problem
stringlengths 18
2.25k
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3. (17 points) In triangle $A B C \quad A B=4, B C=6$, angle $A B C$ is $30^{\circ}, B D-$ is the bisector of triangle $A B C$. Find the area of triangle $A B D$.
|
2.4
|
51. There is a sequence of numbers $1,2,3,5,8,13$, $\qquad$ starting from the 3rd number, each number is the sum of the two preceding ones. The remainder when the 2021st number in this sequence is divided by 3 is $\qquad$.
|
2
|
1. If $a \cdot b \neq 1$, and $3 a^{3}+123456789 a+2=0$, $2 b^{2}+123456789 b+3=0$, then $\frac{a}{b}=$ $\qquad$ .
|
\frac{2}{3}
|
8. There are 9 students participating in a math competition in the same classroom, with seats arranged in 3 rows and 3 columns, represented by a $3 \times 3$ grid, where each cell represents a seat. To prevent cheating, three types of exams, $A$, $B$, and $C$, are used, and it is required that any two adjacent seats (cells sharing a common edge) receive different types of exams. The number of ways to distribute the exams that meet the conditions is $\qquad$ kinds.
|
246
|
In what number system is the following multiplication correct?
$166 \cdot 56=8590$.
|
12
|
Example 10. Find the direction of the greatest increase of the scalar field $u=x y+y z+x z$ at the point $M_{0}(1,1,1)$ and the magnitude of this greatest increase at this point.
|
2\sqrt{3}
|
# Problem 5. (3 points)
In the cells of a $7 \times 7$ table, pairwise distinct non-negative integers are written. It turns out that for any two numbers in the same row or column, the integer parts of their quotients when divided by 8 are different. What is the smallest value that the largest number in the table can take?
|
54
|
Leibniz Gottfried Wilhelm
Consider a numerical triangle:
$$
\begin{array}{cccccc}
1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \ldots & \frac{1}{1993} \\
\frac{1}{2} & \frac{1}{6} & \frac{1}{12} & \ldots & \frac{1}{1992 \cdot 1993} \\
\frac{1}{3} & \frac{1}{12} & \ldots &
\end{array}
$$
(The first row is given, and each element of the subsequent rows is calculated as the difference of the two elements above it). In the 1993rd row, there is one element. Find it.
|
\frac{1}{1993}
|
A sequence $a_{1}, a_{2}, a_{3}, \ldots$ of positive integers satisfies $a_{1}>5$ and $a_{n+1}=5+6+\cdots+a_{n}$ for all positive integers $n$. Determine all prime numbers $p$ such that, regardless of the value of $a_{1}$, this sequence must contain a multiple of $p$.
#
|
2
|
4. 2. 11 * Find the maximum value of $\sin \frac{\theta}{2}(1+\cos \theta)$.
|
\frac{4\sqrt{3}}{9}
|
For $n$ distinct positive integers all their $n(n-1)/2$ pairwise sums are considered. For each of these sums Ivan has written on the board the number of original integers which are less than that sum and divide it. What is the maximum possible sum of the numbers written by Ivan?
|
\frac{(n-1)n(n+1)}{6}
|
4. Given that $f$ is a mapping from the set $M=\{a, b, c\}$ to the set $N=\{-3,-2, \cdots, 3\}$. Then the number of mappings $f$ that satisfy
$$
f(a)+f(b)+f(c)=0
$$
is ( ).
(A) 19
(B) 30
(C) 31
(D) 37
|
37
|
Example 6.21. On average, $85 \%$ of the items coming off the conveyor are of the first grade. How many items need to be taken so that with a probability of 0.997, the deviation of the frequency of first-grade items from 0.85 in absolute value does not exceed 0.01?
|
11475
|
G3.2 It is known that $\sqrt{\frac{50+120+130}{2} \times(150-50) \times(150-120) \times(150-130)}=\frac{50 \times 130 \times k}{2}$. If $t=\frac{k}{\sqrt{1-k^{2}}}$, find the value of $t$.
|
\frac{12}{5}
|
37. Convert $\frac{1}{7}$ to a decimal, then find the sum of the digit in the 2017th position after the decimal point and the digit in the 7102nd position after the decimal point.
|
9
|
II. (50 points) Find the largest real number $m$ such that the inequality
$$
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+m \leqslant \frac{1+x}{1+y}+\frac{1+y}{1+z}+\frac{1+z}{1+x}
$$
holds for any positive real numbers $x$, $y$, $z$ satisfying $x y z=x+y+z+2$.
|
\frac{3}{2}
|
4. If the solution set of the inequality $k x^{2}-2|x-1|+$ $6 k<0$ with respect to $x$ is an empty set, then the range of values for $k$ is $\qquad$.
|
k \geqslant \frac{1+\sqrt{7}}{6}
|
1. [3 points] Find the number of eight-digit numbers, the product of the digits of each of which is equal to 16875. The answer should be presented as an integer.
|
1120
|
2. Let the three interior angles of $\triangle A B C$ be $A, B, C$, and denote the maximum value of $(\sin A \cos B+\sin B \cos C+\sin C \cos A)^{2}$ as $\alpha$. Then the sum of the numerator and denominator of $\alpha$ when written as a simplest fraction is $\qquad$.
|
43
|
You have $8$ friends, each of whom lives at a different vertex of a cube. You want to chart a path along the cube’s edges that will visit each of your friends exactly once. You can start at any vertex, but you must end at the vertex you started at, and you cannot travel on any edge more than once. How many different paths can you take?
|
96
|
Mike leaves home and drives slowly east through city traffic. When he reaches the highway he drives east more rapidly until he reaches the shopping mall where he stops. He shops at the mall for an hour. Mike returns home by the same route as he came, driving west rapidly along the highway and then slowly through city traffic. Each graph shows the distance from home on the vertical axis versus the time elapsed since leaving home on the horizontal axis. Which graph is the best representation of Mike's trip?
|
(B)
|
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{1} \frac{x^{4} \cdot d x}{\left(2-x^{2}\right)^{3 / 2}}
$$
|
\frac{5}{2}-\frac{3\pi}{4}
|
Show that the planes $ACG$ and $BEH$ defined by the vertices of the cube shown in Figure are parallel. What is their distance if the edge length of the cube is $1$ meter?
[img]https://cdn.artofproblemsolving.com/attachments/c/9/21585f6c462e4289161b4a29f8805c3f63ff3e.png[/img]
|
\frac{1}{\sqrt{3}}
|
4. Given $\frac{y+z-x}{x+y+z}=\frac{z+x-y}{y+z-x}=\frac{x+y-z}{z+x-y}$ $=p$. Then $p^{3}+p^{2}+p=$ $\qquad$
|
1
|
1. When $k$ takes any real number, the vertex of the parabola $y=\frac{4}{5}(x-k)^{2}+$ $k^{2}$ lies on the curve ( ).
(A) $y=x^{2}$
(B) $y=-x^{2}$
(C) $y=x^{2}(x>0)$
(D) $y=-x^{2}(x>0)$
(2004, National Junior High School Mathematics Competition Hubei Province Preliminary Contest)
|
A
|
G2.4 Given that $x$ and $y$ are positive integers and $x+y+x y=54$. If $t=x+y$, find the value of $t$.
|
14
|
52. As shown in the figure, in rectangle $A B C D$, $E, F$ are on $C D$ and $B C$ respectively, and satisfy $D E: E C=2: 3$. Connecting $A F, B E$ intersect at point $O$. If $A O: O F=5: 2$, then $B F: F C=$ $\qquad$
|
2:1
|
6. In a mathematics competition, the first round consists of 25 questions. According to the marking rules, each correct answer earns 4 points, and each wrong answer (including unanswered questions) deducts 1 point. If a score of no less than 60 points qualifies a student for the second round, then, how many questions at least must a student answer correctly in the first round to qualify for the second round? $\qquad$
|
17
|
10. (5 points) A rectangular prism, if the length is reduced by 2 cm, the width and height remain unchanged, the volume decreases by 48 cubic cm; if the width is increased by 3 cm, the length and height remain unchanged, the volume increases by 99 cubic cm; if the height is increased by 4 cm, the length and width remain unchanged, the volume increases by 352 cubic cm. The surface area of the original rectangular prism is $\qquad$ square cm.
|
290
|
5. The positive integers $a, b, c, d$ satisfy $a>b>c>d, a+b+c+d=2010$ and $a^{2}-b^{2}+c^{2}-d^{2}=2010$. How many different sets of possible values of $(a, b, c, d)$ are there?
(1 mark)
正整數 $a$ 、 $b$ 、 $、 d$ 滿足 $a>b>c>d$ 、 $a+b+c+d=2010$ 及 $a^{2}-b^{2}+c^{2}-d^{2}=2010$ 。問 $(a, b, c, d)$ 有多少組不同的可能值?
(1 分)
|
501
|
7. On the coordinate plane, consider a figure $M$ consisting of all points with coordinates $(x ; y)$ that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
y+x \geqslant|x-y| \\
\frac{x^{2}-8 x+y^{2}+6 y}{x+2 y-8} \leqslant 0
\end{array}\right.
$$
Sketch the figure $M$ and find its area.
|
8
|
Solve the following equation in the set of real numbers:
$$
\log _{2 x} 4 x+\log _{4 x} 16 x=4
$$
|
x_1=1,\;x_2=\frac{1}{(\sqrt{2})^3}
|
14. A unit spent 500,000 yuan to purchase a piece of high-tech equipment. According to the tracking survey of this model of equipment: after the equipment is put into use, if the maintenance and repair costs are averaged to the first day, the conclusion is: the maintenance and repair cost on the $x$-th day is
$$
\left[\frac{1}{4}(x-i)+5 \sim 0\right] \text { yuan. }
$$
(1) If the total maintenance and repair costs from the start of use to the scrapping of the equipment, as well as the cost of purchasing the equipment, are averaged to each day, this is called the daily average loss. Please express the daily average loss $y$ (yuan) as a function of the usage days $x$ (days);
(2) According to the technical and safety management requirements of this industry, when the average loss of this equipment reaches its minimum value, it should be scrapped. How many days should the equipment be used before it should be scrapped?
[Note]: In solving this problem, you may need to use the following two mathematical points (if needed, you can directly cite the following conclusions):
(A) For any integer $n$, the following equation must hold:
$$
1+2+3+4+\cdots+n=\frac{n(n+1)}{2} \text {. }
$$
(B) For any positive constants $a$ and $b$, and for any positive real number $x$, the following inequality must hold:
$\frac{a}{x}+\frac{x}{b} \geqslant 2 \sqrt{\frac{a x}{x b}}=2 \sqrt{\frac{a}{b}}$. It can be seen that $2 \sqrt{\frac{a}{b}}$ is a constant. That is, the function $y=\frac{a}{x}+\frac{x}{b}$ has a minimum value of $2 \sqrt{\frac{a}{b}}$, and this minimum value is achieved when $\frac{a}{x}=\frac{x}{b}$.
|
2000
|
7.1. The price for a ride on the "Voskhod" carousel in February 2020 was 300 rubles per person. In March, the price was reduced, and the number of visitors increased by $50 \%$, while the revenue increased by $25 \%$. By how many rubles was the price reduced?
|
50
|
Problem 5. Three regular nonagons have a common center, their sides are respectively parallel. The sides of the nonagons are 8 cm and 56 cm. The third nonagon divides the area of the figure enclosed between the first two in the ratio $1: 7$, counting from the smaller nonagon. Find the side of the third nonagon. Answer. $8 \sqrt{7}$.
|
8\sqrt{7}
|
3. Find the remainder when $47^{37^{2}}$ is divided by 7.
Try to find the remainder of $47^{37^{2}}$ when divided by 7.
|
5
|
Problem 11.7. An archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by no more than one bridge. It is known that from each island, no more than 5 bridges lead, and among any 7 islands, there are definitely two connected by a bridge. What is the largest value that $N$ can take?
|
36
|
Let $f_n\ (n=1,\ 2,\ \cdots)$ be a linear transformation expressed by a matrix $\left(
\begin{array}{cc}
1-n & 1 \\
-n(n+1) & n+2
\end{array}
\right)$ on the $xy$ plane. Answer the following questions:
(1) Prove that there exists 2 lines passing through the origin $O(0,\ 0)$ such that all points of the lines are mapped to the same lines, then find the equation of the lines.
(2) Find the area $S_n$ of the figure enclosed by the lines obtained in (1) and the curve $y=x^2$.
(3) Find $\sum_{n=1}^{\infty} \frac{1}{S_n-\frac 16}.$
[i]2011 Tokyo Institute of Technlogy entrance exam, Problem 1[/i]
|
2
|
4. Given that $[x]$ represents the greatest integer not exceeding $x$, the number of integer solutions to the equation $3^{2 x}-\left[10 \cdot 3^{x+1}\right]+ \sqrt{3^{2 x}-\left[10 \cdot 3^{x+1}\right]+82}=-80$ is $\qquad$
|
2
|
27. 9 racing cars have different speeds, and they need to compete to determine who is faster, but there are no timing tools, so they can only race on the track to see who comes first, and each time a maximum of 3 cars can race. Therefore, the minimum number of races needed to guarantee selecting the 2 fastest cars is $\qquad$.
|
5
|
3. (3 points) Anya, Vanya, Danya, and Tanya were collecting apples. It turned out that each of them collected a whole percentage of the total number of apples collected, and all these numbers were different and greater than zero. Then Tanya, who collected the most apples, ate her apples. After this, it turned out that each of the children still had a whole percentage, but now of the remaining number of apples. What is the minimum number of apples that could have been collected?
Answer: 20 : for example $2+3+5+10$
|
20
|
Solve the following system of equations:
$$
\begin{gathered}
x+y-z=-1 \\
x^{2}-y^{2}+z^{2}=1 \\
-x^{3}+y^{3}+z^{3}=-1
\end{gathered}
$$
|
x=-1,y=-1,z=-1
|
One. (20 points) Let the function
$$
f(x)=\cos x \cdot \cos (x-\theta)-\frac{1}{2} \cos \theta
$$
where, $x \in \mathbf{R}, 0<\theta<\pi$. It is known that when $x=\frac{\pi}{3}$, $f(x)$ achieves its maximum value.
(1) Find the value of $\theta$;
(2) Let $g(x)=2 f\left(\frac{3}{2} x\right)$, find the minimum value of the function $g(x)$ on $\left[0, \frac{\pi}{3}\right]$.
|
-\frac{1}{2}
|
4. In the rectangular prism $A^{\prime} C$, $A B=5, B C=4, B^{\prime} B=6$, and $E$ is the midpoint of $A A^{\prime}$. Find the distance between the skew lines $B E$ and $A^{\prime} C^{\prime}$.
|
\frac{60}{\sqrt{769}}
|
Given a positive integer $N$ (written in base $10$), define its [i]integer substrings[/i] to be integers that are equal to strings of one or more consecutive digits from $N$, including $N$ itself. For example, the integer substrings of $3208$ are $3$, $2$, $0$, $8$, $32$, $20$, $320$, $208$, $3208$. (The substring $08$ is omitted from this list because it is the same integer as the substring $8$, which is already listed.)
What is the greatest integer $N$ such that no integer substring of $N$ is a multiple of $9$? (Note: $0$ is a multiple of $9$.)
|
88888888
|
7. Given $\boldsymbol{A} \boldsymbol{B}=(k, 1), \boldsymbol{A} \boldsymbol{C}=(2,3)$. Then the value of $k$ that makes $\triangle A B C$ a right triangle is ( ).
(A) $\frac{3}{2}$
(B) $1-\sqrt{2}$
(C) $1-\sqrt{3}$
(D) $-\sqrt{5}$
|
C
|
2. If $1 \times 2 \times \cdots \times 100=12^{n} M$, where $M$ is a natural number, and $n$ is the largest natural number that makes the equation true, then $M$ ( ).
(A) is divisible by 2 but not by 3
(B) is divisible by 3 but not by 2
(C) is divisible by 4 but not by 3
(D) is not divisible by 3, nor by 2
(1991, National Junior High School Mathematics League)
|
A
|
2. Let $a=\sqrt{x^{2}+x y+y^{2}}, b=p \sqrt{x y}$, $c=x+y$. If for any positive numbers $x$ and $y$, a triangle exists with sides $a$, $b$, and $c$, then the range of the real number $p$ is $\qquad$
|
(2-\sqrt{3}, 2+\sqrt{3})
|
Four, (10 points) Person A and Person B process a batch of parts. If A and B work together for 6 days, and then A continues alone for 5 more days, the task can be completed; if A and B work together for 7 days, and then B continues alone for 5 more days, the task can also be completed. Now, if A first processes 300 parts, and then the two work together for 8 days to complete the task. How many parts are there in total?
|
2700
|
1013. Find the two smallest consecutive natural numbers, each of which has a sum of digits divisible by 7.
|
69999,70000
|
3. On a plane, a certain number of triangles are drawn, the lengths of whose sides are ten-digit natural numbers, containing only threes and eights in their decimal representation. No segment belongs to two triangles, and the sides of all triangles are distinct. What is the maximum number of triangles that can be drawn?
|
341
|
6. If the sum of the squares of two pairs of opposite sides of a spatial quadrilateral are equal, then the angle formed by its two diagonals is
保留了源文本的换行和格式。
|
90
|
2. The bathtub fills up in 23 minutes from the hot water tap, and in 17 minutes from the cold water tap. Pete first opened the hot water tap. After how many minutes should he open the cold water tap so that by the time the bathtub is full, one and a half times more hot water has been added than cold water?
|
7
|
5. The numbers from 1 to 8 are arranged at the vertices of a cube such that the sum of the numbers in any three vertices lying on the same face is at least 10. What is the smallest possible sum of the numbers at the vertices of one face?
|
16
|
4. Let $|a|=1,|b|=2$. If vector $c$ satisfies
$$
|c-(a+b)|=|a-b|,
$$
then the maximum value of $|c|$ is $\qquad$
|
2\sqrt{5}
|
Example 2 Let $n \geqslant 4, \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} ; \beta_{1}, \beta_{2}, \cdots, \beta_{n}$ be two sets of real numbers, satisfying
$$
\sum_{i=1}^{n} \alpha_{i}^{2}<1, \sum_{i=1}^{n} \beta_{i}^{2}<1 .
$$
Let $A^{2}=1-\sum_{i=1}^{n} \alpha_{i}^{2}, B^{2}=1-\sum_{i=1}^{n} \beta_{i}^{2}$,
$$
W=\frac{1}{2}\left(1-\sum_{i=1}^{n} \alpha_{i} \beta_{i}\right)^{2} .
$$
Find all real numbers $\lambda$, such that the equation
$$
x^{n}+\lambda\left(x^{n-1}+\cdots+x^{3}+W x^{2}+A B x+1\right)=0
$$
has only real roots.
|
\lambda=0
|
7. Given the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>1, b>0)$ with focal distance $2 c$, the line $l$ passes through the points $(a, 0)$ and $(0, b)$, and the sum of the distances from the point $(1,0)$ to the line $l$ and from the point $(-1,0)$ to the line $l$ is $s \geqslant \frac{4}{5} c$. Then the range of the eccentricity $e$ of the hyperbola is . $\qquad$
|
[\frac{\sqrt{5}}{2},\sqrt{5}]
|
7. Find the smallest three-digit number with the property that if a number, which is 1 greater, is appended to it on the right, then the result (a six-digit number) will be a perfect square. Answer: 183
|
183
|
Three, (50 points) If the three sides of a triangle are all rational numbers, and one of its interior angles is also a rational number, find all possible values of this interior angle.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
|
60^{\circ}, 90^{\circ}, 120^{\circ}
|
G3.3 If $f(n)=a^{n}+b^{n}$, where $n$ is a positive integer and $f(3)=[f(1)]^{3}+f(1)$, find the value of $a \cdot b$.
|
-\frac{1}{3}
|
1. Determine the area of the figure that is defined in the Cartesian coordinate system by the inequalities:
$$
\begin{aligned}
x^{2}+y^{2} & \leqslant 4(x+y-1) \\
y & \leqslant \sqrt{x^{2}-4 x+4}
\end{aligned}
$$
|
2\pi-4
|
2. Which of the following numbers can be obtained by summing the squares of two integers that are multiples of 3?
(A) 450
(B) 300
(C) 270
(D) 483
(E) 189
|
450
|
Example 8 Find the integer $k$ such that the quadratic equation
$$
k x^{2}+(k+1) x+(k-1)=0
$$
has integer roots.
(1993, 5th Ancestor's Cup Junior Mathematics Contest)
|
k=1
|
2.4. Find all values of $a$ for which the quadratic function $f(x)=a x^{2}+4 a x-1$ takes values, the modulus of which does not exceed 4, at all points of the interval $[-4 ; 0]$. In your answer, specify the total length of the intervals to which the found values of $a$ belong.
|
2
|
6. As shown in Figure 1, stones are arranged in a trapezoidal shape, and the sequence of numbers formed by the "trapezoidal" structure of the stones, $\left\{a_{n}\right\}$: $5,9,14,20, \cdots$, is called the "trapezoidal sequence." According to the construction of the "trapezoid," the value of $a_{624}$ is ( ).
(A) 166247
(B) 196248
(C) 196249
(D) 196250
|
196250
|
Suppose the polynomial $f(x) = x^{2014}$ is equal to $f(x) =\sum^{2014}_{k=0} a_k {x \choose k}$ for some real numbers $a_0,... , a_{2014}$. Find the largest integer $m$ such that $2^m$ divides $a_{2013}$.
|
2004
|
Seven boys and three girls are playing basketball. I how many different ways can they make two teams of five players so that both teams have at least one girl?
|
105
|
4. 34 Find the range of real values of $x$ that satisfy the following relation:
$$|3 x-2|+|3 x+1|=3$$
|
-\frac{1}{3} \leqslant x \leqslant \frac{2}{3}
|
Example 9 In a convex $n$-sided polygon, drawing $(n-3)$ non-intersecting diagonals, how many ways are there to do this?
|
\frac{(2 n-4)!}{(n-1)!(n-2)!}
|
460. Find the sample variance for the given sample distribution of size $n=10$:
$$
\begin{array}{cccc}
x_{i} & 186 & 192 & 194 \\
n_{i} & 2 & 5 & 3
\end{array}
$$
|
8.04
|
3. Stretch out your hands and count, reaching 100 is ( ).
A. Thumb
B. Index finger
C. Middle finger
D. Ring finger
E. Pinky
|
A
|
Example 3 Let $x_{i} \geqslant 0(i=1,2, \cdots, 7)$, and satisfy $x_{1}+x_{2}+\cdots+x_{7}=a$ (a constant), denote
$$
A=\max \left\{x_{1}+x_{2}+x_{3}, x_{2}+x_{3}+x_{4}, \cdots, x_{5}+x_{6}+x_{7}\right\} \text {. }
$$
Try to find $A_{\min }$.
|
\frac{a}{3}
|
8. Given $\sin \alpha \sin \beta=1, \cos (\alpha+\beta)=$
|
-1
|
6.64*. In a regular $n$-gon ( $n \geqslant 3$ ), the midpoints of all sides and diagonals are marked. What is the maximum number of marked points that can lie on one circle?
|
n
|
9.24 How many four-digit numbers divisible by 5 can be formed from the digits $0,1,3,5,7$, if each number must not contain identical digits?
|
42
|
1. Arrange the numbers $1,2, \cdots, 13$ in a row $a_{1}, a_{2}$, $\cdots, a_{13}$, where $a_{1}=13, a_{2}=1$, and ensure that $a_{1}+a_{2}+$ $\cdots+a_{k}$ is divisible by $a_{k+1}(k=1,2, \cdots, 12)$. Then the value of $a_{4}$ $+a_{5}+\cdots+a_{12}$ is $\qquad$ .
|
68
|
What is the digit $a$ in $a 000+a 998+a 999=22$ 997?
|
7
|
Starting with any non-zero natural number, it is always possible to form a sequence of numbers that ends in 1, by repeatedly following the instructions below:
- if the number is odd, add 1;
- if the number is even, divide by 2.
For example, starting with the number 21, the following sequence is formed:
$$
21 \rightarrow 22 \rightarrow 11 \rightarrow 12 \rightarrow 6 \rightarrow 3 \rightarrow 4 \rightarrow 2 \rightarrow 1
$$
This sequence contains nine numbers; therefore, we say it has a length of 9. Additionally, since it starts with an odd number, we say it is an odd sequence.
a) Write the sequence that starts with 37.
b) There are three sequences of length 5, two of which are even and one is odd. Write these sequences.
c) How many even sequences and how many odd sequences are there of length 6? And of length 7?
d) There are a total of 377 sequences of length 15, 233 of which are even and 144 are odd. How many sequences are there of length 16? Of these, how many are even? Do not forget to justify your answer.
|
610
|
5. If the two roots of the equation $x^{2}-2 x+\frac{\sqrt{3}}{2}=0$ are $\alpha, \beta$, and they are also the roots of the equation $x^{4}+p x^{2}+q=0$, then $p=$
|
\sqrt{3}-4
|
1. The price of one pencil is a whole number of eurocents. The total price of 9 pencils is greater than 11, but less than 12 euros, while the total price of 13 pencils is greater than 15, but less than 16 euros. How much does one pencil cost?
|
123
|
Consider a parameterized curve $ C: x \equal{} e^{ \minus{} t}\cos t,\ y \equal{} e^{ \minus{} t}\sin t\ \left(0\leq t\leq \frac {\pi}{2}\right).$
(1) Find the length $ L$ of $ C$.
(2) Find the area $ S$ of the region bounded by $ C$, the $ x$ axis and $ y$ axis.
You may not use the formula $ \boxed{\int_a^b \frac {1}{2}r(\theta)^2d\theta }$ here.
|
\sqrt{2} \left( 1 - e^{-\frac{\pi}{2}} \right)
|
11.030. The base of a right parallelepiped is a parallelogram with sides 1 and 4 cm and an acute angle of $60^{\circ}$. The larger diagonal of the parallelepiped is $5 \mathrm{~cm}$. Determine its volume.
|
4\sqrt{3}\mathrm{~}^{3}
|
What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?
|
672
|
G1.4 If $x$ is an integer satisfying $\log _{1 / 4}(2 x+1)<\log _{1 / 2}(x-1)$, find the maximum value of $x$.
|
3
|
## Task 4 - 280914
Three workpieces $W_{1}, W_{2}, W_{3}$ go through an assembly line with four processing machines $M_{1}, M_{2}, M_{3}, M_{4}$. Each workpiece must pass through the machines in the order $M_{1}, M_{2}, M_{3}, M_{4}$, and the order of the three workpieces must be the same at each machine.
The processing times of the workpieces on the individual machines are (in hours) given in the following table:
| | $M_{1}$ | $M_{2}$ | $M_{3}$ | $M_{4}$ |
| :--- | :---: | :---: | :---: | :---: |
| $W_{1}$ | 4 | 1 | 2 | 1.5 |
| $W_{2}$ | 2 | 2.5 | 1 | 0.5 |
| $W_{3}$ | 2 | 3.5 | 1 | 1 |
Two workpieces can never be processed simultaneously on the same machine. The times for changing the workpieces at the machines are so small that they can be neglected.
Give an order of the three workpieces for passing through the assembly line such that the total time (the time from the entry of the first workpiece into machine $M_{1}$ to the exit of the last workpiece from machine $M_{4}$) is as small as possible! Show that the order you give with its total time is better than any other order!
|
W_{3},W_{1},W_{2}
|
14. Decode the record ${ }_{* *}+{ }_{* * *}={ }_{* * * *}$, if it is known that both addends and the sum will not change if read from right to left.
|
22+979=1001
|
4.034. Find the first three terms of an arithmetic progression, for which the sum of any number of terms is equal to three times the square of this number.
|
3,9,15
|
A1. At a bazaar, you can win a prize by guessing the exact number of ping pong balls in a glass jar. Arie guesses there are 90, Bea guesses there are 97, Cor guesses there are 99, and Dirk guesses there are 101. None of the four win the prize. It turns out that one of the four is off by 7, one is off by 4, and one is off by 3.
How many ping pong balls are in the jar?
|
94
|
Example 4 Given the sequence $\left\{a_{n}\right\}$:
$$
a_{1}=2, a_{n+1}=\frac{5 a_{n}-13}{3 a_{n}-7}(n \geqslant 1) \text {. }
$$
Determine the periodicity of the sequence $\left\{a_{n}\right\}$.
|
3
|
8. Given that $A B C D$ is a square with side length 4, $E, F$ are the midpoints of $A B, A D$ respectively, $G C \perp$ plane $A B C D$, and $G C=2$. Find the distance from point $B$ to plane $E F G$.
|
\frac{2\sqrt{11}}{11}
|
1. Let $A=\sin \left(\sin \frac{3}{8} \pi\right), B=\sin \left(\cos \frac{3}{8} \pi\right), C=\cos \left(\sin \frac{3}{8} \pi\right), D=$ $\cos \left(\cos \frac{3}{8} \pi\right)$, compare the sizes of $A$, $B$, $C$, and $D$.
|
B<C<A<D
|
Consider the set of all rectangles with a given area $S$.
Find the largest value o $ M = \frac{S}{2S+p + 2}$ where $p$ is the perimeter of the rectangle.
|
\frac{S}{2(\sqrt{S} + 1)^2}
|
Let $a$ and $b$ be nonzero real numbers such that $\tfrac{1}{3a}+\tfrac{1}{b}=2011$ and $\tfrac{1}{a}+\tfrac{1}{3b}=1$. What is the quotient when $a+b$ is divided by $ab$?
|
1509
|
1. Given a sequence of positive numbers $\left\{a_{n}\right\}$ satisfying $a_{n+1} \geqslant 2 a_{n}+1$, and $a_{n}<2^{n+1}$ for $n \in \mathbf{Z}_{+}$. Then the range of $a_{1}$ is
|
0<a_{1} \leqslant 3
|
1. In Rt $\triangle A B C$, $\angle C=90^{\circ}, A B=1$, point $E$ is the midpoint of side $A B$, and $C D$ is the altitude on side $A B$. Then the maximum value of $(\overrightarrow{C A} \cdot \overrightarrow{C D}) \cdot(\overrightarrow{C A} \cdot \overrightarrow{C E})$ is $\qquad$ (Li Weiwei provided the problem)
|
\frac{2}{27}
|
4. Find $a+b+c+d+e$ if
$$
\begin{array}{c}
3 a+2 b+4 d=10 \\
6 a+5 b+4 c+3 d+2 e=8 \\
a+b+2 c+5 e=3 \\
2 c+3 d+3 e=4, \text { and } \\
a+2 b+3 c+d=7
\end{array}
$$
|
4
|
[ Divisibility of numbers. General properties ] [ Examples and counterexamples. Constructions ]
A five-digit number is called indivisible if it cannot be factored into the product of two three-digit numbers.
What is the largest number of consecutive indivisible five-digit numbers?
|
99
|
89. Divide 40 chess pieces into 27 piles. The number of chess pieces in each pile is $1, 2$ or 3. If the number of piles with only 1 chess piece is twice the number of the remaining piles, then the number of piles with exactly 2 chess pieces is $\qquad$ piles.
|
5
|
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
|
2. a) $1+3+5+\ldots+2013=\frac{2014 \cdot 1007}{2}=1007^{2}$
$1+3+5+\ldots+105=\frac{106 \cdot 53}{2}=53^{2}$
The equation is equivalent to: $\frac{1007^{2}}{x}=53^{2} \Rightarrow x=\left(\frac{1007}{53}\right)^{2} \Rightarrow x=19^{2} \Rightarrow x=361$
b) $x=361 \Rightarrow(361-1): 10=y^{2} \Rightarrow y^{2}=36$
$S=\{-6,6\}$
|
{-6,6}
|
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