problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
5. In a right-angled triangle, the height drawn to the hypotenuse divides the hypotenuse into two segments whose difference is equal to the length of one of the legs. Determine the angles of this triangle. | \beta=30,\alpha=60 |
7.1. Mihai drew several right-angled triangles on the board, such that any two triangles do not have common points. In each of these triangles, Mihai drew the following important lines (segments): all angle bisectors, all medians, and the altitude to the hypotenuse. Counting, the total number of important lines drawn i... | 6 |
The function $f(x) = \sin^2x + \frac{\sqrt{2}}{2}\cos(2x + \frac{\pi}{4})$
(1) Find the smallest positive period of $f(x)$.
(2) In $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, if $b= \sqrt{3}$ and $f\left(\frac{B}{2} + \frac{\pi}{4}\right) = \frac{1}{8}$, fi... | \frac{3\sqrt{7}}{4} |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. Given that $a_1 + a_2 + a_3 = a_4 + a_5$ and $S_5 = 60$, then $a_{10} =$ （ ）
A: 16
B: 20
C: 24
D: 26 | D |
Example 9 Let $\sin \alpha + \cos \alpha = a (0 \leqslant a \leqslant \sqrt{2})$. Find the expression for $\sin ^{n} \alpha + \cos ^{n} \alpha$ in terms of $a$.
(2005, Fudan University Independent Admission Examination) | \left( \frac{a + \sqrt{2 - a^2}}{2} \right)^n + \left( \frac{a - \sqrt{2 - a^2}}{2} \right)^n |
A1 Determine the value of the parameter $m$ so that the graph of the function $f(x)=(2 m+3) x-(1-2 m)$ intersects the x-axis at 3.
(A) $m=3$
(B) $m=2$
(C) $m=1$
(D) $m=-1$
(E) $m=0$ | -1 |
(1) Let the equation
$$
x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}=0
$$
have real coefficients, and satisfy the condition
$$
0<a_{0} \leqslant a_{1} \leqslant \cdots \leqslant a_{n-1} \leqslant 1
$$
Given that $\lambda$ is a complex root of this equation, and satisfies the condition $|\lambda| \geqslant 1$. Prove: $\... | \lambda^{n+1}=1 |
Given a real number $p>0$, the line $3x-4y+2p=0$ intersects the parabola $x^{2}=2py$ and the circle $x^{2}+\left(y-\frac{p}{2}\right)^{2}= \frac{p^{2}}{4}$ at four points from left to right: A, B, C, D. Calculate the ratio $\frac{AB}{CD}$. | \frac{1}{16} |
5.9 The sequence $101, 104, 116, \cdots$ has the general term $a_{n}=100+n^{2}$, where $n=1$, $2, 3, \cdots$. For each $n$, let $d_{n}$ denote the greatest common divisor of $a_{n}$ and $a_{n+1}$. Find the maximum value of $d_{n}$. Here $n$ takes all positive integers.
(3rd American Invitational Mathematics Examination... | 401 |
Problem 2. Vanya thought of a two-digit number, then swapped its digits and multiplied the resulting number by itself. The result turned out to be four times larger than the number he thought of. What number did Vanya think of? | 81 |
2、During the New Year's Day promotion at the mall, the price of color TVs is reduced by $20 \%$. After the promotion, the mall needs to increase the price by $\qquad$ to return to the original price. | 25 |
Evokimov
A natural number is written on the board. If you erase the last digit (in the units place), the remaining non-zero number will be divisible by 20, and if you erase the first digit, it will be divisible by 21. What is the smallest number that can be written on the board if its second digit is not equal to 0?
... | 1609 |
4.027. The sum of the third and ninth terms of an arithmetic progression is 6, their product is $135 / 16$. Find the sum of the first 15 terms of this progression. | 37.5or52.5 |
Which of the following equations is a quadratic equation?
A: $x^{2}=x+1$
B: $y^{2}+x=1$
C: $2x+1=0$
D: $x+\frac{1}{x}=1$ | A |
6 The line segment connecting two points on a sphere is called a chord of the sphere. For a sphere with a radius of 4, the lengths of two chords $AB$ and $CD$ are $2\sqrt{7}$ and $4\sqrt{3}$, respectively. $M$ and $N$ are the midpoints of $AB$ and $CD$, respectively. The endpoints of the two chords can move on the surf... | A |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $a\tan C=2c\sin A$.
(I) Find the measure of angle $C$;
(II) Find the maximum value of $\sin A+\sin B$. | \sqrt{3} |
1. Given the set
$$
\begin{aligned}
M & =\left\{f(x) \mid f^{2}(x)-f^{2}(y)\right. \\
& =f(x+y) \cdot f(x-y), x, y \in \mathbf{R}\} .
\end{aligned}
$$
The following statements are given:
(1) If $f_{1}(x)=\left\{\begin{array}{cc}1, & x \geqslant 0, \\ -1, & x<0,\end{array}\right.$ then $f_{1}(x) \in$
$M$;
(2) If $f_{2}... | C |
5.66 A deck of cards has a total of 54 cards. The magician divides them into several piles, and the audience writes a natural number on each card, which is equal to the number of cards in the pile where the card is located. Then the magician mixes the cards in a special way and divides them into several piles again. Th... | 3 |
There are 10 different electronic products, among which 2 products are unstable. Technicians test them one by one until both unstable products are identified, and the testing ends. The number of methods to exactly finish the testing in 3 tries is ( )
A: 16
B: 24
C: 32
D: 48 | C |
17. (3 points) Among the natural numbers less than 5000, the numbers that are divisible by 11 and have a digit sum of 13, there are $\qquad$ in total. | 18 |
If $a > b$, then which of the following equations is correct? ( )
A: $a \cdot \lg x > b \cdot \lg x$ ($x > 0$)
B: $a x^2 > b x^2$
C: $a^2 > b^2$
D: $a \cdot 2^x > b \cdot 2^x$ | \text{D} |
A subset $B$ of $\{1, 2, \dots, 2017\}$ is said to have property $T$ if any three elements of $B$ are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property $T$ may contain. | 1009 |
3. (17 points) Find the smallest natural number that is simultaneously twice a perfect cube and three times a perfect square.
# | 432 |
The value of $\cos^2 15^\circ - \sin^2 15^\circ$ is ( )
A: $- \frac { \sqrt {3}}{2}$
B: $\frac { \sqrt {3}}{2}$
C: $- \frac {1}{2}$
D: $\frac {1}{2}$ | B |
3. Find such an integer $\mathrm{n}$ that for any integer $\mathrm{k} \neq 2013$, the number $n-k^{2014}$ is divisible by $2013-k$.
Answer: $2013^{2014}$. | 2013^{2014} |
Given that the line $ax + y - 1 = 0$ is parallel to the line $2x - y + 2 = 0$, find the value of $a$. | a = -2 |
Calculate the expression that equals $2022$.
A: $1-\left(-2021\right)$
B: $1^{2022}$
C: $0-2022$
D: $2022^{-1}$ | A |
11.1. Solve the equation $2 \sin ^{2} x+1=\cos (\sqrt{2} x)$. | 0 |
Given $a=22^{55}$, $b=33^{44}$, $c=55^{33}$, and $d=66^{22}$, arrange $a$, $b$, $c$, and $d$ in ascending order.
A: $a > b > c > d$
B: $a > b > d > c$
C: $b > a > c > d$
D: $a > d > b > c$ | \text{A} |
The slope of the line $x+y=0$ is ( ).
A: $45^{\circ}$
B: $60^{\circ}$
C: $90^{\circ}$
D: $135^{\circ}$ | D |
In the Cartesian coordinate system, with the origin as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. In this polar coordinate system, it is known that circle C passes through point $P(\sqrt{2}, \frac{\pi}{4})$, and the center of the circle is the intersec... | \rho = 2\cos\theta |
Given the function $f(x)=-\frac{1}{2}x^{2}+x$ with a domain that contains an interval $[m,n]$, and its range on this interval is $[3m,3n]$. Find the value of $m+n$ $( \quad )$.
A: $(-2)$
B: $(-3)$
C: $(-4)$
D: $(-5)$ | \text{C} |
3. The vertices of a quadrilateral lie on the sides of a square with side length 1. If the sum of the squares of the four sides is $t$, then the range of values for $t$ is ( ).
(A) $[1,2]$
(B) $[2,4]$
(C) $[1,3]$
(D) $[3,6]$ | B |
Given an ellipse $C$ that shares the same foci with the hyperbola $x^{2}-y^{2}=1$ and has an eccentricity of $\frac{\sqrt{2}}{2}$.
(1) Find the standard equation of ellipse $C$.
(2) A line passing through point $P(0,1)$ intersects the ellipse at points $A$ and $B$. Point $O$ is the origin. If $\overrightarrow{AP}=2\ov... | \frac{\sqrt{126}}{8} |
Problem 9.7. In triangle $ABC$, the bisector $AL$ is drawn. Points $E$ and $D$ are marked on segments $AB$ and $BL$ respectively such that $DL = LC$, $ED \parallel AC$. Find the length of segment $ED$, given that $AE = 15$, $AC = 12$.
. It takes a negligible amount of time for t... | a = 1 |
The power function $y=x^{\alpha}$ must pass through the fixed point ( )
A: (0, 0)
B: (1, 1)
C: (0, 1)
D: (1, 0) | \text{B} |
8. In the school, after classes, someone broke a mirror in the vestibule. The last to leave were 9 students. The school principal asked each of them: "Who broke the mirror?" The answers were as follows. Roland: "It was Percy." Alex: "No, it wasn't Percy." Ted: "I broke the mirror." Fred: "It was either Ted or Walter." ... | Walter |
Read the programs:
Program A:
```
INPUT i=1
S=0
WHILE i ≤ 1000
S = S + i
i = i + 1
WEND
PRINT S
END
```
Program B:
```
INPUT I = 1000
S = 0
DO
S = S + I
I = I - 1
LOOP UNTIL I < 1
PRINT S
END
```
Judging the two programs and their output results, the correct option is ( ).
A: The programs are diffe... | \text{B} |
The equation of the tangent line to the curve $y=2x^{2}-x$ at the point $(1,1)$ is __________.
A: $x-y+2=0$
B: $3x-y+2=0$
C: $x-3y-2=0$
D: $3x-y-2=0$ | D |
In the trihedron $O A B C$, the dihedral angles are right angles. On two edges, we mark $O A=O B=a$, while on the third one, the distance $O C=x$. The value of $x$ is to be determined so that the volume of the tetrahedron $O A B C$ is equal to $\frac{F m}{3}$, where $F$ is the surface area of the tetrahedron and $m$ is... | \leq\frac{}{2}(2-\sqrt{2}) |
Given the propositions $p$: $∃x∈R$, $x-2 > \log _{2}x$, and $q$: $∀x∈R$, $x^{2} > 0$, then $(\ \ \ )$
A: $p∨q$ is a false proposition
B: $p∨(￢q)$ is a false proposition
C: $p∧q$ is a true proposition
D: $p∧(￢q)$ is a true proposition | D |
9. If the acute angle $\theta$ satisfies $\sin (\pi \cos \theta)=\cos (\pi \sin \theta)$, then $\sin 2 \theta=$ | \frac{3}{4} |
Problem 5. Three regular octagons have a common center, their sides are respectively parallel. The sides of two octagons are 7 cm and 42 cm. The third octagon divides the area of the figure enclosed between the first two in the ratio $1: 6$, counting from the smaller octagon. Find the side of the third octagon. | 7\sqrt{6} |
4. Find all real numbers $x$ that satisfies
$$
\frac{x^{4}+x+1}{x^{4}+x^{2}-x-4}=\frac{x^{4}+1}{x^{4}+x^{2}-4} .
$$ | -1,0,1 |
Example 4 Given the family of curves $2(2 \sin \theta-\cos \theta+3)$. $x^{2}-(8 \sin \theta+\cos \theta+1) y=0, \theta$ is a parameter. Find the maximum value of the length of the chord intercepted by the line $y=2 x$ on this family of curves.
(1995, National High School Mathematics Competition) | 8 \sqrt{5} |
The $p\%$ percentile of a set of numbers ($p\in 0,100$) refers to a value that satisfies the following conditions: at least $p\%$ of the data is not greater than this value, and at least $\left(100-p\right)\%$ of the data is not less than this value. Intuitively, the $p\%$ percentile of a set of numbers refers to the v... | 6 |
2. Find the probability that a randomly selected five-digit natural number with non-repeating digits, formed from the digits $1,2,3,4,5,6,7,8$, is divisible by 8 without a remainder. (10 points) | \frac{1}{8} |
Convert the repeating decimal $0.4\overline{31}$ into a fraction. | \frac{427}{990} |
How many non-congruent right triangles are there such that the perimeter in $\text{cm}$ and the area in $\text{cm}^2$ are numerically equal?
$\textbf{(A)} \ \text{none} \qquad \textbf{(B)} \ 1 \qquad \textbf{(C)} \ 2 \qquad \textbf{(D)} \ 4 \qquad \textbf{(E)} \ \text{infinitely many}$ | \textbf{(E)}\ |
Consider the hyperbola given by the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$. If a line passing through the right focus $F$ of the hyperbola with a slope of $1$ intersects one of the asymptotes at exactly one point, find the eccentricity of the hyperbola. | \sqrt{2} |
(1) For what value of the real number $a$ does the inequality system $\left\{\begin{array}{l}x^{2}+y^{2}-2 y \leqslant 1, \\ x+y+a=0\end{array}\right.$, have a unique solution, and find this solution. | =1,{\begin{pmatrix}-1,\\0\end{pmatrix}.;=-3,{\begin{pmatrix}1,\\2\end{pmatrix}.}} |
4. In the triangular prism $A B C-A_{1} B_{1} C_{1}$, the side length of the square $A A_{1} C_{1} C$ is 4, the plane $A B C \perp$ plane $A A_{1} C_{1} C, A B=$ $3, B C=5$. If there is a point $D$ on the line segment $B C_{1}$ such that $A D \perp A_{1} B$, then the value of $\frac{B D}{B C_{1}}$ is | \frac{9}{25} |
The equation $\frac{x^2}{m^2+n}-\frac{y^2}{3m^2-n}=1$ represents a hyperbola, and the distance between the two foci of this hyperbola is $4$. Then, the range of $n$ is $(\ )$.
A: $(-1,3)$
B: $(-1, \sqrt{3})$
C: $(0,3)$
D: $(0, \sqrt{3})$ | A |
There are four students sitting in a row of 7 seats, such that there are exactly two adjacent empty seats. The number of different seating arrangements is $\boxed{480}$. | 480 |
7. $\cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{3 \pi}{15} \cdots \cos \frac{7 \pi}{15}$ The value is $\qquad$ | \frac{1}{128} |
3A. For the non-zero complex number $z$, it holds that $z^{3}=\bar{z}$. Calculate $z^{2004}$. | 1 |
Find the equation of the chord that has point $A(1,-1)$, which lies inside the ellipse $x^{2}+4y^{2}=16$, as its midpoint. | x-4y-5=0 |
Let $ABCDEF$ be a regular hexagon with side length $1$. Denote by $X$, $Y$, and $Z$ the midpoints of sides $\overline {AB}$, $\overline{CD}$, and $\overline{EF}$, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle ACE$ and $\triangle XYZ$?
$\textbf{(A)... | \frac{15}{32}\sqrt{3} |
The distance from the point \\((1,1)\\) to the line \\(x+y-1=0\\) is \\((\\) \\()\\)
A: \\(1\\)
B: \\(2\\)
C: \\( \\dfrac { \\sqrt {2}}{2}\\)
D: \\( \\sqrt {2}\\) | C |
7. If one side of a triangle is twice another side, and one angle is $30^{\circ}$. Then the shape of the triangle is ( ).
(A) right triangle
(B) obtuse triangle
(C) acute triangle
(D) cannot be determined. | D |
A passenger train traveling from $A$ to $B$ has a speed of $60 \mathrm{~km} / \mathrm{h}$. The train, which departs later, is a fast train with a speed of $120 \mathrm{~km} / \mathrm{h}$ and must catch up to the passenger train at $B$. After covering $2 / 3$ of its journey, the passenger train continues at half its ori... | 360\mathrm{~} |
A student's test scores for a certain sports event were as follows (unit: points): $9,7,10,8,10,9,10$. The mode of this data set is
A: $10$
B: $9$
C: $8$
D: $7$ | A |
Let's find the sum of the following series:
$$
2(1 \cdot 2)+2(1 \cdot 3)+\ldots+2 \cdot(1 \cdot n)+2 \cdot(2 \cdot 3)+\ldots+2 \cdot(2 \cdot n)+\ldots+2(n-1) n
$$ | \frac{n(n+1)(3n^{2}-n-2)}{12} |
Determine the sum of the real numbers $x$ for which $\frac{2 x}{x^{2}+5 x+3}+\frac{3 x}{x^{2}+x+3}=1$. | -4 |
Given a function $f(x) = x^2 - 4x + a$ such that $f(x) \geq 0$ holds for all $x \in [0, 1]$, determine the range of values for the real number $a$. | a \in [3, +\infty) |
Let $r$ be the radius of the inscribed circle , $R$ be the radius of the circumscribed circle, $p$ the perimeter and $c$ the length of the hypotenuse of a right triangle.
(a) Show that $\frac{p}{c}-\frac{r}{R}=2$
(b) Of all the right triangles, what is the greatest value the ratio $\frac{r}{R}$ can take ? For which rig... | \frac{1}{\sqrt{2} + 1} |
The line that is tangent to the circle $x^2+y^2=25$ at the point $\left(3,4\right)$ intersects the $x$-axis at $\left(k,0\right)$. What is $k$?
$\text{(A) }\frac{25}{4}\qquad\text{(B) }\frac{19}{3}\qquad\text{(C) }25\qquad\text{(D) }\frac{25}{3}\qquad\text{(E) }-\frac{7}{3}$ | \frac{25}{3} |
On a windless day, a pigeon can fly from Albatrocity to Finchester and back in $3$ hours and $45$ minutes. However, when there is a $10$ mile per hour win blowing from Albatrocity to Finchester, it takes the pigeon $4$ hours to make the round trip. How many miles is it from Albatrocity to Finchester? | 75 |
Given the function $f(x) = 2\sin(\omega x + \frac{\pi}{6})$ ($\omega > 0$), its graph intersects the y-axis at point P and intersects the x-axis at two adjacent points, denoted as A and B. If the area of $\triangle PAB$ equals $\pi$, then $\omega = \_\_\_\_\_\_$. | \frac{1}{2} |
Each cell of an $100\times 100$ board is divided into two triangles by drawing some diagonal. What is the smallest number of colors in which it is always possible to paint these triangles so that any two triangles having a common side or vertex have different colors? | 8 |
Find all integer values can take $n$ such that $$\cos(2x)=\cos^nx - \sin^nx$$ for every real number $x$. | n \in \{2, 4\} |
## 217. Math Puzzle $6 / 83$
In an electrical circuit, the sum of two resistances is $35 \Omega$. The quotient of $2: 5$ is equal to the quotient of $R_{1}$ and $R_{2}$.
How large is each individual resistance? | R_{1}=10\OmegaR_{2}=25\Omega |
140. Find the greatest common divisor of all nine-digit numbers composed of the digits $1,2,3,4,5,6,7,8,9$ (without repetition). | 9 |
Given planes $\alpha$, $\beta$, $\gamma$, and lines $a$, $b$, $c$, which of the following statements is correct?
A: If $a$ is parallel to $\alpha$, $b$ is parallel to $\beta$, and $a$ is parallel to $b$, then $\alpha$ is parallel to $\beta.
B: If $a$ is perpendicular to $\alpha$, $\alpha$ is perpendicular to $\beta$,... | C |
Given $m=\cos(-4)$ and $n=\sin(-4)$, then
A: $m > n$
B: $m < n$
C: $m=n$
D: The magnitude of $m$ and $n$ is uncertain | \text{B} |
369. Prove that the product of $n$ terms of a geometric progression is equal to $a_{1}^{n} q^{\frac{n(n-1)}{2}}$, where $a_{1}$ is the first term of the progression, and $q$ is its common ratio. | a_{1}^{n}q^{\frac{n(n-1)}{2}} |
Given that the function $f(x) = ax + 2a + 1$ has both positive and negative values when $-1 \leqslant x \leqslant 1$, determine the range of values for $a$. The options are:
A: $a \geqslant -\frac{1}{3}$
B: $a \leqslant -1$
C: $-1 < a < -\frac{1}{3}$
D: None of the above | C |
When boiling water, the temperature reaches $100^{\circ}\mathrm{C}$ and the water boils. The table below shows the data of two variables, time $t(\min)$ and temperature $T(^{\circ}\mathrm{C})$, recorded by a student during an experiment on "observing the boiling of water". For every $1$ minute increase in time, the tem... | D |
If $|a-2|+{(b+\frac{1}{2})}^{2}=0$, then $\left(ab\right)^{2023}=\_\_\_\_\_\_.$ | -1 |
7. Margot writes the numbers $1,2,3,4,5,6,7$ and 8 in the top row of a table, as shown. In the second row she plans to write the same set of numbers, in any order.
Each number in the third row is obtained by finding the sum of the two numbers above it. In how many different ways can Margot complete row 2 so that every ... | 576 |
4. Given $\alpha, \beta \in \mathbf{R}$, and $\cos \alpha+\cos \beta=\frac{1}{3}, \tan (\alpha+\beta)=\frac{24}{7}$, then $\sin \alpha+\sin \beta=$ | \frac{1}{4} |
1. Find all values of $p$, for each of which the numbers $|p-3|, 3p+1$, and $9p+10$ are respectively the first, second, and third terms of some geometric progression. | p=-1,p=\frac{29}{18} |
In the final stage of a certain competition, four players, A, B, C, and D, participated. Before the results were announced, three people, A, B, and C, made the following predictions: A said: B is definitely not the champion; B said: the champion is either C or D; C said: A and D are not the champions. After the results... | \text{D} |
7. Highway (from 7th grade, 3 points). A highway running from west to east intersects with $n$ equal roads, numbered from 1 to $n$ in order. Cars travel on these roads from south to north and from north to south. The probability that a car will approach the highway from each of these roads is $\frac{1}{n}$. Similarly, ... | \frac{2kn-2k^{2}+2k-1}{n^{2}} |
1.047. $\left(\frac{\sqrt{561^{2}-459^{2}}}{4 \frac{2}{7} \cdot 0.15+4 \frac{2}{7}: \frac{20}{3}}+4 \sqrt{10}\right): \frac{1}{3} \sqrt{40}$. | 125 |
14. Calculate: $1-3+5-7+9-11+\cdots-1999+2001$ | 1001 |
Given that $a, b, c, d \in (0, +\infty)$, prove that $ac + bd \leq \sqrt{(a^2 + b^2)(c^2 + d^2)}$. | ac + bd \leq \sqrt{(a^2 + b^2)(c^2 + d^2)} |
449. Find the integrals:
1) $\int\left(3 x^{2}-2 x+5\right) d x$;
2) $\int \frac{2 x^{2}+x-1}{x^{3}} d x$;
3) $\int\left(1+e^{x}\right)^{2} d x$
4) $\int \frac{2 x+3}{x^{2}-5} d x$;
5) $\int \frac{x^{2}}{x^{2}+1} d x$;
6) $\int \operatorname{tg}^{2} \varphi d \varphi$. | \begin{aligned}1)&\quadx^{3}-x^{2}+5x+C\\2)&\quad2\ln|x|-\frac{1}{x}+\frac{1}{2x^{2}}+C\\3)&\quadx+2e^{x}+\frac{1}{2}e^{2x}+C\\4)&\quad\ln|x^{2}-5\ |
Let \\(\{a_n\}\) be an arithmetic sequence with a positive common difference. If \\(a_1+a_2+a_3=15\\) and \\(a_1a_2a_3=80\\), then \\(a_{11}+a_{12}+a_{13}=\\) \_\_\_\_\_\_. | 105 |
Circle I is circumscribed about a given square and circle II is inscribed in the given square. If $r$ is the ratio of the area of circle I to that of circle II, then $r$ equals:
$\text{(A) } \sqrt{2} \quad \text{(B) } 2 \quad \text{(C) } \sqrt{3} \quad \text{(D) } 2\sqrt{2} \quad \text{(E) } 2\sqrt{3}$ | 2 |
## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{\sqrt{3 n-1}-\sqrt[3]{125 n^{3}+n}}{\sqrt[5]{n}-n}
$$ | 5 |
If the equation $x^2 + y^2 - 2x - 4y + m = 0$ represents a circle, then the range of values for the real number $m$ is | (-\infty, 5) |
For four distinct points A, B, C, and D in space, consider the following five propositions:
1. If lines AB and CD are coplanar, then lines AC and BD are coplanar as well.
2. If lines AB and CD are skew lines (not coplanar), then lines AC and BD are skew lines as well.
3. If $AB=AC$ and $DB=DC$, then $AD \perp BC... | 1, 2, 3, 4 |
## Problem Statement
Are the vectors $c_{1 \text { and }} c_{2}$, constructed from vectors $a_{\text {and }} b$, collinear?
$$
\begin{aligned}
& a=\{5 ; 0 ;-1\} \\
& b=\{7 ; 2 ; 3\} \\
& c_{1}=2 a-b \\
& c_{2}=3 b-6 a
\end{aligned}
$$ | c_{1}=-\frac{1}{3}\cdotc_{2} |
In convex pentagon $ ABCDE$, denote by
$ AD\cap BE = F,BE\cap CA = G,CA\cap DB = H,DB\cap EC = I,EC\cap AD = J; AI\cap BE = A',BJ%Error. "capCA" is a bad command.
= B',CF%Error. "capDB" is a bad command.
= C',DG\cap EC = D',EH\cap AD = E'.$
Prove that $ \frac {AB'}{B'C}\cdot\frac {CD'}{D'E}\cdot\frac {EA'}{A'B}\cdo... | 1 |
5. Let $0<x<4.5, f(x)=\left(1+\frac{1}{\lg \left(\sqrt{x^{2}+10}+x\right)}\right)\left(1+\frac{2}{\lg \left(\sqrt{x^{2}+10}-x\right)}\right)$, then the minimum value of $f(x)$ is
A. Does not exist
B. $8 \sqrt{2}$
C. $8 \sqrt{3}$
D. $8+4 \sqrt{3}$ | 8+4\sqrt{3} |
21.3.1 * Five boys and two girls stand in a row, with boy A must stand in the middle, and the girls must be adjacent. How many different ways are there to arrange them! | 192 |
If the arithmetic mean of 1 and $a$ is 2, then the value of $a$ is ( )
A: 4
B: 3
C: 1
D: -4 | B |
4. Let the side lengths of $\triangle A B C$ be $6, x, 2x$. Then the maximum value of its area $S$ is $\qquad$ . | 12 |
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