problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
40. On the meadow, there were donkeys grazing. Several boys approached them.
- Get on the donkeys, one per donkey! - commanded the oldest. And one boy was left without a donkey.
- Get on the donkeys, two per donkey, - he commanded again. And one donkey was left without a boy.
How many boys and how many donkeys were ... | 3 |
Definition: Extracting several terms (at least three) from a sequence $\{a_n\}$ and arranging them in the order they appear in $\{a_n\}$ creates a subsequence of $\{a_n\}$. If the subsequence forms an arithmetic (or geometric) progression, it is called an arithmetic (or geometric) subsequence of $\{a_n\}$.
(1) Find the... | q = -1 |
In $\triangle ABC$, $\cos A = -\frac{5}{13}$ and $\sin B = \frac{4}{5}$.
(1) Find the value of $\cos C$.
(2) Given that $BC = 15$, find the area of $\triangle ABC$. | 24 |
5. In the quadrilateral $M A R E$ inscribed in a unit circle $\omega, A M$ is a diameter of $\omega$, and $E$ lies on the angle bisector of $\angle R A M$. Given that triangles $R A M$ and $R E M$ have the same area, find the area of quadrilateral MARE. | \frac{8\sqrt{2}}{9} |
10. Let the function be
$$
f(x)=\cos 2 x+2 \sqrt{3} \sin x \cdot \cos x(x \in \mathbf{R}) \text {. }
$$
If there are ten distinct positive numbers $x_{i}$ satisfying
$$
f\left(x_{i}\right)=2, x_{i}<10 \pi(i=1,2, \cdots, 10),
$$
then $x_{1}+x_{2}+\cdots+x_{10}=$ . $\qquad$ | \frac{140 \pi}{3} |
Given the sets $M=\{x|x^2<3x\}$ and $N=\{x|\ln(x)<0\}$, then the intersection of sets $M$ and $N$ is ( )
A: $(-2,0]$
B: $(0,1)$
C: $(2,3]$
D: $(-2,3)$ | \text{B} |
Given the function $g(x) = g''(1)e^{x-1} - g(0)x + \frac{1}{2}x^2$, and there exists a real number $x_0$ such that the inequality $2m-1 \geqslant g(x_0)$ holds, then the range of $m$ is ( )
A: $\left( -\infty ,2 \right]$
B: $\left[ 1,+\infty \right)$
C: $\left( -\infty ,3 \right]$
D: $\left[ 0,+\infty \right)$ | \text{B} |
The value of $i^{2n-3}+i^{2n-1}+i^{2n+1}+i^{2n+3}$ is ( )
A: -2
B: 0
C: 2
D: 4 | \text{B} |
Given $A=\{1, 2, a^2-3a-1\}$, $B=\{1, 3\}$, and $A \cap B = \{3, 1\}$, then $a=$ ______. | -1 \text{ or } 4 |
Among the numbers 1, 3, 6, 9, the counts of perfect squares, odd numbers, and prime numbers are respectively ( )
A: 2, 3, 1
B: 2, 2, 1
C: 1, 2, 1
D: 2, 3, 2 | \text{A} |
The reciprocal of 0.3 is ____. | \frac{10}{3} |
Let $f_1(x)=x^2-1$, and for each positive integer $n \geq 2$ define $f_n(x) = f_{n-1}(f_1(x))$. How many distinct real roots does the polynomial $f_{2004}$ have? | 2005 |
## Task $9 / 71$
For which real numbers $a$ does the system of equations
$$
\begin{array}{r}
2 x^{3}-2 a y^{3}=(a+1)^{2} \\
x^{3}+a x^{2} y+x y^{2}=1
\end{array}
$$
have exactly one real solution $(x ; y)$ with $x=-y$? | a_{1}=-1,a_{2}=0,a_{3}=+1 |
(Selected Inequality Problems) If the inequality $|x+1|+|x-2|<a$ has no real solution, then the range of $a$ is. | (-\infty, 3] |
Rax $* \star$ Given that the wooden tetrahedron has three pairs of opposite edges all equal, each being $13 \mathrm{~cm}$, $14 \mathrm{~cm}$, and $15 \mathrm{~cm}$. If it is to be carved into a sphere, what is the maximum radius of the sphere? | \frac{3\sqrt{55}}{8}\mathrm{~} |
18. The sum of the squares of four consecutive positive integers is equal to the sum of the squares of the next three consecutive integers. What is the square of the smallest of these integers? | 441 |
An [i]annulus[/i] is the region between two concentric circles. The concentric circles in the figure have radii $ b$ and $ c$, with $ b > c$. Let $ \overline{OX}$ be a radius of the larger circle, let $ \overline{XZ}$ be tangent to the smaller circle at $ Z$, and let $ \overline{OY}$ be the radius of the larger circle t... | \pi a^2 |
For a positive integer $n$ we denote by $d(n)$ the number of positive divisors of $n$ and by $s(n)$ the sum of these divisors. For example, $d(2018)$ is equal to $4$ since $2018$ has four divisors $(1, 2, 1009, 2018)$ and $s(2018) = 1 + 2 + 1009 + 2018 = 3030$.
Determine all positive integers $x$ such that $s(x) \cdot ... | 14, 15, 47 |
Given vectors $\overrightarrow{a}=(\cos \frac{3}{2}x,\sin \frac{3}{2}x)$, $\overrightarrow{b}=(\cos \frac{x}{2},-\sin \frac{x}{2})$, and $x\in[0, \frac{\pi}{2}]$, find:
(Ⅰ) $\overrightarrow{a}\cdot \overrightarrow{b}$ and $|\overrightarrow{a}+ \overrightarrow{b}|$;
(Ⅱ) If the minimum value of $f(x)= \overrightarrow{a... | \lambda= \frac {1}{2} |
Given that the focal length of the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ is $10$, and point $P(2,1)$ is on the asymptote of $C$, find the equation of $C$. | \frac{x^{2}}{20}-\frac{y^{2}}{5}=1 |
2. If the focal distance of the conic section $\frac{x^{2}}{k-2}+\frac{y^{2}}{k+5}=1$ is independent of the real number $k$, then its foci coordinates are $\qquad$ | (0, \pm \sqrt{7}) |
In triangle $ABC$, $AB=20$ and $AC=11$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AC$ and $BM$. The ratio of $CP$ to $PA$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive ... | 51 |
7. Calculate in the most rational way:
\[
3 \frac{1}{117} \cdot 4 \frac{1}{119}-1 \frac{116}{117} \cdot 5 \frac{118}{119}-\frac{5}{119}
\] | \frac{10}{117} |
The area of the figure enclosed by the lines $x = \frac{1}{2}$, $x = 2$, $y = 0$, and the curve $y = \frac{1}{x}$ is ( )
A: $\frac{15}{4}$
B: $\frac{17}{4}$
C: $\frac{1}{2}\ln2$
D: $2\ln2$ | \text{D} |
$\sqrt{9}=\left(\ \ \right)$
A: $\pm 3$
B: $3$
C: $±\sqrt{3}$
D: $\sqrt{3}$ | B |
In the expansion of \\((1{-}x^{3})(1{+}x)^{10}\\), the coefficient of \\(x^{5}\\) is \\(({ })\\)
A: \\({-}297\\)
B: \\({-}252\\)
C: \\(297\\)
D: \\(207\\) | \text{D} |
1. (16 points) There are two circles: one with center at point $A$ and radius 5, and another with center at point $B$ and radius 15. Their common internal tangent touches the circles at points $C$ and $D$ respectively. Lines $A B$ and $C D$ intersect at point $E$. Find $C D$, if $B E=39$. | 48 |
3. Let $y=x^{4}+4 x^{3}+8 x^{2}+8 x+7$, where $x$ is any real number. Then the range of $y$ is ( ).
(A) All real numbers
(B) All positive real numbers
(C) All real numbers greater than or equal to 7
(D) All real numbers greater than or equal to 4 | D |
For which $x$ does the following inequality hold?
$$
\left(\log _{x} 3\right) \cdot\left(\log _{3 x} 3\right) \geq \log _{9 x} 3
$$ | 0<x<3^{-2};\quad3^{-\sqrt{2}}\leqx<3^{-1};\quad1<x\leq3^{\sqrt{2}} |
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ that satisfy
$$
f(-f(x)-f(y))=1-x-y
$$
for all $x, y \in \mathbb{Z}$. | f(x)=x-1 |
4. In $\triangle A B C$, if $\sin A=2 \sin C$, and the three sides $a, b, c$ form a geometric sequence, then the value of $\cos A$ is $\qquad$ . | -\frac{\sqrt{2}}{4} |
Environmentalists have protested against the large volume of logging. The chairman of the forestry company calmed them down as follows: "99% of the trees in the forest are pines. Only pines will be cut, and after the logging, the percentage of pines will remain almost unchanged - pines will be $98\%$". What part of the... | \frac{1}{2} |
Example 6 (1998 National High School Mathematics League Question) If $f(x) (x \in \mathbf{R})$ is an even function with a period of 2. And when $x \in [0,1]$, $f(x) = x^{\frac{1}{110}}$, then the arrangement from smallest to largest of $f\left(\frac{98}{19}\right) \cdot f\left(\frac{101}{17}\right) \cdot f\left(\frac{1... | f(\frac{101}{17})<f(\frac{98}{19})<f(\frac{104}{15}) |
Given a set of data $x_1$, $x_2$, $x_3$ with a variance of 3, find the variance of the data $2x_1+3$, $2x_2+3$, $2x_3+3$. | 12 |
677. $y=\left(\frac{x+1}{x-1}\right)^{2}$, i.e. $y=u^{2}$, where $u=\frac{x+1}{x-1}$. | -\frac{4(x+1)}{(x-1)^{3}} |
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of $\$370$. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of $\$180$. On Wednesday, she sells twelve Stanford sweatshirts... | 300 |
1. For a finite set $A$, there exists a function $f: \mathbf{N}_{+} \rightarrow A$, with the following property: if $i, j \in \mathbf{N}_{+}$ and $|i-j|$ is a prime number, then $f(i) \neq f(j)$. How many elements does the set $A$ have at minimum? | 4 |
13. Given the function $f(x)=a x-\frac{3}{2} x^{2}$ has a maximum value no greater than $\frac{1}{6}$, and when $x \in\left[\frac{1}{4}, \frac{1}{2}\right]$, $f(x) \geqslant \frac{1}{8}$, find the value of $a$. | 1 |
14. In the city of Meow, where mathematics is everywhere, siblings Milli and Geo face a new challenge: they need to select some different odd numbers so that their sum equals 2023. How many odd numbers can they select at most? $\qquad$ | 43 |
Anna walked at a constant rate. The graph shows that she walked 600 metres in 4 minutes. If she continued walking at the same rate, how far did she walk in 6 minutes?
(A) $700 \mathrm{~m}$
(B) $750 \mathrm{~m}$
(C) $800 \mathrm{~m}$
(D) $900 \mathrm{~m}$
(E) $1000 \mathrm{~m}$
Distance-Time Graph
 -199 .
(B) -197 .
(C) -1 .
(D) 197 .
(E) 199 .
(39th American H... | -199 |
4. Depending on the real parameter $a$, solve the equation
$$
x=a-\sqrt{a^{2}-x \sqrt{x^{2}+a^{2}}}
$$
in the set of real numbers. | x\in(-\infty,0]for=0,0\frac{3a}{4}for>0,nosolutionsfor<0 |
2. Let $p_{1}, p_{2}, \ldots, p_{100}$ be one hundred distinct prime numbers. Natural numbers $a_{1}, \ldots, a_{k}$, greater than 1, are such that each of the numbers $p_{1} p_{2}^{3}, p_{2} p_{3}^{3}, \ldots, p_{99} p_{100}^{3}, p_{100} p_{1}^{3}$ is equal to the product of some two of the numbers $a_{1}, \ldots, a_{... | 150 |
## Problem Statement
Calculate the definite integral:
$$
\int_{\sqrt{2}}^{2 \sqrt{2}} \frac{\sqrt{x^{2}-2}}{x^{4}} d x
$$ | \frac{\sqrt{3}}{16} |
50. Three digits can form 6 different three-digit numbers. If the sum of these 6 three-digit numbers is 2886, then the smallest of these three-digit numbers is $\qquad$ .
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 139 |
Find the equation of the circle passing through the three points A(-1, -1), B(-8, 0), and C(0, 6), and identify the radius and center coordinates of the circle. | r = 5 |
8. For the parabola $y^{2}=2 p\left(x-\frac{p}{2}\right)(p>0)$, the minimum distance from a moving point $A$ on the parabola to the point $B(3,0)$ is denoted as $d(p)$. The sum of all real numbers $p$ that satisfy $d(p)=2$ is $\qquad$ . | 13 |
5. Daisies grew along the path. Between every two daisies, a cornflower grew, and then between each cornflower and daisy - a dandelion. It turned out that now 101 flowers grow along the path. How many daisies grow along the path? | 26 |
1. Let the set $X=\{1,2, \cdots, 2017\}$, and the set $S=\{(x, y, z) \mid x, y, z \in X$, and exactly one of the three conditions $x<y<z, y<z<x$, $z<x<y$ holds $\}$. If $(x, y, z) \in S$ and $(z, w, x) \in S$, then which of the following options is correct?
A. $(y, z, w) \in S$ and $(x, y, w) \notin S$
B. $(y, z, w) \i... | B |
Let $ n $ be a positive integer. A sequence $(a_1,a_2,\ldots,a_n)$ of length is called $balanced$ if for every $ k $ $(1\leq k\leq n)$ the term $ a_k $ is equal with the number of distinct numbers from the subsequence $(a_1,a_2,\ldots,a_k).$
a) How many balanced sequences $(a_1,a_2,\ldots,a_n)$ of length $ n $ do exist... | 2^{n-1} |
Given that the even function $f(x)$ is differentiable on $\mathbb{R}$, and $f'(1) = -2$, $f(x+2) = f(x-2)$, then the slope of the tangent line to the curve $y=f(x)$ at $x=-5$ is ( )
A: 2
B: -2
C: 1
D: -1 | A |
Let $ \alpha_1$, $ \alpha_2$, $ \ldots$, $ \alpha_{2008}$ be real numbers. Find the maximum value of \[ \sin\alpha_1\cos\alpha_2 \plus{} \sin\alpha_2\cos\alpha_3 \plus{} \cdots \plus{} \sin\alpha_{2007}\cos\alpha_{2008} \plus{} \sin\alpha_{2008}\cos\alpha_1\] | 1004 |
Given that the derivative of the function $y=f(x)$ is $y=f'(x)$, $x\in R$, and $y=f'(x)$ is a strictly increasing function on $R$, determine the correctness of the following two propositions: ① "$x_{1} \gt x_{2}$" is a necessary and sufficient condition for "$f(x_{1}+1)+f(x_{2}) \gt f(x_{1})+f(x_{2}+1)$"; ② "For any $x... | C |
$5$ high school graduates apply to three key universities, each person can only apply to one university, then the different application methods are $(\ \ \ \ \ \ )$
A: $3^{5}$ kinds
B: $5^{3}$ kinds
C: $60$ kinds
D: $10$ kinds | A |
Problem 10.3. On the side $AD$ of rectangle $ABCD$, a point $E$ is marked. On the segment $EC$, there is a point $M$ such that $AB = BM, AE = EM$. Find the length of side $BC$, given that $ED = 16, CD = 12$.
$.
(1995, Jiangsu Province Junior High School Mathematics Competition)
(A) 3
(B) $\frac{1}{3}$
(C) -1
(D) 3 or -1 | D |
The sequence of positive integers $a_1, a_2, \dots$ has the property that $\gcd(a_m, a_n) > 1$ if and only if $|m - n| = 1$. Find the sum of the four smallest possible values of $a_2$. | 42 |
The ellipse $E$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ has its left and right foci as $F_{1}$ and $F_{2}$, respectively.
(Ⅰ) If the length of the major axis, the length of the minor axis, and the focal distance of ellipse $E$ form an arithmetic sequence, find the eccentricity of ellipse $E$; ... | \dfrac{x^{2}}{5} + \dfrac{y^{2}}{4} = 1 |
Let $c$ be a complex number. Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have
\[
(z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]
Compute the number of distinct possible values of $c$. | 4 |
Simplify: $\overrightarrow{AB}+ \overrightarrow{BC}- \overrightarrow{AD}=$ ______. | \overrightarrow{DC} |
Huidong Shuangyue Bay Beach is an ideal camping site, where many tourists from all over the world choose to camp. An outdoor equipment store is preparing to purchase a batch of tents for sale. It is known that the purchase price of each regular tent is $150, and the purchase price of each sunshade tent is twice that of... | \$2280 |
Observe the following equations:
\\(1^{2}=1\\)
\\(3^{2}=2+3+4\\)
\\(5^{2}=3+4+5+6+7\\)
\\(7^{2}=4+5+6+7+8+9+10\\)
\\(9^{2}=5+6+7+8+9+10+11+12+13\\)
\\(…\\)
\\(n^{2}=100+101+102+…+m\\)
Then \\(n+m=\\) \_\_\_\_\_\_. | 497 |
Let $n$ be a positive integer greater than $2$. Solve in nonnegative real numbers the following system of equations \[x_{k}+x_{k+1}=x_{k+2}^{2}\quad , \quad k=1,2,\cdots,n\] where $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$. | x_i = 2 |
Let $F_m$ be the $m$'th Fibonacci number, defined by $F_1=F_2=1$ and $F_m = F_{m-1}+F_{m-2}$ for all $m \geq 3$. Let $p(x)$ be the polynomial of degree 1008 such that $p(2n+1)=F_{2n+1}$ for $n=0,1,2,\ldots,1008$. Find integers $j$ and $k$ such that $p(2019) = F_j - F_k$.
| (2019, 1010) |
2. Let point $P$ be inside the face $A B C$ of a regular tetrahedron $A B C D$ with edge length 2, and its distances to the faces $D A B$, $D B C$, and $D C A$ form an arithmetic sequence. Then the distance from point $P$ to the face $D B C$ is $\qquad$ | \frac{2\sqrt{6}}{9} |
In the polar coordinate system, the distance from the point ($\sqrt{2}$, $\frac{\pi}{4}$) to the line $\rho\sin(\theta - \frac{\pi}{3}) = -\frac{\sqrt{3}}{2}$ is ( )
A: 1
B: $\frac{1}{2}$
C: $\frac{1}{3}$
D: $\frac{1}{4}$ | B |
What is the value of $\cos \frac {2017\pi}{3}$?
A: $- \frac {1}{2}$
B: $ \frac {1}{2}$
C: $- \frac { \sqrt {3}}{2}$
D: $ \frac { \sqrt {3}}{2}$ | B |
17.51 In $\triangle A B C$, $A C: C B=3: 4, \angle C$'s exterior angle bisector intersects the extension of $B A$ at $P$ ($A$ is between $P$ and $B$), then $P A: A B$ is
(A) $1: 3$.
(B) $3: 4$.
(C) $4: 3$.
(D) $3: 1$.
(E) $7: 1$.
(12th American High School Mathematics Examination, 1961) | 3:1 |
$35 \cdot 22$ A supermarket has 128 boxes of apples. Each box contains between 120 and 144 apples. Boxes with the same number of apples are called a group. What is the minimum number $n$ of boxes in the largest group?
(A) 4 .
(B) 5 .
(C) 6 .
(D) 24 .
(E) 25 .
(27th American High School Mathematics Examination, 1976) | 6 |
In $\triangle ABC$, $B=45^{\circ}$, $C=60^{\circ}$, and $c=1$. The length of the shortest side in $\triangle ABC$ is ( ).
A: $ \dfrac {1}{2}$
B: $ \dfrac { \sqrt {3}}{2}$
C: $ \dfrac { \sqrt {6}}{2}$
D: $ \dfrac { \sqrt {6}}{3}$ | D |
Given $A(-2,0)$, $B(4,a)$, if the distance from points $A$ and $B$ to the line $l:3x-4y+1=0$ are equal, then $a = (\ \ )$
A: $2$
B: $\frac{9}{2}$
C: $2$ or $-8$
D: $2$ or $\frac{9}{2}$ | D |
The curve $y = 3x^5 - 5x^3$ has ________ extreme values. | 2 |
In the "Government Work Report" of 2023, it was pointed out that in the past year, the gross domestic product (GDP) increased by $3\%$, and there were 12.06 million new urban jobs. The number $1206$ million can be expressed in scientific notation as ( ).
A: $0.1206\times 10^{8}$
B: $12.06\times 10^{6}$
C: $1.206\tim... | \text{D} |
In the rectangular coordinate system $xOy$, establish a polar coordinate system with the origin as the pole and the positive semi-axis of $x$ as the polar axis. Given that the polar coordinate equation of the curve $C$ is $ρ=4$, and the parametric equation of the line $l$ is $$\begin{cases} x=a+t\cos{α} \\ y=b+t\sin{α}... | x^2+y^2=8 |
1. Given the equation about $x$
$$
|| x-2|-| x+7||=k
$$
has two real roots. Then the range of $k$ is ( ).
(A) $k \geqslant 0$
(B) $0 \leqslant k<9$
(C) $0<k<9$
(D) $0<k \leqslant 9$ | C |
A circle is defined by the equation $(x-1)^2 + y^2 = 1$. Move the circle one unit to the left to obtain a new curve. Then, for each point on the new curve, keep the y-coordinate the same and multiply the x-coordinate by $\sqrt{3}$ to obtain a new curve C.
(1) Write the parametric equation of curve C.
(2) In a polar coo... | \sqrt{2}/2 |
A5. What sequence do the zeros of the polynomial $p(x)=-2 x^{3}+6 x$ form?
(A) arithmetic sequence
(B) geometric sequence
(C) infinite sequence
(D) constant sequence
(E) alternating sequence | A |
Given the function $f(x) = \frac{1}{3}x^3 - ax^2 + bx$ ($a, b \in \mathbb{R}$), with $f'(0) = f'(2) = 1$.
(1) Find the equation of the tangent line to the curve $y = f(x)$ at the point $(3, f(3))$;
(2) If the function $g(x) = f(x) - 4x$, $x \in [-3, 2]$, find the intervals of monotonicity and the minimum value of $... | -9 |
1. Let $S$ be the set of all integers $n$ such that $\frac{8 n^{3}-96 n^{2}+360 n-400}{2 n-7}$ is an integer. Find the value of $\sum_{n \in S}|n|$. | 50 |
10. (20 points) Find all values of the parameter $c$ such that the system of equations has a unique solution
$$
\left\{\begin{array}{l}
2|x+7|+|y-4|=c \\
|x+4|+2|y-7|=c
\end{array}\right.
$$ | 3 |
1. Determine all natural numbers less than 1000 that are equal to the sum of the factorials of their digits. | 1,2,145 |
1. For the quadrilateral $A B C D$, the following is true: point $A$ has coordinates $(-2,-3)$, point $B$ has coordinates $(1,-5)$, point $C$ is the mirror image of point $E(-6,1)$ when reflected over the coordinate origin, the abscissa of point $D$ is the solution to the equation $t^{2}-6 t+9=0$, and the ordinate of t... | (A,C)=\sqrt{68},\,(B,D)=\sqrt{85} |
8.42 Find the integers $x$, satisfying the inequality $\left|\frac{2}{x-13}\right|>\frac{8}{9}$ | 11;12;14;15 |
## Task $1 / 68$
At point $A$, there are $n$ disks $S_{i}$ with diameters $d_{i}(i=1 ; 2 ; \ldots ; n)$ stacked on top of each other such that $d_{j}<d_{k}$ for $j<k$.
They are to be moved one by one to point $B$, with point $C$ being available as an "intermediate storage".
The condition to be observed is that a lar... | 2^{n}-1 |
Given the functions $f(x)=2\ln x$ and $g(x)=\frac{1}{2}ax^{2}+(2a-1)x$.
(1) Let $h(x)=f(x)-g(x)$. Discuss the monotonic intervals of the function $h(x)$;
(2) If $f(x)-ax=0$ has two distinct real solutions $x_{1}$ and $x_{2}$, prove that $\ln x_{1}+\ln x_{2}>2$. | \ln x_{1}+\ln x_{2}>2 |
In the arithmetic sequence {an}, it is known that a2+a5+a12+a15=36, then S16=( )
A: 288
B: 144
C: 572
D: 72 | \text{B} |
2. Anna has as many nuts as Boris and Vova together. Anna and Boris together have twice as many nuts as Vova. One of them has 43 nuts. How many nuts do they have in total? | 258 |
If $\tan \left( x+\dfrac{{ }{!!}\!\pi\!{ }}{4} \right)=-3$, find the value of $\dfrac{\sin x+2\cos x}{3\sin x+4\cos x}$. | \dfrac{2}{5} |
Example 6 Let the function $f_{3}(x)=|x|, f_{1}(x)=$ $\left|f_{0}(x)-1\right|, f_{2}(x)=\left|f_{1}(x)-2\right|$. Then the area of the closed part of the figure enclosed by the graph of the function $y=f_{2}(x)$ and the $x$-axis is $\qquad$ | 7 |
A1. What is the time 1500 seconds after $14: 35$ ? | 15:00 |
$\sqrt{4^{0}+4^{2}+4^{3}}$ equals
(A) 9
(B) 13
(C) 14
(D) 32
(E) 64 | 9 |
## Task 5 - 210835
Someone withdraws a certain amount of money from their savings account. They receive this amount paid out in a total of 29 banknotes, exclusively in 10-mark notes, 20-mark notes, and 50-mark notes. The number of 10-mark notes is 1 less than the number of 20-mark notes. The number of 50-mark notes is... | 1000\mathrm{M} |
Given the point \\((n,a_n)\\) is on the graph of the function \\(y=2x-13\\), the minimum value of the sum of the first \\(n\\) terms of the sequence \\(\\{a_n\\}\\), \\(S_n\\), is ( ).
A: \\(36\\)
B: \\(-36\\)
C: \\(6\\)
D: \\(-6\\) | \text{B} |
7. The bus left from Zagreb heading to Split with stops in Zadar and Sibenik. In Zadar, $\frac{1}{4}$ of the total number of passengers who were on the bus got off, and in Sibenik, $\frac{2}{5}$ of the number of passengers who arrived in Sibenik got off. How many passengers arrived in Split if two more passengers got o... | 18 |
A $0,1,4,5,7,9$ digits can form how many 4-digit numbers in which at least one digit is repeated? | 780 |
If $f(x)$ is an even function, and when $x \in [0, +\infty)$, $f(x) = \begin{cases} \sin \frac{\pi}{2}x & (0 \leq x \leq 1) \\ x^{2}+\ln x & (x > 1) \end{cases}$, then the solution set of the inequality $f(x-1) < 1$ is ( )
A: $\{x|0 < x < 2\}$
B: $\{x|-1 < x < 1\}$
C: $\{x|0 < x < 1\}$
D: $\{x|-2 < x < 2\}$ | \text{A} |
(2013•Hunan) A school has 500 male and 500 female students. To understand whether there is a significant difference in learning interests and hobbies between male and female students, it is planned to survey 100 students from all students. The appropriate sampling method to use is ( )
A: Lottery method
B: Random numb... | \text{D} |
Define a function $f(x)$ on $\mathbb{R}$ that satisfies $f(x) + f(-x) = 0$. When $x > 0$, $f(x) = -4^{x} + 8 \times 2^{x} + 1$.
(Ⅰ) Find the explicit formula for $f(x)$;
(Ⅱ) When $x \in [-3, -1]$, find the maximum and minimum values of $f(x)$. | \text{minimum} = -17 |
3. The base of a right quadrilateral prism is a rhombus whose diagonals differ by $10 \mathrm{~cm}$. If the larger diagonal is increased by $2 \mathrm{~cm}$, and the smaller one is decreased by $1 \mathrm{~cm}$, the area of the rhombus remains the same.
Calculate the area of the prism if its height is twice the side o... | 1520\mathrm{~}^{2} |
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