problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remai... | 100 |
15. [9] Vijay chooses three distinct integers $a, b, c$ from the set $\{1,2,3,4,5,6,7,8,9,10,11\}$. If $k$ is the minimum value taken on by the polynomial $a(x-b)(x-c)$ over all real numbers $x$, and $l$ is the minimum value taken on by the polynomial $a(x-b)(x+c)$ over all real numbers $x$, compute the maximum possibl... | 990 |
18. Let $A B C$ be an equilateral triangle. Let $\overrightarrow{A B}$ be extended to a point $D$ such that $B$ is the midpoint of $\overline{A D}$. A variable point $E$ is taken on the same plane such that $D E=A B$. If the distance between $C$ and $E$ is as large as possible, what is $\angle B E D$ ? | 15 |
In regression analysis, the larger the value of the coefficient of determination $R^{2}$, the sum of squares of residuals $(\quad)$
A: becomes smaller
B: becomes larger
C: may become larger or smaller
D: none of the above | A |
Which is larger: $1985^{1986}$ or $1986^{1985}$? | 1985^{1986}>1986^{1985} |
[
The numerical function $f$ is such that for any $x$ and $y$, the equality $f(x+y)=f(x)+f(y)+80xy$ holds. Find $f(1)$, if $f(0.25)=2$.
# | 38 |
If $3^{x}=2$ and $3^{y}=4$, then $3^{x-2y}=$____. | \frac{1}{8} |
7. Let $n$ be a positive integer. From the set $\{1,2, \cdots, 2015\}$, the probability that a randomly chosen positive integer $n$ is a solution to the equation
$$
\left[\frac{n}{2}\right]=\left[\frac{n}{3}\right]+\left[\frac{n}{6}\right]
$$
is $\qquad$ ( $[x]$ denotes the greatest integer not exceeding the real numb... | \frac{1007}{2015} |
The opposite of $2023$ is
A: $\frac{1}{{2023}}$
B: $-2023$
C: $-\frac{1}{{2023}}$
D: $3202$ | B |
The graph of the power function $y=x^{\alpha}$ (where $\alpha$ is a constant) must pass through (γγ)
A: the point (0, 0)
B: the point (1, 1)
C: the point (-1, 1)
D: the point (1, -1) | \text{B} |
Let $M$ - be a finite set of numbers. It is known that among any three of its elements, there will be two whose sum belongs to $M$.
What is the maximum number of elements that can be in $M$? | 7 |
Let $n$ be a natural number. Give an explicit formula for the sum $0+1+2+\ldots+n$. | \frac{n(n+1)}{2} |
Given the function $f(x)=e^{x}+ax^{2}-bx-1$ ($a,b\in\mathbb{R}$, $e$ is the base of the natural logarithm).
(I) Let the derivative of $f(x)$ be $g(x)$. Find the minimum value of $g(x)$ in the interval $[0,1]$.
(II) If $f(1)=0$, and the function $f(x)$ has a zero point in the interval $(0,1)$, prove that: $-1 < a < 2-... | -1 < a < 2-e |
2γUse $1, 2, 3, 4$ to form a four-digit number $\overline{a b c d}$, with the following requirements:
οΌ1οΌ $a, b, c, d$ are all different; οΌ2οΌ$b$ is greater than $a$ and $d$, and $c$ is greater than $a$ and $d$. How many such four-digit numbers are there? $\qquad$ | 4 |
Let the sum of the digits of a 1995-digit number divisible by 9 be $a$. We then add the digits of $a$, and let the result be $b$. What can the sum of the digits of $b$ be? | 9 |
1.23 In triangle $ABC$, $AB=33$, $AC=21$, and $BC=m$, where $m$ is a positive integer. If there exists a point $D$ on $AB$ and a point $E$ on $AC$ such that $AD=DE=EC=n$, where $n$ is a positive integer, what value must $m$ take?
(Swedish Mathematical Competition, 1982) | 30 |
If the sum of all integer solutions of the inequality system about $x$ $\left\{{\begin{array}{l}{2x+1>x+a}\\{\frac{x}{2}+1\geq\frac{5}{2}x-9}\end{array}}\right.$ is $14$, then the value of the integer $a$ is ______. | 2 |
Given functions $f(x)=a+2\sin x$ and $g(x)=\log _{2}x$.
$(1)$ Find the value of $a$ such that the function $f(x)$ has exactly two zeros in the interval $\left[0,2\pi \right]$;
$(2)$ If there exist $x_{1}\in \left[0,2\pi \right]$ and $x_{2}\in \left[1,2\right]$, such that $f(x_{1})=g(x_{2})$, determine the range of real... | [-2,3] |
For which values of the real parameters $a$ and $b$ will the maximum of the function $\left|x^{2}+a x+b\right|$ on the interval $[-1,1]$ be the smallest? | =0,b=-\frac{1}{2} |
Given the function $f(x)=kx^{3}+3(k-1)x^{2}-k^{2}+1$, determine the range of values for $k$ such that the function is decreasing on the interval $(0,4)$. | k \leq \frac{1}{3} |
16. Fold a rectangular piece of paper that is $80 \mathrm{~cm}$ long and $60 \mathrm{~cm}$ wide so that the two endpoints of one of its diagonals coincide. The length of the fold line is $\qquad$ $\mathrm{cm}$. | 75 |
39. Find the smallest integer starting with the digit 7 that decreases threefold when this digit is moved to the end. Find all such numbers. | 7241379310344827586206896551 |
Determine all positive integers $a$ for which the equation
$$
7 a n - 3 n! = 2020
$$
has a positive integer solution $n$.
(Hint: For every positive integer $n$: $\quad n! = 1 \cdot 2 \cdot \cdots \cdot n$.)
(Richard Henner) | =68or=289 |
(1) Prove that when $a$, $b$, and $c$ are positive numbers, $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geq 9$
(2) Given $x \in \mathbb{R}$, $a=x^2-1$, $b=2x+2$, prove that at least one of $a$, $b$ is not less than 0. | \text{At least one of } a, b \text{ is not less than } 0 |
3. $\mathbf{C}$ is the set of complex numbers, let the set $A=\left\{z \mid z^{18}=1, z \in \mathbf{C}\right\}, B=\left\{w \mid w^{48}=1, w \in \mathbf{C}\right\}, D=\{z w \mid z \in A$ οΌ $w \in B\}$. Find the number of elements in $D$. | 144 |
2. Among all lines passing through the point $\left(10, \frac{1}{2}\right)$, the number of lines that pass through at least two lattice points is $($ )
A. one
B. none
C. a finite number but no less than 2
D. infinitely many | D |
Given the function $f(x) = -x^2 + 4x + a$, where $x \in [0, 1]$, if $f(x)$ has a minimum value of $-2$, then the maximum value of $f(x)$ is ( )
A: 1
B: 0
C: -1
D: 2 | A |
Let $a_n =\sum_{d|n} \frac{1}{2^{d+ \frac{n}{d}}}$. In other words, $a_n$ is the sum of $\frac{1}{2^{d+ \frac{n}{d}}}$ over all divisors $d$ of $n$.
Find
$$\frac{\sum_{k=1} ^{\infty}ka_k}{\sum_{k=1}^{\infty} a_k} =\frac{a_1 + 2a_2 + 3a_3 + ....}{a_1 + a_2 + a_3 +....}$$ | 4 |
Natural numbers $k, l,p$ and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$ then $a +\frac1b$ and $b + \frac1a$ are the roots of $x^2 -px + q = 0$. What is the sum of all possible values of $q$? | 4 |
Given the polar equation of curve C is $\rho=6\cos\theta$, convert the polar equation of curve C into a Cartesian coordinate equation. | x^2+y^2=6x |
Given the function $f\left( x \right)=\ln \left( {{e}^{x}}+{{e}^{-x}} \right)+{{x}^{2}}$, then the range of $x$ for which $f(2x) > f(x+3)$ holds is $(\ )$
A: Β $\left( -1,3 \right)$
B: $(-\infty ,-3)\cup \left( 3,+\infty \right)$
C: $\left( -3,3 \right)$
D: $(-\infty ,-1)\cup \left( 3,+\infty \right)$ | D |
Given that $a$ and $b$ are nonzero real numbers, and $a < b$, which of the following propositions is true?
A: $a^2 < b^2$
B: $ab^2 < a^2b$
C: $\dfrac{1}{ab^2} < \dfrac{1}{a^2b}$
D: $\dfrac{b}{a} < \dfrac{a}{b}$ | C |
Given that the real numbers $x,y$ satisfy $3\leqslant xy^{2}\leqslant 8,4\leqslant \frac{x^{2}}{y}\leqslant 9$, find the maximum value of $\frac{x^{3}}{y^{4}}$. | 27 |
Question 23 Let $f(n)=\left\{\begin{array}{ll}n-12 & n>2000 ; \\ f[f(n+14)] & n \leqslant 2000\end{array}(n \in \mathbf{N})\right.$. Try to find all the fixed points of $f(n)$. | 1989or1990 |
The range of the function $$y= \frac {1}{x^{2}+2}$$ is \_\_\_\_\_\_ . | (0, \frac {1}{2}] |
A tangent line is drawn from a point on the circle $x^2+y^2=1$ to intersect the positive x-axis and y-axis at points A and B, respectively. The product of $|\overrightarrow{OA}|$ and $|\overrightarrow{OB}|$ is (γγ)
A: Maximum value $\sqrt{2}$
B: Minimum value $\sqrt{2}$
C: Maximum value 2
D: Minimum value 2 | \text{D} |
A random sample of $10$ households was drawn from a residential area, and data on the monthly income ${x_i}$ (in thousands of yuan) and monthly savings ${y_i}$ (in thousands of yuan) of the $i$-th household were obtained. It was calculated that $\sum\limits_{i=1}^{10}{{{x}_{i}}}=80$, $\sum\limits_{i=1}^{10}{{{y}_{i}}}=... | 1.7\text{ (thousand yuan)} |
3. Solve the equation: $[20 x+23]=20+23 x$. Recall that $[a]$ denotes the integer part of the number, that is, the greatest integer not exceeding $a$. (S. S. Koresheva) | \frac{16}{23},\frac{17}{23},\frac{18}{23},\frac{19}{23},\frac{20}{23},\frac{21}{23},\frac{22}{23},1 |
Let $S=\{1,2, \cdots, 2005\}$. If any set of $n$ pairwise coprime numbers in $S$ contains at least one prime number, find the minimum value of $n$. (Tang Lihua) | 16 |
To make $$\frac {x}{8}$$ a simplest proper fraction, and $$\frac {x}{6}$$ an improper fraction, $x$ should be $\boxed{7}$. | 7 |
Find the equation of the ellipse that passes through the point (3, -2) and has the same foci as the ellipse $$\frac {x^{2}}{9}+ \frac {y^{2}}{4}=1$$.
Options:
A: $$\frac {x^{2}}{10}+ \frac {y^{2}}{15}=1$$
B: $$\frac {x^{2}}{15}+ \frac {y^{2}}{10}=1$$
C: $$\frac {x^{2}}{100}+ \frac {y^{2}}{105}=1$$
D: $$\frac {... | \text{B} |
The coefficient of the $x^3$ term in the expansion of $(3-2x-{{x}^{4}}){{\left( 2x-1 \right)}^{6}}$ is
A: $600$
B: $360$
C: $-600$
D: $-360$ | C |
A point $P$ is at the center of a square with a side length of $10 \mathrm{~cm}$. How many points on the edge of the square are $6 \mathrm{~cm}$ away from $P$?
(a) 1
(b) 2
(c) 4
(d) 6
(e) 8 | 8 |
Given the parabola $C$: $y^{2}=4x$ with focus $F$, a line passing through point $F$ intersects the parabola at points $A$ and $B$. If $|AB|=6$, then the x-coordinate of the midpoint $M$ of segment $AB$ is (γ).
A: $2$
B: $4$
C: $5$
D: $6$ | A |
Let $a$, $b$, and $c$ be non-zero real number such that $\tfrac{ab}{a+b}=3$, $\tfrac{bc}{b+c}=4$, and $\tfrac{ca}{c+a}=5$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{abc}{ab+bc+ca}=\tfrac{m}{n}$. Find $m+n$. | 167 |
A certain pharmacy bought 41 boxes of masks from factory A at a price of $a$ yuan per box, and bought the same 59 boxes of masks from factory B at a price of $b$ yuan per box ($a \lt b$). If all these masks are sold at a price of $\frac{{a+b}}{2}$ per box, then this pharmacy will ( )
A: incur a loss
B: make a profit
... | A |
Example 22 Skew lines $a, b, a \perp b, c$ forms a $30^{\circ}$ angle with $a$, find the range of the angle $\theta$ formed between $c$ and $b$. | [60,90] |
In $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a^{2}+c^{2}=b^{2}+ac$.
(1) Find the measure of angle $B$;
(2) If $A= \dfrac {5Ο}{12}$ and $b=2$, find the length of side $c$;
(3) If $a+c=4$, find the minimum value of $b$. | 2 |
2. Solve the equation $\frac{1}{4^{x}+1}+\frac{1}{2^{x} \cdot 3^{x}-1}=\frac{2^{x}}{4^{x} \cdot 3^{x}-2 \cdot 2^{x}-3^{x}}$. | x_{1}=-1x_{2}=1 |
Given a sequence $\{a_n\}$ where all terms are positive numbers and it satisfies the equation $a_n^2 = 4S_n - 2a_n - 1 (n\in \mathbf{N}^*)$, where $S_n$ denotes the sum of the first $n$ terms of the sequence.
(I) Find the values of $a_1$ and $a_2$;
(II) Find the general term formula of the sequence. | a_n = 2n - 1 |
Example 1: If the coordinates of point $(x, y)$ satisfy $x^{2}+y^{2}+5 x \leqslant 0$, find the maximum and minimum values of $3 x+4 y$.
| 5, -20 |
30 Given that the three sides of $\triangle A B C$ are all integers, $\angle A=2 \angle B, \angle C>90^{\circ}$. Find the minimum perimeter of $\triangle A B C$. | 77 |
Given that the solution to the equation $2x + 1 = -x + 4$ is $x = 1$, find the point of intersection for the lines $y = 2x + 1$ and $y = -x + 4$. | (1, 3) |
1. Given $I$ is the incenter of acute $\triangle A B C$, and $A_{1} γ B_{1}$ γ $C_{1}$ are the reflections of point $I$ over sides $B C γ C A γ A B$ respectively. If point $B$ lies on the circumcircle of $\triangle A_{1} B_{1} C_{1}$, then $\angle A B C$ equals $(\quad)$.
(A) $30^{\circ}$
(B) $45^{\circ}$
(C) $60^{\cir... | C |
A two-digit number, subtracting three times the sum of its digits from itself gives $13$; when this two-digit number is divided by the sum of its digits, the quotient is $4$ and the remainder is $6$. Let the digit in the tens place be $x$ and the digit in the ones place be $y$. The system of equations is ( ).
A: $\lef... | A |
In his triangle $ABC$ Serge made some measurements and informed Ilias about the lengths of median $AD$ and side $AC$. Based on these data Ilias proved the assertion: angle $CAB$ is obtuse, while angle $DAB$ is acute. Determine a ratio $AD/AC$ and prove Ilias' assertion (for any triangle with such a ratio). | \frac{AD}{AC} = \frac{1}{2} |
7.3. The numbers $a, b$, and $c$ satisfy the relation $\frac{a+b}{c}=\frac{b+c}{a}=\frac{a+c}{b}$. Find $a+b+c$, given that $b \neq c$. | 0 |
Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's. How far can he number the pages of his scrapbook with these digits?
$\text{(A)}\ 22 \qquad \text{(B)}\ 99 \qquad \text{(C)}\ 112 \qquad \text{(D)}\ 119 \qquad \text{(E)}\ 199$ | 119 |
Given that the value of the algebraic expression $2y^2+3y+7$ is $8$, then the value of the algebraic expression $4y^2+6y-9$ is (γγ)
A: 2
B: 17
C: 7
D: -7 | D |
Given the hyperbola $\frac{x^{2}}{m} + \frac{y^{2}}{n} = 1$ with an eccentricity of $2$, and one of its foci coincides with the focus of the parabola $y = \frac{1}{8}x^{2}$, determine the equation of this hyperbola $(\ \ \ \ )$
A: $\frac{x^{2}}{3} - y^{2} = 1$
B: $y^{2} - \frac{x^{2}}{3} = 1$
C: $\frac{x^{2}}{12} - \f... | B |
## Problem Statement
Calculate the definite integral:
$$
\int_{\sqrt{2}}^{\sqrt{3}} \frac{x}{\sqrt{x^{4}-x^{2}-1}} d x
$$ | \ln\sqrt{\frac{5+2\sqrt{5}}{5}} |
Question 1-1 When any two of the positive divisors of a positive integer $n$ are added together, it is found that these sums are all different, and they are arranged from smallest to largest as $4$, $6$, $8$, $10$, $12$, $14$, $16$, $18$, $20$, $24$, $46$, $48$, $50$, $54$, $60$. Find the value of $n$. | 45 |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a = \sqrt{3}$ and $A = \frac{2\pi}{3}$, find the maximum value of $b + c$. | 2 |
2. Let $n$ be the time in minutes it takes for the slower boy to complete a lap. Then $n>5, n \in \mathbb{N}$. The speed at which the faster boy catches up to the slower one is
$$
\frac{1}{5}-\frac{1}{n}=\frac{n-5}{5 n}\left(\frac{\text { lap }}{\text { min }}\right)
$$
therefore, the time between meetings is
$$
\le... | 6 |
10. Given that $f(x)$ is a function defined on $\mathbf{R}$, $f(1)=1$ and for any $x \in \mathbf{R}$, $f(x+5) \geqslant f(x)+5, f(x+1) \leqslant f(x)+1$. If $g(x)=f(x)+1-x$, then $g(2002)$ $=$ . $\qquad$ | 1 |
Using the systematic sampling method to select 32 people for a questionnaire survey from 960 people, they are randomly numbered 1, 2, β¦, 960. After grouping, the number drawn by simple random sampling in the first group is 29. The number of people among the 32 whose numbers fall within the interval \[200, 480\] is (γγ)... | C |
(5) The sum of the radii of all circles passing through point $A(1505,1008)$ and tangent to the lines $l_{1}: y=0$ and $l_{2}: y=\frac{4}{3} x$ is $\qquad$ . | 2009 |
11. I $\pi$ cchi are animals that live in families of 1, 2, or 3 individuals. Originally, a family of $3 \pi$ cchi was imported to Italy. A family of $n \pi$ cchi reproduces by growing $2n-2$ new individuals, forming a total of $3n-2$, and dividing into new families (not necessarily two families with the same number of... | 253 |
1. A company produces cylindrical tiles. Each tile has a diameter of $6 \mathrm{~cm}$ and a volume of $600 \mathrm{~cm}^{3}$. The company decides to change the diameter of the tiles so that they will have a volume of $700 \mathrm{~cm}^{3}$ and the same height as before. How many $\mathrm{cm}^{2}$ of tile material does ... | 498\mathrm{~}^{2} |
Which combination of three different regular polygons can tile the floor?
A: equilateral triangle, square, pentagon
B: equilateral triangle, square, hexagon
C: equilateral triangle, square, heptagon
D: equilateral triangle, square, octagon | \text{B} |
For real numbers $x, y$ in the interval $[0 ; 12]$:
$$
x y=(12-x)^{2}(12-y)^{2} \text {. }
$$
What is the largest value of the product $x y$? | 81 |
Problem 1. In the bookstore, 8450 notebooks were delivered. On the first day, half of them were sold, and on the second day, 437 fewer notebooks were sold. How many notebooks remained unsold? | 437 |
In the Cartesian coordinate system xOy, define point A at $(a, a)$. Let P be a moving point on the graph of the function $y = \frac{1}{x}$ ($x>0$). If the shortest distance between points P and A is $2\sqrt{2}$, then the set of all real numbers $a$ that satisfy this condition is __________. | a = -1 \text{ or } \sqrt{10} |
13. (12 points) In a cage, there are chickens and rabbits, with a total of 28 heads. The total number of rabbit feet is 12 more than 3 times the total number of chicken feet. How many rabbits are there in the cage? | 18 |
Given the binomial expansion of $\left(x- \dfrac {a}{x^{2}}\right)^{6}$, the coefficient of the $x^{3}$ term is $-12$. Find the value of $a$ and the constant term in the expansion. | 60 |
3. Teacher Wang has 45 candies, and he decides to eat some every day. Since the candies are very delicious, starting from the second day, the number of candies he eats each day is 3 more than the previous day, and he finishes all the candies in 5 days. So, how many candies did Teacher Wang eat on the second day? ( ) | 6 |
Let $P$ be an arbitrary point inside $\triangle ABC$ with sides $3,7,8$. What is the probability that the distance of $P$ to at least one vertices of the triangle is less than $1$?
$ \textbf{(A)}\ \frac{\pi}{36}\sqrt 2
\qquad\textbf{(B)}\ \frac{\pi}{36}\sqrt 3
\qquad\textbf{(C)}\ \frac{\pi}{36}
\qquad\textbf{(D)}\ \fr... | \frac{\pi}{36}\sqrt{3} |
Write a function $f(x)$ that satisfies $f(xy) = f(x) + f(y) - 1$ for $x, y > 0$. | f(x) = \log x + 1 |
## 46. Math Puzzle $3 / 69$
Write down a three-digit number and subtract the number with the reversed digit order. Divide the result by the difference between the 1st and 3rd digit of the original number, then divide by 11 again. If you now take the square root, the result is always 3. This is a neat trick to amaze yo... | 3 |
Find the minimum value of the expression
$$
\sqrt{1-x+x^{2}}+\sqrt{1-\sqrt{3} \cdot x+x^{2}}
$$ | \sqrt{2} |
If the fraction $\frac{x}{2-x}$ is meaningful, then the value of $x$ cannot be:
A: $1$
B: $2$
C: $3$
D: $0$ | B |
Given a sequence $\{a_n\}$ with the sum of its first $n$ terms $S_n=2^n-1$, find the sum of the first $n$ terms of the odd-numbered terms in this sequence.
A: $\dfrac{1}{3}(2^{n+1}-1)$
B: $\dfrac{1}{3}(2^{n+1}-2)$
C: $\dfrac{1}{3}(2^{2n}-1)$
D: $\dfrac{1}{3}(2^{2n}-2)$ | \text{C} |
Two basketball players, player A and player B, have free throw shooting probabilities of $0.7$ and $0.6$ respectively. Each of them takes one shot. Assuming the events "player A makes the shot" and "player B makes the shot" are independent, the probability that at least one of them makes the shot is ______. | 0.88 |
Calculate the quotient of the complex numbers $\frac{1+2i}{1-i}$. | - \frac{1}{2}+ \frac{3}{2}i |
9. It is known that the number of occurrences of a certain character in the text is from $10.5\%$ to $11\%$ of the length of the text (the length of the text is understood as the total number of characters in the text). Find the minimum possible length of the text.
# | 19 |
5. Represent as an algebraic expression (simplify) the sum $8+88+888+8888+\ldots+8 \ldots 8$, if the last term in its notation contains p times the digit eight.
# | \frac{8}{81}(10^{p+1}-10-9p) |
Given the sets $M=\{1, -1\}$, $N=\{1\}$, the number of all non-empty subsets of the set $M \cup N$ is ( )
A: 1
B: 2
C: 3
D: 4 | C |
Subject (3). Consider the triangle $ABC$ with $\mathrm{m}(\angle BAC) = 90^{\circ}$. The bisector of $\angle ACB$ intersects the segment $(AB)$ at point $E$. If there exists $D \in (CE)$ such that $\mathrm{m}(\angle DAC) = \mathrm{m}(\angle BDE) = x^{\circ}$, then calculate $x$.
The statement in the problem is unfortu... | 45 |
G7.1 If the two distinct roots of the equation $a x^{2}-m x+1996=0$ are primes, find the value of $a$. | 2 |
In triangle $ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. Given that $a=1$, $b= \sqrt{3}$, and $\angle C=30^{\circ}$, the area of $\triangle ABC$ is ______. | \frac{\sqrt{3}}{4} |
Given an odd $n\in\mathbb N$. In an $n\times n$ chessboard, you may place many $2\times2$ squares. How many grids, at most, are covered by exactly one square? | n(n-1) |
The equation $x^3+6x^2+11x+6=0$ has:
$\textbf{(A)}\ \text{no negative real roots}\qquad\textbf{(B)}\ \text{no positive real roots}\qquad\textbf{(C)}\ \text{no real roots}\\ \textbf{(D)}\ \text{1 positive and 2 negative roots}\qquad\textbf{(E)}\ \text{2 positive and 1 negative root}$ | B |
\section*{Exercise 4 - V11013}
For a triangle, the following are given: \(a=5 \mathrm{~cm}, \beta=47^{\circ}\) and \(\gamma=55^{\circ}\).
Calculate \(b, c\) and \(\alpha\)! | b\approx3.74\mathrm{~},\,\approx4.19\mathrm{~},\,\alpha=78 |
Consider numbers of the form $1a1$, where $a$ is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?
[i]Note: A palindrome is a number which reads the same from left to right and from right to left. Examples: $353$, $91719$.[/i] | 55 |
Given the function $f(x)=2e^x + \frac{1}{2}ax^2 + ax + 1$ has two extreme values, find the range of real numbers $a$. | (-\infty,-2) |
When $m\neq -1$, the correct judgment about the following system of equations $\begin{cases} mx+y=m+1 \\\\ x+my=2m\\end{cases}$ is ($\quad\quad$).
A: The system of equations has a unique solution
B: The system of equations has a unique solution or has infinitely many solutions
C: The system of equations has no solution... | \text{B} |
Cla
Two circles with centers $O$ and $Q$, intersecting each other at points $A$ and $B$, intersect the bisector of angle $O A Q$ at points $C$ and $D$ respectively. Segments $A D$ and $O Q$ intersect at point $E$, and the areas of triangles $O A E$ and $Q A E$ are 18 and 42 respectively. Find the area of quadrilateral... | 200;3:7 |
The line segment formed by $A(1, 2)$ and $B(3, 3)$ is rotated to the line segment formed by $A'(3, 1)$ and $B'(4, 3)$ about the point $P(r, s)$. What is $|r-s|$?
$\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{2}{3} \qquad \textbf{(E) } 1$ | 1 |
## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(-8 ; 0 ; 7)$
$B(-3 ; 2 ; 4)$
$C(-1 ; 4 ; 5)$ | 2x+2y+z+9=0 |
3.51 On a river with a current speed of $5 \mathrm{km} / \mathrm{h}$, there are piers $A, B$, and $C$ in the direction of the current, with $B$ located midway between $A$ and $C$. From pier $B$, a raft and a boat depart simultaneously, with the raft moving downstream to pier $C$ and the boat heading to pier $A$. The sp... | 5<\nu<15 |
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