problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Problem 4. In triangle $K L M$, the lengths of the sides are $8, 3 \sqrt{17}$, and 13. Find the area of the figure consisting of those and only those points $P$ inside triangle $K L M$ for which the condition $P K^{2}+P L^{2}+P M^{2} \leqslant 145$ is satisfied. | \frac{49\pi}{9} |
31. The general term formula of the sequence $\left\{f_{n}\right\}$ is $f_{n}=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right], n \in \mathbf{N}^{+}$. Let $S_{n}=\mathrm{C}_{n}^{1} f_{1}+\mathrm{C}_{n}^{2} f_{2}+\cdots+\mathrm{C}_{n}^{n} f_{n}$, find all positiv... | 3|n |
Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$ | 12 |
In the arithmetic sequence $\{a_n\}$, if $a_3 + a_7 = 37$, then $a_2 + a_4 + a_6 + a_8 = \_\_\_\_\_\_\_$. | a_2 + a_4 + a_6 + a_8 = 74 |
32. Find the number of ordered 7-tuples of positive integers $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}\right)$ that have both of the following properties:
(i) $a_{n}+a_{n+1}=a_{n+2}$ for $1 \leq n \leq 5$, and
(ii) $a_{6}=2005$. | 133 |
## Task 2 - 180522
Marie-Luise wants to specify a two-digit natural number $z$ that simultaneously meets the following conditions (1), (2), (3):
(1) The number $z$ is not divisible by 10.
(2) If the units digit of the number $z$ is increased by 4, the result is the tens digit of $z$.
(3) If the digits of $z$ are sw... | 51,62 |
3. Given that $S_{n}$ is the sum of the first $n$ terms of an arithmetic sequence with a positive common difference $q$. If $\frac{S_{n}+210}{n}$ reaches its minimum value when $n=6$, then the range of values for $q$ is | [10,14] |
13.9 Given several sets with identical contents, each set consists of 4 objects with integer gram weights. Now, we want to use these objects as weights to measure every integer gram weight from 1 gram to 1985 grams (including 1985 grams). To minimize the total weight of the objects used as weights, how many different w... | 1 |
In the Cartesian coordinate system $xOy$, suppose the terminal side of the obtuse angle $\alpha$ intersects the circle $O: x^{2}+y^{2}=4$ at point $P(x_{1},y_{1})$. If point $P$ moves clockwise along the circle for a unit arc length of $\frac{2\pi}{3}$ to reach point $Q(x_{2},y_{2})$, then the range of values for $y_{1... | \frac{1-3\sqrt{5}}{4} |
Given a three-digit number $N$, the remainders when $N$ is divided by $3, 7, 11$ are 1, 3, and 8, respectively. Try to find the maximum value of $N$.
untranslated part:
```
``` | 976 |
Positive numbers $x, y, z$ satisfy $x^2+y^2+z^2+xy+yz+zy \le 1$.
Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 9 \sqrt6 -19$. | 9\sqrt{6} - 19 |
18.4.13 *Find all natural number solutions to the equation: $7^{x}-3 \cdot 2^{y}=1$.
| (x,y)=(1,1),(2,4) |
For the original proposition "A periodic function is not a monotonic function," the correct statement is ( ).
A: The converse proposition is "A monotonic function is not a periodic function"
B: The negation is "A periodic function is a monotonic function"
C: The contrapositive is "A monotonic function is a periodic... | \text{D} |
## Task 3 - 290623
The image shows a square composed of nine fields. The side length of each individual field is $1 \mathrm{~cm}$.
a) Determine the number of all rectangles that consist of such fields!
b) Determine the sum of the areas of all these rectangles!
 each triangle from $T$ is inscribed in $\omega$; (ii) no two triangles from $T$ have a common interior point. Determine all positive real numbers $t$ such that, for each positive integer $n$, t... | t \in (0,4] |
If line $l_1: ax+2y+6=0$ is parallel to line $l_2: x+(a-1)y+(a^2-1)=0$ and they are not coincident, then the value of $a$ is \_\_\_\_\_\_. | -1 |
$(1)$ Calculate: $(-\frac{1}{3})^{-1}+\sqrt{9}+(2-\pi)^0$;$(2)$ Simplify: $(\frac{1}{a}-1)÷\frac{a^2-2a+1}{a}$. | \frac{1}{1-a} |
5. Given real numbers $x, y$ satisfy
$$
\frac{4}{x^{4}}-\frac{2}{x^{2}}=3, y^{4}+y^{2}=3 \text {. }
$$
Then the value of $\frac{4}{x^{4}}+y^{4}$ is $\qquad$
(2008, "Mathematics Weekly Cup" National Junior High School Mathematics Competition) | 7 |
Given a sequence $\{x_n\}$ with a fixed number of terms $p$ ($p \in \mathbb{N}^*$, $p > 2$), where $x_i \in \{u, v\}$ ($i=1, 2, \ldots, p$). If there exists a positive integer $t$ ($2 \leq t \leq p-1$) such that there are two sets of consecutive $t$ terms in the sequence $\{x_n\}$ that are exactly equal in order, then ... | D |
12.263. The height of a regular truncated triangular pyramid is $H$ and is the mean proportional between the sides of the bases. The lateral edge makes an angle $\alpha$ with the base. Find the volume of the pyramid. | \frac{H^{3}\sqrt{3}}{4\sin^{2}\alpha} |
Given the equation of ellipse C as $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a＞b＞0)$$, its eccentricity is $$\frac { \sqrt {2}}{2}$$, and one of its vertices coincides with the focus of the parabola $$x^{2}=-4 \sqrt {3}y$$.
(Ⅰ) Find the equation of ellipse C;
(Ⅱ) A line passing through the moving point M(0, m) (0＜... | k_{BG} = \frac{\sqrt{6}}{2} |
Which of the following operations is correct?
A: $a^{2}+2a^{2}=3a^{4}$
B: $(-2a^{3})^{2}=-4a^{6}$
C: $a^{2}\cdot a^{4}=a^{8}$
D: $a^{6}\div a^{3}=a^{3}$ | D |
In the Cartesian coordinate system, given $\overrightarrow{a}=(1,-2)$ and $\overrightarrow{b}=(3,4)$.
$(1)$ If $({3\overrightarrow{a}-\overrightarrow{b}})∥({\overrightarrow{a}+k\overrightarrow{b}})$, find the value of the real number $k$.
$(2)$ If $({\overrightarrow{a}-t\overrightarrow{b}})⊥\overrightarrow{b}$, fin... | -\frac{1}{5} |
Given that the perimeter of $\triangle abc$ is $10$, and $\sin B + \sin C = 4\sin A$.
(I) Find the value of side length $a$;
(II) If $bc = 16$, find the cosine value of angle $A$. | \dfrac{7}{8} |
[Example 3.3.9] Let $n$ be a fixed integer, $n \geqslant 2$
(I) Determine the smallest constant $c$ such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{i=1}^{n} x_{i}\right)^{4} ;
$$
(II) For this constant $c$, determine the necessary and suf... | \frac{1}{8} |
Problem 5. Solve the system of equations
$$
\left\{\begin{array}{l}
x^{2}=(y-z)^{2}-8 \\
y^{2}=(z-x)^{2}-16 \\
z^{2}=(x-y)^{2}+32
\end{array}\right.
$$ | (-1,-3,-6),(1,3,6) |
2.4. Gavriila found out that the front tires of the car last for 21,000 km, while the rear tires last for 28,000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km). | 24000 |
If the quadratic equation $mx^{2}+2x+1=0$ has real number solutions, then the range of $m$ is ( )
A: $m\geqslant -1$
B: $m\leqslant 1$
C: $m\geqslant -1$ and $m\neq 0$
D: $m\leqslant 1$ and $m\neq 0$ | \text{D} |
Given the vectors $\overrightarrow{a}=(2,5)$ and $\overrightarrow{b}=(1,y)$, if $\overrightarrow{a} \parallel \overrightarrow{b}$, then the value of the real number $y$ is __________. | \dfrac{5}{2} |
Find the minimum distance from a moving point $Q$ on the circle $x^2 + y^2 - 2x - 2y + 1 = 0$ to the line $3x + 4y + 8 = 0$. | 2 |
The simplification result of the expression $\lg 4 + 2 \lg 5$ is $\underline{\ \ \ \ \ \ \ }$. | 2 |
1. For the system of inequalities about real numbers $x, y$
$$
\left\{\begin{array}{l}
y \geqslant x^{2}+2 a, \\
x \geqslant y^{2}+2 a
\end{array}\right.
$$
to have a unique solution, the values of the parameter $a$ are | \frac{1}{8} |
$\mathbf{F 1 7}$ (39-6, Bulgaria) Let $\mathbf{N}^{*}$ be the set of all positive integers, and the function $f: \mathbf{N}^{*} \rightarrow \mathbf{N}^{*}$ satisfies: for any $s$ and $t$ in $\mathbf{N}^{*}$, we have
$$
f\left(t^{2} f(s)\right)=s(f(t))^{2},
$$
Determine the smallest possible value of $f(1998)$ among al... | 120 |
10.1. Consider functions of the form $y=x^{2}+a x+b$, where $a+b=2021$. Prove that the graphs of all such functions have a common point. | (1;2022) |
[ Arithmetic. Mental calculation, etc.]
Authors: Muroaikin M.Vv, Shapovesovo A.
In six baskets lie pears, plums, and apples. The number of plums in each basket is equal to the number of apples in the other baskets combined, and the number of apples in each basket is equal to the number of pears in the other baskets c... | 31 |
12. Given the function $f(x)=x^{2}-2 \ln x$, if there exist $x_{1}, x_{2}, \cdots, x_{n} \in\left[\frac{1}{\mathrm{e}}, \mathrm{e}\right]$, such that $f\left(x_{1}\right)+f\left(x_{2}\right)+\cdots+f\left(x_{n-1}\right)=f\left(x_{n}\right)$ holds, find the maximum value of $n$. | 6 |
1. The sum of six consecutive natural numbers, none of which is divisible by 7, is divisible by 21, but not by 42. Prove it!
Determine six such numbers, such that their sum is a four-digit number and is the square of some natural number. | 659,660,661,662,663,664 |
If the complex number $z=i(3-2i)$ (where $i$ is the imaginary unit), then $\overset{ .}{z}=$ ()
A: $2-3i$
B: $2+3i$
C: $3+2i$
D: $3-2i$ | A |
The domain of the function $y=\log_{2}(2x+1)$ is $\_\_\_\_\_\_\_.$ | \left(-\frac{1}{2},+\infty\right) |
Problem 9.4. At a round table, 40 knights and 10 samurais are sitting. Exactly 7 knights have a samurai as their right neighbor. What is the maximum number of knights that could be sitting next to two knights? | 32 |
9.1. Four non-zero numbers are written on the board, and the sum of any three of them is less than the fourth number. What is the smallest number of negative numbers that can be written on the board? Justify your answer. | 3 |
4. From a vertex $\mathrm{A}$ of a regular octagon $\mathrm{ABCDEFGH}$ with side length $\mathrm{a}$, draw a perpendicular line $\mathrm{AS}$ to the plane of the octagon. If $\mathrm{AS}=\mathrm{a}$, find the distance from $\mathrm{S}$ to $\mathrm{EF}$, the length of $\mathrm{SE}$, and the angle between $\mathrm{SE}$ a... | \mathrm{SF}=\sqrt{4+2 \sqrt{2}} \mathrm{a}, \angle \mathrm{SEF}=\operatorname{arctg} \sqrt{4+2 \sqrt{2}} |
12 identical chocolates, each can be divided into at most two pieces (not necessarily equal). If these 12 chocolates can be evenly distributed among $n$ students, then $n$ can be ( )
A: 26
B: 23
C: 17
D: 15 | \text{D} |
An arithmetic progression set is given, where the first three terms are in the form of \( \{a_3, a_3 \cdot q, 9\} \), and the sum of the first three terms is 27. What is the value of the common ratio q?
A: 0.1
B: -1/2
C: 1 or -1/2
D: -1 or -1/2 | C |
10. (1991 6th Latin American Mathematical Olympiad) $N$ is a 5-digit number composed of 5 different non-zero digits, and $N$ is equal to the sum of all three-digit numbers formed by taking 3 different digits from these 5 digits. Find all such 5-digit numbers $N$.
| 35964 |
3. Given an integer $n \geqslant 2$. Let non-negative real numbers $a_{1}, a_{2}, \cdots, a_{n}$ satisfy
$$
a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n}, a_{1}+a_{2}+\cdots+a_{n}=n .
$$
Find the minimum value of $a_{1}+a_{1} a_{2}+a_{1} a_{2} a_{3}+\cdots+a_{1} a_{2} \cdots a_{n}$. | n |
$A$ and $B$ are chess players who compete under the following conditions: The winner is the one who first reaches (at least) 2 points; if they both reach 2 points at the same time, the match is a draw. (A win is 1 point, a draw is $1 / 2$ point, a loss is 0 points) -
a) What is the expected number of games, if the pro... | 0.315 |
The monotonic decreasing interval of the function \\(y=3\sin (2x- \dfrac {\pi}{6})+2\\) is \\((\) \\()\\)
A: \\(\left[- \dfrac {\pi}{6}+2k\pi, \dfrac {\pi}{3}+2k\pi\right]\ (k\in\mathbb{Z})\\)
B: \\(\left[ \dfrac {\pi}{3}+2k\pi, \dfrac {5}{6}\pi+2k\pi\right]\ (k\in\mathbb{Z})\\)
C: \\(\left[- \dfrac {\pi}{6}+k\pi, \df... | \text{D} |
If $x > 2$, the minimum value of $y = x + \frac{4}{x - 2}$ is ______. | 6 |
In the sequence $\{a\_n\}$, the sum of the first $n$ terms satisfies $a\_1 + a\_2 + ... + a\_n = 3^n - 1$. Find the value of $\frac{1}{a\_1} + \frac{1}{a\_2} + ... + \frac{1}{a\_n}$. | \frac{3}{4}(1 - \frac{1}{3^n}) |
4. If the product of the first 2011 positive integers
$$
1 \times 2 \times \cdots \times 2011
$$
can be divided by $2010^{k}$, then the maximum value of the positive integer $k$ is | 30 |
A class has 42 female students whose three favorites are: an actor ($S z$), a fencer $(V)$, and a skater $(K)$. The girls collect signed photos of their favorites. Six girls have photos signed by all three. The class collectively has 21 $S z$ photos, 20 $V$ photos, and 18 $K$ photos. When asked who has both an $S z$ ph... | 5 |
A research team obtained some data on the relationship between the speed of sound in air and air temperature from the internet as shown below:
| Temperature ($^\circ$C) | $-20$ | $-10$ | $0$ | $10$ | $20$ | $30$ |
|-------------------------|-------|-------|-----|------|------|------|
| Speed of Sound (m/s) | $318$ ... | D |
8. Given that the three sides and the area of a triangle are all integers, and the perimeter is a perfect square, then the minimum value of such a triangle's perimeter is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly... | 16 |
Determine the symmetry of the graph of the function $f(x) = x^3$.
A. Symmetric about the y-axis
B. Symmetric about the origin
C. Symmetric about the line $y=x$
D. Symmetric about the line $y=-x$ | B |
5. Let $f(x)$ be a function defined on $\mathbf{R}$ with a period of 2, which is even, strictly decreasing on the interval $[0,1]$, and satisfies $f(\pi)=1, f(2 \pi)=2$. Then the solution set of the inequality system
$$
\left\{\begin{array}{l}
1 \leqslant x \leqslant 2, \\
1 \leqslant f(x) \leqslant 2
\end{array}\right... | [\pi-2,8-2\pi] |
12・111 Given integers $m, n$ satisfying $m, n \in\{1,2, \cdots, 1981\}$ and
$$
\left(n^{2}-m n-m^{2}\right)^{2}=1 \text {. }
$$
Find the maximum value of $m^{2}+n^{2}$.
(22nd International Mathematical Olympiad, 1981) | 3524578 |
The function $f(x)=x^{2}-4x+5$ has a maximum value of $10$ and a minimum value of $1$ on the interval $[-1,m]$. The range of the real number $m$ is $(\quad)$
A: $[2,+\infty)$
B: $[2,4]$
C: $[-1,5]$
D: $[2,5]$ | \text{D} |
Given that the function $f(x)$ defined on $\mathbb{R}$ is an odd function, its derivative is $f′(x)$, and for any positive real number $x$, it satisfies $xf′(x) > 2f(-x)$. If $g(x)=x^{2}f(x)$, then the solution set of the inequality $g(x) < g(1-3x)$ is $(\quad\quad)$.
A: $( \frac {1}{4},+∞)$
B: $(-∞, \frac {1}{4})$
C: ... | B |
There are positive integers $b$ and $c$ such that the polynomial $2x^2 + bx + c$ has two real roots which differ by $30.$ Find the least possible value of $b + c.$ | 126 |
Given the function $f(x)=ax^4+bx^2+c$ satisfies $f'(1)=2$, then $f'(-1)$ equals ( )
A: -1
B: -2
C: 2
D: 0 | \text{B} |
Given that $\overline{z} = \frac{i}{1-i}$ is the complex conjugate of the complex number $z$, find $z=$ ( ).
A: $(-\frac{1}{2} - \frac{1}{2}i)$
B: $(-\frac{1}{2} + \frac{1}{2}i)$
C: $(\frac{1}{2} - \frac{1}{2}i)$
D: $(\frac{1}{2} + \frac{1}{2}i)$ | A |
A certain brand of rice is well-known far and wide and deeply loved by consumers. A supermarket purchases a batch of this rice every day at a cost of $4 per kilogram and sells it at a price not lower than the cost but not exceeding $7 per kilogram. When the selling price is $5 per kilogram, 950 kilograms of rice are so... | 2550 |
29-30 Let $a, b$ be positive integers $(a>b)$, $p$ is the greatest common divisor of $a, b$, and $q$ is the least common multiple of $a, b$. Then the size relationship of $p, q, a, b$ is
(A) $p \geqslant q \geqslant a>b$.
(B) $q \geqslant a>b \geqslant p$.
(C) $q \geqslant p \geqslant a>b$.
(D) $p \geqslant a>b \geqsla... | B |
Find the value of $\sin 30^\circ$ and $\cos \frac{11\pi}{4}$. | -\frac{\sqrt{2}}{2} |
18. Prince Gvidon had three sons. Among his descendants, 93 each had two sons and no daughters, while all the others died childless. How many descendants did Prince Gvidon have in total? | 189 |
7. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
\begin{array}{l}
a_{1}=2, a_{2}=6, \\
a_{n+1}=\frac{a_{n}^{2}-2 a_{n}}{a_{n-1}}(n=2,3, \cdots) .
\end{array}
$$
Then $\lim _{n \rightarrow \infty}\left\{\sqrt{a_{n}+n}\right\}=$ $\qquad$ | 1 |
\section*{Problem 3 - 101233}
Let \(f\) be the function defined for all real numbers \(x\) by \(f(x)=\frac{1-x^{2}}{x^{6}+4}\).
It is to be decided whether among all function values \(f(x)\) there is a greatest and a smallest value. These values are to be determined if they exist. | -\frac{1}{12} |
45th Putnam 1984 Problem A4 A convex pentagon inscribed in a circle radius 1 has two perpendicular diagonals which intersect inside the pentagon. What is the maximum area the pentagon can have? Solution | 1+\frac{3\sqrt{3}}{4} |
Find the maximum value of the function $y= \sqrt{2}\sin (x- \frac{π}{4}) (x\in[0,π])$. | \sqrt{2} |
Given the function $f(x)=\cos \omega x \cdot \sin (\omega x- \frac{\pi}{3})+ \sqrt{3}\cos^{2}\omega x- \frac{ \sqrt{3}}{4}(\omega > 0,x\in\mathbb{R})$, and the distance from a center of symmetry of the graph of $y=f(x)$ to the nearest axis of symmetry is $\frac{\pi}{4}$.
(Ⅰ) Find the value of $\omega$ and the equation... | \frac{3+2\sqrt{6}}{3} |
Question 98: As shown in the figure, in the tetrahedron $\mathrm{D}-\mathrm{ABC}$, it is known that $\mathrm{AB}=2, \overrightarrow{\mathrm{AC}} \cdot \overrightarrow{\mathrm{BD}}=-3$, let $\mathrm{AD}=\mathrm{a}, \mathrm{BC}=b$, $\mathrm{CD}=\mathrm{c}$, try to find the minimum value of $\frac{\mathrm{c}^{2}}{\mathrm{... | 2 |
Is there an integer $n$ for which the expression $n^{2}+46 n+4441$ is equal to the fourth power of a natural number? | n_{1}=-180n_{2}=134 |
## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(5, -1, 2)$
$B(2, -4, 3)$
$C(4, -1, 3)$ | 2x+3y-7=0 |
In a high school's second-year research study group, there are two surveys planned:
① From a community of 430 high-income families, 980 middle-income families, and 290 low-income families, randomly select 170 households to investigate a certain indicator of social purchasing power;
② From 12 students in the same gr... | D |
6.4. A square (see figure) was cut into rectangles. It turned out that the areas of all six cut-off rectangles are equal to each other. Find how many times the long side of the shaded rectangle is greater than its short side. Justify your answer. | 1.5 |
If $a$, $b$, $c$ are all positive real numbers, then among the three numbers $a+ \frac {1}{b}$, $b+ \frac {1}{c}$, $c+ \frac {1}{a}$, the number of numbers that are not less than 2 is ( )
A: Can be nonexistent
B: At least 1
C: At least 2
D: At most 2 | \text{B} |
Given point $M(x_{0},1)$, suppose there exists a point $N$ on the circle $O$: $x^{2}+y^{2}=1$ such that $\angle OMN=30^{\circ}$. Determine the range of values for the real number $x_{0}$. | \left[-\sqrt{3}, \sqrt{3}\right] |
6. The sum of ten consecutive even natural numbers is 5,370. Determine all prime divisors of the largest of these numbers. | 2,3,7,13 |
Given the function $f(x)=x^2-ax$, and $g(x)=b+aln(x-1)$, there exists a real number $a$ ($a\ge1$) such that the graph of $y=f(x)$ and the graph of $y=g(x)$ have no common points. The range of the real number $b$ is:
A: $[1, +\infty)$
B: $[1, \frac{3}{4}+\ln2)$
C: $[\frac{3}{4}+\ln2, +\infty)$
D: $(-\infty, \frac{3}... | D |
Given the set $A=\{x|6+5x-x^{2} \gt 0\}$ and the set $B=\{x\left|\right.\left[x-\left(1-a\right)\right]\left[x-\left(1+a\right)\right] \gt 0\}$, where $a \gt 0$.
$(1)$ If $a=2$, find $A\cap (\complement _{R}B)$;
$(2)$ Let proposition $p:x\in A$, proposition $q:x\in B$. If ¬$p$ is a sufficient but not necessary cond... | (0, 2) |
Given the function $f\left(x\right)=1-\frac{a•{3}^{x}}{{3}^{x}+1}(2b-6 \lt x \lt b)$ is an odd function.$(1)$ Find the values of $a$ and $b$;$(2)$ Prove that $f\left(x\right)$ is a decreasing function on the interval $\left(2b-6,b\right)$;$(3)$ If $f\left(m-2\right)+f\left(2m+1\right) \gt 0$, find the range of real num... | (0, \frac{1}{3}) |
If a function $f(x)$ is increasing in its definition domain $D$ on a certain interval $I$, and $F(x) = \frac{f(x)}{x}$ is also increasing on $I$, then $y = f(x)$ is called a "perfect function" on $I$. Given that $g(x) = e^x + x - \ln(x) + 1$, if the function $g(x)$ is a "perfect function" on the interval $\left[\frac{m... | 3 |
Given the function $f(x)=\ln x+ \frac {1}{2}mx^{2}-2$ ($m\in\mathbb{R}$).
$(1)$ The tangent line to the curve $y=f(x)$ at $(1,f(1))$ is perpendicular to the line $2x-y+3=0$, find the value of $m$;
$(2)$ If the inequality $f(x)+2\leqslant mx^{2}+(m-1)x-1$ always holds for $x$, find the minimum integer value of $m$. | 2 |
Evaluate the integral $$∫_{ \frac {1}{e} }^{ e } \frac {1}{x}dx$$. | 2 |
Let the set $A=\{x|2 < x < 4\}$, and $B=\{a < x < 3a\}$.
$(1)$ If $A\cup B=\{x|2 < x < 6\}$, find the value of the real number $a$.
$(2)$ If $A\cap B\neq \varnothing$, find the range of the real number $a$. | \left( \dfrac{2}{3},4 \right) |
Find the sum of all integers $x$, $x \ge 3$, such that $201020112012_x$ (that is, $201020112012$ interpreted as a base $x$ number) is divisible by $x-1$ | 32 |
1. If $\sin ^{3} x+\cos ^{3} x=1$, then $\sin x+\cos x=$ $\qquad$ . | 1 |
Solve the equations:
$(1) 3x-8=x+4$;
$(2) 1-3\left(x+1\right)=2\left(1-0.5x\right)$;
$(3) \frac{1}{6}(3x-6)=\frac{2}{5}x-3$;
$(4) \frac{3y-1}{4}-1=\frac{5y-7}{6}$. | y = -1 |
A function $f(x)$ is defined on $\mathbb{R}$ such that for any $x \in \mathbb{R}$, $f(x+2)=f(x)$. When $x \in (-2,0)$, $f(x)=2^{x}$. Find the value of $f(2017)$. | \frac{1}{2} |
5. Find all positive integers $n$ such that $n^{4}-n^{3}+3 n^{2}+5$ is a perfect square. | 2 |
Given the function $f\left(x\right)=|x-a|+|x+3|$.
$(1)$ When $a=1$, find the solution set of the inequality $f\left(x\right)\geqslant 6$;
$(2)$ If $f\left(x\right) \gt -a$, find the range of values for $a$. | (-\frac{3}{2}, +\infty) |
Problem 3. A dog and a cat simultaneously grabbed a sausage loaf with their teeth from different sides. If the dog bites off its piece and runs away, the cat will get 300 g more than the dog. If the cat bites off its piece and runs away, the dog will get 500 g more than the cat. How much sausage will be left if both bi... | 400 |
A polygon with each interior angle equal to $120^{\circ}$ is a ( ).
A: Pentagon
B: Hexagon
C: Heptagon
D: Octagon | B |
10. As shown in the figure, the volume of tetrahedron $D A B C$ is $\frac{1}{6}$, and it satisfies $\angle A C B=45^{\circ}, A D+B C+\frac{A C}{\sqrt{2}}=3$. Then $C D=$ $\qquad$ | \sqrt{3} |
The equation of the hyperbola that shares its foci with the ellipse $\frac{x^2}{24} + \frac{y^2}{49} = 1$ and has the same asymptotes as the hyperbola $\frac{x^2}{36} - \frac{y^2}{64} = 1$ is ( ).
A: $\frac{x^2}{16} - \frac{y^2}{9} = 1$
B: $\frac{y^2}{16} - \frac{x^2}{9} = 1$
C: $\frac{y^2}{9} - \frac{x^2}{16} = 1$
... | \text{B} |
Example 2. Solve the system of equations:
$$
\left\{\begin{array}{l}
x+y+z=6 \\
x^{2}+y^{2}+z^{2}=14 \\
x^{3}+y^{3}+z^{3}=36
\end{array}\right.
$$ | (1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1) |
In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality? | (AB + BC + CA)^2 \leq 6(AD^2 + BD^2 + CD^2) |
Problem 2. In a pastry shop, each cake costs 32 denars. On the occasion of September 1st (the start of the school year), the pastry chef announced that every child who enters the pastry shop to buy a cake will receive as much money as they have at that moment before paying. On that day, Damjan entered the pastry shop f... | 30 |
Calculate: $2(\sqrt[3]{8}-\sqrt{2})-(\sqrt[3]{27}-\sqrt{2})$. | 1 - \sqrt{2} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.