problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given $f_1(x) = \cos x$, $f_2(x) = f'_1(x)$, $f_3(x) = f'_2(x)$, $f_4(x) = f'_3(x)$, ..., $f_n(x) = f'_{n-1}(x)$, then $f_{2016}(x)$ equals to ( )
A: $\sin x$
B: $-\sin x$
C: $\cos x$
D: $-\cos x$ | A |
Suppose we list the decimal representations of the positive even numbers from left to right. Determine the $2015^{th}$ digit in the list. | 8 |
A hospital has 3 doctors and 6 nurses to be assigned to 3 schools for student physical examinations, with each school being allocated 1 doctor and 2 nurses. The total number of different allocation methods is $\boxed{(\ )}$.
A: $90$
B: $180$
C: $270$
D: $540$ | D |
$3 \cdot 2$ Find $1 \cdot 1!+2 \cdot 2!+3 \cdot 3!+\cdots+(n-1)(n-1)!+n \cdot n$ !, where $n!=n(n-1)(n-2) \cdots 2 \cdot 1$. | (n+1)!-1 |
Example 43 Let $T$ be the set of all positive divisors of $60^{100}$. $S$ is a subset of $T$ in which no number is a multiple of another. Find the maximum value of $|S|$.
| 10201 |
Let $A$ be the set of all sequences from 0’s or 1’s with length 4. What’s the minimal number of sequences that can be chosen, so that an arbitrary sequence from $A$ differs at most in 1 position from one of the chosen? | 4 |
1. In a tennis tournament, 1152 schoolchildren are participating. 1 point is awarded for a win, and 0 points for a loss. Before each round, pairs are formed by drawing lots among participants with the same number of points (those who do not find a pair are awarded a point without playing). A player is eliminated after ... | 14 |
$31 \cdot 1$ A merchant buys $n$ radios for $d$ dollars, where $d$ is a positive integer. Two of them he sells to a charity sale at half cost, and the rest he sells at a profit of 8 dollars each. If the total profit is 72 dollars, then the smallest possible value of $n$ is
(A) 18 .
(B) 16 .
(C) 15 .
(D) 12 .
(E) 11
(18... | 12 |
A stationery factory plans to produce 21,000 exam pens in a week, with an average daily production of 3,000 pens. However, the actual daily production deviates from the plan. The table below shows the production situation for a week (positive for overproduction and negative for underproduction).
| Day of the Week | Mo... | 21,077 \text{ pens} |
If the complex number $x$ satisfies $(3+4i)x=|4+3i|$, then the imaginary part of $x$ is ( )
A: $\frac {4}{5}$
B: -4
C: - $\frac {4}{5}$
D: 4 | C |
3. $A B C D$ is a parallelogram with $\angle D$ obtuse. $M, N$ are the feet of perpendiculars from $D$ to $A B$ and $B C$ respectively. If $D B=D C=50$ and $D A=60$, find $D M+D N$.
$A B C D$ is a parallelogram, where $\angle D$ is obtuse. $M$ and $N$ are the feet of the perpendiculars from $D$ to $A B$ and $B C$ respe... | 88 |
Given three points $A(1,0)$, $B(0, \sqrt {3})$, $C(2, \sqrt {3})$, find the equation of the circumcircle of $\triangle ABC$. | x^{2}+y^{2}-2x- \dfrac {4 \sqrt {3}}{3}y+1=0 |
4. Some students are taking a math contest, in which each student takes one of four tests. One third of the students take one test, one fourth take another test, one fifth take the next test, and 26 students take the last test. How many students are taking the contest in total? | 120 |
57. Point $K$ is the midpoint of edge $A A_{1}$ of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$, point $L$ lies on edge $B C$. Segment $K L$
is tangent to the sphere inscribed in the cube. In what ratio does the point of tangency divide segment $K L$? | \frac{4}{5} |
Let \\(A = \{x \mid \frac{x}{x-2} < 0\}\\) and \\(B = \{x \mid x \in \mathbb{Z}\}\\). Find \\(A \cap B = \) ______. | \{1\} |
In acute triangle $\triangle ABC$, $AB=3$, $AC=4$. If the area of $\triangle ABC$ is $3\sqrt{3}$, then the length of $BC$ is _______. | \sqrt{13} |
\section*{Aufgabe 2 - 171212}
Man ermittle alle reellen Lösungen \((x, y)\) des Gleichungssystems
\[
\begin{aligned}
2 \cdot \sqrt{x+y}+x+y & =8 \\
x^{3}+y^{3} & =40
\end{aligned}
\]
| (2+\sqrt{2},2-\sqrt{2})und(2-\sqrt{2},2+\sqrt{2}) |
Let $[x]$ be the integer part of a number $x$, and $\{x\}=x-[x]$. Solve the equation
$$
[x] \cdot\{x\}=1991 x .
$$ | -\frac{1}{1992} |
4. Kolya has 440 identical cubes with a side length of 1 cm. Kolya assembled a rectangular parallelepiped from them, with all edges having a length of at least 5 cm. Find the total length of all the edges of this parallelepiped | 96 |
Problem 4. A circle is inscribed in a circular sector (the circle touches both radii and the arc of the sector). The radius of the sector is 3 times the radius of the circle. Find the ratio of the areas of the sector and the circle. | \frac{3}{2} |
22. Given a cube $\left.A B C D A_{1} B_{1} C_{1} D_{1} *\right)$ with edge $a$, $K$ is the midpoint of edge $D D_{1}$. Find the angle and the distance between the lines $C K$ and $A_{1} D$. | \arccos\frac{1}{\sqrt{10}},\frac{}{3} |
7. Among the following five numbers, the one that cannot be written as the sum of 50 consecutive natural numbers is ( ).
A. 16275
B. 57825
C. 55575
D. 77755
E. 57625 | D |
In the rectangular coordinate system on a plane, the parametric equations of curve C₁ are $$\begin{cases} x=2+rcos\phi \\ y=rsin\phi \end{cases}$$ (r>0, φ is a parameter), and a polar coordinate system is established with the coordinate origin O as the pole and the positive semi-axis of the x-axis as the polar axis. Th... | \frac{2}{3} |
Find all integers $k \ge 5$ for which there is a positive integer $n$ with exactly $k$ positive divisors
$1 = d_1 <d_2 < ... <d_k = n$ and $d_2d_3 + d_3d_5 + d_5d_2 = n$. | 8, 9 |
# Problem №3 (15 points)
Raindrops fall vertically at a speed of 2 m/s. The mass of one drop is 5 mg. There are 400 drops in one cubic meter of air. How long will it take to completely fill a cylindrical vessel with a height of 20 cm and a base area of 10 cm $^{2}$ with water? The density of water $\rho_{B}=1000 \kapp... | 5\cdot10^{4} |
6th Brazil 1984 Problem 2 Each day 289 students are divided into 17 groups of 17. No two students are ever in the same group more than once. What is the largest number of days that this can be done? | 18 |
Three college graduates, named Jia, Yi, and Bing, decided to start their own businesses. They chose to open a Taobao shop, a WeChat shop, and a physical store, respectively. During a reunion, Jia said: "I opened a Taobao shop, and Yi opened a WeChat shop." Yi said: "Jia opened a WeChat shop, and Bing opened a Taobao sh... | D |
Example 3.35. Find the derivative of the function $u=x^{3} y^{3} z^{3}$ at the point $M(1,1,1)$ in the direction that forms angles of $60^{\circ}, 45^{\circ}, 60^{\circ}$ with the coordinate axes. | \frac{3}{2}(2+\sqrt{2}) |
If the line $ax+3y-4=0$ is parallel to $x+(a+2)y+2=0$, then $a=\left(\ \ \right)$
A: $1$
B: $-3$
C: $1$ or $-3$
D: $-\frac{3}{2}$ | C |
【Question 2】Five people, A, B, C, D, and E, line up in a row, with A and B needing to be adjacent. There are $\qquad$ different ways to arrange them. | 48 |
Let $x_n=2^{2^{n}}+1$ and let $m$ be the least common multiple of $x_2, x_3, \ldots, x_{1971}.$ Find the last digit of $m.$ | 9 |
G3.1 If $a$ is the smallest real root of the equation $\sqrt{x(x+1)(x+2)(x+3)+1}=71$, find the value of $a$. | -10 |
Square $A B C D$ has $A$ and $B$ on the $x$-axis and $C$ and $D$ below the $x$-axis on the parabola with equation $y=x^{2}-4$. What is the area of $A B C D$ ?
 | 24-8\sqrt{5} |
Problem 2. Solve in the set of real numbers the equation $x+\log _{2}\left(1+\sqrt{\frac{5^{x}}{3^{x}+4^{x}}}\right)=4+\log _{1 / 2}\left(1+\sqrt{\frac{25^{x}}{7^{x}+24^{x}}}\right)$. | 2 |
(19) (This question is worth 12 points) In response to the national strategy of "targeted poverty alleviation and industrial support," Harbin City is recruiting volunteer advocates for the "Poverty Alleviation Policy" from the public. From 500 volunteers aged between $[20,45]$, 100 are randomly selected, and their age ... | 1.8 |
Given a sequence $\{a_n\}$ where each term is positive, and $S_n$ is the sum of the first $n$ terms of this sequence, with $a_{1}=1$ and $2S_{n}=a_{n}\cdot a_{n+1}$ for $N\in \mathbb{N}^{*}$. Find the minimum value of the positive integer $n$ that satisfies the inequality $$\log_{2}(1+ \frac {1}{a_{1}})+\log_{2}(1+ \fr... | 32 |
Let the family of lines M be defined by $x\cos\theta + (y-2)\sin\theta = 1$ ($0 \leq \theta \leq 2\pi$). Consider the following four statements:
A. All lines in M pass through a fixed point.
B. There exists a fixed point P that does not lie on any line in M.
C. For any integer $n$ ($n \geq 3$), there exists a reg... | \text{BCD} |
35. Consider integers $\{1,2, \ldots, 10\}$. A particle is initially at 1. It moves to an adjacent integer in the next step. What is the expected number of steps it will take to reach 10 for the first time? | 81 |
Example 1 Given that $p$ is a prime number greater than 3. Find
$$
\prod_{k=1}^{p}\left(1+2 \cos \frac{2 k \pi}{p}\right)
$$
(2012, Tsinghua University Mathematics Autumn Camp Test) | 3 |
Adams plans a profit of $10$ % on the selling price of an article and his expenses are $15$ % of sales. The rate of markup on an article that sells for $ $5.00$ is:
$\textbf{(A)}\ 20\% \qquad \textbf{(B)}\ 25\% \qquad \textbf{(C)}\ 30\% \qquad \textbf{(D)}\ 33\frac {1}{3}\% \qquad \textbf{(E)}\ 35\%$ | D |
8. If for any positive numbers $x, y$, it can be made that $a=x+y$, $b=\sqrt{x^{2}+x y+y^{2}}$, $c=m \sqrt{x y}$ are the three sides of a triangle, then the range of the positive number $m$ is $\qquad$ | 2-\sqrt{3} < m < 2+\sqrt{3} |
Given that $m$ and $n$ are two different lines, and $\alpha$ and $\beta$ are two different planes, a sufficient condition for $m$ to be parallel to $\alpha$ is:
A: $m$ parallel to $\beta$, $\alpha$ parallel to $\beta$
B: $m$ perpendicular to $\beta$, $\alpha$ perpendicular to $\beta$
C: $m$ perpendicular to $n$, $n$... | \text{C} |
Example 3. Find the logarithmic residue of the function
$$
f(z)=\frac{1+z^{2}}{1-\cos 2 \pi z}
$$
with respect to the circle $|z|=\pi$. | -12 |
9. (3 points) At Dongfang Primary School, Grade 6 selected $\frac{1}{11}$ of the girls and 22 boys to participate in the "Spring Welcome Cup" Math Competition. The remaining number of girls is twice the remaining number of boys. It is also known that the number of girls is 2 more than the number of boys. How many stude... | 86 |
9.131. $\log _{|x-1|} 0.5>0.5$. | x\in(0;0.75)\cup(1.25;2) |
Given an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>1\geq b>0)$ with left and right foci $F_{1}$ and $F_{2}$ respectively. A line $l$ (not coinciding with the $x$-axis) passing through point $F_{2}$ intersects $C$ at points $M$ and $N$, and when $M$ is the upper vertex of $C$, the perimeter of $\triangle MNF_{1}$ i... | 8 - 4\sqrt{3} |
In order to cope with the upcoming flood season, a certain flood control headquarters has selected 6 professional personnel (including two individuals, A and B) and divided them into three groups for conducting flood safety inspections on three key water conservancy projects in the area. The probability that A and B ar... | D |
3. Now, subtract the same positive integer $a$ from the numerator and denominator of the fractions $\frac{2018}{2011}$ and $\frac{2054}{2019}$, respectively, to get two new fractions that are equal. What is the positive integer $a$? | 2009 |
Given $z_{1}=(m^{2}+m+1)+(m^{2}+m-4)i$ ($m\in\mathbb{R}$), and $z_{2}=3-2i$, then "$m=1$" is the condition for "$z_{1}=z_{2}$" to be ( ).
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Necessary and sufficient
D: Neither sufficient nor necessary | A |
Given a complex number $z$ satisfying $i(z+1) = -2 + 2i$ (where $i$ is the imaginary unit),
1. Find the imaginary part of $z$.
2. If $\omega = \frac{z}{1-2i}$, find $|\omega|^{2015}$. | 1 |
3. As shown in Figure 2, in two concentric circles with center $O$, $M N$ is the diameter of the larger circle, intersecting the smaller circle at points $P$ and $Q$. The chord $M C$ of the larger circle intersects the smaller circle at points $A$ and $B$. If $O M=2$, $O P=1$, and $M A=A B=B C$, then the area of $\tria... | \frac{3 \sqrt{15}}{8} |
The monotonic increasing interval of the function $f\left(x\right)=\ln x-2x^{2}$ is ( ).
A: $(-\frac{1}{2},\frac{1}{2})$
B: $(0,\frac{1}{2})$
C: $(-∞,-\frac{1}{2}),(\frac{1}{2},+∞)$
D: $(\frac{1}{2},+∞)$ | \text{B} |
Problem 3. The side of a regular polygon is 1 cm long. What is the perimeter of this regular polygon if it has 252 diagonals? | 24\mathrm{~} |
The equation of a tangent line to the curve $y=x^3+ax+1$ is $y=2x+1$. Find the real number $a$. | a = 2 |
## Task 18/86
It is to determine the smallest number $n \in N$ with the following properties:
Its unit digit (in decimal notation) is 7. If this is removed and placed as the highest digit, the result is $5 n$. | 142857 |
9. (12 points) Hixi and Shanshan each have several points cards.
Hixi says to Shanshan: "If you give me 3 cards, my number of cards will be 3 times yours."
Shanshan says to Hixi: "If you give me 4 cards, my number of cards will be 4 times yours."
Hixi says to Shanshan: "If you give me 5 cards, my number of cards will b... | 15 |
An ellipse has the equation $$\frac {x^{2}}{a^{2}}$$+ $$\frac {y^{2}}{b^{2}}$$\=1 (a > b > 0). Its center is O, and its left and right vertices are A and B, respectively. The distance from point A to the right focus is 3, and the eccentricity is $$\frac {1}{2}$$. P is any point on the ellipse that does not coincide wit... | m = \frac{1}{2} |
Suppose $a_1, a_2, a_3, \dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$, compute the maximum possible value of \[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \][i]Proposed by Evan Chen[/i] | 365 |
Among the following curves, the one with an eccentricity of $\dfrac{2\sqrt{2}}{3}$ is
A: $\dfrac{{x}^{2}}{9}-\dfrac{{y}^{2}}{8}=1$
B: $\dfrac{{x}^{2}}{9}-{y}^{2}=1$
C: $\dfrac{{x}^{2}}{9}+\dfrac{{y}^{2}}{8}=1$
D: $\dfrac{{x}^{2}}{9}+{y}^{2}=1$ | D |
4. When a die is rolled three times, the probability that the three numbers facing up can form the side lengths of a triangle whose perimeter is divisible by 3 is $\qquad$ .
| \frac{11}{72} |
1. If $x=\frac{13}{4+\sqrt{3}}$, then $\frac{x^{4}-6 x^{3}-2 x^{2}+18 x+3}{x^{3}-7 x^{2}+5 x+15}=$ $\qquad$ | -5 |
Four points $A, O, B, O^{\prime}$ are aligned in this order on a line. Let $C$ be the circle with center $O$ and radius 2015, and $C^{\prime}$ be the circle with center $O^{\prime}$ and radius 2016. Suppose that $A$ and $B$ are the intersections of two common tangents to the two circles. Calculate $A B$ given that $A B... | 8124480 |
Example 8 Define the sequence $\left\{a_{n}\right\}$:
$$
\begin{array}{l}
a_{1}=0, \\
a_{n}=a_{\left[\frac{n}{2}\right]}+(-1)^{\frac{n(n+1)}{2}}(n=2,3, \cdots) .
\end{array}
$$
For each non-negative integer $k$, find the number of indices $n$ that satisfy $2^{k} \leqslant n<2^{k+1}$ and $a_{n}=0$. | \begin{cases}0&ifkisodd\\\mathrm{C}_{k}^{\frac{k}{2}}& |
Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors. | 162000 |
1. Buratino left Papa Carlo's house and arrived at the Field of Wonders exactly at $22:00$. If his walking speed had been $25\%$ faster, he would have arrived at $21:30$. At what time did he leave the house? | 19:30 |
Find all integer solutions of the equation:
$$
x^{2}+y^{2}=3\left(u^{2}+v^{2}\right)
$$ | (0,0,0,0) |
【Question 7】
A wire 78 cm long, with a red dot painted every 3 cm. The wire is bent at the red dots to form a rectangle. What is the maximum area of the rectangle formed in square centimeters? | 378 |
4. In a football tournament where each team played against each other once, teams A, B, C, D, E, and F participated. Teams received 3 points for a win, 1 point for a draw, and 0 points for a loss. In the end, it turned out that teams A, B, C, D, and E each had 7 points. What is the maximum number of points that team $\... | 7 |
Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the $6$ faces of the cube equal to $2$?
$\text... | 20 |
## Problem Statement
Calculate the definite integral:
$$
\int_{-2}^{0}(x+2)^{2} \cos 3 x \, d x
$$ | \frac{12-2\sin6}{27} |
9.2. Find the minimum value of the expression
$(3 \sqrt{2(1+\cos 2 x)}-\sqrt{8-4 \sqrt{3}} \sin x+2) \cdot(3+2 \sqrt{11-\sqrt{3}} \cos y-\cos 2 y) \cdot$ If the answer is not an integer, round it to the nearest integer. | 4\sqrt{3}-40 |
6. How many kilograms of cherries priced at 18 kn per kilogram should be mixed with $2012 \mathrm{~kg}$ of cherries priced at $25 \mathrm{kn}$ per kilogram so that the resulting mixture of cherries can be sold at 20 kn per kilogram without a difference in profit? | 5030\mathrm{~} |
5. Let the set
$$
S=\{1,2, \cdots, 12\}, A=\left\{a_{1}, a_{2}, a_{3}\right\}
$$
satisfy $a_{1}<a_{2}<a_{3}, a_{3}-a_{2} \leqslant 5, A \subseteq S$. Then the number of sets $A$ that satisfy the conditions is | 185 |
In the figure, all empty white triangles should be filled with integers such that the sum of the three numbers in the white neighboring triangles of each gray triangle is divisible by 5.
The numbers 12 and 3 are given in the left lower and right lower white triangles, respectively.
Determine all integers that can sta... | 0 |
6. The sequence $a_{n}$ is defined as follows:
$a_{1}=1, a_{n+1}=a_{n}+\frac{2 a_{n}}{n}$, for $n \geq 1$. Find $a_{999}$. | 499500 |
In a grade, Class 1, Class 2, and Class 3 each select two students (one male and one female) to form a group of high school students. Two students are randomly selected from this group to serve as the chairperson and vice-chairperson. Calculate the probability of the following events:
- The two selected students are no... | \dfrac{2}{5} |
1.35. Calculate the ratio of the areas of a square, an equilateral triangle, and a regular hexagon inscribed in the same circle.
## Group B | 8:3\sqrt{3}:6\sqrt{3} |
4. The product of three consecutive natural numbers is 65 times greater than their sum. What are these numbers? | 13,14,15 |
\section*{Aufgabe 3 - 181033}
Man ermittle alle diejenigen reellen Zahlen \(a\), für die erstens die Terme, die auf beiden Seiten der Gleichung
\[
\frac{1}{a^{2}-3 a+2}+\frac{1}{a^{2}-5 a+6}+\frac{1}{a^{2}-7 a+12}+\frac{1}{a^{2}-9 a+20}+\frac{1}{a^{2}-11 a+30}+\frac{1}{a^{2}-13 a+42}=\frac{a(a+5)}{a^{2}-8 a+7}
\]
ste... | -6 |
## Task 1 - 230821
Determine all four-digit natural numbers $z$ that satisfy the following conditions:
(1) The two-digit number formed by the first two digits of $z$ in this order is a perfect square.
(2) The number formed by the first and fourth digits of $z$ in this order is also a perfect square.
(3) The number ... | 1646,3646,6494,8161 |
7th Australian 1986 Problem A2 What is the smallest tower of 100s that exceeds a tower of 100 threes? In other words, let a 1 = 3, a 2 = 3 3 , and a n+1 = 3 a n . Similarly, b 1 = 100, b 2 = 100 100 etc. What is the smallest n for which b n > a 100 ? | 99 |
\begin{aligned} & {[\text { Area of a sphere and its parts }]} \\ & {[\underline{\text { Volume helps solve the task. }]}]}\end{aligned}
The base of the pyramid $S A B C$ is a triangle $A B C$, where $A B=15 \sqrt{2}, B C=20$, and the radius of the circle circumscribed around this triangle is $5 \sqrt{5}$. Three spher... | 50(5\sqrt{2}\4) |
If the graph of the power function $y=x^a$ passes through the point $(2, \frac{1}{2})$, then $a=$ ______. | -1 |
6. $M$ is a point inside the convex quadrilateral $A B C D$, and the points symmetric to $M$ with respect to the midpoints of the sides are $P, Q, R, S$. If the area of quadrilateral $A B C D$ is 1, then the area of quadrilateral $P Q R S$ is equal to | 2 |
Let the function $f(x)= \begin{cases} 2^{1-x}, & x\leqslant 1 \\ 1-\log_{2}x, & x > 1\end{cases}$. Then, the range of $x$ that satisfies $f(x)\leqslant 2$ is $(\quad)$.
A: $[-1,2]$
B: $[0,2]$
C: $[1,+\infty)$
D: $[0,+\infty)$ | D |
Zhiqiang, a student, discovered a series of equations in his extracurricular research:
\\(\dfrac{1 + \left(\frac{1}{2}\right)^2}{1 + 2^2} = \left(\dfrac{1 + \frac{1}{2}}{1 + 2}\right)^2\\),
\\(\dfrac{1 + 4^3}{1 + \left(\frac{1}{4}\right)^3} = \left(\dfrac{1 + 4}{1 + \frac{1}{4}}\right)^3\\),
\\(\dfrac{1 + \left(-\frac{... | \dfrac{1 + a^n}{1 + b^n} = \left(\dfrac{1 + a}{1 + b}\right)^n |
14. (3 points) There is a basket of peaches, when counted 4 at a time, 2 are left; when counted 6 at a time, 4 are left; when counted 8 at a time, 2 are short. It is known that the number of peaches is no less than 120 and no more than 150, there are $\qquad$ in total. | 142 |
Given the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1(a > 0,b > 0)$ intersects with the graph of the function $y= \sqrt {x}$ at point $P$. If the tangent line to the graph of $y= \sqrt {x}$ at point $P$ passes through the left focus $F(-2,0)$ of the hyperbola, then the eccentricity of the hyperbola is ( )... | B |
The coefficient of the $x$ term in the expansion of $(1+x)(2-x)^4$ is ( )
A: $-16$
B: $16$
C: $48$
D: $-48$ | \text{A} |
Find all integers $n$ for which $n^{2}+20 n+11$ is a perfect square. | 35 |
Given the ellipse $\frac{x^{2}}{4}+ \frac{y^{2}}{2}=1$.
(1) M is a moving point on the line $l: \frac{x}{4}+ \frac{y}{2}=1$, and N is a moving point on the ellipse. Find the minimum value of |MN|.
(2) A chord AB is drawn through the point $P( \sqrt {2}, \frac{1}{2})$ on the ellipse such that $\overrightarrow {AP}=3 \ov... | 3\sqrt {2}x+8y-10=0 |
Given an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$ with the right focus $F(1,0)$ and a short axis of length $2$. A line $l$ passing through point $F$ and not parallel to the coordinate axes intersects the ellipse $C$ at points $A$ and $B$, with the midpoint of segment $AB$ being $M$.
... | \frac{\sqrt{6}}{2} |
6.1. (12 points) From point $A$ to point $B$, a bus and a cyclist departed simultaneously at 13:00. After arriving at point $B$, the bus, without stopping, headed back and met the cyclist at point $C$ at 13:10. Upon returning to point $A$, the bus again, without stopping, headed to point $B$ and caught up with the cycl... | 40 |
In the interval $[0,\pi]$, a number $x$ is randomly selected. The probability that $\sin x$ falls between $0$ and $\frac{1}{2}$ is ______. | \frac{1}{3} |
$\left[\begin{array}{l}[\text { Combinatorics (other) })] \\ [\text { Extreme Principle (other) })]\end{array}\right.$
In how many ways can the numbers from 1 to 100 be permuted so that adjacent numbers differ by no more than 1? | 2 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $\sin^2 A - \sin^2 B = 2 \sin B \sin C$ and $c = 3b$, find the value of angle $A$. | \frac{\pi}{3} |
Maria Petrovna is walking down the road at a speed of 4 km/h. Seeing a stump, she sits on it and rests for the same whole number of minutes. Mikhail Potapovich is walking down the same road at a speed of 5 km/h, but sits on each stump twice as long as Maria Petrovna. They both set out and arrived at the same time. The ... | 1,3,11,33 |
Given that $\alpha$ is an angle in the third quadrant, simplify $\sqrt{ \frac{1+\cos\left(\frac{9\pi}{2}-\alpha\right)}{1+\sin(\alpha-5\pi)}} - \sqrt{ \frac{1-\cos\left(-\frac{3\pi}{2}-\alpha\right)}{1-\sin(\alpha-9\pi)}}$. | -2\tan(\alpha) |
A coastal tourism company purchased a yacht at the beginning of this year for 490,000 yuan and immediately put it into use. It is expected to generate an annual income of 250,000 yuan. In addition, there will be a certain maintenance cost every year. The maintenance cost for the first year is 40,000 yuan, and from the ... | 7 |
What is the GCD of all numbers of the form $i \times(i+1) \times(i+2)$ with $i \geq 1$ (we are looking for the greatest common divisor of all these numbers). | 6 |
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