problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
28. A teacher wrote down three positive integers on the whiteboard: $1125,2925, N$, and asked her class to compute the least common multiple of the three numbers. One student misread 1125 as 1725 and computed the least common multiple of 1725,2925 and $N$ instead. The answer he obtained was the same as the correct answ... | 2875 |
At the end of the term, the school uses a sum of money to buy prizes. If one pen and two notebooks are considered as one set of prizes, then 60 sets of prizes can be bought; if one pen and three notebooks are considered as one set of prizes, then 50 sets of prizes can be bought. How many pens or notebooks can be bought... | \text{C} |
Given a non-zero common difference arithmetic sequence $\{a_{n}\}$ that satisfies $a_{6}=14$, and $a_{1}$, $a_{3}$, $a_{7}$ are the first three terms of a geometric sequence $\{b_{n}\}$.
(I) Find the general term formulas for both sequences $\{a_{n}\}$ and $\{b_{n}\}$;
(II) Let $c_{n}=a_{n}-b_{n}$, find the sum of the ... | n^2+3n+4-2^{n+2} |
The expressions $A=1\times2+3\times4+5\times6+\cdots+37\times38+39$ and $B=1+2\times3+4\times5+\cdots+36\times37+38\times39$ are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers $A$ and $B$. | 722 |
4.1. Let \( f(x) = x^2 - x + 1 \), \( g(x) = -x^2 + x + 1 \). Find a non-constant polynomial \( h \) such that \( h(f(x)) = h(g(x)) \). | (x-1)^2 |
5. In a right-angled triangle, the sines of the three interior angles form a geometric progression. Then the smallest angle of the triangle is equal to ( ).
(A) $\arcsin \frac{\sqrt{5}-1}{2}$
(B) $\arccos \frac{\sqrt{5}-1}{2}$
(C) $\arcsin \frac{\sqrt{5}+1}{4}$
(D) $\arccos \frac{\sqrt{5}+1}{4}$ | A |
15. Arrange 5 teachers $A, B, C, D, E$ and 25 identical seats in a row, after which 25 students will sit in the seats to take photos with the teachers. Requirements: $A, B, C, D, E$ must appear in alphabetical order from left to right in this row, and there must be at least two seats between each adjacent pair of teach... | 26334 |
We know that
$$
\frac{8^{x}}{2^{x+y}}=64 \text{ and } \frac{9^{x+y}}{3^{4 y}}=243
$$
Determine the value of $2 x y$.
# | 7 |
Let $\{a_{n}\}$ be a sequence with the sum of its first $n$ terms denoted as $S_{n}$, and ${S}_{n}=2{a}_{n}-{2}^{n+1}$. The sequence $\{b_{n}\}$ satisfies ${b}_{n}=log_{2}\frac{{a}_{n}}{n+1}$, where $n\in N^{*}$. Find the maximum real number $m$ such that the inequality $(1+\frac{1}{{b}_{2}})•(1+\frac{1}{{b}_{4}})•⋯•(1... | \frac{3}{4} |
The equation of the tangent line to the curve $y = x^3 - 1$ at $x = 1$ is _____. | y = 3x - 3 |
Example 3 Choose 1 young man and 1 young woman from each of the four groups to perform a circle dance, with the rule that men and women must alternate, and two people from the same group cannot be adjacent. How many different team formation variations are there? | 12 |
An $m\times n\times p$ rectangular box has half the volume of an $(m + 2)\times(n + 2)\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\le n\le p.$ What is the largest possible value of $p$? | 130 |
17.56 Given that the three sides of triangle $P$ are $a, \sqrt{3} a, 2 a$, and the three sides of triangle $Q$ are $3 a, 2 \sqrt{3} a, \sqrt{3} a$, and triangle $R$ is a right triangle with one acute angle equal to $30^{\circ}$, let “c” denote similarity, and “ゆ” denote non-similarity. Then
(A) $P \sim Q$, but $Q \inft... | D |
If $z= \frac {1}{2}+ \frac { \sqrt {3}}{2}i$, and $(x-z)^4=a_0x^4+a_1x^3+a_2x^2+a_3x+a_4$, then $a_2$ equals to ( )
A: $-\frac {1}{2}+ \frac { \sqrt {3}}{2}i$
B: $-3+3 \sqrt {3}i$
C: $6+3 \sqrt {3}i$
D: $-3-3 \sqrt {3}i$ | \text{B} |
12. What is the sum of all possible $\left(\begin{array}{l}i \\ j\end{array}\right)$ subject to the restrictions that $i \geq 10, j \geq 0$, and $i+j \leq 20$ ? Count different $i, j$ that yield the same value separately - for example, count both $\left(\begin{array}{c}10 \\ 1\end{array}\right)$ and $\left(\begin{array... | 27633 |
SG. 4 If $\log _{p} x=2, \log _{q} x=3, \log _{r} x=6$ and $\log _{p q r} x=d$, find the value of $d$. | 1 |
Which of the following operations is correct?
A: $3x^{3}+2x^{3}=5x^{6}$
B: $(x+1)^{2}=x^{2}+1$
C: $x^{8}\div x^{4}=x^{2}$
D: $-1^{2022}-\left(-1\right)^{2023}-\left(-1\right)^{0}=-1$ | D |
Chenguang High School has carried out a variety of cultural activities for clubs. Students A, B, and C were asked whether they had participated in the street dance club, anime club, or instrumental music club.
Student A said: "I have participated in more clubs than B, but I have not participated in the anime club."
St... | \text{street dance club} |
2. From head to tail of the zebra Hippotigris - 360 stripes of the same width. Flea Masha and flea Dasha started crawling from the head of the zebra to its tail. At the same time, flea Sasha started crawling from the tail to the head. Flea Dasha crawls twice as fast as flea Masha. Before meeting flea Sasha, Masha overc... | 240 |
Express the algebraic expression $x^2 - 6x + 10$ in the form of $(x - m)^2 + n$, where $m$ and $n$ are constants. | (x - 3)^2 + 1 |
Let $ n$ be a positive integer. Find all $ x_1,x_2,\ldots,x_n$ that satisfy the relation:
\[ \sqrt{x_1\minus{}1}\plus{}2\cdot \sqrt{x_2\minus{}4}\plus{}3\cdot \sqrt{x_3\minus{}9}\plus{}\cdots\plus{}n\cdot\sqrt{x_n\minus{}n^2}\equal{}\frac{1}{2}(x_1\plus{}x_2\plus{}x_3\plus{}\cdots\plus{}x_n).\] | x_i = 2i^2 \text{ for } i = 1, 2, \ldots, n. |
25. What is the unit digit of $\underbrace{2018 \times 2018 \times \cdots \times 2018}_{2018 \text { 个 } 2018}$? | 4 |
Given that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, $a_{1}=1$, $\frac{S_{n}}{n}-\frac{S_{n-1}}{n-1}=2$ for $n\geq 2$. Find the general formula for the sequence $\{a_{n}\}$; Let $b_{n}=\frac{a_{n}}{2^n}$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$ denoted as $T_{n}$. | 5-\frac{4n+5}{2^n} |
8. $A B$ is the common perpendicular segment of skew lines $a, b$, $A$ is on line $a$, $B$ is on line $b$, $A B=2$, the skew lines $a, b$ form a $30^{\circ}$ angle, and on line $a$ take $A P=4$, then the distance from point $P$ to line $b$ is $\qquad$. | 2\sqrt{2} |
On the number line, which point represents a number that is equidistant from the points representing $-2$ and $4$?
A: The origin
B: $1$
C: $-1$
D: $2$ | B |
15. [9] Let $a, b, c$ be positive integers such that
$$
\frac{a}{77}+\frac{b}{91}+\frac{c}{143}=1 .
$$
What is the smallest possible value of $a+b+c$ ? | 79 |
9. Among the 95 numbers $1^{2}, 2^{2}, 3^{2}, \cdots, 95^{2}$, the numbers with an odd digit in the tens place are $\qquad$ | 19 |
65. How many terms of the sum
$$
1+2+3+\ldots
$$
are needed to obtain a three-digit number, all of whose digits are the same?
20
## Search for solutions | 36 |
Chinese outstanding traditional culture has a long history and is the crystallization of Chinese civilization. "The Mathematical Classic of Sunzi" and "The Mathematical Classic of Zhou Bi" are relatively popular ancient Chinese arithmetic books with many interesting and straightforward problems. The unit price of "The ... | 2136 \text{ yuan} |
8. The number of positive integer solutions $(x, y, z)$ to the equation $x+y+z=2010$ that satisfy $x \leqslant y \leqslant z$ is $\qquad$ . | 336675 |
Calculate
(1) $(2 \frac {1}{4})^{ \frac {1}{2}}-9.6^{0}-(-3 \frac {3}{8})^{- \frac {2}{3}}+(1.5)^{-2}$
(2) $\log_{2}25 \cdot \log_{3}2 \sqrt {2} \cdot \log_{5}9$. | 6 |
If the point $P_{1}(2-m,5)$ is symmetric with respect to the origin to point $P_{2}(3,2n+1)$, then the value of $m-n$ is ( )
A: $6$
B: $-3$
C: $8$
D: $9$ | C |
In a right-angled triangle, $s_{a}$ and $s_{b}$ are the medians to the legs, and $s_{c}$ is the median to the hypotenuse. Determine the maximum value of the expression $\frac{s_{a}+s_{b}}{s_{c}}$. | \sqrt{10} |
Let proposition p be: $x \leq \frac{1}{2}$ or $x \geq 1$, and proposition q be: $(x-a)(x-a-1) \leq 0$. If p is a necessary but not sufficient condition for the negation of q, find the range of the real number a. | [0, \frac{1}{2}] |
5. In an isosceles trapezoid $M N K L$ with bases $M L$, $N K$ the diagonals are perpendicular to the sides $M N, \quad K L$ and intersect at an angle of $22.5^{\circ}$. Find the height of the trapezoid if the length $N Q=3$, where $Q-$ is the midpoint of the larger base. | \frac{3\sqrt{2-\sqrt{2}}}{2} |
On a day during the flood season, the water level of a certain reservoir was $45m$ at $8$ a.m. The water level then rose at a rate of $0.6m$ per hour. At noon, at $12$ p.m., the floodgate was opened to discharge water, causing the water level to decrease at a rate of $0.3m$ per hour. What was the water level of the res... | B |
## Task 2 - 201242
In a fish shop, there are three water containers available for keeping live carp.
At the start of sales, there are exactly 20 carp in each of these three containers. At the end of sales, there are a total of 3 carp left. The sold carp were removed one by one.
No exchange of carp between containers... | 0.108 |
4・194 A book has page numbers from 1 to $n$. When the page numbers of this book were added up, one page number was mistakenly added one extra time, resulting in an incorrect sum of 1986. What is the page number that was added one extra time? | 33 |
1. In the room, there are knights who always tell the truth, and liars who always lie. 10 of them said: "In this room, there are more knights than liars." 12 said: "In this room, there are more liars than knights." The remaining 22 said: "In this room, there are an equal number of liars and knights." How many liars cou... | 22;32 |
Ancient Chinese mathematician Yang Hui posed the following problem in his book "Quick Methods for Multiplication and Division in Similar Fields": "A rectangular field has an area of 864 square steps, with a total length and width of 60 steps. How much longer is the length compared to the width?" After calculation, your... | A |
18. (3 points) There are two three-digit numbers, their sum is 999. If the larger number is placed to the left of the smaller number, the resulting number is exactly 6 times the number formed by placing the smaller number to the left of the larger number. The difference between the two numbers is $\qquad$.
| 715 |
Given that $\{1, 2\} \subseteq X \subseteq \{1, 2, 3, 4, 5\}$, the total number of sets $X$ that satisfy this relation is ( ).
A: 2
B: 6
C: 4
D: 8 | D |
Which of the following pairs of numbers are opposite to each other?
A: $+\left(-5\right)$ and $\frac{1}{5}$
B: $-\left(-8\right)$ and $2^{3}$
C: $-|+3|$ and $3$
D: $+|-3\frac{1}{2}|$ and $3\frac{1}{2}$ | C |
## Example 4. Solve the equation
$$
2 x^{2} y^{\prime \prime}+\left(3 x-2 x^{2}\right) y^{\prime}-(x+1) y=0
$$ | y(x)=Ay_{1}(x)+By_{2}(x) |
A row of 7 seats, with two people, A and B, taking seats. It is required that there is at least one empty seat between A and B. The number of different seating arrangements is
A: 30
B: 28
C: 42
D: 16 | \text{A} |
10. (12 points) Optimus Prime, the Transformer, departs from location $A$ to location $B$ in robot form and can arrive on time; if he transforms into a car from the start, his speed increases by $\frac{1}{4}$, and he can arrive 1 hour earlier; if he travels 150 kilometers in robot form, then transforms into a car, with... | 750 |
Calculate the result of $\frac{a^{2}-1}{a}+\frac{1}{a}$. | a |
Problem 4. Given a square $ABCD$. On the extension of the diagonal $AC$ beyond point $C$, a point $K$ is marked such that $BK=AC$. Find the angle $BKC$. $[6$ points] ( | 30 |
3. Find the set of values of the function $y=f^{[2019]}(x)$, where $f(x)=\log _{0.5}\left(\frac{\sin x}{\sin x+15}\right)$,
 | [4;+\infty) |
2.4. The base of a quadrilateral pyramid is a rectangle with a diagonal equal to $b$, and the angle between the diagonals is $60^{\circ}$. Each of the lateral edges forms an angle of $45^{\circ}$ with the base plane. Find the volume of the pyramid. | \frac{b^{3}\sqrt{3}}{24} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $c=2$, $2\sin A= \sqrt {3}a\cos C$.
(1) Find the measure of angle $C$;
(2) If $2\sin 2A+ \sin (2B+C)= \sin C$, find the area of $\triangle ABC$. | \dfrac {2 \sqrt {3}}{3} |
2. If $x+2 y=14$ and $y=3$, what is the value of $2 x+3 y$ ? | 25 |
$\stackrel{\text { Senderov B.A. }}{ }$.
Find the smallest natural number that cannot be represented in the form $\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a, b, c, d$ are natural numbers. | 11 |
4.5 Try to find the number of positive integer solutions to the indeterminate equation $\frac{1}{x+1}+\frac{1}{y}+\frac{1}{(x+1) y}=\frac{1}{1991}$.
(Japan Mathematical Olympiad, 1991) | 64 |
For positive integer $n$, find all pairs of coprime integers $p$ and $q$ such that $p+q^2=(n^2+1)p^2+q$ | (p, q) = (n + 1, n^2 + n + 1) |
If $a$ and $b$ are real numbers, then "$a\neq b$" is ( ) of "$\frac{a}{b}+\frac{b}{a}＞2$".
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither necessary nor sufficient condition | B |
22) In a golf tournament, 256 participants are competing. The tournament is structured such that in each round, 4 participants compete: the winner advances to the next round while the other 3 participants are eliminated. How many rounds are necessary to determine the absolute winner of the tournament?
(A) 16
(B) 64
$\b... | 85 |
## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(-10 ; 0 ; 9)$
$B(12 ; 4 ; 11)$
$C(8 ; 5 ; 15)$ | -4x+y+4z-76=0 |
12・19 If $a \pm b i(b \neq 0)$ are the complex roots of the equation $x^{3}+q x+r=0$, where $a$, $b, q$ and $r$ are real numbers. Then $q$ expressed in terms of $a, b$ is
(A) $a^{2}+b^{2}$.
(B) $2 a^{2}-b^{2}$.
(C) $b^{2}-a^{2}$.
(D) $b^{2}-2 a^{2}$.
(E) $b^{2}-3 a^{2}$.
(23rd American High School Mathematics Examinati... | b^{2}-3a^{2} |
Select two numbers $x$ and $y$ from the interval $[0,1]$. The probability that $y \geqslant \sqrt{1-x^2}$ is ( ).
A: $\dfrac{\pi}{6}$
B: $\dfrac{\pi}{4}$
C: $1- \dfrac{\pi}{6}$
D: $1- \dfrac{\pi}{4}$ | D |
Vojta bought 24 identical square tiles. The side of each tile measured $40 \mathrm{~cm}$. Vojta wanted to create a rectangular platform in front of the cottage with the smallest possible perimeter. How many meters did the perimeter of the paved rectangle measure, given that no tiles were left over? Vojta did not cut or... | 8 |
6. $\sum_{i=0}^{50} \sum_{j=0}^{50} \mathrm{C}_{50}^{i} \mathrm{C}_{50}^{j}$ modulo 31 is
$\qquad$ . | 1 |
Given the function $f(x)=\sin(\omega x+\varphi)$ (where $\omega\ \ \gt 0$, $|\varphi|＜\frac{π}{2}$) takes its maximum value at $x=\frac{π}{6}$, and the minimum distance between the two symmetric axes is $\pi$. If the line $l$: $y=-x+\frac{2π}{3}$ intersects the curve $y=f(x)$, the number of intersection points is ( ).
... | A |
(5) Find the smallest positive integer $n$ with the following property: If each vertex of a regular $n$-gon is arbitrarily colored with one of three colors (red, yellow, blue), then among these $n$ vertices, there must exist four vertices of the same color that form the vertices of an isosceles trapezoid (a convex quad... | 17 |
Which of the following expressions follows the rules of algebraic notation?
A: $a\times 5$
B: $a7$
C: $3\frac{1}{2}x$
D: $-\frac{7}{8}x$ | D |
Write down the first four terms of the sequences that satisfy the conditions, and deduce the general formula for each sequence:
(1) $a_1=0$, $a_{n+1}=a_n+(2n-1)$ ($n\in\mathbb{N}^*$);
(2) $a_1=3$, $a_{n+1}=3a_n$ ($n\in\mathbb{N}^*$). | a_n=3^n |
2. Solve the equation $8 \sin ^{4}(\pi x)-\sin ^{2} x=\cos ^{2} x-\cos (4 \pi x)$. In your answer, specify the sum of the roots that belong to the interval $[-1 ; 2]$.
points) | 5 |
17. Water makes up 80 per cent of fresh mushrooms. However, water makes up only 20 per cent of dried mushrooms. By what percentage does the mass of a fresh mushroom decrease during drying?
A 60
B 70
С 75
D 80
E 85 | 75 |
2. Given that $a, b, c$ satisfy
$$
\begin{array}{l}
|2 a-4|+|b+2|+\sqrt{(a-3) b^{2}}+a^{2}+c^{2} \\
=2+2 a c .
\end{array}
$$
Then the value of $a-b+c$ is ( ).
(A) 4
(B) 6
(C) 8
(D) 4 or 8 | D |
Given the function $f(x)=\sin 2x-2 \sqrt{3}\sin ^{2}x+ \sqrt{3}+1$.
$(1)$ Find the smallest positive period of $f(x)$ and its intervals of increase;
$(2)$ When $x\in\left[- \frac{\pi}{6}, \frac{\pi}{6}\right]$, find the range of $f(x)$. | \left[1,3\right] |
In triangle $\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to angles $A$, $B$, $C$ respectively, and satisfy $\left(a+b+c\right)\left(a+b-c\right)=3ab$.
$(1)$ Find the measure of angle $C$;
$(2)$ If $\triangle ABC$ is an acute triangle, find the range of $\frac{a+2b}{c}$. | (\frac{4\sqrt{3}}{3}, \frac{2\sqrt{21}}{3}] |
Among the following equations, the solution $x=1$ is for which equation?
A: $x+1=0$
B: $3x=-3$
C: $x-1=2$
D: $2x+2=4$ | D |
【Question 12】
Use $1,2,3,4,5$ to form a five-digit number, such that the difference between any two adjacent digits is at least 2. How many such five-digit numbers are there? $\qquad$个. | 14 |
4. If $[x]$ denotes the greatest integer not exceeding $x$, then $\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\cdots+\left[\frac{2^{1000}}{3}\right]=$ | \frac{1}{3}(2^{1001}-2)-500 |
2. Let the set $A=\left\{x \mid a x^{2}+3 x-2 a=0\right\}$ (where $a$ is a non-zero real number), $B=\left\{x \mid 2 x^{2}-5 x-42 \leqslant 0\right\}$, if $A \cap B=A$, then the range of values for $a$ is $\qquad$ . | (-\infty,-\frac{9}{17}]\cup[\frac{42}{41},+\infty) |
1. (5 points) Find the value of the function $f(x)$ at the point $x_{0}=4000$, if $f(0)=1$ and for any $x$ the equality $f(x+2)=f(x)+3 x+2$ holds. | 11998001 |
2. Masha, Dasha, and Sasha are tasked with harvesting all the currants from the bushes on the garden plot. Masha and Dasha together can collect all the berries in 7 hours 30 minutes, Masha and Sasha - in 6 hours, and Dasha and Sasha - in 5 hours. How many hours will it take the children to collect all the berries if th... | 4 |
Given $f(x)=x+ \frac {a}{x}-\ln x$, where $a\in \mathbb{R}$,
$(1)$ If $a=2$, determine the intervals of monotonicity for $f(x)$;
$(2)$ When $a\leq - \frac {1}{4}$, if $f(x)\geq -\ln 2$ holds for all $x\in [2,e]$, find the range of $a$. | a \in [-4, -\frac{1}{4}] |
7. When $x \in[0,2]$, the function $f(x)=a x^{2}+4(a+1) x-3$ reaches its maximum value at $x=2$, the range of values for $a$ is $\qquad$ . | \geqslant-\frac{1}{2} |
In triangle $\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to angles $A$, $B$, $C$ respectively, and $8ab\sin C = 3(b^{2} + c^{2} - a^{2})$. If $a = \sqrt{10}$ and $c = 5$.
(I) Find $\cos A$.
(II) Find the area $S$ of triangle $\triangle ABC$. | S = \frac{15}{2} \quad \text{or} \quad S = \frac{9}{2} |
In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $a=4$, $\cos A= \frac{3}{4}$, $\sin B= \frac{5\sqrt{7}}{16}$, and $c>4$.
(1) Find the length of $b$.
(2) It is known that the radius $r$ of the incircle of triangle $ABC$ is given by $r= \frac{2S}{l}$, whe... | A = \frac{7\pi}{4} |
If the function $f(x)=a(x-2)e^{x}+\ln x-x$ has a unique extremum, and this extremum is less than $0$, then the range of the real number $a$ is $(\quad)$
A: $\left(- \dfrac {1}{e^{2}}, \dfrac {1}{e^{2}}\right)$
B: $\left(- \dfrac {1}{e}, \dfrac {1}{e}\right)$
C: $\left(- \dfrac {1}{e^{2}},0\right]$
D: $\left(- \... | D |
5. Find an array $\left(a_{1}, a_{2}, \cdots, a_{n}\right)\left(a_{1}<a_{2}<\cdots<a_{n}\right)$ consisting of $n$ distinct positive integers, such that for any non-negative integer $d$, we have
$$
\frac{\left(a_{1}+d\right)\left(a_{2}+d\right) \cdots\left(a_{n}+d\right)}{a_{1} a_{2} \cdots a_{n}} \in \mathbf{Z} .
$$ | (1,2,\cdots,n) |
5. In the diagram, point $P$ is inside quadrilateral $A B C D$. Also, $D A=D P=D C$ and $A P=A B$. If $\angle A D P=\angle C D P=2 x^{\circ}, \angle B A P=(x+5)^{\circ}$, and $\angle B P C=(10 x-5)^{\circ}$, what is the value of $x$ ? | 13 |
$$
\begin{array}{l}
\text { 9. Let } a>0, b>0 \text {, and } \sqrt{a}(\sqrt{a}+2 \sqrt[3]{b}) \\
=\sqrt[3]{b}(\sqrt{a}+6 \sqrt[3]{b}) . \\
\text { Then } \frac{2 a^{4}+a^{3} b-128 a b^{2}-64 b^{3}+b^{4}}{a^{4}+2 a^{3} b-64 a b^{2}-128 b^{3}+2 b^{4}}
\end{array}
$$
9. Let $a>0, b>0$, and $\sqrt{a}(\sqrt{a}+2 \sqrt[3]{b}... | \frac{1}{2} |
Let $a=\log_{3}\frac{1}{4}$, $b=\left(\frac{1}{3}\right)^{0.3}$, $c=\log_{2}(\log_{2}\sqrt{2})$, then ( )
A: $b<c<a$
B: $a<b<c$
C: $c<a<b$
D: $a<c<b$ | \text{D} |
Given $p$: $x=2$, $q$: $0 < x < 3$, then $p$ is the $(\quad)$ of $q$.
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither necessary nor sufficient condition | A |
An unfair coin comes up heads with probability $\frac{4}{7}$ and tails with probability $\frac{3}{7}$. Aerith and Bob take turns flipping the coin until one of them flips tails, with Aerith going first. What is the probability that Aerith wins the game? | \frac{7}{11} |
32. Using the original data from the previous problem, find the probability that music in the disco style is recorded on two randomly taken cassettes in succession. Consider two cases: the DJ, before taking the second cassette,
a) returns the first cassette to the case;
b) does not return the first cassette to the ca... | \frac{4}{9}\approx0.4444\frac{20}{30}\cdot\frac{19}{29}\approx0.4368 |
Given that point \\(A\\) has coordinates \\((1,-1)\\), \\(AB=3\\), and \\(AB\\) is parallel to the \\(y\\)-axis, the coordinates of point \\(B\\) are __________. | (1,-4) \text{ or } (1,2) |
8. $E$ and $F$ are the midpoints of edges $BD$ and $CD$ of the regular tetrahedron $ABCD$, respectively. The cosine value of the dihedral angle formed by plane $ABC$ and plane $AEF$ is $\qquad$ | \frac{5 \sqrt{33}}{33} |
B1. Solve the exponential equation $16 \cdot 16 \sqrt{4^{5 x}}=\sqrt[4]{4^{3 x+10}} \cdot \sqrt[3]{4^{5 x+1}}$. | -14 |
3 [Central symmetry helps solve the task]
What is the maximum number of pawns that can be placed on a chessboard (no more than one pawn per square), if:
1) a pawn cannot be placed on the e4 square;
2) no two pawns can stand on squares that are symmetric relative to the e4 square?
# | 39 |
There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane.
The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of diff erent colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ includin... | 4 |
Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$ shares a focus with the parabola $y^2 = 8x$, and the eccentricity of the hyperbola $C$ is 2, find the equation of the hyperbola $C$. | x^{2} - \frac{y^{2}}{3} = 1 |
Let $A_1$, $\ldots$, $A_{2022}$ be the vertices of a regular $2022$-gon in the plane. Alice and Bob play a game. Alice secretly chooses a line and colors all points in the plane on one side of the line blue, and all points on the other side of the line red. Points on the line are colored blue, so every point in the pla... | 22 |
3. Function
$$
y=\frac{(\sin x-1)(\cos x-1)}{2+\sin 2 x}(x \in \mathbf{R})
$$
The maximum value is ( ).
(A) $\frac{\sqrt{2}}{2}$
(B) 1
(C) $\frac{1}{2}+\frac{\sqrt{2}}{2}$
(D) $\sqrt{2}$ | 1 |
3. Given $P_{1}\left(x_{1}, y_{1}\right), P_{2}\left(x_{2}, y_{2}\right)$ are points on and outside the line $l$ respectively, if the equation of the line $l$ is $f(x, y)=$ 0, then the equation $f(x, y)-f\left(x_{1}, y_{1}\right)-f\left(x_{2}, y_{2}\right)=0$ represents
A. a line coinciding with $l$
B. a line passing t... | C |
$13 \cdot 28$ If real numbers $x, y$ satisfy the condition $2 x^{2}-6 x+y^{2}=0$, then the maximum value of $x^{2}+y^{2}+2 x$ is
(A) 14.
(B) 15.
(C) 16.
(D) Cannot be determined.
(Chinese Jiangsu Province Junior High School Mathematics Competition, 1991) | 15 |
Given the complex number $z= \frac{1-i}{i}$, find the value of $|z|$. | \sqrt{2} |
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