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stringlengths 18
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4. Snow White entered a room where there were 30 chairs around a round table. Some of the chairs were occupied by dwarfs. It turned out that Snow White could not sit down without having someone next to her. What is the minimum number of dwarfs that could have been at the table? (Explain how the dwarfs should have been seated and why, if there were fewer dwarfs, there would be a chair with no one sitting next to it).
|
10
|
3. If $m^{2}+m-1=0$, then the value of $m^{3}+2 m^{2}+2019$ is
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
|
2020
|
Three, (25 points) Try to determine, for any $n$ positive integers, the smallest positive integer $n$ such that at least 2 of these numbers have a sum or difference that is divisible by 21.
|
12
|
2. Given that $x, y$ are positive integers. And they satisfy the conditions $x y+x+y=71, x^{2} y+x y^{2}=880$.
Find the value of $x^{2}+y^{2}$.
(1999, Jiangsu Province Junior High School Mathematics Competition)
|
146
|
2. Let $x, y$ be real numbers. If for any real numbers $\alpha, \beta$ satisfying $\cos \alpha - \cos \beta \neq 0$, we have
$$
\frac{\sin \left(\alpha+\frac{\pi}{6}\right)+\sin \left(\beta-\frac{\pi}{6}\right)}{\cos \alpha-\cos \beta}=x \cot \frac{\alpha-\beta}{2}+y
$$
then $(x, y)$
|
\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)
|
Example 6 Solve the equation $\frac{13 x-x^{2}}{x+1}\left(x+\frac{13-x}{x+1}\right)=42$. (1998, Changchun City, Jilin Province Mathematics Competition (Grade 9))
|
x_{1}=1, x_{2}=6, x_{3}=3+\sqrt{2}, x_{4}=3-\sqrt{2}
|
Let $V$ be the set of all continuous functions $f\colon [0,1]\to \mathbb{R}$, differentiable on $(0,1)$, with the property that $f(0)=0$ and $f(1)=1$. Determine all $\alpha \in \mathbb{R}$ such that for every $f\in V$, there exists some $\xi \in (0,1)$ such that
\[f(\xi)+\alpha = f'(\xi)\]
|
\frac{1}{e - 1}
|
13.192. There are three vessels containing unequal amounts of liquid. To equalize these amounts, three pourings were made. First, $1 / 3$ of the liquid was poured from the first vessel into the second, then $1 / 4$ of the liquid that ended up in the second vessel was poured into the third, and finally, $1 / 10$ of the liquid that ended up in the third vessel was poured into the first. After this, each vessel contained 9 liters of liquid. How much liquid was initially in each vessel
|
12,8,7
|
6. (8 points) Let for positive numbers $x, y, z$ the system of equations holds:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=108 \\
y^{2}+y z+z^{2}=16 \\
z^{2}+x z+x^{2}=124
\end{array}\right.
$$
Find the value of the expression $x y+y z+x z$.
|
48
|
In the interior of the square $ABCD$, the point $P$ lies in such a way that $\angle DCP = \angle CAP=25^{\circ}$. Find all possible values of $\angle PBA$.
|
25^\circ
|
7. Let the set
$$
M=\left\{(x, y) \left\lvert\, \frac{1}{\sqrt{x}}-\frac{1}{\sqrt{y}}=\frac{1}{\sqrt{45}}\right., x, y \in \mathbf{Z}_{+}\right\} \text {. }
$$
Then the number of elements in the set $M$ is ().
(A) 0
(B) 1
(C) 2
(D) 3
|
1
|
Problem 6. (30 points) A regular triangular prism $A B C A_{1} B_{1} C_{1}$ with base $A B C$ and lateral edges $A A_{1}, B B_{1}, C C_{1}$ is inscribed in a sphere of radius 6. Segment $C D$ is a diameter of this sphere. Find the volume of the prism if $A D=4 \sqrt{6}$.
|
48\sqrt{15}
|
A $2$-kg rock is suspended by a massless string from one end of a uniform $1$-meter measuring stick. What is the mass of the measuring stick if it is balanced by a support force at the $0.20$-meter mark?
[asy]
size(250);
draw((0,0)--(7.5,0)--(7.5,0.2)--(0,0.2)--cycle);
draw((1.5,0)--(1.5,0.2));
draw((3,0)--(3,0.2));
draw((4.5,0)--(4.5,0.2));
draw((6,0)--(6,0.2));
filldraw((1.5,0)--(1.2,-1.25)--(1.8,-1.25)--cycle, gray(.8));
draw((0,0)--(0,-0.4));
filldraw((0,-0.4)--(-0.05,-0.4)--(-0.1,-0.375)--(-0.2,-0.375)--(-0.3,-0.4)--(-0.3,-0.45)--(-0.4,-0.6)--(-0.35,-0.7)--(-0.15,-0.75)--(-0.1,-0.825)--(0.1,-0.84)--(0.15,-0.8)--(0.15,-0.75)--(0.25,-0.7)--(0.25,-0.55)--(0.2,-0.4)--(0.1,-0.35)--cycle, gray(.4));
[/asy]
$ \textbf {(A) } 0.20 \, \text{kg} \qquad \textbf {(B) } 1.00 \, \text{kg} \qquad \textbf {(C) } 1.33 \, \text{kg} \qquad \textbf {(D) } 2.00 \, \text{kg} \qquad \textbf {(E) } 3.00 \, \text{kg} $
|
1.33 \, \text{kg}
|
3. Given the sequence
$$
\begin{array}{l}
a_{0}=134, a_{1}=150, \\
a_{k+1}=a_{k-1}-\frac{k}{a_{k}}(k=1,2, \cdots, n-1) .
\end{array}
$$
If $a_{n}=0$, then $n$ is ( ).
(A) 20
(B) 201
(C) 2017
(D) 20101
|
201
|
4. On the school playground, fifth-grade students stood in a row, one next to the other. Then, between each two students, a sixth-grade student inserted themselves. After that, between each two students in the row, a seventh-grade student sat down. Finally, between each two students in the row, an eighth-grade student arrived. At that moment, there were 193 students in the row on the playground. How many sixth-grade students are there?
|
24
|
Question 211, Find the largest integer $k$, such that $\left[\frac{n}{\sqrt{3}}\right]+1>\frac{\mathrm{n}^{2}}{\sqrt{3 \mathrm{n}^{2}-k}}$ holds for all positive integers $n \geq 2$.
---
The translation maintains the original text's format and line breaks.
|
5
|
12.35 $\lim _{x \rightarrow 4} \frac{x+\sqrt{x}-6}{x-5 \sqrt{x}+6}$.
|
-5
|
Question 13: Given that $A$ and $B$ are two subsets of $\{1,2, \ldots, 100\}$, satisfying: $|A|=|B|, A \cap B=\emptyset$, and for any $x \in A, 2 x+2 \in B$. Try to find the maximum value of $|A \cup B|$.
---
The translation maintains the original format and line breaks as requested.
|
66
|
3. Draw 6 chords in a circle, dividing the circle into \( n \) plane parts. The largest \( n \) is ().
(A) 18
(B) 22
(C) 24
(D) 32
|
B
|
Example 8 A point whose both coordinates are integers is called an integer point. If $n$ is a non-negative integer, then, in the region
$$
\{(x, y) \mid | x|+| y | \leqslant n\}
$$
how many integer points are there?
|
2 n^{2}+2 n+1
|
9th APMO 1997 Problem 2 Find an n in the range 100, 101, ... , 1997 such that n divides 2 n + 2. Solution
|
946
|
G5.2 Given that the roots of $x^{2}+a x+2 b=0$ and $x^{2}+2 b x+a=0$ are both real and $a, b>0$. If the minimum value of $a+b$ is $Q$, find the value of $Q$.
|
6
|
14. (1994 Japan 4th Mathematical Olympiad) Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ where the angle between the planes $A B_{1} D_{1}$ and $A_{1} B D$ is $\theta\left(0^{\circ} \leqslant \theta \leqslant 90^{\circ}\right)$, find the value of $\cos \theta$.
|
\frac{1}{3}
|
6.143. $\frac{1}{x^{2}}+\frac{1}{(x+2)^{2}}=\frac{10}{9}$.
|
x_{1}=-3,x_{2}=1
|
Example 3-2 A school only offers three courses: Mathematics, Physics, and Chemistry. It is known that the number of students taking these three courses are 170, 130, and 120, respectively. The number of students taking both Mathematics and Physics is 45; the number of students taking both Mathematics and Chemistry is 20; the number of students taking both Physics and Chemistry is 22; and the number of students taking all three courses is 3. Try to calculate the total number of students in the school.
|
336
|
Example 3 Simplify $\frac{1+\sin \theta+\cos \theta}{1+\sin \theta-\cos \theta}+\frac{1-\cos \theta+\sin \theta}{1+\cos \theta+\sin \theta}$.
|
2\csc\theta
|
7.1. In 7a class, $52\%$ are girls. All students in the class can line up in such a way that boys and girls alternate. How many students are in the class?
|
25
|
64. In how many ways can 3 identical books be distributed among twelve schoolchildren if no more than one book can be given to any one child?
|
220
|
5. The force with which the airflow acts on the sail can be calculated using the formula
$F=\frac{A S \rho\left(v_{0}-v\right)^{2}}{2}$, where $A-$ is the aerodynamic force coefficient, $S-$ is the area of the sail $S$ $=4 \mathrm{m}^{2} ; \rho$ - density of air, $v_{0}$ - wind speed $v_{0}=4.8 \mu / c, v$ - speed of the sailboat. At some moment in time, the instantaneous power of the wind reaches its maximum value. What is the speed of the sailboat at this moment?
Given:
$F=\frac{A S \rho\left(v_{0}-v\right)^{2}}{2}$
$N\left(t^{\prime}\right)=N_{\max }$
$v_{0}=4.8 \frac{M}{c}$
$v\left(t^{\prime}\right)-?$
|
1.6
|
Problem 2. Given that $1580!=a, \quad$ calculate: $1 \cdot 1!+2 \cdot 2!+3 \cdot 3!+\ldots+1580 \cdot 1580!$ $(n!=1 \cdot 2 \cdot 3 \cdot \ldots \cdot n)$
|
1581a-1
|
A4. Any natural number of two digits can be written in reverse. So you can reverse 15 to 51. There are two-digit numbers with the property that the number added to its reverse results in a number that is the square of an integer.
Determine all numbers with this property.
|
29,38,47,56,65,74,83,92
|
4. Given the following two propositions about skew lines:
Proposition I: If line $a$ on plane $\alpha$ and line $b$ on plane $\beta$ are skew lines, and line $c$ is the intersection line of $\alpha$ and $\beta$, then $c$ intersects at most one of $a$ and $b$;
Proposition II: There do not exist infinitely many lines such that any two of them are skew lines. Then
A. Proposition I is correct, Proposition II is incorrect
B. Proposition II is correct, Proposition I is incorrect
C. Both propositions are correct
D. Both propositions are incorrect
|
D
|
Example 20 (2002 Beijing College Entrance Examination for Science) Given that $f(x)$ is a non-zero function defined on $\mathbf{R}$, and for any $a, b \in \mathbf{R}$, it satisfies $f(a b)=a f(b)+b f(a)$.
(1) Find the values of $f(0)$ and $f(1)$;
(2) Determine the odd or even nature of $f(x)$, and prove your conclusion;
(3) Given $f(2)=2, U_{n}=\frac{f\left(2^{-n}\right)}{n}\left(n \in \mathbf{N}^{*}\right)$, find the sum of the first $n$ terms $S_{n}$ of the sequence $\left\{U_{n}\right\}$.
|
S_{n}=(\frac{1}{2})^{n}-1
|
$k$ marbles are placed onto the cells of a $2024 \times 2024$ grid such that each cell has at most one marble and there are no two marbles are placed onto two neighboring cells (neighboring cells are defined as cells having an edge in common).
a) Assume that $k=2024$. Find a way to place the marbles satisfying the conditions above, such that moving any placed marble to any of its neighboring cells will give an arrangement that does not satisfy both the conditions.
b) Determine the largest value of $k$ such that for all arrangements of $k$ marbles satisfying the conditions above, we can move one of the placed marble onto one of its neighboring cells and the new arrangement satisfies the conditions above.
|
k = 2023
|
2.004. $\left(\frac{(a+b)^{-n / 4} \cdot c^{1 / 2}}{a^{2-n} b^{-3 / 4}}\right)^{4 / 3}:\left(\frac{b^{3} c^{4}}{(a+b)^{2 n} a^{16-8 n}}\right)^{1 / 6} ; b=0.04$.
|
0.2
|
7. Given that the ellipse $C$ passes through the point $M(1,2)$, with two foci at $(0, \pm \sqrt{6})$, and $O$ is the origin, a line $l$ parallel to $OM$ intersects the ellipse $C$ at points $A$ and $B$. Then the maximum value of the area of $\triangle OAB$ is $\qquad$
|
2
|
29.52. Compute $\lim _{n \rightarrow \infty} \sum_{k=0}^{n-1} \frac{1}{\sqrt{n^{2}-k^{2}}}$.
See also problem 25.42.
### 29.10. Identities
|
\frac{\pi}{2}
|
Which is the largest positive integer that is 19 times larger than the sum of its digits?
|
399
|
A baker went to the market to buy eggs to make 43 cakes, all with the same recipe, which uses fewer than 9 eggs. The seller notices that if he tries to wrap the eggs the baker bought in groups of 2 or 3 or 4 or 5 or 6 eggs, there is always 1 egg left over. How many eggs does she use in each cake? What is the minimum number of eggs the baker will use to make the 43 cakes?
|
301
|
1.3. Find the greatest negative root of the equation $\sin \pi x = -\sqrt{2} \sin \frac{\pi x}{2}$.
|
-\frac{3}{2}
|
Example 21 As shown in Figure 1.4.23, in isosceles $\triangle ABC$, $AB=AC, \angle A=120^{\circ}$, point $D$ is on side $BC$, and $BD=1, DC=2$. Find the length of $AD$.
|
1
|
(1) Given $x \in \mathbf{R}, y \in \mathbf{R}$, then “ $|x|<1$ and $|y|<1$ ” is “ $|x+y|+ |x-y|<2$ ” ( ).
(A) A sufficient condition but not a necessary condition
(B) A necessary condition but not a sufficient condition
(C) A sufficient and necessary condition
(D) Neither a sufficient condition nor a necessary condition
|
C
|
11. As shown in the figure, the teacher wrote 9 numbers on the blackboard, and asked to select 3 numbers to form a three-digit number, with any two selected numbers not coming from the same row or the same column. A total of $\qquad$ three-digit numbers that can be divided by 4 can be formed.
|
8
|
Four different numbers $a, b, c$, and $d$ are chosen from the list $-1,-2,-3,-4$, and -5 . The largest possible value for the expression $a^{b}+c^{d}$ is
(A) $\frac{5}{4}$
(B) $\frac{7}{8}$
(C) $\frac{31}{32}$
(D) $\frac{10}{9}$
(E) $\frac{26}{25}$
|
\frac{10}{9}
|
Example 4 Find all prime numbers $p$ and positive integers $m$ that satisfy $2 p^{2}+p+8=m^{2}-2 m$. ${ }^{[4]}$
(2010, "Mathematics Weekly Cup" National Junior High School Mathematics Competition).
|
p=5, m=9
|
# 3. The sum of two natural addends is 2016. If the last digit of one of them is erased - the second one is obtained. Find these numbers.
|
1833+183
|
Example 4 Let real numbers $x_{1}, x_{2}, \cdots, x_{1991}$ satisfy the condition
$$
\sum_{i=1}^{1990}\left|x_{i}-x_{i+1}\right|=1991 \text {. }
$$
and $y_{k}=\frac{1}{k} \sum_{i=1}^{k} x_{i}(k=1,2, \cdots, 1991)$. Find the maximum value of $\sum_{i=1}^{1990}\left|y_{i}-y_{i+1}\right|$.
(25th All-Soviet Union Mathematical Olympiad)
|
1990
|
6. From $m$ boys and $n$ girls $(10 \geqslant m>n \geqslant 4)$, 2 people are randomly selected to be class leaders. Let event $A$ represent the selection of 2 people of the same gender, and event $B$ represent the selection of 2 people of different genders. If the probability of $A$ is the same as the probability of $B$, then the possible values of $(m, n)$ are $\qquad$ .
|
(10,6)
|
5.22. Let
$$
f(x)=\left\{\begin{array}{l}
e^{3 x}, \text { if } x<0, \\
a+5 x, \text { if } x \geqslant 0
\end{array}\right.
$$
For which choice of the number $a$ will the function $f(x)$ be continuous?
|
1
|
20. Three equilateral triangles with sides of length 1 are shown shaded in a larger equilateral triangle. The total shaded area is half the area of the larger triangle.
What is the side-length of the larger equilateral triangle?
A $\sqrt{5}$
B $\sqrt{6}$
C $\frac{5}{2}$
D $\frac{3 \sqrt{3}}{2}$
E $1+\sqrt{3}$
|
\sqrt{6}
|
\section*{Problem 2 - 131212}
From a straight circular frustum, a cone is to be cut out whose apex is the center of the (larger) base of the frustum and whose base coincides with the top face of the frustum.
Determine the values of the ratio of the radius of the base to the radius of the top face of the frustum, for which the volume of the remaining body is six times as large as that of the cut-out cone.
|
\frac{r_{1}}{r_{2}}=2
|
5. Given a natural number $x=9^{n}-1$, where $n$ is a natural number. It is known that $x$ has exactly three distinct prime divisors, one of which is 13. Find $x$.
|
728
|
10. How many four-digit numbers can be formed using 3 ones, 2 twos, and 5 threes?
Using the ten digits consisting of 3 ones, 2 twos, and 5 threes, how many four-digit numbers can be formed?
|
71
|
7. $a, b, c$ are natural numbers. (1) Find the solutions $(a, b)$ that satisfy $a>b$ and $(a+b)^{2}=a^{3}+b^{3}$, (2) Find the solutions $(a, b, c)$ that satisfy $a>b>c$ and $(a+b+c)^{2}=a^{3}+b^{3}+c^{3}$.
|
(a, b, c)=(3,2,1)
|
3. $\tan 15^{\circ}+2 \sqrt{2} \sin 15^{\circ}=$
|
1
|
Task B-1.5. Marko and his "band" set off on a tour. On the first day, they headed east, on the second day, they continued north, on the third day, they continued west, on the fourth day, they headed south, on the fifth day, they headed east, and so on. If on the n-th day of the tour they walked $\frac{n^{2}}{2}$ kilometers, how many km were they from the starting point at the end of the fortieth day?
|
580
|
B3. The expression is $Z=5 a^{-x}\left(1-a^{-x}\right)^{-1}-3 a^{-x}\left(1+a^{-x}\right)^{-1}-2 a^{x}\left(a^{2 x}-1\right)^{-1}$, where $a^{x} \neq 0,1,-1$.
a) Simplify the expression $Z$.
b) Calculate the value of the expression $Z$ for $a=9^{b+c} \cdot 3^{2 b+c}: 27^{\frac{4}{3} b+c+\frac{1}{3}}$ and $x=1$.
19th Knowledge Competition
in mathematics for students of secondary technical and vocational schools
National Competition, April 13, 2019
## Problems for 3rd Year
Time for solving: 120 minutes. In section A, we will award three points for each correct answer, and deduct one point for each incorrect answer. Write your answers for section A in the left table, leave the right table blank.
| B1 | B2 | B3 |
| :--- | :--- | :--- |
| | | |
|
-9
|
22. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Then there are ( ) positive integers $n$ not exceeding 1000 such that: $\left[\frac{998}{n}\right]+\left[\frac{999}{n}\right]+\left[\frac{1000}{n}\right]$ is not divisible by 3.
(A) 22
(B) 23
(C) 24
(D) 25
(E) 26
|
22
|
1. Since a syllable consists of two different letters, identical letters can only appear at the junction of syllables.
First, let's find the number of combinations of two syllables with a matching letter at the junction. Such syllables (in terms of the arrangement of vowels and consonants) are either AMMO $(3 \cdot 8 \cdot 3$ variants) or MAAN $(8 \cdot 3 \cdot 8$ variants), totaling 264 variants.
From each such combination, a funny word can be formed in two ways - by adding an arbitrary syllable either at the beginning or at the end. Since the language has 48 syllables $(8 \ldots 3=24$ syllables of the form MA and another 24 syllables of the form AM), each of these methods yields $264 \cdot 48$ words.
However, some words are counted twice. These are words where the letters at the junction of the first syllable with the second and the letters at the junction of the second with the third syllable match. Clearly, all such words have the form AMMOON or MAANNO, and their number is $3 \cdot 8 \cdot 3 \cdot 8 + 8 \cdot 3 \cdot 8 \cdot 3 = 2 \cdot 24^{2}$.
|
24192
|
A certain rectangle had its dimensions expressed in decimeters as whole numbers. Then it changed its dimensions three times. First, it doubled one of its dimensions and changed the other so that it had the same area as at the beginning. Then it increased one dimension by $1 \mathrm{dm}$ and decreased the other by $4 \mathrm{dm}$, while still having the same area as at the beginning. Finally, it decreased its shorter dimension by $1 \mathrm{dm}$ and left the longer one unchanged.
Determine the ratio of the lengths of the sides of the final rectangle.
(E. Novotná)
|
4:1
|
It is given that $\log_{6}a + \log_{6}b + \log_{6}c = 6,$ where $a,$ $b,$ and $c$ are [positive](https://artofproblemsolving.com/wiki/index.php/Positive) [integers](https://artofproblemsolving.com/wiki/index.php/Integer) that form an increasing [geometric sequence](https://artofproblemsolving.com/wiki/index.php/Geometric_sequence) and $b - a$ is the [square](https://artofproblemsolving.com/wiki/index.php/Perfect_square) of an integer. Find $a + b + c.$
|
111
|
10. There is a sequence of numbers +1 and -1 of length $n$. It is known that the sum of every 10 neighbouring numbers in the sequence is 0 and that the sum of every 12 neighbouring numbers in the sequence is not zero.
What is the maximal value of $n$ ?
|
15
|
Exercise 16. Let $k \geqslant 1$ be an integer. What is the smallest integer $n$ such that, no matter how $n$ points are placed in the plane, it is possible to choose a subset $S$ consisting of $k$ of these points that satisfies "for every pair $P, Q$ of points in $S$, the distance between $P$ and $Q$ is less than or equal to 2" or "for every pair $\mathrm{P}, \mathrm{Q}$ of points in $\mathrm{S}$, the distance between $\mathrm{P}$ and $\mathrm{Q}$ is strictly greater than 1."
|
(k-1)^{2}+1
|
7. Given the function $f(x)=\frac{25^{x}}{25^{x}+5}$, then $\sum_{k=1}^{2016} f\left(\frac{k}{2017}\right)=$ $\qquad$
|
1008
|
9. (15 points) A rectangular plot of land $ABCD$ is divided into two rectangles as shown in the figure, which are contracted to households A and B, respectively. The area of household A's vegetable greenhouse is equal to the area of household B's chicken farm, and the remaining part of household A's area is 96 acres more than that of household B. Given that $BF$ $=3 CF$, what is the total area of the rectangle $ABCD$ in acres?
|
192
|
4. $\log _{2} \sin \frac{\pi}{12}+\log _{2} \sin \frac{\pi}{6}+\log _{2} \sin \frac{5 \pi}{12}=$ ( ).
(A) -3
(B) -1
(C) 1
(D) 3
|
A
|
As shown, $U$ and $C$ are points on the sides of triangle MNH such that $MU = s$, $UN = 6$, $NC = 20$, $CH = s$, $HM = 25$. If triangle $UNC$ and quadrilateral $MUCH$ have equal areas, what is $s$?
[img]https://cdn.artofproblemsolving.com/attachments/3/f/52e92e47c11911c08047320d429089cba08e26.png[/img]
|
s = 4
|
8. Given $0<x<\frac{\pi}{2}, \sin x-\cos x=\frac{\pi}{4}$. If $\tan x+\frac{1}{\tan x}$ can be expressed in the form $\frac{a}{b-\pi^{c}}$ ($a$, $b$, $c$ are positive integers), then $a+b+c=$ $\qquad$
|
50
|
A finite sequence of integers $ a_0, a_1, \ldots, a_n$ is called quadratic if for each $ i$ in the set $ \{1,2 \ldots, n\}$ we have the equality $ |a_i \minus{} a_{i\minus{}1}| \equal{} i^2.$
a.) Prove that any two integers $ b$ and $ c,$ there exists a natural number $ n$ and a quadratic sequence with $ a_0 \equal{} b$ and $ a_n \equal{} c.$
b.) Find the smallest natural number $ n$ for which there exists a quadratic sequence with $ a_0 \equal{} 0$ and $ a_n \equal{} 1996.$
|
n = 19
|
4. In the Cartesian coordinate system $x O y$, the curve $y=x^{3}-a x$ has two parallel tangent lines. If the slopes of these two tangent lines are both 1, and the distance between the two tangent lines is 8, then the value of the real number $a$ is $\qquad$.
|
5
|
7,8,9 |
The height $A H$ of triangle $A B C$ is equal to its median $B M$. On the extension of side $A B$ beyond point $B$, point $D$ is marked such that $B D=A B$. Find the angle $B C D$.
|
30
|
4.17 $m$ is the arithmetic mean of $a, b, c, d, e$, $k$ is the arithmetic mean of $a$ and $b$, $l$ is the arithmetic mean of $c, d$, and $e$, $p$ is the arithmetic mean of $k$ and $l$, regardless of how $a, b, c, d, e$ are chosen, then it is always true that
(A) $m=p$.
(B) $m \geqslant p$.
(C) $m>p$.
(D) $m<p$.
(E) None of the above.
(16th American High School Mathematics Examination, 1965)
|
E
|
28.31. Compute the sum $1+2 x+3 x^{2}+\ldots+n x^{n-1}$ by differentiating the equality $1+x+x^{2}+\ldots+x^{n}=\frac{1-x^{n+1}}{1-x}$.
See also problems $14.16,30.14$.
## 28.6. Tangent and Normal
|
\frac{1-(n+1)x^{n}+nx^{n+1}}{(1-x)^{2}}
|
5. The diagonals of quadrilateral $A B C D$ intersect at point $O$. It is known that $A B=B C=$ $=C D, A O=8$ and $\angle B O C=120^{\circ}$. What is $D O ?$
|
8
|
Galperin G.A.
Point $P$ lies inside isosceles triangle $ABC (AB = BC)$, and $\angle ABC = 80^{\circ}, \angle PAC = 40^{\circ}$, $\angle ACP = 30^{\circ}$. Find the angle $BPC$.
|
100
|
Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:
(a) for any integer $n$, $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than $187$.
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$. In every turn, Alice picks a number $k$ from the set $\{1,2,\ldots,187\}$, and Bob will tell Alice the value of $f(k)$. Find the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns.
[i]Proposed by YaWNeeT[/i]
|
187
|
8.3. Given an acute-angled triangle $A B C$. Point $M$ is the intersection point of its altitudes. Find the angle $A$, if it is known that $A M=B C$.
---
The text has been translated while preserving the original formatting and line breaks.
|
45
|
II. (40 points) Let $p$ be a prime number, and the sequence $\left\{a_{n}\right\}(n \geqslant 0)$ satisfies $a_{0}=0, a_{1}=1$, and for any non-negative integer $n$, $a_{n+2}=2 a_{n+1}-p a_{n}$. If -1 is a term in the sequence $\left\{a_{n}\right\}$, find all possible values of $p$.
保留了原文的换行和格式。
|
5
|
Farmer Boso has a busy farm with lots of animals. He tends to $5b$ cows, $5a +7$ chickens, and $b^{a-5}$ insects. Note that each insect has $6$ legs. The number of cows is equal to the number of insects. The total number of legs present amongst his animals can be expressed as $\overline{LLL }+1$, where $L$ stands for a digit. Find $L$.
|
3
|
A circle of length $10$ is inscribed in a convex polygon with perimeter $15$. What part of the area of this polygon is occupied by the resulting circle?
|
\frac{2}{3}
|
Let $\mathcal{F}$ be the set of functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(2x)=f(x)$ for all $x\in\mathbb{R}.$
[list=a]
[*]Determine all functions $f\in\mathcal{F}$ which admit antiderivatives on $\mathbb{R}.$
[*]Give an example of a non-constant function $f\in\mathcal{F}$ which is integrable on any interval $[a,b]\subset\mathbb{R}$ and satisfies \[\int_a^bf(x) \ dx=0\]for all real numbers $a$ and $b.$
[/list][i]Mihai Piticari and Sorin Rădulescu[/i]
|
f(x) = c
|
3. Given that $\mathrm{f}(x)$ is a polynomial of degree 2012, and that $\mathrm{f}(k)=\frac{2}{k}$ for $k=1,2,3, \cdots, 2013$, find the value of $2014 \times \mathrm{f}(2014)$.
|
4
|
Find the minimum value of the expression
$$
\sqrt{1-x+x^{2}}+\sqrt{1-\sqrt{3} \cdot x+x^{2}}
$$
|
\sqrt{2}
|
6. In $\triangle A B C$, it is known that $b c=b^{2}-a^{2}$, and $\angle B-$ $\angle A=80^{\circ}$. Then $\angle C=$ $\qquad$ (answer in degrees).
|
60^{\circ}
|
Rumyantsev V.
The perpendicular line restored at vertex $C$ of parallelogram $A B C D$ to line $C D$ intersects at point $F$ the perpendicular line dropped from vertex $A$ to diagonal $B D$, and the perpendicular line restored from point $B$ to line $A B$ intersects at point $E$ the perpendicular bisector of segment $A C$. In what ratio does side $B C$ divide segment $E F$?
|
1:2
|
2. Roll a die six times, let the number obtained on the $i$-th roll be $a_{i}$. If there exists a positive integer $k$, such that $\sum_{i=1}^{k} a_{i}=6$ has a probability $p=\frac{n}{m}$, where $m$ and $n$ are coprime positive integers. Then
$$
\log _{6} m-\log _{7} n=
$$
|
1
|
Two concentric circles $\omega, \Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\Omega$ and the tangent from $B$ to $\omega$ touches $\omega$ at $D$. What is the length of $AD$.
|
19
|
Find the greatest positive integer $N$ with the following property: there exist integers $x_1, . . . , x_N$ such that $x^2_i - x_ix_j$ is not divisible by $1111$ for any $i\ne j.$
|
1000
|
14. Given
$$
\frac{1}{\log _{2} a}+\frac{1}{\log _{3} a}+\frac{1}{\log _{4} a}=1 \text {. }
$$
Then the value of $a$ is ( ).
(A) 9
(B) 12
(C) 18
(D) 24
(E) 36
|
24
|
19. The integers from 1 to $n$, inclusive, are equally spaced in order round a circle. The diameter through the position of the integer 7 also goes through the position of 23 , as shown. What is the value of $n$ ?
A 30
B 32
C 34
D 36
E 38
|
32
|
Question 8 Find $A^{2}$, where $A$ is the sum of the absolute values of all roots of the equation
$$
x=\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{x}}}}}
$$
(9th American Invitational Mathematics Examination)
|
383
|
4. Let point $P$ be on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0$, $c=\sqrt{a^{2}-b^{2}}$ ), and the equation of line $l$ be $x=-\frac{a^{2}}{c}$. The coordinates of point $F$ are $(-c, 0)$. Draw $P Q \perp l$ at point $Q$. If points $P$, $F$, and $Q$ form an isosceles right triangle, then the eccentricity of the ellipse is ( ).
(A) $\frac{\sqrt{2}}{2}$
(B) $\frac{-1+\sqrt{2}}{2}$
(C) $\frac{-1+2 \sqrt{2}}{3}$
(D) $\frac{-1+\sqrt{5}}{2}$
|
A
|
3. Determine all pairs $(p, m)$ consisting of a prime number $p$ and a positive integer $m$, for which
$$
p^{3}+m(p+2)=m^{2}+p+1
$$
holds.
|
(2,5)
|
In the tetrahedron $A B C D$, $\angle A C B = \angle C A D = 90^{\circ}$ and $C A = C B = A D / 2$ and $C D \perp A B$. What is the angle between the faces $A C B$ and $A C D$?
|
60
|
Find the height of a triangular pyramid, the lateral edges of which are pairwise perpendicular and equal to 2, 3, and 4.
#
|
\frac{12}{\sqrt{61}}
|
Let $z$ be a complex number. If the equation \[x^3 + (4-i)x^2 + (2+5i)x = z\] has two roots that form a conjugate pair, find the absolute value of the real part of $z$.
[i]Proposed by Michael Tang[/i]
|
423
|
10. Point $C$ divides the diameter $A B$ in the ratio $A C: B C=2: 1$. A point $P$ is chosen on the circle. Determine the values that the ratio $\operatorname{tg} \angle P A C: \operatorname{tg} \angle A P C$ can take. In your answer, specify the smallest such value.
|
\frac{1}{2}
|
1. The sum of a set of numbers is the sum of all its elements. Let $S$ be a set of positive integers not exceeding 15, such that the sums of any two disjoint subsets of $S$ are not equal, and among all sets with this property, the sum of $S$ is the largest. Find the sum of the set $S$.
(4th American Invitational Mathematics Examination)
|
61
|
(21) (12 points) Given the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ passes through the point $(0,1)$, and the eccentricity is $\frac{\sqrt{3}}{2}$.
(1) Find the equation of the ellipse $C$;
(2) Suppose the line $l: x=m y+1$ intersects the ellipse $C$ at points $A$ and $B$, and the point $A$ is symmetric to point $A^{\prime}$ about the $x$-axis (where $A^{\prime}$ does not coincide with $B$). Does the line $A^{\prime} B$ intersect the $x$-axis at a fixed point? If so, write down the coordinates of the fixed point and prove your conclusion; if not, explain why.
|
(4,0)
|
$30 \cdot 43$ A two-digit number $N$ minus the number with its digits reversed, the result is a perfect cube, then
(A) $N$ cannot end in 5.
(B) Except for 5, $N$ can end in any digit.
(C) $N$ does not exist.
(D) $N$ has exactly 7 values.
(E) $N$ has exactly 10 values.
(8th American High School Mathematics Examination, 1957)
|
D
|
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