problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
4. There are 8 cards; one side of each is clean, while the other side has the letters: И, Я, Л, З, Г, О, О, О printed on them. The cards are placed on the table with the clean side up, shuffled, and then sequentially flipped over one by one. What is the probability that the letters will form the word ЗОоЛОГИЯ when they... | \frac{1}{6720} |
20. Point $P$ moves on the parabola $y^{2}=a x$, point $Q$ is symmetric to $P$ about $(1,1)$. Then the equation of the trajectory of point $Q$ is $\qquad$. | (y-2)^{2}=a(2-x) |
When phenolphythalein is added to an aqueous solution containing one of the following solutes the solution turns pink. Which solute is present?
${ \textbf{(A)}\ \text{NaCl} \qquad\textbf{(B)}\ \text{KC}_2\text{H}_3\text{O}_2 \qquad\textbf{(C)}\ \text{LiBr} \qquad\textbf{(D)}\ \text{NH}_4\text{NO}_3 } $ | \text{B} |
If $(x+1)^n = x^n + \ldots + px^2 + qx + 1$ (where $n \in \mathbb{N}^*$), and $p + q = 6$, then $n = \ $. | 3 |
Calculate: $|\sqrt{25}-3|+\sqrt[3]{-27}-(-2)^{3}$. | 7 |
## Task 6 - 180736
Determine all rational numbers $a$ with the following property:
The product of the number $a$ and its absolute value is equal to the sum of the number $a$ and its absolute value. | 02 |
Given the equation of the line $l$ as $2x+y+1=0$.
$(1)$ Find the equation of the line $l_{1}$ passing through point $A(3,2)$ and perpendicular to line $l$.
$(2)$ Find the equation of line $l_{2}$ that satisfies the following conditions:
① Parallel to line $l$;
② The distance from point $P(3,0)$ to line $l_{2}$ ... | 2x + y - 11 = 0 |
Three of the $16$ squares from a $4 \times 4$ grid of squares are selected at random. The probability that at least one corner square of the grid is selected is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
$ \begin{tabular}{ | l | c | c | r| }
\hline
& & & \\ \hli... | 45 |
In the geometric sequence $\{a_n\}$, where the common ratio $q \neq 1$, there exists an arithmetic sequence $\{b_n\}$ satisfying $b_1 = a_1 = 3$, $b_4 = a_2$, and $b_{13} = a_3$.
(I) Find the general formulae for the sequences $\{a_n\}$ and $\{b_n\}$.
(II) Let $c_n = (-1)^n b_n + a_n$. Calculate the sum $S_{2n}$ of... | 2n + \frac{1}{2} (3^{2n+1} - 3) |
We call a set $ A$ a good set if it has the following properties:
1. $ A$ consists circles in plane.
2. No two element of $ A$ intersect.
Let $ A,B$ be two good sets. We say $ A,B$ are equivalent if we can reach from $ A$ to $ B$ by moving circles in $ A$, making them bigger or smaller in such a way that during thes... | 2 < \text{growth ratio of } a_n < 4 |
In an isosceles triangle $A B C(A B=B C)$, the medians $A M$ and $C N$ intersect at point $D$ at a right angle. Find all the angles of triangle $A B C$ and the area of quadrilateral $N B M D$, if the base $A C=1$.
# | \angleA=\angleC=\operatorname{arctg}3;\angleB=\pi-2\operatorname{arctg}3;\frac{1}{4} |
On the first day of entering the seventh grade, the homeroom teacher wanted all 40 students in the class to get to know each other. They were required to shake hands and introduce themselves to each other. The total number of handshakes among the students was . | 780 |
If $a$ and $b$ are rational numbers, and $b < 0$, then the order of $a$, $a-b$, $a+b$ is ( )
A: $a < a+b < a-b$
B: $a < a-b < a+b$
C: $a+b < a < a-b$
D: $a-b < a+b < a$ | \text{C} |
Solve the following equation:
$$
\sqrt{x+2+4 \sqrt{x-2}}+\sqrt{x+7-6 \sqrt{x-2}}=5
$$ | 2\leqx\leq11 |
1-116 The sum of the squares of two consecutive integers can equal the square of another integer (such as $3^{2}+4^{2}=5^{2}$).
(1) Prove: When $m=3,4,5,6$, the sum of the squares of $m$ consecutive integers cannot be the square of another integer.
(2) Find an example where the sum of the squares of 11 consecutive inte... | 121 |
[ Product of the lengths of the chord segments and the lengths of the secant segments ]
A line passing through point $A$, located outside the circle at a distance of 7 from its center, intersects the circle at points $B$ and $C$. Find the radius of the circle, given that $AB=3$, $BC=5$.
# | 5 |
8. The Lonely Ball (from 8th grade, 2 points). In 100 boxes, exactly 100 balls are placed randomly. What is the probability that the last box will contain only one ball? | 0.370 |
Asahanov $H . X$.
The number $x$ is such that among the four numbers $a=x-\sqrt{2}, b=x-\frac{1}{x}, c=x+\frac{1}{x}, d=x^{2}+2 \sqrt{2}$ exactly one is not an integer.
Find all such $x$. | \sqrt{2}-1 |
8. (10 points) The calculation result of the expression $2013 \times \frac{2015}{2014}+2014 \times \frac{2016}{2015}+\frac{4029}{2014 \times 2015}$ is ( )
A. 4027
B. 4029
C. 2013
D. 2015 | 4029 |
12. The maximum value of the function $S=\sqrt{x^{4}-5 x^{2}-8 x+25}-\sqrt{x^{4}-3 x^{2}+4}$ is | \sqrt{17} |
3. Given $x^{2}-x-1=0$. Then, the value of the algebraic expression $x^{3}$ $-2 x+1$ is $\qquad$ . | 2 |
The opposite of $\sqrt{3}-2$ is ______, and its absolute value is ______. | 2-\sqrt{3} |
Let $n$ be a positive integer.
Consider sequences $a_0, a_1, ..., a_k$ and $b_0, b_1,,..,b_k$ such that $a_0 = b_0 = 1$ and $a_k = b_k = n$ and such that for all $i$ such that $1 \le i \le k $, we have that $(a_i, b_i)$ is either equal to $(1 + a_{i-1}, b_{i-1})$ or $(a_{i-1}; 1 + b_{i-1})$.
Consider for $1 \le i \... | n^2 - 1 |
Ishari places the letters $A, B, C, D, E, F$ clockwise around a circle, starting with $A$ at the top. Jaxon writes a list beginning with $A$, and then, moving clockwise around the circle, writes down every third letter that he has not yet written. Doing this, Jaxon's list is $A, D, B, F, C, E$. Katharina mixes up the l... | L,R,O,M,S,Q,N,P |
Given that all vertices of a regular triangular prism are on the surface of a sphere, and the edge length of the base of the prism is $\sqrt{3}$, with the lateral edge length being 2, the surface area of the sphere is \_\_\_\_\_\_. | 8\pi |
8. Given $\lg 2=0.30103$, then the sum of the first and last digits of $2^{1000}$ is ().
(A) 3.
(B) 5.
(C) 8.
(D) 10. | B |
Given that $a$ and $b$ are both positive real numbers, and $\frac{1}{a} + \frac{1}{b} = 2$, find the maximum value of $\frac{1}{b}(\frac{2}{a} + 1)$. | \frac{25}{8} |
11. (20 points) As shown in Figure 1, given
$\odot G:(x-2)^{2}+y^{2}=r^{2}$ is the incircle of the inscribed $\triangle A B C$ in the ellipse $\frac{x^{2}}{16}+y^{2}=1$, where $A$ is the left vertex of the ellipse.
(1) Find the radius $r$ of $\odot G$;
(2) Two tangents are drawn from point $M(0,1)$ to $\odot G$, inters... | \frac{2}{3},thelineEFistangentto\odotG |
Problem 6. (30 points) A regular triangular prism $A B C A_{1} B_{1} C_{1}$ is inscribed in a sphere with the base $A B C$ and lateral edges $A A_{1}, B B_{1}, C C_{1}$. Segment $C D$ is the diameter of this sphere, point $K$ and $L$ are the midpoints of edge $A A_{1}$ and $A B$ respectively. Find the volume of the pri... | 12\sqrt{3} |
2. Through the right focus of the hyperbola $x^{2}-\frac{y^{2}}{2}=1$, a line $l$ intersects the hyperbola at points $A$ and $B$. If a real number $\lambda$ makes $|A B|=\lambda$ such that there are exactly 3 lines $l$, then $\lambda=$
(Proposed by the Problem Committee) | 4 |
Among the following equations, the one that represents factoring from left to right is:
A: $x\left(x+y\right)=x^{2}+xy$
B: $x^{2}-2xy=x\left(x-2y\right)$
C: $24=2\times 2\times 2\times 3$
D: $x^{2}+x+1=x\left(x+1\right)+1$ | B |
Given that the terminal side of angle $α$ passes through a point $P(3a, 4a)$, where $a \neq 0$, find $\sin α$, $\cos α$, and $\tan α$. | \tan α = \dfrac{4}{3} |
8.51 Let the function $y=f(x)$ satisfy for all real numbers $x$
$$
f(3+x)=f(3-x),
$$
and the equation $f(x)=0$ has exactly 6 distinct real roots, then the sum of these 6 roots is
(A) 18.
(B) 12.
(C) 9.
(D) 0.
(China High School Mathematics League, 1991) | 18 |
Question 25, Given that the graph of the function $y=f(x)$ is centrally symmetric about the point $(1,1)$ and axially symmetric about the line $x+y=0$. If $x \in(0,1)$, $f(x)=\log _{2}(x+1)$, then the value of $f\left(\log _{2} 10\right)$ is $\qquad$ - | \frac{17}{5} |
Given that $f(x-1)=3x-1$, find $f(x)$ _____. | 3x+2 |
Jeffrey rolls fair three six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i] | 101 |
Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and
\[
a_{n} = 2 a_{n-1}^2 - 1
\]
for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality
\[
a_0 a_1 \d... | 167 |
Example 2 There are 100 points on a plane, and the distance between any two points is not less than 3. Now, line segments are drawn between every two points whose distance is exactly 3. Try to prove: such line segments will not exceed 300. | 300 |
In the Cartesian coordinate system $xOy$, the point $(2, -2)$ is transformed by the matrix $M=\begin{pmatrix}0 & 1 \\ a & 0\end{pmatrix}$ to the point $(-2, 4)$. The curve $C: x^2+y^2=1$ is transformed by the matrix $M$ to the curve $C'$. Find the equation of the curve $C'$. | x^2+\frac{y^2}{4}=1 |
Determine the intervals of monotonicity and the extreme values of the function $f(x) = xe^{-x}$. | \frac{1}{e} |
Among the following groups of vectors, the ones that can serve as a basis are ( )
A: $\overrightarrow {e_{1}}=(0，0)$, $\overrightarrow {e_{2}}=(1，-2)$
B: $\overrightarrow {e_{1}}=(-1，-2)$, $\overrightarrow {e_{2}}=(3，6)$
C: $\overrightarrow {e_{1}}=(3，-5)$, $\overrightarrow {e_{2}}=(6，10)$
D: $\overrightarrow ... | C |
# Problem 7.
Inside a convex 13-gon, 200 points are placed such that no three of these 213 points (including the vertices of the polygon) are collinear. The polygon is divided into triangles, with the vertices of each triangle being any three of the 213 points. What is the maximum number of triangles that could result... | 411 |
The kindergarten received cards for reading lessons: some have "MA" written on them, and others have "NYA". Each child took three cards and started forming words from them. It turned out that 20 children can form the word "MAMA", 30 children can form the word "NYANYA", and 40 children can form the word "MANYA". How man... | 10 |
Given that \\(\alpha\\) and \\(\beta\\) are acute angles, and \\(\sin \alpha= \dfrac {3}{5}\\), \\(\tan (\alpha-\beta)=- \dfrac {1}{3}\\).
\\((1)\\) Find the value of \\(\sin (\alpha-\beta)\\);
\\((2)\\) Find the value of \\(\cos \beta\\). | \dfrac {9 \sqrt {10}}{50} |
8. There is an unlimited number of test tubes of three types - A, B, and C. Each test tube contains one gram of a solution of the same substance. Test tubes of type A contain a $10\%$ solution of this substance, type B $-20\%$ solution, and type C $-90\%$ solution. Sequentially, one after another, the contents of the t... | 73 |
Problem 7.4. In triangle $ABC$, the median $CM$ and the bisector $BL$ were drawn. Then, all segments and points were erased from the drawing, except for points $A(2 ; 8)$, $M(4 ; 11)$, and $L(6 ; 6)$. What were the coordinates of point $C$?
 |
\section*{Problem 6 - 081236}
All real numbers \(a\) are to be determined for which the equation
\[
\sin ^{6} x+\cos ^{6} x=a\left(\sin ^{4} x+\cos ^{4} x\right)
\]
has at least one real solution. Furthermore, all solutions for \(a=\frac{5}{6}\) are to be specified. | (2k+1)\frac{\pi}{8}\quad(k\in\mathbb{Z}) |
382. What is the smallest number of weights and of what weight can be used to weigh any whole number of pounds from 1 to 40 on a balance scale, given that during weighing, weights can be placed on both pans of the scale. | 4 |
Which of the following calculations is correct?
A: $7a+a=7a^{2}$
B: $5y-3y=2$
C: $x^{3}-x=x^{2}$
D: $2xy^{2}-xy^{2}=xy^{2}$ | D |
If the statement "All shirts in this store are on sale." is false, then which of the following statements must be true?
I. All shirts in this store are at non-sale prices.
II. There is some shirt in this store not on sale.
III. No shirt in this store is on sale.
IV. Not all shirts in this store are on sale.
$\tex... | D |
3. Two wagons start simultaneously from places $A$ and $B$ towards each other. Each of them, when they reach the opposite place, immediately turns back. The wagons first met $50 \mathrm{~km}$ from place $A$, and the second time they met $30 \mathrm{~km}$ from place $B$. How long is the track between places $A$ and $B$?... | 120 |
9,10
Someone has arranged a 10-volume collection of works in a random order. Let's call a disorder a pair of volumes where the volume with the higher number is to the left. For this arrangement of volumes, the number $S$ of all disorders has been calculated. What values can $S$ take? | Anyintegervaluefrom0to45 |
\left.\begin{array}{l}\text { [Sum of angles in a triangle. Theorem about the exterior angle.] } \\ {[\text { Concurrency of altitudes. Angles between altitudes. ] }}\end{array}\right]
The altitudes of triangle $ABC$, drawn from vertices $A$ and $C$, intersect at point $M$. Find $\angle AMC$, if $\angle A=$ $70^{\circ... | 150 |
Consider a function $f(x)$ defined on $\mathbb{R}$ where $f(0) \neq 0$, $f(1)=2$, for all $x > 0$ we have $f(x) > 1$, and for any $a, b \in \mathbb{R}$, the function satisfies $f(a+b) = f(a) \cdot f(b)$.
1. Find the value of $f(0)$.
2. Prove that for any $x \in \mathbb{R}$, we have $f(x) > 0$.
3. If $f(x)$ is an increa... | (-\infty, \frac{1}{2}) |
Find the coefficient of $x^{3}$ in the expansion of ${\left( \dfrac{1}{x}-x\right)}^{9}$.
A: $84$
B: $-84$
C: $126$
D: $-126$ | A |
In a box, there are 5 cards of the same shape and size, marked with the numbers 1, 2, 3, 4, 5. Players A and B each randomly draw one card from the box without replacement.
(Ⅰ) List all possible outcomes and calculate the probability that the sum of the numbers on the cards drawn by A and B is even.
(Ⅱ) Using the remai... | \dfrac{3}{10} |
6. (4 points) The students form a square array for the radio calisthenics performance. Xiao Hai's position is the 5th from the front, from the back, from the left, and from the right. The total number of students participating in the radio calisthenics performance is $\qquad$ people. | 81 |
Example 8 Find the minimum value of $\frac{a}{\sin \theta}+\frac{b}{\cos \theta}\left(a, b>0, \theta \in\left(0, \frac{\pi}{2}\right)\right)$. | (\sqrt[3]{^{2}}+\sqrt[3]{b^{2}})^{4} |
2. The sequence, formed from the sequence of primes by rounding each to the nearest ten, begins $0,0,10,10,10,10,20,20,20,30, \ldots$.
When continued, how many terms in this sequence are equal to 40 ?
A 1
B 2
C 3
D 4
E 5 | 3 |
8. (10 points) Among the three given phrases “尽心尽力”, “可拔山”, and “山穷水尽”, each Chinese character represents a number between 1 and 8, with the same character representing the same number and different characters representing different numbers. If the sum of the numbers represented by the characters in each phrase is 19, ... | 7 |
9.5. At the exchange office, only the following operations can be performed:
1) exchange 2 gold coins for three silver coins and one copper coin;
2) exchange 5 silver coins for three gold coins and one copper coin.
Nikolai had only silver coins. After several visits to the exchange office, he had fewer silver coins, ... | 10 |
$ABCD$ is a cyclic quadrilateral such that $AB = BC = CA$. Diagonals $AC$ and $BD$ intersect at $E$. Given that $BE = 19$ and $ED = 6$, find the possible values of $AD$. | 10 \text{ or } 15 |
An ellipse (E) is defined by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$. Point $O$ is the coordinate origin, point $A$ has coordinates $(a, 0)$, and point $B$ has coordinates $(0, b)$. Point $M$ lies on segment $AB$ and satisfies $|BM| = 2|AM|$. The slope of line $OM$ is $\frac{\sqrt{5}}{10}$.
1.... | \frac{x^2}{45} + \frac{y^2}{9} = 1 |
João is playing a game where the only allowed operation is to replace the natural number $n$ with the natural number $a \cdot b$ if $a+b=n$, with $a$ and $b$ being natural numbers. For example, if the last number obtained was 15, he can replace it with $56=7 \cdot 8$, since $7+8=15$ and both are natural numbers.
a) St... | 2014 |
# 3. CONDITION
Prove that in the product $P=1! \times 2! \times \ldots \times 100!$, one of the factors can be erased so that the product of the remaining factors is a perfect square.
Proof. From the equality $(2k)! = (2k-1)! \times 2k$, it follows that the given product can be rewritten as $(1!)^2 \times 2 \times (3... | 50! |
A certain scenic spot shop sells a kind of souvenir, with a purchase price of $40$ yuan per item. According to market research, when the selling price of the souvenir is $50$ yuan per item, 200 items can be sold per day. When the selling price increases by $1$ yuan per item, the daily sales quantity decreases by 10 ite... | 2250 \text{ yuan} |
3. Given $\alpha=3+2 \sqrt{2}, \beta=3-2 \sqrt{2}$. If $\alpha^{10}+\beta^{10}$ is a positive integer, then its last digit is ( ).
(A) 2
(B) 4
(C) 6
(D) 8 | B |
18. There were 36 participants in a party, some of whom shook hands with each other, such that any two participants shook hands with each other at most once. Each participant then recorded the number of handshakes made, and it was found that no two participants with the same number of handshakes made had shaken hands w... | 546 |
The Youth League Committee of a school plans to organize student volunteers to go tree planting in the wild. The number of volunteers from the first and second grades of high school are 150 and 100, respectively. For ease of management, the committee decides to select 5 volunteers as temporary leaders from these two gr... | \frac{3}{5} |
Observe the following equations: $2^1=2$, $2^2=4$, $2^3=8$, $2^4=16$, $2^5=32$, $2^6=64$, $2^7=128$, ... By observing, determine what the units digit of $2^{2010}$ is using the pattern you find. How do you know? | 4 |
The distance from a point \\(P(2,m)\\) on the parabola \\(y^{2}=2px(p > 0)\\) to its focus \\(F\\) is \\(4\\), then \\(p=\\)_______. | 4 |
Six consecutive prime numbers have sum $p$. Given that $p$ is also prime, determine all possible values of $p$. | 41 |
Example 4 Given that $x, y, z$ are real numbers, and satisfy
$$x+y+z=xy+yz+zx \text{.}$$
Find the minimum value of $\frac{x}{x^{2}+1}+\frac{y}{y^{2}+1}+\frac{z}{z^{2}+1}$. | -\frac{1}{2} |
Given the set $A=\{1,2,3\}$, and $B=\{y|y=2x-1, x\in A\}$, then $A\cap B= \_\_\_\_\_\_$. | \{1,3\} |
8・165 Given a sequence of natural numbers $\left\{x_{n}\right\}$ that satisfies
$$x_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, n=1,2,3, \cdots$$
If one of the terms in the sequence is 1000, what is the smallest possible value of $a+b$? | 10 |
## Problem A1
Find all possible values for the sum of the digits of a square.
| any\non-negative\integer\=0,1,4\or\7\\bmod\9 |
Let $a=2^{1.5}$, $b=\log_{\frac{1}{2}}1.5$, $c=\left(\frac{1}{2}\right)^{1.5}$, then the relationship among $a$, $b$, $c$ is ( )
A: $a>c>b$
B: $c>a>b$
C: $a>b>c$
D: $b>a>c$ | A |
\section*{Problem 5 - 061045}
Let \(a\) be any given real number. Determine all real \(x\) that satisfy the equation:
\[
\sqrt{a+x}-\sqrt{\frac{a^{2}}{a+x}}=\sqrt{2 a+x}
\] | -2afor0,anypositiverealxfor0 |
1. (6 points) Calculate: $234+432-4 \times 8+330 \div 5=$ | 700 |
Given that points $A$, $B$, and $C$ move on the circle $x^{2}+y^{2}=4$, and the midpoint of $B$ and $C$ is $D(1,0)$. If the coordinates of point $P$ are $(5,0)$, then the maximum value of $|{\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}}|$ is ______. | 15 |
A path is 80 meters long, and it is planned to plant plane trees on both sides of the path, with one tree planted every 4 meters (including at both ends). How many trees can be planted in total? | 42 |
Isabella has a sheet of paper in the shape of a right triangle with sides of length 3, 4, and 5. She cuts the paper into two pieces along the altitude to the hypotenuse, and randomly picks one of the two pieces to discard. She then repeats the process with the other piece (since it is also in the shape of a right trian... | 64 |
2. Find all even natural numbers $n$ for which the number of divisors (including 1 and $n$ itself) is equal to $\frac{n}{2}$. (For example, the number 12 has 6 divisors: $\left.1,2,3,4,6,12.\right)$ | {8,12} |
Example 6 Find the largest positive integer $n$ that satisfies the following condition: $n$ is divisible by all positive integers less than $\sqrt[3]{n}$. (1998, Asia Pacific Mathematical Olympiad) | 420 |
If the complex number $(1+bi) \cdot i = 1+i$ (where $i$ is the imaginary unit), then the real number $b=$ __ . | -1 |
3. Non-zero integers $a, b, c$ are three consecutive terms of a decreasing arithmetic progression. All six quadratic equations, whose coefficients are the numbers $2a, 2b, c$, taken in any order, have two roots. Find the greatest possible value of the common difference of the progression and the corresponding numbers $... | d_{\max}=-5;=4,b=-1,=-6 |
The function $f(x)=(m^2-m-1)x^{m^2-2m-3}$ is a power function, and it is a decreasing function on $x\in(0,+\infty)$. Find the real number $m$. | 2 |
Given that the sequence $\{a_n\}$ is a geometric sequence, consider the proposition P: "If $a_1 < a_2 < a_3$, then the sequence $\{a_n\}$ is an increasing sequence". How many propositions are correct among P, its converse, its inverse, and its contrapositive? ( )
A: 1
B: 2
C: 3
D: 4 | \text{D} |
The line $ax+my-2a=0$ ($m \neq 0$) passes through the point $(1,1)$. The angle of inclination of this line is ______. | 135^{\circ} |
Given ten points in space, where no four points lie on the same plane. Some points are connected by line segments. If the resulting figure contains no triangles and no spatial quadrilaterals, determine the maximum number of line segments that can be drawn. ${ }^{[1]}$
(2016, National High School Mathematics Joint Compe... | 15 |
Let $D$ be a point in the plane of $\triangle ABC$, and $\overrightarrow{AB}=2\overrightarrow{DC}$. Then
A: $\overrightarrow{BD}=\overrightarrow{AC}- \dfrac{3}{2}\overrightarrow{AB}$
B: $\overrightarrow{BD}= \dfrac{3}{2}\overrightarrow{AC}-\overrightarrow{AB}$
C: $\overrightarrow{BD}= \dfrac{1}{2}\overrightarrow{AC}... | \text{A} |
Given $(1-2x)^7 = a_0 + a_1x + a_2x^2 + \ldots + a_7x^7$, then $a_1 + a_2 + \ldots + a_7 =$ _______. | -2 |
Three, (16 points) In acute $\triangle ABC$, it is known that $AB=$ 4, $AC=5, BC=6, AD, BE, CF$ are the altitudes to sides $BC, CA, AB$ respectively, with $D, E, F$ being the feet of the perpendiculars. Find $\frac{S_{\triangle DEF}}{S_{\triangle ABC}}$ | \frac{27}{256} |
In the sequence $\{a_{n}\}$, $a_{1}=1$, $a_{2}=2$, and $a_{n+2}-a_{n}=1-(-1)^{n}$, where $n\in N^{*}$. Find $S_{100}$.
A: $0$
B: $1300$
C: $2600$
D: $2650$ | C |
Let $A_{1} A_{2} A_{3}$ be a triangle. We construct the following points:
- $B_{1}, B_{2}$, and $B_{3}$ are the midpoints of $A_{1} A_{2}, A_{2} A_{3}$, and $A_{3} A_{1}$.
- $C_{1}, C_{2}$, and $C_{3}$ are the midpoints of $A_{1} B_{1}, A_{2} B_{2}$, and $A_{3} B_{3}$.
- $D_{1}$ is the intersection of $\left(A_{1} C... | \frac{25}{49} |
Suppose $\mathcal{T}=A_0A_1A_2A_3$ is a tetrahedron with $\angle A_1A_0A_3 = \angle A_2A_0A_1 = \angle A_3A_0A_2 = 90^\circ$, $A_0A_1=5, A_0A_2=12$ and $A_0A_3=9$. A cube $A_0B_0C_0D_0E_0F_0G_0H_0$ with side length $s$ is inscribed inside $\mathcal{T}$ with $B_0\in \overline{A_0A_1}, D_0 \in \overline{A_0A_2}, E_0 \in... | \frac{180}{71} |
Given that $α \in (0,π)$, and $\cos (α+ \frac{π}{3})=-\frac{\sqrt{2}}{2}$, find $\tan 2α=(\ \ \ )$.
A: $\frac{\sqrt{3}}{3}$
B: $-\sqrt{3}$ or $-\frac{\sqrt{3}}{3}$
C: $-\frac{\sqrt{3}}{3}$
D: $-\sqrt{3}$ | C |
Calculate: $-\left[-\left(+2\right)\right]=$______, $-|-2|=\_\_\_\_\_\_$. | -2 |
1.66 Is the number $\sin \frac{\pi}{18} \sin \frac{3 \pi}{18} \sin \frac{5 \pi}{18} \sin \frac{7 \pi}{18} \sin \frac{9 \pi}{18}$ a rational number? | \frac{1}{16} |
The graph of the function $y=f(x)$ is symmetric about the line $x=1$, and it is monotonically decreasing on $[1,+\infty)$, $f(0)=0$. Then the solution set of $f(x+1) > 0$ is
A: $(1,+\infty)$
B: $(-1,1)$
C: $(-\infty,-1)$
D: $(-\infty,-1)\cup(1,+\infty)$ | B |
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