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86,593 | From the 30 numbers $1, 2, 3, 4, \cdots, 30$, if we randomly select 10 consecutive numbers, how many situations are there where exactly 2 of the selected numbers are prime? | 4 |
86,781 | The base of pyramid \( S A B C D \) is a rectangle \( A B C D \), and its height is the edge \( S A = 25 \). Point \( P \) lies on the median \( D M \) of face \( S C D \), point \( Q \) lies on the diagonal \( B D \), and lines \( A P \) and \( S Q \) intersect. Find the length of \( P Q \) if \( B Q : Q D = 3 : 2 \). | 10 |
86,811 | Shapovvoov A. ·
On a sausage, thin rings are drawn across. If you cut along the red rings, you get 5 pieces; if along the yellow ones, you get 7 pieces; and if along the green ones, you get 11 pieces. How many pieces of sausage will you get if you cut along the rings of all three colors? | 21 |
86,892 | The $p\%$ percentile of a set of numbers ($p\in 0,100$) refers to a value that satisfies the following conditions: at least $p\%$ of the data is not greater than this value, and at least $\left(100-p\right)\%$ of the data is not less than this value. Intuitively, the $p\%$ percentile of a set of numbers refers to the value at the $p\%$ position when the numbers are arranged in ascending order. For example, the median is a $50\%$ percentile. In March 2023, in order to create a civilized city, Hohhot city randomly selected $10$ residents from a certain community to investigate their satisfaction with their current living conditions. The satisfaction index number ranges from $1$ to $10$, with $10$ indicating the highest satisfaction. The satisfaction index numbers of these $10$ residents are $8$, $4$, $5$, $6$, $9$, $8$, $9$, $7$, $10$, $10$. What is the $25\%$ percentile of this data set? | 6 |
86,964 | Calculate: $|-5|-202{3}^{0}+tan45°+\sqrt{9}$. | 8 |
87,120 | Let \( n! = 1 \times 2 \times \cdots \times n \) and \([x]\) represent the greatest integer less than or equal to the real number \( x \). Then the equation
$$
\left[\frac{x}{1!}\right] + \left[\frac{x}{2!}\right] + \cdots + \left[\frac{x}{10!}\right] = 3468
$$
has a positive integer solution of \(\qquad\) | 2020 |
87,148 | 2. (10 points) In each cell of a $50 \times 50$ square, a number is written that is equal to the number of $1 \times 16$ rectangles (both vertical and horizontal) in which this cell is an end cell. In how many cells are numbers greater than or equal to 3 written? | 1600 |
87,301 | The number of lines that are equidistant from point A (-3, 2) and B (1, 1) and have a distance of 2 is \_\_\_\_\_\_. | 4 |
87,302 | Given that $i$ is the imaginary unit, and the complex number $z = (x^2 - 1) + (x + 1)i$ is purely imaginary. Find the value(s) of the real number $x$. | 1 |
87,337 | All natural numbers from 1 to 100 inclusive are divided into 2 groups - even and odd. Determine in which of these groups the sum of all the digits used to write the numbers is greater and by how much. | 49 |
87,354 | When the product
$$
\left(2021 x^{2021}+2020 x^{2020}+\cdots+3 x^{3}+2 x^{2}+x\right)\left(x^{2021}-x^{2020}+\cdots+x^{3}-x^{2}+x-1\right)
$$
is expanded and simplified, what is the coefficient of \(x^{2021}\)? | -1011 |
87,382 | If the angles $A$ and $C$ in $\triangle ABC$ satisfy the equation $5(\cos A+\cos C)+4(\cos A \cos C+1)=0$, then $\tan \frac{A}{2} \cdot \tan \frac{C}{2} = \ \qquad$ . | 3 |
87,415 | The sum of five consecutive odd numbers is 130. What is the middle number? (Solve by setting up an equation) | 26 |
87,523 | Over all real numbers $x$ and $y$ such that $$x^{3}=3 x+y \quad \text { and } \quad y^{3}=3 y+x$$ compute the sum of all possible values of $x^{2}+y^{2}$. | 15 |
87,608 | There are 13 shapes in total, consisting of squares and triangles, with a total of 47 edges. How many triangles and how many squares are there? | 8 |
87,617 | A certain plant's main stem grows a certain number of branches, and each branch grows the same number of small branches. The total number of main stems, branches, and small branches is 57. How many small branches does each branch grow? | 7 |
87,621 | Say that a polynomial with real coefficients in two variables, $x,y$, is \emph{balanced} if
the average value of the polynomial on each circle centered at the origin is $0$.
The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$.
Find the dimension of $V$. | 2020050 |
87,627 | Given the real number \( x \), \([x] \) denotes the integer part that does not exceed \( x \). Find the positive integer \( n \) that satisfies:
\[
\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right] = 1994
\] | 312 |
87,666 | Given an arithmetic sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $2\vec{OC} = a_4\vec{OA} + a_8\vec{OB}$, and points A, B, C are not collinear (the line does not pass through point O), find $S_{11}$. | 11 |
87,714 | The line $l_1$: $x+my+6=0$ is parallel to the line $l_2$: $(m-2)x+3y+2m=0$. Find the value of $m$. | -1 |
87,729 | Among the four-digit numbers composed of the ten digits 0, 1, 2, ..., 9 without any repetition, find the number of such numbers where the absolute difference between the units digit and the hundreds digit equals 8, and the tens digit is an even number. | 104 |
87,813 | In the geometric sequence $\{a_n\}$, $S_n$ represents the sum of the first $n$ terms, $a_3 = 2S_2 + 1$, $a_4 = 2S_3 + 1$, then the common ratio $q$ is. | 3 |
87,818 | Let $b$ and $c$ represent two lines, and $\alpha$ and $\beta$ represent two planes. The following statements are given:
① If $b \subset \alpha$ and $c \parallel \alpha$, then $b \parallel c$;
② If $b \subset \alpha$ and $b \parallel c$, then $c \parallel \alpha$;
③ If $c \parallel \alpha$ and $\alpha \perp \beta$, then $c \perp \beta$;
④ If $c \parallel \alpha$ and $c \perp \beta$, then $\alpha \perp \beta$.
Among these statements, the correct ones are . (Write down the numbers of all correct statements) | 4 |
87,946 | A cylinder with a volume of 9 is inscribed in a cone. The plane of the upper base of this cylinder truncates the original cone, forming a frustum with a volume of 63. Find the volume of the original cone. | 64 |
87,981 | In the sequence $\{a\_n\}$, $a\_1=32$, $a_{n+1}=a\_n-4$. Determine the value of $n=$\_\_\_\_\_\_ when the sum of the first $n$ terms, $S\_n$, reaches its maximum value, and find the maximum value itself. | 144 |
87,986 | How many square columns are there where the edge length measured in cm is an integer, and the surface area measured in $\mathrm{cm}^{2}$ is equal to the volume measured in $\mathrm{cm}^{3}$? | 4 |
88,010 | The "Rapid Advance" ride at the Western Suburbs Zoo comes in two models, one with a capacity of 7 people, priced at 65 yuan; the other with a capacity of 5 people, priced at 50 yuan. Now, a tour group of 73 people plans to take the "Rapid Advance" ride. The minimum amount they need to spend on tickets is yuan. | 685 |
88,039 | Expanding the expression \((1+\sqrt{7})^{207}\) using the binomial theorem, we obtain terms of the form \(C_{207}^{k}(\sqrt{7})^{k}\). Find the value of \(k\) for which this term has the greatest value. | 150 |
88,053 | 3. Find the successive convergents and their values of the finite simple continued fraction $\langle 2,1,2,1,1,4,1,1,6,1,1,8\rangle$, and compare them with the value of the base of the natural logarithm $e$. | 2.718283229 |
88,075 | A unit square in the first quadrant on the coordinate plane ( $0 \leq x, y \leq 1$ ) is divided into smaller squares with a side length of $2 \cdot 10^{-4}$. How many grid points of this division (inside the unit square) lie on the parabola $y=x^{2}$? | 49 |
88,144 | Given that the sequence $\{a\_n\}$ is a geometric sequence with the sum of the first $n$ terms denoted as $S\_n$, and it is known that $a\_5 = 4S\_4 + 3$, $a\_6 = 4S\_5 + 3$. Find the common ratio $q$ of this geometric sequence. | 5 |
88,146 | Given that $y=f(x)$ is a continuous and differentiable function on $\mathbb{R}$, and $xf'(x)+f(x) > 0$, the number of zeros of the function $g(x)=xf(x)+1(x > 0)$ is _____. | 0 |
88,163 | A $\frac 1p$ -array is a structured, infinite, collection of numbers. For example, a $\frac 13$ -array is constructed as follows:
\begin{align*} 1 \qquad \frac 13\,\ \qquad \frac 19\,\ \qquad \frac 1{27} \qquad &\cdots\\ \frac 16 \qquad \frac 1{18}\,\ \qquad \frac{1}{54} \qquad &\cdots\\ \frac 1{36} \qquad \frac 1{108} \qquad &\cdots\\ \frac 1{216} \qquad &\cdots\\ &\ddots \end{align*}
In general, the first entry of each row is $\frac{1}{2p}$ times the first entry of the previous row. Then, each succeeding term in a row is $\frac 1p$ times the previous term in the same row. If the sum of all the terms in a $\frac{1}{2008}$ -array can be written in the form $\frac mn$, where $m$ and $n$ are relatively prime positive integers, find the remainder when $m+n$ is divided by $2008$. | 1 |
88,195 | If $3$ different math books and $3$ different Chinese books are placed on the same shelf, the number of ways to arrange them such that books of the same subject are not adjacent is ____. | 72 |
88,257 | Given $a$, $b$, $c \in \{1, 2, 3, 4, 5, 6\}$, if the lengths $a$, $b$, and $c$ can form an isosceles (including equilateral) triangle, then there are \_\_\_\_\_\_ such triangles. | 27 |
88,263 | Calculate the limit of the function:
\[
\lim _{x \rightarrow 0} \frac{\sin ^{2} (x) - \tan ^{2} (x)}{x^{4}}
\] | -1 |
88,301 | If line $l_1: x + (1 + k)y = 2 - k$ is parallel to line $l_2: kx + 2y + 8 = 0$, then the value of $k$ is \_\_\_\_\_. | 1 |
88,305 | The last two digits of $\left[\frac{10^{93}}{10^{31}+3}\right]$ are $\qquad$ (where [x] denotes the greatest integer less than or equal to $x$). | 08 |
88,306 | In an opaque bag, there are a total of 50 glass balls in red, black, and white colors. Except for the color, everything else is identical. After several trials of drawing balls, Xiao Gang found that the probability of drawing a red or black ball stabilized at 15% and 45%, respectively. What could be the possible number of white balls in the bag? | 20 |
88,346 | Given an arithmetic sequence $\{a\_n\}$ with a common difference $d∈(0,1)$, and $\frac {\sin ^{2}a_{3}-\sin ^{2}a_{7}}{\sin (a_{3}+a_{7})}=-1$, if $a_{1}∈(-\frac {5π}{4},-\frac {9π}{8})$ then the minimum value of $n$ for which the sum of the first $n$ terms of the sequence, $S_{n}$, is minimized is $\_\_\_\_\_\_$. | 10 |
88,354 | Znayka cut out a semicircle from paper. Neznayka marked a point \( C \) on the diameter \( AB \) of this semicircle and cut out two smaller semicircles with diameters \( AC \) and \( CB \) from Znayka's semicircle. Find the area of the remaining figure, if the length of the chord passing through point \( C \) perpendicular to \( AB \) inside the figure is 8. Round the answer to two decimal places if necessary. | 12.57 |
88,389 | How many hundreds of millions, tens of millions, and millions are there in 1,234,000,000? | 4 |
88,476 | Calculate the number of different arrangements for 3 male students and 3 female students in a row, given that male student A must not be adjacent to the other two male students. Answer with a specific number. | 288 |
88,490 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b\cos C+c\cos B=3a\cos B$, $b=2$, and the area of $\triangle ABC$ is $\frac{3\sqrt{2}}{2}$, then $a+c=$ \_\_\_\_\_\_. | 4 |
88,496 | The lines tangent to a circle with center $O$ at points $A$ and $B$ intersect at point $M$. Find the chord $AB$ if the segment $MO$ is divided by it into segments equal to 2 and 18. | 12 |
88,597 | The MathMatters competition consists of 10 players $P_1$, $P_2$, $\dots$, $P_{10}$ competing in a ladder-style tournament. Player $P_{10}$ plays a game with $P_9$: the loser is ranked 10th, while the winner plays $P_8$. The loser of that game is ranked 9th, while the winner plays $P_7$. They keep repeating this process until someone plays $P_1$: the loser of that final game is ranked 2nd, while the winner is ranked 1st. How many different rankings of the players are possible? | 512 |
88,611 | The $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers satisfies the following:
[list](i) $1\le a_1<a_2<\cdots < a_n\le 50$
(ii) for each $n$-tuple $(b_1,b_2,\ldots,b_n)$ of positive integers, there exist a positive integer $m$ and an $n$-tuple $(c_1,c_2,\ldots,c_n)$ of positive integers such that \[mb_i=c_i^{a_i}\qquad\text{for } i=1,2,\ldots,n. \] [/list]Prove that $n\le 16$ and determine the number of $n$-tuples $(a_1,a_2,\ldots,a_n$) satisfying these conditions for $n=16$. | 1 |
88,622 | Alexa wrote the first $16$ numbers of a sequence:
\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\]
Then she continued following the same pattern, until she had $2015$ numbers in total.
What was the last number she wrote?
| 1344 |
88,740 | A set $A$ contains exactly $n$ integers, each of which is greater than $1$ and every of their prime factors is less than $10$. Determine the smallest $n$ such that $A$ must contain at least two distinct elements $a$ and $b$ such that $ab$ is the square of an integer. | 17 |
88,762 | Cynthia loves Pokemon and she wants to catch them all. In Victory Road, there are a total of $80$ Pokemon. Cynthia wants to catch as many of them as possible. However, she cannot catch any two Pokemon that are enemies with each other. After exploring around for a while, she makes the following two observations:
1. Every Pokemon in Victory Road is enemies with exactly two other Pokemon.
2. Due to her inability to catch Pokemon that are enemies with one another, the maximum number of the Pokemon she can catch is equal to $n$.
What is the sum of all possible values of $n$? | 469 |
88,803 | A set of 1990 persons is divided into non-intersecting subsets in such a way that
1. No one in a subset knows all the others in the subset,
2. Among any three persons in a subset, there are always at least two who do not know each other, and
3. For any two persons in a subset who do not know each other, there is exactly one person in the same subset knowing both of them.
(a) Prove that within each subset, every person has the same number of acquaintances.
(b) Determine the maximum possible number of subsets.
Note: It is understood that if a person $A$ knows person $B$, then person $B$ will know person $A$; an acquaintance is someone who is known. Every person is assumed to know one's self. | 398 |
88,868 | $DEB$ is a chord of a circle such that $DE=3$ and $EB=5$. Let $O$ be the centre of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$. (See diagram). Given $EC=1$, find the radius of the circle.
[asy]
size(6cm);
pair O = (0,0), B = dir(110), D = dir(30), E = 0.4 * B + 0.6 * D, C = intersectionpoint(O--2*E, unitcircle);
draw(unitcircle);
draw(O--C);
draw(B--D);
dot(O);
dot(B);
dot(C);
dot(D);
dot(E);
label("$B$", B, B);
label("$C$", C, C);
label("$D$", D, D);
label("$E$", E, dir(280));
label("$O$", O, dir(270));
[/asy] | 16 |
88,895 | In triangle $ABC$, $D$ is a point on $AB$ between $A$ and $B$, $E$ is a point on $AC$ between $A$ and $C$, and $F$ is a point on $BC$ between $B$ and $C$ such that $AF$, $BE$, and $CD$ all meet inside $\triangle ABC$ at a point $G$. Given that the area of $\triangle ABC$ is $15$, the area of $\triangle ABE$ is $5$, and the area of $\triangle ACD$ is $10$, compute the area of $\triangle ABF$. | 3 |
88,916 | [b](a)[/b] Sketch the diagram of the function $f$ if
\[f(x)=4x(1-|x|) , \quad |x| \leq 1.\]
[b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$
[b](c)[/b] Let $g$ be a function such that
\[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\right.\]
Is the function $g$ continuous in the point $x=0 \ ?$
[b](d)[/b] Sketch the diagram of $g.$ | 4 |
88,947 | In the following figure, the bigger wheel has circumference $12$m and the inscribed wheel has circumference $8 $m.
$P_{1}$ denotes a point on the bigger wheel and $P_{2}$ denotes a point on the smaller wheel. Initially $P_{1}$ and $P_{2}$ coincide as in the figure. Now we roll the wheels on a smooth surface and the smaller wheel also rolls in the bigger wheel smoothly. What distance does the bigger wheel have to roll so that the points will be together again? | 24 |
88,951 | p1. Determine the smallest natural number that has the property that it's cube ends in $888$.
p2. Triangle $ABC$ is isosceles with $AC = BC$. The angle bisector at $A$ intercepts side $BC$ at the point $D$ and the bisector of the angle at $C$ intercepts side $AB$ at $E$. If $AD = 2CE$, find the measure of the angles of triangle $ABC$.
p3.In the accompanying figure, the circle $C$ is tangent to the quadrant $AOB$ at point $S$. AP is tangent to $C$ at $P$ and $OQ$ is tangent at $Q$. Calculate the length $AP$ as a function of the radius$ R$ of the quadrant $AOB$.
[img]https://cdn.artofproblemsolving.com/attachments/3/c/d34774c3c6c33d351316574ca3f7ade54e6441.png[/img]
p4. Let $A$ be a set with seven or more natural numbers. Show that there must be two numbers in $A$ with the property that either its sum or its difference is divisible by ten. Show that the property can fail if set $A$ has fewer than seven elements.
p5. An $n$-sided convex polygon is such that there is no common point for any three of its diagonals. Determine the number of triangles that are formed such that two of their vertices are vertices of the polygon and the third is an intersection of two diagonals. | 192 |
88,986 | For how many integers $k$ does the following system of equations has a solution other than $a=b=c=0$ in the set of real numbers? \begin{align*} \begin{cases} a^2+b^2=kc(a+b),\\ b^2+c^2 = ka(b+c),\\ c^2+a^2=kb(c+a).\end{cases}\end{align*} | 2 |
89,026 | Given a positive integer $ k$, there is a positive integer $ n$ with the property that one can obtain the sum of the first $ n$ positive integers by writing some $ k$ digits to the right of $ n$. Find the remainder of $ n$ when dividing at $ 9$. | 1 |
89,086 | The number $201212200619$ has a factor $m$ such that $6 \cdot 10^9 <m <6.5 \cdot 10^9$. Find $m$. | 6490716149 |
89,196 | [u]Round 5[/u]
[b]5.1.[/b] A triangle has lengths such that one side is $12$ less than the sum of the other two sides, the semi-perimeter of the triangle is $21$, and the largest and smallest sides have a difference of $2$. Find the area of this triangle.
[b]5.2.[/b] A rhombus has side length $85$ and diagonals of integer lengths. What is the sum of all possible areas of the rhombus?
[b]5.3.[/b] A drink from YAKSHAY’S SHAKE SHOP is served in a container that consists of a cup, shaped like an upside-down truncated cone, and a semi-spherical lid. The ratio of the radius of the bottom of the cup to the radius of the lid is $\frac23$ , the volume of the combined cup and lid is $296\pi$, and the height of the cup is half of the height of the entire drink container. What is the volume of the liquid in the cup if it is filled up to half of the height of the entire drink container?
[u]Round 6[/u]
[i]Each answer in the next set of three problems is required to solve a different problem within the same set. There is one correct solution to all three problems; however, you will receive points for any correct answer regardless whether other answers are correct.[/i]
[b]6.1.[/b] Let the answer to problem $2$ be $b$. There are b people in a room, each of which is either a truth-teller or a liar. Person $1$ claims “Person $2$ is a liar,” Person $2$ claims “Person $3$ is a liar,” and so on until Person $b$ claims “Person $1$ is a liar.” How many people are truth-tellers?
[b]6.2.[/b] Let the answer to problem $3$ be $c$. What is twice the area of a triangle with coordinates $(0, 0)$, $(c, 3)$ and $(7, c)$ ?
[b]6.3.[/b] Let the answer to problem $ 1$ be $a$. Compute the smaller zero to the polynomial $x^2 - ax + 189$ which has $2$ integer roots.
[u]Round 7[/u]
[b]7.1. [/b]Sir Isaac Neeton is sitting under a kiwi tree when a kiwi falls on his head. He then discovers Neeton’s First Law of Kiwi Motion, which states:
[i]Every minute, either $\left\lfloor \frac{1000}{d} \right\rfloor$ or $\left\lceil \frac{1000}{d} \right\rceil$ kiwis fall on Neeton’s head, where d is Neeton’s distance from the tree in centimeters.[/i]
Over the next minute, $n$ kiwis fall on Neeton’s head. Let $S$ be the set of all possible values of Neeton’s distance from the tree. Let m and M be numbers such that $m < x < M$ for all elements $x$ in $S$. If the least possible value of $M - m$ is $\frac{2000}{16899}$ centimeters, what is the value of $n$?
Note that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, and $\lceil x \rceil$ is the least integer greater than or equal to $x$.
[b]7.2.[/b] Nithin is playing chess. If one queen is randomly placed on an $ 8 \times 8$ chessboard, what is the expected number of squares that will be attacked including the square that the queen is placed on? (A square is under attack if the queen can legally move there in one move, and a queen can legally move any number of squares diagonally, horizontally or vertically.)
[b]7.3.[/b] Nithin is writing binary strings, where each character is either a $0$ or a $1$. How many binary strings of length $12$ can he write down such that $0000$ and $1111$ do not appear?
[u]Round 8[/u]
[b]8.[/b] What is the period of the fraction $1/2018$? (The period of a fraction is the length of the repeated portion of its decimal representation.) Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input.
$$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.1 |I|}, 13 - \frac{|I-X|}{0.1 |I-2X|} \right\} \right\rceil \right\}$$
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2765571p24215461]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]. | 84 |
89,284 | A circle with area $40$ is tangent to a circle with area $10$. Let R be the smallest rectangle containing both circles. The area of $R$ is $\frac{n}{\pi}$. Find $n$.
[asy]
defaultpen(linewidth(0.7)); size(120);
real R = sqrt(40/pi), r = sqrt(10/pi);
draw(circle((0,0), R)); draw(circle((R+r,0), r));
draw((-R,-R)--(-R,R)--(R+2*r,R)--(R+2*r,-R)--cycle);[/asy] | 240 |
89,361 | For a special event, the five Vietnamese famous dishes including Phở, (Vietnamese noodle), Nem (spring roll), Bún Chả (grilled pork noodle), Bánh cuốn (stuffed pancake), and Xôi gà (chicken sticky rice) are the options for the main courses for the dinner of Monday, Tuesday, and Wednesday. Every dish must be used exactly one time. How many choices do we have? | 150 |
89,448 | 1. Over three years, Marina did not invest funds from her Individual Investment Account (IIS) into financial instruments and therefore did not receive any income from the IIS. However, she still acquired the right to an investment tax deduction for contributing personal funds to the IIS (Article 291.1 of the Tax Code of the Russian Federation).
The tax deduction is provided in the amount of funds deposited in the tax period into the IIS, but not exceeding 400,000 rubles in total per year.
Marina is entitled to receive 13% of the amount deposited into the IIS from the state in the form of a refunded personal income tax (NDFL) that she paid to the treasury from her income.
The amount of NDFL deducted from Marina's annual salary = 30000 * 12 months * 0.13 = 46800 rubles.
The tax deduction for the first year = 100000 * 0.13 = 13000 rubles. This amount does not exceed the amount of NDFL deducted from Marina's annual salary.
The tax deduction for the second year = 400000 * 0.13 = 52000 rubles. This amount exceeds the amount of NDFL deducted from Marina's annual salary. Therefore, the tax deduction will be limited to the amount of NDFL paid for the second year of the IIS's existence, or 46800 rubles.
The tax deduction for the third year = 400000 * 0.13 = 52000 rubles. This amount also exceeds the amount of NDFL deducted from Marina's annual salary. Therefore, the tax deduction will be limited to the amount of NDFL paid for the third year, or 46800 rubles.
The total tax deduction over 3 years = 13000 + 46800 + 46800 = 106600 rubles.
The return on Marina's operations over 3 years = 106600 / 1000000 * 100% = 10.66%.
The annual return on Marina's operations = 10.66% / 3 = 3.55%. | 3.55 |
89,451 | 2. (15 points) A satellite is launched vertically from the pole of the Earth at the first cosmic speed. To what maximum distance from the Earth's surface will the satellite travel? (The acceleration due to gravity at the Earth's surface $g=10 \mathrm{m} / \mathrm{c}^{2}$, radius of the Earth $R=6400$ km). | 6400 |
89,454 | XLIV OM - I - Problem 11
In six different cells of an $ n \times n $ table, we place a cross; all arrangements of crosses are equally probable. Let $ p_n $ be the probability that in some row or column there will be at least two crosses. Calculate the limit of the sequence $ (np_n) $ as $ n \to \infty $. | 30 |
89,535 | 30. It is known that for pairwise distinct numbers $a, b, c$, the equality param1 holds. What is the smallest value that the expression $a+b+c$ can take?
| param1 | Answer |
| :---: | :---: |
| $a^{3}(b-c)+b^{3}(c-a)+c^{3}(a-b)+2\left(a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)\right)=0$ | -2 |
| $a^{3}(b-c)+b^{3}(c-a)+c^{3}(a-b)+4\left(a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)\right)=0$ | -4 |
| $a^{3}(b-c)+b^{3}(c-a)+c^{3}(a-b)+6\left(a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)\right)=0$ | -6 |
| $a^{3}(b-c)+b^{3}(c-a)+c^{3}(a-b)+8\left(a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)\right)=0$ | -8 |
| $a^{3}(b-c)+b^{3}(c-a)+c^{3}(a-b)+10\left(a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)\right)=0$ | -10 | | -10 |
89,564 | 4. (15 points) The efficiency of an ideal heat engine is $50 \%$. What will it become if the temperature of the heater is increased by $50 \%$, and the temperature of the cooler is decreased by $50 \%$? | 83 |
89,572 | 3. The equation $x^{2}+a x+4=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,
$$
x_{1}^{2}-\frac{20}{3 x_{2}^{3}}=x_{2}^{2}-\frac{20}{3 x_{1}^{3}}
$$
Find all possible values of $a$. | -10 |
89,583 | 4. On a line, several points were marked, including points $A$ and $B$. All possible segments with endpoints at the marked points are considered. Vasya calculated that point $A$ is inside 40 of these segments, and point $B$ is inside 42 segments. How many points were marked? (The endpoints of a segment are not considered its internal points.) | 14 |
89,597 | 5. A person is walking parallel to a railway track at a constant speed. A train also passes by him at a constant speed. The person noticed that depending on the direction of the train, it passes by him either in $t_{1}=1$ minute or in $t_{2}=2$ minutes. Determine how long it would take the person to walk from one end of the train to the other.
## $(15$ points) | 4 |
89,647 | 5. If the angle at the vertex of a triangle is $40^{\circ}$, then the bisectors of the other two angles of the triangle intersect at an angle of $70^{\circ}$. | 70 |
89,648 | 1. Given a parallelogram $A B C D$. It is known that the centers of the circles circumscribed around triangles $A B C$ and $C D A$ lie on the diagonal $B D$. Find the angle $D B C$, if $\angle A B D=35^{\circ}$. | 55 |
89,665 | 5. A product that initially contained $98\%$ water dried over some time and began to contain $95\%$ water. By what factor did it shrink (i.e., reduce its weight)? | 2.5 |
89,719 | 1. To qualify for the competition, wrestler Vladimir had to conduct three bouts and win at least two in a row. His opponents were Andrey (A) and Boris (B). Vladimir could choose the sequence of matches: ABA or BAB. The probability of Vladimir losing a single bout to Boris is 0.3, and to Andrey is 0.4; the probabilities are constant. Which sequence gives a higher probability of qualifying for the competition, and what is this probability? | 0.588 |
89,786 | 2-4. Find the minimum value of the function
$$
f(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-104)^{2}
$$
If the result is a non-integer, round it to the nearest integer and write it in the answer. | 49608 |
89,796 | 1. Petya's watch gains 5 minutes per hour, while Masha's watch loses 8 minutes per hour. At 12:00, they set their watches to the school clock (which is accurate) and agreed to go to the rink together at half past six. How long will Petya wait for Masha if each arrives at the rink exactly at $18-30$ by their own watches | 1.5 |
89,859 | 7. Find the smallest natural number ending in the digit 2 that doubles when this digit is moved to the beginning. | 105263157894736842 |
89,912 | # Problem 7.
Find all values of $a$ for each of which the equation $x^{2}+2 a x=8 a$ has two distinct integer roots. In the answer, write the product of all such $a$, rounding to the hundredths if necessary. | 506.25 |
89,955 | 7.1. (14 points) Misha drew a triangle with a perimeter of 11 and cut it into parts with three straight cuts parallel to the sides, as shown in the figure. The perimeters of the three shaded figures (trapezoids) turned out to be 5, 7, and 9. Find the perimeter of the small triangle that resulted from the cutting.
# | 10 |
89,962 | Task 3. (15 points) Laboratory engineer Sergei received an object for research consisting of about 200 monoliths (a container designed for 200 monoliths, which was almost completely filled). Each monolith has a specific name (sandy loam or clayey loam) and genesis (marine or lake-glacial deposits). The relative frequency (statistical probability) that a randomly selected monolith will be sandy loam is $\frac{1}{9}$. The relative frequency that a randomly selected
monolith will be marine clayey loam is $\frac{11}{18}$. How many monoliths of lake-glacial genesis does the object contain, if there are no marine sandy loams among the sandy loams? | 77 |
89,983 | 1. Calculation of EMF
## TASK 4
## SOLUTION
The amount of potassium hydroxide and nitric acid in the solutions:
$$
\begin{gathered}
v(\mathrm{KOH})=\omega \rho V / M=0.062 \cdot 1.055 \text { g/ml } \cdot 22.7 \text { ml / } 56 \text { g/ mol }=0.0265 \text { mol. } \\
v\left(\mathrm{HNO}_{3}\right)=\text { C }_{\mathrm{M}} \cdot \mathrm{V}=2.00 \text { mol/l } \cdot 0.0463 \text { l }=0.0926 \text { mol. }
\end{gathered}
$$
Since $v(\mathrm{KOH})<v(\mathrm{HNO} 3)$, and the coefficients of these substances in the reaction equation are the same and equal to one, potassium hydroxide is in deficiency, and its amount will determine the heat effect of the reaction, which will be:
$$
\text { Q = 55.6 kJ/mol } \cdot \text { 0.0265 mol = 1.47 kJ. }
$$ | 1.47 |
89,988 | 5. Given a natural number $x=8^{n}-1$, where $n$ is a natural number. It is known that $x$ has exactly three distinct prime divisors, one of which is 31. Find $x$. | 32767 |
89,989 | 6.3. Let $f(x)=x^{2}+p x+q$. It is known that the inequality $|f(x)|>\frac{1}{2}$ has no solutions on the interval $[3 ; 5]$. Find $\underbrace{f(f(\ldots f}_{2017}\left(\frac{7+\sqrt{15}}{2}\right)) \ldots)$. Round your answer to the nearest hundredth if necessary. | 1.56 |
90,055 | 2. To walk 2 km, ride 3 km on a bicycle, and drive 20 km by car, Uncle Vanya needs 1 hour 6 minutes. If he needs to walk 5 km, ride 8 km on a bicycle, and drive 30 km by car, it will take him 2 hours 24 minutes. How much time will Uncle Vanya need to walk 4 km, ride 5 km on a bicycle, and drive 80 km by car? | 2 |
90,085 | 2. Usually, Dima leaves home at $8:10$ AM, gets into Uncle Vanya's car, who delivers him to school by a certain time. But on Thursday, Dima left home at 7:20 and ran in the opposite direction. Uncle Vanya waited for him and at $8:20$ drove after him, caught up with Dima, turned around, and delivered him to school 26 minutes late. How many times faster was Uncle Vanya's car speed compared to Dima's running speed? | 8.5 |
90,119 | 2. Let $x_{1}$ and $x_{2}$ be the largest roots of the polynomials
$$
\begin{gathered}
f(x)=1-x-4 x^{2}+x^{4} \\
\text { and } \\
g(x)=16-8 x-16 x^{2}+x^{4}
\end{gathered}
$$
respectively. Find $\frac{x_{2}}{x_{1}}$. | 2 |
90,136 | 1. The function $y=f(x)$ is defined on the set $(0,+\infty)$ and takes positive values on it. It is known that for any points $A$ and $B$ on the graph of the function, the areas of the triangle $A O B$ and the trapezoid $A B H_{B} H_{A}$ are equal to each other $\left(H_{A}, H_{B}\right.$ - the bases of the perpendiculars dropped from points $A$ and $B$ to the x-axis; $O$ - the origin). Find all such functions. Given $f(1)=4$, write the number $f(4)$ in the answer. | 1 |
90,193 | 8.4. From identical isosceles triangles, where the angle opposite the base is $45^{\circ}$ and the lateral side is 1, a figure was formed as shown in the diagram. Find the distance between points $A$ and $B$. | 2 |
90,246 | 5. During the shooting practice, each soldier fired 10 times. One of them completed the task successfully and scored 90 points. How many times did he score 7 points, if he hit the bullseye 4 times, and the results of the other hits were sevens, eights, and nines? There were no misses at all. | 1 |
90,281 | 2.1. A metal weight has a mass of 20 kg and is an alloy of four metals. The first metal in this alloy is one and a half times more than the second, the mass of the second metal is to the mass of the third as $3: 4$, and the mass of the third to the mass of the fourth - as $5: 6$. Determine the mass of the fourth metal. Give the answer in kilograms, rounding to hundredths if necessary. | 5.89 |
90,282 | 9. Patrick and Slippers. Every day, the dog Patrick gnaws one slipper from the existing supply in the house. With a probability of 0.5, Patrick wants to gnaw a left slipper, and with a probability of 0.5 - a right slipper. If the desired slipper is not available, Patrick gets upset. How many pairs of identical slippers need to be bought so that with a probability of at least 0.8, Patrick does not get upset for a whole week (7 days)? | 5 |
90,290 | 5. There are several technologies for paying with bank cards: chip, magnetic stripe, paypass, cvc. Arrange the actions performed with a bank card in the order corresponding to the payment technologies.
1 - tap
2 - pay online
3 - swipe
4 - insert into terminal | 4312 |
90,304 | 18. The figure shows a track scheme for karting. The start and finish are at point $A$, and the kart driver can make as many laps as they want, returning to the starting point.
The young driver Yura spends one minute on the path from $A$ to $B$ or back. Yura also spends one minute on the loop. The loop can only be driven counterclockwise (arrows indicate possible directions of movement). Yura does not turn back halfway and does not stop. The race duration is 10 minutes. Find the number of possible different routes (sequences of passing sections). # | 34 |
90,324 | # 2. Inflation over two years will be
$$
\left((1+0.025)^{\wedge 2-1}\right)^{*} 100 \% = 5.0625 \%
$$
The real return on a bank deposit, taking into account the extension for the second year, will be $(1.06 * 1.06 / (1+0.050625) - 1) * 100 = 6.95 \%$
## Evaluation Criteria:
A maximum of 25 points for a correct and justified solution. Of these:
2 points for the correct calculation of the investment amount.
5 points for the correct calculation of the nominal return on the bank deposit. If this value was calculated incorrectly, the participant could receive points for the following steps in the solution:
- 3 points for calculating the amount the family can withdraw after closing the deposit, taking into account its extension for the second year.
- 1 point for calculating the amount the family can withdraw after closing the deposit, without correctly accounting for its extension for the second year.
7 points for the correct calculation of the expected return on investment in construction. If this value was calculated incorrectly, the participant could receive points for the following steps in the solution:
- a maximum of 2 points for conceptually incorrect consideration of the probabilities of different outcomes.
- 2 points for correctly accounting for the 13% income tax.
1 point for the correct choice of investment method. If the correct choice was made based on incorrect calculations of the return in any of the investment options, the participant did not receive a point for this part of the solution.
10 points for the correct calculation of the return. If this value was calculated incorrectly, the participant could receive points for the following steps in the solution:
- 4 points for a logically correct method of calculating the return in %, provided that the real return was calculated incorrectly.
- 8 points for calculating the return in % as the difference between the nominal return and inflation over 2 years, provided that the nominal return was calculated correctly.
- 1 point for incorrect accounting of inflation over two years.
If the entire logic of the calculations is followed, but the participant makes only arithmetic errors, then -1 point for each such error.
## Task 5
The banking sector has always been a tempting target for criminals, so commercial and government banks are implementing more and more protection technologies.
You head the information security department of a large commercial bank. Propose your protection strategy for the following objects of the banking infrastructure (at least one existing method for each object):
- employee profiles;
- bank cards;
- ATMs;
- online banking;
- online store payments;
- central automated banking system. | 6.95 |
90,331 | Task 13. (8 points)
Natalia Petrovna has returned from her vacation, which she spent traveling through countries in North America. She has a certain amount of money left in foreign currency.
Natalia Petrovna familiarized herself with the exchange rates at the nearest banks: "Rebirth" and "Garnet." She decided to take advantage of the most favorable offer. What amount will she receive in rubles for exchanging 120 US dollars, 80 Canadian dollars, and 10 Mexican pesos at one of the two banks?
| Type of Currency | Exchange Rate | |
| :--- | :---: | :---: |
| | Rebirth | Garnet |
| US Dollar | 74.9 rub. | 74.5 rub. |
| Canadian Dollar | 59.3 rub. | 60.1 rub. |
| Mexican Peso | 3.7 rub. | 3.6 rub. |
In your answer, provide only the number without units of measurement! | 13784 |
90,363 | Problem 5. An electric kettle heats water from room temperature $T_{0}=20^{\circ} \mathrm{C}$ to $T_{m}=100^{\circ} \mathrm{C}$ in $t=10$ minutes. How long will it take $t_{1}$ for all the water to boil away if the kettle is not turned off and the automatic shut-off system is faulty? The specific heat capacity of water $c=4200$ J/kg $\cdot$ K. The specific latent heat of vaporization of water $L=2.3$ MJ/kg. Round the answer to the nearest whole number of minutes. | 68 |
90,366 | 3. In July, Volodya and Dima decided to start their own business producing non-carbonated bottled mineral water called "Dream," investing 1,500,000 rubles, and used these funds to purchase equipment for 500,000 rubles. The technical passport for this equipment indicates that the maximum production capacity is 100,000 bottles.
At the end of July, Dima and Volodya decided to launch a trial batch of water production and received 200 bottles, 5 of which were not full. In August, the equipment started operating at full capacity, and 15,000 bottles of water were produced. In the 20th of September, the equipment broke down and was idle for several days, but 12,300 bottles of water were produced over 20 days of this month.
In October, the friends decided to stop producing water, as it would not be in demand during the winter season, and decided to sell the equipment.
a) Determine the total depreciation of the equipment.
b) Determine the residual value of the equipment at the time of sale.
c) For what amount should the equipment be sold to achieve a profit of 10,000 rubles? (20 points) | 372500 |
90,379 | 4.3. Two identical cylindrical vessels are connected at the bottom by a small cross-section pipe with a valve. While the valve was closed, water was poured into the first vessel, and oil into the second, so that the level of the liquids was the same and equal to \( h = 40 \) cm. At what level will the water in the first vessel stabilize if the valve is opened? The density of water is 1000 kg/ \(\mathrm{m}^{3}\), and the density of oil is 700 kg/ \(\mathrm{m}^{3}\). Neglect the volume of the connecting pipe. Give the answer in centimeters. | 34 |
90,413 | 6. On graph paper, a polygon with a perimeter of 2014 is drawn, with its sides running along the grid lines. What is the maximum area it can have? | 253512 |
90,417 | Problem 5.5. A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of the square is 24, and the perimeter of the small rectangle is 16. What is the perimeter of the large rectangle?
The perimeter of a figure is the sum of the lengths of all its sides.
 | 52 |
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