problem
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5. Mr. Patrick is the math teacher of 15 students. After a test, he found that the average score of the rest of the students, excluding Peyton, was 80 points, and the average score of the entire class, including Peyton, was 81 points. What was Peyton's score in this test? ( ) points.
(A) 81
(B) 85
(C) 91
(D) 94
(E) 95
|
95
|
11. How many integers between 1 and 2005 (inclusive) have an odd number of even digits?
|
1002
|
6.13 Find the natural numbers forming an arithmetic progression if the products of the first three and the first four of its terms are 6 and 24, respectively.
|
1;2;3;4
|
Task 3. A team of workers was working on pouring the rink on the large and small fields, with the area of the large field being twice the area of the small field. In the part of the team that worked on the large field, there were 4 more workers than in the part that worked on the small field. When the pouring of the large rink was completed, the part of the team that was on the small field was still working. What is the maximum number of workers that could have been in the team
|
10
|
PROBLEM 4. We say that a number of the form $\overline{0, a b c d e}$ has property $(P)$ if the digits $a, b, c, d, e$ belong to the set $\{4,6\}$.
a) Show that the solution to the equation $x+0,46646=1,1111$ has property $(P)$;
b) Determine how many different numbers of the form $\overline{0, a b c d e}$ have property $(P)$;
c) Show that from any 17 different numbers of the form $\overline{0, a b c d e}$ that have property $(P)$, two can be chosen whose sum is 1,1111 .[^0]
## NATIONAL MATHEMATICS OLYMPIAD Local stage - 5.03. 2016 GRADING GUIDE - 6th Grade
|
32
|
Anna and Kati are celebrating their birthdays today. Three years from now, Anna will be four times as old as Kati was when Anna was two years older than Kati is now. How old is Anna if Kati is currently in high school?
|
25
|
## Task 5 - 230935
In a triangle $A B C$, a line parallel to $A B$, whose position is otherwise not specified, intersects the side $A C$ at a point $A_{1}$ between $A$ and $C$, and it intersects the side $B C$ at $B_{1}$. Furthermore, let $P$ be a point on $A B$ between $A$ and $B$, whose position is also not otherwise specified.
The area of triangle $A B C$ is $F_{0}$, and the area of triangle $A_{1} B_{1} C$ is $F_{1}$.
Determine the area $F$ of the quadrilateral $A_{1} P B_{1} C$ in terms of $F_{0}$ and $F_{1}$!
|
\sqrt{F_{1}\cdotF_{0}}
|
11. Let $a$ and $b$ be integers for which $\frac{a}{2}+\frac{b}{1009}=\frac{1}{2018}$. Find the smallest possible value of $|a b|$.
|
504
|
30. Let $f(x)=4 x-x^{2}$. Given $x_{0}$, consider how many real numbers $x_{0}$, make the sequence $x_{0}, x_{1}, x_{2}, \cdots H$ take infinitely many different values.
(A)0.(B)1 or 2.(C)3, 1, or 6.
(D)More than 6 but finite. (E)Infinitely many.
|
E
|
(4) As shown in the figure, in the tetrahedron $D-ABC$, it is known that $DA \perp$ plane $ABC$, and $\triangle ABC$ is an equilateral triangle with a side length of 2. Then, when the tangent value of the dihedral angle $A-BD-C$ is 2, the volume $V$ of the tetrahedron $D-ABC$ is $\qquad$
|
2
|
# Problem №1 (15 points)
Two identical cars are driving in the same direction. The speed of one is $36 \kappa \mu / h$, and the other is catching up at a speed of $54 \mathrm{km} / h$. It is known that the reaction time of the driver of the rear car to the activation of the brake lights of the front car is 2 seconds. What should be the distance between the cars so that they do not collide if the first driver decides to brake sharply? For a car of this make, the braking distance is 40 meters at a speed of $72 \kappa \mu / h$.
|
42.5
|
1. (16 points) There are two circles: one with center at point $A$ and radius 5, and another with center at point $B$ and radius 15. Their common internal tangent touches the circles at points $C$ and $D$ respectively. Lines $A B$ and $C D$ intersect at point $E$. Find $C D$, if $B E=39$.
|
48
|
5. Rewrite the equation as
$$
5^{(x-1)^{2}} \log _{7}\left((x-1)^{2}+2\right)=5^{2|x-a|} \log _{7}(2|x-a|+2)
$$
Let $f(t)=5^{t} \log _{7}(t+2)$. This function is increasing. Therefore,
$$
f\left((x-1)^{2}\right)=f(2|x-a|) \Longleftrightarrow(x-1)^{2}=2|x-a|
$$
There will be three solutions in the case of tangency for $a=1 / 2, a=3 / 2$ and when $a=1$, since the vertices of the parabola $y=(x-1)^{2}$ and the curve $y=|x-1|$ coincide.
|
1/2;1;3/2
|
110) Let $p$ be a given odd prime, and let the positive integer $k$ be such that $\sqrt{k^{2}-p k}$ is also a positive integer. Then $k=$ $\qquad$
|
\frac{(p+1)^{2}}{4}
|
A rectangle is divided by two vertical and two horizontal segments into nine rectangular parts. The areas of some of the resulting parts are indicated in the figure. Find the area of the upper right part.
| 30 | | $?$ |
| :---: | :---: | :---: |
| 21 | 35 | |
| | 10 | 8 |
|
40
|
9.2. In a kindergarten, each child was given three cards, each of which had either "MA" or "NYA" written on it. It turned out that 20 children could form the word "MAMA" from their cards, 30 children could form the word "NYANYA," and 40 children could form the word "MANYA." How many children had all three cards the same?
|
10
|
17.5 Given that the area of $\triangle A B C$ is $S$, points $D$, $E$, and $F$ are the midpoints of $B C$, $C A$, and $A B$ respectively, $I_{1}$, $I_{2}$, and $I_{3}$ are the incenters of $\triangle A E F$, $\triangle B F D$, and $\triangle C D E$ respectively, and the area of $\triangle I_{1} I_{2} I_{3}$ is $S^{\prime}$. Find the value of $S^{\prime} / S$.
|
\frac{1}{4}
|
3. Among the four functions $y=\sin |x|, y=\cos |x|, y=|\cot x|, y=\lg |\sin x|$, the even function that is periodic with $\pi$ and monotonically increasing in $\left(0, \frac{\pi}{2}\right)$ is
A. $y=\sin |x|$
B. $y=\cos |x|$
C. $y=|\cot x|$
D. $y=\lg |\sin x|$
|
D
|
2. The sum of the ages of three children, Ava, Bob and Carlo, is 31 . What will the sum of their ages be in three years' time?
A 34
B 37
С 39
D 40
E 43
|
40
|
3. Given $a>1$. Then the minimum value of $\log _{a} 16+2 \log _{4} a$ is . $\qquad$
|
4
|
21. A triangle whose angles are $A, B, C$ satisfies the following conditions
$$
\frac{\sin A+\sin B+\sin C}{\cos A+\cos B+\cos C}=\frac{12}{7},
$$
and
$$
\sin A \sin B \sin C=\frac{12}{25} .
$$
Given that $\sin C$ takes on three possible values $s_{1}, s_{2}$ and $s_{3}$, find the value of $100 s_{1} s_{2} s_{3}$ -
|
48
|
3. Solve the system of equations:
$$
\left\{\begin{array}{l}
x^{100}-y^{100}=2^{99}(x-y), \\
x^{200}-y^{200}=2^{199}(x-y)
\end{array}(x \neq y) .\right.
$$
|
(x,y)=(2,0)or(0,2)
|
20. Find the total number of positive integers $n$ not more than 2013 such that $n^{4}+5 n^{2}+9$ is divisible by 5 .
|
1611
|
4. Solve the inequality
$$
\frac{9^{x}-5 \cdot 15^{x}+4 \cdot 25^{x}}{-9^{x}+8 \cdot 15^{x}-15 \cdot 25^{x}}<0
$$
|
x\in(-\infty,\log_{\frac{3}{5}}5)\cup(\log_{\frac{3}{5}}4,\log_{\frac{3}{5}}3)\cup(0,\infty)
|
1. (2 points) In a row without spaces, all natural numbers are written in ascending order: $1234567891011 . .$. What digit stands at the 2017th place in the resulting long number?
|
7
|
2. Given that $a$ and $b$ are non-zero constants, and
$$
\frac{a^{2}}{\sin ^{2} \theta}+\frac{b^{2}}{\cos ^{2} \theta}=(a+b)^{2} \text {. }
$$
Then $\frac{a^{3}}{\sin ^{4} \theta}+\frac{b^{3}}{\cos ^{4} \theta}=$ (express in terms of $a$ and $b$).
|
(a+b)^{3}
|
Example 1. An isosceles trapezoid is inscribed in a semicircle with a radius of 1, where the lower base is the diameter of the semicircle. (1) Try to find the functional relationship between its perimeter \( \mathrm{y} \) and the leg length \( \mathrm{x} \), and write down the domain; (2) When the leg length is how much, does the perimeter have a maximum value? (This example is similar to the 1982 Tianjin Junior High School Mathematics Exam)
|
y=5 \text{ when } x=1
|
Let's determine the numbers $A, B, C, D$ such that
$$
\frac{A}{x+1}+\frac{B}{(x+1)^{2}}+\frac{C x+D}{x^{2}+x+1}=\frac{1}{(x+1)^{2} \cdot\left(x^{2}+x+1\right)}
$$
holds for all $x$ different from $-1$.
|
A=1,B=1,C=-1,D=-1
|
B1. Simplify the expression: $\frac{2-x}{x^{3}-x^{2}-x+1}:\left(\frac{1}{x-1} \cdot \frac{x}{x+1}-\frac{2}{x+1}\right)$.
|
\frac{1}{x-1}
|
Find the period of the repetend of the fraction $\frac{39}{1428}$ by using [i]binary[/i] numbers, i.e. its binary decimal representation.
(Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2\cdots a_ka_1a_2\cdots a_ka_1a_2 \cdots a_k \cdots$. Here the repeated sequence $a_1a_2\cdots a_k$ is called the [i]repetend[/i] of the fraction, and the smallest length of the repetend, $k$, is called the [i]period[/i] of the decimal number.)
|
24
|
11. According to the "Personal Income Tax Law of the People's Republic of China," citizens do not need to pay tax on the portion of their monthly salary and wages that does not exceed 800 yuan. The portion exceeding 800 yuan is the monthly taxable income, which is taxed according to the following table in segments:
\begin{tabular}{|c|c|}
\hline Monthly taxable income & Tax rate \\
\hline The portion not exceeding 500 yuan & $5 \%$ \\
\hline The portion exceeding 500 yuan to 2000 yuan & $10 \%$ \\
\hline The portion exceeding 2000 yuan to 5000 yuan & $15 \%$ \\
\hline$\ldots .$. & $\ldots .$. \\
\hline
\end{tabular}
A person's tax for January is 30 yuan. Then his monthly salary and wages for that month are $\qquad$ yuan.
|
1350
|
11. Given that the odd function $f(x)$ is a monotonically decreasing function on $[-1,0]$, and $\alpha, \beta$ are two interior angles of an acute triangle, the relationship in size between $f(\cos \alpha)$ and $f(\sin \beta)$ is $\qquad$ .
|
f(\cos\alpha)>f(\sin\beta)
|
59. Find the non-negative integer values of $n$ for which $\frac{30 n+2}{12 n+1}$ is an integer.
|
0
|
B1. We consider numbers of two or more digits where none of the digits is 0. We call such a number thirteenish if every two adjacent digits form a multiple of 13. For example, 139 is thirteenish because $13=1 \times 13$ and $39=3 \times 13$.
How many thirteenish numbers of five digits are there?
|
6
|
11. A belt drive system consists of the wheels $K, L$ and $M$, which rotate without any slippage. The wheel $L$ makes 4 full turns when $K$ makes 5 full turns; also $L$ makes 6 full turns when $M$ makes 7 full turns.
The perimeter of wheel $M$ is $30 \mathrm{~cm}$. What is the perimeter of wheel $K$ ?
A $27 \mathrm{~cm}$
B $28 \mathrm{~cm}$
C $29 \mathrm{~cm}$
D $30 \mathrm{~cm}$
E $31 \mathrm{~cm}$
|
28\mathrm{~}
|
11. (20 points) Given the parabola $P: y^{2}=x$, with two moving points $A, B$ on it, the tangents at $A$ and $B$ intersect at point $C$. Let the circumcenter of $\triangle A B C$ be $D$. Is the circumcircle of $\triangle A B D$ (except for degenerate cases) always passing through a fixed point? If so, find the fixed point; if not, provide a counterexample.
保留源文本的换行和格式如下：
11. (20 points) Given the parabola $P: y^{2}=x$, with two moving points $A, B$ on it, the tangents at $A$ and $B$ intersect at point $C$. Let the circumcenter of $\triangle A B C$ be $D$. Is the circumcircle of $\triangle A B D$ (except for degenerate cases) always passing through a fixed point? If so, find the fixed point; if not, provide a counterexample.
|
(\frac{1}{4},0)
|
A chessboard $1000 \times 1000$ is covered by dominoes $1 \times 10$ that can be rotated. We don't know which is the cover, but we are looking for it. For this reason, we choose a few $N$ cells of the chessboard, for which we know the position of the dominoes that cover them.
Which is the minimum $N$ such that after the choice of $N$ and knowing the dominoed that cover them, we can be sure and for the rest of the cover?
(Bulgaria)
|
100,000
|
13. How many five-digit numbers are exactly divisible by $6$, $7$, $8$, and $9$?
|
179
|
7. To color 8 small squares on a $4 \times 4$ chessboard black, such that each row and each column has exactly two black squares, there are $\qquad$ different ways (answer with a number).
|
90
|
Find all positive integers $n$ for which $(x^n+y^n+z^n)/2$ is a perfect square whenever $x$, $y$, and $z$ are integers such that $x+y+z=0$.
|
n = 1, 4
|
## Task 4 - 100734
According to the legend, the Bohemian Queen Libussa made the granting of her hand dependent on the solution of a riddle that she gave to her three suitors:
"If I give the first suitor half of the contents of this basket of plums and one more plum, the second suitor half of the remaining contents and one more plum, and the third suitor half of the now remaining contents and three more plums, the basket would be emptied. Name the number of plums that the basket contains!"
|
30
|
$[$ [extension of a tetrahedron to a parallelepiped]
Segment $A B(A B=1)$, being a chord of a sphere with radius 1, is positioned at an angle of $60^{\circ}$ to the diameter $C D$ of this sphere. The distance from the end $C$ of the diameter to the nearest end $A$ of the chord $A B$ is $\sqrt{2}$. Find $B D$.
#
|
1
|
590. The last two digits of the decimal representation of the square of some natural number are equal and different from zero. What are these digits? Find all solutions.
|
44
|
A soccer team has $22$ available players. A fixed set of $11$ players starts the game, while the other $11$ are available as substitutes. During the game, the coach may make as many as $3$ substitutions, where any one of the $11$ players in the game is replaced by one of the substitutes. No player removed from the game may reenter the game, although a substitute entering the game may be replaced later. No two substitutions can happen at the same time. The players involved and the order of the substitutions matter. Let $n$ be the number of ways the coach can make substitutions during the game (including the possibility of making no substitutions). Find the remainder when $n$ is divided by $1000$.
|
122
|
2 Let $X=\{1,2, \cdots, 1995\}, A$ be a subset of $X$, if for any two elements $x$ 、 $y(x<y)$ in $A$, we have $y \neq 15 x$, find the maximum value of $|A|$.
untranslated text remains unchanged.
|
1870
|
In $\vartriangle ABC$ points $D, E$, and $F$ lie on side $\overline{BC}$ such that $\overline{AD}$ is an angle bisector of $\angle BAC$, $\overline{AE}$ is a median, and $\overline{AF}$ is an altitude. Given that $AB = 154$ and $AC = 128$, and $9 \times DE = EF,$ fi nd the side length $BC$.
|
94
|
Example 11 (2003 University of South Carolina High School Competition Problem) As shown in Figure 5-10, extend the three sides of $\triangle ABC$ so that $BD=\frac{1}{2} AB, CE=\frac{1}{2} BC, AF=\frac{1}{2} CA$. Then the ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$ is ( ).
A. $3: 1$
B. $13: 4$
C. $7: 2$
D. $14: 15$
E. $10: 3$
|
\frac{13}{4}
|
A cell can divide into 42 or 44 smaller cells. How many divisions are needed to obtain, starting from one cell, exactly 1993 cells?
## - Solutions -
|
48
|
5. Given a function $f(x)$ defined on $\mathbf{R}$ that satisfies:
(1) $f(1)=1$;
(2) When $00$;
(3) For any real numbers $x, y$,
$$
f(x+y)-f(x-y)=2 f(1-x) f(y) \text {. }
$$
Then $f\left(\frac{1}{3}\right)=$ . $\qquad$
|
\frac{1}{2}
|
7. In triangle $A B C$, angles $A$ and $B$ are $45^{\circ}$ and $30^{\circ}$ respectively, and $C M$ is the median. The incircles of triangles $A C M$ and $B C M$ touch segment $C M$ at points $D$ and $E$. Find the radius of the circumcircle of triangle $A B C$ if the length of segment $D E$ is $4(\sqrt{2}-1)$.
|
8
|
Example. Evaluate the double integral
$$
\iint_{D}\left(54 x^{2} y^{2}+150 x^{4} y^{4}\right) d x d y
$$
where the region $D$ is bounded by the lines $x=1, y=x^{3}$ and $y=-\sqrt{x}$.
|
11
|
23. Tom and Geri have a competition. Initially, each player has one attempt at hitting a target. If one player hits the target and the other does not then the successful player wins. If both players hit the target, or if both players miss the target, then each has another attempt, with the same rules applying. If the probability of Tom hitting the target is always $\frac{4}{5}$ and the probability of Geri hitting the target is always $\frac{2}{3}$, what is the probability that Tom wins the competition?
A $\frac{4}{15}$
B $\frac{8}{15}$
C $\frac{2}{3}$
D $\frac{4}{5}$
E $\frac{13}{15}$
|
\frac{2}{3}
|
Call a pair of integers $(a,b)$ [i]primitive[/i] if there exists a positive integer $\ell$ such that $(a+bi)^\ell$ is real. Find the smallest positive integer $n$ such that less than $1\%$ of the pairs $(a, b)$ with $0 \le a, b \le n$ are primitive.
[i]Proposed by Mehtaab Sawhney[/i]
|
299
|
Find all positive integer pairs $(a, b)$ such that $\frac{2^{a+b}+1}{2^a+2^b+1}$ is an integer.
[b]General[/b]
Find all positive integer triples $(t, a, b)$ such that $\frac{t^{a+b}+1}{t^a+t^b+1}$ is an integer.
|
(2, 1, 1)
|
What are the last $2$ digits of the number $2018^{2018}$ when written in base $7$?
|
44
|
1.017. $\frac{0.128: 3.2+0.86}{\frac{5}{6} \cdot 1.2+0.8} \cdot \frac{\left(1 \frac{32}{63}-\frac{13}{21}\right) \cdot 3.6}{0.505 \cdot \frac{2}{5}-0.002}$.
|
8
|
7. Given the function $f(x)=x^{3}-2 x^{2}-3 x+4$, if $f(a)=f(b)=f(c)$, where $a<b<c$, then $a^{2}+b^{2}+c^{2}=$
|
10
|
Problem 11.2. Let $\alpha$ and $\beta$ be the real roots of the equation $x^{2}-x-2021=0$, with $\alpha>\beta$. Denote
$$
A=\alpha^{2}-2 \beta^{2}+2 \alpha \beta+3 \beta+7
$$
Find the greatest integer not exceeding $A$.
|
-6055
|
8. (5 points) Fill the five numbers $2015, 2016, 2017, 2018, 2019$ into the five squares labeled “$D, O, G, C, W$” in the figure, such that $D+O+G=C+O+W$. The total number of different ways to do this is $\qquad$.
|
24
|
15.8. Find the smallest positive integer $x$ such that the sum of $x, x+3, x+6$, $x+9$, and $x+12$ is a perfect cube.
|
19
|
2. Given the lengths of the four sides of a quadrilateral are $m$, $n$, $p$, and $q$, and they satisfy $m^{2}+n^{2}+p^{2}+q^{2}=2 m n+2 p q$. Then this quadrilateral is ( ).
(A) parallelogram
(B) quadrilateral with perpendicular diagonals
(C) parallelogram or quadrilateral with perpendicular diagonals
(D) quadrilateral with equal diagonals
|
C
|
7.5. In triangle $A B C$, we construct $A D \perp B C$, where $D \in (B C)$. Points $E$ and $F$ are the midpoints of segments $A D$ and $C D$, respectively. Determine the measure in degrees of angle $B A C$, knowing that $B E \perp A F$.
|
90
|
Problem 11.8. Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$. A point $X$ is chosen on the edge $A_{1} D_{1}$, and a point $Y$ is chosen on the edge $B C$. It is known that $A_{1} X=5, B Y=3, B_{1} C_{1}=14$. The plane $C_{1} X Y$ intersects the ray $D A$ at point $Z$. Find $D Z$.
|
20
|
3. Let $x$ and $y$ be two distinct non-negative integers, and satisfy $x y + 2x + y = 13$. Then the minimum value of $x + y$ is $\qquad$
|
5
|
Compose the equation of the line passing through the point $A(0 ; 7)$ and tangent to the circle $(x-15)^{2}+(y-2)^{2}=25$.
#
|
7or-\frac{3}{4}x+7
|
2. (2 points) Point $M$ lies on the side of a regular hexagon with side length 10. Find the sum of the distances from point $M$ to the lines containing the other sides of the hexagon.
|
30\sqrt{3}
|
3. If $\log _{a} 3>\log _{b} 3>0$, then the following equation that holds is $(\quad)$.
A. $a>b>1$
B. $b>a>1$
C. $a<b<1$
D. $b<a<1$
|
B
|
2. The cities $A$ and $B$ are $106 \mathrm{~km}$ apart. At noon, from city $A$, Antonio set off on a bicycle heading towards $B$ at a constant speed of $30 \mathrm{~km} / \mathrm{h}$. Half an hour later, from city $B$, Bojan set off on foot heading towards $A$ at a constant speed of $5 \mathrm{~km} / \mathrm{h}$. At the moment Antonio started heading towards $B$, a fly took off from his nose and flew towards $B$ at a constant speed of $50 \mathrm{~km} / \mathrm{h}$. When the fly met Bojan, it landed on his nose, and immediately flew back to Antonio at the same speed, landed on his nose, and immediately flew back to Bojan, and so on until Antonio and Bojan met. How many kilometers did the fly fly?
|
155
|
Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37}\\ b(a\plus{}d)\equiv b\pmod {37}\\ c(a\plus{}d)\equiv c\pmod{37}\\ bc\plus{}d^2\equiv d\pmod{37}\\ ad\minus{}bc\equiv 1\pmod{37}\end{cases}\]
|
1
|
For any positive integer $n$ , let $s(n)$ denote the number of ordered pairs $(x,y)$ of positive integers for which $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$ . Determine the set of positive integers for which $s(n) = 5$
|
\{p^2 \mid p \text{ is a prime number}\}
|
(JBMO 2002 - Shorlist, Problem G3), treated in class
Let $A B C$ be an isosceles triangle at $A$, with $\widehat{C A B}=20^{\circ}$. Let $D$ be a point on the segment $[A C]$, such that $A D=B C$. Calculate the angle $\widehat{B D C}$.
|
30
|
1.012. $\frac{3 \frac{1}{3} \cdot 1.9+19.5: 4 \frac{1}{2}}{\frac{62}{75}-0.16}: \frac{3.5+4 \frac{2}{3}+2 \frac{2}{15}}{0.5\left(1 \frac{1}{20}+4.1\right)}$.
|
4
|
3. Circles $\omega_{1}$ and $\omega_{2}$ with centers $O_{1}$ and $O_{2}$ respectively intersect at point $A$. Segment $O_{2} A$ intersects circle $\omega_{1}$ again at point $K$, and segment $O_{1} A$ intersects circle $\omega_{2}$ again at point $L$. The line passing through point $A$ parallel to $K L$ intersects circles $\omega_{1}$ and $\omega_{2}$ again at points $C$ and $D$ respectively. Segments $C K$ and $D L$ intersect at point $N$. Find the angle between the lines $O_{1} A$ and $O_{2} N$.
|
90
|
11. (20 points) If the equation about $x$
$$
x^{2}+k x-12=0 \text { and } 3 x^{2}-8 x-3 k=0
$$
have a common root, find all possible values of the real number $k$.
|
1
|
There are $10000$ trees in a park, arranged in a square grid with $100$ rows and $100$ columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump.
|
2500
|
2. In the summer, Ponchik eats honey cakes four times a day: instead of morning exercise, instead of a daytime walk, instead of an evening run, and instead of a nighttime swim. The quantities of cakes eaten instead of exercise and instead of a walk are in the ratio of $3: 2$; instead of a walk and instead of a run - as $5: 3$; and instead of a run and instead of a swim - as $6: 5$. By how many more or fewer cakes did Ponchik eat instead of exercise compared to instead of swimming on the day when a total of 216 cakes were eaten?
|
60
|
Jirka drew a square grid with 25 squares, see the image. Then he wanted to color each square so that squares of the same color do not share any vertex.
How many colors did Jirka need at least?
(M. Dillingerová)
Hint. Start in one of the corner squares.
|
4
|
6. Set $R$ represents the set of points $(x, y)$ in the plane, where $x^{2}+6 x+1+y^{2}+6 y+1 \leqslant 0$, and $x^{2}+6 x+1-\left(y^{2}+\right.$ $6 y+1) \leqslant 0$. Then, the area of the plane represented by set $R$ is closest to ( ).
(A) 25
(B) 24
(C) 23
(D) 22
|
25
|
Problem 3. If we divide the numbers 701 and 592 by the same natural number, we get remainders of 8 and 7, respectively. By which number did we divide the given numbers?
|
9
|
4. Six congruent rhombuses, each of area $5 \mathrm{~cm}^{2}$, form a star. The tips of the star are joined to draw a regular hexagon, as shown. What is the area of the hexagon?
A $36 \mathrm{~cm}^{2}$
B $40 \mathrm{~cm}^{2}$
C $45 \mathrm{~cm}^{2}$
D $48 \mathrm{~cm}^{2}$
E $60 \mathrm{~cm}^{2}$
|
45\mathrm{~}^{2}
|
There are positive integers $x$ and $y$ that satisfy the system of equations
\begin{align*} \log_{10} x + 2 \log_{10} (\text{gcd}(x,y)) &= 60\\ \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) &= 570. \end{align*}
Let $m$ be the number of (not necessarily distinct) prime factors in the prime factorization of $x$, and let $n$ be the number of (not necessarily distinct) prime factors in the prime factorization of $y$. Find $3m+2n$.
|
880
|
7. A point $M$ is chosen on the diameter $A B$.
Points $C$ and $D$, lying on the circle on the same side of $AB$ (see figure), are chosen such that $\angle A M C = \angle B M D = 30^{\circ}$. Find the diameter of the circle if it is known that
$C D=12$.
|
8\sqrt{3}
|
5. $[x]$ represents the greatest integer not exceeding the real number $x$, for example $[3]=3$, $[2.7]=2,[-2.2]=-3$, then the last two digits of $\left[\frac{10^{93}}{10^{31}+3}\right]$ are . $\qquad$
|
8
|
3. In 2018, Pavel will be as many years old as the sum of the digits of the year he was born. How old is Pavel?
|
10
|
13. The smallest natural number with 10 different factors is
|
48
|
## Task A-3.4.
How many ordered pairs of natural numbers $(a, b)$ satisfy
$$
\log _{2023-2(a+b)} b=\frac{1}{3 \log _{b} a} ?
$$
|
5
|
6. A die is rolled twice. If the number of dots that appear on the first roll is $x$, and the number of dots that appear on the second roll is $y$, then the probability that the point $M(x, y)$ determined by $x$ and $y$ lies on the hyperbola $y=\frac{6}{x}$ is . $\qquad$
|
\frac{1}{9}
|
5. If in an acute scalene triangle three medians, three angle bisectors, and three altitudes are drawn, they will divide it into 34 parts.
|
34
|
Gus has to make a list of $250$ positive integers, not necessarily distinct, such that each number is equal to the number of numbers in the list that are different from it. For example, if $15$ is a number from the list so the list contains $15$ numbers other than $15$. Determine the maximum number of distinct numbers the Gus list can contain.
|
21
|
5. Let $a, b$ be integers greater than 1,
$$
m=\frac{a^{2}-1}{b+1}, n=\frac{b^{2}-1}{a+1} \text {. }
$$
If $m+n$ is an integer, then the integer(s) among $m, n$ is(are) ).
(A) $m, n$
(B) $m$
(C) $n$
(D) none
|
A
|
Problem 9.3. Given a convex quadrilateral $ABCD$, $X$ is the midpoint of diagonal $AC$. It turns out that $CD \parallel BX$. Find $AD$, if it is known that $BX=3, BC=7, CD=6$.
|
14
|
Example 1 In $\triangle A B C$, $a, b, c$ are the sides opposite to angles $A, B, C$ respectively. Given $a+c=2 b, A-C=\frac{\pi}{3}$, find the value of $\sin B$.
|
\frac{\sqrt{39}}{8}
|
3. The smallest positive period of the function $f(x)=|\sin (2 x)+\sin (3 x)+\sin (4 x)|$ =
untranslated portion:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
translated portion:
Please translate the text above into English, preserving the line breaks and format of the original text, and output the translation directly.
|
2\pi
|
7. The function
$$
y=\frac{1}{\sqrt{x^{2}-6 x+8}}+\log _{2}\left(\frac{x+3}{x-1}-2\right)
$$
has the domain $\qquad$
|
(1,2)\cup(4,5)
|
22. The numbers $x$ and $y$ satisfy both of the equations
$$
23 x+977 y=2023 \quad \text { and } \quad 977 x+23 y=2977 .
$$
What is the value of $x^{2}-y^{2}$ ?
A 1
B 2
C 3
D 4
E 5
|
5
|
$6 \cdot 90$ Let the numbers $x_{1}, \cdots, x_{1991}$ satisfy the condition
$$\left|x_{1}-x_{2}\right|+\cdots+\left|x_{1990}-x_{1991}\right|=1991,$$
and $y_{k}=\frac{1}{k}\left(x_{1}+\cdots+x_{k}\right), k=1, \cdots, 1991$. Try to find the maximum value that the following expression can achieve.
$$\left|y_{1}-y_{2}\right|+\cdots+\left|y_{1990}-y_{1991}\right|$$
|
1990
|
7.091. $\log _{2}\left(4 \cdot 3^{x}-6\right)-\log _{2}\left(9^{x}-6\right)=1$.
|
1
|
3. Given $a^{2}+b^{2}+c^{2}=1$. Then the range of $a b+b c+a c$ is . $\qquad$
|
\left[-\frac{1}{2}, 1\right]
|
3.247. $\left(1-\operatorname{ctg}^{2}\left(\frac{3}{2} \pi-2 \alpha\right)\right) \sin ^{2}\left(\frac{\pi}{2}+2 \alpha\right) \operatorname{tg}\left(\frac{5}{4} \pi-2 \alpha\right)+\cos \left(4 \alpha-\frac{\pi}{2}\right)$.
|
1
|
1. How many positive numbers are there among the 2014 terms of the sequence: $\sin 1^{\circ}, \sin 10^{\circ}$, $\sin 100^{\circ}, \sin 1000^{\circ}, \ldots ?$
|
3
|
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