problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Among the transformations from left to right below, the one that represents factorization is
A: $$(a+1)(a-1)=a^{2}-1$$
B: $$(x-y)(m-n)=(y-x)(m-n)$$
C: $$ab-a-b+1=(a-1)(b-1)$$
D: $$m^{2}-2m-3=m(m-2)-3$$ | C |
Given vectors $\vec{m} = (\sin x, 1)$ and $\vec{n} = (\sqrt{3}A\cos x, \frac{A}{2}\cos 2x)$ with $A > 0$, the maximum value of the function $f(x) = \vec{m} \cdot \vec{n}$ is $6$.
1. Find $A$;
2. Translate the graph of the function $y = f(x)$ to the left by $\frac{\pi}{12}$ units, then compress the $x$-coordinates of a... | [-3, 6] |
Find the number of positive integers $x$ such that \[ \left[ \frac{x}{99} \right] = \left[ \frac{x}{101} \right] . \] | 2499 |
Determine the graph of the function $y=\sin \left(2x-\frac{\pi }{3}\right)$, $x\in \mathbb{R}$, by transforming the graph of the function $y=\sin 2x$, $x\in \mathbb{R}$. The transformation is:
A: A left shift by $\frac{\pi }{6}$ units
B: A right shift by $\frac{\pi }{6}$ units
C: A left shift by $\frac{\pi }{3}$ units
... | B |
[ [motion problems ]
A swimmer is swimming upstream against the current of the Neva River. Near the Palace Bridge, he lost an empty flask. After swimming another 20 minutes against the current, he noticed the loss and turned back to catch up with the flask; he caught up with it near the Lieutenant Schmidt Bridge. What... | 3 |
14. $C$ is the graph of $x y=1$, and the symmetric image of $C$ about the line $y=2 x$ is $C^{\prime}$. It is known that $C^{\prime}$ can be expressed as
$$
12 x^{2}+b x y+c y^{2}+d=0
$$
Use this form to find the value of $b c$ | 84 |
Example 1-24 Find the number of 5-digit numbers that contain at least one 6 and are divisible by 3. For a $k$-digit number $a_{k-1} a_{k-2} \cdots a_{1} a_{0}$ $\equiv a_{k} \times 10^{k-1}+a_{k-2} \times 10^{k-2}+\cdots+a_{1} \times 10+a_{0}$, since $10 \equiv 1 \bmod 3,10^{2} \equiv 1 \bmod 3, \cdots$, $10^{k} \equiv... | 12504 |
Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$ | 3 |
When $x \in (0, +\infty)$, the power function $y = (m^2 - m - 1)x^m$ is a decreasing function. The value of the real number $m$ is ($\quad\quad$).
A: $0$
B: $1$
C: $2$
D: $-1$ | D |
Task B-2.2. In a circle with radius $r=10 \mathrm{~cm}$, two parallel chords of lengths $16 \mathrm{~cm}$ and $12 \mathrm{~cm}$ are drawn. If the center of the circle is inside the trapezoid whose bases are these chords, calculate the perimeter of the trapezoid. | 4(7+5\sqrt{2}) |
## Problem Statement
Calculate the definite integral:
$$
\int_{5 / 2}^{10 / 3} \frac{\sqrt{x+2}+\sqrt{x-2}}{(\sqrt{x+2}-\sqrt{x-2})(x-2)^{2}} d x
$$ | \frac{9}{4}+\ln2 |
11. Class A and Class B have an equal number of students, some of whom participate in an extracurricular astronomy club. The number of students from Class A who participate in the astronomy club is exactly one-third of the number of students from Class B who do not participate. The number of students from Class B who p... | \frac{8}{9} |
\(\dfrac{3+i}{1+i}=\) \(\)
A: \(1+2i\)
B: \(1-2i\)
C: \(2+i\)
D: \(2-i\) | D |
1.3. A ballpoint pen costs 10 rubles, a fountain pen costs 60 rubles, and a gel pen costs 70 rubles. What is the maximum number of fountain pens that can be bought if you need to buy exactly 25 pens in total and among them there must be pens of all three types, and you need to spend exactly 1000 rubles on them? | 9 |
Given functions $f(x)$ and $g(x)$, both have $f'(x) < g'(x)$ in the interval $[a, b]$. Which of the following relationships is correct?
A: $f(x) + f(b) \geq g(x) + g(b)$
B: $f(x) - f(b) \geq g(x) - g(b)$
C: $f(x) \geq g(x)$
D: $f(a) - f(b) \geq g(b) - g(a)$ | \text{B} |
2.45. Through the vertices $A, C$ and $D_{1}$ of a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, a plane is drawn, forming a dihedral angle of $60^{\circ}$ with the base plane. The sides of the base are 4 and 3 cm. Find the volume of the parallelepiped. | \frac{144\sqrt{3}}{5} |
If $n$ is a positive integer, the symbol $n$ ! (read " $n$ factorial") represents the product of the integers from 1 to $n$. For example, $4 !=(1)(2)(3)(4)$ or $4 !=24$. Determine
$$
\frac{1}{\log _{2} 100 !}+\frac{1}{\log _{3} 100 !}+\frac{1}{\log _{4} 100 !}+\cdots+\frac{1}{\log _{99} 100 !}+\frac{1}{\log _{100} 100... | 1 |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 1} \frac{1+\cos \pi x}{\tan^{2} \pi x}$ | \frac{1}{2} |
Express $F_{m+n}$ in terms of $F_{m-1}, F_{m}, F_{n}$, and $F_{n+1}$. | F_{-1}F_{n}+F_{}F_{n+1}=F_{+n} |
4. A, B, and C went to the orchard to pick oranges, and they picked a total of 108 oranges. A first gave half of his oranges to B, then B gave $\frac{1}{5}$ of his oranges at that time to C, and finally C gave $\frac{1}{7}$ of his oranges at that time to A. If at this point the three of them had the same number of oran... | 60,15,33 |
Given that the terminal side of angle $\alpha$ passes through point $P(5a, -12a)$, where $a < 0$. Find:
(1) $\tan\alpha$;
(2) $\sin\alpha + \cos\alpha$. | \frac{7}{13} |
2. If the positive root of the quadratic equation $x^{2}-p x-q=0$ $\left(p, q \in \mathbf{N}_{+}\right)$ is less than 3, then the number of such quadratic equations is ( ).
(A) 5
(B) 6
(C) 7
(D) 8 | C |
Three cubes have edge lengths $3 \mathrm{~cm}, 12 \mathrm{~cm}$, and $x \mathrm{~cm}$. The average volume of the three cubes is $700 \mathrm{~cm}^{3}$. The value of $x$, rounded to the nearest integer, is
(A) 6
(B) 10
(C) 8
(D) 9
(E) 7 | 7 |
3.12. $\left\{\begin{array}{l}x+\frac{3 x-y}{x^{2}+y^{2}}=3, \\ y-\frac{x+3 y}{x^{2}+y^{2}}=0 .\end{array}\right.$ | x_1=2,y_1=1x_2=1,y_2=-1 |
Let the function $f'(x) = 2x^3 + ax^2 + x$, and $f'(1) = 9$, then $a=$ __ . | 6 |
Given the universal set $U=\{0,1,2,3,4\}$, set $M=\{0,1\}$, and set $N=\{2,3\}$, then $(\complement_U M) \cap N=$ ? | \{2,3\} |
Calculate: $-2\times \sqrt[3]{-27}+|1-\sqrt{3}|-(\frac{1}{2})^{-2}$ | 1 + \sqrt{3} |
22. The unit digit of the product $3^{76} \times 7^{33}$ is $(\quad)$.
(A ) 1
( B ) 3
(C) 7
(D) 9 | 7 |
Example 3 Given $A \cup B \cup C=\{1,2, \cdots, 6\}$, and $A \cap B=\{1,2\},\{1,2,3,4\} \subseteq B \cup C$.
Then the number of $(A, B, C)$ that satisfy the conditions is $\qquad$ groups (different orders of $A, B, C$ are considered different groups). | 1600 |
20. (6 points) As shown in the figure, the upper base $AD$ of trapezoid $ABCD$ is 12 cm long, the height $BD$ is 18 cm long, and $BE=2DE$. Then the length of the lower base $BC$ is
cm. | 24 |
## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(1+\tan^{2} x\right)^{\frac{1}{\ln \left(1+3 x^{2}\right)}}
$$ | e^{\frac{1}{3}} |
Given $a \geq 0$, $b \geq 0$, prove that: $a^6 + b^6 \geq ab(a^4 + b^4)$. | a^6 + b^6 \geq ab(a^4 + b^4) |
7、Xiaoming and Xiaohong are running on a 600-meter circular track. The two start from the same starting point at the same time and run in opposite directions. The time interval between the first and second meeting is 50 seconds. It is known that Xiaohong's speed is 2 meters/second slower than Xiaoming's, so Xiaoming's ... | 7 |
1. Find all natural numbers $n$ such that the number
$$
1!+2!+3!+\ldots+n!
$$
is a perfect square. (Here: $k!=1 \cdot 2 \cdot \ldots \cdot k, 1!=1$) | 1,3 |
Task 1. A vacuum robot is programmed to move on the floor according to the law:
$\left\{\begin{array}{l}x=t(t-6)^{2} \\ y=0,0 \leq t \leq 7 ; y=(t-7)^{2}, t \geq 7\end{array}\right.$
where the axes are parallel to the walls and the movement starts from the origin. Time $t$ is measured in minutes, and the coordinates ... | 71;\sqrt{445} |
I2.1 Suppose $P$ is an integer and $5<P<20$. If the roots of the equation $x^{2}-2(2 P-3) x+4 P^{2}-14 P+8=0$ are integers, find the value of $P$. | 12 |
Task 3. Find all solutions to the equation $x^{2}-[x]=1$. Here $[x]$ is the integer part of $x$, that is, the greatest integer not exceeding the given number. For example, $[2.9]=2 ;[-2.9]=-3$. | \sqrt{2} |
1. Given the function $f(x)=a \ln x+x^{2}$ has an extremum at $x=1$, then the value of the real number $a$ is
A. -2
B. -1
C. 1
D. 2 | -2 |
Given that the means of two sets of data for group A and B are $X_{甲} = 80$ and $X_{乙} = 90$, with variances $S_{甲}^2 = 10$ and $S_{乙}^2 = 5$, respectively, compare these two sets of data. Which of the following statements is correct?
A: Group A's data is better.
B: Group B's data is better.
C: Group A's data range is ... | D |
4. Petya wrote down $n$ consecutive two-digit numbers $(n \geqslant 2)$ in a row on the board, the first of which does not contain the digit 4, and the last one does not contain the digit 7. Vasya thought that this was the decimal representation of a natural number $x$, and he factored $x$ into prime factors. It turned... | 2021 |
3. Cutting a rectangular block of tofu with 3 cuts, it is impossible to get ( ) pieces.
A. 4
B. 5
C. 6
D. 8
E. 9 | E |
3. If the sum of the terms of the infinite geometric sequence $\left\{a_{n}\right\}$ is 1, and the sum of the absolute values of the terms is 2, then the value of the first term $a_{1}$ is . $\qquad$ | \frac{4}{3} |
$1 \cdot 110$ The distance from $A$ to $B$ is 120 miles, a car travels from $A$ to $B$ at a speed of 30 miles per hour; and from $B$ back to $A$ at a speed of 40 miles per hour, then, the average speed for the round trip is
(A) $33 \mathrm{mph}$.
(B) $34 \mathrm{mph}$.
(C) $35 \mathrm{mph}$.
(D) $36 \mathrm{mph}$.
(E) ... | 34\mathrm{} |
One. (This question is worth 40 points) Let real numbers $a_{1}, a_{2}, \cdots, a_{2016}$ satisfy $9 a_{i}>11 a_{i+1}^{2}(i=1,2, \cdots, 2015)$. Find the maximum value of $\left(a_{1}-a_{2}^{2}\right) \cdot\left(a_{2}-a_{3}^{2}\right) \cdots \cdot\left(a_{2015}-a_{2016}^{2}\right) \cdot\left(a_{2016}-a_{1}^{2}\right)$. | \frac{1}{4^{2016}} |
3. Among the 4 functions $y=\sin |x|, y=\cos |x|, y=|\cot x|, y=\lg |\sin x|$, the even function that is periodic with $\pi$ and monotonically increasing in $\left(0, \frac{\pi}{2}\right)$ is ().
A. $y=\sin |x|$
B. $y=\cos |x|$
C. $y=|\cot x|$
D. $y=\lg |\sin x|$ | D |
In △ABC, the sides opposite to angles A, B, C are a, b, c, respectively, and $$\sqrt {3}$$asinC=c+ccosA.
(1) Find the measure of angle A;
(2) If a=2$$\sqrt {3}$$, the area of △ABC is $$\sqrt {3}$$, find the perimeter of △ABC. | 2 \sqrt {3}+2 \sqrt {6} |
We call an arrangement each way of seating $n$ couples at a round table with $2n$ seats. We say that a couple is together if its two members are sitting next to each other. What is the average number of couples together per arrangement. | \frac{2n}{2n-1} |
When $m=$____, the function $y=(m+2)x^{m^2-3}$ is a direct proportion function. | 2 |
【Question 1】Calculate: $5 \times 13 \times 31 \times 73 \times 137=$ | 20152015 |
3. Determine all pairs $(a, b)$ of natural numbers such that their difference $a-b$ is the fifth power of some prime number and for which $a-4 \sqrt{b}=b+4 \sqrt{a}$.
The written part of the school round in category C takes place
on Thursday, January 25, 2007
so that it starts in the morning and the contestants have... | (36,4) |
3. In a $999 \times 999$ grid, some cells are white, and the others are red. Let $T$ be the number of cell groups $\left(C_{1}, C_{2}, C_{3}\right)$ such that $C_{1}$ and $C_{2}$ are in the same row, $C_{2}$ and $C_{3}$ are in the same column, and $C_{1}$ and $C_{3}$ are white, while $C_{2}$ is red. Find the maximum va... | 148 \times 999^{3} |
In the rectangular coordinate system, the distance from point A(1, 0, -2) to point B(-2, 4, 3) is _____. | 5\sqrt{2} |
$32 \cdot 28$ is less than 1000 and the number of positive integers that cannot be divided by 5 and 7 is
(A) 688.
(B) 686.
(C) 684.
(D) 658.
(E) 630.
(17th American High School Mathematics Examination, 1966) | 686 |
11.1. $\quad$ Find the smallest period of the function $y=\cos ^{10} x+\sin ^{10} x$. | \frac{\pi}{2} |
12. What digits do the decimal representations of the following numbers end with:
1) $135^{x}+31^{y}+56^{x+y}$, if $x \in N, y \in N$
2) $142+142^{2}+142^{3}+\ldots+142^{20}$
3) $34^{x}+34^{x+1}+34^{2 x}$, if $x \in N$. | 6 |
B2. The difference in the perimeters of two squares is $8 \mathrm{~cm}$, and the difference in their areas is $16 \mathrm{~cm}^{2}$. Calculate the sum of their areas. | 34\mathrm{~}^{2} |
Let $S_{n}$ denote the sum of the digits of the number $n$ in the decimal system. Determine all numbers $n$ for which $n=S^{2}(n)-S(n)+1$ | 1,13,43,91,157 |
Given the set $P = \{x \,|\, x^2 - x - 2 = 0\}$, and the set $T = \{x \,|\, -1 < x \leq 2\}$, find $P \cap T$. | \{2\} |
A rational number written in base eight is $\underline{ab} . \underline{cd}$, where all digits are nonzero. The same number in base twelve is $\underline{bb} . \underline{ba}$. Find the base-ten number $\underline{abc}$. | 321 |
Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?
$\textbf{(A) }\text{Aaron... | \textbf{(A)} |
Given that the slant height of a certain cone is $3$, then when the volume of the cone is maximized, the radian measure of the central angle of its lateral surface development is ( )
A: $\frac{2\sqrt{6}}{3}\pi$
B: $\frac{\sqrt{6}}{3}\pi$
C: $\frac{2\sqrt{3}}{3}\pi$
D: $\frac{\sqrt{3}}{3}\pi$ | A |
## Task 3 - V10823
In the number .378., the two dots are to be replaced by digits so that the resulting number is divisible by 72.
How did you determine the missing digits? | 53784 |
Given the function $f(x) = 6x - 4$ ($x = 1, 2, 3, 4, 5, 6$) has a range of set $A$, and the function $g(x) = 2x - 1$ ($x = 1, 2, 3, 4, 5, 6$) has a range of set $B$. For any $a \in A \cup B$, the probability that $a \in A \cap B$ is. | 0 |
Given that the areas of the top and bottom surfaces of a certain circular truncated cone are $3\pi$ and $12\pi$ respectively, and the slant height is $2$, what is the volume of this circular truncated cone?
A: $6\pi$
B: $18\pi$
C: $7\pi$
D: $21\pi$ | C |
4. Let $S$ be a set of $n$ distinct real numbers, and $A_{s}$ be the set of all distinct averages of pairs of elements from $S$. For a given $n \geqslant 2$, what is the minimum number of elements in $A_{s}$?
(1993 Putnam Competition) | 2n-3 |
Given $9^{a}=3$, $\ln x=a$, then $x=$ ______. | \sqrt {e} |
Let the function $f(x) = 2\lg(2x - 1)$. The value of $f^{-1}(0)$ is ( )
A: 0
B: 1
C: 10
D: Does not exist | \text{B} |
14. Let $a_{1}=2006$, and for $n \geq 2$,
$$
a_{1}+a_{2}+\cdots+a_{n}=n^{2} a_{n} .
$$
What is the value of $2005 a_{2005}$ ? | 2 |
3. Given a triangle $A B C, \angle B=90^{\circ}$. On the sides $A C, B C$ points $E$ and $D$ are chosen respectively, such that $A E=E C, \angle A D B=\angle E D C$. Find the ratio $C D: B D$. | 2:1 |
Given the function $f(x)=\begin{cases}f\left(x+2\right), & x < 3 \\ \left( \frac{1}{2}\right)^{x}, & x\geqslant 3\end{cases}$, then $f(-4)=$ ( )
A: $ \frac{1}{16} $
B: $ \frac{1}{8} $
C: $ \frac{1}{4} $
D: $ \frac{1}{2} $ | A |
2. The cold water tap fills the bathtub in 17 minutes, and the hot water tap in 23 minutes. The hot water tap was opened. After how many minutes should the cold water tap be opened so that by the time the bathtub is completely filled, there is an equal amount of cold and hot water in it? | 3 |
Given the function $f(x)=\frac{a+\ln x}{x}$ where $a\in\mathbb{R}$:
1. If $a=4$, find the equation of the tangent line to the curve $f(x)$ at the point $(1,4)$.
2. If the graph of the function $f(x)$ intersects with the graph of the function $g(x)=1$ in the interval $(0, e^2]$, find the range of values for the real nu... | [1,+\infty) |
4. A route on the $3 \times 3$ board shown consists of a number of steps. Each step is from one square to an adjacent square of a different colour.
How many different routes are there from square $\mathrm{S}$ to square $\mathrm{T}$ which pass through every other square exactly once?
A 0
B 1
C 2
D 3
E 4 | 2 |
Given the parabola \\(C\\): \\(y^{2}=4x\\) with focus \\(F\\), and a point \\(A(x_{0},y_{0})\\) on the parabola \\(C\\), if \\(|AF|=\frac{5}{4}x_{0}\\), then the value of \\(x_{0}\\) is \\((\\) \\()\\)
A: \\(8\\)
B: \\(4\\)
C: \\(2\\)
D: \\(1\\) | \text{B} |
The function $f(x) = x^2 - 2ax + a$ has a minimum value in the interval $(-\infty, 1)$. The range of $a$ is ( )
A: $a 1$
D: $a \geq 1$ | A |
It is given that the roots of the polynomial $P(z) = z^{2019} - 1$ can be written in the form $z_k = x_k + iy_k$ for $1\leq k\leq 2019$. Let $Q$ denote the monic polynomial with roots equal to $2x_k + iy_k$ for $1\leq k\leq 2019$. Compute $Q(-2)$. | \frac{-1 - 3^{2019}}{2^{2018}} |
$23$ frat brothers are sitting in a circle. One, call him Alex, starts with a gallon of water. On the first turn, Alex gives each person in the circle some rational fraction of his water. On each subsequent turn, every person with water uses the same scheme as Alex did to distribute his water, but in relation to themse... | a = 0, b = 0, c = 1 |
If three of the roots of $x^4+ax^2+bx+c=0$ are $1$, $2$, and $3$ then the value of $a+c$ is:
$\text{(A) } 35 \quad \text{(B) } 24 \quad \text{(C) } -12 \quad \text{(D) } -61 \quad \text{(E) } -63$ | -61 |
Use the Horner's Rule to calculate the value of $f(x) = 5x^6 + 4x^5 + x^4 + 3x^3 - 81x^2 + 9x - 1$ when $x=2$. The number of addition and multiplication operations required are ( )
A: 6, 11
B: 6, 6
C: 7, 5
D: 6, 13 | \text{B} |
3. The product of the areas of the six faces of a rectangular prism is 14641, then the volume of the rectangular prism is
保留源文本的换行和格式，直接输出翻译结果。 | 11 |
689. Find the following improper integrals:
1) $\int_{0}^{+\infty} e^{-x} d x$
2) $\int_{-\infty}^{+\infty} \frac{d x}{x^{2}+1}$
3) $\int_{0}^{1} \frac{d x}{x}$
4) $\int_{-1}^{2} \frac{d x}{\sqrt[3]{(x-1)^{2}}}$
Explain the solutions geometrically. | 1,\pi,+\infty,3(\sqrt[3]{2}+1) |
## SUBJECT I
The sequence of numbers is given: $1,4,7,10,13,16,19$
a) Determine whether the 22nd term of the sequence is a perfect square.
b) Find the 2015th term of the sequence. | 6043 |
In the Cartesian coordinate system, there is a line $l$ represented by the parametric equations:
$$
\begin{cases}
x=1+ \frac { \sqrt {2}}{2}t \\
y= \frac { \sqrt {2}}{2}t
\end{cases}
$$
where $t$ is a parameter. Using the origin $O$ as the pole and the positive $x$-axis as the polar axis, a polar coordinate system i... | \sqrt{14} |
3. Find the geometric locus of points $P(u, v)$ which are equally distant from the curve $x^{2}+4 x+y^{2}=0$ and the point $M(2,0)$. | u^{2}-\frac{v^{2}}{3}=1 |
If a polynomial minus $3x$ equals $5x^2-3x-5$, then the polynomial is ( )
A: $5x^2+5$
B: $5x^2-5$
C: $5x^2-6x-5$
D: $-5x^2-6x+5$ | \text{B} |
Let P and Q be points on the line $$\begin{cases} x=t \\ y=6-2t \end{cases}$$ (where t is the parameter) and on the curve C: $$\begin{cases} x=1+\sqrt{5}\cos\theta \\ y=-2+\sqrt{5}\sin\theta \end{cases}$$ (where $\theta$ is the parameter), respectively. Find the minimum value of the distance |PQ|. | \frac{\sqrt{5}}{5} |
3. As shown in Figure 4, in the "dart-shaped" quadrilateral $ABCD$, $AB=4\sqrt{3}$, $BC=8$, $\angle A=\angle B=\angle C=30^{\circ}$. Then the distance from point $D$ to $AB$ is $\qquad$ | 1 |
The rectangular coordinates of point $M$ are $(-1,-\sqrt{3})$, then its polar coordinates are $(\quad)$
A: $(2,\dfrac{\pi }{3})$
B: $(2,\dfrac{2}{3}\pi )$
C: $(2,\dfrac{5}{3}\pi )$
D: $(2,\dfrac{4}{3}\pi )$ | D |
4. In $\triangle A B C$, let the largest angle $\angle A$ be twice the smallest angle $\angle C$, and $A B=2, A C=3$. Then the perimeter of $\triangle A B C$ is ( ).
(A) $9-\sqrt{13}$
(B) $5 \sqrt{3}-\sqrt{10}$
(C) $4 \sqrt{2}$
(D) $5+\sqrt{10}$ | D |
In △ABC, the sides opposite to angles A, B, C are a, b, c respectively. If a=2, b=3, C=2A, then the value of cosC is _______. | \frac {1}{4} |
10. From the set $\{1,2, \cdots, 37\}$, select two numbers such that the sum of the remaining 35 numbers is the product of these two numbers. Then the difference between the two numbers is ( ).
(A) 5
(B) 7
(C) 8
(D) 9
(E) 10 | 10 |
10. (12th IMO Problem) Let real numbers $x_{1}, x_{2}, \cdots, x_{1997}$ satisfy the following conditions:
(1) $-\frac{1}{\sqrt{3}} \leqslant x_{i} \leqslant \sqrt{3}$, where $i=1,2, \cdots, 1997$.
(2) $x_{1}+x_{2}+\cdots+x_{1997}=-318 \sqrt{3}$.
Find: $x_{1}^{12}+x_{2}^{12}+\cdots+x_{1997}^{12}$'s maximum value. | 189548 |
11. (3 points) Cutting a cake, cutting 1 time can make at most 2 pieces, cutting 2 times can make at most 4 pieces, cutting 3 times can make at most 7 pieces, continuing this way, cutting 5 times can make at most $\qquad$ pieces. | 16 |
Given a function $f(x)$ with domain $\mathbb{R}$ that is monotonically decreasing in the interval $(-\infty, 5)$, and for any real number $t$, it holds that
$f(5 + t) = f(5 - t)$, then among the following expressions, the one that must hold is:
A: $f(-1) < f(9) < f(13)$
B: $f(13) < f(9) < f(-1)$
C: $f(9) < f(-1... | \text{C} |
3. As shown in Figure 2, fold $\triangle A B C$ along the dotted line $D E$ to get a heptagon $A D E C F G H$. If the area ratio of the heptagon to the original triangle is $2: 3$, and the area of the overlapping part after folding is 4, then the area of the original $\triangle A B C$ is | 12 |
Given that the image of $(x,y)$ under the mapping $f$ is $(x-y,x+y)$, then the image of $(3,5)$ under $f$ is \_\_\_\_\_\_\_\_\_, and the preimage is \_\_\_\_\_\_\_\_\_\_\_\_\_. | (4,1) |
10. As shown in the figure, in rectangle $A B C D$, connect diagonal $A C$. If $O$ is the incenter of $\triangle A B C$, and draw $O E \perp$ $A D$ at $E$, and draw $O F \perp C D$ at $F$. Then the ratio of the area of rectangle $O F D E$ to the area of rectangle $A B C D$ is ( ).
(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C... | A |
Given the function $f\left(x\right)=\ln x-x^{2}$, which of the following statements is correct?
A: $f\left(x\right)$ attains its maximum value at $x=\frac{{\sqrt{2}}}{2}$.
B: $f\left(x\right)$ is monotonically increasing on $({\frac{1}{2}},{\frac{{\sqrt{2}}}{2}})$.
C: $f\left(x\right)$ has two distinct zeros.
D: $f... | ABD |
Calculate: $\left(\pi -3\right)^{0}-\sqrt{8}+2\sin 45^{\circ}+(\frac{1}{2})^{-1}$. | 3-\sqrt{2} |
If there are several heaps of stones on the table, it is said that there are $\textit{many}$ stones on the table, if we can find $50$ piles and number them with the numbers from $1$ to $50$ so that the first pile contains at least one stone, the second - at least two stones,..., the $50$-th has at least $50$ stones. Le... | 5099 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.