problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
30. [14] Alice has an equilateral triangle $A B C$ of area 1. Put $D$ on $B C, E$ on $C A$, and $F$ on $A B$, with $B D=D C, C E=2 E A$, and $2 A F=F B$. Note that $A D, B E$, and $C F$ pass through a single point $M$. What is the area of triangle $E M C$ ? | \frac{1}{6} |
A certain unit needs to participate in three meetings: A, B, and C. Meeting A requires 2 participants, while meetings B and C each require 1 participant. From a group of 10 people, 4 individuals are to be selected to attend these three meetings. The number of different arrangements possible is ( )
A: 1260
B: 2025
C: 25... | C |
3. If there are five consecutive natural numbers, and each number is a composite number, then the minimum value of the sum of these five consecutive natural numbers is
$\qquad$ . | 130 |
12. Let the set $M=\{1,2, \cdots, 1000\}$, and for any non-empty subset $X$ of $M$, let $\alpha_{X}$ denote the sum of the largest and smallest numbers in $X$. Then, the arithmetic mean of all such $\alpha_{X}$ is $\qquad$ . | 1001 |
The sum of the squares of the first $n$ natural numbers is divisible by $n$ when? | 6k\1 |
1. [3] Travis is hopping around on the vertices of a cube. Each minute he hops from the vertex he's currently on to the other vertex of an edge that he is next to. After four minutes, what is the probability that he is back where he started? | \frac{7}{27} |
In the sequence $\{a_n\}$, it is known that $a_1 = 2$, $a_2 = 7$, and $a_{n+2}$ is equal to the units digit of the product of $a_n$ and $a_{n+1}$ (where $n \in \mathbb{N}^*$). The value of $a_{2013}$ is (γγ)
A: 8
B: 6
C: 4
D: 2 | C |
In the expansion of the binomial $(1-2x)^n$, the sum of the coefficients of the even terms is 128. Then, the coefficient of the middle term in the expansion is (γγ)
A: -960
B: 960
C: 1120
D: 1680 | C |
## Task 3 - 311223
Determine all triples $(a ; b ; c)$ of natural numbers such that the function
$$
f(x)=\frac{x^{2}+a x+b}{x^{2}+c x+b}
$$
satisfies the following conditions:
(1) The function $f$ is defined for all real $x$.
(2) It holds that $1<f(2)<f(1)<2$.
(3) The function $f$ has two distinct real roots. | (3;1;1) |
Given the function $f(x)=x^{3}+\sin x$, if $\alpha\in[0,\pi]$ and $\beta\in\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$ and $f\left(\frac{\pi}{2}-\alpha\right)=f(2\beta)$, then $\cos\left(\frac{\alpha}{2}+\beta\right)=$ ______. | \frac{\sqrt{2}}{2} |
13.063. In the club's auditorium, there were 320 seats arranged in identical rows. After the number of seats in each row was increased by 4 and one more row was added, the auditorium had 420 seats. How many rows are there in the auditorium now? | 21 |
Example 2 Arrange all positive integers that are coprime with 105 in ascending order, and find the 1000th term of this sequence. | 2186 |
The number of zeros of the function $f(x)=|x-2|-\ln x$ within its domain is (Β Β Β Β )
A: $0$
B: $1$
C: $2$
D: $3$ | C |
In equilateral triangle $ABC$ with side length $2$, let the parabola with focus $A$ and directrix $BC$ intersect sides $AB$ and $AC$ at $A_1$ and $A_2$, respectively. Similarly, let the parabola with focus $B$ and directrix $CA$ intersect sides $BC$ and $BA$ at $B_1$ and $B_2$, respectively. Finally, let the parabola w... | 12\sqrt{3} - 18 |
Given that point $A(m,-3)$ is symmetric to point $B(2,n)$ with respect to the $y$-axis, then $m+n=\_\_\_\_\_\_$. | -5 |
Three lines $m$, $n$, and $\ell$ lie in a plane such that no two are parallel. Lines $m$ and $n$ meet at an acute angle of $14^{\circ}$, and lines $m$ and $\ell$ meet at an acute angle of $20^{\circ}$. Find, in degrees, the sum of all possible acute angles formed by lines $n$ and $\ell$.
[i]Ray Li[/i] | 40 |
Problem 2. In a children's summer camp, children flew in 4 shifts. In the first shift, 128 students were registered, and in the second shift, 2 times more than in the first. In the third shift, 69 more students registered than in the second, and in the fourth shift, 3 times fewer than the sum of the first and third shi... | 848 |
# Problem 4. Maximum 20 points
Svetlana is convinced that any feline can be interesting to her for two purposes - catching mice and conducting cat therapy. Based on these considerations, Svetlana has decided to acquire 2 kittens, each of which has an equal probability of being either a male or a female. The weekly pro... | 3 |
Example 4 Let $D=\{1,2, \cdots, 10\}, f(x)$ be a one-to-one mapping from $D$ to $D$, and let $f_{n+1}(x)=$ $f\left(f_{n}(x)\right), n \in \mathbf{N}_{+} ; f_{1}(x)=f(x)$. Try to find a permutation $x_{1}, x_{2}, \cdots, x_{10}$ of $D$, such that $\sum_{i=1}^{10} x_{i} f_{2520}(i)=220$. | 10,9,8,7,6,5,4,3,2,1 |
Given the function $f(x)=e^{2x}-4ae^{x}+(4a-2)x$, where $a\geqslant 1$.
- (I) Discuss the monotonicity of $f(x)$;
- (II) If there exists $x$ such that $f(x)+f(-x)=0$, find the range of the real number $a$;
- (III) If $f(x)\geqslant f(-x)$ holds for all $x\geqslant 0$, find the range of the real number $a$. | [1,2] |
## Task 1
An LPG is picking up seed potatoes. The first wagon already has 35 full sacks. Nine full sacks are still prepared on the ramp. Two wagons are to be loaded with 40 sacks of seed potatoes each.
How many sacks still need to be filled? | 36 |
## Subject 4.
Consider the sets $A=\{a, b, c, d\} \subset \mathbb{N}$ and $S=\{s \in \mathbb{N} \mid s=x+y, x \in A, y \in A, x \neq y\}$.
It is known that $S=\{82,96,104,112,126\}$.
a) Calculate the sum of the elements in set $A$;
b) Determine the elements of set $A$.
Prof. Mircea Fianu, Bucharest
| Details of s... | 37,45,59,67 |
Example 2 If the linear function $f(x)$ satisfies $f(f(x))=4x+1$, then $f(x)$ | f(x)=2x+\frac{1}{3}\text{or}f(x)=-2x-1 |
In an acute triangle $ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively, and it is given that $\sqrt{3}a - 2c\sin A = 0$. If $c=2$, then the maximum value of $a+b$ is \_\_\_\_\_\_. | 4 |
Given $(1+i)(x-yi)=2$, where $x$, $y$ are real numbers and $i$ is the imaginary unit, then $x+y=$ _____
A: $1$
B: $\sqrt{2}$
C: $\sqrt{3}$
D: $2$ | D |
Let vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $(\overrightarrow{a} - \overrightarrow{b}) \cdot (\overrightarrow{a} + 2\overrightarrow{b}) = -8$, and $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$. Find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} |
If the focus of the parabola $y^{2}=2px(p > 0)$ coincides with the right focus of the hyperbola $x^{2}-y^{2}=2$, then the value of $p$ is (γγ).
A: $\sqrt {2}$
B: $2$
C: $4$
D: $2\sqrt {2}$ | C |
Determine the interval(s) where the function $f(x)=e^{x}-\ln (x+1)$ is strictly increasing. | (0,+\infty) |
9. (14 points) Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
\frac{a_{n+1}+a_{n}-1}{a_{n+1}-a_{n}+1}=n\left(n \in \mathbf{N}_{+}\right) \text {, and } a_{2}=6 \text {. }
$$
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$.
(2) Let $b_{n}=\frac{a_{n}}{n+c}\left(n \in \mathbf{N}_{+}, c\ri... | 4-\frac{n+2}{2^{n-1}} |
9. Two circles are concentric. The area of the ring between them is $A$. In terms of $A$, find the length of the longest chord contained entirely within the ring. | 2\sqrt{\frac{A}{\pi}} |
XXII OM - III - Task 3
How many locks at least need to be placed on the treasury so that with a certain distribution of keys among the 11-member committee authorized to open the treasury, any 6 members can open it, but no 5 can? Determine the distribution of keys among the committee members with the minimum number of ... | 462 |
11. A and B repeatedly play a game. In each game, the players take turns tossing a fair coin, and the first player to toss heads wins. In the first game, A tosses first, and in subsequent games, the loser of the previous game starts first. The probability that A wins the 6th game is $\qquad$. | \frac{1}{2}(1-\frac{1}{3^{6}}) |
1. If $a, b$ are integers, and $x^{2}-x-1$ is a factor of $a x^{17}$ $+b x^{16}+1$, then $a=$ $\qquad$ | 987 |
2. Kolya is twice as old as Olya was when Kolya was as old as Olya is now. And when Olya is as old as Kolya is now, their combined age will be 36 years. How old is Kolya now?
ANS: 16 years. | 16 |
Equilateral triangle $ABC$ has side length $6$. Circles with centers at $A$, $B$, and $C$ are drawn such that their respective radii $r_A$, $r_B$, and $r_C$ form an arithmetic sequence with $r_A<r_B<r_C$. If the shortest distance between circles $A$ and $B$ is $3.5$, and the shortest distance between circles $A$ and $C... | \frac{29\pi}{24} |
Petr had a rectangle with a width of $2 \mathrm{~cm}$ and an unknown length. Radka had a rectangle with a width of $2 \mathrm{~cm}$, whose length was equal to the perimeter of Petr's rectangle. When they placed the rectangles together by their widths, they obtained a new rectangle with a perimeter of $63 \mathrm{~cm}$.... | 17\mathrm{~}^{2} |
8. Determine all natural numbers $n \geq 2$ such that for all integers $0 \leq i, j \leq n$:
$$
i+j \equiv\binom{n}{i}+\binom{n}{j} \quad(\bmod 2)
$$
Answer: The solutions are all numbers of the form $n=2^{k}-2$ with $k \geq 2$. | 2^{k}-2 |
4. Dana wrote a three-digit number on paper, which when divided by seven gives a remainder of 2. By swapping the first two digits, a three-digit number was formed that when divided by seven gives a remainder of 3. The number formed by swapping the last two digits of the original number gives a remainder of 5 when divid... | 6 |
## Task 4 - 190724
A driver drove his car from A to B. After 20 minutes of driving, he had a breakdown that took 30 minutes to fix. After another 12 minutes of driving, he had to wait 4 minutes at a closed railway crossing. By then, he had covered $40 \mathrm{~km}$. The journey from the railway crossing to B started a... | 122.5\mathrm{~} |
In 2022, the Ministry of Education issued the "Compulsory Education Curriculum Plan" and the curriculum standard (2022 edition), separating labor from the original comprehensive practical activities course. A certain middle school opened a farming garden to allow students to experience agricultural labor. They need to ... | 60 |
6. Let $|\overrightarrow{A B}|=10$, if a point $P$ on the plane satisfies, for any $t \in \mathbf{R}$, $|\overrightarrow{A P}-t \overrightarrow{A B}| \geq 3$, then the minimum value of $\overrightarrow{P A} \cdot \overrightarrow{P B}$ is
$\qquad$ , and at this time $|\overrightarrow{P A}+\overrightarrow{P B}|=$ $\qquad... | -16,6 |
Point $O$ is a point on the plane of $\triangle ABC$, satisfying $\overrightarrow{OA} \cdot \overrightarrow{OB} = \overrightarrow{OB} \cdot \overrightarrow{OC} = \overrightarrow{OC} \cdot \overrightarrow{OA}$. Then point $O$ is the ($\quad\quad$) of $\triangle ABC$.
A: Circumcenter of the triangle
B: Centroid of the t... | D |
9.6. Find the sum $1+2 x+3 x^{2}+\ldots+(n+1) x^{n}$. | \frac{(n+1)x^{n+2}-(n+2)x^{n+1}+1}{(x-1)^{2}} |
For any integer $n>1$, the number of prime numbers greater than $n!+1$ and less than $n!+n$ is:
$\text{(A) } 0\quad\qquad \text{(B) } 1\quad\\ \text{(C) } \frac{n}{2} \text{ for n even, } \frac{n+1}{2} \text{ for n odd}\quad\\ \text{(D) } n-1\quad \text{(E) } n$ | 0 |
If point $P$ is a moving point on the ellipse $\frac{x^2}{2}+y^2=1$, then the maximum distance from $P$ to the line $l: y=x+1$ is _______. | \frac{\sqrt{6} + \sqrt{2}}{2} |
If the function $f(x)=x^{3}-3bx+3b$ has a local minimum, then ()
A: $b\geqslant 0$
B: $b\leqslant 1$
C: $b\geqslant 1$
D: $b \gt 0$ | D |
## Problem Statement
Let $k$ be the coefficient of similarity transformation with the center at the origin. Is it true that point $A$ belongs to the image of plane $a$?
$A(2; -3; 1)$
$a: x + y - 2z + 2 = 0$
$k = \frac{5}{2}$ | 2\neq0 |
Define $ n!!$ to be $ n(n\minus{}2)(n\minus{}4)\ldots3\cdot1$ for $ n$ odd and $ n(n\minus{}2)(n\minus{}4)\ldots4\cdot2$ for $ n$ even. When $ \displaystyle \sum_{i\equal{}1}^{2009} \frac{(2i\minus{}1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $ 2^ab$ with $ b$ odd. Find $ \displaystyle... | 401 |
$f:\mathbb{R}^2 \to \mathbb{R}^2$ is injective and surjective. Distance of $X$ and $Y$ is not less than distance of $f(X)$ and $f(Y)$. Prove for $A$ in plane:
\[ S(A) \geq S(f(A))\]
where $S(A)$ is area of $A$ | S(A) \geq S(f(A)) |
Which of the following equations always holds true? (γγ)
A: $\sin(\alpha+\beta)=\sin\alpha+\sin\beta$
B: $\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$
C: $\sin(\alpha-\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$
D: $\cos(\alpha+\beta)=\cos\alpha\sin\beta-\sin\alpha\cos\beta$ | B |
Express $\sqrt[3]{5^{2}}$ in the form of a fractional exponent as ( ).
A: $5^{\frac{2}{3}}$
B: $5^{\frac{3}{2}}$
C: $5^{3}$
D: $5^{2}$ | A |
Using the numbers 0, 1, 2, 3, 4,
(1) How many five-digit numbers can be formed?
(2) How many five-digit numbers without repeated digits can be formed?
(3) How many five-digit odd numbers without repeated digits can be formed?
(4) Among the five-digit numbers without repeated digits, what is the position of 4213... | 32 |
Let's find two numbers whose sum is 316 and whose least common multiple is 4560. | 240,76 |
16. Let the function $f(x)=p x-\frac{p}{x}-2 \ln x$.
(1) If $f(x)$ is a monotonically increasing function in its domain, find the range of the real number $p$;
(2) Let $g(x)=\frac{2 \mathrm{e}}{x}$, and $p>0$. If there exists at least one point $x_{0}$ in the interval $[1, \mathrm{e}]$ such that $f\left(x_{0}\right)>g\... | p \in\left(\frac{4 \mathrm{e}}{\mathrm{e}^{2}-1},+\infty\right) |
In the arithmetic sequence ${a_n}$, $a_n > 0$ and $a_4 = 5$. Find the minimum value of $\frac{1}{a_2} + \frac{16}{a_6}$. | \frac{5}{2} |
The increasing [geometric sequence](https://artofproblemsolving.com/wiki/index.php/Geometric_sequence) $x_{0},x_{1},x_{2},\ldots$ consists entirely of [integral](https://artofproblemsolving.com/wiki/index.php/Integer) powers of $3.$ Given that
$\sum_{n=0}^{7}\log_{3}(x_{n}) = 308$ and $56 \leq \log_{3}\left ( \sum_{n=0... | 91 |
24th Putnam 1963 Problem B3 R is the reals. Find all f : R β R which are twice differentiable and satisfy: f(x) 2 - f(y) 2 = f(x + y) f(x - y). | f(Y)=AYorf(Y)=A\sinhaY |
For positive rational $x$, if $x$ is written in the form $p/q$ with $p, q$ positive relatively prime integers, define $f(x)=p+q$. For example, $f(1)=2$.
a) Prove that if $f(x)=f(mx/n)$ for rational $x$ and positive integers $m, n$, then $f(x)$ divides $|m-n|$.
b) Let $n$ be a positive integer. If all $x$ which satisf... | n |
4. If $x, y, z$ are real numbers, satisfying
$$
x+\frac{1}{y}=2 y+\frac{2}{z}=3 z+\frac{3}{x}=k \text{, and } x y z=3 \text{, }
$$
then $k=$ . $\qquad$ | 4 |
Given the set $B=\{-1, 1, 4\}$, the number of sets $M$ that satisfy the condition $\emptyset \subset M \subseteq B$ is (γγ)
A: 3
B: 6
C: 7
D: 8 | C |
1. Senya thought of two numbers, then subtracted the smaller from the larger, added both numbers and the difference, and got 68. What was the larger of the numbers Senya thought of? | 34 |
Let $n$ be a positive integer. Consider an $n\times n$ grid of unit squares. How many ways are there to partition the horizontal and vertical unit segments of the grid into $n(n + 1)$ pairs so that the following properties are satisfi ed?
(i) Each pair consists of a horizontal segment and a vertical segment that shar... | (n+1)! |
9. Given that $\left\{a_{n}\right\}$ is a geometric sequence with the first term 1 and common ratio 2, and $\left\{b_{n}\right\}$ is an arithmetic sequence with the first term 2 and common difference 5, the numbers that appear in both sequences are arranged in ascending order to form the sequence $\left\{x_{n}\right\}$... | 2^{397} |
4. Let the natural number $n \geqslant 5, n$ different natural numbers $a_{1}, a_{2}, \cdots$, $a_{n}$ have the following property: for any two different non-empty subsets $A, B$ of the set $S=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$, the sum of the numbers in $A$ and the sum of the numbers in $B$ will not be equal.... | 2-\frac{1}{2^{n-1}} |
## Task 12/69
All pairs $(x ; y)$ are to be found that satisfy the system of equations
$$
\begin{aligned}
& |y-x|=|x+1| \\
& \frac{y-3}{4}=\left[\frac{x-1}{5}\right]
\end{aligned}
$$
where $[a]$ is an integer with $a-1<[a] \leq a$. | (1;3)(x_{0};-1)with-4\leqx_{0}<1 |
30. A Thriving Business. A businessman initially invested $2000 in his business. Every 3 years, he increased his capital by $50\%$. What was the amount of his capital after 18 years? | 22781.25 |
946. Find two natural numbers whose product does not exceed their sum. List all solutions. | one\\is\1,\the\other\is\any\natural\;\2,2 |
## Task Condition
Find the derivative.
$y=\frac{e^{2 x}(2-\sin 2 x-\cos 2 x)}{8}$ | e^{2x}\cdot\sin^{2}x |
Determine the range of the function $f(x) = x^2 - 2x - 3$ on the interval $[0, 3)$. | [-4, 0) |
Given that $$f(x)= \frac {(x+1)^{2}+sinx}{x^{2}+1}$$, where $f''(x)$ is the derivative of $f(x)$, find the value of $f(2019) + f''(2019) + f(-2019) - f''(-2019) = (\text{ })$
A: 8056
B: 4028
C: 1
D: 2 | D |
If for any real number $k$, the line $y-2=k(x+1)$ always passes through a fixed point $M$, then the coordinates of $M$ are (γγ)
A: $(1,2)$
B: $(1,-2)$
C: $(-1,2)$
D: $(-1,-2)$ | C |
Example 3 As shown in Figure 20-3, given that $A B$ is a slant line segment of plane $\alpha$, $\boldsymbol{n}$ is the normal vector of plane $\alpha$. Prove: The distance from $A$ to plane $\alpha$ is $\frac{|\overrightarrow{A B} \cdot n|}{|n|}$. | \frac{|\overrightarrow{AB}\cdot\boldsymbol{n}|}{|\boldsymbol{n}|} |
Given the quadratic function $y=n(n+1)x^{2}-(2n+1)x+1$, when $n$ takes the values $1$, $2$, $3$, $4$, ..., $10$ in turn, the total length of the segments cut by its graph on the $x$-axis is ( ).
A: $1$
B: $\dfrac {10}{11}$
C: $\dfrac {12}{11}$
D: $\dfrac {11}{12}$ | B |
15. In the diagram shown, $P Q=S Q=Q R$ and $\angle S P Q=2 \times \angle R S Q$. What is the size of angle $Q R S$ ?
A $20^{\circ}$
B $25^{\circ}$
C $30^{\circ}$
D $35^{\circ}$
E $40^{\circ}$ | 30 |
How many of the following propositions are correct: "If $x=3$, then $x^2-7x+12=0$" and its converse, inverse, and contrapositive propositions?
A: 0
B: 1
C: 2
D: 3 | \text{C} |
Given the function $f(x)=x^{3}-3ax^{2}+3x+1$.
(I) When $a=2$, find the monotonic intervals of $f(x)$;
(II) If $f(x)$ has at least one extreme value point in the interval $(2,3)$, find the range of $a$. | \frac {5}{4} < a < \frac {5}{3} |
6. Four prime numbers $a, b, c, d$ satisfy $a+2=b+6=c+11=d$, then $a b c d+18=$ | 2020 |
If in a scatter plot, all the sample points fall on a straight line with a non-zero real number as the slope, then the correlation coefficient $R^2=$ ___. | 1 |
If the sum $1 + 2 + 3 + \cdots + K$ is a perfect square $N^2$ and if $N$ is less than $100$, then the possible values for $K$ are:
$\textbf{(A)}\ \text{only }1\qquad \textbf{(B)}\ 1\text{ and }8\qquad \textbf{(C)}\ \text{only }8\qquad \textbf{(D)}\ 8\text{ and }49\qquad \textbf{(E)}\ 1,8,\text{ and }49$ | \textbf{(E)} |
Given an isosceles triangle with sides of $5$ and $10$, the perimeter of the triangle is ______. | 25 |
If $a^{-1}=(-1)^{0}$, then $a=$ \_\_\_\_\_\_. | 1 |
Exactly three of the interior angles of a convex [polygon](https://artofproblemsolving.com/wiki/index.php/Polygon) are obtuse. What is the maximum number of sides of such a polygon?
$\mathrm{(A)\ } 4 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ } 6 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ }8$ | 6 |
Given that the area of $\triangle ABC$ is $S= \sqrt {3}$, and $\angle A= \dfrac {\pi}{3}$, find the dot product of $\overrightarrow{AB}$ and $\overrightarrow{AC}$. | 2 |
[131 The equation $x^{2}-m x+2 m-2=0$ with respect to $x$ has a solution in the interval $\left[0, \frac{3}{2}\right]$, find the range of real number $m$.
] | [-\frac{1}{2},4-2\sqrt{2}] |
53. Let $\sum_{i=1}^{n} x_{i}=1, x_{i}>0$, prove: $n \sum_{i=1}^{n} x_{i}^{2}-\sum_{i \leqslant i<j \leqslant n} \frac{\left(x_{i}-x_{j}\right)^{2}}{x_{i}+x_{j}} \leqslant 1$. (2007 Zhejiang Province Mathematics Competition Problem) | 1 |
$1-2-3+4+5-6-7+8+9-10-11+\cdots + 1992+1993-1994-1995+1996=$
$\text{(A)}\ -998 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 998$ | 0 |
11. If $2^{6}+2^{9}+2^{n}$ is a perfect square, then the positive integer $n=$ $\qquad$ . | 10 |
Determine the value of the definite integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\sin x + \cos x) dx$. | 2 |
Given a triangle with side lengths 15, 20, and 25, find the triangle's shortest altitude.
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 12.5 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 15$ | 12 |
Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings:
(1) $\sum_{v\in V} f(v)=|E|$;
(2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the... | n \leq m(G) \leq n! |
Given the one-variable quadratic equation $x^{2}+\left(2m-1\right)x+m^{2}=0$ has two distinct real roots.$(1)$ Find the range of values for $m$;$(2)$ Let the two real roots of the equation be $\alpha $ and $\beta $, and $\alpha ^{2}+\beta ^{2}=1$, find the value of $m$. | m = 0 |
## Task Condition
Calculate the areas of figures bounded by lines given in polar coordinates.
$$
r=\cos \phi-\sin \phi
$$ | \frac{\pi}{2} |
Henry starts with a list of the first 1000 positive integers, and performs a series of steps on the list. At each step, he erases any nonpositive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty?
[i]Proposed by... | 11 |
Let $y=f(x)$ be a quadratic function, the equation $f(x)=0$ has two equal real roots, and $f'(x)=2x+2$, find the analytical expression of $f(x)$. | f(x)=x^2+2x+1 |
Example 1 As shown in Figure 4, in tetrahedron $A B C D$, $\triangle A B C$ is an equilateral triangle, $A D=B D=2, A D \perp$ $B D, A D \perp C D$. Then the distance from point $D$
Figure 4 to plane $A B C$ is
[1] | \frac{2\sqrt{3}}{3} |
8. Compute
$$
\int_{0}^{\pi / 3} x \tan ^{2}(x) d x
$$ | \frac{\pi\sqrt{3}}{3}-\frac{\pi^{2}}{18}-\ln2 |
Given the integral $\int_{0}^{1}(kx+1)dx=k$, then the real number $k=$ (__).
A: $2$
B: $-2$
C: $1$
D: $-1$ | A |
Alexandra draws a letter A which stands on the $x$-axis.
[list=a][*]The left side of the letter A lies along the line with equation $y=3x+6$. What is the $x$-intercept of the line with equation $y=3x+6$?
[*]The right side of the letter A lies along the line $L_2$ and the leter is symmetric about the $y$-axis. What is t... | 2 |
1. Let $m=\sqrt{5}+1$. Then the integer part of $m+\frac{1}{m}$ is $\qquad$ . | 3 |
Let set $A=\{x \mid \frac{x+3}{x-1} \geq 0\}$, set $B=\{x \mid x^2-x-2 \leq 0\}$, and set $C=\{x \mid x \geq a^2-2\}$.
(1) Find $A \cap B$.
(2) If $B \cup C = C$, find the range of values for the real number $a$. | [-1, 1] |
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