problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
2. Given three linear functions: $y=a x+b+3, y=-b x+a-2, y=2 x-8$. It is known that the graphs of the first and second intersect on the y-axis, and the graphs of the second and third - on the x-axis. Find the point of intersection of the graphs of the first and third functions. | (-3,-14) |
4. The function
$$
y=\frac{1}{|\sin x|}+\frac{1}{|\cos x|}+\frac{1}{|\tan x|}+\frac{1}{|\cot x|}
$$
has a minimum value of | 2(\sqrt{2}+1) |
Given that the sum of the first $n$ terms of a geometric sequence $\{a_n\}$ is $S_n$, where $a_2 = 2$ and $a_5 = 16$, find the minimum value of $\frac{S_{2n} + S_n + 18}{2^n}$. | 9 |
1. The sequence $a_{n}$ is constructed as follows: $a_{0}=4, a_{2 n+1}=a_{2 n}-3, a_{2 n}=-2 a_{2 n-1}$. Find $a_{100}$.
If the answer requires an exponentiation operation, use the symbol «^» | 2^{51}+2 |
## Task A-2.4.
The vertices of the pentagon $A B C D E$ lie on the same circle. If $\varangle C A D=50^{\circ}$, determine
$$
\varangle A B C+\varangle A E D \text {. }
$$ | 230 |
The graph of the function $f(x)= \sqrt {3}\sin x-\cos x$ is translated $m$ units to the left $(m > 0)$. If the resulting graph corresponds to an even function, find the smallest value of $m$. | \frac {2}{3}\pi |
The left most digit of an integer of length 2000 digits is 3. In this integer, any two consecutive digits must be divisible by 17 or 23 . The 2000th digit may be either ' $a$ ' or ' $b$ '. What is the value of $a+b$ ?
(A) 3
(B) 7
(C) 4
(D) 10
(E) 17 | 7 |
Given $2a=3b=k$ ($k \neq 1$), and $2a+b=ab$, the value of the real number $k$ is
A: 6
B: 9
C: 12
D: 18 | \text{D} |
4. (3 points) In the grid paper in the figure, there are five small squares numbered $1,2,3,4,5$. By shading two of them, the shaded parts in the figure can exactly form the net of a cube. The numbers of these two squares can be ( )
A. 1,2
B. 2,3
C. 3,4
D. 4,5 | D |
To obtain the graph of the function $y=\sin \left(3x+\frac{\pi }{4}\right)$, it is only necessary to shift the graph of the function $y=\sin 3x$ by
A: Shift to the right by $\frac{\pi}{12}$ units
B: Shift to the left by $\frac{\pi}{3}$ units
C: Shift to the left by $\frac{\pi}{12}$ units
D: Shift to the right by $\frac... | C |
## 53. Math Puzzle 10/69
At the beginning of a group meeting, the Pioneers greet each other with handshakes. A total of 66 handshakes are exchanged.
How many friends attended the meeting? | 12 |
60. Riding in windy weather. A cyclist takes 3 minutes to travel 1 km if the wind is at his back, and 4 minutes if he has to ride against a headwind. How long will it take him to travel 1 km if the wind dies down? Someone might say that since the arithmetic mean of 3 and 4 is $3 \frac{1}{2}$, the cyclist will need $3 \... | 3\frac{3}{7} |
If a number eight times as large as $x$ is increased by two, then one fourth of the result equals
$\text{(A)} \ 2x + \frac{1}{2} \qquad \text{(B)} \ x + \frac{1}{2} \qquad \text{(C)} \ 2x+2 \qquad \text{(D)}\ 2x+4 \qquad \text{(E)}\ 2x+16$ | 2x+\frac{1}{2} |
[ Properties of medians. The centroid of a triangle.] [ The ratio of areas of similar triangles ]
In triangle $F G H$, angle $G$ is right, $F G=8, G H=2$. Point $D$ lies on side $F H$, $A$ and $B$ are the points of intersection of the medians of triangles $F G D$ and $D G H$. Find the area of triangle $G A B$. | \frac{16}{9} |
Simplify: \\( \dfrac {1}{\cos 80 ^{\circ} }- \dfrac {\sqrt {3}}{\sin 80 ^{\circ} }=\)\_\_\_\_\_\_． | 4 |
Let $a$, $b$, $c$ be positive real numbers, and they satisfy $a+b+c=1$. Find the minimum value of $$\frac {bc}{a}+ \frac {ac}{b}+ \frac {ab}{c}$$. | 1 |
The graph of the function $y=a^x+2$ ($a>0$ and $a\neq 1$) must pass through the point ( )
A: (0, 1)
B: (0, 3)
C: (1, 0)
D: (3, 0) | \text{B} |
Question 111, Try to find the value of $\sum_{\mathrm{k}=0}^{2022} \mathrm{C}_{2022}^{\mathrm{k}} \cos \frac{(1011-\mathrm{k}) \pi}{2}$. | 2^{1011} |
10. 12 people gather for dinner at the same restaurant every week, each time divided into three tables, with four people at each table. It is required that any two people sit at the same table at least once. Question: What is the minimum number of weeks such dinners need to be held? | 5 |
The number $2^{29}$ has a $9$-digit decimal representation that contains all but one of the $10$ (decimal) digits. Determine which digit is missing | 4 |
3. A circle $\omega$ is circumscribed around triangle $ABC$. A line tangent to $\omega$ at point $C$ intersects the ray $BA$ at point $P$. On the ray $PC$ beyond point $C$, a point $Q$ is marked such that $PC = QC$. The segment $BQ$ intersects the circle $\omega$ again at point $K$. On the smaller arc $BK$ of the circl... | 30 |
1. Find the sum of the digits of the number $A$, if $A=2^{63} \cdot 4^{25} \cdot 5^{106}-2^{22} \cdot 4^{44} \cdot 5^{105}-1$. Answer: 959. | 959 |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{2 \pi} \sin ^{2}\left(\frac{x}{4}\right) \cos ^{6}\left(\frac{x}{4}\right) d x
$$ | \frac{5}{64}\pi |
The function $y=f\left(x\right)$ is a monotonically increasing function defined on $R$. If $t\neq 0$, then ()
A: $f\left(t\right) \gt f\left(2t\right)$
B: $f(t^{2}) \gt f\left(t\right)$
C: $f(t^{2}+t) \gt f\left(t\right)$
D: $f(t^{2}+t) \gt f\left(t+1\right)$ | C |
25. Anna, Bridgit and Carol run in a $100 \mathrm{~m}$ race. When Anna finishes, Bridgit is $16 \mathrm{~m}$ behind her and when Bridgit finishes, Carol is $25 \mathrm{~m}$ behind her. The girls run at constant speeds throughout the race. How far behind was Carol when Anna finished?
A $37 \mathrm{~m}$
B $41 \mathrm{~m}... | 37\mathrm{~} |
Given an geometric sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, satisfying $S_4=2a_4-1$ and $S_3=2a_3-1$.
$(1)$ Find the general formula for $\{a_n\}$;
$(2)$ Let $b_n=\log_{\sqrt{2}}\left(\frac{16}{S_n+1}\right)$, find the maximum value of $b_1+b_2+\ldots+b_n$. | 12 |
In $\triangle A B C$, the three sides $a, b, c$ form an arithmetic sequence. Find the value of $5 \cos A-4 \cos A \cos C+5 \cos C$. | 4 |
1. Let the pair (1,1) be a solution of the equation $\frac{x}{a}+\frac{y}{b}=1$, where $a$ and $b$ are non-zero real parameters. Determine $a$ and $b$ if $a^{2}+b^{2}=\frac{45}{4}$. | (,b)=(3,\frac{3}{2}),(\frac{3}{2},3),(\frac{-5+\sqrt{65}}{4},\frac{-5-\sqrt{65}}{4}),(\frac{-5-\sqrt{65}}{4},\frac{-5+\sqrt{65}}{4}) |
## Problem Statement
Find the derivative.
$$
y=\ln \frac{\sqrt{5}+\tan \frac{x}{2}}{\sqrt{5}-\tan \frac{x}{2}}
$$ | \frac{\sqrt{5}}{6\cos^{2}\frac{x}{2}-1} |
Given a geometric sequence $\{a_n\}$, the sum of its first $n$ terms is $S_n$. It is known that $a_1+a_4=-\frac{7}{16}$, and for any $n\in\mathbb{N}^*$, $S_n$, $S_{n+2}$, $S_{n+1}$ form an arithmetic sequence;
(Ⅰ) Find the general formula for the sequence $\{a_n\}$;
(Ⅱ) Given $b_n=n$ ($n\in\mathbb{N}_+$), let $T_n=... | \left[\frac{1}{7},+\infty\right) |
$p>2$ persons participate at a chess tournament, two players play at most one game against each other. After $n$ games were made, no more game is running and in every subset of three players, we can find at least two that havem't played against each other.
Show that $n \leq \frac{p^{2}}4$. | n \leq \frac{p^2}{4} |
7. Find all positive integers $m, n$, such that $2^{m}+3^{n}$ is a perfect square. | (4,2) |
The number of subsets of the set $M=\{1,2,3,4,5\}$ is ____. | 32 |
1. If the equation
$$
x^{2}+(1-2 \mathrm{i}) x+3 m-\mathrm{i}=0(m \in \mathbf{R})
$$
has one real root, then its other root is ( ).
(A) $-\frac{3}{2}+2$ i
(B) $\frac{3}{2}-2 \mathrm{i}$
(C) $-\frac{1}{2}+2$ i.
(D) $\frac{1}{2}-2 \mathrm{i}$ | C |
There are the following statements about two variables $x$, $y$ and their correlation coefficient $r$:
(1) If $r>0$, then as $x$ increases, $y$ also increases correspondingly;
(2) If $r<0$, then as $x$ increases, $y$ also increases correspondingly;
(3) If $r=1$ or $r=-1$, then the relationship between $x$ and $y$... | \text{C} |
138. Solve the equation: $x^{4}-2 x^{2}-400 x=9999$ (solved elementarily). | x_1=11,\,x_2=-9,\,x^2+2x+101= |
Given that the slope of line $l_1$ is 2, and $l_1 \parallel l_2$, line $l_2$ passes through the point (-1,1) and intersects the y-axis at point P, then the coordinates of point P are ( )
A: (3,0)
B: (－3,0)
C: (0,－3)
D: (0,3) | \text{D} |
If the chord of the ellipse $\frac{x^2}{36} + \frac{y^2}{9} = 1$ is bisected by the point $(4,2)$, then the equation of the line on which this chord lies is _________. | x + 2y - 8 = 0 |
For all positive integers $n$, let $f(n)$ return the smallest positive integer $k$ for which $\tfrac{n}{k}$ is not an integer. For example, $f(6) = 4$ because $1$, $2$, and $3$ all divide $6$ but $4$ does not. Determine the largest possible value of $f(n)$ as $n$ ranges over the set $\{1,2,\ldots, 3000\}$. | 11 |
For any real numbers $a$, $b$, and $c$, consider the following propositions:
1. "$a=b$" is a necessary and sufficient condition for "$ac=bc$".
2. "$a+5$ is an irrational number" is a necessary and sufficient condition for "$a$ is an irrational number".
3. "$a>b$" is a sufficient condition for "$a^2>b^2$".
4. "$a<5$" is... | 2, 4 |
A four-digit number, composed of the digits 2, 3, 4, 5, 6 without repetition, and where 5 and 6 are adjacent, how many odd numbers are there in total? | 14 |
There are four statements below: (1) The result of $1+2-3+4-5+6-7+8-9+10-\ldots-2001+2002$ is an even number; (2) $(\text{odd} \times \text{odd}) \times (\text{odd} - \text{odd}) = \text{odd}$; (3) The sum of 2002 consecutive natural numbers is necessarily even; (4) There exist integers $a, b$ such that $(a+b)(a-b)=200... | \text{A} |
1. Find the value of the expression $\left(\left(\frac{3}{a-b}+\frac{3 a}{a^{3}-b^{3}} \cdot \frac{a^{2}+a b+b^{2}}{a+b}\right)\right.$ ? $\left.\frac{2 a+b}{a^{2}+2 a b+b^{2}}\right) \cdot \frac{3}{a+b}$ when $a=2023, b=2020$ | 3 |
3. The book is sewn from 12 identical booklets. Each booklet consists of several double sheets nested within each other. The booklets of the book are sewn sequentially one after another. All pages of the book are numbered, starting from 1. The sum of the numbers of four pages of one of the double sheets in the fourth b... | 288 |
4-0. Kisa and Busya came to the cafeteria during the break, where only muffins and doughnuts were sold, costing a whole number of rubles. Kisa bought 8 muffins and 3 doughnuts, spending less than 200 rubles, while Busya bought 4 muffins and 5 doughnuts, spending more than 150 rubles. Name the highest possible price of ... | 19 |
1. A car was moving at a speed of $V$. Upon entering the city, the driver reduced the speed by $x \%$, and upon leaving the city, increased it by $0.5 x \%$. It turned out that this new speed was $0.6 x \%$ less than the speed $V$. Find the value of $x$. | 20 |
Initially, there are $n$ red boxes numbered with the numbers $1,2,\dots ,n$ and $n$ white boxes numbered with the numbers $1,2,\dots ,n$ on the table. At every move, we choose $2$ different colored boxes and put a ball on each of them. After some moves, every pair of the same numbered boxes has the property of either t... | n = 11m |
Given four numbers, the sum of every three of them is respectively 24, 36, 28, 32. Find the average of these four numbers. | 10 |
$$
\begin{array}{l}
\frac{a}{a^{3}+a^{2} b+6 b^{2}+b^{3}}+\frac{b}{a^{3}-a^{2} b+a b^{2}-b^{3}} \\
+\frac{1}{a^{2}-b^{2}}-\frac{1}{b^{2}+a^{2}}-\frac{a^{2}+3 b^{2}}{a^{4}-b^{4}} .
\end{array}
$$ | 0 |
Let the universal set $U=\{1,2,3,4,5,6\}$, and $M=\{1,3,4\}$. Then, the complement of $M$ in $U$, denoted as $\complement_U M$ is ( )
A: $\{3,5,6\}$
B: $\{1,3,5\}$
C: $\{2,5,6\}$
D: $U$ | \text{C} |
3. The boat traveled 165 km upstream against the current and then returned. On the return trip, it took 4 hours less than the trip there. Find the boat's own speed if the river's current speed is 2 km/h. | 13 |
3. Find the area of the region defined by the inequality: $|y-| x-2|+| x \mid \leq 4$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 32 |
Problem 5. Small Island and Big Island have a rectangular shape and are divided into rectangular counties. In each county, a road is laid along one of the diagonals. On each island, these roads form a closed path that does not pass through any point more than once. Here is how Small Island is organized, with a total of... | 9 |
A telephone pole that is $10 \mathrm{~m}$ tall was struck by lightning and broken into two pieces. The top piece, $A B$, has fallen down. The top of the pole is resting on the ground, but it is still connected to the main pole at $B$. The pole is still perpendicular to the ground at $C$. If the angle between $A B$ and ... | \frac{10}{3}\mathrm{~} |
5. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=2, a_{2}=6$, and $\frac{a_{n+2}+a_{n}}{a_{n+1}+1}=2$. Then $\lim _{n \rightarrow+\infty} \frac{\sqrt{a_{1}}+\sqrt{a_{2}}+\cdots+\sqrt{a_{n}}}{n^{2}}$ $=$ . $\qquad$ | \frac{1}{2} |
Sure, here is the translated text:
```
II. (50 points) Let $x_{1}, x_{2}, \cdots, x_{2000}$ be 2000 real numbers, satisfying $x_{i} \in [0,1] (i=1,2, \cdots, 2000)$. Define
$$
F_{i}=\frac{x_{i}^{2000}}{\sum_{j=1}^{2000} x_{j}^{3999} - x_{i}^{3999} + 2000}.
$$
Find the maximum value of $\sum_{i=1}^{2000} F_{i}$, and p... | \frac{2000}{3999} |
2. Given $\log _{\sqrt{7}}(5 a-3)=\log _{\sqrt{a^{2}+1}} 5$. Then the real number
$$
a=
$$
. $\qquad$ | 2 |
Find all positive integers $n>1$ such that
\[\tau(n)+\phi(n)=n+1\]
Which in this case, $\tau(n)$ represents the amount of positive divisors of $n$, and $\phi(n)$ represents the amount of positive integers which are less than $n$ and relatively prime with $n$.
[i]Raja Oktovin, Pekanbaru[/i] | n = 4 |
11. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{n+1}=-\frac{1}{2} a_{n}+\frac{1}{3^{n}}\left(n \in \mathbf{Z}_{+}\right) \text {. }
$$
Find all values of $a_{1}$ such that $\left\{a_{n}\right\}$ is a monotonic sequence, i.e., $\left\{a_{n}\right\}$ is either an increasing sequence or a decreasing sequenc... | a_{1}=\frac{2}{5} |
Five students named Fred, Gail, Henry, Iggy, and Joan are seated around a circular table in that order. To decide who goes first in a game, they play "countdown”. Henry starts by saying ' 34 ', with Iggy saying ' 33 '. If they continue to count down in their circular order, who will eventually say ' 1 '?
(A) Fred
(B) G... | A |
8. The integer sequence $\left\{a_{i, j}\right\}(i, j \in \mathbf{N})$, where,
$$
\begin{array}{l}
a_{1, n}=n^{n}\left(n \in \mathbf{Z}_{+}\right), \\
a_{i, j}=a_{i-1, j}+a_{i-1, j+1}(i 、 j \geqslant 1) .
\end{array}
$$
Then the unit digit of $a_{128,1}$ is $\qquad$ | 4 |
5. Let $a$ and $b$ be real numbers such that
$$
a^{3}-3 a b^{2}=44 \text{ and } b^{3}-3 a^{2} b=8
$$
Determine $a^{2}+b^{2}$. | 10\sqrt[3]{2} |
Example 5. (Hungarian Mathematical Olympiad Problem) Given the area of $\triangle ABC$ is $S$, and $\angle C=\alpha$. What kind of triangle will make the side $c$ opposite to $\angle C$ the shortest? (Figure 4) | c^{2}=4 S \frac{1-\cos \alpha}{\sin \alpha} |
In the three-dimensional Cartesian coordinate system, if $P(3,-2,1)$, then the coordinates of the point symmetric to $P$ with respect to the $xOz$ plane are:
A: $(-3,-2,-1)$
B: $(3,2,1)$
C: $(-3,2,-1)$
D: $(3,-2,-1)$ | \text{B} |
8.3. The square of the sum of the digits of the number $A$ is equal to the sum of the digits of the number $A^{2}$. Find all such two-digit numbers $A$ and explain why there are no others. | 10,20,11,30,21,12,31,22,13 |
Solve the following inequality:
$$
\frac{2+7 x-15 x^{2}}{5-x+6 x^{2}}>0
$$ | -\frac{1}{5}<x<\frac{2}{3} |
Question 97, In $\triangle ABC$, $AB=\sqrt{2}, \angle ACB=\frac{\pi}{4}, O$ is the circumcenter of $\triangle ABC$, try to find the range of $\overrightarrow{OC} \cdot \overrightarrow{AB} + \overrightarrow{CA} \cdot \overrightarrow{CB}$. | (-1,3) |
Given $log_x(2x^2+x-1) > log_x 2 - 1$, the range of $x$ is
A: $x 1$ and $x \neq 1$
C: $x > 1$
D: $0 < x < 1$ | \text{B} |
90. The school plans to organize a movie for the students on January 4th or January 10th. After the date is determined, the teacher informs the class president, but due to the similar pronunciation of “four” and “ten,” there is a $10 \%$ chance that the class president will hear it incorrectly (mistaking 4 for 10 or 10... | 82 |
Given sets $A=\left\{{x|1＜{3^x}≤9}\right\}$, $B=\left\{{x|\frac{{x+2}}{{x-1}}≤0}\right\}$, then $A\cap B=\left(\ \ \right)$
A: $\left[0,1\right]$
B: $(0,1]$
C: $(0,1)$
D: $\left[0,1\right)$ | C |
In the geometric sequence $\{a\_n\}$, $a\_1=-16$, $a\_4=8$, find $a\_7=(\;\;\;)$. | -4 |
[ Properties and characteristics of an isosceles triangle.] [Sum of the angles of a triangle. Theorem of the exterior angle.]
Triangle $A B C$ is isosceles $(A B=B C)$. Segment $A M$ divides it into two isosceles triangles with bases $A B$ and $M C$. Find angle $B$. | 36 |
3. Given non-zero real numbers $a, b, c$ satisfy
$$
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c} \text {. }
$$
Then for positive integer $k$ such that
(1) $\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^{2 k+1}=\frac{1}{a^{2 k+1}}+\frac{1}{b^{2 k+1}}+\frac{1}{c^{2 k+1}}$,
(2) $\frac{1}{a^{2 k+1}}+\frac{1}{b^{... | B |
4. Given a regular quadrilateral pyramid $S-A B C D$ with a height of 2 and a base edge length of $\sqrt{2}$, points $P, Q$ lie on segments $B D, S C$ respectively. The shortest distance between $P$ and $Q$ is
A. $]$
B. $\frac{2 \sqrt{3}}{3}$
C. $\frac{2 \sqrt{5}}{5}$
D. $\frac{\sqrt{10}}{5}$ | \frac{2\sqrt{5}}{5} |
1. Calculate: $1-2+3-4+\cdots+1993-1994+1995$ $=$ | 998 |
13.007. The price of the product was first reduced by $20 \%$, then the new price was reduced by another $15 \%$, and finally, after recalculation, a further reduction of $10 \%$ was made. By what total percentage was the original price of the product reduced? | 38.8 |
4. After swapping the last two digits of a three-digit number, a new three-digit number is obtained. The sum of this three-digit number and the original three-digit number is a four-digit number with 1 in the thousands place, 7 in the hundreds place, and 3 in the tens place. Find the last digit of this four-digit numbe... | 2 |
Determine all numbers $x, y$ and $z$ satisfying the system of equations
$$\begin{cases} x^2 + yz = 1 \\ y^2 - xz = 0 \\ z^2 + xy = 1\end{cases}$$ | x = y = z = \pm \frac{\sqrt{2}}{2} |
If $A-B+C=0$, then the line $Ax+By+C=0$ must pass through ( )
A: $(0,1)$
B: $(1,0)$
C: $(1,-1)$
D: $(-1,-1)$． | C |
Problem 8.3. In the class, there are 29 students: some are excellent students and some are troublemakers. Excellent students always tell the truth, while troublemakers always lie.
All students in this class sat around a round table.
- Several students said: “There is exactly one troublemaker next to me.”
- All other ... | 10 |
## Subject I
Consider the real numbers:
$$
\begin{aligned}
& x=\sqrt{2}+\sqrt{2^{2}}+\sqrt{2^{3}}+\ldots .+\sqrt{2^{2014}} \\
& y=2+2^{2}+2^{3}+\ldots .+2^{2014}
\end{aligned}
$$
a) Show that $\quad x=(2+\sqrt{2})\left(2^{1007}-1\right)$
b) Show that the number $n=2\left(\frac{y}{x} \cdot \frac{\sqrt{2}+1}{\sqrt{2}... | (2^{504})^{2} |
Given that $(x^2 - 1) + (x^2 + 3x + 2)i$ is a purely imaginary number, find the value of the real number $x$. | 1 |
[b]Q[/b] Let $n\geq 2$ be a fixed positive integer and let $d_1,d_2,...,d_m$ be all positive divisors of $n$. Prove that:
$$\frac{d_1+d_2+...+d_m}{m}\geq \sqrt{n+\frac{1}{4}}$$Also find the value of $n$ for which the equality holds.
[i]Proposed by dangerousliri [/i] | n = 2 |
If traveling east for $3km$ is denoted as $+3km$, then traveling west for $2km$ should be denoted as:
A: $+2km$
B: $-2km$
C: $+3km$
D: $-3km$ | B |
When $|a| \leqslant 1$ and $|x| \leqslant 1$, the inequality $|x^2 - ax - a^2| \leqslant m$ always holds. Then, the range of the real number $m$ is $(\ )$.
A: $\left(\frac{3}{4}, +\infty\right)$
B: $\left(\frac{5}{4}, +\infty\right)$
C: $\left(\frac{3}{2}, +\infty\right)$
D: $\left(\frac{5}{2}, +\infty\right)$ | \text{B} |
5. In a certain figure skating competition, a referee bribery incident occurred. The competition committee decided to increase the number of referees from the original 9 to 14, but only 7 of the 14 referees' scores would be considered valid. If 2 of the 14 referees are corrupt, the probability that the valid scores do ... | \frac{3}{13} |
Given that $\{a_n\}$ is an arithmetic sequence with a non-zero common difference $d$, and $a_3 + a_9 = a_{10} - a_8$, find the value of $a_5$.
A. $-1$
B. $0$
C. $1$
D. $2$ | B |
6. Using the four digits 6, 7, 8, and 9, many four-digit numbers without repeated digits can be formed. If they are arranged in ascending order, 9768 is the th. | 21 |
Find all $ n > 1$ such that the inequality \[ \sum_{i\equal{}1}^nx_i^2\ge x_n\sum_{i\equal{}1}^{n\minus{}1}x_i\] holds for all real numbers $ x_1$, $ x_2$, $ \ldots$, $ x_n$. | n \le 5 |
Given the complex number $z=-2+i$, find the value of $\frac{z \cdot \overline{z}}{i}$ . | -5i |
7. In the expansion of $(\sqrt{x}+2)^{2 n+1}$, the sum of the coefficients of the terms where the power of $x$ is an integer is $\qquad$ | \frac{1}{2}(3^{2n+1}+1) |
4. Given $S_{\triangle M B C}=4, 3 A B=2 B C$, draw the perpendicular from point $C$ to the angle bisector $B E$ of $\angle A B C$, and let the foot of the perpendicular be $D$. Then $S_{\triangle B D C}=$ $\qquad$ | 3 |
4. As shown in the figure, the area of parallelogram $A B C D$ is $60, D P$ intersects $A C$ at $M, A M=\frac{1}{4} A C$. The area of $\triangle A P D$ is _. $\qquad$ | 10 |
5. [4] Find the smallest positive integer that is twice a perfect square and three times a perfect cube. | 648 |
In triangle $ABC$, $\tan B= \sqrt {3}$, $AB=3$, and the area of triangle $ABC$ is $\frac {3 \sqrt {3}}{2}$. Find the length of $AC$. | \sqrt {7} |
Given the function $f(x) = \begin{cases} 2^{x+1}, & x \leqslant 0 \\ -\log_{2}(x+1) + 2, & x > 0 \end{cases}$, and $f(a) = -1$, find $f(6 - a) = (\quad)$.
A: $1$
B: $2$
C: $3$
D: $4$ | A |
Given a circle $C: (x+1)^2 + (y-3)^2 = 9$ on which there exist two points $P$ and $Q$. These points are symmetric with respect to the line $x + my + 4 = 0$. Find the real number $m=$_______. | m = -1 |
A small island has $\frac{2}{5}$ of its surface covered by forest and $\frac{1}{4}$ of the remainder of its surface by sand dunes. The island also has 90 hectares covered by farm land. If the island is made up of only forest, sand dunes and farm land, what is the total area of the island, to the nearest hectare?
(A) 16... | 200 |
Given a geometric sequence $\{a_n\}$ with a common ratio not equal to 1, where $a_1=1$, $a_2=a$, and $a_{n+1}=k(a_n+a_{n+2})$ holds for any positive integer $n$. If for any three consecutive terms $a_m$, $a_{m+1}$, $a_{m+2}$, they can be arranged in some order to form an arithmetic sequence, then $k= \boxed{-\frac{2}{5... | -\frac{2}{5} |
8. In triangle $A B C$ with $\angle B=120^{\circ}$, the angle bisectors $A A_{1}, B B_{1}, C C_{1}$ are drawn. Segment $A_{1} B_{1}$ intersects the angle bisector $C C_{1}$ at point M. Find $\angle C B M$.
# | 30 |
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