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Select $k$ edges and diagonals from the faces of a cube such that any two chosen line segments lie on skew lines. What is the maximum value of $k$? | 4 | 0.125 |
The sequence \(101, 104, 109, 116, \cdots\) has the general term \(a_{n} = 100 + n^{2}\). For each \(n\), let \(d_{n}\) represent the greatest common divisor (GCD) of \(a_{n}\) and \(a_{n+1}\). Determine the maximum value of \(d_{n}\). | 401 | 0.625 |
Ksyusha runs twice as fast as she walks (both speeds are constant).
On Tuesday, when she left home for school, she first walked, and then, when she realized she was late, she started running. The distance Ksyusha walked was twice the distance she ran. As a result, she reached the school from home in exactly 30 minutes.
On Wednesday, Ksyusha left home even later, so she had to run twice the distance she walked. How many minutes did it take her to get from home to school on Wednesday? | 24 | 0.875 |
In how many ways can a rectangular board of size \(2 \times 13\) be covered with rectangular tiles of size \(1 \times 2\)? (The tiles are placed such that they do not overlap and completely fit on the board.) | 377 | 0.625 |
The base of the quadrilateral prism $A B C D A_{1} B_{1} C_{1} D_{1}$ is a rhombus $A B C D$, where $B D=3$ and $\angle A D C=60^{\circ}$. A sphere passes through the vertices $D, C, B, B_{1}, A_{1}, D_{1}$.
a) Find the area of the circle obtained in the cross-section of the sphere by the plane passing through the points $A_{1}, C_{1}$, and $D_{1}$.
b) Find the angle $B_{1} C_{1} A$.
c) It is additionally known that the radius of the sphere is 2. Find the volume of the prism. | 3\sqrt{3} | 0.375 |
Let \( w, x, y, \) and \( z \) be positive real numbers such that
\[
\begin{aligned}
0 & \neq \cos w \cos x \cos y \cos z \\
2 \pi & = w + x + y + z \\
3 \tan w & = k(1 + \sec w) \\
4 \tan x & = k(1 + \sec x) \\
5 \tan y & = k(1 + \sec y) \\
6 \tan z & = k(1 + \sec z)
\end{aligned}
\]
(Here \( \sec t \) denotes \( \frac{1}{\cos t} \) when \( \cos t \neq 0 \).) Find \( k \). | \sqrt{19} | 0.5 |
In the sequence \(\{a_n\}\), \(a_1 = 1\), \(a_{n+1} > a_n\), and \(a_{n+1}^2 + a_n^2 + 1 = 2(a_{n+1}a_n + a_{n+1} + a_n)\), find \(a_n\). | n^2 | 0.875 |
If the value of the 5th term in the expansion of $\left(x \sqrt{x}-\frac{1}{x}\right)^{6}$ is $\frac{15}{2}$, then $\lim _{n \rightarrow \infty}\left(x^{-1}+x^{-2}+\cdots+x^{-n}\right)=$ | 1 | 0.875 |
Calculate the sum:
(1) $\cos ^{2} 1^{\circ}+\cos ^{2} 3^{\circ}+\cdots+\cos ^{2} 89^{\circ}$
(2) Given $f(x)=\frac{4^{x}}{4^{x}+2}$, find $f\left(\frac{1}{1001}\right)+f\left(\frac{2}{1001}\right)+\cdots+f\left(\frac{1000}{1001}\right)$ | 500 | 0.875 |
The probability of an event occurring in each of 900 independent trials is 0.5. Find the probability that the relative frequency of the event will deviate from its probability by no more than 0.02. | 0.7698 | 0.25 |
In triangle \(ABC\), it is known that \(AB = c\), \(BC = a\), \(AC = b\). Point \(O\) is the center of the circle that touches side \(AB\) and the extensions of sides \(AC\) and \(BC\). Point \(D\) is the intersection of ray \(CO\) with side \(AB\). Find the ratio \(\frac{CO}{OD}\). | \frac{a+b}{c} | 0.5 |
For any positive integer \( n \), let \( f(n) \) denote the index of the highest power of 2 which divides \( n! \). For example, \( f(10) = 8 \) since \( 10! = 2^8 \times 3^4 \times 5^2 \times 7 \). Find the value of \( f(1) + f(2) + \cdots + f(1023) \). | 518656 | 0.5 |
Given a positive integer \( n \leq 500 \) with the following property: If one randomly selects an element \( m \) from the set \(\{1, 2, \cdots, 500\}\), the probability that \( m \) divides \( n \) is \(\frac{1}{100}\). Determine the maximum value of \( n \). | 81 | 0.875 |
In a math competition consisting of problems $A$, $B$, and $C$, among the 39 participants, each person answered at least one problem correctly. Among the people who answered $A$ correctly, the number of people who only answered $A$ is 5 more than the number of people who answered other problems as well. Among the people who did not answer $A$ correctly, the number of people who answered $B$ is twice the number of people who answered $C$. Additionally, the number of people who only answered $A$ equals the sum of the number of people who only answered $B$ and the number of people who only answered $C$. What is the maximum number of people who answered $A$ correctly? | 23 | 0.75 |
Given the ellipse \(\frac{y}{4} + x^{2} = 1\), let \(P\) be an arbitrary point on the ellipse. Draw lines through point \(P\) that are parallel to \(l_1: y=2x\) and \(l_2: y=-2x\). These lines intersect the lines \(l_2\) and \(l_1\) at points \(M\) and \(N\) respectively. Find the maximum value of \(|MN|\). | 2 | 0.625 |
Given the sequence $\left\{a_{n}\right\}$ where $a_{1}=1, a_{2}=2,$ and the recurrence relation $a_{n} a_{n+1} a_{n+2}=a_{n}+a_{n+1}+a_{n+2}$ with the condition $a_{n+1} a_{n+2} \neq 1$, find $S_{1999}=\sum_{n=1}^{1999} a_{n}$. | 3997 | 0.625 |
In two weeks three cows eat all the grass on two hectares of land, together with all the grass that regrows there during the two weeks. In four weeks, two cows eat all the grass on two hectares of land, together with all the grass that regrows there during the four weeks.
How many cows will eat all the grass on six hectares of land in six weeks, together with all the grass that regrows there over the six weeks?
(Assume: - the quantity of grass on each hectare is the same when the cows begin to graze,
- the rate of growth of the grass is uniform during the time of grazing,
- and the cows eat the same amount of grass each week.) | 5 | 0.75 |
Let circle $\Gamma_{1}$ be the circumscribed circle of $\triangle ABC$. Points $D$ and $E$ are the midpoints of arcs $\overparen{AB}$ (excluding point $C$) and $\overparen{AC}$ (excluding point $B$), respectively. The incircle of $\triangle ABC$ touches sides $AB$ and $AC$ at points $F$ and $G$, respectively. The lines $EG$ and $DF$ intersect at point $X$. If $X D = X E$, show that $AB = AC$. | AB = AC | 0.875 |
Determine the share of the Japanese yen in the currency structure of the National Wealth Fund (NWF) as of 01.12.2022 using one of the following methods:
First method:
a) Find the total amount of NWF funds placed in Japanese yen as of 01.12.2022:
\[ J P Y_{22} = 1388.01 - 41.89 - 2.77 - 309.72 - 554.91 - 0.24 = 478.48 \text{ (billion rubles)} \]
b) Determine the share of the Japanese yen in the currency structure of NWF funds as of 01.12.2022:
\[ \alpha_{22}^{J P Y} = \frac{478.48}{1388.01} \approx 34.47\% \]
c) Calculate by how many percentage points and in what direction the share of the Japanese yen in the currency structure of NWF funds changed during the period considered in the table:
\[ \Delta \alpha^{J P Y} = \alpha_{22}^{J P Y} - \alpha_{21}^{J P Y} = 34.47 - 47.06 = -12.59 \approx -12.6 \text{ (percentage points)} \]
Second method:
a) Determine the share of the euro in the currency structure of NWF funds as of 01.12.2022:
\[ \alpha_{22}^{E U R} = \frac{41.89}{1388.01} \approx 3.02\% \]
b) Determine the share of the Japanese yen in the currency structure of NWF funds as of 01.12.2022:
\[ \alpha_{22}^{J P Y} = 100 - 3.02 - 0.2 - 22.31 - 39.98 - 0.02 = 34.47\% \]
c) Calculate by how many percentage points and in what direction the share of the Japanese yen in the currency structure of NWF funds changed during the period considered in the table:
\[ \Delta \alpha^{J P Y} = \alpha_{22}^{J P Y} - \alpha_{21}^{J P Y} = 34.47 - 47.06 = -12.59 \approx -12.6 \text{ (percentage points)} \] | -12.6 | 0.375 |
Calculate the value of the total differential of the function \( z = \operatorname{arcctg} \frac{x}{y} \) at \( x = 1, y = 3, dx = 0.01, dy = -0.05 \). | -0.008 | 0.875 |
Let \( f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z} \) be a function with the following properties:
(i) \( f(1) = 0 \),
(ii) \( f(p) = 1 \) for all prime numbers \( p \),
(iii) \( f(x y) = y f(x) + x f(y) \) for all \( x, y \) in \( \mathbb{Z}_{>0} \).
Determine the smallest integer \( n \geq 2015 \) that satisfies \( f(n) = n \). | 3125 | 0.625 |
Since \(\left(1-\frac{1}{k^2}\right)=\left(1-\frac{1}{k}\right)\left(1+\frac{1}{k}\right)=\frac{k-1}{k} \cdot \frac{k+1}{k}\), find the value of the product
\[ \left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right) \cdots \left(1-\frac{1}{2021^2}\right)\left(1-\frac{1}{2022^2}\right). \] | \frac{2023}{4044} | 0.375 |
A positive integer sequence has its first term as 8 and its second term as 1. From the third term onwards, each term is the sum of the two preceding terms. What is the remainder when the 2013th term in this sequence is divided by 105? | 16 | 0.375 |
Find the area enclosed by the graph \( x^{2}+y^{2}=|x|+|y| \) on the \( xy \)-plane. | \pi + 2 | 0.625 |
Simplify the expression \(\frac{\sin 4\alpha + \sin 5\alpha + \sin 6\alpha}{\cos 4\alpha + \cos 5\alpha + \cos 6\alpha}\). | \tan 5\alpha | 0.75 |
Given two circles, we want to draw a common tangent - a line that touches both circles. How many such lines exist? (The answer depends on the ratio of the circles' radii and the distance between their centers.) | 4 | 0.5 |
A gear mechanism has a smaller gear with 12 teeth and a larger gear with 32 teeth. Due to a manufacturing defect, one tooth on the smaller gear and one slot on the larger gear stick and make noise upon contact. At what intervals do we hear the squeaks if the larger gear takes 3 seconds for one full revolution? How can the squeaking be eliminated, assuming all other tooth-slot pairings are flawless and silent? | 9 \text{ seconds} | 0.625 |
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(1 ,-1 , 2) \)
\( A_{2}(2 , 1 , 2) \)
\( A_{3}(1 , 1 , 4) \)
\( A_{4}(6 ,-3 , 8) \) | 3\sqrt{6} | 0.125 |
The sum of the integer parts of all positive real solutions \( x \) that satisfy \( x^{4}-x^{3}-2 \sqrt{5} x^{2}-7 x^{2}+\sqrt{5} x+3 x+7 \sqrt{5}+17=0 \) is | 5 | 0.625 |
In the diagram, \(\triangle ABC\) is right-angled at \(C\). Point \(D\) is on \(AC\) so that \(\angle ABC = 2 \angle DBC\). If \(DC = 1\) and \(BD = 3\), determine the length of \(AD\). | \frac{9}{7} | 0.5 |
What will the inflation be over two years:
$$
\left((1+0,015)^{\wedge} 2-1\right)^{*} 100 \%=3,0225 \%
$$
What will be the real yield of a bank deposit with an extension for the second year:
$$
(1,07 * 1,07 /(1+0,030225)-1) * 100 \%=11,13 \%
$$ | 11.13\% | 0.5 |
Calculate the line integral
$$
\int_{L}(x-y) d x+d y+z d z
$$
from the point \( M(2,0,4) \) to the point \( N(-2,0,4) \) (where \( y \geq 0 \)) along the curve \( L \), formed by the intersection of the paraboloid \( z=x^{2}+y^{2} \) and the plane \( z=4 \). | 2\pi | 0.625 |
A sculpture in the Guggenheim Museum in New York is shaped like a cube. A bug that has landed on one of the vertices wants to inspect the sculpture as quickly as possible so it can move on to other exhibits (for this, it only needs to reach the opposite vertex of the cube). Which path should it choose? | \sqrt{5} | 0.125 |
Let the function \( f(x) = x - \ln(a x + 2a + 1) + 2 \). If for any \( x \geq -2 \), \( f(x) \geq 0 \) always holds, then the range of the real number \( a \) is \(\quad\). | [0, 1] | 0.75 |
Let the function \( f(x) \) have a derivative \( f'(x) \) on \( \mathbf{R} \), such that for any \( x \in \mathbf{R} \), \( f(x) + f(-x) = x^2 \). In the interval \( (0, +\infty) \), \( f'(x) > x \). If \( f(1+a) - f(1-a) \geq 2a \), then the range of the real number \( a \) is ______. | [0,+\infty) | 0.875 |
In the unit cube \(ABCD-A_1B_1C_1D_1\), points \(E\) and \(F\) are the midpoints of edges \(AB\) and \(BC\) respectively. Find the distance from point \(D\) to the plane \(B_1EF\). | 1 | 0.75 |
An integer with four digits is a multiple of 5. When this integer is divided by 11, 7, and 9, the remainders are 7, 4, and 4 respectively. What is the smallest such integer? | 2020 | 0.75 |
The cross-section of a regular triangular pyramid passes through the midline of the base and is perpendicular to the base. Find the area of the cross-section if the side of the base is 6 and the height of the pyramid is 8. | 9 | 0.125 |
On the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$, different points $A\left(x_{1}, y_{1}\right), B\left(4, \frac{9}{5}\right), C\left(x_{2}, y_{2}\right)$ are such that their distances to the focus $F(4,0)$ form an arithmetic sequence. If the perpendicular bisector of segment $A C$ intersects the $x$-axis at point $T$, find the slope $k$ of the line $BT$. $\qquad$ . | \frac{5}{4} | 0.75 |
On the side $BC$ of an acute-angled triangle $ABC$ ($AB \neq AC$), a semicircle is constructed with $BC$ as its diameter. The semicircle intersects the altitude $AD$ at point $M$. Given $AD = a$, $MD = b$, and $H$ is the orthocenter of the triangle $ABC$, find $AH$. | \frac{a^2 - b^2}{a} | 0.375 |
Six pirates - a captain and five crew members - are sitting around a campfire facing the center. They need to divide a treasure of 180 gold coins. The captain proposes a way to divide the coins (i.e., how many coins each pirate should receive: each pirate must receive a whole non-negative number of coins; different pirates may receive different amounts of coins). After this, the other five pirates vote on the captain's proposal. A pirate will vote "yes" only if he receives more coins than each of his two neighbors. The proposal is accepted if at least three out of the five crew members vote "yes".
What is the maximum number of coins the captain can receive under these rules? | 59 | 0.375 |
A function \( f: A \rightarrow A \) is called idempotent if \( f(f(x)) = f(x) \) for all \( x \in A \). Let \( I_{n} \) be the number of idempotent functions from \(\{1, 2, \ldots, n\}\) to itself. Compute
\[
\sum_{n=1}^{\infty} \frac{I_{n}}{n!}.
\] | e^e - 1 | 0.375 |
In the figure given, there is a point \( P \) within \(\angle M A N\), and it is known that \( \tan \angle M A N = 3 \). The distance from point \( P \) to the line \( A N \) is \( P D = 12 \) and \( A D = 30 \). A line passing through \( P \) intersects \( A N \) and \( A M \) at points \( B \) and \( C \) respectively. Find the minimum area of \(\triangle A B C\). | 624 | 0.625 |
A first-grader has one hundred cards with natural numbers from 1 to 100 written on them, as well as a large supply of "+" and "=" signs. What is the maximum number of correct equations they can form? (Each card can be used no more than once, each equation can have only one "=" sign, and cards cannot be flipped or combined to create new numbers.) | 33 | 0.5 |
Given the set $\{1,2,3,4,5,6,7,8,9,10\}$, determine the number of subsets of this set that contain at least 2 elements, such that the absolute difference between any two elements in each subset is greater than 1. | 133 | 0.625 |
If a complex number \( z \) satisfies \( |z|=1 \), and \( z^2 = a + bi \), where \( a \) and \( b \) are real numbers, then the maximum value of \( a + b \) is ______. | \sqrt{2} | 0.75 |
Find the natural number that has six natural divisors (including one and the number itself), two of which are prime, and the sum of all its natural divisors is equal to 78. | 45 | 0.875 |
A \(4 \times 4\) Sudoku grid is filled with digits so that each column, each row, and each of the four \(2 \times 2\) sub-grids that compose the grid contains all of the digits from 1 to 4.
Find the total number of possible \(4 \times 4\) Sudoku grids. | 288 | 0.625 |
In the diagram below, points $E$ and $F$ are located on sides $AB$ and $BD$ of triangle $\triangle ABD$ such that $AE = AC$ and $CD = FD$. If $\angle ABD = 60^\circ$, determine the measure of angle $\angle ECF$. | 60^\circ | 0.125 |
Find all positive solutions to the system of equations
\[
\left\{
\begin{array}{l}
x_{1} + x_{2} = x_{3}^{2} \\
x_{2} + x_{3} = x_{4}^{2} \\
x_{3} + x_{4} = x_{5}^{2} \\
x_{4} + x_{5} = x_{1}^{2} \\
x_{5} + x_{1} = x_{2}^{2}
\end{array}
\right.
\] | (2,2,2,2,2) | 0.25 |
Julia possesses a deck of 18 cards, numbered from 1 to 18. After shuffling, she distributes the cards face down in 3 rows and 6 columns.
Julia chooses a sum \(X\) and flips two cards. If the sum is \(X\), she removes the pair from the table, but if it is not \(X\), she returns the cards to their original positions. She repeats this process until all pairs with a sum equal to \(X\) are flipped.
a) If she chooses \(X=8\), which pairs will be removed from the table?
b) For which value of \(X\) will all the cards be removed from the table?
c) For a specific \(X\), exactly 2 cards remain on the table. How many possible values of \(X\) are there? | 4 | 0.5 |
Given the equation:
\[ A=\operatorname{tg}\left(\frac{7 \pi}{4}+\frac{1}{2} \arccos \frac{2 a}{b}\right)+\operatorname{tg}\left(\frac{7 \pi}{4}-\frac{1}{2} \arccos \frac{2 a}{b}\right) \]
Find the value of \(A\). | A = -\frac{b}{a} | 0.875 |
Somewhere in the universe, \(n\) students are taking a 10-question math competition. Their collective performance is called laughable if, for some pair of questions, there exist 57 students such that either all of them answered both questions correctly or none of them answered both questions correctly. Compute the smallest \(n\) such that the performance is necessarily laughable. | 253 | 0.25 |
A convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. For the \( V \) vertices, each vertex has \( T \) triangular faces and \( P \) pentagonal faces intersecting. Find the value of \( P + T + V \). | 34 | 0.875 |
Integers \( n \) and \( m \) satisfy the inequalities \( 3n - m < 5 \), \( n + m > 26 \), and \( 3m - 2n < 46 \). What values can \( 2n + m \) take? List all possible options. | 36 | 0.875 |
Fill in the numbers 1, 2, 3, ..., 15, 16 into the 16 squares in the table below, and satisfy the following conditions.
\[
\begin{array}{|c|c|c|c|}
\hline
A & B & C & D \\
\hline
E & F & G & H \\
\hline
J & K & L & M \\
\hline
N & P & Q & R \\
\hline
\end{array}
\]
1. \( A + C + F = 10 \)
2. \( B + H = R \)
3. \( D - C = 13 \)
4. \( E \times M = 126 \)
5. \( F + G = 21 \)
6. \( G \div J = 2 \)
7. \( H \times M = 36 \)
8. \( J \times P = 80 \)
9. \( K - N = Q \)
What is \( L \)? | 6 | 0.375 |
Find all functions \( f \) from the reals to the reals such that
\[ f(f(x) + y) = 2x + f(f(y) - x) \]
for all real \( x \) and \( y \). | f(x) = x + c | 0.75 |
Each of \( A \) and \( B \) is a four-digit palindromic integer, \( C \) is a three-digit palindromic integer, and \( A - B = C \). What are the possible values of \( C \)? [A palindromic integer reads the same 'forwards' and 'backwards'.] | 121 | 0.5 |
Given that point \( O \) is the origin of coordinates, the curves \( C_{1}: x^{2} - y^{2} = 1 \) and \( C_{2}: y^{2} = 2px \) intersect at points \( M \) and \( N \). If the circumcircle of \( \triangle OMN \) passes through the point \( P\left(\frac{7}{2}, 0\right) \), then what is the equation of the curve \( C_{2} \)? | y^2 = \frac{3}{2}x | 0.5 |
Consider the sequence of Fibonacci numbers defined by the recursion $f_{1}=f_{2}=1, f_{n}=f_{n-1}+f_{n-2} \text{ for } n \geq 3$. Suppose that for positive integers $a$ and $b$, the fraction $\frac{a}{b}$ is smaller than one of the fractions $\frac{f_{n}}{f_{n-1}}$ or $\frac{f_{n+1}}{f_{n}}$ and larger than the other. Show that $b \geq f_{n+1}$. | b \geq f_{n + 1} | 0.75 |
ABC is a triangle. The bisector of ∠C meets AB at D. Show that \(CD^2 < CA \cdot CB\). | CD^2 < CA \cdot CB | 0.625 |
In a country, some cities are connected by direct bus routes. The residents of this country consider an odd natural number \( n \) to be unlucky, so there does not exist a cyclic route consisting of exactly \( n \) trips (and not entering any city twice). Nonetheless, a path containing exactly 100 trips (and not passing through any city twice) exists from any city to any other city. For which odd \( n \) is this possible? | n = 101 | 0.25 |
A sealed horizontal cylindrical vessel of length \( l \) is divided into two parts by a movable partition. On one side of the partition, there is 1 mole of oxygen, and on the other side, there is 1 mole of helium and 1 mole of oxygen, with the partition being initially in equilibrium. At a certain moment, the partition becomes permeable to helium but remains impermeable to oxygen. Find the displacement of the partition. The temperatures of the gases are the same and do not change during the process. | \frac{l}{6} | 0.875 |
1. Find the sum: $\cos ^{2} 1^{\circ}+\cos ^{2} 3^{\circ}+\cdots+\cos ^{2} 89^{\circ}$
2. Let $f(x)=\frac{4^{x}}{4^{x}+2}$, find $f\left(\frac{1}{1001}\right)+f\left(\frac{2}{1001}\right)+\cdots+f\left(\frac{1000}{1001}\right)$ | 500 | 0.75 |
In trapezoid \( ABCD \), point \( X \) is taken on the base \( BC \) such that segments \( XA \) and \( XD \) divide the trapezoid into three similar but pairwise unequal, non-isosceles triangles. The side \( AB \) has a length of 5. Find \( XC \cdot BX \). | 25 | 0.375 |
The pages in a book are numbered as follows: the first sheet contains two pages (numbered 1 and 2), the second sheet contains the next two pages (numbered 3 and 4), and so on. A mischievous boy named Petya tore out several consecutive sheets: the first torn-out page has the number 185, and the number of the last torn-out page consists of the same digits but in a different order. How many sheets did Petya tear out? | 167 | 0.75 |
Let \( a \) and \( b \) be real numbers, and there exists a complex number \( z \) such that \( |z| \leq 1 \), and \( z + \bar{z}|z| = a + b \mathrm{i} \). Find the maximum value of \( ab \). | \frac{1}{8} | 0.75 |
Let us denote the product of all natural numbers from 1 to $m$ by $m!$, i.e., $m! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot (m-1) \cdot m$. Find all values of $m$ for which the sum $1! + 2! + 3! + \ldots + m!$ is a perfect square. | m=1, \, m=3 | 0.375 |
Let \([x]\) be the largest integer not greater than \(x\). If \( A = \left\lfloor \frac{2008 \times 80 + 2009 \times 130 + 2010 \times 180}{2008 \times 15 + 2009 \times 25 + 2010 \times 35} \right\rfloor \), find the value of \(A\). | 5 | 0.625 |
Find the possible value of $x + y$ given that $x^3 + 6x^2 + 16x = -15$ and $y^3 + 6y^2 + 16y = -17$. | -4 | 0.875 |
Find the largest natural number \( n \) that satisfies the inequality \( n^{300} < 3^{500} \). | 6 | 0.875 |
Let \(ABCDE\) be a convex pentagon such that \(\angle ABC = \angle ACD = \angle ADE = 90^\circ\) and \(AB = BC = CD = DE = 1\). Compute the length \(AE\). | 2 | 0.375 |
The teacher wrote a two-digit number on the board. Each of the three boys made two statements.
- Andrey: "This number ends in the digit 6" and "This number is divisible by 7."
- Borya: "This number is greater than 26" and "This number ends in the digit 8."
- Sasha: "This number is divisible by 13" and "This number is less than 27."
It is known that each of the boys told the truth once and made a mistake once. What number could have been written on the board? List all possible options. | 91 | 0.875 |
There are 200 candies. What is the minimum number of schoolchildren that these candies can be distributed to so that, no matter how the candies are distributed, there are always at least two schoolchildren who receive the same number of candies (possibly none)? | 21 | 0.5 |
Let \( X = \{1,2, \cdots, 100\} \). For any non-empty subset \( M \) of \( X \), define the characteristic of \( M \), denoted as \( m(M) \), as the sum of the maximum and minimum elements of \( M \). Find the average value of the characteristics of all non-empty subsets of \( X \). | 101 | 0.875 |
Given $m$ and $n$ are non-negative integers, the sets $A=\{1, n\}$ and $B=\{2, 4, m\}$, and the set $C=\{c \mid c=xy, x \in A, y \in B\}$. If $|C|=6$ and the sum of all elements in $C$ is 42, find the value of $m+n$. | 6 | 0.5 |
Given a positive integer \( N \) (written in base 10), define its integer substrings to be integers that are equal to strings of one or more consecutive digits from \( N \), including \( N \) itself. For example, the integer substrings of 3208 are \( 3, 2, 0, 8, 32, 20, 320, 208 \), and 3208. (The substring 08 is omitted from this list because it is the same integer as the substring 8, which is already listed.) What is the greatest integer \( N \) such that no integer substring of \( N \) is a multiple of 9? (Note: 0 is a multiple of 9.) | 88,888,888 | 0.5 |
Let \( O \) be the circumcenter of triangle \( ABC \), and let us draw the reflections of \( O \) with respect to the sides of the triangle. Show that the resulting triangle \( A_1 B_1 C_1 \) is congruent to triangle \( ABC \). Determine the orthocenter of triangle \( A_1 B_1 C_1 \). | O | 0.75 |
Calculate
$$
\operatorname{tg} \frac{\pi}{47} \cdot \operatorname{tg} \frac{2 \pi}{47} + \operatorname{tg} \frac{2 \pi}{47} \cdot \operatorname{tg} \frac{3 \pi}{47} + \ldots + \operatorname{tg} \frac{k \pi}{47} \cdot \operatorname{tg} \frac{(k+1) \pi}{47} + \ldots + \operatorname{tg} \frac{2021 \pi}{47} \cdot \operatorname{tg} \frac{2022 \pi}{47}
$$ | -2021 | 0.5 |
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 \( AOB \) and the trapezoid \( ABH_BH_A \) are equal (\( H_A, H_B \) are the bases of the perpendiculars dropped from points \( A \) and \( B \) to the x-axis; \( O \) is the origin). Find all such functions. Given that \( f(1) = 4 \), find the value of \( f(4) \). | 1 | 0.5 |
Consider the set \( S = \{1, 2, 3, \cdots, 2010, 2011\} \). A subset \( T \) of \( S \) is said to be a \( k \)-element RP-subset if \( T \) has exactly \( k \) elements and every pair of elements of \( T \) are relatively prime. Find the smallest positive integer \( k \) such that every \( k \)-element RP-subset of \( S \) contains at least one prime number. | 16 | 0.375 |
Among the unseen beasts that left tracks on unknown paths, there was a herd of one-headed 34-legged creatures and three-headed Dragons. There are a total of 286 legs and 31 heads in the herd. How many legs does a three-headed Dragon have? | 6 | 0.625 |
As shown in the figure, in a trapezoid $ABCD$, $AD$ is parallel to $BC$, and the ratio of $BC$ to $AD$ is 5:7. Point $F$ is on segment $AD$, and point $E$ is on segment $CD$, satisfying the ratios $AF:FD=4:3$ and $CE:ED=2:3$. If the area of quadrilateral $ABEF$ is 123, what is the area of $ABCD$? | 180 | 0.5 |
Alice encounters one of the brothers in the forest who says: "The true owner of the rattle is telling the truth today." Humpty Dumpty, nearby, claims that the chances the speaker is the true owner of the rattle are exactly 13 out of 14. How did Humpty Dumpty arrive at these numbers? | \frac{13}{14} | 0.625 |
Let \( A \) be the sum of the digits of \( 2012^{2012} \). Let \( B \) be the sum of the digits of \( A \), and \( C \) the sum of the digits of \( B \). Determine \( C \). | 7 | 0.375 |
In triangle $ABC$, the internal angle bisector from $C$ intersects the median from $B$ at point $P$ and side $AB$ at point $T$. Show that
$$
\frac{CP}{PT} - \frac{AC}{BC} = 1
$$ | 1 | 0.75 |
In the parallelogram \(KLMN\), side \(KL\) is equal to 8. A circle tangent to sides \(NK\) and \(NM\) passes through point \(L\) and intersects sides \(KL\) and \(ML\) at points \(C\) and \(D\) respectively. It is known that \(KC : LC = 4 : 5\) and \(LD : MD = 8 : 1\). Find the side \(KN\). | 10 | 0.5 |
In a town of \( n \) people, a governing council is elected as follows: each person casts one vote for some person in the town, and anyone that receives at least five votes is elected to council. Let \( c(n) \) denote the average number of people elected to council if everyone votes randomly. Find \( \lim _{n \rightarrow \infty} \frac{c(n)}{n} \). | 1 - \frac{65}{24e} | 0.25 |
Calculate the mass of the tetrahedron bounded by the planes \(x=0\), \(y=0\), \(z=0\), and \(\frac{x}{10} + \frac{y}{8} + \frac{z}{3} = 1\), if the mass density at each point is given by the function \(\rho = \left(1 + \frac{x}{10} + \frac{y}{8} + \frac{z}{3}\right)^{-6}\). | m = 2 | 0.875 |
Given that \( A \) is a two-digit number and the remainder when \( A^2 \) is divided by 15 is 1, find the number of such \( A \). ( ) | 24 | 0.125 |
Given a natural number \( x = 9^n - 1 \), where \( n \) is a natural number. It is known that \( x \) has exactly three distinct prime divisors, one of which is 11. Find \( x \). | 59048 | 0.625 |
Given the equation \(x + 11y + 11z = n\) where \(n \in \mathbf{Z}_{+}\), there are 16,653 sets of positive integer solutions \((x, y, z)\). Find the minimum value of \(n\). | 2014 | 0.375 |
The points \( S, T, U \) lie on the sides of the triangle \( PQR \) such that \( QS = QU \) and \( RS = RT \). Given that \(\angle TSU = 40^\circ\), what is \(\angle TPU\)?
A) \(60^\circ\)
B) \(70^\circ\)
C) \(80^\circ\)
D) \(90^\circ\)
E) \(100^\circ\) | 100^\circ | 0.125 |
A rectangular table of size \( x \) cm \( \times 80 \) cm is covered with identical sheets of paper measuring 5 cm \( \times 8 \) cm. The first sheet is placed at the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet touches the top-right corner. What is the length \( x \) in centimeters? | 77 | 0.875 |
20 phones are connected with wires such that each wire connects two phones, each pair of phones is connected by at most one wire, and each phone has at most two wires connected to it. We need to paint the wires (each wire entirely with one color) so that the wires connected to each phone are of different colors. What is the minimum number of colors needed for such painting? | 3 | 0.75 |
How many integer solutions does the inequality
$$
|x| + |y| < 1000
$$
have, where \( x \) and \( y \) are integers? | 1998001 | 0.25 |
Find all continuous functions on the entire number line that satisfy the identity \(2 f(x+y) = f(x) f(y)\) and the condition \(f(1) = 10\). | f(x) = 2 \cdot 5^x | 0.75 |
After walking \( \frac{2}{5} \) of the length of a narrow bridge, a pedestrian noticed that a car was approaching the bridge from behind. He then turned back and met the car at the beginning of the bridge. If the pedestrian had continued walking forward, the car would have caught up with him at the end of the bridge. Find the ratio of the car's speed to the pedestrian's speed. | 5 | 0.625 |
In a convex quadrilateral \(ABCD\), angles \(A\) and \(C\) are both equal to \(100^\circ\). Points \(X\) and \(Y\) are chosen on sides \(AB\) and \(BC\) respectively such that \(AX=CY\). It is found that line \(YD\) is parallel to the bisector of angle \(ABC\). Find angle \(AXY\). | 80^\circ | 0.625 |
Find all prime numbers \( p \) and \( q \) such that \( p + q = (p - q)^3 \). | p = 5, \, q = 3 | 0.375 |
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