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Let \( n \) be an integer. Determine the largest possible constant \( C \) such that for all \( a_1, \ldots, a_n \geq 0 \), we have \(\sum a_i^2 \geq C \sum_{i < j} a_i a_j\).
|
\frac{2}{n-1}
| 0.875 |
Points \(M\) and \(N\) lie on edges \(BC\) and \(AA_1\) of the parallelepiped \(ABCD A_1 B_1 C_1 D_1\). Construct the intersection point of line \(MN\) with the base plane \(A_1 B_1 C_1 D_1\).
|
P
| 0.125 |
In five pots standing in a row, Rabbit poured three kilograms of honey (not necessarily into each pot and not necessarily in equal amounts). Winnie the Pooh can take any two pots standing next to each other. What is the maximum amount of honey Winnie the Pooh will be able to eat for sure?
|
1 \text{ kg}
| 0.5 |
Determine the coordinates of the focus of the parabola $y=-\frac{1}{6} x^{2}$ and write the equation of its directrix.
|
y = 1.5
| 0.25 |
Find the sum of the first 10 elements that appear both in the arithmetic progression $\{4, 7, 10, 13, \ldots\}$ and the geometric progression $\{10, 20, 40, 80, \ldots\}$. (10 points)
|
3495250
| 0.125 |
At the namesake festival, 45 Alexanders, 122 Borises, 27 Vasily, and several Gennady attended. At the beginning of the festival, all of them lined up so that no two people with the same name stood next to each other. What is the minimum number of Gennadys that could have attended the festival?
|
49
| 0.5 |
Given that \( m \) and \( n \) are positive integers, with \( n \) being an odd number, find the greatest common divisor of \( 2^m + 1 \) and \( 2^n - 1 \).
|
1
| 0.75 |
Find the area of the figure defined on the coordinate plane by the inequalities \( |x| - 1 \leq y \leq \sqrt{1 - x^2} \).
|
\frac{\pi}{2} + 1
| 0.125 |
It is known that for three consecutive natural values of the argument, the quadratic function \( f(x) \) takes the values 13, 13, and 35 respectively. Find the smallest possible value of \( f(x) \).
|
\frac{41}{4}
| 0.375 |
The legs of a right triangle are 3 and 4. Find the distance between the centers of the circumscribed and inscribed circles.
|
\frac{\sqrt{5}}{2}
| 0.875 |
Compare the numbers \( x = 2 \cdot 20212021 \cdot 1011 \cdot 202320232023 \) and \( y = 43 \cdot 47 \cdot 20232023 \cdot 202220222022 \).
|
x = y
| 0.375 |
a) How many zeros are at the end of the number \( A = 2^{5} \times 3^{7} \times 5^{7} \times 11^{3} \) ?
b) How many zeros are at the end of the number \( B = 1 \times 2 \times 3 \times 4 \times \cdots \times 137 \) ?
|
33
| 0.75 |
Find the smallest possible value of \(x+y\) where \(x, y \geq 1\) and \(x\) and \(y\) are integers that satisfy \(x^{2}-29y^{2}=1\).
|
11621
| 0.25 |
The diagram shows a sector \( OAB \) of a circle, center \( O \) and radius \( 8 \text{ cm} \), in which \( \angle AOB = 120^\circ \). Another circle of radius \( r \text{ cm} \) is to be drawn through the points \( O \), \( A \), and \( B \). Find the value of \( r \).
|
8
| 0.625 |
Will the equation \(x^{2019}+2 x^{2018}+3 x^{2017}+\cdots+2019 x+2020=0\) have integer roots?
|
\text{No}
| 0.875 |
For what smallest natural number \( a \) are there exactly 50 perfect squares in the numerical interval \( (a, 3a) \)?
|
4486
| 0.125 |
In triangle \( \triangle ABC \), if \( \tan A \tan B = \tan A \tan C + \tan C \tan B \), then \( \frac{a^2 + b^2}{c^2} = \) ?
|
3
| 0.75 |
Let \(\alpha, \beta\) be the roots of \(x^{2}+bx-2=0\). If \(\alpha>1\) and \(\beta<-1\), and \(b\) is an integer, find the value of \(b\).
|
0
| 0.875 |
In the village where Glafira lives, there is a small pond that is filled by springs at the bottom. Glafira discovered that a herd of 17 cows completely drank this pond in 3 days. After some time, the springs refilled the pond, and then 2 cows drank it in 30 days. How many days will it take for one cow to drink this pond?
|
75
| 0.875 |
Find all values of \( x \) for which the minimum of the numbers \( 8 - x^{2} \) and \( \operatorname{ctg} x \) is not less than -1. In the answer, record the total length of the found intervals on the number line, rounding it to the nearest hundredth if necessary.
|
4.57
| 0.25 |
Let \( q(n) \) be the sum of the digits of the natural number \( n \). Determine the value of
$$
q\left(q\left(q\left(2000^{2000}\right)\right)\right)
$$
|
4
| 0.5 |
If \( A = 10^{9} - 987654321 \) and \( B = \frac{123456789 + 1}{10} \), what is the value of \( \sqrt{AB} \)?
|
12345679
| 0.875 |
In a $5 \times 5$ grid, place an "L" shape composed of 4 small squares, which can be rotated and flipped. No two "L" shapes can overlap. What is the maximum number of "L" shapes that can be placed?
|
6
| 0.25 |
What is the minimum area of a triangle that contains a unit square?
|
2
| 0.5 |
There are 200 matches. How many ways are there to form, using all the matches, a square and (separately) an equilateral triangle? (Different ways are distinguished by the sizes of the square and the triangle).
|
16
| 0.5 |
Let \( x < y \) be positive real numbers such that
\[ \sqrt{x} + \sqrt{y} = 4 \]
and
\[ \sqrt{x+2} + \sqrt{y+2} = 5. \]
Compute \( x \).
|
\frac{49}{36}
| 0.25 |
In a circle with radius \( R \), two chords \( AB \) and \( AC \) are drawn. A point \( M \) is taken on \( AB \) or its extension such that the distance from \( M \) to the line \( AC \) is equal to \( |AC| \). Similarly, a point \( N \) is taken on \( AC \) or its extension such that the distance from \( N \) to the line \( AB \) is equal to \( |AB| \). Find the distance \( |MN| \).
|
2R
| 0.5 |
Given \( a \in \mathbf{Z} \), and \( x^{6} - 33x + 20 \) is divisible by \( x^{2} - x + a \), determine the value of \( a \).
|
4
| 0.875 |
In an isosceles triangle, the side is divided by the point of tangency of the inscribed circle in the ratio 7:5 (starting from the vertex). Find the ratio of the side to the base.
|
\frac{6}{5}
| 0.75 |
There are 15 stones placed in a line. In how many ways can you mark 5 of these stones so that there are an odd number of stones between any two of the stones you marked?
|
77
| 0.75 |
Given that $A$, $B$, and $C$ are points on the circle $\odot O$, and
$$
\overrightarrow{A O}=\frac{1}{2}(\overrightarrow{A B}+\overrightarrow{A C}),
$$
find the dot product $\overrightarrow{A B} \cdot \overrightarrow{A C}$.
|
0
| 0.375 |
Find the sum of the digits in the decimal representation of the integer part of the number $\sqrt{\underbrace{11 \ldots 11}_{2018} \underbrace{55 \ldots 55}_{2017} 6}$.
|
6055
| 0.375 |
From the vertex of the obtuse angle \( A \) of triangle \( ABC \), a perpendicular \( AD \) is drawn. From point \( D \), a circle is drawn with radius equal to \( AD \), intersecting the sides \( AB \) and \( AC \) at points \( M \) and \( N \), respectively. Find the side \( AC \), given that \( AB = c \), \( AM = m \), and \( AN = n \).
|
\frac{mc}{n}
| 0.5 |
A circle with a radius of 15 is tangent to two adjacent sides \( AB \) and \( AD \) of square \( ABCD \). On the other two sides, the circle intercepts segments of 6 and 3 cm from the vertices, respectively. Find the length of the segment that the circle intercepts from vertex \( B \) to the point of tangency.
|
12
| 0.75 |
Given a line \( l \) tangent to the unit circle \( S \) at point \( P \), point \( A \) lies on the same side of \( l \) as circle \( S \), and the distance from \( A \) to \( l \) is \( h \) where \( h > 2 \). From point \( A \), two tangents to \( S \) are drawn, intersecting \( l \) at points \( B \) and \( C \). Find the product of the lengths of line segments \( PB \) and \( PC \).
|
\frac{h}{h-2}
| 0.25 |
Under one of the squares of an $8 \times 8$ board, there is a treasure buried. Under each of the remaining squares, there is a tag indicating the minimum number of steps it takes to reach the treasure from that square (one step consists of moving to an adjacent square along one of its sides). What is the minimum number of squares that need to be dug to ensure retrieval of the treasure?
|
3
| 0.5 |
They multiplied several natural numbers and obtained 224, with the smallest number being exactly half of the largest number. How many numbers were multiplied?
|
3
| 0.5 |
Fill 2 $a$'s and 2 $b$'s into the 16 squares as shown in the diagram, with each square containing at most one letter. If the same letters cannot be in the same row or column, how many different ways can this be done? (Answer with a number.)
|
3960
| 0.125 |
A certain store sells a brand of socks for 4.86 yuan per pair. Now they are running a promotion: "buy five, get one free". What is the actual price per pair of socks now?
|
4.05
| 0.875 |
Given a cyclic quadrilateral \(ABCD\) with side lengths \(AB = 2\), \(BC = 6\), and \(CD = DA = 4\), find the area of the quadrilateral \(ABCD\).
|
8\sqrt{3}
| 0.875 |
Let \( D \) be a point inside the acute triangle \( \triangle ABC \) such that
\[
\angle ADB = \angle ACB + 90^\circ
\]
and \( AC \cdot BD = AD \cdot BC \). Find the value of \(\frac{AB \cdot CD}{AC \cdot BD}\).
|
\sqrt{2}
| 0.375 |
Vasya thought of three natural numbers with a sum of 1003. After calculating their product, Vasya noticed that it ends in $N$ zeros. What is the maximum possible value of $N$?
|
7
| 0.625 |
A set consisting of a finite number of points in the plane has the following property: for any two points A and B in this set, there exists a point C in this set such that triangle ABC is equilateral. How many points can such a set contain?
|
3
| 0.875 |
Given the quadratic function $y=f(x)=x^{2}+bx+c$ whose graph passes through the point $(1,13)$, and the function $y=f\left(x-\frac{1}{2}\right)$ is an even function:
(1) Find the explicit expression for $f(x)$;
(2) Determine whether there exists a point on the graph of $y=f(x)$ whose x-coordinate is a positive integer and y-coordinate is a perfect square. If such a point exists, find its coordinates; if not, provide an explanation.
|
(10, 121)
| 0.5 |
Let \( A \) be the sum of the digits of the decimal number \( 4444^{4444} \), and let \( B \) be the sum of the digits of \( A \). Find the sum of the digits of \( B \), where all numbers mentioned are in decimal form.
|
7
| 0.625 |
Misha wrote on the board 2004 pluses and 2005 minuses in some order. From time to time, Yura comes to the board, erases any two signs, and writes one in their place. If he erases two identical signs, he writes a plus; if the signs are different, he writes a minus. After several such actions, only one sign remains on the board. What is the final sign?
|
-
| 0.875 |
Triangle \(A B C\) has a right angle at \(C\), and \(D\) is the foot of the altitude from \(C\) to \(A B\). Points \(L\), \(M\), and \(N\) are the midpoints of segments \(A D, D C\), and \(C A\), respectively. If \(C L=7\) and \(B M=12\), compute \(B N^{2}\).
|
193
| 0.625 |
An n-digit number has the property that if you cyclically permute its digits, the result is always divisible by 1989. What is the smallest possible value of n? What is the smallest such number?
|
48
| 0.875 |
Gena went to the shooting range with his dad. They agreed that Gena makes 5 shots and for each hit, he gets the right to make 2 more shots. In total, Gena made 17 shots. How many times did he hit the target?
|
6
| 0.375 |
Determine the largest square number that is not divisible by 100 and, when its last two digits are removed, is also a square number.
|
1681
| 0.625 |
Given the functions \( f(x) \) and \( g(x) \) as defined in Theorem 1, if \( T_1 = 1 \) and \( T_2 = \frac{1}{m} \) (with \( m \in \mathbb{N} \) and \( m > 1 \)), then the smallest positive period of the function \( h(x) = f(x) + g(x) \) is \( \frac{1}{k} \), where \( k = 1 \) or \( k \in \mathbb{N} \), \( k \) is not a multiple of \( m \), and \( m \) is not a multiple of \( k \).
|
1
| 0.625 |
27 If the sum of all positive divisors of a positive integer equals twice the number itself, the number is called a perfect number. Find all positive integers \( n \) such that \( n-1 \) and \( \frac{n(n+1)}{2} \) are both perfect numbers.
|
7
| 0.75 |
The base of a right prism is a right triangle with a hypotenuse of length \( c \) and an acute angle of \( 30^\circ \). A plane is drawn through the hypotenuse of the lower base and the vertex of the right angle of the upper base, forming an angle of \( 45^\circ \) with the base's plane. Determine the volume of the triangular pyramid cut off from the prism by this plane.
|
\frac{c^3}{32}
| 0.25 |
If a positive integer \( n \) makes the equation \( x^{3} + y^{3} = z^{n} \) have positive integer solutions \((x, y, z)\), then \( n \) is called a "good number". Find the number of "good numbers" not exceeding 2019.
|
1346
| 0.375 |
Two small and large cubes are glued together to form a three-dimensional shape. The four vertices of the smaller cube's glued face are at the quarter points (not the midpoints) of the edges of the larger cube's glued face. If the edge length of the larger cube is 4, what is the surface area of this three-dimensional shape?
|
136
| 0.75 |
Let $a$ and $b$ be natural numbers, $a > b$, and the numbers $a+b$ and $a-b$ are relatively prime. Find all values of the greatest common divisor of the numbers $a$ and $b$.
|
1
| 0.75 |
Let \( \triangle ABC \) have centroid \( S \), midpoint of segment \( AS \) be \( H \), and midpoint of side \( AB \) be \( Q \). Let the line parallel to \( BC \) through \( H \) intersect \( AB \) at \( P \) and line \( CQ \) at \( R \). What is the ratio of the areas of triangles \( PQR \) and \( APH \)?
|
1
| 0.5 |
Stringing 6 red balls, 1 green ball, and 8 yellow balls into a necklace, how many possible different types can appear (balls of the same color are not distinguished)?
|
1519
| 0.125 |
There are 6 different numbers from 6 to 11 on the faces of a cube. The cube was rolled twice. The sum of the numbers on the four lateral faces was 36 in the first roll and 33 in the second roll. What number is on the face opposite to the face with the number 10?
|
8
| 0.625 |
Bethany, Chun, Dominic, and Emily go to the movies. They choose a row with four consecutive empty seats. If Dominic and Emily must sit beside each other, in how many different ways can the four friends sit?
|
12
| 0.875 |
Take a clay sphere of radius 13, and drill a circular hole of radius 5 through its center. Take the remaining "bead" and mold it into a new sphere. What is this sphere's radius?
|
12
| 0.75 |
Show that the equation
$$
a x^{2} + b x + c = 0
$$
can only be satisfied by three distinct numbers $\left( x_{1}, x_{2}, x_{3} \right)$ if $a = b = c = 0$.
|
a = b = c = 0
| 0.75 |
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
| 0.75 |
Represent the expression \(2x^{2} + 2y^{2}\) as a sum of two squares.
|
(x+y)^2 + (x-y)^2
| 0.25 |
Find the smallest positive integer \( n \) such that for any positive integer \( k \geqslant n \), in the set \( M = \{1, 2, \cdots, k\} \), for any \( x \in M \), there always exists another number \( y \in M \) (with \( y \neq x \)) such that \( x + y \) is a perfect square.
|
7
| 0.875 |
Given the sequences \( \left\{a_{n}\right\} \) and \( \left\{b_{n}\right\} \) with general terms \( a_{n}=2^{n} \) and \( b_{n}=3n+2 \), respectively, arrange the common terms of \( \left\{a_{n}\right\} \) and \( \left\{b_{n}\right\} \) in ascending order to form a new sequence \( \left\{c_{n}\right\} \). Determine the general term of \( \left\{c_{n}\right\} \).
|
c_n = 2^{2n+1}
| 0.25 |
A three-digit number \( A \) is such that the difference between the largest three-digit number and the smallest three-digit number that can be formed using the digits of \( A \) is still \( A \). What is the value of this number \( A \)?
|
495
| 0.5 |
Find the value of $\cos \frac{2 \pi}{7} + \cos \frac{4 \pi}{7} + \cos \frac{6 \pi}{7}$.
|
-\frac{1}{2}
| 0.875 |
Find all prime numbers \( p \) such that the numbers \( p + 4 \) and \( p + 8 \) are also prime.
|
3
| 0.75 |
The Fibonacci numbers are defined recursively by \( F_{0}=0 \), \( F_{1}=1 \), and \( F_{i}=F_{i-1}+F_{i-2} \) for \( i \geq 2 \). Given 15 wooden blocks of weights \( F_{2}, F_{3}, \ldots, F_{16} \), compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the blue blocks.
|
32
| 0.125 |
Towers are placed on an $n \times n$ chessboard such that if the square $(i, j)$ is empty, there are at least $n$ towers on the $i$-th row and the $j$-th column. Show that there are at least $n^{2} / 2$ towers on the chessboard.
|
\frac{n^2}{2}
| 0.25 |
Given nonzero real numbers \(a, b, c, d\) and the function \(f(x)=\frac{ax+b}{cx+d}\) for \(x \in \mathbb{R}\) such that \(f(19) = 19\) and \(f(97) = 97\). If for any real number \(x \neq -\frac{d}{c}\), it holds that \(f[f(x)] = x\), find the unique number that is outside the range of \(f(x)\).
|
58
| 0.625 |
For a homework assignment, Tanya was asked to come up with 20 examples of the form \( * + * = * \), where different natural numbers need to be inserted in place of \( * \) (i.e., a total of 60 different numbers should be used). Tanya loves prime numbers very much, so she decided to use as many of them as possible while still getting correct examples. What is the maximum number of prime numbers Tanya can use?
|
41
| 0.625 |
Find the minimum value of the expression
$$
\frac{|a-3b-2| + |3a-b|}{\sqrt{a^2 + (b+1)^2}}
$$
for \(a, b \geq 0\).
|
2
| 0.875 |
On the sides $AB$, $BC$, and $AC$ of triangle $ABC$, whose area is 75, points $M$, $N$, and $K$ are respectively located. It is known that $M$ is the midpoint of $AB$, the area of triangle $BMN$ is 15, and the area of triangle $AMK$ is 25. Find the area of triangle $CNK$.
|
15
| 0.25 |
0 < k < 1 is a real number. Define \( f: [0, 1] \to [0, 1] \) by
\[ f(x) =
\begin{cases}
0 & \text{for } x \leq k, \\
1 - \left( \sqrt{kx} + \sqrt{(1-k)(1-x)} \right)^2 & \text{for } x > k.
\end{cases}
\]
Show that the sequence \( 1, f(1), f(f(1)), f(f(f(1))), \ldots \) eventually becomes zero.
|
0
| 0.5 |
How many terms of the sum
$$
1+2+3+\ldots
$$
are needed for the result to be a three-digit number in which all digits are the same?
|
36
| 0.5 |
In an arithmetic progression with 12 terms, the sum of the terms is 354. The ratio of the sum of the terms with even indices to the sum of the terms with odd indices is 32:27. Determine the first term and the common difference of the progression.
|
a = 2, \, d = 5
| 0.375 |
Find all functions \( f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+} \) such that for any positive integers \( m \) and \( n \), \( f(m) + f(n) - mn \) is non-zero and divides \( mf(m) + nf(n) \).
|
f(n) = n^2
| 0.375 |
Let points \( A_{1}, A_{2}, A_{3}, A_{4}, A_{5} \) be located on the unit sphere. Find the maximum value of \( \min \left\{A_{i} A_{j} \mid 1 \leq i < j \leq 5 \right\} \) and determine all cases where this maximum value is achieved.
|
\sqrt{2}
| 0.75 |
Find all natural numbers \( x \) such that the product of all digits in the decimal representation of \( x \) is equal to \( x^{2} - 10x - 22 \).
|
12
| 0.375 |
Given a right triangle \(ABC\) with legs \(BC = 30\) and \(AC = 40\). Points \(C_1\), \(A_1\), and \(B_1\) are chosen on the sides \(AB\), \(BC\), and \(CA\), respectively, such that \(AC_1 = BA_1 = CB_1 = 1\). Find the area of triangle \(A_1 B_1 C_1\).
|
554.2
| 0.375 |
In the expression below, each letter has replaced a certain digit in the base-6 numbering system:
$$
F A R E S = (F E E)^{2}
$$
(identical letters represent identical digits). Restore the original digits.
|
15324
| 0.125 |
A state issues car license plates consisting of 6 digits (each digit ranging from $0$ to $9$), with the condition that any two license plate numbers must differ in at least two places. (For example, license numbers 027592 and 020592 cannot both be used). Determine the maximum number of license plate numbers possible under this condition. Provide a proof.
|
100000
| 0.5 |
In the sequence $\left\{a_{n}\right\}$, $a_{1}=1, a_{n+1}>a_{n}$, and $a_{n+1}^{2}+a_{n}^{2}+1 = 2(a_{n+1} a_{n} + a_{n+1} + a_{n})$. Determine $\lim _{n \rightarrow +\infty} \frac{S_{n}}{n a_{n}} = \qquad$.
|
\frac{1}{3}
| 0.875 |
Seven fishermen are standing in a circle. Each fisherman has a professional habit of exaggerating numbers, with a distinct measure of exaggeration (an integer) indicating by how many times the number mentioned by the fisherman exceeds the true value. For example, if a fisherman with an exaggeration measure of 3 catches two fish, he will claim to have caught six fish. When asked: "How many fish did your left neighbor catch?" the answers were (not necessarily in the seated order) $12, 12, 20, 24, 32, 42,$ and $56$. When asked: "How many fish did your right neighbor catch?" six of the fishermen answered $12, 14, 18, 32,$ $48,$ and $70$. What did the seventh fisherman answer?
|
16
| 0.125 |
Find all prime numbers for which \( p^{3} + p^{2} + 11p + 2 \) is also prime.
|
3
| 0.25 |
For how many integers \( x \) is the expression \(\frac{\sqrt{75-x}}{\sqrt{x-25}}\) equal to an integer?
|
5
| 0.75 |
In the city of Omsk, a metro line was built in a straight line. On this same line is the house where Nikita and Yegor live. Every morning, they simultaneously leave the house for their lessons, after which Yegor runs to the nearest metro station at a speed of 12 km/h, while Nikita walks along the metro line to another station at a speed of 6 km/h. Despite this, Nikita always manages to reach his lesson on time, whereas Yegor does not, even though he doesn't delay anywhere. Find the maximum possible speed of the metro trains, given that it is constant and an integer. (Assume the school is located directly at a certain metro station, distinct from the given ones).
|
23
| 0.25 |
Given the sequence \( b_{k} \) defined as:
$$
b_{k}=\left\{\begin{array}{cc}
0 & k<l \\
a_{k-l} & k \geqslant l,
\end{array} \text{ then } B(x)=x^{l} A(x)
\right.
$$
|
B(x) = x^l A(x)
| 0.875 |
The sum of positive numbers \(a, b, c,\) and \(d\) does not exceed 4. Find the maximum value of the expression
\[
\sqrt[4]{a^{2}(a+b)}+\sqrt[4]{b^{2}(b+c)}+\sqrt[4]{c^{2}(c+d)}+\sqrt[4]{d^{2}(d+a)}
\]
|
4 \sqrt[4]{2}
| 0.875 |
Which values of \( x \) satisfy the following system of inequalities?
\[
\frac{x}{6} + \frac{7}{2} > \frac{3x + 29}{5}, \quad x + \frac{9}{2} > \frac{x}{8}, \quad \frac{11}{3} - \frac{x}{6} < \frac{34 - 3x}{5}.
\]
|
\text{No solution}
| 0.625 |
Find the largest natural number that cannot be represented as the sum of two composite numbers.
|
11
| 0.625 |
Find the sum of all natural numbers that have exactly four natural divisors, three of which are less than 15, and the fourth is not less than 15.
|
649
| 0.375 |
Let $A$ and $B$ be two opposite vertices of a cube with side length 1. What is the radius of the sphere centered inside the cube, tangent to the three faces that meet at $A$ and to the three edges that meet at $B$?
|
2 - \sqrt{2}
| 0.625 |
Two squares \(A B C D\) and \(B E F G\) share vertex \(B\), with \(E\) on the side \(B C\) and \(G\) on the side \(A B\). The length of \(C G\) is \(9 \text{ cm}\) and the area of the shaded region is \(47 \text{ cm}^2\). Calculate the perimeter of the shaded region.
|
32 \text{ cm}
| 0.125 |
A circle with a radius of \(1 + \sqrt{2}\) is circumscribed around an isosceles right triangle. Find the radius of a circle that is tangent to the legs of this triangle and internally tangent to the circumscribed circle.
|
2
| 0.5 |
Calculate the value of the product \(\left(1-\frac{1}{4}\right) \cdot\left(1-\frac{1}{9}\right) \cdot\left(1-\frac{1}{16}\right) \cdot \ldots \cdot\left(1-\frac{1}{2020^{2}}\right)\).
|
\frac{2021}{4040}
| 0.375 |
Given that one root of the quadratic trinomial \(a x^{2}+b x+b\) and one root of the quadratic trinomial \(a x^{2}+a x+b\) have a product equal to 1, determine the sum of the squares of these two roots.
|
3
| 0.875 |
Is it possible to find two lucky tickets among ten consecutive tickets? A ticket is considered lucky if the sum of the first three digits is equal to the sum of the last three digits.
|
\text{Yes}
| 0.625 |
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