problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
$22 \cdot 3$ Given a rectangular parallelepiped $A B C D-A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, among the following twelve lines: $A B^{\prime}, B A^{\prime}, C D^{\prime}, D C^{\prime}, A D^{\prime}, D A^{\prime}, B C^{\prime}$, $C B^{\prime}, A C, B D, A^{\prime} C^{\prime}, B^{\prime} D^{\prime}$, the number ... | 30 |
4. The line given by the equation $2 a x+4 y-6=0$, together with the coordinate axes, forms a triangle with an area of 4.5. What can $a$ be? | \\frac{1}{2} |
2. [5 points] Solve the equation $\frac{\cos 8 x}{\cos 3 x+\sin 3 x}+\frac{\sin 8 x}{\cos 3 x-\sin 3 x}=\sqrt{2}$. | \frac{\pi}{44}+\frac{2\pik}{11},k\neq11p+4,k\in\mathbb{Z},p\in\mathbb{Z} |
6. (3 points) Starting with 100, perform the operations "add 15, subtract 12, add 3, add 15, subtract 12, add 3, ... " in a repeating sequence. After 26 steps (1 step refers to each "add" or "subtract" operation), the result is $\qquad$ . | 151 |
Anzhans A.
There are 100 silver coins, ordered by weight, and 101 gold coins, also ordered by weight. It is known that all coins have different weights. We have a two-pan balance that allows us to determine which of any two coins is heavier. How can we find the coin that ranks 101st in weight among all the coins with ... | 8 |
The short tandem repeat sequence (STR, a segment of nucleotide sequence) on human chromosome 21 can be used as a genetic marker for rapid genetic diagnosis of Trisomy 21 syndrome. There is a child with Trisomy 21 syndrome, whose genotype of this genetic marker is ++-. The genotype of this genetic marker for the father ... | C |
In the polar coordinate system, point $M$ moves on the circle $\rho =2\cos \theta$. Determine the minimum distance from point $M$ to the line $\rho \sin (\theta +\frac{\pi }{4})=-\frac{\sqrt{2}}{2}$. | \sqrt{2}-1 |
A sequence ${S_n}$ is the sum of the first $n$ terms of an arithmetic progression ${a_n}$ with a non-zero common difference. If ${a_1, a_2, a_4}$ form a geometric progression, find the value of $\frac{S_4}{S_2}$. | \frac{10}{3} |
5. Lucia wants to write three positive integers $a, b, c$ such that each of them is a divisor of $30$ and that the greatest common divisors between two consecutive terms (i.e., $\operatorname{MCD}(a, b)$ and $\operatorname{MCD}(b, c)$) are prime numbers. In how many ways can she do this?
(A) 69
(B) 72
(C) 105
(D) $2^{7... | 105 |
The solution set of the inequality system about $x$ $\left\{\begin{array}{c}x-1＜0\\ x＜a+3\end{array}\right.$ is $x \lt 1$. Find the range of values for $a$: ______. | a \geqslant -2 |
How many integers must one select from the set $\{1,2, \ldots, 20\}$ to ensure that this selection includes two integers $a$ and $\mathrm{b}$ such that $a-b=2$? | 11 |
A hotel has 1 triple room and 2 double rooms available. Three adults come to stay with two children. Children should not stay in a room alone (they must be accompanied by an adult), and it is not required that all rooms are occupied. There are \_\_\_\_\_\_\_\_\_ different arrangements for their accommodation. | 60 |
On a board the numbers $(n-1, n, n+1)$ are written where $n$ is positive integer. On a move choose 2 numbers $a$ and $b$, delete them and write $2a-b$ and $2b-a$. After a succession of moves, on the board there are 2 zeros. Find all possible values for $n$.
Proposed by Andrei Eckstein | 3^k |
Problem 4. Three lines intersect to form 12 angles, and $n$ of them turn out to be equal. What is the maximum possible value of $n$? | 6 |
2. The algebraic expression $\left[(\sqrt{6}+\sqrt{5})^{6}\right]=$ $\qquad$
(2004, "Five Sheep Cup" Mathematics Competition (Grade 9)) | 10581 |
In the garden, four times as many cabbages grew as broccoli and three times as many radishes as cabbages. The total weight of the broccoli was $5 \mathrm{~kg}$. How many pieces of vegetables grew in the garden if each broccoli weighed 250 g? (No other vegetables grew there.) (L. Černíček) | 340 |
If the complex number $z$ satisfies $(1+3i)z=i-3$, then $z$ equals
A: $i$
B: $\dfrac{4-3i}{5}$
C: $-i$
D: $\dfrac{5}{2}i$ | A |
If the sum of all binomial coefficients in the binomial expansion of $\( (x-\frac{1}{x})^n \)$ is $64$, then the constant term is $\underline{\hspace{1cm}}$ (answer with a number) | -20 |
33. Find the minimum value of $\sum_{k=1}^{100}|n-k|$, where $n$ ranges over all positive integers. | 2500 |
Given points M(2, 5), N(3, -2), point P lies on the directed line segment $\overrightarrow{MN}$ and satisfies $\overrightarrow{MP} = 3\overrightarrow{PN}$. Find the coordinates of point P. | \left(\frac{11}{4}, -\frac{1}{4}\right) |
Which of the following functions is an odd function?
A: $f\left(x\right)=1+\cos x$
B: $f\left(x\right)=x+\sin x$
C: $f\left(x\right)=x+\cos x$
D: $f\left(x\right)=1+\sin x$ | B |
The monotonic decreasing interval of the function $f(x) = x^3 - 3x^2 + 4$ is ( )
A: $(-\infty, 0)$
B: $(-2, 0)$
C: $(0, 2)$
D: $(2, +\infty)$ | C |
Task 13. (8 points)
Maxim Viktorovich returned from a trip to Asian countries. He has a certain amount of money in foreign currency left.
Maxim Viktorovich familiarized himself with the exchange rates at the nearest banks: "Voskhod" and "Elfa". He decided to take advantage of the most favorable offer. What amount wil... | 7540 |
To obtain the graph of the function $y=\sin (2x- \frac {\pi }{3})$, it is only necessary to shift the graph of the function $y=\sin (2x+ \frac {\pi }{6})$ by
A: Shift to the left by $\frac {\pi }{4}$ units
B: Shift to the left by $\frac {\pi }{2}$ units
C: Shift to the right by $\frac {\pi }{4}$ units
D: Shift to the r... | \text{C} |
Let $E(n)$ denote the largest integer $k$ such that $5^k$ divides $1^{1}\cdot 2^{2} \cdot 3^{3} \cdot \ldots \cdot n^{n}.$ Calculate
$$\lim_{n\to \infty} \frac{E(n)}{n^2 }.$$ | \frac{1}{8} |
Given two points A (2, 1) and B (m, 4), find
(1) The slope of line AB and the equation of line AB;
(2) Given $m \in [2- \sqrt{3}, 2+3\sqrt{3}]$, find the range of the inclination angle $\alpha$ of line AB. | \left[\frac{\pi}{6}, \frac{2\pi}{3}\right] |
3. The following 4 propositions are given:
(1) In $\triangle A B C$, $\sin A+\sin B-\sin C$ is always positive;
(2) In $\triangle A B C$, $\cos A+\cos B+\cos C$ is always positive;
(3) In $\triangle A B C$, $\cot A+\cot B+\cot C$ is always positive;
(4) In a non-right $\triangle A B C$, $\tan A+\tan B+\tan C$ is always... | B |
If $a=2^{3}$ and $b=3^{2}$ evaluate $\frac{(a-b)^{2015}+1^{2015}}{(a-b)^{2015}-1^{2015}}$. | 0 |
19 In a carriage, any $m(m \geqslant 3)$ passengers have a unique common friend (if A is a friend of B, then B is also a friend of A, and no one is a friend of themselves), how many friends does the person with the most friends have in this carriage?
| m |
5. (10 points) From the 16 integers from 1 to 16, at least how many numbers must be taken to ensure that there are two numbers, one of which is twice the other. | 12 |
8. The diagram shows eight small squares. Six of these squares are to be shaded so that the shaded squares form the net of a cube.
In how many different ways can this be done?
A 10
B 8
C 7
D 6
E 4 | 6 |
Which of the following operations is correct?
A: $2a^{2}-a^{2}=2$
B: $a^{2}\cdot a^{4}=a^{6}$
C: $(a^{2})^{3}=a^{5}$
D: $a^{6}\div a^{2}=a^{3}$ | B |
The irrational number among the following real numbers is ( )
A: -1
B: 0.101001
C: $$\sqrt {2}$$
D: $$\frac {3}{2}$$ | \text{C} |
2. Given the set $A=\{k+1, k+2, \cdots, k+n\}, k, n$ are positive integers, if the sum of all elements in set $A$ is 2019, then when $n$ takes the maximum value, the set $A=$ | {334,335,336,337,338,339} |
Let $\{a\_n\}$ be a geometric sequence with a common ratio of $q$ ($|q| > 1$). Define $b\_n = a\_n + 1 (n=1,2,...)$. If the sequence $\{b\_n\}$ has four consecutive terms in the set $\{-53, -23, 19, 37, 82\}$, find the value of $6q$. | -9 |
Is it true that there are $130$ consecutive natural numbers, such that each of them has exactly $900$ natural divisors? | \text{no} |
2. Circles $k(S ; 6 \mathrm{~cm})$ and $l(O ; 4 \mathrm{~cm})$ have an internal tangency at point $B$. Determine the lengths of the sides of triangle $A B C$, where point $A$ is the intersection of line $O B$ with circle $k$ and point $C$ is the intersection of circle $k$ with the tangent from point $A$ to circle $l$. | 6\sqrt{3}\mathrm{~} |
In the $xOy$ plane with rectangular coordinates, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{\sqrt{3}}{2}t\\ y=\frac{1}{2}t\end{array}\right.$ (where $t$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis.... | \frac{\sqrt{15}}{3} |
Define: a line that does not coincide with the coordinate axis intersects the coordinate axis at points $A$ and $B$ (where $A$ and $B$ are not coincident). If a parabola $L$ passes through points $A$ and $B$, then this parabola is called the "friendship line" of the line.
$(1)$ If the parabola $L$ is the "friendship ... | b \geqslant 1 |
9. (10 points) Given that 7 red balls and 5 white balls weigh 43 grams, and 5 red balls and 7 white balls weigh 47 grams, then 4 red balls and 8 white balls weigh $\qquad$ grams. | 49 |
In triangle $ABC$, with sides $a$, $b$, and $c$ opposing angles $A$, $B$, and $C$, respectively, it is given that $\cos B \cos C - \sin B \sin C = \frac{1}{2}$ and $a = 2\sqrt{3}$.
(1) Find the measure of angle $A$.
(2) If $b = 2$, find the length of side $c$.
(3) If $b + c = 4$, find the area of $\triangle ABC$. | \sqrt{3} |
Example 1. Let the integer part of $\frac{1}{3-\sqrt{7}}$ be $a$, and the decimal part be $b$, find the value of $a^{2}+(1+\sqrt{7}) a b$. | 10 |
Given an arithmetic sequence $\{a\_n\}$ with a non-zero common difference $d$, and $a\_1=1$, if $a\_1$, $a\_3$, $a\__{13}$ form a geometric sequence, then the common difference $d=$ ( ).
A: $1$
B: $2$
C: $3$
D: $5$ | B |
4. On the side $AC$ of triangle $ABC$, a point $K$ is taken which divides $AC$ in the ratio $1:3$, and on the side $BC$ a point $L$ is taken which divides $BC$ in the ratio $1:4$. Let $M$ be the intersection point of segments $AL$ and $BK$. Determine the ratio of segments $KM$ and $MB$. | 1:1 |
6. (20 points) Each of the two baskets contains white and black balls, and the total number of balls in both baskets is 25. A ball is randomly drawn from each basket. It is known that the probability that both drawn balls will be white is 0.54. Find the probability that both drawn balls will be black. | 0.04 |
Subject 3. On the plane of isosceles triangle $\mathrm{ABC}$, perpendicular $\mathrm{AM}$ is raised. If $\mathrm{AM}=\mathrm{AC}=\mathrm{b}, \mathrm{AB}=\mathrm{c}$ and $\left(b^{2}+c^{2}-6\right)^{2}+\left(b^{2}-c^{2}+4\right)^{2}=0$, calculate the distance from M to BC. | \frac{\sqrt{195}}{10} |
The problem involves finding the value of the expression $\left( \frac{1+i}{ \sqrt {2}}\right)^{2016}$. | 1 |
Given vectors $\overrightarrow {OA} = (1, -2)$, $\overrightarrow {OB} = (-3, 2)$, then $\frac {1}{2} \overrightarrow {AB} = $ ______. | (-2, 2) |
Find the least possible cardinality of a set $A$ of natural numbers, the smallest and greatest of which are $1$ and $100$, and having the property that every element of $A$ except for $1$ equals the sum of two elements of $A$. | 9 |
Let $ABCD$ be a cyclic quadrilateral, and suppose that $BC = CD = 2$. Let $I$ be the incenter of triangle $ABD$. If $AI = 2$ as well, find the minimum value of the length of diagonal $BD$. | 2\sqrt{3} |
13.012. The working day has been reduced from 8 to 7 hours. By what percentage does labor productivity need to increase to ensure that, with the same rates, the wage increases by $5 \%$? | 20 |
Given that the perimeter of a sector of a circle is $4$ cm and its area is $1$ cm$^{2}$, determine the measure of the central angle of the sector in radians. | \alpha = 2 |
## Task B-1.5.
Let $A, B, C$ be points on a circle with center at point $S$ such that $\angle S B C=30^{\circ}$, and $\angle B C A=50^{\circ}$. What is the measure of the angle $\angle A B C$? (Point $S$ is inside the triangle $A B C$). | 70 |
Task 2. Write all five-digit numbers of the form $\overline{a b c d a}$ divisible by 45, where the digit in the hundreds place is the largest one-digit prime number. | 50715,51705,51795,52785,54765,56745,58725,59715 |
Identify an interval within which the function $f(x) = \log_2 x + x - 5$ has a root.
A. $(1, 2)$
B. $(2, 3)$
C. $(3, 4)$
D. $(4, 5)$ | C |
Given the parabola $y=a\left(x-3\right)^{2}+2$ passes through the point $\left(1,-2\right)$.
$(1)$ Find the value of $a$.
$(2)$ If points $A(m$,$y_{1})$ and $B(n$,$y_{2})(m \lt n \lt 3)$ both lie on the parabola, compare the sizes of $y_{1}$ and $y_{2}$. | y_1 < y_2 |
Ranu starts with one standard die on a table. At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step. After 2013 such steps, let $D$ be the number of dice on the table. What is the expected value (ave... | 10071 |
Let $X$ be a set containing $n$ elements. Find the number of ordered triples $(A,B, C)$ of subsets of $X$ such that $A$ is a subset of $B$ and $B$ is a proper subset of $C$. | 4^n - 3^n |
If point $A$ lies on line $a$, line $a$ lies in plane $\alpha$, and point $B$ lies in plane $\alpha$, it can be represented as:
A: $A\subset a$, $a\subset \alpha$, $B\in \alpha$
B: $A\in a$, $a\subset \alpha$, $B\in \alpha$
C: $A\subset a$, $a\in \alpha$, $B\subset \alpha$
D: $A\in a$, $a\in \alpha$, $B\in \alpha$ | \text{B} |
Given the expansion of $(\frac{1}{2} + 2x)^n$.
(1) If the binomial coefficients of the 5th, 6th, and 7th terms form an arithmetic sequence, find the binomial coefficient of the term with the largest coefficient in the expansion.
(2) If the sum of the binomial coefficients of the first three terms is 79, find the term w... | 16896x^{10} |
2. Let $A_{k}=\left\{x \left\lvert\, x=k t+\frac{1}{k t}\right., \frac{1}{k^{2}} \leqslant t \leqslant 1\right\}$, where $k=2,3, \cdots, 2012$. Then the intersection of all $A_{k}$ is ( ).
(A) $\varnothing$
(B) $\{2\}$
(C) $\left[2, \frac{5}{2}\right]$
(D) $\left[2, \frac{2012^{2}+1}{2012}\right]$ | C |
Example 4. In the acute triangle $\triangle A B C$, the lengths of the three sides $a, b, c$ are all integers, and $a>b>c, a+b+c=20$. Find $\angle A + \angle C$. | 120^{\circ} |
4. Given that the graph of a quadratic function is tangent to the lines containing the three sides of $\triangle A B C$. If the coordinates of the vertices of the triangle are $A\left(\frac{1}{4}, \frac{3}{2}\right), B\left(\frac{1}{2}, 1\right), C\left(\frac{3}{4}, 1\right)$, then the expression of the quadratic funct... | y=x^{2}-2 x+2 |
Given that the expression of line $l_{1}$ is $y=-2x+b$, if line $l_{1}$ is symmetric about the $y$-axis with line $l_{2}$ passing through the point $(1,6)$, then the value of $b$ is ( ).
A: $8$
B: $4$
C: $-8$
D: $-4$ | B |
Let $k$ be a positive integer. Find the smallest positive integer $n$ for which there exists $k$ nonzero vectors $v_1,v_2,…,v_k$ in $\mathbb{R}^n$ such that for every pair $i,j$ of indices with $|i-j|>1$ the vectors $v_i$ and $v_j$ are orthogonal.
[i]Proposed by Alexey Balitskiy, Moscow Institute of Physics and Techno... | \left\lceil \frac{k}{2} \right\rceil |
How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy \[1-3\sin\theta+5\cos3\theta = 0?\]
$\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8$ | 6 |
Given the parabola $y=x^{2}+bx-2$ passes through the point $(1,3)$, intersects the $y$-axis at point $A$, and has its vertex at point $B$. Let $k$ be the abscissa of the intersection point of the parabola $y=x^{2}+bx-2$ with the $x$-axis.
$(1)$ Find the area of $\triangle OAB$;
$(2)$ Find the value of the algebraic... | \frac{1}{107} |
3. For the cyclic quadrilateral $ABCD$, the lengths of the four sides in sequence are $AB=2, BC=7, CD=6, DA=9$. Then the area of the quadrilateral is $\qquad$ . | 30 |
Given that $α$ is an acute angle, and $\cos α= \frac{1}{3}$, find the value of $\sin \left( \frac{π}{4}-α \right)$. | \frac{ \sqrt{2}-4}{6} |
Six, find the largest constant $M(M>0)$, such that for any positive integer $n$, there exist positive real number sequences $a_{1}, a_{2}, \cdots, a_{n}$ and $b_{1}, b_{2}, \cdots, b_{n}$ satisfying
$$
\begin{array}{l}
\text { (1) } \sum_{k=1}^{n} b_{k}=1, \\
2 b_{k} \geqslant b_{k-1}+b_{k+1}(k=2,3, \cdots, n-1) ; \\
\... | \frac{3}{2} |
Given that $f(x)$ is an odd function on $\mathbb{R}$, and $f(x) = 2^x$ when $x > 0$, find $f(-1)=$ _____. | -2 |
Example 1: A and B participate in an election, A gets $m$ votes, B gets $n$ votes, $m>n$. Question: In the process of counting the $m+n$ votes one by one, how many possible vote counting records are there where A's vote count is always leading? | \frac{-n}{+n}C_{+n}^{} |
1. Square $C A S H$ and regular pentagon $M O N E Y$ are both inscribed in a circle. Given that they do not share a vertex, how many intersections do these two polygons have? | 8 |
The graph of $y^2+2xy+40|x|=400$ partitions the plane into several regions. What is the area of the bounded region? | 800 |
Let the complex numbers $z_{1}$ and $z_{2}$ correspond to points $Z_{1}$ and $Z_{2}$ in the complex plane. If $|z_{1}|\leqslant 2$ and $z_{2}=3-4i$, then the maximum value of $|\overrightarrow{{Z}_{1}{Z}_{2}}|$ is ______. | 7 |
Enlarge each edge of a graph by four times its original size. This is equivalent to enlarging the graph by a scale of \_\_\_\_\_\_. | 4:1 |
6.3. How many natural numbers $\mathrm{N}$ greater than 700 exist such that among the numbers $3 \mathrm{~N}, \mathrm{~N}-$ $700, N+35,2 N$ exactly two are four-digit numbers? | 5229 |
Let $S=\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, and let $A = \{a_1, a_2, a_3\}$ be a subset of $S$ such that $a_1 < a_2 < a_3$ and $a_3 - a_2 \leq 6$. Determine the number of possible subsets $A$ that satisfy these conditions. | 83 |
Given the function $f\left(x\right)=ae^{x-1}-\ln x+\ln a$.
$(1)$ When $a=e$, find the area of the triangle formed by the tangent line to the curve $y=f\left(x\right)$ at the point $\left(1,f\left(1\right)\right)$ and the two coordinate axes.
$(2)$ If $f\left(x\right)\geqslant 1$, find the range of values for $a$. | [1, +\infty) |
4. In $\triangle A B C$, the sides opposite to angles $A, B, C$ are denoted as $a, b, c (b \neq 1)$, respectively, and $\frac{C}{A}, \frac{\sin B}{\sin A}$ are both roots of the equation $\log _{\sqrt{b}} x=\log _{b}(4 x-4)$. Then $\triangle A B C$
A. is an isosceles triangle, but not a right triangle
B. is a right tri... | B |
2. Find all such triples of numbers $m, n, k$, such that each of the equations $m x^{2}+n=0, n x^{2}+k=0$ and $k x^{2}+m=0$ has at least one solution. | =n=k=0 |
Let the function $f(x)=2\sin \pi x$ and the function $y= \frac {1}{1-x}$ have their graphs intersect at points in the interval $[-2,4]$ with the x-coordinates of these points being $x_{1}$, $x_{2}$, …, $x_{n}$, respectively. Then, $\sum\limits_{i=1}^{n}x_{i}=$ ()
A: $4$
B: $6$
C: $8$
D: $10$ | C |
Find the equation of the line that passes through the intersection point of the two lines $l_{1}: x+y-4=0$ and $l_{2}: x-y+2=0$, and is respectively
$(1)$ parallel to the line $2x-y-1=0$;
$(2)$ perpendicular to the line $2x-y-1=0$. | x+2y-7=0 |
10. A rectangular floor measuring 17 feet by 10 feet is covered with 170 square tiles, each $1 \times 1$ square foot. A bug crawls from one corner to the opposite corner. Starting from the first tile it leaves, how many tiles does it cross in total?
(A) 17
(B) 25
(C) 26
(D) 27
(E) 28 | 26 |
9. There are 1000 lamps and 1000 switches, each switch controls all lamps whose numbers are multiples of its own, initially all lamps are on. Now pull the $2, 3, 5$ switches, then the number of lamps that are still on is $\qquad$.
| 499 |
On February 28, 2023, the National Bureau of Statistics announced that the per capita Gross Domestic Product (GDP) of China in 2022 was approximately 86,000 yuan. The number 86,000 can be expressed in scientific notation as ( ).
A: $86\times 10^{3}$
B: $8.6\times 10^{4}$
C: $8.6\times 10^{5}$
D: $0.86\times 10^{5}$ | B |
Given the set $A=\{2,3,4,5,6\}$ and $B=\{x|x^{2}-8x+12\geqslant 0\}$, then $A\cap (\complement _{R}B)=\left(\ \ \right)$
A: $\{2,3,4,5\}$
B: $\{2,3,4,5,6\}$
C: $\{3,4,5\}$
D: $\{3,4,5,6\}$ | C |
Given vectors $\overset{→}{a} =(4,3)$, $\overset{→}{b}=(−1,2)$,
(1) Find the cosine value of the angle between $\overset{→}{a}$ and $\overset{→}{b}$;
(2) If $\overset{→}{c}$ is in the same direction as $\overset{→}{a}$ and $\left| \overset{→}{c} \right|=10$, find the coordinates of $\overset{→}{c}$. | (8, 6) |
2. Find the area of a right triangle if the height drawn from the right angle divides it into two triangles, the radii of the inscribed circles of which are 3 and 4. | 150 |
Elective 4-4: Special Topics on Inequalities
Let the solution set of the inequality $-2 < |x-1|-|x+2| < 0$ be $M$, and $a,b \in M$.
(1) Prove that: $\left| \frac{1}{3}a + \frac{1}{6}b\right| < \frac{1}{4}$; (2) Compare the magnitude of $\left|1-4ab\right|$ and $2\left|a-b\right|$, and explain the reason. | |1-4ab| > 2|a-b| |
Given two-dimensional vectors $\overrightarrow {a} = (1, -2)$ and $\overrightarrow {b} = (-2, x)$. If $\overrightarrow {a}$ is parallel to $\overrightarrow {b}$, find the value of $x$.
Options:
A: 4
B: -4
C: -1
D: 2 | A |
Given that $a=\ln \dfrac {1}{2}$, $b=\left( \dfrac {1}{3}\right)^{0.8}$, and $c=2^{\frac {1}{3}}$, arrange $a$, $b$, and $c$ in ascending order. | a < b < c |
Provide all continuous ${ }^{1}$ solutions of the Cauchy equation on $\mathbb{R}$. | f(x)=ax |
4. The function is
$$
f(x)=\frac{a x+2}{3 x-\frac{1}{a}}
$$
Determine all possible values of the real parameter $a$ such that for all real values of $x$ for which $f(x)$ is defined, it is also true that $f(f(x))$ is defined and $f(f(x))=x$. | 1-1 |
What is the remainder when $2^{33}$ is divided by 9? | 8 |
Let $n$ be positive integer. Define a sequence $\{a_k\}$ by
\[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\]
(1) Find $a_2$ and $a_3$.
(2) Find the general term $a_k$.
(3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$.
50 point... | \ln 2 |
13. Given that $p$, $q$, $\frac{2 q-1}{p}$, and $\frac{2 p-1}{q}$ are all integers, and $p>1$, $q>1$. Find the value of $p+q$. | 8 |
Define a function $f(x)$ on $\mathbb{R}$. For any $x_1, x_2 \in \mathbb{R}$ ($x_1 \neq x_2$), it holds that $$\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}} < 0.$$ Then, which of the following is true?
A: $f(3) < f(1) < f(2)$
B: $f(1) < f(2) < f(3)$
C: $f(2) < f(1) < f(3)$
D: $f(3) < f(2) < f(1)$ | \text{D} |
Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?
$\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }12 \qquad \textbf{(D) }1... | 12 |
In \\(\triangle ABC\\), if \\(0 < \tan A\tan B < 1\\), then \\(\triangle ABC\\) is a \_\_\_\_\_\_ triangle. (Fill in "obtuse", "acute", or "right") | \text{obtuse} |
Subsets and Splits
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