problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
5. Given three triangles $\triangle, \triangle_{1}, \triangle_{2}$ with perimeters $p, p_{1}, p_{2}$ respectively. If $\triangle \backsim \triangle_{1} \backsim \triangle_{2}$, and the smaller two triangles $\triangle_{1}$ and $\triangle_{2}$ can be placed inside the larger triangle $\triangle$ without overlapping. The... | \sqrt{2} p |
Given $a=\log_{3}2$, $b=(\log_{3}2)^{2}$, $c=\log_{4}\frac{2}{3}$, then ( )
A: $a<c<b$
B: $c<b<a$
C: $a<b<c$
D: $b<a<c$ | \text{B} |
2. A circle with radius 5 is tangent to the $x$-axis, the $y$-axis, and the line $4 x-3 y+10=0$. Find its center. | (-5,5) |
1. In $\triangle A B C$, $\frac{2}{a}=\frac{1}{b}+\frac{1}{c}$. Then $\angle A$ ).
(A) must be an acute angle
(B) must be a right angle
(C) must be an obtuse angle
(D) none of the above answers is correct | A |
Bogdanov I.I.
In a certain country, there are 100 cities (consider the cities as points on a plane). In a directory, for each pair of cities, there is a record of the distance between them (a total of 4950 records).
a) One record has been erased. Can it always be uniquely restored from the remaining ones?
b) Suppose... | 96 |
Given that the binomial coefficients of the third and seventh terms in the expansion of $(x+2)^n$ are equal, find the value of $n$. Additionally, if the coefficient of the $(k+1)^{th}$ term in the expansion is maximum, find the value of $k$. | 5 \text{ or } 6 |
Which of the following operations is correct?
A: $x+x=2x^{2}$
B: $2y^{3}+3y^{2}=5y^{5}$
C: $2x-x=1$
D: $4x^{3}y^{2}-\left(-2\right)^{2}x^{3}y^{2}=0$ | D |
Find all natural numbers $ x,y$ such that $ y| (x^{2}+1)$ and $ x^{2}| (y^{3}+1)$. | (x, y) = (1, 1) |
1. As shown in Figure $4, \angle B C A$ $=90^{\circ}, C D$ is the altitude. It is known that the three sides of the right triangle $\triangle A B C$ are all integers, and $B D=11^{3}$. Find the ratio of the perimeters of the right triangles $\triangle B C D$ and $\triangle A C D$.
(2002, National Junior High School Mat... | \frac{11}{60} |
2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation. | 18 |
The number 2018 has exactly two divisors that are prime numbers. The sum of these two prime numbers is
(A) 793
(B) 1011
(C) 38
(D) 209
(E) 507 | 1011 |
11.1. It is known that $\operatorname{tg} A+\operatorname{tg} B=2$ and $\operatorname{ctg} A+\operatorname{ctg} B=3$. Find $\operatorname{tg}(A+B)$. | 6 |
5. Petya writes on a piece of paper a string of 6 zeros and 29 ones, arranged in a completely random order. Find the mathematical expectation of the random variable - the number of zeros written before the first one appears. | \frac{1}{5} |
4. A4 (KOR) Let $a_{1}, a_{2}, \ldots, a_{n}$ be nonnegative real numbers, not all zero.
(a) Prove that $x^{n}-a_{1} x^{n-1}-\cdots-a_{n-1} x-a_{n}=0$ has precisely one positive real root.
(b) Let $A=\sum_{j=1}^{n} \alpha_{j}, B=\sum_{j=1}^{n} j a_{j}$, and let $R$ be the positive real root of the equation in part (a).... | A^{A}\leqR^{B} |
3. Given the sequence $\left\{x_{n}\right\}$ satisfies
$$
x_{1}=2,(n+1) x_{n+1}=x_{n}+n \text {. }
$$
Then the general term of the sequence $\left\{x_{n}\right\}$ is $x_{n}=$ $\qquad$ . | x_{n}=1+\frac{1}{n!} |
Given that the positive real numbers $x$ and $y$ satisfy the equation $\ln(x + 2y) = \ln x + \ln y$, find the minimum value of $2x + y$. The value of $x$ is ($\,\,$).
A: 5
B: 3
C: 2
D: 1 | B |
Example 1 Find all $n$ such that there exist $a$ and $b$ satisfying
$$
S(a)=S(b)=S(a+b)=n .
$$ | n=9k |
G7.3 The equation of the line through $(4,3)$ and $(12,-3)$ is $\frac{x}{a}+\frac{y}{b}=1$. Find $a$. | 8 |
2. It is known that the numbers $x, y, z$ form an arithmetic progression in the given order with a common difference $\alpha=\arccos \left(-\frac{1}{3}\right)$, and the numbers $\frac{1}{\cos x}, \frac{3}{\cos y}, \frac{1}{\cos z}$ also form an arithmetic progression in the given order. Find $\cos ^{2} y$. | \frac{4}{5} |
Given an integer $n \ge 2$, determine the number of ordered $n$-tuples of integers $(a_1, a_2,...,a_n)$ such that
(a) $a_1 + a_2 + .. + a_n \ge n^2$ and
(b) $a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1$ | a_i = n \text{ for all } 1 \le i \le n |
If the equation $kx^{2}+2x+1=0$ has real roots with respect to $x$, then the range of real number $k$ is ()
A: $k\neq 0$
B: $k\leqslant 1$
C: $k\geqslant 1$
D: $k\leqslant 1$ and $k\neq 0$ | B |
7. If $\tan 4 x=$
$$
\frac{\sqrt{3}}{3} \text {, then } \frac{\sin 4 x}{\cos 8 x \cos 4 x}+\frac{\sin 2 x}{\cos 4 x \cos 2 x}+\frac{\sin x}{\cos 2 x \cos x}+\frac{\sin x}{\cos x}=
$$
$\qquad$ | \sqrt{3} |
6.221. $\left\{\begin{array}{l}x+y+z=2, \\ 2 x+3 y+z=1, \\ x^{2}+(y+2)^{2}+(z-1)^{2}=9 .\end{array}\right.$ | (-1;0;3),(3;-2;1) |
4. The Little Fairy has the same number of magic balls and magic wands. If the number of magic balls doubles, and the number of magic wands is halved, the magic balls will be 9 more than the magic wands. The Little Fairy originally had $\qquad$ magic balls. | 6 |
3. Given the function $f(x)=\log _{a}\left(a x^{2}-x+\frac{1}{2}\right)$ is always positive on $[1,2]$, then the range of the real number $a$ is ( )
A. $\left(\frac{1}{2},+\infty\right)$
B. $\left(\frac{1}{2}, \frac{5}{8}\right) \cup(1,+\infty)$
C. $\left(\frac{1}{2}, 1\right) \cup\left(\frac{3}{2},+\infty\right)$
D. $... | D |
On August 31, 2013, the 12th National Sports Games were held in Liaoning Province. A sports team has 6 male athletes and 4 female athletes. If 5 people are to be chosen to participate in the competition, how many ways are there to select at least 1 female athlete? | 246 |
Example 1. Suppose that $f(x)$ is a function continuous on the entire real line and periodic with period $2 \pi$. Derive the formulas for computing the coefficients of its Fourier series. | \alpha_{0}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x),\alpha_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cosnx,\beta_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sinnx |
Given that $\alpha$ is an angle in the second quadrant, and $|\cos\frac{\alpha}{3}|=-\cos\frac{\alpha}{3}$, then $\frac{\alpha}{3}$ is an angle in the
(A) first quadrant
(B) second quadrant
(C) third quadrant
(D) fourth quadrant | \text{B} |
Which of the following is not a linear equation in one variable?
A: $5x+3=3x-7$
B: $1+2x=3$
C: $\frac{|2x|}{3} + \frac{5}{x} = 3$
D: $x-7=0$ | C |
A company has two R&D teams, A and B. To compare their R&D levels, the results of randomly selected new product developments from these two teams in previous years are as follows:
(a, b), (a, $$\overline {b}$$), (a, b), ($$\overline {a}$$, b), ($$\overline {a}$$, $$\overline {b}$$), (a, b), (a, b), (a, $$\overline {b... | \frac {7}{15} |
Ricardo has $2020$ coins, some of which are pennies ($1$-cent coins) and the rest of which are nickels ($5$-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least amounts of money that Ricardo can have?
$\textbf{(A) }\text{806} \qquad \tex... | 8072 |
Find the maximum number of pairwise disjoint sets of the form
$S_{a,b} = \{n^{2}+an+b | n \in \mathbb{Z}\}$, $a, b \in \mathbb{Z}$. | 2 |
What is the area of the set of points $P(x ; y)$ in the Cartesian coordinate system for which $|x+y|+|x-y| \leq 4 ?$ | 16 |
# Problem 5. (3 points)
In trapezoid $A B C D$, the lateral side $C D$ is equal to the diagonal $A C$. On the smaller arc $B C$ of the circumscribed circle of triangle $B C D$, a point $E$ is chosen such that $C D=C E$. Find the angle $\angle A E B$. | 90 |
In a football preliminary competition, a certain group has $5$ teams playing a round-robin tournament (each pair of teams plays two matches). It is known that winning one match earns $3$ points, drawing one match earns $1$ point, and losing one match earns $0$ points. The top two teams with the most points can advance ... | \text{C} |
6. An isosceles triangle $P Q T$ with base $P Q$ is inscribed in a circle $\Omega$. Chords $A B$ and $C D$, parallel to the line $P Q$, intersect the side $Q T$ at points $L$ and $M$ respectively, and $Q L=L M=M T$. Find the radius of the circle $\Omega$ and the area of triangle $P Q T$, if $A B=2 \sqrt{14}, C D=2 \sqr... | R=\frac{15}{4},S=18 |
Example 2 Consider a permutation $\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ of $\{1,2, \cdots, n\}$. If
\[
\begin{array}{l}
a_{1}, a_{1}+a_{2}, a_{1}+a_{2}+a_{3}, \cdots, \\
a_{1}+a_{2}+\cdots+a_{n}
\end{array}
\]
contains at least one perfect square, then the permutation is called a "quadratic permutation."
Find al... | n=\frac{(3+2 \sqrt{2})^{k}+(3-2 \sqrt{2})^{k}-2}{4} |
Find the range of real numbers $k$ for which the inequality $|x+1|+k<x$ has solutions. | k<-1 |
Given the function $f(x)$ with derivative $f'(x) = a(x+1)(x-a)$, if $f(x)$ has a local maximum at $x=a$, then the range of $a$ is. | (-1, 0) |
7. Two detectives met. Here is their dialogue:
- Do you have two sons?
- Yes, they are young, they don't go to school.
- By the way, the product of their ages equals the number of pigeons near us.
- That's not enough information.
- And I named the older one after you.
- Now I know how old they are.
How old are the so... | 1 |
## Problem Statement
Calculate the area of the parallelogram constructed on vectors $a$ and $b$.
$a=3 p+q$
$b=p-3 q$
\[
\begin{aligned}
|p| & =7 \\
|q| & =2
\end{aligned}
\]
\[
(\widehat{p, q})=\frac{\pi}{4}
\] | 70\sqrt{2} |
In $\triangle ABC$, if $AB=4$, $AC=5$, and $\cos C= \frac{4}{5}$, then $\sin B=$ `____`. | \sin B = \frac{3}{4} |
5. As shown in Figure 1, the plane containing
square $A B C D$ and the plane containing
square $A B E F$ form a $45^{\circ}$ dihedral angle.
Then the angle formed by the skew lines $A C$ and
$B F$ is $\qquad$ | \arccos \frac{2-\sqrt{2}}{4} |
Given the curve $f(x) = -e^{x+1}$, it intersects the $y$-axis at point $P$. Determine the equation of the tangent line to the curve at point $P$. | y = -ex - e |
During the Beijing Olympics, sports venues had to conduct security checks on spectators. Suppose that at the start of the security check, there were already several spectators waiting outside the venue, and after the security check started, the number of spectators arriving at the sports venue increased at a fixed rate... | C |
The maximum value of the quadratic function $y=x^{2}-2x+1$ in the range $-5\leqslant x\leqslant 3$ is ____. | 36 |
## Task 4 - 331234
Determine the number of all pairs $(x, y)$ of integers $x$, y, for which $\sqrt{x}+\sqrt{y}=1993$ holds. | 1994 |
Example 4 Find the last three digits of $2013^{2013^{2013}}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 053 |
In the expansion of the algebraic expression $({\sqrt{x}-\frac{1}{{{x^2}}}})^5$, the constant term is ______. | -5 |
Convert -885° into the form of $\alpha + k \cdot 360°$ ($0° \leq \alpha < 360°$, $k \in \mathbb{Z}$). The correct option is:
A: $-165° + (-2) \cdot 360°$
B: $195° + (-3) \cdot 360°$
C: $195° + (-2) \cdot 360°$
D: $165° + (-3) \cdot 360°$ | \text{B} |
Given that $α$ is an angle in the fourth quadrant and $\sin α + \cos α = \frac{1}{5}$, find the value of $\tan α$. | -\frac{3}{4} |
Given that the general formula of the sequence $\{a_{n}\}$ is $a_n = {(-1)}^n$, where $n \in \mathbb{N}_+$, then the $8$th term of the sequence is ______. | 1 |
Can the signs in the expression $\pm 1 \pm 2 \pm \ldots \pm 100$ be chosen such that the value of the expression is exactly 1982? | 1982 |
There are $38$ people in the California Baseball League (CBL). The CBL cannot start playing games until people are split into teams of exactly $9$ people (with each person in exactly one team). Moreover, there must be an even number of teams. What is the fewest number of people who must join the CBL such that the CBL c... | 16 |
The graph of the function f(x) = 2sin(2x - $\frac{π}{4}$) is translated $\frac{π}{4}$ units to the right to obtain the graph of function g(x). What is g(0) equal to?
A: $\sqrt{2}$
B: 2
C: $- \sqrt{2}$
D: 0 | \text{C} |
Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ such that$$f(-x)+\int \limits_0^xtf(x-t)dt=x,\ \forall\ x\in \mathbb{R}$$ | f(x) = -\sinh(x) |
In the complex plane, the complex number $z$ satisfies $z(1+i)=|1+ \sqrt{3}i|$. The point corresponding to $\overline{z}$ is located in ( ).
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | A |
8. In the sequence $\left\{a_{n}\right\}$, if $a_{1}=1, a_{n} \neq 0\left(n \in \mathbf{N}_{+}\right)$, and $a_{n}+2 a_{n} a_{n-1}-a_{n-1}=$ $0\left(n \geqslant 2, n \in \mathbf{N}_{+}\right)$, then $a_{n}=$ $\qquad$ | a_{n}=\frac{1}{2n-1} |
The function $f(x)=\frac{{3}^{x+1}-1}{{3}^{x}-1}+a{(sinx+cosx)}^{2}$ is an odd function. Find the real number $a$. | -2 |
2. Determine the pairs of complex numbers $(u, v)$ with $|u|=|v|=1$ such that
$$
|1-u|+\left|v^{2}+1\right|=|1-v|+\left|u^{2}+1\right|=\sqrt{2}
$$
Marius Damian, Brăila | (u,v)\in{(i,i),(-i,i),(i,-i),(-i,-i)} |
In an arithmetic sequence $\{a_{n}\}$ with a positive common difference, $a_{1}$, $a_{4}$, and $a_{7}+12$ form a geometric sequence, and $S_{n}$ is the sum of the first $n$ terms, satisfying $S_{3}=15$.$(1)$ Find the general formula and the sum of the first $n$ terms of the sequence $\{a_{n}\}$;$(2)$ If ___, find the s... | T_{n}=(n-1)\cdot 2^{n+1}+2 |
Given the statements $p: |4x-3| \leqslant 1$ and $q: x^2 - (2a+1)x + a(a+1) \leqslant 0$, if the statement "$\neg p \Rightarrow \neg q$" is false and "$\neg q \Rightarrow \neg p$" is true, find the range of real number values for $a$. | a \in [0, \frac{1}{2}] |
Problem 4. Point $O$ is the center of the circumcircle of triangle $ABC$ with sides $AB=5, AC=8$, and $BC=4$. Find the length of the vector $\overrightarrow{O A}-4 \overrightarrow{O B}+3 \overrightarrow{O C}$. | 10 |
Given the function $f\left(x\right)=\frac{{\ln x}}{{1-x}}$, $x\in D$, where $D=\left(0,1\right)\cup \left(1,+\infty \right)$. $(1)$ Find the equation of the tangent line to the function $f\left(x\right)$ at the point $({\frac{1}{2}},f({\frac{1}{2}}))$; $(2)$ If $g\left(x\right)=-\frac{a}{{\sqrt{x}}}$, and $\forall x\in... | [1, +\infty) |
Given the equation of circle $C$ as $x^{2}+y^{2}+x-6y+m=0$, and line $l:x+2y-3=0$.
(1) Find the range of values for $m$.
(2) If circle $C$ intersects with line $l$ at points $P$ and $Q$, and the circle that has $PQ$ as its diameter passes through the origin, find the value of the real number $m$. | 3 |
Task B-4.8. A circle is described around the focus of the parabola $y^{2}=16 x$ which touches the directrix of the parabola. At what angle do these two curves intersect? | 45 |
6. All positive integer solutions $(a, b)$ of the equation $a^{3}-a^{2} b+a^{2}+2 a+2 b+1=0$ are $\qquad$ . | (5,7) |
Given the function $f(x) = x\ln x + x^{2}$, $x_{0}$ is a critical point of the function $f(x)$. Which of the following conclusions is correct?
A: $0 \lt x_{0} \lt \dfrac{1}{e}$
B: $x_{0} \gt \dfrac{1}{e}$
C: $f(x_{0}) + 2x_{0} \lt 0$
D: $f(x_{0}) + 2x_{0} \gt 0$ | D |
A certain class of students is divided into $x$ groups. If each group has 8 students, there will be 2 students left over. If each group has 9 students, there will be 4 students short. Find the value of $x$. | 6 |
Given a point Q(2, 0) on the Cartesian coordinate plane and a circle C: $x^2+y^2=1$, the length of the tangent from a moving point M to circle C equals the sum of the radius of circle C and the distance |MQ|. Find the equation of the trajectory of the moving point M. | 3x^2-y^2-8x+5=0 |
10. There are 8 red, 8 white, and 8 yellow chopsticks. Without looking, how many chopsticks must be taken out to ensure that at least two pairs of chopsticks are of different colors? $\qquad$ | 11 |
# Problem 6. (3 points)
Three runners are moving along a circular track at constant equal speeds. When two runners meet, they instantly turn around and start running in opposite directions.
At some point, the first runner meets the second. After 15 minutes, the second runner meets the third for the first time. Anothe... | 80 |
B2. The expression is $2-\frac{x-2}{5}+\frac{1+x}{4}$.
A. Calculate the value of the expression for $x=5$.
B. Determine the value of $x$ such that the value of the expression is 4.
C. Determine the value of $x$ such that the value of the expression is at least 2.
D. Find the smallest natural number $x$ such that th... | 7 |
In the expansion of $$(x- \frac {1}{ 3x })^{24}$$, the number of terms with integer exponents of $x$ is ( )
A: 3 terms
B: 4 terms
C: 5 terms
D: 9 terms | \text{D} |
Consider a regular $2 n+1$-gon $P$ in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $S E$ contains no other points that lie on the sides of $P$ except $S$. We want to color the sides of $P$ in 3 co... | n=1 |
Given the ellipse $C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1 (a>b>0)$, its eccentricity is the reciprocal of the eccentricity of an equilateral hyperbola. The line $l: x-y+ \sqrt {2}=0$ is tangent to the circle centered at the origin with the radius equal to the semi-minor axis of the ellipse $C$.
(Ⅰ) Find the e... | (- \frac {1}{2}, -1) |
8.020. $1-\sin 3 x=\left(\sin \frac{x}{2}-\cos \frac{x}{2}\right)^{2}$.
8.020. $1-\sin 3x=\left(\sin \frac{x}{2}-\cos \frac{x}{2}\right)^{2}$. | x_{1}=\pin;x_{2}=\frac{\pi}{4}(2k+1),\quadn,k\inZ |
Given that $\lg a + \lg b = 1$, find the value of $ab$. | 10 |
1768. From a batch of 5000 electric lamps, 300 were selected using a simple random sampling without replacement. The average burning duration of the lamps in the sample was 1450 hours, and the variance was 40000. What is the probability that the average burning duration of the lamps in the entire batch is within the ra... | 0.99964 |
Let $z_1=18+83i,~z_2=18+39i,$ and $z_3=78+99i,$ where $i=\sqrt{-1}.$ Let $z$ be the unique complex number with the properties that $\frac{z_3-z_1}{z_2-z_1}~\cdot~\frac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$. | 56 |
8. The positive integer solution of the system of equations $\left\{\begin{array}{l}x^{3}-y^{3}-z^{3}=3 x y z, \\ x^{2}=2(y+z)\end{array}\right.$ is | x=2, y=1, z=1 |
Given the function f(x) = (x + a)² + (e^x + $\frac{a}{e}$)², if there exists an x₀ such that f(x₀) ≤ $\frac{4}{e^{2}+1}$, find the value of the real number a. | a = \frac{e^{2}-1}{e^{2}+1} |
1. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1}=1, a_{n}=5 a_{n-1}+3^{n-1}(n \geqslant 2) \text {. }
$$
Find $a_{n}$ and the highest power of 2 that divides $a_{k}\left(k=2^{2019}\right)$ | 2021 |
【Question 4】
Three candles can burn for $30$, $40$, and $50$ minutes respectively (but they are not lit at the same time). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. Then, the time when exactly two candles are burning simultaneously is $\... | 35 |
177. "Common Divisor". Here is a puzzle that readers often ask me (of course, the specific numbers in it are different). A correspondent of a provincial newspaper reported that many teachers had undermined their health in vain attempts to solve it! Probably, he exaggerated a bit, because the question is actually simple... | 79 |
The solutions to the equation $x(2x+1)=2x+1$ are. | x_1 = -\frac{1}{2}, x_2 = 1 |
It is given that one root of $2x^2 + rx + s = 0$, with $r$ and $s$ real numbers, is $3+2i (i = \sqrt{-1})$. The value of $s$ is:
$\textbf{(A)}\ \text{undetermined}\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ -13\qquad\textbf{(E)}\ 26$ | 26 |
48 blacksmiths need to shoe 60 horses. What is the minimum time they will spend on the work if each blacksmith spends 5 minutes on one horseshoe
# | 25 |
13.072. The city $C$, located between points $A$ and $B$, is supplied with gas from these points, the distance between which is 500 km. From reservoir $A$, 10000 $\mathrm{M}^{3}$ of gas is pumped out every minute, and from reservoir $B$ - 12% more. In this case, the gas leakage in each pipeline is 4 m $^{3}$ per kilome... | 100 |
[Inscribed angle, based on the diameter] [Pythagorean theorem (direct and inverse).]
In triangle $ABC$, $AB = BC = 6$. A circle is constructed on side $AB$ as the diameter, intersecting side $BC$ at point $D$ such that $BD: DC = 2: 1$.
Find $AC$. | 2\sqrt{6} |
1. The number of positive integer values of $n$ that satisfy $\left|\sqrt{\frac{n}{n+2009}}-1\right|>\frac{1}{1005}$ is $\qquad$ | 1008015 |
A bottle of oil was consumed by $$\frac {1}{2}$$ the first time, and $$\frac {1}{5}$$ of the remainder was consumed the second time. There are still 30 kilograms of oil left in the bottle. How many kilograms of oil were there originally? | 75 |
10.122. The larger base of the trapezoid has a length of 24 cm (Fig. 10.113). Find the length of its smaller base, given that the distance between the midpoints of the diagonals of the trapezoid is 4 cm. | 16 |
Problem 1. On the parabola $y=x^{2}$, four points $A, B, C, D$ are chosen such that the segments $A B$ and $C D$ intersect on the y-axis. Find the abscissa of point $D$, if the abscissas of points $A, B$ and $C$ are equal to $a, b$ and $c$ respectively. | \frac{}{} |
Given an odd function $f\left(x\right)$ defined on the set of real numbers $R$, and $f\left(x\right)$ is a decreasing function on $\left(-\infty ,+\infty \right)$, is there a real number $m$ such that $f\left(4m-2m\cos \theta \right)+f(2\cos ^{2}\theta -4) \gt f\left(0\right)$ holds for all $θ∈[{-\frac{π}{3},\frac{π}{2... | m \in (-\infty, 1) |
Example 12. Find the inverse function of $y=\sqrt{x+5}(x \geqslant-5)$.
| y=x^{2}-5 |
1. Let $\left\{a_{n}\right\}$ be an arithmetic sequence with the sum of the first $n$ terms denoted as $S_{n}$. If $S_{2}=10, S_{5}=55$, then a direction vector of the line passing through points $P\left(n, a_{n}\right)$ and $Q(n+2$, $a_{n+2})$ can be ( ).
(A) $\left(2, \frac{1}{2}\right)$
(B) $\left(-\frac{1}{2},-2\ri... | B |
4. A square with an area of $400 \mathrm{~cm}^{2}$ is divided by two lines into two squares and two rectangles. Calculate the perimeter of each square and each rectangle, if the lengths of the adjacent sides of the two rectangles differ by $60 \mathrm{~mm}$. | o_{1}=52\mathrm{~},o_{2}=28\mathrm{~} |
5. In the coordinate plane, a point whose both coordinates are integers is called an integer point. We use $I$ to denote the set of all lines, $M$ to denote the set of lines passing through exactly one integer point, $N$ to denote the set of lines not passing through any integer point, and $P$ to denote the set of line... | D |
Three. (50 points) Let $M=\{1,2, \cdots, 65\}, A \subseteq M$ be a subset. If $|A|=33$, and there exist $x, y \in A, x<y$, $x \mid y$, then $A$ is called a "good set". Find the largest $a \in M$ such that any 33-element subset containing $a$ is a good set. | 21 |
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