problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
2. Given $S=1^{2}-2^{2}+3^{2}-1^{2}+\cdots+99^{2}-$ $100^{2}+101^{2}$. Then the remainder when $S$ is divided by 103 is $\qquad$ | 1 |
# Task 1. Maximum 15 points
In a small town, three main straight highways converge pairwise at points A, B, and C. The town has a circular ring road that also passes through points A, B, and C and is represented by a circle. The lengths of the highways inside the circle are 7, 8, and 9 kilometers. At the center of thi... | 1024.7 |
If $a > 1$, let $f(x) = a^x + x - 4$ and its zero point is $x\_1$, and $g(x) = \log\_a x + x - 4$ and its zero point is $x\_2$. What is the range of $\frac{1}{x\_1} + \frac{1}{x\_2}$?
A: $[3.5, +\infty)$
B: $[1, +\infty)$
C: $[4, +\infty)$
D: $[4.5, +\infty)$ | B |
1. Given a regular tetrahedron \(ABCD\) with the midpoints of the opposite edges \(AB, CD\) being \(M, N\) respectively. Find the angle between \(MN\) and \(BC\). | 45^{\circ} |
Given three numbers $a=0.7^{6}$, $b=6^{0.7}$, and $c=\log _{0.7}6$, determine their order of magnitude.
A: $a < c < b$
B: $a < b < c$
C: $c < b < a$
D: $c < a < b$ | D |
In the geometric sequence $\{a\_n\}$, $a\_n > 0$, and $a\_3a\_5 + a\_2a\_10 + 2a\_4a\_6 = 100$, find the value of $a\_4 + a\_6$. | 10 |
$19422 *$ Find the largest integer $x$ that makes $4^{27}+4^{1000}+4^{x}$ a perfect square. | 1972 |
2. Given $a=\frac{1}{2+\sqrt{3}}$. Then the value of $\sqrt{a^{2}-2 a+1}$ is ( ).
(A) $\sqrt{3}-1$
(B) $1-\sqrt{3}$
(C) $\frac{1}{1+\sqrt{3}}$
(D) $\frac{1}{1-\sqrt{3}}$ | A |
Given that $a$, $b$, $c$, and $d$ are distinct positive integers, and $abcd = 441$, then the value of $a+b+c+d$ is ( )
A: 30
B: 32
C: 31
D: 36 | B |
182. Perform the operations: a) $(2+3 i)^{2} ;$ b) $(3-5 i)^{2}$; c) $(5+$ $+3 i)^{3}$ | -5+12i,-16-30i,-10+198i |
Calculate: $3+(-2)=$____. | 1 |
13. (14 points) To welcome the "World Women's Conference", a school has formed a ceremonial guard consisting of 48 boys and 32 girls. They need to be arranged in $n$ rows $(n>1)$, with the number of boys and girls in each row being equal. How many different ways can they be arranged? How many boys and girls are there i... | 4 |
4. Find a fraction with a denominator of 20, whose value is between $-\frac{5}{13}$ and $-\frac{4}{13}$. | -\frac{7}{20} |
Given that the sequence $\{a\_n\}$ is a geometric sequence, and the sum of its first $n$ terms $S\_n=3^{n-1}+t$, determine the value of $t$ $(\ \ \ )$
A: $-1$
B: $-3$
C: $- \frac {1}{3}$
D: $1$ | C |
Example 8. Calculate the volume of the body bounded above by the surface $z=x y^{2}$, and below by the rectangle $0 \leqslant x \leqslant 1,0 \leqslant y \leqslant 2$. | \frac{4}{3} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ have an angle of $120^{\circ}$ between them, $|\overrightarrow{a}|=1$, and $|\overrightarrow{b}|=3$, then $|\overrightarrow{a}-\overrightarrow{b}|=$ ()
A: $2\sqrt{3}$
B: $\sqrt{15}$
C: $4$
D: $\sqrt{13}$ | D |
3.3.5 ** The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+a_{n}, n \in \mathbf{N}^{*}, b_{n}=\frac{1}{1+a_{n}}$,
$$
S_{n}=b_{1}+b_{2}+\cdots+b_{n}, P_{n}=b_{1} b_{2} \cdots b_{n},
$$
Find the value of $2 P_{n}+S_{n}$. | 2 |
Simplify $\sqrt{\frac{1}{3}}$ to ______. | \frac{\sqrt{3}}{3} |
9.092. $\log _{0.5}^{2} x+\log _{0.5} x-2 \leq 0$. | x\in[0.5;4] |
Which of the following statements are correct?
A: When tossing a coin $1000$ times, there will definitely be $500$ times "heads up."
B: If the variance of data set A is $0.03$ and the variance of data set B is $0.1, then data set A is more stable than data set B.
C: The median of a data set $1, 2, 5, 5, 5, 3, 3$ is ... | BC |
Let $S$ be a set of real numbers with mean $M$. If the means of the sets $S\cup \{15\}$ and $S\cup \{15,1\}$ are $M + 2$ and $M + 1$, respectively, then how many elements does $S$ have? | 4 |
Example 24 (2004 Hunan Province Competition Question) The schematic diagram of a computer device is shown in Figure 19, $J_{1} 、 J_{2}$ represent data inputs, $C$ is the output for the calculation result. The calculation process involves inputting natural numbers $m$ and $n$ from $J_{1}$ and $J_{2}$ respectively, and a... | 2^{2001}+16 |
Compute: $\dfrac{\cos 10^{\circ} + \sqrt{3}\sin 10^{\circ}}{\sqrt{1-\cos 80^{\circ}}}=$ \_\_\_\_\_\_. | \sqrt{2} |
Computer viruses spread quickly. If a computer is infected, after two rounds of infection, there will be 121 computers infected. If on average one computer infects $x$ computers in each round of infection, then $x=\_\_\_\_\_\_$. | 10 |
The absolute value of $2-\sqrt{5}$ is ____. | \sqrt{5}-2 |
9. If $0<a<\sqrt{3} \sin \theta, \theta \in\left[\frac{\pi}{4}, \frac{5 \pi}{6}\right]$, then the minimum value of $f(a, \theta)=\sin ^{3} \theta+\frac{4}{3 a \sin ^{2} \theta-a^{3}}$ is $\qquad$ | 3 |
7. 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} |
If one pair of corresponding sides of two congruent angles are parallel, then:
A. The other pair of corresponding sides is also parallel
B. The other pair of corresponding sides is not parallel
C. The other pair of corresponding sides cannot be perpendicular
D. None of the above is correct | D |
In the arithmetic sequence $\{a_n\}$, if $a_9= \frac{1}{2}a_{12}+6$, then the sum of the first $11$ terms of the sequence $S_{11}=$ ()
A: $24$
B: $48$
C: $12$
D: $132$ | D |
Task 1. (5 points) Calculate
$$
\left(\frac{10001}{20232023}-\frac{10001}{20222022}\right) \cdot 4090506+\sqrt{4092529}
$$ | 2022 |
Define: $max\left\{{a,b}\right\}=\left\{{\begin{array}{l}{a(a≥b)}\\{b(a＜b)}\end{array}}\right.$, if the function $y=\max (-x-1$,$x^{2}-2x-3)$, then the minimum value of this function is ( )
A: $-1$
B: $0$
C: $-3$
D: $3$ | C |
The function $f(x)$ is an odd function defined on the real numbers $\mathbb{R}$, and when $x \in (0, +\infty)$, $f(x) = 2^x$. Calculate $f(-1)$. | -2 |
10. A school program will randomly start between 8:30AM and 9:30AM and will randomly end between 7:00PM and 9:00PM. What is the probability that the program lasts for at least 11 hours and starts before 9:00AM? | \frac{5}{16} |
Given the sample 9, 10, 11, x, y has an average of 10 and a variance of 4, find the value of $x \cdot y$. | 91 |
The equation of the axis of symmetry of the graph of the function $f(x)=\sin \left(x- \frac {\pi}{3}\right)$ is ( )
A: $x= \frac {5\pi}{6}+k\pi$, $k\in\mathbb{Z}$
B: $x= \frac {5\pi}{6}+2k\pi$, $k\in\mathbb{Z}$
C: $x=- \frac {\pi}{6}+2k\pi$, $k\in\mathbb{Z}$
D: $x= \frac {\pi}{3}+k\pi$, $k\in\mathbb{Z}$ | A |
50. How many five-digit numbers are there that are divisible by 5 and do not have any repeated digits in their representation? | 5712 |
I. Fill in the Blanks (8 points each, total 64 points)
1. Among the positive integers less than 20, choose three different numbers such that their sum is divisible by 3. The number of different ways to choose these numbers is $\qquad$. | 327 |
For some positive integer $n$, there exists $n$ different positive integers $a_1, a_2, ..., a_n$ such that
$(1)$ $a_1=1, a_n=2000$
$(2)$ $\forall i\in \mathbb{Z}$ $s.t.$ $2\le i\le n, a_i -a_{i-1}\in \{-3,5\}$
Determine the maximum value of n. | 1996 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $c^2 = a^2 + b^2 + ab$.
$(1)$ Find the measure of angle $C$;
$(2)$ If $\sin A = \frac{2}{3}$ and $b=2$, find the area of triangle $ABC$. | \frac{12\sqrt{5} + 8\sqrt{3}}{11} |
Given that a hyperbola passes through the point $(2,3)$ and one of its asymptotes is given by the equation $y= \sqrt {3}x$, find the standard equation of the hyperbola. | x^{2} - \frac{y^{2}}{3} = 1 |
8.4. It is known that $a^{2}+2=b^{4}, b^{2}+2=c^{4}, c^{2}+2=a^{4}$. What is the value of the product $\left(a^{2}-1\right)\left(b^{2}-1\right)\left(c^{2}-1\right)$? Find all possible values and prove that there are no others. | 1 |
Given the sets $M=\{x\mid x \geq -1\}$ and $N=\{x\mid 2-x^2 \geq 0\}$, then the union of $M$ and $N$, denoted $M \cup N$, is:
A: $[-\sqrt{2}, +\infty)$
B: $[-1, \sqrt{2}]$
C: $[-1, +\infty)$
D: $(-\infty, -\sqrt{2}] \cup [-1, +\infty)$ | A |
Petya bought a "Constructor" set, which contained 100 sticks of different lengths. The instructions for the "Constructor" state that any three sticks from the set can form a triangle. Petya decided to test this statement by forming triangles with the sticks. The sticks are arranged in the constructor in ascending order... | 1 |
2. The working day at the enterprise lasted 8 hours. During this time, the labor productivity was as planned for the first six hours, and then it decreased by $25 \%$. The director (in agreement with the labor collective) extended the shift by one hour. As a result, it turned out that again the first six hours were wor... | 8 |
The unit-sided square $ABCD$ has its midpoint $F$ on side $AB$. A circle centered at $F$ passing through $C$ intersects the half-line $AB$ at $G$. The circle with diameter $AG$ intersects the segment $BC$ at $H$. Let the point $E$ be the closer quarter point of the diagonal $AC$ to $C$, and the lengths of segments $BE$... | 6.9 |
8. As shown in Figure 3, in quadrilateral $A B C D$, $A D / / B C, A B=A C$, a perpendicular line is drawn from point $D$ to $B C$, with the foot of the perpendicular being $F, D F$ intersects $A C$ at point $E$. If $B E=2 D F$, and $B E$ bisects $\angle A B C$, then $\angle B A D=$ $\qquad$ . (China) | 144 |
The derivative of the function $y=3\sin x-4\cos x$ is $({\ })$
A: $3\cos x+4\sin x$
B: $3\cos x-4\sin x$
C: $-3\cos x+4\sin x$
D: $-3\cos x-4\sin x$ | A |
For an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), the distances from any point on the ellipse to the two foci are $d_1$ and $d_2$, respectively, and the focal distance is $2c$. If $d_1$, $2c$, and $d_2$ form an arithmetic sequence, then the eccentricity of the ellipse is ( ). | \frac{1}{2} |
7. From the set $\{1,2, \cdots, 100\}$, randomly select elements $m$ and $n$ (which can be the same). Then the probability that the unit digit of $2^{m}+3^{n}$ is 3 is . $\qquad$ | \frac{3}{16} |
5. Garden workers need to dig holes at equal distances along the edge of a circular flower bed with a circumference of 300 meters. They started digging holes every 3 meters along the edge of the flower bed. When they had dug 30 holes, they received instructions from their superiors to change to planting trees every 5 m... | 54 |
Question 4 Given that $x, y, z$ are all positive numbers, and
$$
x y z=\frac{1}{2}, x^{2}+y^{2}+z^{2} \leqslant 2 \text{. }
$$
Then the maximum value of $x^{4}+y^{4}+z^{4}$ is | \frac{3}{2} |
Given that $\overrightarrow{a}=(-3,2,5)$ and $\overrightarrow{b}=(1,m,3)$, if $\overrightarrow{a} \perp \overrightarrow{b}$, then the constant $m=$ ( ).
A: $-6$
B: $6$
C: $-9$
D: $9$ | A |
4. A password lock's password setting involves assigning one of the two numbers, 0 or 1, to each vertex of a regular $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$, and coloring each vertex with one of two colors, red or blue, such that for any two adjacent vertices, at least one of the number or color is the same. How m... | a_{n}=3^{n}+2+(-1)^{n} |
1. Problem: Find all integers $a, b$ such that
$$
a^{2}+b=b^{2022} \text {. }
$$ | (0,0)(0,1) |
Given that the chord of the parabola $C: x^{2}=4y$ cut by the line $l$ passing through the focus $F$ of $C$ has a length of $8$, then the distance from the origin $O$ to $l$ is ( )
A: $\frac{{\sqrt{6}}}{3}$
B: $\frac{{\sqrt{3}}}{3}$
C: $\frac{{\sqrt{2}}}{2}$
D: $\frac{1}{2}$ | C |
84. As shown in the figure, Jia and Yi are two cylindrical containers, with the ratio of their base areas being $2: 3$. In the Jia container, there is an iron ball with a volume of 30 cubic centimeters, at which point the water levels in the two containers differ by $x$ centimeters; if the iron ball is moved from the J... | 1 |
2. Calculate the value of the expression $\frac{a^{2}}{a b+b^{2}}$ if the ratio of the numbers $a$ and $b$ is $2: 5$. | \frac{4}{35} |
Given two arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$ with the sum of their first $n$ terms being $S\_n$ and $T\_n$ respectively. If $\frac{S\_n}{T\_n} = \frac{7n + 1}{4n + 27}$ for all positive integers $n$, find the value of $\frac{a\_7}{b\_7}$. | \frac{92}{79} |
When negating an existential proposition, the existential quantifier becomes a universal quantifier. ( )
A: Correct
B: Incorrect
C:
D: | \text{A} |
(2) Given that the solution set of the inequality $a x^{2}+b x+c>0$ with respect to $x$ is $(-3,4)$, find the solution set of the inequality $c x^{2}-b x+a>0$ with respect to $x$. | (-\infty,-\frac{1}{4})\cup(\frac{1}{3},+\infty) |
Given that the lengths of all three sides of a triangle satisfy the equation $x^{2}-6x+8=0$, the perimeter of this triangle is ( )
A: $8$
B: $10$ or $8$
C: $10$
D: $6$ or $12$ or $10$ | D |
Problem 5.5. On some trees in the magical forest, coins grow. The number of trees that do not grow any coins at all is twice as many as the trees that grow three coins. On three trees, two coins grow, on four trees - four coins, and no tree grows more than four coins. By how much is the total number of coins in the mag... | 15 |
Given vectors $\overrightarrow{m}=(\sin A,\sin B)$, $\overrightarrow{n}=(\cos B,\cos A)$, and $\overrightarrow{m} \cdot \overrightarrow{n}=\sin 2C$, where $A$, $B$, and $C$ are the angles opposite to sides $a$, $b$, and $c$, respectively, in $\triangle ABC$.
(I) Find the measure of angle $C$;
(II) If $\sin A,\sin C,\... | c=6 |
381. Given the curve $y=-x^{2}+4$. Draw a tangent to it at the point where the abscissa is $x=-1$. | 2x+5 |
Given the complex number $z= \frac {-3+i}{2+i}$, determine the quadrant in which the point corresponding to its conjugate complex number lies.
A: Quadrant I
B: Quadrant II
C: Quadrant III
D: Quadrant IV | C |
The set $\{x|8 < x < 12, x \in \mathbb{N}\}$ can be expressed by listing method as __________. | \{9, 10, 11\} |
Given a set of data: $3$, $4$, $5$, $5$, $6$, the mode of this set of data is ______. | 5 |
Express the solution set $\{x|x\leq1\}$ as an interval.
A: $(-\infty, 1)$
B: $(-\infty, 1]$
C: $[1, +\infty)$
D: $[1, +\infty]$ | B |
Given the function $f(x)=\ln x+(a+1)x+1$, $a\in \mathbb{R}$, and $g(x)=xe^{x}$.
$(1)$ If the slope of the tangent line to the curve $f(x)$ at the point $(1,f(1))$ is $3$, find the value of $a$.
$(2)$ For $x\geqslant -2$, if the function $y=g(x)-m+2$ has two distinct zeros, find the range of values for $m$.
$(3)$ ... | a \in (-\infty, 0] |
9. (16 points) Let the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ have its right focus at $F$, and let $P$ be a point outside the hyperbola. Draw two tangents from point $P$ to the hyperbola, touching it at points $A$ and $B$. If $A B \perp P F$, find the locus of point $P$. | \frac{^{2}}{\sqrt{^{2}+b^{2}}} |
Given point $P(3m+6, m-3)$, find the coordinates of point $P$ according to the following conditions:$(1)$ Point $P$ lies on the angle bisector in the first and third quadrants;$(2)$ The ordinate of point $P$ is $5$ greater than the abscissa;$(3)$ Point $P$ lies on the line passing through point $A(3, -2)$ and parallel ... | (3, -4) |
5. In the Cartesian coordinate system, grid points are numbered as follows: $(0,0)$ is No. 1, $(1,0)$ is No. 2, $(1,1)$ is No. 3, $(0,1)$ is No. 4, $(0,2)$ is No. 5, $(1,2)$ is No. 6, $(2,2)$ is No. 7, $(2,1)$ is No. 8, $(2,0)$ is No. 9, $\cdots$ (as shown in Figure 7-1). Following the order of the arrows in the figure... | (44,19) |
Determine the number of integers $n$ such that $1 \leqslant n \leqslant 10^{10}$, and such that for all $k=1,2, \ldots, 10$, the integer $n$ is divisible by $k$. | 3968253 |
7) Which of the inequalities below are satisfied for every positive real value of $x$?
a) $x^{2}+\frac{1}{x^{2}} \geq 2$
(A) All three (B) only
b) $x^{2} \geq 2 x-2$
(D) only a) (E) only c).
(C) only b) and c)
c) $x^{4}-2 x^{3}+x^{2} \geq 0$. | A |
Initially $20^{\circ}$. There are 800 points on the circumference, labeled in a clockwise direction as $1,2, \cdots, 800$. They divide the circumference into 800 gaps. Now, choose one point and color it red, then follow the rule to color other points red one by one: if the $k$-th point has been colored red, then move $... | 25 |
Determine all triplets $(p, n, k)$ of strictly positive integers such that $p$ is prime and satisfy the following equation:
$$
144 + p^n = k^2
$$ | (5,2,13),(2,8,20),(3,4,15) |
On a plane, there are 3 consecutive points on a line: $A, C, B$. $AB = a$, $AC = 2x$, and the midpoint of $AC$ is denoted as $\mathrm{O}$. Draw a semicircle around $\mathrm{O}$ with radius $OA = OC = x$. The point of tangency from $B$ to the semicircle is denoted as $D$, and the intersection of this tangent and the per... | \frac{(+1)-\sqrt{1-2m}}{+4} |
Example 3 There is a stone arch bridge in the shape of a parabola. At normal water level, the width of the water surface under the bridge is 20 meters, and the distance from the arch top to the water surface is 4 meters.
(1) Based on the normal water level, when the water level rises by $h$ meters, the width of the wat... | 0.76 |
## Task 2 - 140522
Anita and Peter were supposed to get 7 bottles of sparkling water for their group from the grocery store. They had an amount of money that would have been exactly enough for this. However, they could only get soda, which cost 15 pennies more per bottle than sparkling water. For their entire amount o... | 1.40 |
Let $P(z)=z^3+az^2+bz+c$, where a, b, and c are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w+3i$, $w+9i$, and $2w-4$, where $i^2=-1$. Find $|a+b+c|$. | 136 |
If the inequality $x^2 + ax + a > 0$ holds for all $x \in \mathbb{R}$, then the range of the real number $a$ is. | 0 < a < 4 |
(IMO 1999, P4)
Find all prime numbers $p$ and all integers $x$ such that $1 \leqslant x \leqslant 2 p$ and $x^{p-1}$ divides $(p-1)^{x}+1$
## Solutions | (2,2),(3,3),(p,1) |
$31 \cdot 49$ satisfies the system of equations $\left\{\begin{array}{l}x y+x z=255 \\ x y+y z=31\end{array}\right.$, the number of positive integer solutions $(x, y, z)$ is
(A) 3.
(B) 2.
(C) 1.
(D) 0.
(Chinese Hubei Province Huanggang Junior High School Mathematics Competition, 1993) | 2 |
Task B-4.2. Determine the number of sides of a polygon for which the measure of the smallest angle is $132^{\circ}$, and each subsequent angle is $2^{\circ}$ larger. | 9 |
Given a sequence of positive terms, the sum of the first $n$ terms is denoted as $S_n$, with $a_1=1$ and $a_{n+1}^2=S_{n+1}+S_n$.
(1) Find the general formula for the sequence $\{a_n\}$.
(2) Let $b_n=a_{2n-1}\cdot 2^{a_n}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$. | (2n-3)\cdot 2^{n+1}+6 |
Let $\alpha ,\ \beta$ be the distinct positive roots of the equation of $2x=\tan x$.
Evaluate the following definite integral.
\[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \] | 0 |
4. In triangle $A B C$, the median $B M$ was drawn. It turned out that $A B=2 B M$ and $\angle M B A=40^{\circ}$. Find $\angle C B A$.
# | 110 |
2. In an acute triangle $ABC (AB \neq AC)$ with angle $\alpha$ at vertex $A$, point $E$ is the Euler center, and $P$ is a point on segment $AE$. If $\angle ABP = \angle ACP = x$, prove that $x = 90^\circ - 2\alpha$.
(Dušan Đukić) | 90-2\alpha |
The function $f(x)=x^{3}+ax^{2}+(a+6)x+1$ has extreme values on $R$. Determine the range of values for the real number $a$.
A: $-3\leqslant a\leqslant 6$
B: $a\geqslant 6$ or $a\leqslant -3$
C: $-3 6$ or $a < -3$ | D |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. Given that $a_2 + a_{12} = 18$, find $S_{13}$. | 117 |
The equation of the tangent line to the curve $y=e^{x}+2$ at point $P(0,3)$ is __________. | x-y+3=0 |
4. Given that $A D$ is the median of $\triangle A B C$ on side $B C$, and $A D < \frac{1}{2} B C$, then $\triangle A B C$ is ( ).
(A) acute triangle
(B) right triangle
(C) obtuse triangle
(D) cannot be determined | C |
Given the sequence $1 \times \frac{1}{2}$, $2 \times \frac{1}{4}$, $3 \times \frac{1}{8}$, $4 \times \frac{1}{16}$, …
(1) Find the general formula for the sequence.
(2) Find the sum of the first $n$ terms of this sequence. | 2- \frac{1}{2^{n-1}} - \frac{n}{2^n} |
Exercise 6. Let $n$ be a non-zero positive integer. Suppose there exists a set of $n$ distinct lines in the plane such that each one intersects exactly ten others. What are the possible values for $n$? | {11;12;15;20} |
10.3. In a row from left to right, all natural numbers from 1 to 37 are written in such an order that each number, starting from the second to the 37th, divides the sum of all numbers to its left: the second divides the first, the third divides the sum of the first and second, and so on, the last divides the sum of the... | 2 |
Given that $f\left(x\right)$ is an odd function on $R$, when $x \gt 0$, $f\left(x\right)=x^{2}-2x+4$. Find the value of $f\left(-3\right)$.
A: $19$
B: $7$
C: $-7$
D: $-19$ | C |
(4) Given $a>b, ab=1$, then the minimum value of $\frac{a^{2}+b^{2}}{a-b}$ is ( ).
(A) $2 \sqrt{2}$
(B) $\sqrt{2}$
(C) 2
(D) 1 | 2\sqrt{2} |
In the rectangular coordinate system $xOy$, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of circle $C$ is $\rho^2 - 2m\rho\cos\theta + 4\rho\sin\theta = 1 - 2m$.
(1) Find the rectangular coordi... | 2 + \sqrt{3} |
$6 \cdot 116$ Let $A=\{1,2,3, \cdots, 17\}$, for the mapping $f: A \rightarrow A$, denote
$$f^{[1]}(x)=f(x), f^{[k+1]}(x)=f\left(f^{[k]}(x)\right), k \in N .$$
Suppose the one-to-one mapping $f$ from $A$ to $A$ satisfies the condition: there exists a natural number $M$ such that:
(1) When $m<M, 1 \leqslant i \leqslant... | 8 |
If the graph of the power function $f(x)$ passes through point $A\left( \frac{1}{4}, \frac{1}{2}\right)$, then the equation of its tangent line at point $A$ is ( )
A: $4x+4y+1=0$
B: $4x-4y+1=0$
C: $2x-y=0$
D: $2x+y=0$ | \text{B} |
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