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3. Let the bivariate curve $C$ be given by the equation: $3 y^{2}-4(x+1) \cdot y+12(x-2)=0$. Then the graph of $C$ is (.
A. Ellipse
B. Hyperbola
C. Circle
D. None of the above
|
B
|
2. The lengths of two sides of a triangle are $20$ and $15$. Which of the following ( ) cannot be the perimeter of the triangle.
(A) 52
(B) 57
(C) 62
(D) 67
(E) 72
|
72
|
10. (15 points) Solve the system of equations
$$
\left\{\begin{array}{l}
a b+c+d=3 \\
b c+d+a=5 \\
c d+a+b=2 \\
d a+b+c=6
\end{array}\right.
$$
|
(a, b, c, d)=(2,0,0,3)
|
A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that 25% of these fish are no longer in the lake on September 1 (because of death and emigrations), that 40% of the fish were not in the lake May 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the September 1 sample are representative of the total population. What does the biologist calculate for the number of fish in the lake on May 1?
|
840
|
1. In an aquarium shaped like a rectangular prism, with a length of $75 \mathrm{~cm}$, a width of $28 \mathrm{~cm}$, and a height of $40 \mathrm{~cm}$, water is present up to a height of $25 \mathrm{~cm}$. A stone is dropped into the aquarium, and the water level rises by a quarter of its height. Fish are then added to the aquarium, occupying 1 \% of the aquarium's volume. The allowed water level in the aquarium must be 10 \% lower than the height of the aquarium to prevent the fish from jumping out. How many liters of water can still be added to the aquarium after the stone and fish have been added?
|
9.135
|
A circle with area $36 \pi$ is cut into quarters and three of the pieces are arranged as shown. What is the perimeter of the resulting figure?
(A) $6 \pi+12$
(B) $9 \pi+12$
(C) $9 \pi+18$
(D) $27 \pi+12$
(E) $27 \pi+24$
|
9\pi+12
|
15. (14 points) Let the ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, with the minor axis being 4, and the eccentricity being $e_{1}$. The hyperbola $\frac{x^{2}}{m}-\frac{y^{2}}{n}=1$ $(m>0, n>0)$ has asymptotes $y= \pm x$, and the eccentricity is $e_{2}$, and $e_{1} e_{2}=1$.
(1) Find the equation of the ellipse $E$.
(2) Let the right focus of the ellipse $E$ be $F$, and draw a line through the point $G(4,0)$ with a non-zero slope, intersecting the ellipse $E$ at points $M$ and $N$. Let the slopes of the lines $F M$ and $F N$ be $k_{1}$ and $k_{2}$, respectively. Determine whether $k_{1}+k_{2}$ is a constant value. If it is a constant value, find the value; if not, explain the reason.
|
k_{1}+k_{2}=0
|
Let $x$ be the smallest positive integer that simultaneously satisfies the following conditions: $2 x$ is the square of an integer, $3 x$ is the cube of an integer, and $5 x$ is the fifth power of an integer. Find the prime factorization of $x$.
#
|
2^{15}3^{20}5^{24}
|
10. For any positive integer $n \geqslant 1, 1+2+\cdots+n$ when expressed as a decimal, the last digit of the number can take which of the following values?
|
0,1,3,5,6,8
|
Four, $n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns;
$$
\begin{array}{llllll}
a_{11} & a_{12} & a_{13} & a_{14} & \cdots & a_{1 n} \\
a_{21} & a_{22} & a_{23} & a_{24} & \cdots & a_{2 n} \\
a_{31} & a_{32} & a_{33} & a_{34} & \cdots & a_{3 n} \\
a_{41} & a_{42} & a_{43} & a_{44} & \cdots & a_{4 n} \\
\cdots \cdots
\end{array}
$$
where the numbers in each row form an arithmetic sequence, and the numbers in each column form a geometric sequence, with all common ratios being equal. Given $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$, find
$$
a_{11}+a_{22}+a_{33}+a_{44}+\cdots+a_{n n} .
$$
|
2-\frac{1}{2^{n-1}}-\frac{n}{2^{n}}
|
17. Last year's match at Wimbledon between John Isner and Nicolas Malut, which lasted 11 hours and 5 minutes, set a record for the longest match in tennis history. The fifth set of the match lasted 8 hours and 11 minutes.
Approximately what fraction of the whole match was taken up by the fifth set?
A $\frac{1}{5}$
B $\frac{2}{5}$
C $\frac{3}{5}$
D $\frac{3}{4}$
E $\frac{9}{10}$
|
\frac{3}{4}
|
3. Let $x_{1}$ and $x_{2}$ be the roots of the equation
$$
p^{2} x^{2}+p^{3} x+1=0
$$
Determine $p$ such that the expression $x_{1}^{4}+x_{2}^{4}$ has the smallest value.
|
\\sqrt[8]{2}
|
$33 \cdot 54$ Let $F=0.48181 \cdots \cdots$ be an infinite repeating decimal, where 8 and 1 are the repeating digits. When $F$ is written as a simplified fraction, the denominator exceeds the numerator by
(A) 13.
(B) 14.
(C) 29.
(D) 57.
(E) 126.
(21st American High School Mathematics Examination, 1970)
|
57
|
One, (20 points) If the two sides of a right triangle, $x, y$, are both prime numbers, and make the algebraic expressions $\frac{2 x-1}{y}$ and $\frac{2 y+3}{x}$ both positive integers, find the inradius $r$ of this right triangle.
|
2
|
4. If real numbers $x, y$ satisfy $x^{3}+y^{3}+3 x y=1$, then the minimum value of $x^{2}+y^{2}$ is $\qquad$ .
|
\frac{1}{2}
|
24. Define $\operatorname{gcd}(a, b)$ as the greatest common divisor of integers $a$ and $b$. Given that $n$ is the smallest positive integer greater than 1000 and satisfies:
$$
\begin{array}{l}
\operatorname{gcd}(63, n+120)=21, \\
\operatorname{gcd}(n+63,120)=60 .
\end{array}
$$
Then the sum of the digits of $n$ is ( ).
(A) 12
(B) 15
(C) 18
(D) 21
(E) 24
|
18
|
Example 5 Try to find the maximum value of the following expression $x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}$.
|
1
|
Problem 8.1. Solve the equation
$$
|| x-\frac{5}{2}\left|-\frac{3}{2}\right|=\left|x^{2}-5 x+4\right|
$$
Ivan Tonov
|
x_{1}=1,x_{2}=4
|
92. A class selects representatives from four students to serve as environmental protection volunteers (there is no limit on the number of slots, and it is also possible to select none), then there are $\qquad$ ways of selection.
|
16
|
4. Red, blue, green, and white four dice, each die's six faces have numbers $1, 2, 3, 4, 5, 6$. Simultaneously roll these four dice so that the product of the numbers facing up on the four dice equals 36, there are $\qquad$ possible ways.
|
48
|
Example 3 For any $a>0, b>0$, find $\max \left\{\min \left\{\frac{1}{a}, \frac{1}{b}, a^{2}+b^{2}\right\}\right\}$.
(2002, Beijing Middle School Mathematics Competition)
|
\sqrt[3]{2}
|
3. [5] A polygon $\mathcal{P}$ is drawn on the 2D coordinate plane. Each side of $\mathcal{P}$ is either parallel to the $x$ axis or the $y$ axis (the vertices of $\mathcal{P}$ do not have to be lattice points). Given that the interior of $\mathcal{P}$ includes the interior of the circle $x^{2}+y^{2}=2022$, find the minimum possible perimeter of $\mathcal{P}$.
|
8\sqrt{2022}
|
\section*{Problem 2 - 071022}
Given a rectangle \(A B C D\). The midpoint of \(A B\) is \(M\). Connect \(C\) and \(D\) with \(M\) and \(A\) with \(C\). The intersection point of \(A C\) and \(M D\) is \(S\).
Determine the ratio of the area of the rectangle \(A B C D\) to the area of the triangle \(\triangle S M C\)!
|
6:1
|
Let $A B C D$ be a square, $P$ inside $A B C D$ such that $P A=1, P B=2, P C=3$. Calculate $\widehat{A P B}$.
|
135
|
23. In a basketball game at a certain middle school, A scored $\frac{1}{4}$ of the total points, B scored $\frac{2}{7}$ of the total points, C scored 15 points, and the remaining seven players each scored no more than 2 points. Then the total score of the remaining seven players is ( ).
(A) 10
(B) 11
(C) 12
(D) 13
(E) 14
|
11
|
3. As shown in Figure $1, \Omega$ is the geometric body obtained after the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$ is cut by the plane $E F G H$, removing the geometric body $E F G H B_{1} C_{1}$. Here, $E$ and $F$ are points on the line segments $A_{1} B_{1}$ and $B B_{1}$, respectively, different from $B_{1}$, and $E H / / A_{1} D_{1}$. Which of the following conclusions is incorrect? ( ).
(A) $E H / / F G$
(B) Quadrilateral $E F G H$ is a rectangle
(C) $\Omega$ is a prism
(D) $\Omega$ is a frustum
|
D
|
15. A1 (POL) ${ }^{\mathrm{IMO} 2}$ Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such that the inequality
$$ \sum_{i<j} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq C\left(\sum_{i} x_{i}\right)^{4} $$
holds for every $x_{1}, \ldots, x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.
|
\frac{1}{8}
|
Marta chose a 3-digit number with different non-zero digits and multiplied it by 3. The result was a 3-digit number with all digits equal to the tens digit of the chosen number. What number did Marta choose?
#
|
148
|
24. Find the number of permutations $a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}$ of the six integers from 1 to 6 such that for all $i$ from 1 to $5, a_{i+1}$ does not exceed $a_{i}$ by 1 .
|
309
|
2. Given two linear functions $f(x)$ and $g(x)$ such that the graphs $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. It is known that the graph of the function $y=(f(x))^{2}$ touches the graph of the function $y=20 g(x)$. Find all values of $A$ such that the graph of the function $y=(g(x))^{2}$ touches the graph of the function $y=\frac{f(x)}{A}$.
|
-0.05
|
Problem 5. The numbers $a, b$, and $c$ satisfy the equation $\sqrt{a}=\sqrt{b}+\sqrt{c}$. Find $a$, if $b=52-30 \sqrt{3}$ and $c=a-2$.
|
27
|
1. If the set $M=\left\{x \left\lvert\, x^{2}-\frac{5}{4} x+a1\right., x \in \mathbf{R}\right\}
$
is an empty set, then the range of values for $a$ is ( ).
(A) $a \leqslant-\frac{3}{2}$
(B) $a \geqslant-\frac{3}{2}$
(C) $a \leqslant \frac{1}{4}$
(D) $a \geqslant \frac{1}{4}$
|
D
|
9. (16 points) Given $a, b \in [1,3], a+b=4$.
Find the maximum value of $f(a, b)=\left|\sqrt{a+\frac{1}{b}}-\sqrt{b+\frac{1}{a}}\right|$.
|
2-\frac{2}{\sqrt{3}}
|
141. Find all values of the digit $a$, if the number $\overline{a 719}$ is divisible by 11.
|
4
|
7.2. The two adjacent sides of a rectangle are in the ratio 3:7. What is the area of the rectangle if its perimeter is 40 cm?
|
84\mathrm{~}^{2}
|
Let $\mathbb{N}$ denote the set of positive integers, and let $S$ be a set. There exists a function $f :\mathbb{N} \rightarrow S$ such that if $x$ and $y$ are a pair of positive integers with their difference being a prime number, then $f(x) \neq f(y)$. Determine the minimum number of elements in $S$.
|
4
|
3. Find all real functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{2}-y^{2}\right)=(x-y)[f(x)+f(y)]
$$
|
f(x)=kx
|
31. In one month, three Wednesdays fell on even dates. What date will the second Sunday of the month be?
|
13
|
Problem 4. Let $\left(a_{n}\right)_{n \in N}$ be a sequence of real numbers with the property that for any $m, n \in N, m \geq n$,
$2\left(a_{m+n}+a_{m-n}\right)=a_{2 m}+a_{2 n}$ and $a_{1}=1$. Determine a formula for the general term of the sequence $\left(a_{n}\right)_{n \in N}$.
|
a_{n}=n^{2}
|
(1) For $ a>0,\ b\geq 0$, Compare
$ \int_b^{b\plus{}1} \frac{dx}{\sqrt{x\plus{}a}},\ \frac{1}{\sqrt{a\plus{}b}},\ \frac{1}{\sqrt{a\plus{}b\plus{}1}}$.
(2) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{1}{\sqrt{n^2\plus{}k}}$.
|
1
|
If $x, y$ are real numbers, and $1 \leqslant x^{2}+4 y^{2} \leqslant 2$. Find the range of $x^{2}-2 x y+4 y^{2}$.
|
\left[\frac{1}{2}, 3\right]
|
C2. The integers $a, b, c, d, e, f$ and $g$, none of which is negative, satisfy the following five simultaneous equations:
$$
\begin{array}{l}
a+b+c=2 \\
b+c+d=2 \\
c+d+e=2 \\
d+e+f=2 \\
e+f+g=2 .
\end{array}
$$
What is the maximum possible value of $a+b+c+d+e+f+g$ ?
|
6
|
C2. The natural numbers from 1 to 50 are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?
|
25
|
## Task 7
All 2nd grade classes of the Ernst-Thälmann-Oberschule are participating in the ABC newspaper's call "Good morning, hometown!"
22 Young Pioneers help keep the green spaces in the city district clean, 19 Young Pioneers help design a wall newspaper in the residential area. 18 Young Pioneers explore what is produced in the businesses of their hometown.
How many Young Pioneers have taken on assignments?
|
59
|
9. (5 points) $1^{2013}+2^{2013}+3^{2013}+4^{2013}+5^{2013}$ divided by 5, the remainder is $\qquad$ . ( $a^{2013}$ represents 2013 $a$s multiplied together)
|
0
|
## Task B-1.2.
When asked how many minutes she spends on social networks daily, Iva answered: "The nonuple of that number is between 1100 and 1200, and the tridecuple is between 1500 and 1600." How many minutes does Iva spend on social networks daily?
|
123
|
Problem 2. How many solutions in integers does the equation
$$
\frac{1}{2022}=\frac{1}{x}+\frac{1}{y} ?
$$
|
53
|
Example 2 For any natural number $k$, if $k$ is even, divide it by 2, if $k$ is odd, add 1 to it, this is called one operation. Let the number of numbers that become 1 after exactly $n$ operations be $a_{n}$, try to find $a_{15}$.
|
610
|
234*. Once in a room, there were several inhabitants of an island where only truth-tellers and liars live. Three of them said the following.
- There are no more than three of us here. All of us are liars.
- There are no more than four of us here. Not all of us are liars.
- There are five of us. Three of us are liars.
How many people are in the room and how many of them are liars?
|
4
|
What is the maximum number of squares on an $8 \times 8$ chessboard on which pieces may be placed so that no two of these squares touch horizontally, vertically, or diagonally?
|
16
|
The distance between points $A$ and $B$ is 40 km. A pedestrian left $A$ at 4 o'clock. When he had walked half the distance, he was caught up by a cyclist who had left $A$ at 7:20. An hour after this, the pedestrian met another cyclist who had left $B$ at 8:30. The speeds of the cyclists are the same. Determine the speed of the pedestrian.
|
5
|
3. If $\left(x^{2}-x-2\right)^{3}=a_{0}+a_{1} x+\cdots+a_{6} x^{6}$, then $a_{1}+a_{3}+a_{5}=$ $\qquad$
|
-4
|
In how many ways can we place two rooks of different colors on a chessboard so that they do not attack each other?
|
3136
|
9.5. For what minimum $\boldsymbol{n}$ in any set of $\boldsymbol{n}$ distinct natural numbers, not exceeding 100, will there be two numbers whose sum is a prime number?
|
51
|
4. Given that $P$ is a point on the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$, $F_{1}$ is its left focus, and $Q$ is on $P F_{1}$ such that
$$
\overrightarrow{O Q}=\frac{1}{2}\left(\overrightarrow{O P}+\overrightarrow{O F_{1}}\right),|\overrightarrow{O Q}|=3 \text {. }
$$
Then the distance from point $P$ to the left directrix of the ellipse is
|
5
|
69. This year, the sum of Dan Dan's, her father's, and her mother's ages is 100 years. If 6 years ago, her mother's age was 4 times Dan Dan's age, and 13 years ago, her father's age was 15 times Dan Dan's age, then Dan Dan is $\qquad$ years old this year.
|
15
|
Let's simplify the following fraction:
$$
\frac{x^{8}+x^{7}-x^{5}-x^{4}+x^{3}-1}{x^{6}-x^{4}+x-1}
$$
|
x^{2}+x+1
|
Consider a sequence $F_0=2$, $F_1=3$ that has the property $F_{n+1}F_{n-1}-F_n^2=(-1)^n\cdot2$. If each term of the sequence can be written in the form $a\cdot r_1^n+b\cdot r_2^n$, what is the positive difference between $r_1$ and $r_2$?
|
\frac{\sqrt{17}}{2}
|
Example 5 In $\triangle A B C$, it is known that $\angle B A C=45^{\circ}$, $A D \perp B C$ at point $D$. If $B D=2, C D=3$, then $S_{\triangle A B C}$ $=$ $\qquad$
(2007, Shandong Province Junior High School Mathematics Competition)
|
15
|
2. Given $(a+b i)^{2}=3+4 i$, where $a, b \in \mathbf{R}, i$ is the imaginary unit, then the value of $a^{2}+b^{2}$ is $\qquad$
|
5
|
28. The teacher wrote down the counting numbers starting from 1 continuously on the blackboard: $1,3,5,7,9,11 \cdots$ After writing, the teacher erased two of the numbers, dividing the sequence of odd numbers into 3 segments. If the sums of the first two segments are 961 and 1001 respectively, then the sum of the two odd numbers erased by the teacher is $(\quad)$
A. 154
B. 156
C. 158
D. 160
|
154
|
3.13 A three-digit number ends with the digit 2. If it is moved to the beginning of the number, the resulting number will be 18 more than the original. Find the original number.
|
202
|
5. A $7 \times 7$ board has a chessboard coloring. In one move, you can choose any $m \times n$ rectangle of cells and repaint all its cells to the opposite color (black cells become white, white cells become black). What is the minimum number of moves required to make the board monochromatic?
Answer: in 6 moves.
|
6
|
5. Given that $D, F$ are points on the sides $A B, A C$ of $\triangle A B C$ respectively, and $A D: D B=C F: F A=2: 3$, connect $D F$ to intersect the extension of side $B C$ at point $E$, find $E F: F D$.
|
EF:FD=2:1
|
I2.3 There are less than $4 Q$ students in a class. In a mathematics test, $\frac{1}{3}$ of the students got grade $A$, $\frac{1}{7}$ of the students got grade $B$, half of the students got grade $C$, and the rest failed. Given that $R$ students failed in the mathematics test, find the value of $R$.
|
1
|
4. In $\triangle A B C$, the sides opposite to angles $A, B, C$ are denoted as $a, b, c (b \neq 1)$, respectively, and $\frac{C}{A}, \frac{\sin B}{\sin A}$ are both roots of the equation $\log _{\sqrt{6}} x=\log _{6}(4 x-4)$. Then $\triangle A B C$
(A) is an isosceles triangle, but not a right triangle;
(B) is a right triangle, but not an isosceles triangle;
(C) is an isosceles right triangle;
(D) is neither an isosceles triangle nor a right triangle.
|
B
|
(i) Find the number of odd subsets and even subsets of $X$;
(ii) Find the sum of the sums of elements of all odd subsets of $X$.
|
2^{(n-3)}n(n+1)
|
10. Let $f(n)$ be the integer closest to $\sqrt[4]{n}$. Then $\sum_{k=1}^{2018} \frac{1}{f(k)}=$ $\qquad$
|
\frac{2823}{7}
|
Example 10 Calculation
$$
\begin{array}{l}
\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2006}\right)\left(1+\frac{1}{2}+\cdots+\frac{1}{2005}\right) \\
\left(1+\frac{1}{2}+\cdots+\frac{1}{2006}\right)\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2005}\right)
\end{array}
$$
|
\frac{1}{2006}
|
## Zadatak B-1.1.
Ako je
$$
\left(\frac{1}{1+\frac{1}{1+\frac{1}{a}}}-\frac{1}{1+\frac{1}{1-\frac{1}{a}}}\right) \cdot\left(2+\frac{2}{1+\frac{1}{a-1}}\right)=\frac{1}{504}
$$
koliko je $2 a+1$ ?
|
2016
|
12. For any positive integer $n$, define the function $\mu(n)$:
$$
\mu(1)=1 \text {, }
$$
and when $n=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{t}^{\alpha_{4}} \geqslant 2$,
$$
\mu(n)=\left\{\begin{array}{ll}
(-1)^{t}, & \alpha_{1}=\alpha_{2}=\cdots=\alpha_{t}=1 ; \\
0, & \text { otherwise, }
\end{array}\right.
$$
where, $t \geqslant 1, p_{1}, p_{2}, \cdots, p_{t}$ are distinct primes.
If we denote $A=\left\{x_{1}, x_{2}, \cdots, x_{k}\right\}$ as the set of all distinct positive divisors of 12, then $\sum_{i=1}^{k} \mu\left(x_{i}\right)=$ $\qquad$
|
0
|
10.6. A square $100 \times 100$ is divided into squares $2 \times 2$. Then it is divided into dominoes (rectangles $1 \times 2$ and $2 \times 1$). What is the smallest number of dominoes that could end up inside the squares of the division?
(C. Berlov)
#
|
100
|
13.181. A team of workers was supposed to manufacture 7200 parts per shift, with each worker making the same number of parts. However, three workers fell ill, and therefore, to meet the entire quota, each of the remaining workers had to make 400 more parts. How many workers were in the team?
|
9
|
Example 3 Given $x \in \mathbf{R}$, let $M=\max \{1-x, 2 x-$ $3, x\}$. Find $M_{\min }$.
|
\frac{1}{2}
|
9.141. $\sqrt{9^{x}-3^{x+2}}>3^{x}-9$.
9.141. $\sqrt{9^{x}-3^{x+2}}>3^{x}-9$.
|
x\in(2;\infty)
|
2. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{R}$ such that $f(1)=\frac{5}{2}$ and
$$
f(x) f(y)=f(x+y)+f(x-y), \text { for all } x, y \in \mathbb{Z}
$$
|
f(x)=\frac{2^{2x}+1}{2^{x}}
|
5. The integer part $[x]$ of a number $x$ is defined as the greatest integer $n$ such that $n \leqslant x$, for example, $[10]=10,[9.93]=9,\left[\frac{1}{9}\right]=0,[-1.7]=-2$. Find all solutions to the equation $\left[\frac{x+3}{2}\right]^{2}-x=1$.
|
0
|
$1 \cdot 31$ Let $\alpha_{n}$ denote the integer closest to $\sqrt{n}$, find the sum $\frac{1}{\alpha_{1}}+\frac{1}{\alpha_{2}}+\cdots+\frac{1}{\alpha_{1980}}$
|
88
|
6. (2000 "Hope Cup" Invitational Competition Question) Let $a>b>c$, and $\frac{1}{a-b}+\frac{1}{b-c} \geqslant \frac{n}{a-c}$ always holds, then what is the maximum value of $n$?
|
4
|
9. How many pairs of integers solve the system $|x y|+|x-y|=2$ if $-10 \leq x, y \leq 10 ?$
|
4
|
Problem 2. Vanya thought of a two-digit number, then swapped its digits and multiplied the resulting number by itself. The result turned out to be four times larger than the number he thought of. What number did Vanya think of?
|
81
|
G1.3 If 90 ! is divisible by $10^{k}$, where $k$ is a positive integer, find the greatest possible value of $k$.
|
21
|
90. In a stationery box, there are 38 watercolor pens of the same size, including 10 white, yellow, and red pens each, as well as 4 blue pens, 2 green pens, and 2 purple pens. How many pens must be taken out to ensure that at least 4 pens of the same color are taken out?
|
17
|
3. In $\triangle A B C$, $D$ is a point on side $B C$. It is known that $A B=13, A D=12, A C=15, B D=5$, find $D C$.
|
9
|
100. Use 10 $1 \times 2$ small rectangles (horizontal or vertical) to cover a $2 \times 10$ grid, there are $\qquad$ different ways to do so.
|
89
|
Task 1. Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$, the sum of the first $m$ terms or the sum of the last $m$ terms of the sequence is an integer. Determine the minimum number of integers in a complete sequence of $n$ numbers.
|
2
|
3. After teacher Mary Ivanovna moved Vovochka from the first row to the second, Vanechka from the second row to the third, and Mashenka from the third row to the first, the average age of students sitting in the first row increased by one week, those sitting in the second row increased by two weeks, and those sitting in the third row decreased by four weeks. It is known that there are 12 people sitting in the first and second rows. How many people are sitting in the third row?
|
9
|
7.045. $\log _{5}(3 x-11)+\log _{5}(x-27)=3+\log _{5} 8$.
|
37
|
9.2. It is known that $\frac{a^{2} b^{2}}{a^{4}-2 b^{4}}=1$. Find all possible values of the expression $\frac{a^{2}-b^{2}}{a^{2}+b^{2}}$.
|
\frac{1}{3}
|
Problem 5. In the confectionery Coconut, there are tables and chairs grouped into sets as follows: several sets of one table with four legs and four chairs with three legs, and several sets of one table with three legs and four chairs with four legs. The total number of legs is 264. How many children can sit in the confectionery?
|
60
|
3. Find the smallest positive number $x$, for which the following holds: If $a, b, c, d$ are any positive numbers whose product is 1, then
$$
a^{x}+b^{x}+c^{x}+d^{x} \geqq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}
$$
(Pavel Novotný)
|
3
|
Proivolov V.V.
In an isosceles triangle $ABC (AB = AC)$, the angle $A$ is equal to $\alpha$. On the side $AB$, a point $D$ is taken such that $AD = AB / n$. Find the sum of $n-1$ angles under which the segment $AD$ is seen from the points dividing the side $BC$ into $n$ equal parts:
a) for $n=3$
b) for an arbitrary $n$.
|
\alpha/2
|
XXXIV OM - I - Problem 1
$ A $ tosses a coin $ n $ times, $ B $ tosses it $ n+1 $ times. What is the probability that $ B $ will get more heads than $ A $?
|
\frac{1}{2}
|
11.2. Solve the system of equations
$$
\left\{\begin{array}{l}
x y+x=15 \\
x^{2}+y^{2}+x y+y=41
\end{array}\right.
$$
|
(3;4),(5;2)
|
(2) Let the internal angles $A, B, C$ of $\triangle ABC$ have opposite sides $a, b, c$ respectively, and satisfy $a \cos B - b \cos A = \frac{3}{5} c$. Then the value of $\frac{\tan A}{\tan B}$ is $\qquad$.
|
4
|
4.1. The sea includes a bay with more saline water. The salinity of the water in the sea is 120 per mille, in the bay 240 per mille, in the part of the sea not including the bay - 110 per mille. How many times is the volume of water in the sea larger than the volume of water in the bay? The volume of water is considered, including the volume of salt. Per mille - one-thousandth of a number; salinity is determined as the ratio of the volume of salt to the total volume of the mixture.
|
13
|
## 24. Math Puzzle $5 / 67$
A pedestrian walking along a tram line noticed that a tram overtook him every 12 minutes and that a tram of this line approached him every 4 minutes. The pedestrian walked at a constant speed, as did the tram on this section of the route.
At what time interval do the trams operate on this tram line?
|
6
|
Exercise 3. Consider a number $N$ that is written in the form $30 x 070 y 03$, with $x, y$ being digits between 0 and 9. For which values of $(x, y)$ is the integer $N$ divisible by 37?
|
(x,y)=(8,1),(4,4),(0,7)
|
Determine the positive integers $a, b$ such that $a^{2} b^{2}+208=4\{lcm[a ; b]+gcd(a ; b)\}^{2}$.
|
(a, b) \in\{(2,12) ;(4,6) ;(6,4) ;(12 ; 2)\}
|
4. If $p, q \in N^{+}$ and $p+q>2017, 0<p<q \leq 2017, (p, q)=1$, then the sum of all fractions of the form $\frac{1}{p q}$ is $\qquad$
|
\frac{1}{2}
|
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