problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Solve the following equations using appropriate methods:$(1)\left(x-1\right)^{2}=25$;$(2)x^{2}-4x+3=0$;$(3)\left(2x+1\right)^{2}=2\left(2x+1\right)$;$(4)2x^{2}-5x+3=0$. | x_1 = 1, x_2 = 1.5 |
In the parallelepiped $ABCD-{A'}{B'}{C'}{D'}$, the base $ABCD$ is a square with side length $2$, the length of the side edge $AA'$ is $3$, and $\angle {A'}AB=\angle {A'}AD=60^{\circ}$. Find the length of $AC'$. | \sqrt{29} |
Given that an object is in a state of equilibrium under the action of three forces $\overrightarrow{F}_{1}=(1,2)$, $\overrightarrow{F}_{2}=(-1,-3)$, and $\overrightarrow{F}_{3}$, find the value of $\overrightarrow{F}_{3}$.
A: $(0,1)$
B: $(-1,0)$
C: $(0,-1)$
D: $(-1,1)$ | A |
Given that the function $f(x)$ is an even function defined on $\mathbb{R}$, and when $x \in (0, +\infty)$, $f(x) = x^{2} + x$, which of the following statements is correct?
A: $f(-1) = 2$
B: $f(x)$ is an increasing function on the domain $\mathbb{R}$
C: When $x \in (-\infty, 0)$, $f(x) = x^{2} - x$
D: The solution ... | ACD |
Given the function $f(x)=x^{3}-3a^{2}x+a$ ($a > 0$), its maximum value is positive, and its minimum value is negative. Find the range of $a$. | \left(\dfrac{\sqrt{2}}{2},+\infty\right) |
844. Solve the equation in natural numbers
$$
x^{2}+x y+y^{2}=x+y+9
$$ | (3;1),(1;3) |
12th Swedish 1972 Problem 1 Find the largest real number a such that x - 4y = 1, ax + 3y = 1 has an integer solution. | 1 |
10. Given the set $A=\{3 k+2 \mid 0 \leqslant k \leqslant 667, k \in \mathbf{Z}\}$. If $n$ numbers are randomly selected from $A$, it is always possible to find 2 different numbers $a$ and $b$ such that $a+b=2104$, then the minimum value of $n$ is $\qquad$. | 352 |
10. Let $I$ be the universal set, non-empty sets $P, Q$ satisfy $P \varsubsetneqq Q \varsubsetneqq I$. If a set operation expression containing $P, Q$ results in the empty set $\varnothing$, then this operation expression can be $\qquad$ (just write one expression). | P\cap\complement_{1}Q=\varnothing |
【Question 6】
A bottle contains 1001 blue balls, 1000 red balls, and 1000 green balls. At the same time, Xiaoming has enough balls of these three colors in his hand. Next, Xiaoming takes out two balls from the bottle each time, and then puts one or two balls back into the bottle according to the following rules:
(1) If ... | red |
4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-8.5,8.5]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 306 |
Which of the following pairs of shapes may not necessarily be similar?
A: Square
B: Isosceles right triangle
C: Rectangle
D: Equilateral triangle | \text{C} |
The real function $f$ is defined for $\forall$ $x\in \mathbb{R}$ and $f(0)=0$. Also $f(9+x)=f(9-x)$ and $f(x-10)=f(-x-10)$ for $\forall$ $x\in \mathbb{R}$. What’s the least number of zeros $f$ can have in the interval $[0;2014]$? Does this change, if $f$ is also continuous? | 107 |
B3. At the Pythagoras Lyceum, all classes have a maximum of 25 students. In one of the classes, a survey is conducted about all sorts of things, and everyone fills out all the questions. In this class, it turns out that $31 \%$ of the students have a black bicycle and that $94 \%$ of the students live more than $1 \mat... | 16 |
Given the functions $f(x)=\log_{2}x$ and $g(x)=x^{2}+2x$, a sequence $\{a_{n}\}$ has its first $n$ terms sum denoted as $S_{n}$, and $b_{n}$ represents the general term of the sequence $\{b_{n}\}$, where $n \in \mathbb{N}^*$. The points $(b_{n},n)$ and $(n,S_{n})$ lie on the graphs of $f(x)$ and $g(x)$, respectively.
(... | \frac {n}{2n+1} |
Among the following angles: ① the angle formed by two skew lines; ② the angle formed between a line and a plane; ③ the dihedral angle's plane angle; those that can be obtuse are ( )
A: 0
B: 1
C: 2
D: 3 | B |
Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n=n^2$, and $\{b_n\}$ be a geometric sequence, with $a_1=b_1$, and $b_2(a_2-a_1)=b_1$.
$(1)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.
$(2)$ Let $c_n=a_n \cdot b_n$, find the sum of the first $n$ terms of the sequence $\{c_... | T_n=6-\frac{2n+3}{2^{n-1}} |
1. Find all prime numbers $p, q, r$ and $s$ such that
$$
6 p+7 p q+8 p q r+9 p q r s=2018
$$ | (2,17,2,2) |
56. In triangle $ABC$, $R$ and $r$ are the radii of the circumcircle and the incircle, respectively. Let $A_{1}, B_{1}$, $C_{1}$ be the points of intersection of the angle bisectors of triangle $ABC$ with the circumcircle. Find the ratio of the areas of triangles $ABC$ and $A_{1} B_{1} C_{1}$. | \frac{2r}{R} |
Given vectors $\overrightarrow{m}=( \sqrt{3}\sin \frac{x}{4},1)$ and $\overrightarrow{n}=(\cos \frac{x}{4},\cos^2 \frac{x}{4})$, let $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$.
$(1)$ If $f(x)=1$, determine the value of $\cos (x+ \frac{\pi}{3})$;
$(2)$ In an acute-angled $\triangle ABC$, with sides $a$, $b$, and... | \left( \frac{\sqrt{3}+1}{2}, \frac{3}{2}\right] |
Given that the function f(x) = sin(ωx + φ) (ω > 0, φ ∈ R) is an even function, and the point (1, 0) is the center of symmetry of the graph of the function y = f(x). Find the minimum value of ω. | \frac{π}{2} |
Given that the coefficient of the $x$ term in the expansion of $\left(\frac{\sqrt{x}}{2}-\frac{a}{\sqrt{x}}\right)^{8}$ is $-14$, find the value of $a$. | 2 |
In an apartment, the length of room I and room II is equal, and the width of room II and room III is equal, finally, the diagonal of room I and room III is equal. (The base of the rooms is, of course, rectangular.) The floor area of the rooms is $a, b, c$ square meters, respectively. Calculate the dimensions of the ind... | \sqrt[4]{\frac{b^{2}(b^{2}-^{2})}{b^{2}-^{2}}},\quad\frac{}{x},\quad\frac{b}{x},\quadu=\frac{}{b} |
Among the following statements, the correct one is $({\ })$
A: Through two parallel lines, there is exactly one plane.
B: If two lines are parallel to the same plane, then these two lines are parallel.
C: Three points determine a unique plane.
D: If three non-collinear points in one plane are equidistant from another p... | A |
4. Given $-\frac{\pi}{2}<\alpha<\frac{\pi}{2}$,
$$
2 \tan \beta=\tan 2 \alpha, \tan (\beta-\alpha)=-2 \sqrt{2} \text {. }
$$
Then $\cos \alpha=(\quad)$.
(A) $\frac{\sqrt{3}}{2}$
(B) $\frac{\sqrt{2}}{2}$
(C) $\frac{\sqrt{3}}{3}$
(D) $\frac{\sqrt{2}}{3}$ | C |
Let $k$ be a positive integer. How many non-negative integer solutions does the equation
$$
\left[\frac{x}{k}\right]=\left[\frac{x}{k+1}\right]
$$
have? | \frac{k(k+1)}{2} |
Among the following four propositions, the correct one(s) is(are) （ ）
p1: There exists an $x_0 > 0$ such that $\ln x_0 > x_0 - 1$;
p2: For all $x \in \mathbb{R}$, $x^2 - x + 1 > 0$;
p3: There exists an $x_0 > 0$ such that $\ln \frac{1}{x_0} > -x_0 + 1$;
p4: For all $x \in (0, +\infty)$, $(\frac{1}{2})^x > \log... | C |
If $C^p_n=\frac{n!}{p!(n-p)!} (p \ge 1)$, prove the identity
\[C^p_n=C^{p-1}_{n-1} + C^{p-1}_{n-2} + \cdots + C^{p-1}_{p} + C^{p-1}_{p-1}\]
and then evaluate the sum
\[S = 1\cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdots + 97 \cdot 98 \cdot 99.\] | 23527350 |
For the function $y=f(x)$, where $x \in \mathbb{R}$, "the graph of $y=|f(x)|$ is symmetric about the y-axis" is a(n) ______ condition for "$y=f(x)$ is an odd function".
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Necessary and sufficient
D: Neither sufficient nor necessary | \text{B} |
8. Find the sum of all integer values of c for which the equation $15|p-1|+|3 p-| p+c||=4$ has at least one root with respect to p
# | -15 |
3. Zoka, Joka, and Coka, three housewives, received 9 closed milk bottles at the market, containing, in order: $2,5,8,11,14,17,20,23$ and 26 deciliters of milk. In how many ways can they divide these bottles among themselves (without opening the bottles) so that each gets the same number of bottles and the same amount ... | 12 |
On the number line, points $A$ and $B$ represent the numbers $-1$ and $5$ respectively. Point $P$ is also on this number line, and the distance from $P$ to points $A$ and $B$ is equal. What number does point $P$ represent? | 2 |
Given two lines $l_1: 2x+my=0$ and $l_2: y=3x-1$, determine the value of $m$ for which the two lines are parallel. | -\frac{2}{3} |
394. Solve the equation:
$$
\sqrt[3]{x}+\sqrt[3]{x+19}=5
$$ | 8 |
25. Let $n$ be the smallest integer for which $7 n$ has 2016 digits.
What is the units digit of $n$ ?
A 0
B 1
C 4
D 6
E 8 | 6 |
Given the function $f(x)=2^{x}$ with its inverse function $f^{-1}(x)$,
(1) If $f^{-1}(x)-f^{-1}(1-x)=1$, find the value of the real number $x$;
(2) If the equation $f(x)+f(1-x)-m=0$ has solutions in the interval $[0,2]$, find the range of values for the real number $m$. | [2 \sqrt {2}, \frac {9}{2}] |
44th Putnam 1983 Problem B5 Define ||x|| as the distance from x to the nearest integer. Find lim n→∞ 1/n ∫ 1 n ||n/x|| dx. You may assume that ∏ 1 ∞ 2n/(2n-1) 2n/(2n+1) = π/2. Solution | \ln(4/\pi) |
9. A water pool has three inlet pipes and one outlet pipe. When the outlet and any two of the inlet pipes are opened simultaneously, it takes 6 hours, 5 hours, and 4 hours respectively to fill the entire pool; when the outlet and all three inlet pipes are opened simultaneously, it takes 3 hours to fill the entire pool.... | 2\frac{14}{23} |
Given that $S_{n}$ is the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$ with a common difference of $d$ and $S_{5} - S_{2} = 195$.
(1) If $d = -2$, find the general term formula for the sequence $\{a_{n}\}$.
(2) If in a geometric sequence $\{b_{n}\}$, $b_{1} = 13$ and $b_{2} = a_{4}$, find the sum ... | \frac{13(5^{n} - 1)}{4} |
Given the function $f(x) = -x^2 + 2x + 3$, find the sum of the maximum and minimum values of $f$ on the interval $[0, 3]$. | 4 |
1. Find all integer values of $x, y$ that satisfy the condition $\frac{x+y}{x^{2}-x y+y^{2}}=\frac{3}{7}$.
The above text has been translated into English, preserving the original text's line breaks and format. | (4,5),(5,4) |
Convert $-150^{\circ}$ into radians. | - \frac{5}{6}\pi |
Given two non-zero vectors $\vec{a}$ and $\vec{b}$ that satisfy $|\vec{a}| = |\vec{a} + \vec{b}| = 1$ and the angle between $\vec{a}$ and $\vec{b}$ is $120^\circ$, find the value of $|\vec{b}|$. | |\vec{b}| = 1 |
88. Calculate: $\frac{\left(2^{2}+4^{2}+6^{2}+\cdots+100^{2}\right)-\left(1^{2}+3^{2}+5^{2}+\cdots+99^{2}\right)}{1+2+3+\cdots+9+10+9+\cdots+3+2+1}=$ | 50.5 |
Given the function $f(x) = \begin{cases} 2x+a, & x<1 \\ -x-2a, & x\geq1 \end{cases}$, if $f(1-a)=f(1+a)$, then the value of $a$ is \_\_\_\_\_\_. | -\frac{3}{4} |
Find all pairs $ (x, y)$ of integers such that $ x^3 \minus{} y^3 \equal{} 2xy \plus{} 8$. | (0, -2), (2, 0) |
Solve the inequality: $5x-12\leqslant 2\left(4x-3\right)$. | x \geqslant -2 |
In $\triangle ABC$ the median from $A$ is given perpendicular to the median from $B$. If $BC=7$ and $AC=6$, find the length of $AB$.
$\textbf{(A)}\ 4\qquad \textbf{(B)}\ \sqrt{17} \qquad \textbf{(C)}\ 4.25\qquad \textbf{(D)}\ 2\sqrt{5} \qquad \textbf{(E)}\ 4.5$ | \sqrt{17} |
[ Fundamental Theorem of Arithmetic. Prime Factorization]
Author: Blinkov A. d:
The product of five different integers is 2022. What can their sum be? If there are multiple answers, list them all.
# | 338,336,-332,-342 |
4. Given point $A(-3,4)$, moving point $B$ on the $y$-axis, moving point $C$ on the line $l: x+y=0$, then the minimum value of the perimeter of $\triangle A B C$ is
A. $5 \sqrt{2}$
B. $4+\sqrt{10}$
C. $3+2 \sqrt{2}$
D. 7 | 5\sqrt{2} |
Given the function $f(x)=ae^{x}-x+\ln a-2$.
$(Ⅰ)$ If $x=0$ is a critical point of $f(x)$, find the minimum value of $f(x)$.
$(Ⅱ)$ If the function $g(x)=f(x)+x-\ln(x+2)$ has two zeros, find the range of values for $a$. | (0, e) |
6. Let $[x]$ denote the greatest integer not exceeding $x$. If $p, q, r$ are positive, find the minimum value of:
$$
\left[\frac{p+q}{r}\right]+\left[\frac{q+r}{p}\right]+\left[\frac{r+p}{q}\right]
$$ | 4 |
Given sets A={3, 2^a} and B={a, b}, if A ∩ B = {2}, find the value of a+b. | a+b=3 |
(4 points) In the three-dimensional Cartesian coordinate system, find a point C on the z-axis such that the distance from point C to point A (1, 0, 2) and point B (1, 1, 1) is equal. The coordinates of point C are ______. | (0, 0, 1) |
At a bus station, there are three buses departing to a school between 6:30 AM and 7:30 AM each day. The ticket prices for the buses are the same, but the comfort levels vary. Xiao Jie, a student, observes before boarding. When the first bus arrives, he does not get on but carefully observes its comfort level. If the co... | \frac{1}{2} |
10. Given that $A$ and $B$ are the left and right vertices of the hyperbola $E$, point $M$ lies on the hyperbola, and $\triangle ABM$ is an isosceles triangle with the vertex angle being $120^{\circ}$. Then the eccentricity of the hyperbola is ( ).
(A) $\sqrt{5}$
(B) 2
(C) $\sqrt{3}$
(D) $\sqrt{2}$ | D |
11.4. Nyusha has 2022 coins, and Barash has 2023. Nyusha and Barash toss all their coins simultaneously and count how many heads each of them gets. The one who gets more heads wins, and in case of a tie, Nyusha wins. $C$ What is the probability that Nyusha wins? | 0.5 |
The graph of the inverse proportion function $y=\frac{k}{x}$ passes through two points $A\left(3,m\right)$ and $B\left(m-1,6\right)$. What is the value of $k$?
A: $4$
B: $6$
C: $9$
D: $12$ | B |
Given that the focus of the ellipse $x^{2}+ky^{2}=2$ lies on the $y$-axis, if the focal length of the ellipse is $4$, then the value of $k$ is ______. | \frac{1}{3} |
In the diagram, the size of $\angle A C B$ is
(A) $57^{\circ}$
(B) $37^{\circ}$
(C) $47^{\circ}$
(D) $60^{\circ}$
(E) $17^{\circ}$
 | 37 |
Given a complex number $z=\frac{5+i}{1+i}$, which of the following statements is correct?
A: $|z|=13$
B: The imaginary part of $z$ is $-2$
C: The point in the complex plane corresponding to $z$ is in the fourth quadrant
D: The conjugate complex number of $z$ is $-3-2i$ | BC |
## Problem Statement
Find the point of intersection of the line and the plane.
$\frac{x-5}{-2}=\frac{y-2}{0}=\frac{z+4}{-1}$
$2 x-5 y+4 z+24=0$ | (3;2;-5) |
Given the function $f(x) = \log_{2}x + x - 2$, its root is in the interval $(n, n+1)$ $(n \in \mathbb{Z})$, then $n=$ | 1 |
# 2. CONDITION
Solve the equation in natural numbers
$$
\text { LCM }(a ; b)+\text { GCD }(a ; b)=a b .
$$
(GCD - greatest common divisor, LCM - least common multiple).
# | =b=2 |
2.12. Compute surface integrals of the first kind:
a) $\iint_{\sigma}|x| d S, \quad \sigma: x^{2}+y^{2}+z^{2}=1, \quad z \geqslant 0$
b) $\iint_{\sigma}\left(x^{2}+y^{2}\right) d S, \quad \sigma: x^{2}+y^{2}=2 z, \quad z=1$;
c) $\iint_{\sigma}\left(x^{2}+y^{2}+z^{2}\right) d S$, where $\sigma$ is the part of the cone... | \pi,\frac{4\pi}{15}(1+15\sqrt{3}),3\sqrt{2}\pi |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 5$, $|\overrightarrow{b}| = 3$, and $|\overrightarrow{a} - \overrightarrow{b}| = 7$, find the value of $\overrightarrow{a} \cdot \overrightarrow{b}$. | \overrightarrow{a} \cdot \overrightarrow{b} = -\frac{15}{2} |
7.3. Two spheres are inscribed in a dihedral angle, touching each other. The radius of one sphere is four times that of the other, and the line connecting the centers of the spheres forms an angle of $60^{\circ}$ with the edge of the dihedral angle. Find the measure of the dihedral angle. Write the cosine of this angle... | 0.04 |
3. Let $p, q \in \mathbf{N}$, and $1 \leqslant p<q \leqslant n$, where $n$ is a natural number not less than 3. Then the sum $S$ of all fractions of the form $\frac{p}{q}$ is $\qquad$ | \frac{1}{4} n(n-1) |
Let $x$ be the number
\[0.\underbrace{0000...0000}_{1996\text{ zeros}}1,\]
where there are 1996 zeros after the decimal point. Which of the following expressions represents the largest number?
$\text{(A)}\ 3+x \qquad \text{(B)}\ 3-x \qquad \text{(C)}\ 3\cdot x \qquad \text{(D)}\ 3/x \qquad \text{(E)}\ x/3$ | D |
6.247. Determine the values of $m$ for which one of the roots of the equation $z^{3}-\left(m^{2}-m+7\right) z-\left(3 m^{2}-3 m-6\right)=0$ is -1. Find the other two roots of the equation for these values of $m$. | m_{1}=-2,m_{2}=3;z_{1}=-1,z_{2}=-3,z_{3}=4 |
Example 2 (2002 National High School Competition Question) If $x \in\left[-\frac{5 \pi}{12},-\frac{\pi}{3}\right]$, then the maximum value of $y=\tan \left(x+\frac{2 \pi}{3}\right)-$ $\tan \left(x+\frac{\pi}{6}\right)+\cos \left(x+\frac{\pi}{6}\right)$ is ( ).
A. $\frac{12}{5} \sqrt{2}$
B. $\frac{11}{6} \sqrt{2}$
C. $\... | \frac{11}{6}\sqrt{3} |
If $a\neq b$, which of the following fractions is simplified correctly?
A: $\frac{a+2}{b+2}=\frac{a}{b}$
B: $\frac{a-2}{b-2}=\frac{a}{b}$
C: $\frac{{a}^{2}}{{b}^{2}}=\frac{a}{b}$
D: $\frac{\frac{1}{2}a}{\frac{1}{2}b}=\frac{a}{b}$ | D |
The [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $n$ is the smallest [positive](https://artofproblemsolving.com/wiki/index.php/Positive) [multiple](https://artofproblemsolving.com/wiki/index.php/Multiple) of $15$ such that every [digit](https://artofproblemsolving.com/wiki/index.php/Digit) of $n$ i... | 592 |
7.5. Vera has 27 cubes with an edge of 1 cm: 9 red and 18 blue. She assembled them into a cube with an edge of 3 cm. Can the number of red squares with a side of 1 cm on the surface of the cube be equal to the number of such blue squares? | No |
The graph of the [polynomial](https://artofproblemsolving.com/wiki/index.php/Polynomial)
\[P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e\]
has five distinct $x$-intercepts, one of which is at $(0,0)$. Which of the following [coefficients](https://artofproblemsolving.com/wiki/index.php/Coefficient) cannot be zero?
$\text{(A)... | D |
Given the directrix of a parabola is $y= \frac {1}{2}$, then its standard equation is ( )
A: $y^{2}=2x$
B: $x^{2}=-2y$
C: $y^{2}=-x$
D: $x^{2}=-y$ | \text{B} |
If $f(x)$ and $g(x)$ are odd and even functions defined on $\mathbb{R}$, respectively, and $f(x) + g(x) = x^3 - x^2 + x - 3$, find $f(x)$. | f(x) = x^3 + x |
A train departs from $A$ and moves towards $B$ at a constant speed. 11 minutes later, a train departs from $B$ on the same route, also at a constant speed, heading towards $A$. From the moment they meet, it takes 20 and 45 minutes respectively for the trains to reach $B$ and $A$. In what ratio does the meeting point $T... | \frac{9}{5} |
Let the function $f(x) = k(x-1) - 2\ln x$ ($k > 0$).
(1) If the function $f(x)$ has exactly one zero point, find the value of the real number $k$;
(2) Let the function $g(x) = xe^{1-x}$ (where $e$ is the base of the natural logarithm), if for any given $s \in (0, e)$, there exist two different $t_i \in \left( \frac{1... | \left[ \frac{3}{e-1}, \frac{3e^2}{e^2-1} \right] |
Given that $f(x)= \dfrac {1}{ \sqrt { \log _{ \frac {1}{2}}(2x-1)}}$, determine the domain of $f(x)$.
A: $(\dfrac {1}{2},1)$
B: $(\dfrac {1}{2},1]$
C: $(\dfrac {1}{2},+∞)$
D: $(1,+∞)$ | \text{A} |
A two-digit number has a sum of its tens and units digits equal to $13$. If the digits are reversed, the new number is $27$ less than the original number. What is the original number? | 85 |
Given the set $M = \{x \mid 2x - 3 < 1\}$, and the set $N = \{x \mid -1 < x < 3\}$, then the intersection $M \cap N$ is:
A: $M$
B: $N$
C: $\{x \mid -1 < x < 2\}$
D: $\{x \mid x < 3\}$ | C |
2.203. $\frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2} \cdot\left(2 x+\sqrt{x^{2}-1}\right)}{\sqrt{(x+1)^{3}}-\sqrt{(x-1)^{3}}}$.
2.203. $\frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2} \cdot\left(2 x+\sqrt{x^{2}-1}\right)}{\sqrt{(x+1)^{3}}-\sqrt{(x-1)^{3}}}$.
The expression is already in a... | \frac{1}{\sqrt[4]{x^{2}-1}} |
Given that the sum of the first $n$ terms of a geometric sequence $\{a\_n\}$ is $S\_n$, if $S\_5$, $S\_4$, and $S\_6$ form an arithmetic sequence, then the common ratio $q$ of the geometric sequence $\{a\_n\}$ is equal to $\boxed{\text{answer}}$. | -2 |
Example 2 If $x$, $y$, $z$ are all real numbers, and $x^{2}+y^{2}+z^{2}$ $=1$, then the maximum value of $\sqrt{2} x y+y z$ is $\qquad$ | \frac{\sqrt{3}}{2} |
Example 10 Let $p, q$ be given positive integers, $n=2^{p} \cdot 3^{q}$, find the number of positive integers less than $n$ and not divisors of $n$ among the positive divisors of $n^{2}$ (when $p=31, q=19$, this problem is from the 13th American Invitational Mathematics Examination) | pq |
Example 2 Let the perfect square $y^{2}$ be the sum of the squares of 11 consecutive integers. Then the minimum value of $|y|$ is $\qquad$ . | 11 |
11.5. Let $M$ be some set of pairs of natural numbers $(i, j), 1 \leq i<j \leq n$ for a fixed $n \geq 2$. If a pair $(i, j)$ belongs to $M$, then no pair $(j, k)$ belongs to it. What is the largest set of pairs that can be in the set $M$? | \frac{n^{2}}{4}forevenn,\frac{n^{2}-1}{4}foroddn |
Example 3. If $a=\sqrt{17}-1$, find the value of $\left(a^{5}+2 a^{4}-17 a^{3}\right.$ $\left.-a^{2}+18 a-17\right)^{1993}$.
(Adapted from the 1987 Chongqing Junior High School Mathematics Invitational Competition) | -1 |
Given a set of data: $3$, $x$, $6$, $5$, $4$, if the mode of this data set is $4$, then the value of $x$ is ______. | 4 |
Let $ m, n \geq 1$ be two coprime integers and let also $ s$ an arbitrary integer. Determine the number of subsets $ A$ of $ \{1, 2, ..., m \plus{} n \minus{} 1\}$ such that $ |A| \equal{} m$ and $ \sum_{x \in A} x \equiv s \pmod{n}$. | \frac{1}{n} \binom{m+n-1}{m} |
ii. (40 points) Find all positive integers $(m, n)$ such that
$$
\frac{m^{3}+n^{3}-m^{2} n^{2}}{(m+n)^{2}}
$$
is a non-negative integer. | (2,2) |
## Task Condition
Find the derivative.
$$
y=\frac{1}{2} \cdot(x-4) \sqrt{8 x-x^{2}-7}-9 \arccos \sqrt{\frac{x-1}{6}}
$$ | \sqrt{8x-x^{2}-7} |
8. A sequence of numbers starts from 0 and follows the rule below, continuing up to 2020. This sequence contains $\qquad$ numbers.
$$
0 \rightarrow 1 \rightarrow 3 \rightarrow 6 \rightarrow 10 \rightarrow 11 \rightarrow 13 \rightarrow 16 \rightarrow 20 \rightarrow 21 \rightarrow 23 \rightarrow 26 \text {..... }
$$ | 809 |
2.069. $\frac{a^{7 / 3}-2 a^{5 / 3} b^{2 / 3}+a b^{4 / 3}}{a^{5 / 3}-a^{4 / 3} b^{1 / 3}-a b^{2 / 3}+a^{2 / 3} b}: a^{1 / 3}$. | ^{1/3}+b^{1/3} |
The express train between Station A and Station B takes 3 hours and 30 minutes to travel. The first train departs from both stations at 6:00 AM, and thereafter, a train departs every hour. The number of times the express train departing from Station A at 9:00 AM will meet the train departing from Station B is ( )
... | C |
Given that $\{a_n\}$ is an arithmetic sequence with a non-zero common difference, and $S_n$ is the sum of its first $n$ terms, with $S_3=9$, and $a_2$, $a_5$, $a_{14}$ form a geometric sequence. The sum of the first $n$ terms of the sequence $\{b_n\}$ is $T_n= \frac{3^{n+1}-3}{2}$.
$(1)$ Find the general formula for t... | M_n=2 -\frac{n+2}{3^n} |
Find the limit of the sequence $\lim_{n\to\infty} \frac{2n+3}{n+1}$. | 2 |
Point $B$ is due east of point $A$. Point $C$ is due north of point $B$. The distance between points $A$ and $C$ is $10\sqrt 2$, and $\angle BAC = 45^\circ$. Point $D$ is $20$ meters due north of point $C$. The distance $AD$ is between which two integers?
$\textbf{(A)}\ 30\ \text{and}\ 31 \qquad\textbf{(B)}\ 31\ \text{... | B |
## Task B-3.2.
For the angles $\alpha, \beta$ and $\gamma$ of the acute-angled triangle $ABC$, it holds that $\operatorname{tg} \alpha: \operatorname{tg} \beta: \operatorname{tg} \gamma=1: 3: 12$. Calculate
$$
\sin \alpha \cdot \sin \beta \cdot \sin \gamma
$$ | \frac{32}{65} |
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