problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given that the focus of the parabola $y^{2}=-4x$ is $F$, and a line $l$ passing through point $F$ with a slope of $1$ intersects the parabola at points $A$ and $B$.
(1) Find the value of $|AB|$.
(2) Find the value of $\dfrac{1}{|AF|}+\dfrac{1}{|BF|}$. | 1 |
Given the ellipse $\dfrac{{{x}^{2}}}{4}+\dfrac{{{y}^{2}}}{3}=1$, find the equation of the line containing the chord with midpoint $p(-1,1)$. | 3x - 4y + 7 = 0 |
3. If a two-digit number $\bar{x}$ and a three-digit number $\overline{3 y z}$ have a product of 29,400, then, $x+y+z=$ $\qquad$ | 18 |
Given a set of points on a plane $P=\left\{P_{1}, P_{2}, \cdots, P_{1994}\right\}$, where no three points in $P$ are collinear, divide all the points in $P$ into 83 groups arbitrarily, such that each group has at least 3 points, and each point belongs to exactly one group. Then, connect any two points in the same group... | 168544 |
Daniel rode his bicycle at a constant speed. After 40 minutes, he cycled $24 \mathrm{~km}$. How far did he cycle in 30 minutes?
(A) $12 \mathrm{~km}$
(B) $14 \mathrm{~km}$
(C) $16 \mathrm{~km}$
(D) $18 \mathrm{~km}$
(E) $20 \mathrm{~km}$
## Part B: Each correct answer is worth 6. | 18\mathrm{~} |
If $α$ is an angle in the third quadrant, then the quadrant in which $\frac{α}{2}$ lies is __( )__.
A: First or second quadrant
B: Second or fourth quadrant
C: First or third quadrant
D: Second or third quadrant | B |
A [triangular](https://artofproblemsolving.com/wiki/index.php/Triangle) [array](https://artofproblemsolving.com/wiki/index.php/Array) of [squares](https://artofproblemsolving.com/wiki/index.php/Square) has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 11... | 640 |
9. The three-digit number $\overline{a b c}$ consists of three non-zero digits. The sum of the other five three-digit numbers formed by rearranging $a, b, c$ is 2017. Find $\overline{a b c}$.
三位數 $\overline{a b c}$ 由三個非零數字組成。若把 $a 、 b 、 c$ 重新排列, 則其餘五個可組成的三位數之和是 2017 。求 $\overline{a b c}$ 。 | 425 |
## Problem Statement
Calculate the area of the figure bounded by the lines given by the equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=t-\sin t \\
y=1-\cos t
\end{array}\right. \\
& y=1(0<x<2 \pi, y \geq 1)
\end{aligned}
$$ | \frac{\pi}{2}+2 |
$20 \cdot 55$ In an equilateral triangle, the height equals the radius of a circle, and another equilateral triangle is inscribed in this circle. Then the ratio of the perimeters of the two triangles is
(A) $1: 2$.
(B) $1: 3$.
(C) $1: \sqrt{3}$.
(D) $\sqrt{3}: 2$.
(E) $2: 3$.
(3rd American High School Mathematics Exam... | 2:3 |
The area of square $A B C D$ is $300 \mathrm{~cm}^{2}$. In the figure, $M$ is the midpoint of $D C$ and point $F$ lies on the line passing through $B$ and $C$.
(a) What is the area of triangle $\triangle A B F$?
(b) What is the area of triangle $\triangle A F D$?
 16
(B) 86
(C) 34
(D) 68
(E) 128 | 68 |
1. The solution set of the inequality $\sqrt{\log _{2} x-1}+\frac{1}{2} \log _{\frac{1}{2}} x^{3}+2>0$ is | 2\leqslantx<4 |
In the arithmetic sequence $\{2-3n\}$, the common difference $d$ is equal to $(\ \ \ \ )$
A: $2$
B: $3$
C: $-1$
D: $-3$ | D |
3. The range of the function $y=\sqrt{7-x}+\sqrt{9+x}$ is the interval | [4,4 \sqrt{2}] |
$4 \cdot 3$ Given the equation $2 x^{2}-9 x+8=0$, find a quadratic equation whose
one root is the reciprocal of the sum of the roots of the original equation, and the other root is the square of the difference of the roots of the original equation. | 36x^{2}-161x+34=0 |
Compare the magnitude: $\log_{2}0.3$ and $2^{0.3}$. | \log_{2}0.3 < 2^{0.3} |
Condition A "a > 1" is the ( ) of condition B "a > $\sqrt{a}$".
A: Neither sufficient nor necessary condition
B: Necessary and sufficient condition
C: Sufficient but not necessary condition
D: Necessary but not sufficient condition | \text{B} |
The minimum positive period of the function $g(x)=\tan \left( \frac {\pi}{3}x- \frac {\pi}{6}\right)$ is $M$, then the range of $f(x)=M\sin (2x- \frac {\pi}{6})$ in the interval $\left[0, \frac {\pi}{2}\right]$ is ______. | \left[- \frac {3}{2},3\right] |
Let $p(x) = x^{2} + bx + c$, where $b$ and $c$ are integers.
If $p(x)$ is a factor of both $x^{4} + 6x^{2} + 25$ and $3x^{4} + 4x^{2} + 28x + 5$, what is $p(1)$?
$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 8$ | 4 |
3. A rectangle of size $1 \times k$ for any natural number $k$ will be called a strip. For which natural numbers $n$ can a rectangle of size $1995 \times n$ be cut into pairwise distinct strips? | 3989 |
17.2.1 * How many positive integers $n$ are there, which are divisors of at least one of the numbers $10^{40}, 20^{30}$? | 2301 |
In the arithmetic sequence $\{a_n\}$, $a_4 = 2$ and $a_7 = -4$. Suppose we randomly select numbers from the first 10 terms of $\{a_n\}$, replacing the number after each selection. We perform this selection process three times, assuming each selection is independent. What is the probability that among the three selected... | \frac{6}{25} |
How many types of regular polyhedra are there? What are their names? | 5 |
A certain university has two math modeling clubs, Club A and Club B. Club A has 40 members, while Club B has 50 members. After analyzing the scores of one exam, it was found that the average score of Club A was 90 points, and the average score of Club B was 81 points. What is the average score of the math modeling club... | 85 |
19. Let $x$ be a positive real number. What is the maximum value of $\frac{2022 x^{2} \log (x+2022)}{(\log (x+2022))^{3}+2 x^{3}}$ ? | 674 |
The arithmetic sequence 1, -1, -3, -5, …, -89 has how many terms? | 46 |
3. Find all three-digit numbers that decrease by 5 times after the first digit is erased. | 125,250,375 |
9.3. On the board, there are $N$ prime numbers (not necessarily distinct). It turns out that the sum of any three numbers on the board is also a prime number. For what largest $N$ is this possible | 4 |
The digits 2, 4, 6 and 8 are each used once to create two 2-digit numbers. What is the largest possible difference between the two 2-digit numbers?
(A) 66
(B) 62
(C) 58
(D) 44
(E) 36 | 62 |
The monotonic increasing interval of the function $y=\log_{\frac{1}{2}}(x^2+4x-12)$ is \_\_\_\_\_\_. | (-\infty, -6) |
A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up $36\%$ of the area of the flag, what percent of the area of t... | 2 |
3. From a point on the Earth's surface, a large number of small balls are launched in all directions at the same speed of 10 m/s. Among all the balls that land at a distance from the starting point not closer than $96 \%$ of the distance at which the farthest flying ball lands, find the one that will spend the most tim... | 1.6 |
7. Toss a die twice, observe the numbers facing up, the probability that the product of the two numbers is a multiple of 3 is | \frac{5}{9} |
Given two vectors in the plane, $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(-2,m)$, and $\overrightarrow{a} \parallel \overrightarrow{b}$, then $3\overrightarrow{a}+2\overrightarrow{b}$ equals to ( )
A: $(-2,-4)$
B: $(-1,-2)$
C: $(-4,-8)$
D: $(1,2)$ | \text{B} |
For the proposition "The incircle of an equilateral triangle touches the midpoints of its three sides," we can analogously conjecture the position where the insphere of a regular tetrahedron touches each of the triangular faces. What is this position?
A: A point inside each equilateral triangle
B: A point on a certain ... | \text{C} |
A magician has a hundred cards, numbered from 1 to 100. He places each of them into one of three boxes - a red box, a white box, and a blue box - in such a way that each box contains at least one card.
A member of the audience selects two of the three boxes and draws one card from each, then announces the sum of the n... | 12 |
Which of the following functions of $y$ with respect to $x$ must be a quadratic function?
A: $y=\left(a+2\right)x^{2}+1$
B: $y=\frac{1}{x^{2}}+1$
C: $y=\left(x+2\right)\left(x+1\right)-x^{2}$
D: $y=2x^{2}+3x$ | D |
$(1)-{1}^{2023}+\sqrt{9}-{(\pi-3)}^{0}+|\sqrt{2}-1|$.
$(2)\left\{\begin{array}{c}\frac{3x+1}{2}≥\frac{4x+3}{3}\\ 2x+7≥5x-17\end{array}\right.$. | 3 \leqslant x \leqslant 8 |
Steph scored $15$ baskets out of $20$ attempts in the first half of a game, and $10$ baskets out of $10$ attempts in the second half. Candace took $12$ attempts in the first half and $18$ attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the sam... | 9 |
A certain fruit planting base responded to the government's call to vigorously plant high-quality fruits. A supermarket sees the market value of two high-quality fruits, A and B. After investigation, the cost price and selling price of these two fruits are as shown in the table below:
| Fruit Type | Cost Price (元/$/kg... | 1.2 |
1. If four consecutive natural numbers are all composite, then the smallest possible value of the sum of these four numbers is ( ).
(A) 100
(B) 101
(C) 102
(D) 103 | 102 |
Galoycin A.i.
In the basket, there are 30 mushrooms - russulas and boletus. It is known that among any 12 mushrooms, there is at least one russula, and among any 20 mushrooms - at least one boletus. How many russulas and how many boletus are in the basket?
# | 19 |
2. Let $A, B$ be sets of positive integers such that a sum of arbitrary two different numbers from $A$ is in $B$ and a ratio of arbitrary two different numbers from $B$ (greater one to smaller one) is in $A$. Find the maximum number of elements in $A \cup B$.
(Martin Panák)
| 5 |
Create a five-digit number $a_1a_2a_3a_4a_5$ using the digits 1, 2, 3, 4, 5 without repetition, such that $a_1 a_3$, $a_3 a_5$. How many such five-digit numbers are there? | 16 |
Given the universal set $U=\{1,2,3,4,5,6\}$, and the set $A=\{3,4,5\}$, then $\complement_U A=$ （ ）
A: $\{1,2,6\}$
B: $\{3,4,5\}$
C: $\{1,2,3,4,5,6\}$
D: $\emptyset$ | \text{A} |
Given the parabola $C$: $y^{2}=4x$ with its focus $F$ and $O$ as the coordinate origin. Point $P$ lies on the parabola $C$ and $PF \perp OF$, then $| \overrightarrow{OF}- \overrightarrow{PF}|=$ _____ . | \sqrt {5} |
If the quadratic equation $x(kx+1)-x^{2}+3=0$ has real roots, then the largest integer value of $k$ is ()
A: $2$
B: $1$
C: $0$
D: $-1$ | C |
6. Probabilistic Voting (6-9). In the final of a play competition for March 8, two plays made it to the finals. In the first play, $n$ students from 5th grade A participated, and in the second play, $n$ students from 5th grade B participated. At the play, $2n$ mothers of all $2n$ students were present. The best play is... | 1-(\frac{1}{2})^{n} |
12. (18 points) The inverse function of $f(x)$ is $y=\frac{x}{1+x}$, $g_{n}(x)+\frac{1}{f_{n}(x)}=0$, let $f_{1}(x)=f(x)$, and for $n>1\left(n \in \mathbf{N}_{+}\right)$, $f_{n}(x)=f_{n-1}\left(f_{n-1}(x)\right)$. Find the analytical expression for $g_{n}(x)\left(n \in \mathbf{N}_{+}\right)$. | g_{n}(x)=2^{n-1}-\frac{1}{x} |
The six edges of a tetrahedron $ABCD$ measure $7, 13, 18, 27, 36$ and $41$ units. If the length of edge $AB$ is $41$, then the length of edge $CD$ is
$\textbf{(A)}\ 7\qquad \textbf{(B)}\ 13\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 27\qquad \textbf{(E)}\ 36$ | 13 |
If $8210=8.21 \times 10^{\square}$, then the value that should go in the $\square$ is
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5 | 3 |
Emelyanov L.A.
A square is cut into $n$ rectangles of size $a_{i} \times b_{i}, i=1, \ldots, n$.
For what smallest $n$ can all the numbers in the set $\left\{a_{1}, b_{1}, \ldots, a_{n}, b_{n}\right\}$ be distinct? | 5 |
Given $a > 0, b > 0, a+b=1$, the minimum value of $\dfrac{1}{a}+ \dfrac{2}{b}$ is ( )
A: $4 \sqrt{2}$
B: $6$
C: $3+2 \sqrt{2}$
D: $\dfrac{20}{3}$ | \text{C} |
Given $(1-2x)^4 = a + a_1x + a_2x^2 + a_3x^3 + a_4x^4$, then the value of $(a + a_2 + a_4)^2 - (a_1 + a_3)^2$ is ( )
A: 1
B: 16
C: 41
D: 81 | D |
Calculate the following:$(1)\left(ab\right)^{6}\div \left(ab\right)^{2}\cdot \left(ab\right)^{4}$;$(2)(3x^{3})^{2}\cdot x^{5}-(-x^{2})^{6}\div x$. | 8x^{11} |
Given the real numbers $a$ and $b$ that satisfy $\log_{2}a + \log_{2}b = -2$, find the minimum value of $a + b$ $(\quad)$.
A: $\frac{1}{4}$
B: $\frac{1}{2}$
C: $1$
D: $4$ | C |
13. (13 points) As shown in Figure 2, in the pyramid $P-ABCD$, $PA \perp$ base $ABCD$, $BC=CD=2$, $AC=4$, $\angle ACB = \angle ACD = \frac{\pi}{3}$, $F$ is the midpoint of $PC$, and $AF \perp PB$. Find:
(1) the length of $PA$;
(2) the sine of the dihedral angle $B-AF-D$.
| 2 \sqrt{3}, \frac{3 \sqrt{7}}{8} |
From the numbers \\(5\\), \\(6\\), \\(7\\), \\(8\\), if we randomly pick \\(2\\) different numbers, the probability that the absolute difference between these \\(2\\) numbers is \\(2\\) is ______. | \dfrac{1}{3} |
Given the function $f(x)=2\sin (\pi x+\phi)+1(0 < \phi < \pi)$, find the value of $\phi$ if the function is even. | \dfrac{\pi}{2} |
Problem 5.4. The school principal, the caretaker, and the parent committee, failing to agree with each other, each bought a carpet for the school auditorium, which is $10 \times 10$. After thinking about what to do, they decided to place all three carpets as shown in the picture: the first carpet $6 \times 8$ - in one ... | 6 |
3. A train passes a pole in 5 s. How long (in seconds) will it take for the train and an electric train to pass each other, if the speed of the train is twice the speed of the electric train, and the length of the train is three times the length of the electric train | \frac{40}{9} |
If the function $f(x) = x^2 - 2ax + b$ ($a > 1$) has both its domain and range as $[1, a]$, then the real number $b = \_\_\_\_\_\_$. | 5 |
2. There are 4 cards each of red and blue, and each color of cards is marked with the numbers $1,2,3,4$. Arrange all 8 cards into a 2-row, 4-column matrix, requiring that cards with the same number are in the same column. The number of different arrangements is $(\quad)$.
(A) $\left(\mathrm{A}_{4}^{4} \mathrm{~A}_{4}^{... | B |
A circle intersecting the lateral sides $A C$ and $C B$ of an isosceles triangle $A C B$ at points $P$ and $Q$, respectively, is circumscribed around triangle $A B Q$. Segments $A Q$ and $B P$ intersect at point $D$ such that $A Q: A D=4$ : 3. Find the area of triangle $D Q B$ if the area of triangle $P Q C$ is 3. | \frac{9}{2} |
10.208. Perpendiculars are drawn from the vertex of the acute angle of a rhombus to the lines containing the sides of the rhombus to which this vertex does not belong. The length of each perpendicular is 3 cm, and the distance between their bases is $3 \sqrt{3}$ cm. Calculate the lengths of the diagonals of the rhombus... | 6 |
12. A. Do there exist prime numbers $p$ and $q$ such that the quadratic equation $p x^{2}-q x+p=0$ has rational roots? | p=2, q=5 |
Given the function $f(x)=ax^{2}+bx-2\ln x$ ($a > 0$, $b\in\mathbb{R}$), if for any $x > 0$, $f(x)\geqslant f(2)$ holds, then
A: $\ln a > -b-1$
B: $\ln a\geqslant -b-1$
C: $\ln a < -b-1$
D: $\ln a\leqslant -b-1$ | C |
Example 4 Let $D$ be a point on the side $AB$ of $\triangle ABC$, point $D$ moves along a direction parallel to $BC$ to point $E$ on side $AC$; then from point $E$ along a direction parallel to $AB$ to point $F$ on side $BC$; then from point $F$ along a direction parallel to $CA$ to point $G$ on side $AB$, $\cdots \cdo... | 6 |
4.2. In rectangle $A B C D$, point $E$ is marked on the extension of side $C D$ beyond point $D$. The bisector of angle $A B C$ intersects side $A D$ at point $K$, and the bisector of angle $A D E$ intersects the extension of side $A B$ at point $M$. Find $B C$, if $M K=9$ and $A B=4$. | \sqrt{65} |
5. Fill the numbers $1,2, \cdots, 9$ into a $3 \times 3$ grid such that the sum of the absolute differences of adjacent (sharing a common edge) numbers is maximized. Then this maximum value is ( ).
(A) 57
(B) 58
(C) 59
(D) 60 | B |
Let $c_1,c_2,\ldots,c_{6030}$ be 6030 real numbers. Suppose that for any 6030 real numbers $a_1,a_2,\ldots,a_{6030}$, there exist 6030 real numbers $\{b_1,b_2,\ldots,b_{6030}\}$ such that \[a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}\] and \[b_n = \sum_{d\mid n} c_d a_{n/d}\] for $n=1,2,\ldots,6030$. Find $c_{6030}$.
[i]Victor... | 528 |
Example. Calculate the area of the parallelogram constructed on the vectors $\vec{a}=3 \vec{p}+2 \vec{q}$ and $\vec{b}=2 \vec{p}-\vec{q}$, given that $|\vec{p}|=4,|\vec{q}|=3$ and the angle between the vectors $\vec{p}$ and $\vec{q}$ is $3 \pi / 4$. | 42\sqrt{2} |
9. Given a positive integer $n \geqslant 3$, the function $f:\{1,2, \cdots, n\} \rightarrow\{1,2, \cdots, n\}$ satisfies $f(f(k))=(f(k))^{3}-$ $6(f(k))^{2}+12 f(k)-6, \forall k \in\{1,2, \cdots, n\}$, find the number of all such functions. | 3^{n-3}+3\cdot2^{n-2}+3 |
If the value of the fraction $\frac{x-1}{x+3}$ is $0$, then $x=$____. | 1 |
Given a sequence $\{a_n\}$ with its partial sum denoted as $S_n$, for any $n\in\mathbb{N}^*$, the point $(a_n, S_n)$ lies on the graph of the function $f(x)=-\frac {1}{2}x+ \frac {1}{2}$.
(I) Find the general term formula of the sequence $\{a_n\}$.
(II) Let $b_n= \frac {3}{2}\log_{3}(1-2S_{n})+10$, calculate the ma... | T_{6}= \frac {57}{2} |
(15) A $0, 1$ matrix with 20 rows and several columns satisfies: all columns are distinct and the number of rows where any two columns both have a 1 is at most 2. Find the minimum number of 1's in the matrix when the number of columns is maximized. | 3820 |
A function $f(x)$ satisfies: for all $x \in \mathbb{R}^+$, $f'(x) = \frac{3}{x}f(x)$, and $f(2^{2016}) \neq 0$. The value of $\frac{f(2^{2017})}{f(2^{2016})}$ is:
A: 0.125
B: 0.8
C: 1
D: 8 | D |
3. $6^{11}+C_{11}^{1} 6^{10}+C_{11}^{2} 6^{9}+\cdots+C_{11}^{10} 6-1$ when divided by 8 yields a remainder of $\qquad$ . | 5 |
Problem 1. In two tubs, there are $4 \frac{1}{6}$ and $3 \frac{1}{2}$ liters of water respectively. How much water needs to be transferred from the first tub to the second tub so that after the transfer, both tubs have the same amount of water? Explain your answer! | \frac{1}{3} |
Task 2.
Maximum 15 points
Solve the inequality:
$$
\sqrt{2 x+8}+\sqrt{10-2 x} \geq \log _{2}\left(4 x^{2}-4 x+65\right)
$$ | \frac{1}{2} |
In the triangular pyramid $P-ABC$, point $M$ on segment $PC$ satisfies $PM=\frac{1}{3}PC$, and point $N$ on segment $PB$ satisfies $PN=\frac{2}{3}PB$. The ratio of the volumes of the triangular pyramids $P-AMN$ and $P-ABC$ is ( ).
A: $\frac{1}{9}$
B: $\frac{2}{9}$
C: $\frac{1}{3}$
D: $\frac{4}{9}$ | \text{B} |
A school is preparing to participate in the 2004 National High School Mathematics Competition and has 10 spots to allocate among 8 senior classes, with each class receiving at least one spot. The number of different allocation schemes is $\boxed{36}$. | 36 |
## Task 24/83
Let $p_{1}$ and $p_{2}$ be two consecutive prime numbers with $p_{1}<p_{2}$ and $f(x)$ a polynomial of degree $n$ in $x$ with integer coefficients $a_{i}(i=0 ; 1 ; 2 ; \ldots ; n)$. Determine $p_{1}$ and $p_{2}$ from $f\left(p_{1}\right)=1234$ and $f\left(p_{2}\right)=4321$. | p_{1}=2,p_{2}=3 |
10) As shown in the figure, the volume of tetrahedron $D A B C$ is $\frac{1}{6}$, and it satisfies $\angle A C B=45^{\circ}, A D+B C+$ $\frac{A C}{\sqrt{2}}=3$, then $C D=$ $\qquad$ . | \sqrt{3} |
2. The solution to the equation $10^{x} \times 100^{2 x}=1000^{5}$ is ( ).
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5 | C |
## Problem 2.
Solve the system: $2014 \cdot \lg [x]+\{\lg y\}=2014 \cdot \lg [y]+\{\lg z\}=2014 \cdot \lg [z]+\{\lg x\}=0$. | 1,1,1 |
If the real number $x$ satisfies $C_{18}^{x}=C_{18}^{3x-6}$, then the set of possible values for $x$ is \_\_\_\_\_\_. | \{3,6\} |
2. The cells of a $100 \times 100$ table are painted white. In one move, it is allowed to select any 99 cells from one row or one column and repaint each of them to the opposite color - from white to black, and from black to white. What is the minimum number of moves required to obtain a table with a checkerboard patte... | 100 |
In a shooting training, the probability that a soldier scores 9 rings (including 9 rings) or above in one shot is 0.18. The probability of scoring between 8 and 9 rings (including 8 rings) is 0.51. The probability of scoring between 7 and 8 rings (including 7 rings) is 0.15. The probability of scoring between 6 and 7 r... | 0.93 |
Calculate: $$\frac {\cos 2^\circ}{\sin 47^\circ} + \frac {\cos 88^\circ}{\sin 133^\circ}$$. | \sqrt{2} |
(1) Given the sequence $\left\{a_{n}\right\}$ with the sum of the first $n$ terms $S_{n}=n^{2}+3 n+4\left(n \in \mathbf{N}^{*}\right)$, then $a_{1}+a_{3}+a_{5}+\cdots+a_{21}=$ $\qquad$ . | 268 |
Given $A=\{1, 2, 3, k\}$, $B=\{4, 7, a^4, a^2+3a\}$, where $a\in\mathbb{N}^*$, $x\in A$, $y\in B$, and $f: x\rightarrow y=3x+1$ is a function from the domain $A$ to the codomain $B$. Find the values of $a$ and $k$. | a=2, k=5 |
Konagin S.
Find $x_{1000}$, if $x_{1}=4, x_{2}=6$, and for any natural $n \geq 3$, $x_{n}$ is the smallest composite number greater than $2 x_{n-1}-x_{n-2}$. | 501500 |
16. (15 points) Uncle Li contracted 12 mu of paddy fields, with a yield of 660 kilograms per mu. Aunt Lin contracted 2 mu less of paddy fields than Uncle Li, and the total yield of rice is 420 kilograms less than Uncle Li's. Questions:
(1) What is Uncle Li's total yield of rice in kilograms?
(2) How much less is Uncle ... | 90 |
4. For any $a, b \in \mathbf{R}$,
$$
\max \{|a+b|,|a-b|,|1-b|\}
$$
the minimum value is $\qquad$. | \frac{1}{2} |
The coefficient of the $x^{6}y^{2}$ term in the expansion of $(x- \sqrt {2}y)^{8}$ is ( ).
A: $56$
B: $-56$
C: $28$
D: $-28$ | A |
7. Given that for any real numbers $x, y$, the function $f(x)$ satisfies $f(x)+f(y)=f(x+y)+xy$.
If $f(1)=m$, then the number of positive integer pairs $(m, n)$ that satisfy $f(n)=2019$ is $\qquad$. | 8 |
## Problem Statement
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(2 ; -1 ; 2) \)
\( A_{2}(1 ; 2 ; -1) \)
\( A_{3}(3 ; 2 ; 1) \)
\( A_{4}(-4 ; 2 ; 5) \) | 3\sqrt{\frac{11}{2}} |
Estimate the population using random sampling. Among the following statements, the correct one is $(\ \ )$
A: The sample result is the population result
B: The larger the sample size, the more precise the estimation
C: The standard deviation of the sample can approximately reflect the average state of the population
D:... | B |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.