problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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10.1. Let these numbers be A and B. Let B be the smallest of these numbers. Then the larger number $\mathrm{A}=10 \mathrm{~B}+\mathrm{n}$, where $\mathrm{n}$ is the crossed-out digit, and
$0 \leq \mathrm{n} \leq 9$. According to the condition, $\mathrm{A}+\mathrm{B}=11 \mathrm{~B}+\mathrm{n}=2022$. Therefore, $\mathrm... | B=183,A=1839 |
$2 \cdot 73$ Let positive integers $a, b$ make $15a + 16b$ and $16a - 15b$ both squares of positive integers. Find the smallest value that the smaller of these two squares can take. | 481^2 |
Let the equation of line $l$ be $2x + (k - 3)y - 2k + 6 = 0 \ (k \neq 3)$. If the sum of intercepts of line $l$ on the $x$-axis and $y$-axis is 0, find the value of $k$. | 1 |
Task 1. For which values of the variables $x$ and $y$, the difference between the expressions $\frac{2 x+15}{8}$ and $1 \frac{1}{3} \cdot(y-1)$ will be 3 times smaller than the expression $2 \cdot(5-2 y)$, and the expression $\frac{x+5 \frac{3}{4}}{2}$ will be 0.125 greater than $3 y?$ | \frac{1}{2},1 |
# Problem 8.
Let $A(n)$ denote the greatest odd divisor of the number $n$. For example, $A(21)=21$, $A(72)=9, A(64)=1$. Find the sum $A(111)+A(112)+\ldots+A(218)+A(219)$. | 12045 |
Problem 9.3. Given a convex quadrilateral $ABCD$, $X$ is the midpoint of diagonal $AC$. It turns out that $CD \parallel BX$. Find $AD$, if it is known that $BX=3, BC=7, CD=6$.
 | 14 |
Given that $A$ is a point on the curve $y=e^{x}$, and $B$ is a point on the curve $y=\ln x$, where $|AB|\geqslant a$ always holds, the range of real number $a$ is ______. | (-\infty, \sqrt{2}] |
B1. For which values of the real number $a$ does the expression $(a-0.\overline{3})(-3)+(-4a)$ have a value of at least -6? Present the solution graphically as well. | \leq1 |
356. Find the sum of all products of the form $1 \cdot 2, 1 \cdot 3, \ldots$ $n(n-1)$, that can be formed from the numbers $1, 2, 3, \ldots n$. | \frac{n(n+1)(n-1)(3n+2)}{24} |
Let the complex number $z = m^2 - 2m - 3 + (m^2 + 3m + 2)i$. Find the values of the real number $m$ for which:
1. $z$ is a pure imaginary number?
2. The point corresponding to $z$ is located in the second quadrant of the complex plane? | -1 < m < 3 |
Let $(2x - 1)^5 = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5$. Find:
(1) $a_0 + a_1 + a_2 + a_3 + a_4$
(2) $(a_0 + a_2 + a_4)^2 - (a_1 + a_3 + a_5)^2$. | -243 |
How many different pairs $(m, n)$ can be formed using numbers from the list of integers $\{1,2,3, \ldots, 20\}$ such that $m<n$ and $m+n$ is even?
(A) 55
(B) 90
(C) 140
(D) 110
(E) 50 | 90 |
An equilateral triangle is tiled with $n^2$ smaller congruent equilateral triangles such that there are $n$ smaller triangles along each of the sides of the original triangle. For each of the small equilateral triangles, we randomly choose a vertex $V$ of the triangle and draw an arc with that vertex as center connect... | \frac{(n-2)(n-1)}{1458} |
A foreign investor plans to invest in 3 different projects among 4 candidate cities, with no more than 2 projects in the same city. How many different investment plans are there? | 60 |
Given $k < -1$, the curve represented by the equation $(1-k)x^{2}+y^{2}=k^{2}-1$ regarding $x$ and $y$ is ( )
A: A hyperbola with its real axis on the $x$-axis
B: A hyperbola with its real axis on the $y$-axis
C: An ellipse with its major axis on the $x$-axis
D: An ellipse with its major axis on the $y$-axis | \text{D} |
Prove that in a planar bipartite graph $E \geq 2 F$, if $E \geq 2$ ( $E$ - number of edges, $F$ - number of faces).
# | E\geq2F |
Two points $A, B$ are randomly chosen on a circle with radius $100.$ For a positive integer $x$, denote $P(x)$ as the probability that the length of $AB$ is less than $x$. Find the minimum possible integer value of $x$ such that $\text{P}(x) > \frac{2}{3}$. | 174 |
The monotonic decreasing interval of $y=2\cos \left( \frac {\pi}{4}-2x\right)$ is ( )
A: $\left[k\pi+ \frac {\pi}{8},k\pi+ \frac {5}{8}\pi\right](k\in\mathbb{Z})$
B: $\left[- \frac {3}{8}\pi+k\pi, \frac {\pi}{8}+k\pi\right](k\in\mathbb{Z})$
C: $\left[ \frac {\pi}{8}+2k\pi, \frac {5\pi}{8}+2k\pi\right](k\in\mathbb{Z})$... | \text{A} |
Problem 4.3. Zhenya drew a square with a side of 3 cm, and then erased one of these sides. A figure in the shape of the letter "P" was obtained. The teacher asked Zhenya to place dots along this letter "P", starting from the edge, so that the next dot was 1 cm away from the previous one, as shown in the picture, and th... | 31 |
A rectangular parallelepiped has edges of length 1, 2, and 3 units extending from vertex $A$. The endpoints of these edges, different from $A$, form a triangle. How far is point $A$ from the plane of the triangle? | \frac{6}{7} |
2. Determine all triples of composite natural numbers, each of which is divisible by the product of the superdivisors of the remaining two numbers. (A superdivisor of a number $n>1$ is its greatest divisor $d$ with the property $d<n$.)
(Jaromír Šimša) | (p^2,p^2,p^2) |
Task B-2.7. Let $z_{1}=3\left(\frac{1+i}{1-i}\right)^{2}-2\left(\frac{1-i}{1+i}\right)^{3}$. Determine all complex numbers $z \in \mathbb{C}$ for which
$$
\left|z_{1} \cdot \bar{z}+z \cdot \overline{z_{1}}\right|=4, \quad \operatorname{Im} \frac{z_{1} \cdot z}{1-i}=0
$$ | \frac{2}{7}-\frac{10}{7}ior-\frac{2}{7}+\frac{10}{7}i |
6. In the Cartesian coordinate system $x O y$, the ellipse $\Gamma: \frac{x^{2}}{4}+y^{2}=1$, point $L$ is in the third quadrant and $O L=3 \cdot \sqrt{2}$, and the angle between the negative half-axis of the $x$-axis and $O L$ is $\frac{\pi}{4}$. Two tangent lines $l_{1} 、 l_{2}$ are drawn through point $L$ to $\Gamma... | 9 |
If $z=\frac{2-7i}{4-i}$, then the point corresponding to $z$ in the complex plane is located in which quadrant?
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | D |
23. The quality control department detected five defective books in a batch of 100 randomly selected books. Find the relative frequency of the appearance of defective books. | 0.05 |
4. Find all non-negative real solutions $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ of the system of equations
$$
x_{i+1}=x_{i}^{2}-\left(x_{i-1}-1\right)^{2}, \quad i=1,2, \ldots, n
$$
(Indices are taken cyclically modulo $n$.) | x_{1}=x_{2}=\cdots=x_{n}=1 |
4. 148 Given
$$
\left\{\begin{array}{l}
\frac{x^{2}}{2^{2}-1^{2}}+\frac{y^{2}}{2^{2}-3^{2}}+\frac{z^{2}}{2^{2}-5^{2}}+\frac{\omega^{2}}{2^{2}-7^{2}}=1, \\
\frac{x^{2}}{4^{2}-1^{2}}+\frac{y^{2}}{4^{2}-3^{2}}+\frac{z^{2}}{4^{2}-5^{2}}+\frac{\omega^{2}}{4^{2}-7^{2}}=1, \\
\frac{x^{2}}{6^{2}-1^{2}}+\frac{y^{2}}{6^{2}-3^{2}... | 36 |
Given that the probability mass function of the random variable $\xi$ is $P(\xi=k)= \frac{c}{k+1}$, where $k=0$, $1$, $2$, $3$, find the value of $c$. | c= \frac{12}{25} |
Calculate the function value of $f(x) = x^5 - 2x^4 + 3x^3 - 7x^2 + 6x - 3$ at $x = 2$ using the Horner's method. What is the result of the third step, $v_3$?
A: 3
B: -1
C: 6
D: 0 | B |
6. Given that for all $x \in[-1,1]$, we have $x^{3}-a x+1 \geqslant 0$.
Then the range of the real number $a$ is $\qquad$ | \left[0, \frac{3 \sqrt[3]{2}}{2}\right] |
Which one statisfies $n^{29} \equiv 7 \pmod {65}$?
$ \textbf{(A)}\ 37 \qquad \textbf{(B)}\ 39 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 46 \qquad \textbf{(E)}\ 55$ | 37 |
8. Let the rational number $r=\frac{p}{q} \in(0,1)$, where $p, q$ are coprime positive integers, and $pq$ divides 3600. The number of such rational numbers $r$ is $\qquad$ . | 112 |
We write 1s at the two endpoints of a segment. Then, step by step, we halve the resulting segments and write the sum of the numbers at the two ends of the segments to the midpoint. Repeating the halving of the segments one million times, what will be the sum of the numbers written? | 3^{10^{6}}+1 |
12 Let $E$ be the intersection of the diagonals $AC$ and $BD$ of the cyclic quadrilateral $ABCD$. Given: $AC=BC, AD=5, BE=12, DE=3$, find $\angle BCD$. | 90 |
Given that the function $f(x)=a\sin x-\cos x$ has a symmetric center at $(\frac{\pi}{4},0)$, if the vertical coordinates of the function $f(x)$ remain unchanged, the horizontal coordinates are reduced to $\frac{1}{3}$ of the original, and the resulting graph is shifted to the right by $\frac{\pi}{12}$ units to obtain t... | D |
5. Let $a$ and $b$ be skew lines with an angle of $30^{\circ}$ between them. Then the planes $\alpha$ and $\beta$ that satisfy the condition “$a \subseteq \alpha, b \subseteq \beta$, and $\alpha \perp \beta$” ( ).
(A) do not exist
(B) exist and are unique
(C) exist and there are exactly two pairs
(D) exist and there ar... | D |
Calculate the result of $\sqrt{2} \times \sqrt{6}$. | 2\sqrt{3} |
Which of the following statements is correct? ( )
A: If a coin is flipped 10 times and lands heads up 3 times, then the probability of landing heads up is $$\frac {3}{10}$$.
B: If the probability of winning a certain lottery is $$\frac {1}{1000}$$, then buying 1000 lottery tickets guarantees a win.
C: If the weather f... | D |
7. If the union of two sets $A$ and $B$ has two elements, and $f(A)$ denotes the number of elements in $A$, $f(B)$ denotes the number of elements in $B$, then the number of such pairs $(f(A), f(B))$ is $\qquad$ pairs. | 6 |
Let $A=(2,3)$, $B=(4,k)$, and $AB$ is parallel to the x-axis. Find $k$. | 3 |
Given $x\log_{3}2=1$, then $4^{x}-2^{x}=$ ______. | 6 |
The equation of the trajectory of a moving point that is equidistant from the two coordinate axes is ( )
A: y=
B: y=|x|
C: $y^2=x^2$
D: $x^2+y^2=0$ | \text{C} |
In the equation below, $A$ and $B$ are consecutive positive integers, and $A$, $B$, and $A+B$ represent number bases: \[132_A+43_B=69_{A+B}.\]
What is $A+B$?
$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 17$ | 13 |
[ [Assist
In triangle $ABC$, the height $BD$ is 11.2 and the height $AE$ is 12. Point $E$ lies on side $BC$ and $BE$ : $EC=5: 9$. Find the side $AC$.
# | 15 |
Given the sets $A=\{x|x>-2\}$ and $B=\{x|1-x>0\}$, then $A\cap B=$ ? | (-2,1) |
1. A professor from the Department of Mathematical Modeling at FEFU last academic year gave 6480 twos, thereby overachieving the commitments taken at the beginning of the year. In the next academic year, the number of professors increased by 3, and each of them started giving more twos. As a result, a new record was se... | 5 |
Problem 7.1.1. Let $A, B, C, D, E, F, G, H$ - be different digits from 0 to 7 - satisfy the equation
$$
\overline{A B C}+\overline{D E}=\overline{F G H}
$$
Find $\overline{D E}$, if $\overline{A B C}=146$.
(The notation $\overline{A B C}$ represents a three-digit number consisting of digits $A, B, C$, similarly cons... | 57 |
Given that the graph of a linear function passes through the points $\left(1,-1\right)$ and $\left(-2,8\right)$. Find the analytical expression of $y$ in terms of $x$, and calculate the value of $y$ when $x=-10$. | 32 |
Let 7 be the first term and the common difference of an arithmetic sequence both be non-negative integers, the number of terms is no less than 3, and the sum of all terms is $97^{2}$. How many such sequences are there? | 4 |
1. Let $-1<a<0, \theta=\arcsin a$, then the solution set of the inequality $\sin x<a$ is
A. $\{x \mid 2 n \pi+\theta<x<(2 n+1) \pi-\theta, n \in \mathbf{Z}\}$
B. $\{x \mid 2 n \pi-\theta<x<(2 n+1) \pi+\theta, n \in \mathbf{Z}\}$
C. $\{x \mid(2 n-1) \pi+\theta<x<2 n \pi-\theta, n \in \mathbf{Z}\}$
D. $\{x \mid(2 n-1) \p... | D |
Calculate the sum
$$
1+11+111+1111+\cdots+\underbrace{1111 \ldots 11}_{n \text { ones }}
$$ | \frac{10}{81}(10^{n}-1)-\frac{n}{9} |
## Task 1 - 200821
Mr. Schäfer had bought two dogs. However, he had to sell them soon. For each dog, he received 180 Marks.
As Mr. Schäfer found out, with one of the dogs, he had made a profit of $20 \%$ of its original purchase price, while he sold the other dog at a loss of $20 \%$ of its original purchase price.
... | 15 |
G8.2 From 1 to 121 , there are $b$ numbers which are not divisible by 5 nor 7 . Find $b$. | 83 |
Let $M=\{x|x^{2}+x-6=0\}$, $N=\{x|ax+2=0, a\in R\}$ and $M\cap N=N$. Then the real number $a$ can be ( )
A: $-1$
B: $1$
C: $\frac{2}{3}$
D: $0$ | ACD |
4. Problem: Determine all prime numbers $p$ such that $p^{2}-6$ and $p^{2}+6$ are both prime numbers. | 5 |
## Task 5 - 301245
Determine a polynomial
$$
f(x)=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{n} x^{n} \quad\left(a_{0}, a_{1}, \ldots, a_{n} \text { real } ; a_{n} \neq 0\right)
$$
that satisfies the conditions
$$
f(-4)=0, f(-3)=1, f(-2)=0, f(-1)=-1, f(0)=0, f(1)=1, f(2)=0, f(3)=-1, f(4)=0
$$
and has the lowest possible ... | -\frac{1}{630}x^{7}+\frac{1}{18}x^{5}-\frac{26}{45}x^{3}+\frac{32}{21}x |
Example 8 Choose $k$ numbers from $1,2, \cdots, 2004$, such that among the chosen $k$ numbers, there are definitely three numbers that can form the side lengths of a triangle (here it is required that the three side lengths of the triangle are all different). Then the minimum value of $k$ that satisfies the condition i... | 17 |
Problem 9. At a children's party, the children brought treats from home: cookies, candies, and waffles. The class teacher noticed that among those who brought cookies, exactly $1 / 3$ also brought candies, and exactly $1 / 4$ also brought waffles; among those who brought candies, exactly $1 / 7$ also brought cookies, a... | \frac{7}{12} |
8. In the right trapezoid $A B C D$, it is known that $A D \perp A B$, $A B / / D C, A B=4, A D=D C=2$. Let $N$ be the midpoint of side $D C$, and $M$ be a moving point within or on the boundary of trapezoid $A B C D$. Then the maximum value of $\overrightarrow{A M} \cdot \overrightarrow{A N}$ is $\qquad$ | 6 |
In the arithmetic sequence $\{a_n\}$, $a_2=-1$, $2a_1+a_3=-1$.
(Ⅰ) Find the general formula for the sequence $\{a_n\}$.
(Ⅱ) Let $S_n$ be the sum of the first $n$ terms of $\{a_n\}$. If $S_k=-99$, find $k$. | k=11 |
117. Two Ferries. Two ferries travel between two opposite banks of a river at constant speeds. Upon reaching a bank, each immediately starts moving in the opposite direction. The ferries departed from opposite banks simultaneously, met for the first time 700 meters from one of the banks, continued to their respective b... | 1700 |
A parabola $C\_1:y= \frac{1}{2p}x^{2}(p > 0)$ has its focus connected to the right focus of a hyperbola $C\_2: \frac{x^{2}}{8}-y^{2}=1$ by a line intersecting $C\_1$ at the first quadrant point $M$. If the tangent line of $C\_1$ at point $M$ is parallel to one asymptote of $C\_2$, then $p=$ ( ).
A: $ \frac{7 \sqrt{2}}... | D |
For given integer $n \geq 3$, set $S =\{p_1, p_2, \cdots, p_m\}$ consists of permutations $p_i$ of $(1, 2, \cdots, n)$. Suppose that among every three distinct numbers in $\{1, 2, \cdots, n\}$, one of these number does not lie in between the other two numbers in every permutations $p_i$ ($1 \leq i \leq m$). (For exampl... | 2^{n-1} |
Given $0 < \alpha < \frac{\pi}{2}$, and $\sin\alpha = \frac{4}{5}$.
(Ⅰ) Find the value of $\tan\alpha$;
(Ⅱ) Calculate the value of $\frac{\sin(\alpha+\pi)-2\cos(\frac{\pi}{2}+\alpha)}{-\sin(-\alpha)+\cos(\pi+\alpha)}$. | 4 |
The angle that has the same terminal side as \\(- \dfrac {\pi}{3}\\) is \\((\quad)\)
A: \\( \dfrac {5\pi}{6}\\)
B: \\( \dfrac {\pi}{3}\\)
C: \\( \dfrac {11\pi}{6}\\)
D: \\( \dfrac {5\pi}{3}\\) | D |
Example 1 Let $\left\{a_{n}\right\}$ be an arithmetic sequence with the first term $a_{1}$ and common difference $d$. The sequence $\left\{a_{n}\right\}$ is grouped according to the rule of $3k$ numbers in the $k$-th group as follows:
$$
\left(a_{1}, a_{2}, a_{3}\right),\left(a_{4}, a_{5}, \cdots, a_{9}\right),\left(a_... | k a_{1}+\frac{3 k\left(k^{2}+2 k-1\right) d}{2} |
Among the four numbers $-1$, $0$, $1$, and $2$, the smallest number is:
A: $0$
B: $-1$
C: $1$
D: $2$ | B |
3. Let $E_{i} F_{i}(i=1,2,3,4)$ be line segments parallel to the upper and lower bases of a trapezoid and whose endpoints lie on the legs of the trapezoid. If $E_{1} F_{1}$ divides the trapezoid into two equal areas, $E_{2} F_{2}$ is the midline of the trapezoid, $E_{3} F_{3}$ divides the trapezoid into two similar fig... | B |
57. Cut a line segment of length 14 into two segments, such that the two resulting segments and a line segment of length 10 can form a right triangle. Then the area of this right triangle is $\qquad$ - | 24 |
The focal length of the ellipse $\dfrac {x^{2}}{5}+ \dfrac {y^{2}}{4}=1$ is $(\ \ \ )$
A: $1$
B: $2$
C: $4$
D: $8$ | \text{B} |
Example 7. Find $\int x^{2}(2 x-1)^{31} d x$. | \frac{x^{2}}{64}(2x-1)^{32}-\frac{x}{2112}(2x-1)^{33}+\frac{(2x-1)^{34}}{143616}+C |
Given the sequence $\{a_n\}$ satisfies $a_{1}=1, a_{2}=4, a_{n+2}+2a_{n}=3a_{n+1}$ for $n\in \mathbb{N}^{*}$:
1. Prove that the sequence $\{a_{n+1}-a_n\}$ is a geometric sequence and find the general term for $\{a_n\}$.
2. Let $S_n$ denote the sum of the first $n$ terms of $\{a_n\}$. Find the smallest integer $n$ such ... | n = 4 |
Task 1. Given the fraction $\frac{57}{71}$. What number needs to be subtracted from the numerator and the same number added to the denominator so that the value of the fraction after simplification is $\frac{1}{3}$? | 25 |
4 male students and 5 female students are lined up in a row, where the number of different arrangements with alternating male and female students is ( )
A: 126
B: 3024
C: 15120
D: 2880 | \text{D} |
A student's family runs a small store, and to study the impact of temperature on the sales of hot drinks, they have established a linear relationship between the number of hot drinks sold and the temperature of the day. The regression equation is $$\hat y=-2.35x+155.47$$. If the temperature one day is 4°C, then the app... | B |
Which of the following statements is correct? ( )
A: The entropy of a spontaneous reaction always increases, and the entropy of a non-spontaneous reaction always decreases.
B: An endothermic reaction with an increase in entropy can always proceed spontaneously.
C: If the reaction \\(\rm{2Mg(s)+CO_{2}(g)=C(s)+2MgO(s)... | \text{D} |
In the pyramid $A B C D$, the angle $A B C$ is equal to $\alpha$, the orthogonal projection of point $D$ onto the plane $A B C$ is point $B$. Find the angle between the planes $A B D$ and $C B D$. | \alpha |
6. Given the lengths of the diagonals of three faces of a rectangular prism are $6, 4, x$, then the range of values for $x$ is $\qquad$ | 2\sqrt{5}<x<2\sqrt{13} |
Problem 5. A group of tourists is dividing cookies. If they evenly distribute two identical packs, one extra cookie will remain. But if they evenly distribute three such packs, 13 extra cookies will remain. How many tourists are in the group? [7 points] (I.V. Raskina) | 23 |
Convert $110101_{(2)}$ to a decimal number. | 53 |
2. Given $5 \sin 2 \alpha=\sin 2^{\circ}$, then the value of $\frac{\tan \left(\alpha+1^{\circ}\right)}{\tan \left(\alpha-1^{\circ}\right)}$ is | -\frac{3}{2} |
Exercise 6. Find all strictly positive integers $p, q$ such that
$$
\mathrm{p} 2^{\mathrm{q}}=\mathrm{q} 2^{\mathrm{p}}
$$ | (p,p)forp\in\mathbb{N}^{*}(2,1),(1,2) |
Given a geometric sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and it is known that $a_5 = 4S_4 + 3$, $a_6 = 4S_5 + 3$. Find the common ratio $q$ of this sequence. | 5 |
Due to the outbreak of H1N1 flu, the epidemic prevention station conducted a health survey on students by using stratified sampling. The school has a total of 1,600 students, and a sample of 200 students was taken. It is known that 20 fewer females were sampled compared to males. The number of female students in the sc... | 720 |
172. The sum of positive numbers $x_{1}, x_{2}, \ldots, x_{n}$ is 1. Let $s$ be the largest of the numbers
$$
\frac{x_{1}}{1+x_{1}}, \frac{x_{2}}{1+x_{1}+x_{2}}, \ldots, \frac{x_{n}}{1+x_{1}+x_{2}+\ldots+x_{n}}
$$
Find the smallest possible value of $s$. For which values of $x_{1}, x_{2}, \ldots, x_{n}$ is it achieve... | 1-2^{-1/n} |
Given that the sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n$, and for any positive integer $n$, $S\_n = 3a\_n - 5n$.
1. Find the first term of the sequence $\{a\_n\}$.
2. Prove that the sequence $\{a\_n + 5\}$ is a geometric sequence and find the general term of the sequence $\{a\_n\}$.
3. If the seq... | \frac{88}{45} |
Given the function $f(x)=\log_{2}\frac{x}{2}\cdot\log_{2}\frac{x}{8}$, if $f(x_{1})=f(x_{2})$ (where $x_{1}\neq x_{2}$), then the minimum value of $\frac{13}{x_{1}}+\frac{16}{x_{2}}$ is:
A: $\sqrt{13}$
B: $2\sqrt{13}$
C: $4$
D: $8$ | \text{B} |
1. Given $x, y, z \in \mathbf{R}^{+}$ and satisfy $x y z(x+y+z)=1$. Then the minimum value of $(x+y)(y+z)$ is ( ).
(A) 2
(B) 3
(C) $\frac{4 \sqrt{3}}{3}$
(D) 1 | A |
If $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ are unit vectors, and $\overrightarrow{a} \cdot \overrightarrow{b} = 0$, $(\overrightarrow{a} - \overrightarrow{c}) \cdot (\overrightarrow{b} - \overrightarrow{c}) \leq 0$, then the maximum value of $|\overrightarrow{a} + \overrightarrow{b} - \over... | 1 |
Solve the following system of equations:
$$
\begin{gathered}
\left(x^{2}+11\right) \sqrt{21+y^{2}}=180, \quad\left(y^{2}+21\right) \sqrt{z^{2}-33}=100 \\
\left(z^{2}-33\right) \sqrt{11+x^{2}}=96
\end{gathered}
$$ | \begin{aligned}&x,y,+5,+2,+7;\quad+5,+2,-7;\quad+5,-2,+7;\quad+5,-2,-7\\&-5,+2,+7;\quad-5,+2,-7;\quad-5,-2,+7;\quad-5,-2,-7\end{aligned} |
The domain of the odd function $f(x)$ is $\mathbb{R}$. If $f(x+3)$ is an even function, and $f(1)=1$, then $f(6)+f(11)=$ \_\_\_\_.
A: $-2$
B: $-1$
C: $0$
D: $1$ | \text{B} |
Given a point P(-4, 3) on the terminal side of angle x, find the value of $$\frac {\cos( \frac {\pi}{2}+x)\sin(-\pi-x)}{\cos( \frac {\pi}{2}-x)\sin( \frac {9\pi}{2}+x)}$$. | \frac {3}{4} |
8. Let $a>0, b>0$. Then the inequality that does not always hold is ( ).
(A) $\frac{2}{\frac{1}{a}+\frac{1}{b}} \geqslant \sqrt{a b}$
(B) $\frac{1}{a}+\frac{1}{b} \geqslant \frac{4}{a+b}$
(C) $\sqrt{|a-b|} \geqslant \sqrt{a}-\sqrt{b}$
(D) $a^{2}+b^{2}+1>a+b$ | A |
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and they satisfy the equation $2acosC + ccosA = b$.
(I) Find the measure of angle $C$;
(II) Find the maximum value of $sinAcosB + sinB$. | \frac{5}{4} |
The square $A B C D$ drawn in the figure below has a side of $3 \mathrm{~cm}$.
The points $P$ and $Q$ can be moved along the segments $A B$ and $A D$ respectively such that the length of th... | \frac{27}{4}\mathrm{~}^{2} |
Let $A B C D E F$ be a regular hexagon. The points $\mathrm{M}$ and $\mathrm{N}$ are internal points of the sides $\mathrm{DE}$ and $\mathrm{DC}$ respectively, such that $\angle A M N=90^{\circ}$ and $A N=\sqrt{2} \cdot C M$. Find the measure of the angle $\angle B A M$. | 75^{\circ} |
6. For a regular tetrahedron $P-ABC$ with all edge lengths equal to $1$, $L, M, N$ are the midpoints of edges $PA, PB, PC$ respectively. The area of the section formed by the plane $LMN$ cutting through the circumsphere of the tetrahedron is $\qquad$ | \frac{\pi}{3} |
If the lengths of the sides of a triangle are all different integers and the longest side is $6$, then the number of triangles that satisfy this condition is:
A: $4$
B: $5$
C: $6$
D: $7$ | A |
2. In $\triangle A B C$, $A B=A C, A D$ is the altitude on side $B C$, the midpoint of $A D$ is $M$, and the extension of $C M$ intersects $A B$ at $K$. Prove: $A B=3 A K$.
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This translation maintains the original format and structure of the text, including the use of mathematical symbols and notation. | AB=3AK |
Let $n=9867$. If you calculated $n^{3}-n^{2}$, what would be the unit digit found?
(a) 0
(b) 2
(c) 4
(d) 6
(e) 8 | 4 |
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