of the volume of the cone $(M-ABC)$, what is the probability? | \frac{19}{27} |
The equation of the tangent line to the curve \\(y=x^{2}+2x\\) at the point \\((1,3)\\) is \\((\\) \\()\\)
A: \\(4x-y-1=0\\)
B: \\(3x-4y+1=0\\)
C: \\(3x-4y+1=0\\)
D: \\(4y-3x+1=0\\) | \text{A} |
Let proposition $p$: The function $y=\sin 2x$ has a minimum positive period of $\dfrac{\pi}{2}$; proposition $q$: The graph of the function $y=\cos x$ is symmetric about the line $x= \dfrac{\pi}{2}$. Then, the correct judgment among the following is $(\quad)$
A: $p$ is true
B: $\neg q$ is false
C: $p \land q$ is false
... | C |
The number of sets $M$ that satisfy $\{1, 2\} \subsetneq M \subseteq \{1, 2, 3, 4, 5, 6\}$ is ______. | 15 |
Calculate the probability of drawing at least two consecutive numbers when selecting 3 numbers at random from the set {1, 2, 3, ..., 10}. | \frac{8}{15} |
【Question 15】
Suppose there is a convex polygon, and the sum of all other interior angles except one is $2015^{\circ}$. Try to find the degree measure of this interior angle. | 145 |
2. After a reduction of $12.5 \%$ the price of the gaming console is 2044 kn. What was the price of the gaming console before the reduction? | 2336 |
The sequence $\{a_n\}$ satisfies $a_{n+1}+(-1)^{n}a_{n}=2n-1$. Find the sum of the first $60$ terms of $\{a_n\}$. | 1830 |
In a shopping mall, an escalator is moving upwards at a constant speed. A boy and a girl are climbing up this escalator. It is known that the boy climbs at a speed twice that of the girl. The boy climbed 27 steps to reach the top, and the girl climbed 18 steps to reach the top. The number of steps of the escalator from... | B |
3. In a convex 12-sided polygon, all angles are equal. It is known that the lengths of some ten of its sides are 1, and the length of one more is 2. What can the area of this 12-sided polygon be? (M. A. Evdokimov) Solution. Consider the 12-sided polygon $A_{1} A_{2} \ldots A_{12}$, satisfying the condition of the probl... | 8+4\sqrt{3} |
Let $a$ and $b$ be two real numbers. We set $s=a+b$ and $p=ab$. Express $a^{3}+b^{3}$ in terms of $s$ and $p$ only. | ^{3}+b^{3}=^{3}-3sp |
A permutation of $\{1,2, \ldots, n\}$ is drawn at random. What is the average number of fixed points | 1 |
Joãozinho lies on Tuesdays, Thursdays, and Saturdays, and tells the truth the rest of the days. One day, Pedrinho meets Joãozinho and they have the following conversation:
- Pedrinho asks: What day is it today?
- Joãozinho replies: Saturday.
- Pedrinho asks: And what day will it be tomorrow?
- Joãozinho replies: Wedne... | Thursday |
247. Сколько весит рыба? Крэкхэмы задумали остановиться во время своего путешествия в каком-нибудь месте, где есть хорошая рыбная ловля, поскольку дядя Джейбз был заядлым рыболовом и они хотели доставить ему удовольствие. Они выбрали очаровательное местечко и, воспользовавшись случаем, устроили там пикник. Когда дядя п... | 72 |
33 Let the line $y=m x$ have no common points with the curve $y=\frac{|x|-1}{|x-1|}$, then the range of values for $m$ is $\qquad$ | -1\leqslant<-3+2\sqrt{2} |
8.2. Through the vertices $A$ and $C$ of triangle $ABC$, lines are drawn perpendicular to the bisector of angle $ABC$ and intersect the lines $CB$ and $BA$ at points $K$ and $M$ respectively. Find $AB$, if $BM=10, KC=2$. | 8or12 |
Given that $m$ and $n$ are lines, and $\alpha$ and $\beta$ are planes, a sufficient condition for $m \perp \alpha$ could be ( )
A: $\alpha \perp \beta$, $\alpha \cap \beta = n$, $m \perp n$
B: $\alpha || \beta$, $m \perp \beta$
C: $\alpha \perp \beta$, $m || \beta$
D: $n \subset \alpha$, $m \perp n$ | B |
7. Suppose $A B C D$ is an isosceles trapezoid in which $\overline{A B} \| \overline{C D}$. Two mutually externally tangent circles $\omega_{1}$ and $\omega_{2}$ are inscribed in $A B C D$ such that $\omega_{1}$ is tangent to $\overline{A B}, \overline{B C}$, and $\overline{C D}$ while $\omega_{2}$ is tangent to $\over... | \frac{3}{7} |
Given proposition $p$: The equation $x^2+mx+1=0$ has two distinct negative real roots. If the proposition $\neg p$ is a false proposition, then the range of values for $m$ is ______. | m > 2 |
Given that $\left\{{\begin{array}{l}x=1\\ y=1\end{array}}\right.$ and $\left\{{\begin{array}{l}x=2\\ y=3\end{array}}\right.$ are solutions of the linear equation $y=kx+b$ with respect to $x$ and $y$, then the values of $k$ and $b$ are ( )
A: $k=1$, $b=0$
B: $k=-1$, $b=2$
C: $k=2$, $b=-1$
D: $k=-2$, $b=1$ | C |
Example 4.23 Make all permutations of 5 distinct elements $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$, where $a_{1}$ is not in the 1st or 2nd position, $a_{2}$ is not in the 2nd or 3rd position, $a_{3}$ is not in the 5th position, $a_{4}$ is not in the 4th or 5th position, and $a_{5}$ is not in the 3rd or 4th position. How man... | 16 |
Example 2 Let $a \in \mathbf{N}_{+}$. Find the function $f: \mathbf{N}_{+} \rightarrow \mathbf{R}$, such that for any $x, y \in \mathbf{N}_{+}, x y>a$, we have
$$
f(x+y)=f(x y-a) .
$$ | f(x)=f(1) |
I bought a lottery ticket, the sum of the digits of its five-digit number turned out to be equal to the age of my neighbor. Determine the number of this ticket, given that my neighbor solved this problem without difficulty.
# | 99999 |
If $a > 0$, $b > 0$, and $a+b=1$, then the minimum value of $$\frac {4}{a}+ \frac {9}{b}$$ is ( )
A: 24
B: 25
C: 36
D: 72 | \text{B} |
To enter a park, a group with two men, four women, and two children paid 226 reais, while a group with three men, three women, and one child paid 207 reais.
a) How much would a group with 8 men, 10 women, and 4 children pay to enter the park?
b) If the ticket prices are all natural numbers, how many possible prices a... | 640 |
Point $P(a,b)$ is in the fourth quadrant, and $|a| \gt |b|$. Then point $Q(a+b,a-b)$ is in which quadrant?
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | \text{A} |
Example 6. Find the length of the arc of the curve $y=\arcsin \sqrt{x}-\sqrt{x-x^{2}}$ from $x_{1}=0$ to $x_{2}=1$. | 2 |
3. Solve the inequality:
$$
\frac{2|2 x-1|+2}{3}+\frac{6}{1+|2 x-1|} \leq 4-\sqrt{16 x^{4}-8 x^{2}+1}
$$ | -0.5 |
7. As shown in Figure 3, in
$\square A B C D$, $B C$ $=2 A B, C E \perp A B$, $E$ is the foot of the perpendicular, $F$ is the midpoint of $A D$. If $\angle A E F=54^{\circ}$, then $\angle B=$ ( ).
$\begin{array}{lll}\text { (A) } 54^{\circ} & \text { (B) } 60^{\circ}\end{array}$
(C) $66^{\circ}$
(D) $72^{\circ}$ | D |
Among the six-digit numbers formed by the digits 0, 1, 2, 3, 4, 5 without repetition, there are a total of such numbers that are multiples of 5 (answer with a number). | 216 |
The sum of the first 2011 terms of a geometric series is 200. The sum of the first 4022 terms of the same series is 380. Find the sum of the first 6033 terms of the series. | 542 |
Given $$\frac {sinα-2cosα}{sin\alpha +cos\alpha } = -1,$$ find $tanα = \_\_\_\_\_\_$. | \frac {1}{2} |
## 7. Potencije
Neka je $a$ znamenka jedinica broja $7^{456}, b$ znamenka jedinica broja $8^{567}$ i neka je $c$ znamenka jedinica broja $9^{678}$. Odredi broj $\overline{a b c}$.
Rezultat: 121
| 121 |
10. Fill in each box in the following equation with “+” or “-”, what is the sum of all different calculation results?
$$
625 \square 125 \square 25 \square 5 \square 1
$$ | 10000 |
Is it true that
$$
\frac{1}{2}+\frac{2}{2^{2}}+\frac{3}{2^{3}}+\ldots+\frac{30}{2^{30}}<2 ?
$$ | <2 |
# Problem 8. II variant.
On the coordinate plane $O a b$, plot the set of points for which the system $\left\{\begin{array}{l}x^{2}+y^{2}=a^{2}, \\ x+|y|=b\end{array}\right.$
has at least one solution. | -||\leqb\leq\sqrt{2}|| |
$33 \cdot 219^{89}+89^{19}$ The unit digit is
(A) 0 .
(B) 2 .
(C) 8 .
(D) 9 .
(1st "Five Sheep Cup" Junior High School Mathematics Competition, 1989) | 8 |
Let's determine the maximum or minimum value of the function
$$
y=\left(x+\frac{a}{2}\right)^{2}-(3 x+2 a)^{2}
$$
knowing that $y$ takes its extreme value when
$$
x=-\frac{11}{8}
$$ | \frac{1}{8} |
Example 6 If $x=\sqrt{19-8 \sqrt{3}}$. Then the fraction $\frac{x^{4}-6 x^{3}-2 x^{2}+18 x+23}{x^{2}-8 x+15}=$
Preserve the original text's line breaks and format, output the translation result directly. | 5 |
A pentagon is divided into 5 equal sections, as shown. An arrow is attached to the centre of the pentagon. The arrow is spun once. What is the probability that the arrow stops in the section numbered 4 ?
(A) $\frac{3}{5}$
(D) $\frac{1}{4}$
(B) $\frac{1}{2}$
(E) $\frac{1}{5}$
(C) $\frac{4}{5}$
, x_{1}=a, x_{2}=b$, and let $S_{n}=x_{1}+x_{2}+\cdots$ $+x_{n}$, then the correct conclusion is ( ).
A. $x_{100}=-a, S_{100}=2 b-a$
B. $x_{100}=-b, S_{100}=2 b-a$
C. $x_{100}=-b, S_{100}=b-a$
D. $x_{100}=-a, S_{100}=b-a$ | x_{100}=-,S_{100}=2b- |
If $\sin x = \frac{1}{3}$, and $x \in \left[\frac{\pi}{2}, \frac{3\pi}{2}\right]$, then $x = $ ______. (Express the result using the inverse trigonometric function) | \pi - \arcsin \frac{1}{3} |
In how many different orders can the characters $P \ U \ M \ \alpha \ C$ be arranged such that the $M$ is to the left of the $\alpha$ and the $\alpha$ is to the left of the $C?$ | 20 |
Given that $x, y, z$ are positive numbers greater than 1, $m > 0$, and $\log_x m = 24$, $\log_y m = 40$, $\log_{xyz} m = 12$, find the value of $\log_z m$. | 60 |
## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(2 ;-4 ;-3)$
$M_{2}(5 ;-6 ; 0)$
$M_{3}(-1 ; 3 ;-3)$
$M_{0}(2 ;-10 ; 8)$ | \frac{73}{\sqrt{83}} |
The positive integers are colored with black and white such that:
- There exists a bijection from the black numbers to the white numbers,
- The sum of three black numbers is a black number, and
- The sum of three white numbers is a white number.
Find the number of possible colorings that satisfies the above conditions... | 2 |
Given $A=3b^{2}-2a^{2}+5ab$, $B=4ab+2b^{2}-a^{2}$.
$(1)$ Simplify: $2A-3B$;
$(2)$ When $a=-1$, $b=2$, find the value of $2A-3B$. | 3 |
Gukhanov $H . X$.
Given quadratic trinomials $f_{1}(x), f_{2}(x), \ldots, f_{100}(x)$ with the same coefficients for $x^{2}$, the same coefficients for $x$, but different constant terms; each of them has two roots. From each trinomial $f_{i}(x)$, one root was chosen and denoted as $x_{i}$. What values can the sum $f_{... | 0 |
10. (2006 Polish Mathematical Olympiad) Given a prime number $p$ and a positive integer $n(p \geqslant n \geqslant 3)$. The set $A$ consists of sequences of length $n$ whose elements are taken from the set $\{1,2, \cdots, p\}$. If for any two sequences $\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ and $\left(y_{1}, y_{2},... | p^{n-2} |
39. Objects $A, B, C$ and $D$ need to be compared by weight using a two-pan balance. When $A$ and $B$ were placed on one pan and $C$ and $D$ on the other, equilibrium was established. However, when $C$ and $A$ were moved from one pan to the other, the pan with $D$ and $A$ sank. When $B$ was placed on one pan and $A$ an... | C,A,B,D |
A sequence $\{a_n\}$ satisfies the equation $S_n = \frac{3}{2}a_n - \frac{1}{2}a_1$, where $S_n$ is the sum of the first $n$ terms. Also, $a_1, a_2 + 6, a_3$ form an arithmetic sequence.
(I) Find the general formula for the sequence $\{a_n\}$.
(II) Let $b_n = \frac{a_{n+1}}{S_n S_{n+1}}$, find the sum of the first $n... | \frac{2}{3} \left( \frac{1}{2} - \frac{1}{3^{n+1}-1} \right) |
Let set $A = \{x | x^2 + 2x - 8 < 0\}$, and set $B = \{x | 5 - m < x < 2m - 1\}$. If $U = \mathbb{R}$, and $A \cap (\complement_U B) = A$, then the range of the real number $m$ is ______. | (-\infty, 3] |
21. Find the greatest integer less than or equal to $(2+\sqrt{3})^{3}$. | 51 |
If $M = 10a^2 + 2b^2 - 7a + 6$, $N = a^2 + 2b^2 + 5a + 1$, then the relationship between $M$ and $N$ is ( )
A: $M > N$
B: $M < N$
C: $M \geq N$
D: $M \leq N$ | \text{A} |
For the quadratic equation $x^{2}-6x+m=0$, if one root is $2$, then the other root is ______. | 4 |
For the quadratic function $y=x^2-2mx-3$, the following statements are made:
① Its graph intersects the x-axis at two points;
② If $y$ decreases as $x$ increases when $x \leq 1$, then $m=1$;
③ If its graph is translated 3 units to the left and then passes through the origin, then $m=-1$;
④ If the function value when $x... | 2 |
## Task B-2.4.
Determine all natural numbers $x$ that satisfy the system of inequalities
$$
x-\frac{6}{x} \geq 1, \quad \frac{1}{x-6} \leq 0
$$ | x\in{3,4,5} |
5. At some moment, Charles noticed that the tip of the minute hand, the tip of the hour hand, and a suitable point of the circumference of his watch formed three vertices of an equilateral triangle. A period $t$ of time elapsed before the next occurrence of this phenomenon. Find the greatest possible $t$ for a given w... | \frac{4}{11} |
At the entrance B1 of the Shangrao High-speed Railway Station, there are 3 ticket-checking gates. If a family of 3 people goes through the ticket-checking process, and if the same person chooses a different gate, or several people choose the same gate but in a different order, each scenario is considered a different wa... | 60 |
Given that $y=\left(m+1\right)x^{3-|m|}+2$ is a linear function of $x$, and $y$ decreases as $x$ increases, find the value of $m$. | -2 |
The negation of the proposition "If $x \neq 3$ and $x \neq 2$, then $x^2 - 5x + 6 \neq 0$" is ( )
A: If $x = 3$ and $x = 2$ then $x^2 - 5x + 6 = 0$
B: If $x \neq 3$ and $x \neq 2$ then $x^2 - 5x + 6 = 0$
C: If $x = 3$ or $x = 2$ then $x^2 - 5x + 6 = 0$
D: If $x = 3$ or $x = 2$ then $x^2 - 5x + 6 \neq 0$ | C |
Let $a,b$ and $c$ be real numbers such that $| (a-b) (b-c) (c-a) | = 1$. Find the smallest value of the expression $| a | + | b | + | c |$. (K.Satylhanov ) | \sqrt[3]{4} |
During a class meeting, it was decided to vote on whether to add a new class committee member. The voting method that best reflects the true will of all students is ( ).
A: By a show of hands of those who agree to add the new member.
B: By a show of hands of those who disagree to add the new member.
C: By secret ballot... | C |
$2 \cdot 10$ In the known sequence $1,4,8,10,16,19,21,25,30,43$, how many groups of consecutive numbers have a sum that is divisible by 11.
(China High School Mathematics League, 1985) | 7 |
$4 \cdot 34$ Try to find the range of real values of $x$ that satisfy the following relation:
$$
|3 x-2|+|3 x+1|=3 \text{. }
$$ | -\frac{1}{3}\leqslantx\leqslant\frac{2}{3} |
6. Place point $C$ at the origin of the Cartesian coordinate system, and point $B$ on the x-axis, as shown in the figure. Then the points $C, B, A$ will have coordinates $C(0,0), B(2,0), A(1, \sqrt{3})$. The geometric locus of points $K(x, y)$ such that $B K: C K=1: 2$ is a circle. Indeed,
$\frac{B K^{2}}{C K^{2}}=\fr... | \frac{16}{3} |
Given $a=\log_2 3$, $b=4^{ \frac {1}{3}}$, $c=\log_2 0.3$, the size relationship among these three numbers is $(\ )$
A: $b < a < c$
B: $a < b < c$
C: $c < a < b$
D: $c < b < a$ | C |
[ Decimal numeral system ]
Top student Polycarp has formed a huge number by writing down natural numbers from 1 to 500 in a row:
123...10111213...499500. Poor student Kolya erased the first 500 digits of this number. What do you think, with which digit does the remaining number start?
# | 3 |
3. (12 points) There are four weights of different masses. Katya weighs the weights in pairs. As a result, she got 1800, 1970, 2110, 2330, and 2500 grams. How many grams does the sixth weighing variant weigh? | 2190 |
Given that the terminal side of angle $\alpha$ passes through the point $P(x, -6)$, and $\tan\alpha= -\frac{3}{5}$, find the value of $x$. | 10 |
Given the function $f(x) = a^{x+1} - 2$ ($a > 1$), its inverse function is denoted as $y = f^{-1}(x)$. Find the value of $f^{-1}(-1)$. | -1 |
Given $$\overrightarrow {a}=(2,2), \overrightarrow {b}=(-1,3),$$ then $$2 \overrightarrow {a}- \overrightarrow {b}$$ equals \_\_\_\_\_\_\_\_\_\_\_\_. | (5,1) |
Problem 9.8. 73 children are standing in a circle. A mean Father Frost walks around the circle clockwise and distributes candies. At first, he gave one candy to the first child, then skipped 1 child, gave one candy to the next child, then skipped 2 children, gave one candy to the next child, then skipped 3 children, an... | 36 |
. For which $k$ do there exist $k$ pairwise distinct primes $p_{1}, p_{2}, \ldots, p_{k}$ such that
$$
p_{1}^{2}+p_{2}^{2}+\cdots+p_{k}^{2}=2010 ?
$$ | 7 |
6. (10 points) Person A and Person B work together on a project, which can be completed in several days. If Person A completes half of the project alone, it will take 10 days less than if Person A and Person B work together to complete the entire project; if Person B completes half of the project alone, it will take 15... | 60 |
Given that the function $f(x)$ is an even function with a period of $2$, and when $0 < x < 1$, $f(x)=\log_{0.5}x$, then ( )
A: $f\left( \dfrac {7}{5}\right) < f\left( \dfrac {4}{3}\right) < f\left(- \dfrac {1}{2}\right)$
B: $f\left( \dfrac {4}{3}\right) < f\left(- \dfrac {1}{2}\right) < f\left( \dfrac {7}{5}\right)$
C... | C |
Given that $F$ is the focus of the parabola $y^2=x$, and points $A$ and $B$ are on the parabola and on either side of the $x$-axis, with $\overrightarrow{OA} \cdot \overrightarrow{OB}=2$ (where $O$ is the origin), find the minimum value of the sum of the areas of $\triangle ABC$ and $\triangle AFO$.
A: $2$
B: $\sqrt{10... | D |
suppose that polynomial $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$ does not have a real root. what is the maximum number of coefficients to be $-1$?(14 points) | 1005 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $2c=3b$, $\sin A=2\sin B$, find the value of $\frac{\cos A}{\cos B}$. | -\frac{2}{7} |
The hyperbola $\dfrac {x^{2}}{n}-y^{2}=1(n > 1)$ has two foci $F_{1}$ and $F_{2}$. A point $P$ is on the hyperbola and satisfies $|PF_{1}|+|PF_{2}|=2 \sqrt {n+2}$. The area of $\triangle PF_{1}F_{2}$ is ______. | 1 |
In a tennis tournament, $n$ players want to make $2$ vs $2$ matches such that each player has each of the other players as opponents exactly once. Find all possible values of $n$. | n = 8k + 1 |
Given $f(\ln x + 1) = x$, then $f(3) = \, ?$
A: $e$
B: $e^2$
C: $e^3$
D: $\ln 3 + 1$ | B |
The radii $R$ and $r$ of two circles are the roots of the equation $x^{2}-3x+2=0$, and the distance between the centers of the circles is $d=3$. The positional relationship between the two circles is ( ).
A: Externally tangent
B: Internally tangent
C: Externally separate
D: Intersecting | \text{A} |
Given: $mn=m+3$, find the value of $3m-3mn+10$. | 1 |
7. Highway (from 7th grade, 3 points). A highway running from west to east intersects with $n$ equal roads, numbered from 1 to $n$ in order. Cars travel on these roads from south to north and from north to south. The probability that a car will approach the highway from each of these roads is $\frac{1}{n}$. Similarly, ... | \frac{2kn-2k^{2}+2k-1}{n^{2}} |
5.2.3. $B L$ is the bisector of triangle $A B C$. Find its area, given that $|A L|=2$, $|B L|=\sqrt{30},|C L|=5$. | \frac{7\sqrt{39}}{4} |
\left.\begin{array}{l}{[\text { Coordinate Method in Space }]} \\ {[\text { Parametric Equations of a Line }]\end{array}\right]
Derive the parametric equations of the line of intersection of the planes $2 x-y-3 z+5=0$ and $x+y-2=0$. | ,2-,1+ |
22. China has currently built 9 tidal power stations, the $\qquad$ tidal power station, completed in 1980, is China's first bidirectional tidal power station, and it is also one of the largest bidirectional tidal power stations in the world.
A. Xiamen
B. Jiangxia
C. Humen
D. Sanya | B |
2. For the lengths $a, b$ of the legs of a right triangle, it holds that
$$
\log \frac{a-b}{2}=\frac{1}{2}(\log a+\log b-\log 2)
$$
Determine the acute angles of this triangle. | \alpha=75,\beta=15 |
# Problem 3.
Find all functions $f(x)$ defined on the entire number line and satisfying the condition
$$
f(x-y)=f(x) \cdot f(y) \text { for all } x, y.
$$
# | f(x)\equiv0,f(x)\equiv1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.