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Given an equilateral triangle with side length \( a \), find the segment connecting a vertex of the triangle to a point that divides the opposite side in the ratio \( 2:1 \). | \frac{a\sqrt{7}}{3} | 0.25 |
Find the smallest natural number $A$ that satisfies the following conditions:
a) Its notation ends with the digit 6;
b) By moving the digit 6 from the end of the number to its beginning, the number increases fourfold. | A = 153846 | 0.75 |
These points form a new triangle with angles \(45^{\circ}, 60^{\circ},\) and \(75^{\circ}\). Find the ratio of the areas of the original and the new triangles. | \sqrt{3} - 1 | 0.25 |
There is a natural number written on the board. Five students made the following statements about this number:
- Petya: "This number is greater than 10."
- Kolya: "This number is at least 11."
- Vasya: "This number is greater than 12."
- Dima: "This number is less than 12."
- Oleg: "This number is at most 12."
Find the maximum possible number of correct statements. | 4 | 0.75 |
There are four weights of different masses. Katya weighs the weights in pairs. As a result, she gets $1700,1870,2110,2330,$ and 2500 grams. How many grams does the sixth pair of weights weigh? | 2090 | 0.625 |
Car A departs from point $A$ heading towards point $B$ and returns; Car B departs from point $B$ at the same time heading towards point $A$ and returns. After the first meeting, Car A continues for 4 hours to reach $B$, and Car B continues for 1 hour to reach $A$. If the distance between $A$ and $B$ is 100 kilometers, what is Car B's distance from $A$ when Car A first arrives at $B$? | 100 | 0.25 |
The sequence $\{a_n\}$ satisfies $a_1 = 2, a_{n+1} = \frac{2(n+2)}{n+1} a_n$ for $n \in \mathbf{N}^*$. Find $\frac{a_{2014}}{a_1 + a_2 + \cdots + a_{2013}}$. | \frac{2015}{2013} | 0.875 |
There are 2018 playing cards on the table (2018 piles with one card each). Petka wants to combine them into one deck of 2018 cards in 2017 operations. Each operation involves combining two piles into one. When Petka combines piles of $a$ and $b$ cards, Vasily Ivanovich pays Petka $a \cdot b$ rubles. What is the maximum amount of money Petka can earn by performing all 2017 operations? | 2035153 | 0.5 |
Calculate the number of zeros at the end of 2014!. | 501 | 0.625 |
The sequence $\left\{a_{n}\right\}$ has its first term as 2, and the sum of the first $n$ terms is denoted by $S_{n}$. The sequence $\left\{S_{n}\right\}$ forms a geometric sequence with a common ratio of $\frac{1}{3}$.
1. Find the general term formula for the sequence $\left\{a_{n}\right\}$.
2. Let $b_{n}=a_{n} S_{n}$, find the sum of the terms of the sequence $\left\{S_{n}\right\}$. | 3 | 0.625 |
Given a positive integer \( n \), find the number of ordered quadruples of integers \( (a, b, c, d) \) such that \( 0 \leq a \leq b \leq c \leq d \leq n \). | \binom{n+4}{4} | 0.5 |
Determine all positive integer solutions \( x \) and \( y \) for the equation \( x^2 - 2 \cdot y! = 2021 \). | (45, 2) | 0.375 |
In the coordinate plane, a square $K$ with vertices at points $(0,0)$ and $(10,10)$ is given. Inside this square, illustrate the set $M$ of points $(x, y)$ whose coordinates satisfy the equation
$$
[x] < [y]
$$
where $[a]$ denotes the integer part of the number $a$ (i.e., the largest integer not exceeding $a$; for example, $[10]=10,[9.93]=9,[1 / 9]=0,[-1.7]=-2$). What portion of the area of square $K$ does the area of set $M$ constitute? | 0.45 | 0.5 |
Two circles touch each other internally. It is known that two radii of the larger circle, which form an angle of $60^\circ$ between them, are tangent to the smaller circle. Find the ratio of the radii of the circles. | 3 | 0.5 |
An ant starts at the origin, facing in the positive \( x \)-direction. Each second, it moves 1 unit forward, then turns counterclockwise by \( \sin^{-1}\left(\frac{3}{5}\right) \) degrees. What is the least upper bound on the distance between the ant and the origin? The least upper bound is the smallest real number \( r \) that is at least as big as every distance that the ant ever is from the origin. | \sqrt{10} | 0.75 |
In triangle \( ABC \), \( AB = 33 \), \( AC = 21 \), and \( BC = m \), where \( m \) is a positive integer. If point \( D \) can be found on \( AB \) and point \( E \) can be found on \( AC \) such that \( AD = DE = EC = n \), where \( n \) is a positive integer, what must the value of \( m \) be? | 30 | 0.125 |
The figure below shows a target on a wall that is fixed and cannot be rotated. It is divided into 10 parts, including a central circle, a smaller ring, and a larger (outer) ring. We must distribute the numbers 1 through 10, assigning one number to each part, which will correspond to the scores obtained by hitting each part.
a) In how many ways can we distribute the numbers among the parts of the target?
b) In how many ways can we distribute the numbers such that numbers closer to the center are not smaller than numbers farther from the center?
c) In how many ways can we distribute the numbers so that the sum of the numbers in the outer ring is equal to the sum of the numbers in the smaller ring? | 34,560 | 0.25 |
Cat food is sold in large and small packages (with more food in the large package than in the small one). One large package and four small packages are enough to feed a cat for exactly two weeks. Is one large package and three small packages necessarily enough to feed the cat for 11 days? | Yes | 0.875 |
Arseniy sat down at the computer between 16:00 and 17:00 when the hour and minute hands were pointing in opposite directions, and got up the same day between 22:00 and 23:00 when the hands overlapped. How long did Arseniy sit at the computer? | 6 \text{ hours} | 0.75 |
Once in winter, 43 children were throwing snowballs. Each of them threw exactly one snowball at someone else. It is known that:
- The first child threw a snowball at the one who threw a snowball at the second child,
- The second child threw a snowball at the one who threw a snowball at the third child,
- The forty-third child threw a snowball at the one who threw a snowball at the first child.
What is the number of the person who threw a snowball at the third child? | 24 | 0.625 |
Given that \( g \) is a twice differentiable function over the positive real numbers such that
\[ g(x) + 2x^3 g'(x) + x^4 g''(x) = 0 \quad \text{for all positive reals } x \]
and
\[ \lim_{x \to \infty} x g(x) = 1, \]
find the real number \(\alpha > 1\) such that \( g(\alpha) = \frac{1}{2} \). | \frac{6}{\pi} | 0.375 |
Find the integral \( \int \frac{dx}{\sin x \cos x} \). | \ln |\tan x| + C | 0.875 |
Find the only value of \( x \) in the open interval \((- \pi / 2, 0)\) that satisfies the equation
$$
\frac{\sqrt{3}}{\sin x} + \frac{1}{\cos x} = 4.
$$ | -\frac{4\pi}{9} | 0.875 |
In a $3 \times 3$ grid, there are four $2 \times 2$ subgrids. Kasun wants to place an integer from 1 to 4 inclusive in each cell of the $3 \times 3$ grid so that every $2 \times 2$ subgrid contains each integer exactly once. For example, the grid below on the left satisfies the condition, but the grid below on the right does not. In how many ways can Kasun place integers in the grid so that they satisfy the condition?
| 1 | 2 | 3 |
| :--- | :--- | :--- |
| 3 | 4 | 1 |
| 1 | 2 | 3 |
| 1 | 3 | 2 |
| :--- | :--- | :--- |
| 2 | 4 | 1 |
| 1 | 3 | 3 | | 72 | 0.125 |
Find the maximum value of the expression \((\sqrt{36-4 \sqrt{5}} \sin x-\sqrt{2(1+\cos 2 x)}-2) \cdot (3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y)\). If the answer is not an integer, round it to the nearest whole number. | 27 | 0.5 |
Given a regular triangular pyramid \( S A B C \). Point \( S \) is the apex of the pyramid, \( AB = 1 \), \( AS = 2 \), \( BM \) is the median of triangle \( ABC \), and \( AD \) is the angle bisector of triangle \( SAB \). Find the length of segment \( DM \). | \frac{\sqrt{31}}{6} | 0.375 |
Find the last two digits of \(1032^{1032}\). Express your answer as a two-digit number. | 76 | 0.75 |
The distance between the centers of two circles with radii 1 and 9 is 17. A third circle is tangent to these two circles and also to their common external tangent. Find the radius of the third circle. | \frac{225}{64} | 0.125 |
Compute the volumes of the bodies bounded by the surfaces:
\[ z = 4x^2 + 9y^2 \]
\[ z = 6 \] | 3\pi | 0.875 |
Mother decides to take Xiaohua to 10 cities for a vacation by car. Xiaohua checked the map and was surprised to find that any three of these 10 cities have either all highways between them or only two of the cities have no highway between them. How many highways are there at least between these 10 cities? (Note: There can be at most one highway between any two cities.) | 40 | 0.375 |
Find the largest positive integer $n$ such that there exist $n$ real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying the inequality $\left(1+x_{i} x_{j}\right)^{2} \leqslant 0.99\left(1+x_{i}^{2}\right)\left(1+x_{j}^{2}\right)$ for any $1 \leq i < j \leq n$. | 31 | 0.125 |
In a math class, each dwarf needs to find a three-digit number without any zero digits, divisible by 3, such that when 297 is added to the number, the result is a number with the same digits in reverse order. What is the minimum number of dwarfs that must be in the class so that there are always at least two identical numbers among those found? | 19 | 0.75 |
Let \( x_{1}, y_{1}, x_{2}, y_{2} \) be real numbers satisfying the equations \( x_{1}^{2}+5 x_{2}^{2}=10 \), \( x_{2} y_{1}-x_{1} y_{2}=5 \) and \( x_{1} y_{1}+5 x_{2} y_{2}=\sqrt{105} \). Find the value of \( y_{1}^{2}+5 y_{2}^{2} \). | 23 | 0.75 |
The Small and Big Islands have a rectangular shape and are divided into rectangular counties. In each county, a road is laid along one of the diagonals. On each island, these roads form a closed path that does not pass through any point twice. Here is how the Small Island is arranged, where there are only six counties:
Draw how the Big Island can be arranged if it has an odd number of counties. How many counties did you get? | 9 | 0.875 |
On September 10, 2005, the following numbers were drawn in the five-number lottery: 4, 16, 22, 48, 88. All five numbers are even, exactly four of them are divisible by 4, three by 8, and two by 16. In how many ways can five different numbers with these properties be selected from the integers ranging from 1 to 90? | 15180 | 0.5 |
On the base \(AB\) of an isosceles trapezoid \(ABCD\), point \(P\) is chosen such that the base is divided in the ratio \(AP : BP = 4 : 1\). Given that \(\angle CPD = \angle PAD\), find the ratio \(PD / PC\). | 2 | 0.75 |
In triangle \( \triangle ABC \), with \( \angle A \leq \angle B \leq \angle C \), suppose
$$
\frac{\sin A + \sin B + \sin C}{\cos A + \cos B + \cos C} = \sqrt{3},
$$
find the value of \( \sin B + \sin 2B \). | \sqrt{3} | 0.875 |
Find the analytic function \( w = f(z) \) given its imaginary part \( v(x, y) = 3x + 2xy \) under the condition that \( f(-i) = 2 \). | f(z) = z^2 + 3iz | 0.875 |
A triangle has a perimeter \( 2p \) and an inscribed circle. A tangent to this circle, parallel to one side of the triangle, is drawn. Find the maximum possible length of the segment of this tangent that lies within the triangle. | \frac{p}{4} | 0.625 |
A square \(ABCD\) has a side length of 40 units. Point \(F\) is the midpoint of side \(AD\). Point \(G\) lies on \(CF\) such that \(3CG = 2GF\). What is the area of triangle \(BCG\)? | 320 | 0.875 |
Let \([x]\) denote the greatest integer less than or equal to \(x\). Determine the number of elements in the set \(\{ [x] + [2x] + [3x] \mid x \in \mathbb{R} \} \cap \{1, 2, \ldots, 100\}\). | 67 | 0.375 |
Determine the pairs of positive integers \((a_1, a_2)\) for which the sequence defined by the recurrence relation \(a_{n+2} = \frac{a_n + a_{n+1}}{\gcd(a_n, a_{n+1})} \, (n \geq 1)\) is periodic. | (2, 2) | 0.875 |
The angle bisectors $\mathrm{AD}$ and $\mathrm{BE}$ of the triangle $\mathrm{ABC}$ intersect at point I. It turns out that the area of triangle $\mathrm{ABI}$ is equal to the area of quadrilateral $\mathrm{CDIE}$. Find the maximum possible value of angle $\mathrm{ACB}$. | 60^\circ | 0.625 |
Given an isosceles triangle \(ABC\) where \(AB = AC\) and \(\angle ABC = 53^\circ\), point \(K\) is such that \(C\) is the midpoint of segment \(AK\). Point \(M\) is chosen such that:
- \(B\) and \(M\) lie on the same side of line \(AC\);
- \(KM = AB\);
- the angle \(\angle MAK\) is the maximum possible.
How many degrees is the angle \(\angle BAM\)? | 44 | 0.125 |
Find the volume common to two cylinders: \( x^{2} + y^{2} = a^{2} \) and \( y^{2} + z^{2} = a^{2} \) (bounded by these cylindrical surfaces). | \frac{16}{3} a^3 | 0.5 |
Let \( f \) be a function from \(\mathbb{R}\) to \(\mathbb{R}\) that satisfies the following functional equation for all real numbers \(x\) and \(y\):
$$
(y+1) f(x) + f(x f(y) + f(x+y)) = y
$$
1. Show that \( f \) is bijective.
2. Find all functions that satisfy this equation. | f(x) = -x | 0.375 |
In a convex 13-gon, all diagonals are drawn. They divide it into polygons. Consider the polygon with the largest number of sides among them. What is the greatest number of sides that it can have? | 13 | 0.125 |
Behind a thin lens with a focal length \( F \) and a diameter \( D \), a flat screen is placed perpendicularly to its optical axis at its focal point. A point light source is placed on the main optical axis at a distance \( d > F \) from the lens. Determine the diameter of the light spot on the screen. | \frac{F D}{d} | 0.25 |
Find the integer closest to the sum $S$ obtained by taking the base 10 logarithm of each proper divisor of $1,000,000$ and adding these logarithm values together. | 141 | 0.5 |
Let \(\mathbb{N}\) be the set of positive integers, and let \(f: \mathbb{N} \rightarrow \mathbb{N}\) be a function satisfying:
- \(f(1) = 1\)
- For \(n \in \mathbb{N}\), \(f(2n) = 2f(n)\) and \(f(2n + 1) = 2f(n) - 1\).
Determine the sum of all positive integer solutions to \(f(x) = 19\) that do not exceed 2019. | 1889 | 0.375 |
Given the positive integers \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \). Any set of four numbers selected and summed results in the set \(\{44, 45, 46, 47\}\). Determine these five numbers. | 10, 11, 11, 12, 13 | 0.75 |
What is the sum of all four-digit numbers that are equal to the cube of the sum of their digits (leading zeros are not allowed)? | 10745 | 0.75 |
Express the number one million using numbers that contain only the digit 9 and the algebraic operations of addition, subtraction, multiplication, division, powers, and roots. Determine at least three different solutions. | 1000000 | 0.25 |
In an office, each computer was connected by cables to exactly 5 other computers. After a virus affected some of the computers, all the cables from the infected computers were disconnected (a total of 26 cables had to be disconnected). Now, each of the uninfected computers is connected by cables to only 3 other computers. How many computers were affected by the virus? | 8 | 0.25 |
Let \( n \) be an integer greater than 3. Let \( R \) be the set of lattice points \( (x, y) \) such that \( 0 \leq x, y \leq n \) and \( |x-y| \leq 3 \). Let \( A_n \) be the number of paths from \( (0,0) \) to \( (n,n) \) that consist only of steps of the form \( (x, y) \rightarrow (x, y+1) \) and \( (x, y) \rightarrow (x+1, y) \) and are contained entirely within \( R \). Find the smallest positive real number that is greater than \( \frac{A_{n+1}}{A_n} \) for all \( n \). | 2+\sqrt{2} | 0.25 |
There are six wooden sticks, each 50 cm long. They are to be connected end to end in sequence, with each connection section measuring 10 cm. After nailing them together, what is the total length of the wooden sticks? ( ) cm. | 250 | 0.5 |
For the function \( f(x) \), the condition \( f(f(f(x))) + 3 f(f(x)) + 9 f(x) + 27 x = 0 \) is satisfied. Find \( f(f(f(f(2)))) \). | 162 | 0.875 |
The sequence \( x_{n} \) is defined by the following conditions: \( x_{1}=3 \) and \( x_{1}+x_{2}+\ldots+x_{n-1}+\frac{4}{3}x_{n}=4 \). Find \( x_{2015} \). | \frac{3}{4^{2014}} | 0.375 |
\( A_1, A_2, A_3, A_4 \) are consecutive vertices of a regular \( n \)-gon. Given the equation \( \frac{1}{A_1A_2} = \frac{1}{A_1A_3} + \frac{1}{A_1A_4} \), what are the possible values of \( n \)? | n = 7 | 0.875 |
Vendelín lives between two bus stops, at three-eighths of their distance. Today he left home and discovered that whether he ran to one or the other stop, he would arrive at the stop at the same time as the bus. The average speed of the bus is $60 \mathrm{~km} / \mathrm{h}$.
What is the average speed at which Vendelín is running today? | 15 \text{ km/h} | 0.375 |
Given the circle \( x^2 + y^2 = 1 \), a tangent line intersects the \( x \)-axis and \( y \)-axis at points \( A \) and \( B \) respectively. What is the minimum value of \( |AB| \)? | 2 | 0.875 |
There is a point $A$, a line $b$ through it, the first image of a line $f$ (which also passes through $A$) is $f'$, an angle $\alpha$, and two distances $r$ and $2s$. Construct a triangle $ABC$ where one vertex is $A$, side $b$ lies on line $b$, the angle bisector lies on line $f$, one angle is $\alpha$, the perimeter is $2s$, and the circumradius is $r$. | ABC | 0.25 |
Find the product of all the roots of the equation \( x^{4} + 4x^{3} - 2015x^{2} - 4038x + 2018 = 0 \). | 2018 | 0.625 |
Let \( A \) and \( B \) be points on the curve \( xy = 1 \) (where \( x > 0 \) and \( y > 0 \)) in the Cartesian coordinate system \( xOy \). Given the vector \( \vec{m} = (1, |OA|) \), find the minimum value of the dot product \( \vec{m} \cdot \overrightarrow{OB} \). | 2 \sqrt[4]{2} | 0.125 |
A pedestrian walks along a highway at a speed of 5 km/h. Buses travel along this highway in both directions at the same speed, meeting every 5 minutes. At 12 o'clock, the pedestrian noticed that the buses met near him and, continuing to walk, began counting the passing and overtaking buses. At 2 o'clock, the buses met again near him. It turned out that during this time, the pedestrian encountered four more buses than those overtaking him. Find the speed of the bus. | 30 \text{ km/h} | 0.875 |
Sergey, while being a student, worked part-time at a student cafe throughout the year. Sergey's salary was 9000 rubles per month. In the same year, Sergey paid 100000 rubles for his treatment at a healthcare institution and purchased medication prescribed by a doctor for 20000 rubles (eligible for deduction). The following year, Sergey decided to claim a social tax deduction. What amount of personal income tax paid is subject to refund to Sergey from the budget? (Answer as a whole number, without spaces or units of measurement.) | 14040 | 0.375 |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(1+\operatorname{tg}^{2} x\right)^{\frac{1}{\ln \left(1+3 x^{2}\right)}}
$$ | e^{\frac{1}{3}} | 0.625 |
All natural numbers starting from 1 were written consecutively, forming a sequence of digits, as follows.
$$
1234567891011121314151617181920212223 \ldots
$$
What is the digit that appears in the 206788th position? | 7 | 0.375 |
On a board, the numbers $1, 2, 3, \ldots, 235$ were written. Petya erased several of them. It turned out that among the remaining numbers, no number is divisible by the difference of any two others. What is the maximum number of numbers that could remain on the board? | 118 | 0.375 |
Given the vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\) satisfy the following conditions:
$$
\begin{array}{l}
|a - b| = 3, \\
|a + 2b| = 6, \\
a^{2} + a \cdot b - 2b^{2} = -9.
\end{array}
$$
Find the magnitude of \(\boldsymbol{b}\), \(|b| =\). | \sqrt{7} | 0.75 |
In a tennis tournament, 512 schoolchildren participate. For a win, 1 point is awarded, for a loss, 0 points. Before each round, pairs are drawn from participants with an equal number of points (those who do not have a pair are awarded a point without playing). The tournament ends as soon as a sole leader is determined. How many schoolchildren will finish the tournament with 6 points? | 84 | 0.25 |
Let real numbers \( x_{1}, x_{2}, \ldots, x_{1997} \) satisfy the following two conditions:
1. \( -\frac{1}{\sqrt{3}} \leq x_{i} \leq \sqrt{3} \quad(i=1, 2, \ldots, 1997) \);
2. \( x_{1} + x_{2} + \cdots + x_{1997} = -318 \sqrt{3} \).
Find the maximum value of \( x_{1}^{12} + x_{2}^{12} + \cdots + x_{1997}^{12} \) and explain the reasoning. | 189548 | 0.375 |
The sum of five consecutive integers is \(10^{2018}\). What is the middle number?
A) \(10^{2013}\)
B) \(5^{2017}\)
C) \(10^{2017}\)
D) \(2^{2018}\)
E) \(2 \times 10^{2017}\) | 2 \times 10^{2017} | 0.375 |
Let \( x_{1}, x_{2} \) be the roots of the equation \( x^{2} - x - 3 = 0 \). Find \(\left(x_{1}^{5} - 20\right) \cdot \left(3 x_{2}^{4} - 2 x_{2} - 35\right)\). | -1063 | 0.5 |
A six-digit number begins with digit 1 and ends with digit 7. If the digit in the units place is decreased by 1 and moved to the first place, the resulting number is five times the original number. Find this number. | 142857 | 0.625 |
Masha has 2 kg of "Lastochka" candies, 3 kg of "Truffle" candies, 4 kg of "Ptichye Moloko" candies, and 5 kg of "Citron" candies. What is the maximum number of New Year gifts she can make if each gift must contain 3 different types of candies, with 100 grams of each type? | 45 | 0.25 |
Let \( A(x_{1}, y_{1}) \) and \( B(x_{2}, y_{2}) \) be two points on the curve \( C: x^{2}-y^{2}=2 \) (where \( x > 0 \)). Then the minimum value of \( f = \overrightarrow{OA} \cdot \overrightarrow{OB} \) is ______. | 2 | 0.375 |
Four people, A, B, C, and D, each have a different number of playing cards.
A says: "I have 16 more cards than C."
B says: "D has 6 more cards than C."
C says: "A has 9 more cards than D."
D says: "If A gives me 2 cards, I will have 3 times as many cards as C."
It is known that the person with the fewest cards is lying, and the others are telling the truth. How many cards does D have? | 10 | 0.875 |
Let \( x, y \) be nonnegative integers such that \( x + 2y \) is a multiple of 5, \( x + y \) is a multiple of 3, and \( 2x + y \geq 99 \). Find the minimum possible value of \( 7x + 5y \). | 366 | 0.25 |
In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse." How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646. | 646 | 0.75 |
Appending three digits at the end of 2007, one obtains an integer \(N\) of seven digits. In order to get \(N\) to be the minimal number which is divisible by 3, 5, and 7 simultaneously, what are the three digits that one would append? | 075 | 0.125 |
Determine the numeral system in which the following multiplication is performed: \(352 \cdot 31 = 20152\). | 6 | 0.375 |
Find the four-digit number that gives a remainder of 112 when divided by 131 and a remainder of 98 when divided by 132. | 1946 | 0.375 |
The reals \( x_1, x_2, \ldots, x_{n+1} \) satisfy \( 0 < x_i < \frac{\pi}{2} \) and \(\sum_{i=1}^{n+1} \tan \left( x_i - \frac{\pi}{4} \right) \geq n-1 \). Show that \(\prod_{i=1}^{n+1} \tan x_i \geq n^{n+1} \). | n^{n+1} | 0.125 |
Given the sequence $\{a_{n}\}$, where $a_{n}$ are integers, and for $n \geq 3, n \in \mathbf{N}$, the relation $a_{n} = a_{n-1} - a_{n-2}$ holds. If the sum of the first 1985 terms of the sequence is 1000, and the sum of the first 1995 terms is 4000, then what is the sum of the first 2002 terms of the sequence? | 3000 | 0.375 |
Over the course of three years, Marina did not invest the funds in her Individual Investment Account (IIA) into financial instruments and therefore did not receive any income from it. However, she gained the right to an investment tax deduction for depositing her own money into the IIA.
The tax deduction is provided for the amount of money deposited into the IIA during the tax period, but not more than 400,000 rubles in total per year.
Marina is entitled to receive 13% of the amount deposited into the IIA as a refund of the Personal Income Tax (PIT) that she paid on her income.
The PIT amount deducted from Marina's annual salary is equal to \( 30,000 \text{ rubles} \times 12 \text{ months} \times 0.13 = 46,800 \text{ rubles} \).
The tax deduction for the first year is \( 100,000 \text{ rubles} \times 0.13 = 13,000 \text{ rubles} \). This amount does not exceed the PIT deducted from Marina's annual salary.
The tax deduction for the second year is \( 400,000 \text{ rubles} \times 0.13 = 52,000 \text{ rubles} \). This amount exceeds the PIT deducted from Marina's annual salary. Therefore, the tax deduction will be limited to the PIT paid for the second year, or 46,800 rubles.
The tax deduction for the third year is \( 400,000 \text{ rubles} \times 0.13 = 52,000 \text{ rubles} \). This amount also exceeds the PIT deducted from Marina's annual salary. Therefore, the tax deduction will be limited to the PIT paid for the third year, or 46,800 rubles.
The total amount of tax deduction for 3 years is \( 13,000 + 46,800 + 46,800 = 106,600 \text{ rubles} \).
The return on Marina's transactions over 3 years is \( \frac{106,600}{1,000,000} \times 100 \% = 10.66 \% \).
The annual return on Marina's transactions is \( \frac{10.66 \%}{3} = 3.55 \% \). | 3.55\% | 0.625 |
An event is a hit or a miss. The first event is a hit, the second is a miss. Thereafter, the probability of a hit equals the proportion of hits in the previous trials. For example, the probability of a hit in the third trial is 1/2. What is the probability of exactly 50 hits in the first 100 trials? | \frac{1}{99} | 0.625 |
Determine the largest natural number \( n \) such that
\[ 4^{995} + 4^{1500} + 4^{n} \]
is a square number. | 2004 | 0.875 |
Determine the mass \( m \) of helium needed to fill an empty balloon with mass \( m_{1} = 10 \) g so that the balloon can ascend. Assume the temperature and pressure of the gas in the balloon are equal to the temperature and pressure of the air. The molar mass of helium \( M_{\mathrm{r}} = 4 \) g/mol and the molar mass of air \( M_{\mathrm{B}} = 29 \) g/mol. | 1.6 \, \text{g} | 0.125 |
The natural number \( A \) has the following property: the number \( 1+2+\cdots+A \) can be written (in base 10) as the number \( A \) followed by three other digits. Find \( A \). | 1999 | 0.625 |
A circle is inscribed in triangle \(ABC\). On the longest side \(AC\) of the triangle, points \(E\) and \(F\) are marked such that \(AE = AB\) and \(CF = CB\). The segment \(BE\) intersects the inscribed circle at points \(P\) and \(Q\), with \(BP = 1\) and \(PQ = 8\). What is the length of the segment \(EF\)? | 6 | 0.625 |
Find all pairs of integers \((a, n)\) such that \(n\) divides \((a + 1)^{n} - a^{n}\). | (a, 1) | 0.5 |
The corridors of a maze are the sides and diagonals of an n-sided convex polygon. What is the minimum number of lanterns we need to place in the maze in order to illuminate every corridor? | n-1 | 0.75 |
Let $b_{n}$ be the sum of the numbers in the $n$-th diagonal of Pascal's triangle. Show that $b_{n} = a_{n}$ where $a_{n}$ is the $n$-th term of the Fibonacci sequence
$$
a_{0} = 0, \quad a_{1} = 1, \quad a_{2} = 1, \quad a_{3} = 2, \ldots
$$ | b_n = a_n | 0.375 |
Evaluate the sum: $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{m^{2} n}{3^{m}\left(n \cdot 3^{m}+m \cdot 3^{n}\right)}$. | \frac{9}{32} | 0.375 |
A right circular cone with a base radius \( R \) and height \( H = 3R \sqrt{7} \) is laid sideways on a plane and rolled in such a manner that its apex remains stationary. How many rotations will its base make until the cone returns to its original position? | 8 | 0.75 |
Three cyclists started simultaneously: the first and second from point A, and the third towards them from point B. After 1.5 hours, the first cyclist was equidistant from the other two, and 2 hours after departure, the third cyclist was equidistant from the first and second. How many hours after departure was the second cyclist equidistant from the first and third? | 3 | 0.75 |
Given three positive numbers \( a, b, \mathrm{and} c \) satisfying \( a \leq b+c \leq 3a \) and \( 3b^2 \leq a(a+c) \leq 5b^2 \), what is the minimum value of \(\frac{b-2c}{a}\)? | -\frac{18}{5} | 0.875 |
Several different points are marked on a line, and all possible line segments are constructed between pairs of these points. One of these points lies on exactly 80 of these segments (not including any segments of which this point is an endpoint). Another one of these points lies on exactly 90 segments (not including any segments of which it is an endpoint). How many points are marked on the line? | 22 | 0.625 |
Let $ABC$ be a triangle. $K$ is the foot of the angle bisector of $\widehat{ABC}$ and $L$ is the foot of the angle bisector of $\widehat{CAB}$. Let $M$ be the point of intersection of the perpendicular bisector of the segment $[KB]$ and the line $(AL)$. The line parallel to $(KL)$ passing through $L$ intersects $(BK)$ at $N$. Show that $LN = NA$. | LN = NA | 0.875 |
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