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|---|---|---|
Calculate the sum
$$
S=\frac{2014}{2 \cdot 5}+\frac{2014}{5 \cdot 8}+\frac{2014}{8 \cdot 11}+\ldots+\frac{2014}{2012 \cdot 2015}
$$
In the answer, indicate the remainder when dividing by 5 the even number closest to the obtained value of $S$.
|
1
| 0.625 |
Natural numbers are placed in the cells of a chessboard such that each number is equal to the arithmetic mean of its neighbors. The sum of the numbers in the corners of the board is 16. Find the number in the cell at $e2$.
|
4
| 0.625 |
Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.
|
16
| 0.5 |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(\frac{\operatorname{arctg} 3 x}{x}\right)^{x+2}
$$
|
9
| 0.875 |
Elsa uses ice blocks to make ice sculptures. One ice block can make a small sculpture, and three ice blocks can make a large sculpture. The leftover fragments from making 2 small sculptures or 1 large sculpture can perfectly form 1 new ice block. Given 30 ice blocks, and ensuring that the number of small sculptures made is greater than the number of large sculptures made, what is the maximum number of large sculptures that can be made?
|
11
| 0.625 |
Let $\{a_{n}\}$ be an integer sequence such that for any $n \in \mathbf{N}^{*}$, the condition \((n-1) a_{n+1} = (n+1) a_{n} - 2 (n-1)\) holds. Additionally, \(2008 \mid a_{2007}\). Find the smallest positive integer \(n \geqslant 2\) such that \(2008 \mid a_{n}\).
|
501
| 0.25 |
In square \( A B C D \), points \( F \) and \( E \) are the midpoints of sides \( A B \) and \( C D \), respectively. The point \( E \) is connected to vertices \( A \) and \( B \), and the point \( F \) is connected to vertices \( C \) and \( D \), as shown in the figure. Determine the area of the rhombus \( F G E H \) formed in the center if the side of the square \( A B = 4 \).
|
4
| 0.75 |
The largest diagonal of a regular hexagonal prism is \(d\) and forms a \(30^\circ\) angle with the lateral edge of the prism. Find the volume of the prism.
|
\frac{9 d^3}{64}
| 0.75 |
What is the smallest natural number that leaves a remainder of 2 when divided by 3, a remainder of 4 when divided by 5, and a remainder of 4 when divided by 7?
|
74
| 0.75 |
First, select $n$ numbers from the set $ \{1, 2, \cdots, 2020\} $. Then, from these $n$ numbers, choose any two numbers $a$ and $b$, such that $a$ does not divide $b$. Find the maximum value of $n$.
|
1010
| 0.5 |
In the triangles \( \triangle ABC \) and \( \triangle AEF \), \( B \) is the midpoint of \( EF \). Given that \( AB = EF = 1 \), \( BC = 6 \), \( CA = \sqrt{33} \), and \( \overrightarrow{AB} \cdot \overrightarrow{AE} + \overrightarrow{AC} \cdot \overrightarrow{AF} = 2 \), find the cosine of the angle between \( \overrightarrow{EF} \) and \( \overrightarrow{BC} \).
|
\frac{2}{3}
| 0.375 |
In $\triangle ABC$, find the integer part of $S=\sqrt{3 \tan \frac{A}{2} \tan \frac{B}{2}+1}+\sqrt{3 \tan \frac{B}{2} \tan \frac{C}{2}+1}+\sqrt{3 \tan \frac{C}{2} \tan \frac{A}{2}+1}$.
|
4
| 0.875 |
In the NBA Finals between the Los Angeles Lakers and the Boston Celtics, the series is played in a best-of-seven format, meaning the first team to win 4 games will be crowned the champion. The games are divided into home and away games. Since the Los Angeles Lakers had a better regular-season record, games 1, 2, 6, and 7 are played in Los Angeles, while games 3-5 are played in Boston. The Lakers eventually win the championship on their home court. How many possible win-loss sequences are there during the series?
|
30
| 0.75 |
Cheburashka and Gena ate a cake. Cheburashka ate twice as slow as Gena, but he started eating one minute earlier. In the end, they ate an equal amount of cake. How long would it take for Cheburashka to eat the entire cake by himself?
|
4
| 0.875 |
If \( P = 2 \sqrt[4]{2007 \cdot 2009 \cdot 2011 \cdot 2013 + 10 \cdot 2010 \cdot 2010 - 9} - 4000 \), find the value of \( P \).
|
20
| 0.875 |
What is the value of the expression \(x^{2000} + x^{1999} + x^{1998} + 1000x^{1000} + 1000x^{999} + 1000x^{998} + 2000x^{3} + 2000x^{2} + 2000x + 3000\) (where \(x\) is a real number), if \(x^{2} + x + 1 = 0\)? Vasya calculated the answer to be 3000. Is Vasya correct?
|
3000
| 0.75 |
ABC is a triangle. Show that the medians BD and CE are perpendicular if and only if \( b^2 + c^2 = 5a^2 \).
|
b^2 + c^2 = 5a^2
| 0.875 |
What could be the minimum length of a cycle in a graph where no single vertex is connected to all others, any two non-adjacent vertices have a common neighbor, and if the number of vertices is denoted by $n$, the sum of the squares of the degrees is $n^2 - n$?
|
5
| 0.375 |
A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is 12 meters. What is the area (in square meters) of the room?
|
18\pi
| 0.5 |
Let \( p \) be an odd prime that satisfies \( p \equiv 1 \pmod{4} \). Compute the value of \( \sum_{k=1}^{p-1} \left\{ \frac{k^2}{p} \right\} \), where \( \{ x \} = x - \lfloor x \rfloor \) and \(\lfloor x \rfloor\) is the greatest integer less than or equal to \( x \).
|
\frac{p-1}{2}
| 0.625 |
Simplify the expression
$$
\frac{\sin 11^{\circ} \cos 15^{\circ}+\sin 15^{\circ} \cos 11^{\circ}}{\sin 18^{\circ} \cos 12^{\circ}+\sin 12^{\circ} \cos 18^{\circ}}
$$
|
2 \sin 26^{\circ}
| 0.75 |
Determine the maximum value of \( m^{2} + n^{2} \). Here, \( m \) and \( n \) are integers, and \( m, n \in \{1, 2, \cdots, 1981\} \), such that \((n^{2} - mn - m^{2})^{2} = 1\).
|
3524578
| 0.625 |
A single burger is not enough to satisfy a guy's hunger. The five guys go to Five Guys' Restaurant, which has 20 different meals on the menu. Each meal costs a different integer dollar amount between $1 and $20. The five guys have \$20 to split between them, and they want to use all the money to order five different meals. How many sets of five meals can the guys choose?
|
7
| 0.75 |
When choosing, without replacement, 4 out of 30 labelled balls that are marked from 1 to 30, how many combinations are possible? Find the value of \( r \).
|
27405
| 0.75 |
The set $\{[x]+[2x]+[3x] \mid x \in \mathbf{R}\} \bigcap \{1, 2, \cdots, 100\}$ contains how many elements, where $[x]$ represents the greatest integer less than or equal to $x$.
|
67
| 0.5 |
In the triangle \(ABC\), \(AB = 8\), \(BC = 7\), and \(CA = 6\). Let \(E\) be the point on \(BC\) such that \(\angle BAE = 3 \angle EAC\). Find \(4AE^2\).
|
135
| 0.375 |
Calculate the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty}\left(\frac{3 n^{2}-5 n}{3 n^{2}-5 n+7}\right)^{n+1}
\]
|
1
| 0.875 |
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2029\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution. Find the minimum possible value of \(c\).
|
1015
| 0.375 |
Let \( T = \left\{ |9^k| \mid k \in \mathbf{Z}, 0 \leqslant k \leqslant 4000 \right\} \). It is known that \( 9^{4000} \) is a 3817-digit number, and its first digit is 9. How many numbers in \( T \) have 9 as their first digit?
|
184
| 0.125 |
\[
\frac{b^{-1 / 6} \cdot \sqrt{a^{3} b} \cdot \sqrt[3]{a^{3} b}-\sqrt{a^{3} b^{2}} \cdot \sqrt[3]{b^{2}}}{\left(2 a^{2}-b^{2}-a b\right) \cdot \sqrt[6]{a^{9} b^{4}}} : \left( \frac{3 a^{3}}{2 a^{2}-a b-b^{2}} - \frac{a b}{a-b} \right)
\]
|
\frac{1}{a(3a+b)}
| 0.75 |
For which natural numbers \( n \) is the number \( 3^{2n+1} - 2^{2n+1} - 6^n \) composite?
|
n \geq 2
| 0.875 |
Given positive integers \( a, b, c, \) and \( d \) such that \( a > b > c > d \) and \( a + b + c + d = 2004 \), as well as \( a^2 - b^2 + c^2 - d^2 = 2004 \), what is the minimum value of \( a \)?
|
503
| 0.625 |
From the set \( M = \{1, 2, \cdots, 2008\} \) of the first 2008 positive integers, a \( k \)-element subset \( A \) is chosen such that the sum of any two numbers in \( A \) cannot be divisible by the difference of those two numbers. What is the maximum value of \( k \)?
|
670
| 0.5 |
Simplify the following expression:
$$
\frac{\frac{a+b}{1-ab} + \frac{c-a}{1+ca}}{1-\frac{a+b}{1-ab} \cdot \frac{c-a}{1+ca}}
$$
Explain the result using trigonometric identities.
|
\frac{b+c}{1-bc}
| 0.875 |
Suppose \(\overline{a_{1} a_{2} \ldots a_{2009}}\) is a 2009-digit integer such that for each \(i=1,2, \ldots, 2007\), the 2-digit integer \(\overline{a_{i} a_{i+1}}\) contains 3 distinct prime factors. Find \(a_{2008}\).
(Note: \(xyz \ldots\) denotes an integer whose digits are \(x, y, z, \ldots\))
|
6
| 0.375 |
For which (real) values of the variable $x$ are the following equalities valid:
a) $\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=\sqrt{2}$,
b) $\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=1$,
c) $\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=2$.
(The square root without a sign always means the non-negative square root!)
|
\frac{3}{2}
| 0.375 |
Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{n \sqrt[6]{n}+\sqrt[3]{n^{10}+1}}{(n+\sqrt[4]{n}) \sqrt[3]{n^{3}-1}}$$
|
\infty
| 0.875 |
In a rectangular coordinate system, a circle centered at the point $(1,0)$ with radius $r$ intersects the parabola $y^2 = x$ at four points $A$, $B$, $C$, and $D$. If the intersection point $F$ of diagonals $AC$ and $BD$ is exactly the focus of the parabola, determine $r$.
|
\frac{\sqrt{15}}{4}
| 0.375 |
Place the numbers $1-8$ on the eight vertices of a cube, then at the midpoint of each edge write the average of the two numbers at the endpoints of that edge. If the numbers at the four midpoints of the top face and the four midpoints of the bottom face are all integers, how many of the numbers at the other four midpoints are not integers?
|
4
| 0.625 |
Zhenya had 9 cards with numbers from 1 to 9. He lost the card with the number 7. Is it possible to arrange the remaining 8 cards in a row so that any two adjacent cards form a number divisible by 7?
|
\text{No}
| 0.125 |
The diameter of the red circle is 6 decimeters. What should be the minimum diameter (to the nearest $\frac{1}{2}$ decimeter) of five discs such that they can completely cover the circle?
|
4
| 0.375 |
Given the coordinates of the vertices of triangle $\triangle O A B$ are $O(0,0), A(4,4 \sqrt{3}), B(8,0)$, with its incircle center being $I$. Let the circle $C$ pass through points $A$ and $B$, and intersect the circle $I$ at points $P$ and $Q$. If the tangents drawn to the two circles at points $P$ and $Q$ are perpendicular, then the radius of circle $C$ is $\qquad$ .
|
2\sqrt{7}
| 0.75 |
Petya is creating a password for his smartphone. The password consists of 4 decimal digits. Petya wants the password to not contain the digit 7, and it must have at least two (or more) identical digits. How many ways can Petya create such a password?
|
3537
| 0.625 |
How many ordered pairs of integers \((a, b)\) satisfy all of the following inequalities?
\[
\begin{array}{l}
a^{2}+b^{2}<16 \\
a^{2}+b^{2}<8a \\
a^{2}+b^{2}<8b
\end{array}
\]
|
6
| 0.875 |
In three-dimensional space, let \( S \) be the region of points \( (x, y, z) \) satisfying \( -1 \leq z \leq 1 \). Let \( S_{1}, S_{2}, \ldots, S_{2022} \) be 2022 independent random rotations of \( S \) about the origin \( (0,0,0) \). The expected volume of the region \( S_{1} \cap S_{2} \cap \cdots \cap S_{2022} \) can be expressed as \( \frac{a \pi}{b} \), for relatively prime positive integers \( a \) and \( b \). Compute \( 100a+b \).
|
271619
| 0.25 |
On a table, there are three cones standing on their bases, touching each other. The radii of their bases are 10, 15, and 15. A truncated cone is placed on the table with its smaller base down, which has a common slant height with each of the other cones. Find the area of the smaller base of the truncated cone.
|
4\pi
| 0.375 |
Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the point $X$ is chosen on the edge $A_{1} D_{1}$ and the point $Y$ is chosen on the edge $B C$. It is known that $A_{1} X=5$, $B Y=3$, and $B_{1} C_{1}=14$. The plane $C_{1} X Y$ intersects the ray $D A$ at point $Z$. Find $D Z$.
|
20
| 0.875 |
For which natural numbers \( n \) is the sum \( 5^n + n^5 \) divisible by 13? What is the smallest \( n \) that satisfies this condition?
|
12
| 0.375 |
A basket of apples is divided into two parts, A and B. The ratio of the number of apples in A to the number of apples in B is $27: 25$. Part A has more apples than Part B. If at least 4 apples are taken from A and added to B, then Part B will have more apples than Part A. How many apples are in the basket?
|
156
| 0.625 |
For a positive integer \( n \), let \( p(n) \) denote the product of the positive integer factors of \( n \). Determine the number of factors \( n \) of 2310 for which \( p(n) \) is a perfect square.
|
27
| 0.375 |
Given that \( f(x) \) is a function defined on \( \mathbf{R} \) with \( f(1) = 1 \) and for any \( x \in \mathbf{R} \), \( f(x+5) \geq f(x) + 5 \) and \( f(x+1) \leq f(x) + 1 \). If \( g(x) = f(x) + 1 - x \), find \( g(2002) \).
|
1
| 0.75 |
In triangle \( FGH \), angle \( G \) is right, \( FG = 8 \), \( GH = 2 \). Point \( D \) lies on side \( FH \), and \( A \) and \( B \) are the centroids of triangles \( FGD \) and \( DGH \) respectively. Find the area of triangle \( GAB \).
|
\frac{16}{9}
| 0.875 |
If the sum of the digits of a natural number \( n \) is subtracted from \( n \), the result is 2016. Find the sum of all such natural numbers \( n \).
|
20245
| 0.875 |
A regular triangular prism \( A B C A_{1} B_{1} C_{1} \) is inscribed in a sphere. The base of the prism is \( A B C \), and the lateral edges are \( A A_{1}, B B_{1}, C C_{1} \). The segment \( C D \) is a diameter of this sphere, and the point \( K \) is the midpoint of the edge \( A A_{1} \). Find the volume of the prism, given that \( C K = 2 \sqrt{6} \) and \( D K = 4 \).
|
36
| 0.125 |
Given \( a \) and \( b \) are two orthogonal unit vectors, and \( c \cdot a = c \cdot b = 1 \), find the minimum value of \( \left|c+t a+\frac{1}{t} b\right| \) for any positive real number \( t \).
|
2\sqrt{2}
| 0.75 |
The librarian of a physics and mathematics high school noticed that if the number of geometry textbooks in the school library is increased by several (whole number) times and the obtained number is added to the number of algebra textbooks, the result is 2015. If the number of algebra textbooks is increased by the same number of times and the obtained number is added to the number of geometry textbooks, the result is 1580. How many algebra textbooks are in the library?
|
287
| 0.625 |
Today is January 30th, so we first write down 130. The rule for writing subsequent numbers is: if the last number written is even, divide it by 2 and then add 2; if the last number written is odd, multiply it by 2 and then subtract 2. Thus the sequence obtained is: $130, 67, 132, 68, \cdots$. What is the 2016th number in this sequence?
|
6
| 0.375 |
An isosceles trapezoid \(ABCD\) is inscribed in a circle with radius \(2 \sqrt{7}\), whereby its base \(AD\) is the diameter, and angle \(\angle BAD\) is \(60^\circ\). Chord \(CE\) intersects the diameter \(AD\) at point \(P\) such that the ratio \(AP:PD = 1:3\). Find the area of triangle \(BPE\).
|
3\sqrt{3}
| 0.625 |
The first term of an arithmetic sequence is 9, and the 8th term is 12. How many of the first 2015 terms of this sequence are multiples of 3?
|
288
| 0.75 |
Given \( 0 < a < b \), lines \( l \) and \( m \) pass through two fixed points \( A(a, 0) \) and \( B(b, 0) \), respectively, intersecting the parabola \( y^2 = x \) at four distinct points. When these four points lie on a common circle, determine the locus of the intersection point \( P \) of lines \( l \) and \( m \).
|
x = \frac{a + b}{2}
| 0.375 |
Masha has an integer multiple of toys compared to Lena, and Lena has the same multiple of toys compared to Katya. Masha gave 3 toys to Lena, and Katya gave 2 toys to Lena. After that, the number of toys each girl had formed an arithmetic progression. How many toys did each girl originally have? Provide the total number of toys the girls had initially.
|
105
| 0.875 |
Given a quadratic polynomial \( f(x) \) such that the equation \( (f(x))^3 - 4f(x) = 0 \) has exactly three solutions. How many solutions does the equation \( (f(x))^2 = 1 \) have?
|
2
| 0.875 |
Calculate the area of the figure enclosed by the lines given by the equations:
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=6(t-\sin t) \\
y=6(1-\cos t)
\end{array}\right. \\
& y=6(0<x<12 \pi, y \geq 6)
\end{aligned}
$$
|
18\pi + 72
| 0.25 |
A girl, who is an acquaintance of a young man sitting by the window of a moving tram, was walking towards the tram. Eight seconds after she passed by the window, the young man got off the tram and started following her. How much time passed from this moment until he caught up with the girl? The speed of the young man is twice the speed of the girl and one-fifth of the speed of the tram.
|
88 \text{ s}
| 0.625 |
Several oranges (not necessarily of equal mass) were picked from a tree. On weighing them, it turned out that the mass of any three oranges taken together is less than 5% of the total mass of the remaining oranges. What is the minimum number of oranges that could have been picked?
|
64
| 0.5 |
In triangle \(ABC\), \(AB = 13\) and \(BC = 15\). On side \(AC\), point \(D\) is chosen such that \(AD = 5\) and \(CD = 9\). The angle bisector of the angle supplementary to \(\angle A\) intersects line \(BD\) at point \(E\). Find \(DE\).
|
7.5
| 0.125 |
A regular triangular pyramid \(SABC\) is given, with the edge of its base equal to 1. Medians of the lateral faces are drawn from the vertices \(A\) and \(B\) of the base \(ABC\), and these medians do not intersect. It is known that the edges of a certain cube lie on the lines containing these medians. Find the length of the lateral edge of the pyramid.
|
\frac{\sqrt{6}}{2}
| 0.5 |
In how many non-empty subsets of the set $\{1, 2, 3, \ldots, 10\}$ are there no two consecutive numbers?
|
143
| 0.5 |
An acute angle at the base of a trapezoid inscribed in a circle with a radius of 13 is $30^{\circ}$. The leg (non-parallel side) of the trapezoid is 10. Find the median (the midsegment) of the trapezoid.
|
12
| 0.125 |
Agronomist Bilbo noticed that if the length of his rectangular field were increased by 20 meters, then the perimeter of the field would be twice as much. However, if the width of the field were twice as much, then the perimeter of the field would be 18 meters more. What is the area of the field?
|
99
| 0.875 |
Three cones are standing on their bases on a table, touching each other. The radii of their bases are $2r$, $3r$, and $10r$. A truncated cone with the smaller base down is placed on the table, sharing a slant height with each of the other cones. Find $r$ if the radius of the smaller base of the truncated cone is 15.
|
29
| 0.125 |
A and B are taking turns shooting with a six-shot revolver that has only one bullet. They randomly spin the cylinder before each shot. A starts the game. Find the probability that the gun will fire while A is holding it.
|
\frac{6}{11}
| 0.875 |
Yura has an unusual clock with several minute hands moving in different directions. Yura calculated that over the course of an hour, the minute hands coincided exactly 54 times. What is the maximum number of minute hands that Yura's clock could have?
|
28
| 0.375 |
Given a sequence \(\left\{a_{n}\right\}\) of nonzero terms with the sum of the first \(k\) terms denoted by \(S_{k}\), where
\[ S_{k} = \frac{1}{2} a_{k} a_{k+1} \quad (k \in \mathbb{N}_{+}), \]
and \(a_{1} = 1\).
(1) Find the general term formula of the sequence \(\left\{a_{n}\right\}\).
(2) For any given positive integer \(n (n \geqslant 2)\), the sequence \(\left\{b_{n}\right\}\) satisfies:
\[ \frac{b_{k+1}}{b_{k}} = \frac{k-n}{a_{k+1}} \quad (k=1,2,3, \cdots, n), \quad b_{1}=1. \]
Find the sum \(b_{1}+b_{2}+\cdots+b_{n}\).
|
\frac{1}{n}
| 0.625 |
There are two squares. The side length of the larger square is 4 decimeters longer than that of the smaller square. The area of the larger square is 80 square decimeters greater than that of the smaller square. The sum of the areas of the larger and smaller squares is ____ square decimeters.
|
208
| 0.875 |
Cheburashka spent his money to buy as many mirrors from Galya's store as Gena bought from Shapoklyak's store. If Gena were buying from Galya, he would have 27 mirrors, and if Cheburashka were buying from Shapoklyak, he would have 3 mirrors. How many mirrors would Gena and Cheburashka have bought together if Galya and Shapoklyak agreed to set a price for the mirrors equal to the average of their current prices? (The average of two numbers is half of their sum, for example, for the numbers 22 and 28, the average is 25.)
|
18
| 0.75 |
Compute the lengths of the arcs of curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\left(t^{2}-2\right) \sin t+2 t \cos t \\
y=\left(2-t^{2}\right) \cos t+2 t \sin t
\end{array}\right. \\
& 0 \leq t \leq \pi
\end{aligned}
$$
|
\frac{\pi^3}{3}
| 0.75 |
In a particular group of people, some always tell the truth, the rest always lie. There are 2016 in the group. One day, the group is sitting in a circle. Each person in the group says, "Both the person on my left and the person on my right are liars."
What is the difference between the largest and smallest number of people who could be telling the truth?
|
336
| 0.375 |
There are $n$ people standing in a circle, including two individuals $A$ and $B$. How many ways are there to arrange them so that there are exactly $r$ people standing between $A$ and $B$ $\left(r<\frac{n}{2} - 1\right)$?
|
2(n-2)!
| 0.375 |
In an isosceles trapezoid \(ABCD\), the angle bisectors of angles \(B\) and \(C\) intersect at the base \(AD\). Given that \(AB=50\) and \(BC=128\), find the area of the trapezoid.
|
5472
| 0.125 |
Find the maximum value of the expression \(\cos(x+y)\) given that \(\cos x - \cos y = \frac{1}{4}\).
|
\frac{31}{32}
| 0.25 |
Let " $\sum$ " denote the cyclic sum. If $a, b, c$ are given distinct real numbers, then
$$
f(x)=\sum \frac{a^{2}(x-b)(x-c)}{(a-b)(a-c)}
$$
Simplify the expression to obtain...
|
x^2
| 0.5 |
Given a triangle \( \triangle KLM \), point \( A \) lies on the extension of \( LK \). Construct a rectangle \( ABCD \) such that points \( B \), \( C \), and \( D \) lie on the lines containing \( KM \), \( KL \), and \( LM \), respectively.
|
ABCD
| 0.625 |
$7.63 \log _{2} 3 \cdot \log _{3} 4 \cdot \log _{4} 5 \ldots \log _{n}(n+1)=10, n \in \mathbb{N}$.
|
2
| 0.625 |
Given complex numbers \(z_{1} = 2 + 3i\) and \(z_{2} = 5 - 7i\), find:
a) \(z_{1} + z_{2}\);
b) \(z_{1} - z_{2}\);
c) \(z_{1} \cdot z_{2}\).
|
31 + i
| 0.125 |
What is the minimum number of vertices in a graph that contains no cycle of length less than 6 and where every vertex has a degree of 3?
|
14
| 0.375 |
In a regular 1000-sided polygon, all the diagonals are drawn. What is the maximum number of diagonals that can be chosen so that among any three chosen diagonals, at least two of them have the same length?
|
2000
| 0.625 |
Find the minimum value of the function
$$
f(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-102)^{2}
$$
If you obtain a non-integer number, round the result to the nearest whole number.
|
46852
| 0.125 |
Let set \( A = \{a \mid a = 3k + 2, k \leqslant 2000, k \in \mathbf{N}_{+}\} \) and set \( B = \{b \mid b = 4k - 1, k \leqslant 2000, k \in \mathbf{N}_{+}\} \). How many elements are in \( A \cap B \)?
|
500
| 0.875 |
Given the equation with respect to \( x \)
\[
x^{2} - 34x + 34k - 1 = 0
\]
that has at least one positive integer root, find the values of the positive integer \( k \) that satisfy this condition.
|
1
| 0.25 |
A and B are jogging on a 300-meter track. They start at the same time and in the same direction, with A starting behind B. A overtakes B for the first time after 12 minutes, and for the second time after 32 minutes. Assuming both maintain constant speeds, how many meters behind B did A start?
|
180 \text{ meters}
| 0.75 |
The height of a cone is $h$. The development of the lateral surface of this cone is a sector with a central angle of $120^\circ$. Calculate the volume of the cone.
|
\frac{\pi h^3}{24}
| 0.875 |
People are standing in a circle - there are liars, who always lie, and knights, who always tell the truth. Each of them said that among the people standing next to them, there is an equal number of liars and knights. How many people are there in total if there are 48 knights?
|
72
| 0.25 |
The numbers \( x \) and \( y \) are such that the equalities \( \operatorname{ctg} x - \operatorname{ctg} y = 2 \) and \( 5 \sin (2x - 2y) = \sin 2x \sin 2y \) hold. Find \( \operatorname{tg} x \operatorname{tg} y \).
|
-\frac{6}{5}
| 0.875 |
Find the smallest positive integer \( b \) such that \( 1111_b \) (1111 in base \( b \)) is a perfect square. If no such \( b \) exists, write "No solution".
|
7
| 0.5 |
We placed a cube on the center of a potter's wheel, with a natural number written on each of its faces. Just before we spun the wheel, from our point of view, we could see three faces of the cube with numbers totaling 42. After rotating the potter's wheel by $90^{\circ}$, we observed three faces with numbers totaling 34, and after another $90^{\circ}$ rotation, we still saw three numbers totaling 53.
1. Determine the sum of the three numbers we will see from our point of view if the wheel is rotated another $90^{\circ}$.
2. The cube always rested on the face with the number 6. Determine the maximum possible sum of all six numbers on the cube.
|
100
| 0.75 |
Find all ordered 4-tuples of integers \((a, b, c, d)\) (not necessarily distinct) satisfying the following system of equations:
\[
\begin{aligned}
a^2 - b^2 - c^2 - d^2 &= c - b - 2, \\
2ab &= a - d - 32, \\
2ac &= 28 - a - d, \\
2ad &= b + c + 31.
\end{aligned}
\]
|
(5, -3, 2, 3)
| 0.375 |
Let's call a number "small" if it is a 10-digit number and there is no smaller 10-digit number with the same sum of digits. How many small numbers exist?
|
90
| 0.5 |
Three marksmen shoot at a target independently of each other. The probability of hitting the target for the first marksman is 0.6, for the second marksman is 0.7, and for the third marksman is 0.75. Find the probability that at least one of them hits the target if each marksman makes one shot.
|
0.97
| 0.875 |
Let \( P \) be a moving point on the ellipse \(\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1\) in the first quadrant. From point \( P \), two tangents \( PA \) and \( PB \) are drawn to the circle \( x^{2} + y^{2} = 9 \), with points of tangency \( A \) and \( B \) respectively. The line \( AB \) intersects the \( x \)-axis and \( y \)-axis at points \( M \) and \( N \) respectively. Find the minimum area of \( \triangle MON \).
|
\frac{27}{4}
| 0.875 |
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