problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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Task B-3.3. Determine all integers $k$ for which the equation $x^{2014} + k x^{2} + 36 = 0$ has at least one integer solution. | k\in{-37,-4^{1006}-9,-9^{1006}-4,-36^{1006}-1} |
Given the curve $y= \frac {1}{3}x^{3}+x$.
(1) Find the equation of the tangent line to the curve at point P($1$, $\frac {4}{3}$).
(2) Determine the range of values for the slope angle of the tangent line to this curve. | \left[ \frac {\pi}{4}, \frac {\pi}{2}\right) |
Problem 2.4. Consider an alphabet of $n$ letters, with which we will form words. We will say that a word contains a palindrome if a segment of that word, of more than one letter, reads the same forwards as backwards. For example, the word OLIMPIADAS contains the palindrome ADA. Given an integer $k$ greater than 2, dete... | n^{2}(n-1)^{k-2} |
6. Black and white balls are arranged in a circle, with black balls being twice as many as white ones. It is known that among pairs of adjacent balls, there are three times as many monochromatic pairs as polychromatic ones. What is the smallest number of balls that could have been arranged? (B. Trushin) | 24 |
3-2. Calculate the sum
$$
S=\frac{2013}{2 \cdot 6}+\frac{2013}{6 \cdot 10}+\frac{2013}{10 \cdot 14}+\ldots+\frac{2013}{2010 \cdot 2014}
$$
In your answer, specify the remainder when the nearest even number to the obtained value of $S$ is divided by 5. | 2 |
Given the function $f(x)= \begin{cases} 2x, & x\geqslant 0 \\ x^{2}, & x < 0 \end{cases}$, then the value of $f(f(-2))$ is __( )__.
A: $4$
B: $-4$
C: $8$
D: $-8$ | C |
Given \\(x > 0\\), the minimum value of the function \\(y=x+\\frac{2}{2x+1}-\\frac{3}{2}\\) is \\((\\) \\()\\).
A: \\(0\\)
B: \\(\\frac{1}{2}\\)
C: \\(1\\)
D: \\(\\frac{3}{2}\\) | \text{A} |
The subsets of the set $\{0, 1\}$ are , the total number of subsets of the set $\{a, b, c\}$ is , the number of non-empty subsets is , and the number of proper subsets is . | 7 |
3. (20 points) We will say that a number has the form $\overline{a b a}$ if its first and third digits are the same; the second digit does not have to be different. For example, 101 and 222 have this form, while 220 and 123 do not. Similarly, we define the form of the number $\overline{\overline{b a b c}}$. How many nu... | 180 |
Express the number $929000$ in scientific notation as ______. | 9.29 \times 10^{5} |
Given an arithmetic sequence $\\{a_{n}\\}$, let $S_{n}$ denote the sum of its first $n$ terms. If $a_{1}= \dfrac {1}{2}$ and $S_{4}=20$, find $S_{6}=(\ )$
A: $12$
B: $24$
C: $48$
D: $96$ | C |
B2. In a bag, there are 5 red and 5 green coins. Each coin has the number 0 on one side and the number 1 on the other side. All coins are fair, meaning the probability of landing on 0 is equal to the probability of landing on 1. We randomly select 6 coins from the bag and toss them. What is the probability that the sum... | \frac{89}{336} |
7. (3 points) Three piles of balls have a total of 2012 balls. If the same number of balls is taken from each pile, the second pile will have 17 balls left, and the number of balls left in the first pile is twice the number of balls left in the third pile. How many balls were originally in the third pile? $\qquad$
u... | 665 |
Quadrilateral $ABCD$ is inscribed in a circle of radius $R$. Its diagonals are perpendicular to each other and intersect at point $P$.
Find $AP^{2}+BP^{2}+CP^{2}+DP^{2}$ and $AB^{2}+BC^{2}+CD^{2}+AD^{2}$. | 4R^{2},8R^{2} |
Given that the corresponding element of $(x, y)$ under the mapping $f$ is $(x+y, x-y)$, then the corresponding element of $(4, 6)$ under the mapping $f$ is \_\_\_\_\_\_. | (10, -2) |
Calculate the Minkowski dimensions of the following objects: a point, the interval $[0,1]$, the rationals in $[0,1]$, the triadic Cantor set $C(3)$. | \frac{\log(2)}{\log(3)} |
Given the function $f(x) = -x^2 + 4x$ has its range on the interval $[m, n]$ as $[-5, 4]$, then the range of values for $m+n$ is ( )
A: $[1, 7]$
B: $[1, 6]$
C: $[-1, 1]$
D: $[0, 6]$ | A |
For $ n \equal{} 1,\ 2,\ 3,\ \cdots$, let $ (p_n,\ q_n)\ (p_n > 0,\ q_n > 0)$ be the point of intersection of $ y \equal{} \ln (nx)$ and $ \left(x \minus{} \frac {1}{n}\right)^2 \plus{} y^2 \equal{} 1$.
(1) Show that $ 1 \minus{} q_n^2\leq \frac {(e \minus{} 1)^2}{n^2}$ to find $ \lim_{n\to\infty} q_n$.
(2) Find ... | 1 |
Given a continuous odd function $f(x)$ defined on $\mathbb{R}$, which is increasing on $(0, +\infty)$, if $f(x) > f(2-x)$, then the range of $x$ is ( )
A: $x > 1$
B: $x < 1$
C: $0 < x < 2$
D: $1 < x < 2$ | \text{A} |
# 2. Option 1
On a sheet, two rectangles are drawn. It is known that both the length and the width of the second rectangle are 3 cm larger than the length and width of the first rectangle, and the area of the second rectangle is 48 cm ${ }^{2}$ larger than the area of the first rectangle. Find the perimeter of the sec... | 38 |
Jolana is preparing refreshments for her friends - open-faced sandwiches. She spreads them with potato salad and wants to add more toppings: ham, hard cheese, a slice of egg, and a strip of pickled pepper. However, she does not want any two of her sandwiches to have the exact same combination of toppings. What is the m... | 14 |
Find all positive integers $n$ for which $n^{n-1} - 1$ is divisible by $2^{2015}$, but not by $2^{2016}$. | n = 2^{2014}k - 1 |
Factorize the expression: $4-4x+x^{2}=\_\_\_\_\_\_.$ | (2-x)^{2} |
4. Take a random 8-bit number consisting of 0s and 1s, the probability that the sum of the digits in the even positions is equal to the sum of the digits in the odd positions is
$\qquad$ . | \frac{35}{128} |
1. (10 points) As shown in the figure, a square pond with a side length of 12 meters is surrounded by grass. There are wooden stakes at points $A$, $B$, $C$, and $D$ on the pond's edge, and $A B=B C=C D=3$ meters. Now, a 4-meter-long rope is used to tie a sheep to one of these wooden stakes. To maximize the area where ... | B |
The complex number $\dfrac{3+2i}{2-3i}=$ _________________ | i |
If the function $f(x)=2x^{3}-3mx^{2}+6x$ is increasing on the interval $(2,+\infty)$, then the range of values for the real number $m$ is ( )
A: $(-\infty,2)$
B: $(-\infty,2]$
C: $(-\infty, \dfrac{5}{2})$
D: $(-\infty, \dfrac{5}{2}]$ | \text{D} |
3 Let $M$ be a set of $n$ points in the plane I, satisfying:
(1) $M$ contains 7 points that are the vertices of a heptagon;
(2) For any 5 points in $M$, these 5 points are the vertices of a convex pentagon, and this pentagon contains at least one point from $M$ inside it.
Find the minimum value of $n$. (Leng Gangsong) | 11 |
Translate the parabola $y=-5x^{2}+1$ one unit to the left and then two units down, the resulting parabola is ( )
A: $y=-5\left(x+1\right)^{2}-1$
B: $y=-5\left(x-1\right)^{2}-1$
C: $y=-5\left(x+1\right)^{2}+3$
D: $y=-5\left(x-1\right)^{2}+3$ | \text{A} |
## 108. Math Puzzler $5 / 74$
"Monkeys were frolicking, divided into two parties. The eighth part squared was merrily romping in the woods, while 12 monkeys filled the fresh air with their cries.
Tell me, how many monkeys were there in total!" | 48or16 |
5. Given $x=2007^{2008}$. Then the remainder when $x$ is divided by 10 is ( ).
(A) 1
(B) 3
(C) 7
(D) 9 | A |
10.4. The incircle of triangle $A B C$ touches side $A C$ at point $D$. A second circle passes through point $D$, touches ray $B A$ at point $A$, and also touches the extension of side $B C$ beyond point $C$. Find the ratio $A D: D C$. | 3:1 |
Find the nearest integer to the number $x$, if $x=\frac{1}{\sqrt[4]{\frac{5}{4}+1}-\sqrt[4]{\frac{5}{4}-1}}$. | 2 |
7. Given in $\triangle A B C$, $a=2 b, \cos B=\frac{2 \sqrt{2}}{3}$, then $\sin \frac{A-B}{2}+\sin \frac{C}{2}=$ $\qquad$ | \frac{\sqrt{10}}{3} |
29th VMO 1991 Problem A1 Find all real-valued functions f(x) on the reals such that f(xy)/2 + f(xz)/2 - f(x) f(yz) ≥ 1/4 for all x, y, z. | f(x)=\frac{1}{2}forallx |
4. If three angles $\alpha, \beta, \gamma$ form an arithmetic sequence with a common difference of $\frac{\pi}{3}$, then $\tan \alpha \cdot \tan \beta + \tan \beta \cdot \tan \gamma + \tan \gamma \cdot \tan \alpha$ $=$ . $\qquad$ | -3 |
Let the universal set $U=\{1,2,3,4,5\}$, $A=\{1,3,5\}$, and $B=\{2,4\}$. Then $A\cup C_{U}B$ is ( )
A: $\{2\}$
B: $\{1,3\}$
C: $\{3\}$
D: $\{1,3,5\}$ | \text{D} |
Given the proposition: If $a+b\geq 2$, then at least one of $a$ or $b$ is not less than 1. The truthfulness of the original proposition and its converse is ( )
A: The original proposition is true, the converse is false
B: The original proposition is false, the converse is true
C: Both the original proposition and its c... | \text{A} |
Example 5 Let real numbers $a \geqslant b \geqslant c \geqslant d>0$. Find the minimum value of the function
$$
\begin{array}{l}
f(a, b, c, d) \\
=\left(1+\frac{c}{a+b}\right)\left(1+\frac{d}{b+c}\right)\left(1+\frac{a}{c+d}\right)\left(1+\frac{b}{d+a}\right)
\end{array}
$$ | \left(\frac{3}{2}\right)^{4} |
$$
(2 \sqrt{12}-4 \sqrt{27}+3 \sqrt{75}+7 \sqrt{8}-3 \sqrt{18}) \cdot(4 \sqrt{48}-3 \sqrt{27}-5 \sqrt{18}+2 \sqrt{50})=?
$$ | 97 |
Given a quadratic equation $x^{2}-2023x+1=0$ with two roots $x_{1}$ and $x_{2}$, the value of $\frac{1}{{{x_1}}}+\frac{1}{{{x_2}}}$ is ____. | 2023 |
For $a \geqslant 1$, calculate the sum of the infinite series
$$
\begin{array}{l}
\frac{a}{a+1}+\frac{a^{2}}{(a+1)\left(a^{2}+1\right)} \\
+\frac{a^{4}}{\left.(a+1) a^{2}+1\right)\left(a^{4}+1\right)} \\
+\frac{a^{8}}{(a+1)\left(a^{2}+1\right)\left(a^{4}+1\right)\left(a^{8}+1\right)}+\cdots
\end{array}
$$ | 1 |
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms $S_n = n^2 - 2n + 3$, then the first three terms of this sequence are ( )
A: $-1, 1, 3$
B: $2, 3, 6$
C: $6, 1, 3$
D: $2, 1, 3$ | \text{D} |
A sufficient but not necessary condition for "lgm < 1" to be true is ( )
A: $m \in (0, +\infty)$
B: $m \in \{1, 2\}$
C: $0 < m < 10$
D: $m < 1$ | \text{B} |
The equations of the sides of a triangle are
$$
y=x+1, \quad y=-\frac{1}{2} x-2, \quad y=3 x-9
$$
Determine the equation of the circumscribed circle. | (x-2)^{2}+(y-2)^{2}=25 |
1. (2002 National High School Competition Question) The function $f(x)=\frac{x}{1-2^{x}}-\frac{x}{2}(\quad)$.
A. is an even function but not an odd function
B. is an odd function but not an even function
C. is both an even function and an odd function
D. is neither an even function nor an odd function | A |
Given circle $C$: $\begin{cases} x=1+\cos \theta \\ y=\sin \theta \end{cases}(\theta$ is a parameter $)$ and line $l$: $\begin{cases} x=2+t\cos \alpha \\ y=\sqrt {3}+t\sin \alpha \end{cases}(\text{where } t \text{ is a parameter, } \alpha \text{ is the angle of inclination of line } l )$
(1) When $\alpha= \dfrac {2π}{3... | [\dfrac {\pi}{6}, \dfrac {\pi}{2}] |
Find the smallest positive period of the function $y=-2\sin(3x-\frac{\pi}{6})$. | T=\frac{2\pi}{3} |
Given that the initial side of angle $\theta$ coincides with the non-negative half-axis of the x-axis, and the terminal side lies on $y= \frac {1}{2}x$, then $tan2\theta= \_\_\_\_\_\_$. | \frac {4}{3} |
If the inequality $2x+a\leqslant 1$ has only $2$ positive integer solutions for $x$, then the range of values for $a$ is:
A: $-5 \lt a \lt -3$
B: $-5\leqslant a \lt -3$
C: $-5 \lt a\leqslant -3$
D: $-5\leqslant a\leqslant -3$ | C |
Let $\mathcal{P}$ be the set of all polynomials $p(x)=x^4+2x^2+mx+n$, where $m$ and $n$ range over the positive reals. There exists a unique $p(x) \in \mathcal{P}$ such that $p(x)$ has a real root, $m$ is minimized, and $p(1)=99$. Find $n$.
[i]Proposed by [b]AOPS12142015[/b][/i] | 56 |
4. Decipher the message RTOLK TOIK, which was enciphered using the affine transformation $C \equiv 3 P+24(\bmod 26)$. | PHONE HOME |
Problem 10.8. Rectangle $ABCD$ is such that $AD = 2AB$. Point $M$ is the midpoint of side $AD$. Inside the rectangle, there is a point $K$ such that $\angle AMK = 80^{\circ}$ and ray $KD$ is the bisector of angle $MKC$. How many degrees does angle $KDA$ measure?
 Equilateral triangles $A B C$ and $A_{1} B_{1} C_{1}$ with side length 12 are inscribed in circle $S$ such that point $A$ lies on arc $B_{1} C_{1}$, and point $B$ lies on arc $A_{1} B_{1}$. Find $A A_{1}^{2}+B B_{1}^{2}+C C_{1}^{2}$. | 288 |
3. Given the sequences $\left\{a_{n}\right\} 、\left\{b_{n}\right\}$, for any $n \in \mathbf{Z}_{+}$, it holds that $a_{n}>b_{n}$. As $n \rightarrow+\infty$, the limits of the sequences $\left\{a_{n}\right\} 、\left\{b_{n}\right\}$ are $A 、 B$ respectively. Then ( ).
(A) $A>B$
(B) $A \geqslant B$
(C) $A \neq B$
(D) The r... | B |
Subject III
A natural number of the form $\overline{a b c d}$ is called superb if $4 \cdot \overline{a b}=\overline{c d}$.
a) How many superb numbers exist?
b) Show that any superb number is divisible by 13. | 15 |
Example 6 Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=a_{2}=1$, and $a_{n+2}=3 a_{n+1}+18 a_{n}+2^{n}\left(n \in \mathbf{N}_{+}\right)$.
Find $a_{n}$. | a_{n}=\frac{6^{n}}{12}-\frac{(-3)^{n}}{5}-\frac{2^{n}}{20} |
In the year 1275, the Chinese mathematician Yang Hui of the Southern Song Dynasty posed the following problem: a rectangular field measures 650 paces in area, with the width being 1 pace less than the length. The question is, what are the width and length of the field? If we let the length be $x$ paces, then the equati... | x(x - 1) = 650 |
Given that the function $f(x)=ax^{2}+ \frac{2}{x}$ (where $a\in R$) is an odd function.
1. Compare the sizes of $f(\log _{2}3)$, $f(\log _{3}8)$, and $f(\log _{3}26)$, and explain the reasons (Hint: $\log _{2}3\approx1.59$);
2. If $t > 0$, and $f(t+x^{2})+f(1-x-x^{2}-2^{x}) > 0$ holds true for $x\in[2,3]$, find the ran... | 0 < t < 5 |
Consider a sequence $\{a_n\}$ satisfying $a_1 = 3$ and $a_{n+1} = \frac{2}{a_n + 1}$.
(1) Prove that the sequence $\left\{\frac{a_n - 1}{a_n + 2}\right\}$ forms a geometric progression;
(2) If $a_n - t^2 - mt \geq 0$ holds for all $n \in \mathbb{N}^*$ and $m \in [-1, 1]$, find the range of the real number $t$. | \left[\frac{1 - \sqrt{3}}{2}, \frac{\sqrt{3} - 1}{2}\right] |
95. Three points $A, B$ and $C$ are taken at random on a circle. What is the probability that triangle $A B C$ will be acute? | \frac{1}{4} |
11. (2001 China Western Mathematical Olympiad) Find all real numbers $x \in\left[0, \frac{\pi}{2}\right]$, such that $(2-\sin 2 x)$ $\sin \left(x+\frac{\pi}{4}\right)=1$, and prove your conclusion. | \frac{\pi}{4} |
30. $\left\{\begin{array}{l}\log _{a} x+\log _{a} y=2 \\ \log _{b} x-\log _{b} y=4\end{array}\right.$ | ^{2},\frac{}{b^{2}} |
The distribution of the random variable $X$ is given by $P(X=k)=a\left( \frac {1}{2}\right)^{k}$ $(k=1,2,3)$. The value of $a$ is __( )__.
A: $1$
B: $\frac {8}{7}$
C: $\frac {4}{7}$
D: $\frac {6}{7}$ | B |
Let $f(x) = \sin(\omega x + \varphi)$ ($\omega > 0$, $-\frac{π}{2} < \varphi < \frac{π}{2}$) have a minimum positive period of $T$, and $f(T) = \frac{\sqrt{3}}{2}$. If the graph of $y = f(x)$ is shifted to the right by $\frac{π}{6}$ units and becomes symmetric about the $y$-axis, then the minimum value of $\omega$ is (... | D |
Given an arithmetic sequence $\{a_n\}$, where the arithmetic mean of $a_2$ and $a_6$ is 5, and the arithmetic mean of $a_3$ and $a_7$ is 7, find the general formula for $a_n$. | 2n - 3 |
Given the parabola $E$: $x^{2}=2py(p > 0)$, and the line $y=kx+2$ intersects $E$ at points $A$ and $B$, and $\overrightarrow{OA} \cdot \overrightarrow{OB}=2$, where $O$ is the origin.
$(1)$ Find the equation of the parabola $E$;
$(2)$ The coordinates of point $C$ are $(0,-2)$, let the slopes of lines $CA$ and $CB$ ... | 16 |
Let $n \ge 2$ be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$. | 8 |
6. A point on a circle inscribed in a square is 1 and 2 units from the two closest sides of the square. Find the area of the square. | 100 |
Given the universal set $U={x|x}$ is a positive integer less than $9${}$, $M={1,3,5,7}${}, and $N={5,6,7}${}, find the complement of $(M∪N)$ in $U$, denoted as $∁_{U}(M∪N)=($ $)$
A: ${5,7}${}
B: ${2,4}${}
C: ${2,4,8}${}
D: ${1,3,5,6,7}${} | C |
There are 11 children, consisting of 5 boys and 6 girls, playing the following game: First, 4 children (not all boys) stand at 4 equidistant positions on a circle. If two adjacent children are of the same gender, a boy is placed between them; otherwise, a girl is placed between them. Then, the original 4 children tempo... | \text{C} |
14. The largest even number that cannot be expressed as the sum of two odd composite numbers is what?
Will the translated text be used for further discussion or do you need more information on this topic? | 38 |
The negation of the proposition "For all $x > 1$, $(\frac{1}{2})^{x} 1$, $(\frac{1}{2})^{x} \geq \frac{1}{2}$
B: For all $x \leq 1$, $(\frac{1}{2})^{x} \geq \frac{1}{2}$
C: There exists an $x_0 > 1$, $(\frac{1}{2})^{x_0} \geq \frac{1}{2}$
D: There exists an $x_0 \leq 1$, $(\frac{1}{2})^{x_0} \geq \frac{1}{2}$ | C |
The general term formula of the sequence {an} is an\=n(sin $$\frac {nπ}{2}$$+cos $$\frac {nπ}{2}$$), and its sum of the first n terms is Sn. Find S2019. | -2020 |
71*. a) Banach's problem ${ }^{1}$. A person simultaneously bought two boxes of matches and put them in his pocket. After that, every time he needed to light a match, he randomly took one or the other box. After some time, upon emptying one of the boxes, the person discovered that it was empty. What is the probability ... | 2^{2n} |
5. (20 points) The clock shows the time as four hours and fifteen minutes. Determine the angle between the minute and hour hands at this moment. | 37.5 |
Which of the following sets of three line segments can form a triangle?
A: $2$, $3$, $5$
B: $5$, $6$, $10$
C: $1$, $1$, $3$
D: $3$, $4$, $9$ | B |
Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and $S_n=n(n+1)$,
(1) Find the general term formula $a_n$ of the sequence ${a_n}$.
(2) The general term formula of the sequence ${b_n}$ is $b_n= \frac {1}{a_n \cdot a_{n+2}}$, find the sum of the first $n$ terms of the sequence ${b_n}$, denoted... | T_n = \frac {1}{8}( \frac {3}{2}- \frac {1}{n+1}- \frac {1}{n+2}) |
5. (3 points) Anya, Vanya, Danya, Sanya, and Tanya were collecting apples. It turned out that each of them collected a whole percentage of the total number of apples collected, and all these numbers are different and greater than zero. What is the minimum number of apples that could have been collected?
Answer: 20 | 20 |
Given an arithmetic sequence $\{a_n\}$ where $a_1 > 0$, $S_{13} > 0$, and $S_{14} < 0$, the sum of the first few terms is the largest.
A: $13$
B: $14$
C: $7$
D: $8$ | C |
2. If the function
$$
y=\frac{x+k}{(k+1) x^{2}+(2 k+2) x+2 k-3}
$$
has the domain of the independent variable $x$ as all real numbers, then the range of values for $k$ is $\qquad$ | k \leqslant-1 \text{ or } k>4 |
## Task 4.
Determine all natural numbers $n \geqslant 3$ for which the product of the first $n$ natural numbers divides the product of all sums of distinct pairs of prime numbers not greater than $n$, i.e., for which
$$
n! \mid \prod_{\substack{p<q \leqslant n \\ p, q \text{ prime }}}(p+q)
$$ | 7 |
Solve the following inequalities:
(1) $x + |2x + 3| \geqslant 2$; (2) $|x - 1| - |x - 5| < 2$. | (-\infty, 4) |
In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the area of the square?
$\textbf{(A) }85\qquad\textbf{(B) }93\qquad\textbf{(C) }100\qquad\textbf{(D) }117\qqu... | 117 |
Two numbers are randomly selected from interval $I = [0, 1]$. Given $\alpha \in I$, what is the probability that the smaller of the two numbers does not exceed $\alpha$?
Is the answer $(100 \alpha)$%, it just seems too easy. :| | 2\alpha - \alpha^2 |
4. Pete is trying to lay out a square on a table using identical cardboard rectangles measuring $14 \times 10$. Will he be able to do this? Propose your version of constructing such a square. What is the minimum number of rectangles he will need? | 35 |
[Problems on percentages and ratios]
In the club, there are girls, but boys make up more than $94\%$ of the composition. What is the minimum number of people that can be in the club? | 17 |
Let the function $f(x)=(x^{2}-2x)\ln x+(a- \frac {1}{2})x^{2}+2(1-a)x+a$.
- (I) Discuss the monotonicity of $f(x)$;
- (II) When $a < -2$, discuss the number of zeros of $f(x)$. | 3 |
14) How many angles greater than $90^{\circ}$ can a quadrilateral (non-intersecting) have?
(A) It always has at least one
(B) it has at most one
(C) it has at most two
(D) it has at most three
(E) it can have four. | D |
10. There are 12 pencils of pairwise different lengths. In how many ways can they be placed in a box in 2 layers of 6 pencils each, so that in each layer the pencils are arranged in increasing order of length (from left to right), and each pencil in the upper layer lies strictly above a pencil in the lower layer and is... | 132 |
If $$\frac {\sin(\pi-\alpha)+\sin( \frac {\pi}{2}-\alpha)}{\sin\alpha-\cos\alpha} = \frac {1}{2}$$, then $\tan2\alpha$ is ( )
A: $-\frac {3}{4}$
B: $\frac {3}{4}$
C: $-\frac {4}{3}$
D: $\frac {4}{3}$ | \text{B} |
1.35. On the lateral sides $AB$ and $CD$ of trapezoid $ABCD$, points $M$ and $N$ are taken such that segment $MN$ is parallel to the bases and divides the area of the trapezoid in half. Find the length of $MN$, if $BC=a$ and $AD=b$. | \sqrt{\frac{^2+b^2}{2}} |
In the diagram, $P Q R$ is a line segment, $\angle P Q S=125^{\circ}$, $\angle Q S R=x^{\circ}$, and $S Q=S R$. What is the value of $x$ ?
(A) 60
(B) 70
(C) 80
(D) 110
(E) 125
 | 70 |
Example 6 Given 2014 real numbers $x_{1}, x_{2}, \cdots, x_{2014}$ satisfy the system of equations
$$
\sum_{k=1}^{2014} \frac{x_{k}}{n+k}=\frac{1}{2 n+1}(n=1,2, \cdots, 2014) .
$$
Try to calculate the value of $\sum_{k=1}^{2014} \frac{x_{k}}{2 k+1}$. | \frac{1}{4}(1-\frac{1}{4029^{2}}) |
The coordinates of the vertex of the parabola $y=\left(x+3\right)^{2}-1$ are ( ).
A: $(3,-1)$
B: $(3,1)$
C: $(-3,1)$
D: $(-3,-1)$ | \text{D} |
20. Let $V$ be the vertex of the parabola, and $F$ be the focus of the parabola. If a point $A$ on the parabola satisfies $A F=20, A V=21$, then the sum of all possible values of the length of $F V$ is ( ).
(A) 13
(B) $\frac{40}{3}$
(C) $\frac{41}{3}$
(D) 14
(E) $\frac{43}{3}$ | \frac{40}{3} |
## Task Condition
Find the derivative.
$$
y=\ln \left(5 x+\sqrt{25 x^{2}+1}\right)-\sqrt{25 x^{2}+1} \cdot \operatorname{arctg} 5 x
$$ | -\frac{25x\cdot\operatorname{arctg}5x}{\sqrt{25x^{2}+1}} |
Given that $\sqrt{2+ \frac{2}{3}}=2 \sqrt{ \frac{2}{3}}$, $\sqrt{3+ \frac{3}{8}}=3 \sqrt{ \frac{3}{8}}$, $\sqrt{4+ \frac{4}{15}}=4 \sqrt{ \frac{4}{15}}$, $\sqrt{5+ \frac{5}{24}}=5 \sqrt{ \frac{5}{24}},\cdots, \sqrt{10+ \frac{a}{b}}=10 \sqrt{ \frac{a}{b}}$, then by reasoning, we get $a+b=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_... | 109 |
Given $X\sim B\left(n,p\right)$, and $3E\left(X\right)=10D\left(X\right)$, then $p=\left(\ \ \right)$
A: $0.3$
B: $0.4$
C: $0.7$
D: $0.8$ | C |
If the line $(2t-3)x+2y+t=0$ does not pass through the second quadrant, then the range of values for the real number $t$ is $\_\_\_\_\_\_\_\_\_.$ | 0 \leq t \leq \frac{3}{2} |
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