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|---|---|---|
There are several points on a plane, all pairwise distances between which are different. Each of these points is connected to the nearest one. Can a closed polygonal chain result from this?
|
\text{No}
| 0.625 |
In a list of five numbers, the first number is 60 and the last number is 300. The product of the first three numbers is 810000, the product of the three in the middle is 2430000, and the product of the last three numbers is 8100000. Which number is third in the list?
|
150
| 0.75 |
Given a positive term sequence $\left\{a_{n}\right\}$ with the sum of the first $n$ terms being $S_{n}$, both $\left\{a_{n}\right\}$ and $\left\{\sqrt{S_{n}}\right\}$ are arithmetic sequences with a common difference $d$. Find $S_{n}$.
|
S_n=\frac{n^2}{4}
| 0.875 |
Find all the roots of the equation
\[ 1 - \frac{x}{1} + \frac{x(x-1)}{2!} - \frac{x(x-1)(x-2)}{3!} + \frac{x(x-1)(x-2)(x-3)}{4!} - \frac{x(x-1)(x-2)(x-3)(x-4)}{5!} + \frac{x(x-1)(x-2)(x-3)(x-4)(x-5)}{6!} = 0 \]
(Where \( n! = 1 \cdot 2 \cdot 3 \cdots n \))
In the answer, specify the sum of the found roots.
|
21
| 0.5 |
Let \( a, b, c > 0 \) such that \( abc = 1 \). Show that
\[
\frac{1}{a+b+1} + \frac{1}{b+c+1} + \frac{1}{a+c+1} \leq 1.
\]
|
1
| 0.875 |
On the board, there are 101 numbers written: \(1^2, 2^2, \ldots, 101^2\). In one operation, it is allowed to erase any two numbers and write the absolute value of their difference instead.
What is the smallest number that can be obtained after 100 operations?
|
1
| 0.75 |
Find a four-digit number that is a perfect square, where the first two digits are the same and the last two digits are also the same.
|
7744
| 0.5 |
What is the locus of the points of intersection of mutually perpendicular tangents drawn to the circle $x^{2}+y^{2}=32$?
|
x^2 + y^2 = 64
| 0.625 |
There are four distinct codes $A, B, C, D$ used by an intelligence station, with one code being used each week. Each week, a code is chosen randomly with equal probability from the three codes that were not used the previous week. Given that code $A$ is used in the first week, what is the probability that code $A$ is also used in the seventh week? (Express your answer as a simplified fraction.)
|
\frac{61}{243}
| 0.375 |
Show that \(a^{2}+b^{2}+c^{2} \geq ab+bc+ca\). Find the cases of equality.
|
a = b = c
| 0.375 |
In triangle \(ABC\), \(AC < AB\), and the median \(AF\) divides the angle at \(A\) in the ratio \(1: 2\). The perpendicular from \(B\) to \(AB\) intersects line \(AF\) at point \(D\). Show that \(AD = 2 AC\).
|
AD = 2AC
| 0.875 |
In how many ways can the number 1500 be represented as the product of three natural numbers (variations where the factors are the same but in different orders are considered identical)?
|
32
| 0.375 |
Find the number of natural numbers \( k \), not exceeding 454500, such that \( k^2 - k \) is divisible by 505.
|
3600
| 0.25 |
How many eight-digit numbers can be written using only the digits 1, 2, and 3 such that the difference between any two adjacent digits is 1?
|
32
| 0.125 |
Given the sequence \(\left\{a_{n}\right\}\) satisfying \(a_{1}=a > 2\), \(a_{2017} = 2017\), and for any positive integer \(n\), \(a_{n+1} = a_{n}^{2} - 2\). Compute \(\left\lfloor \frac{\sqrt{a-2}}{10^{6}} a_{1} a_{2} \cdots a_{2017} \right\rfloor\), where \(\lfloor x \rfloor\) denotes the greatest integer not exceeding the real number \(x\).
|
2
| 0.625 |
Young marketer Masha was supposed to interview 50 customers in an electronics store throughout the day. However, there were fewer customers in the store that day. What is the maximum number of customers Masha could have interviewed if, according to her data, 7 of those interviewed made an impulsive purchase, $75\%$ of the remaining respondents bought electronics influenced by advertising, and the number of customers who chose a product based on the advice of a sales consultant is one-third of the number who chose a product influenced by advertising?
|
47
| 0.875 |
The diagonals of the cyclic quadrilateral $ABCD$ intersect at point $O$. Within triangle $AOB$, a point $K$ is chosen such that line $KO$ is the angle bisector of $\angle CK$. Ray $DK$ intersects the circumcircle of triangle $COK$ again at point $L$, and ray $CK$ intersects the circumcircle of triangle $DOK$ again at point $M$. Find the ratio of the areas of triangles $ALO$ and $BMO$.
|
1
| 0.875 |
Given a sequence of positive integers $\left\{a_{n}\right\}$ satisfying
$$
a_{n+2}=a_{n+1}+a_{n}\ \ (n \in \mathbf{Z}_{+}),
$$
and knowing that $a_{11}=157$, find the value of $a_{1}$.
|
3
| 0.875 |
Given the sequence $\left\{a_{n}\right\}$ such that $a_{1}=1$ and $a_{n+1}=\frac{1}{8} a_{n}^{2}+m$ for $n \in \mathbf{N}^{*}$, if $a_{n}<4$ for any positive integer $n$, find the maximum value of the real number $m$.
|
2
| 0.875 |
Find all functions \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) such that for all positive natural numbers \( m \) and \( n \),
$$
f\left(f^{2}(m)+2 f^{2}(n)\right)=m^{2}+2 n^{2}
$$
|
f(n) = n
| 0.625 |
Find a natural number \( N \) that is divisible by 5 and 49, and has exactly 10 divisors, including 1 and \( N \).
|
12005
| 0.375 |
One thousand points form the vertices of a convex 1000-sided polygon, with an additional 500 points inside such that no three of the 500 are collinear. This 1000-sided polygon is triangulated in such a way that all the given 1500 points are vertices of the triangles and these triangles have no other vertices. How many triangles are formed by this triangulation?
|
1998
| 0.75 |
In $\triangle ABC$, $\angle A = 60^{\circ}$, $AB > AC$, point $O$ is the circumcenter, and the altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ lie on segments $BH$ and $HF$ respectively, such that $BM = CN$. Find the value of $\frac{MH + NH}{OH}$.
|
\sqrt{3}
| 0.125 |
Let \( n \) be a natural number. Roger has a square garden of size \( (2n + 1) \times (2n + 1) \). He erects fences to divide it into rectangular beds. He wants exactly two horizontal \( k \times 1 \) beds and exactly two vertical \( 1 \times k \) beds for every even number \( k \) between 1 and \( 2n + 1 \), as well as a single square bed of size \( 1 \times 1 \) when he is finished. How many different ways can Roger divide his garden?
|
2^n
| 0.125 |
What is the minimum number of cells that need to be marked in a $7 \times 7$ grid so that in each vertical or horizontal $1 \times 4$ strip there is at least one marked cell?
|
12
| 0.125 |
A horse stands at the corner of a chessboard, on a white square. With each jump, the horse can move either two squares horizontally and one vertically or two vertically and one horizontally, like a knight moves. The horse earns two carrots every time it lands on a black square, but it must pay a carrot in rent to the rabbit who owns the chessboard for every move it makes. When the horse reaches the square on which it began, it can leave. What is the maximum number of carrots the horse can earn without touching any square more than twice?
|
0
| 0.25 |
Let \( P \) be any point inside a regular tetrahedron \( ABCD \) with side length \( \sqrt{2} \). The distances from point \( P \) to the four faces are \( d_1, d_2, d_3, d_4 \) respectively. What is the minimum value of \( d_1^2 + d_2^2 + d_3^2 + d_4^2 \)?
|
\frac{1}{3}
| 0.75 |
Calculate: $\frac{4}{7} \times 9 \frac{3}{4} + 9.75 \times \frac{2}{7} + 0.142857 \times 975 \% = $
|
9.75
| 0.625 |
There is an $n \times n$ table in which $n-1$ cells contain ones, and the remaining cells contain zeros. The following operation is allowed with the table: choose a cell, subtract one from the number in that cell, and add one to all the other numbers in the same row or column as the chosen cell. Is it possible to obtain a table where all numbers are equal using these operations?
|
\text{No}
| 0.375 |
In quadrilateral \(ABCD\), points \(X, Y, Z\) are the midpoints of segments \(AB, AD, BC\), respectively. It is known that \(XY\) is perpendicular to \(AB\), \(YZ\) is perpendicular to \(BC\), and the measure of angle \(ABC\) is \(100^\circ\). Find the measure of angle \(ACD\).
|
90^\circ
| 0.125 |
An isosceles trapezoid \(ABCD\) is inscribed in a circle with diameter \(AD\) and center at point \(O\). A circle with center at point \(I\) is inscribed in the triangle \(BOC\). Find the ratio of the areas of triangles \(AID\) and \(BIC\) given that \(AD = 15\) and \(BC = 5\).
|
9
| 0.875 |
In the full permutation of 4 $x$, 3 $y$, and 2 $z$, determine the number of permutations where the patterns $x x x x$, $y y y$, and $z z$ do not appear. Let $A_{1}$ represent the set of permutations of 9 characters where the pattern $x x x x$ appears, $A_{2}$ represent the set where the pattern $y y y$ appears, and $A_{3}$ represent the set where the pattern $z z$ appears.
|
871
| 0.25 |
Into how many parts do the face planes divide the space of a) a cube; b) a tetrahedron?
|
15
| 0.375 |
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x \in \mathbb{R}^{*} \) (nonzero real numbers) and all \( y \in \mathbb{R} \),
$$
f\left(x^{2} + y\right) \geq \left(\frac{1}{x} + 1\right) f(y)
$$
|
f(x) = 0
| 0.625 |
Two Hungarian teams, Bolyai TC and Eötvös TK, have reached the top 16 teams in the European Cup. What is the probability that they will play against each other? (The competition follows a single-elimination format; in each match, one team progresses while the other is eliminated.)
|
\frac{1}{8}
| 0.25 |
Given a triangle \(ABC\) with the midpoints of sides \(BC\), \(AC\), and \(AB\) denoted as \(D\), \(E\), and \(F\) respectively, it is known that the medians \(AD\) and \(BE\) are perpendicular to each other, with lengths \(\overline{AD} = 18\) and \(\overline{BE} = 13.5\). Calculate the length of the third median \(CF\) of this triangle.
|
22.5
| 0.625 |
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 (where \( H_A \) and \( H_B \) are the bases of the perpendiculars dropped from points \( A \) and \( B \) onto the x-axis; \( O \) is the origin). Find all such functions.
Given \( f(1) = 4 \), find the value of \( f(4) \).
|
1
| 0.375 |
A perfect square greater than 1 sometimes has the property that the sum of all its positive divisors is also a perfect square. For example, the sum of all positive divisors of \(9^{2}\) (which are \(1, 3, 9, 27, 81\)) is 121, which equals \(11^{2}\). Find another perfect square with the aforementioned property.
|
400
| 0.5 |
Find all prime numbers \( p, q, r \) such that the equation \( p^3 = p^2 + q^2 + r^2 \) holds.
|
p = 3, q = 3, r = 3
| 0.25 |
Bag $A$ contains 2 ten-yuan bills and 3 one-yuan bills, and bag $B$ contains 4 five-yuan bills and 3 one-yuan bills. If two bills are randomly drawn from each bag, what is the probability that the sum of the denominations of the remaining bills in bag $A$ is greater than the sum of the denominations of the remaining bills in bag $B$?
|
\frac{9}{35}
| 0.5 |
Let \( S \) be the set of all rational numbers in \(\left(0, \frac{5}{8}\right)\). For each reduced fraction \(\frac{q}{p} \in S\) where \(\gcd(p, q) = 1\), define the function \( f(q p) = \frac{q+1}{p} \). Determine the number of solutions to \( f(x) = \frac{2}{3} \) in \( S \).
|
5
| 0.375 |
If the equation \( x^{3} - 3x^{2} - 9x = a \) has exactly two different real roots in the interval \([-2, 3]\), then the range of the real number \( a \) is \(\quad\) .
|
[-2, 5)
| 0.375 |
Let \( n \) (\( \geqslant 3 \)) be a positive integer, and \( M \) be an \( n \)-element set. Find the largest positive integer \( k \) that satisfies the following condition:
There exists a family \( \psi \) of \( k \) three-element subsets of \( M \) such that the intersection of any two elements in \( \psi \) (note: the elements of \( \psi \) are all three-element sets) is nonempty.
|
\binom{n-1}{2}
| 0.125 |
In an isosceles trapezoid \(ABCD\) (\(BC \parallel AD\)), the angles \(ABD\) and \(DBC\) are \(135^{\circ}\) and \(15^{\circ}\) respectively, and \(BD = \sqrt{6}\). Find the perimeter of the trapezoid.
|
9 - \sqrt{3}
| 0.25 |
Vitya and Masha were born in the same year in June. Find the probability that Vitya is at least one day older than Masha.
|
\frac{29}{60}
| 0.875 |
Neznaika wrote several different natural numbers on the board and divided (in his mind) the sum of these numbers by their product. After that, Neznaika erased the smallest number and divided (again in his mind) the sum of the remaining numbers by their product. The second result turned out to be 3 times larger than the first. What number did Neznaika erase?
|
4
| 0.5 |
Let the set
\[ \left\{\left.\frac{3}{a}+b \right\rvert\, 1 \leqslant a \leqslant b \leqslant 4\right\} \]
have maximum and minimum elements denoted by \( M \) and \( N \) respectively. Find \( M - N \).
|
7 - 2\sqrt{3}
| 0.875 |
Calculate:
$$\frac{\left(1+\frac{1}{2}\right)^{2} \times\left(1+\frac{1}{3}\right)^{2} \times\left(1+\frac{1}{4}\right)^{2} \times\left(1+\frac{1}{5}\right)^{2} \times \cdots \times\left(1+\frac{1}{10}\right)^{2}}{\left(1-\frac{1}{2^{2}}\right) \times\left(1-\frac{1}{3^{2}}\right) \times\left(1-\frac{1}{4^{2}}\right) \times\left(1-\frac{1}{5^{2}}\right) \times \cdots \times\left(1-\frac{1}{10^{2}}\right)}$$
|
55
| 0.625 |
Juquinha marks points on the circumference and draws triangles by connecting 3 of these points. The lengths of the arcs between 2 consecutive points are equal.
a) By marking 4 points on the circumference, how many triangles can he draw?
b) By marking 5 points on the circumference, how many equilateral triangles can he draw?
c) How many right triangles can he draw if he marks 6 points?
|
12
| 0.5 |
30 students from five courses created 40 problems for the olympiad, with students from the same course creating the same number of problems, and students from different courses creating different numbers of problems. How many students created exactly one problem?
|
26
| 0.375 |
There are 13 weights arranged in a row on a table, ordered by mass (the lightest on the left, the heaviest on the right). It is known that the mass of each weight is an integer number of grams, the masses of any two adjacent weights differ by no more than 5 grams, and the total mass of the weights does not exceed 2019 grams. Find the maximum possible mass of the heaviest weight under these conditions.
|
185
| 0.25 |
Given a real number \( t \), find all functions \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that
\[ f(x + t + f(y)) = f(f(x)) + f(t) + y. \]
|
f(x) = x
| 0.875 |
\[
\frac{\sqrt{c-d}}{c^{2} \sqrt{2c}} \cdot \left( \sqrt{\frac{c-d}{c+d}} + \sqrt{\frac{c^{2}+cd}{c^{2}-cd}} \right) \quad \text{where} \quad c=2 \quad \text{and} \quad d=\frac{1}{4}.
\]
|
\frac{1}{3}
| 0.875 |
Given a strictly increasing function \( f: \mathbb{N}_0 \to \mathbb{N}_0 \) (where \( \mathbb{N}_0 \) is the set of non-negative integers) that satisfies the relation \( f(n + f(m)) = f(n) + m + 1 \) for any \( m, n \in \mathbb{N}_0 \), find all possible values of \( f(2023) \).
|
2024
| 0.875 |
In triangle \( \triangle ABC \), if \( \frac{\cos A}{\sin B} + \frac{\cos B}{\sin A} = 2 \), and the perimeter of \( \triangle ABC \) is 12, find the maximum possible value of its area.
|
36(3 - 2\sqrt{2})
| 0.125 |
Find the smallest integer \( n \) such that the expanded form of \( (xy - 7x - 3y + 21)^n \) has 2012 terms.
|
44
| 0.5 |
In a rectangular parallelepiped \( ABCD A_{1} B_{1} C_{1} D_{1} \), the edge lengths are given by \( AB = 42 \), \( AD = 126 \), and \( AA_{1} = 42 \). Point \( E \) is marked at the midpoint of edge \( A_{1}B_{1} \), and point \( F \) is marked at the midpoint of edge \( B_{1}C_{1} \). Find the distance between the lines \( AE \) and \( BF \).
|
36
| 0.625 |
The second term of a geometric sequence is \( b_{2} = 24.5 \), and the fifth term is \( b_{5} = 196 \). Find the third term and the sum of the first four terms.
|
183.75
| 0.125 |
The orthogonal projections of the triangle \(ABC\) onto two mutually perpendicular planes are equilateral triangles with sides of length 1. Find the perimeter of triangle \(ABC\), given that \(AB = \frac{\sqrt{5}}{2}\).
|
\sqrt{2} + \sqrt{5}
| 0.25 |
Calculate \( \sin ^{2} 20^{\circ}+\cos ^{2} 50^{\circ}+\sin 20^{\circ} \cos 50^{\circ}= \)
|
\dfrac{3}{4}
| 0.625 |
Find all quadratic trinomials \( p(x) \) that attain a minimum value of \(-\frac{49}{4}\) at \(x=\frac{1}{2}\), and the sum of the fourth powers of its roots equals 337.
|
p(x) = x^2 - x - 12
| 0.875 |
Given that \( c \) cannot be divided by the square of a prime number, if \( b^{2} c \) can be divided by \( a^{2} \), then \( b \) can be divided by \( a \).
|
a \mid b
| 0.125 |
A ball was thrown vertically upward from a balcony. It is known that it fell to the ground after 3 seconds. Knowing that the initial speed of the ball was 5 m/s, determine the height of the balcony. The acceleration due to gravity is $10 \mathrm{~m/s}^2$.
|
30 \text{ m}
| 0.875 |
Show that the perimeter of a triangle is related to the perimeter of its pedal triangle as \( R : r \), where \( R \) is the radius of the circumcircle of the original triangle, and \( r \) is the radius of the inscribed circle of the triangle.
|
K:k = R:r
| 0.5 |
The regular octagon \( A B C D E F G H \) is inscribed in a circle. Points \( P \) and \( Q \) are on the circle, with \( P \) between \( C \) and \( D \), such that \( A P Q \) is an equilateral triangle. It is possible to inscribe a regular \( n \)-sided polygon, one of whose sides is \( P D \), in the circle. What is the value of \( n \)?
|
24
| 0.875 |
In an equilateral triangle $\triangle ABC$ with unit area, external equilateral triangles $\triangle APB$, $\triangle BQC$, and $\triangle CRA$ are constructed such that $\angle APB = \angle BQC = \angle CRA = 60^\circ$.
1. Find the maximum area of triangle $\triangle PQR$.
2. Find the maximum area of the triangle whose vertices are the incenters of $\triangle APB$, $\triangle BQC$, and $\triangle CRA$.
|
1
| 0.125 |
The real-valued function \( f \) is defined on the reals and satisfies \( f(xy) = x f(y) + y f(x) \) and \( f(x + y) = f(x \cdot 1993) + f(y \cdot 1993) \) for all \( x \) and \( y \). Find \( f(\sqrt{5753}) \).
|
0
| 0.5 |
On the coordinate plane with origin at point \( O \), a parabola \( y = x^2 \) is drawn. Points \( A \) and \( B \) are marked on the parabola such that \( \angle AOB \) is a right angle. Find the minimum possible area of triangle \( AOB \).
|
1
| 0.625 |
Given non-negative real numbers $a_{1}, a_{2}, \cdots, a_{2008}$ whose sum equals 1, determine the maximum value of $a_{1} a_{2} + a_{2} a_{3} + \cdots + a_{2007} a_{2008} + a_{2008} a_{1}$.
|
\frac{1}{4}
| 0.75 |
In the center of a circular field, there is a geologists' hut. From it, 6 straight roads radiate out, dividing the field into 6 equal sectors. Two geologists leave their hut at a speed of 4 km/h, each choosing a road at random. Determine the probability that the distance between them after one hour will be at least 6 km.
|
0.5
| 0.625 |
In tetrahedron \(ABCD\), \(\triangle ABD\) is an equilateral triangle, \(\angle BCD = 90^\circ\), \(BC = CD = 1\), \(AC = \sqrt{3}\). Points \(E\) and \(F\) are the midpoints of \(BD\) and \(AC\) respectively. Determine the cosine of the angle between lines \(AE\) and \(BF\).
|
\frac{\sqrt{2}}{3}
| 0.25 |
Given that \( f(n) = \sin \frac{n \pi}{4} \), where \( n \) is an integer. If \( c = f(1) + f(2) + \ldots + f(2003) \), find the value of \( c \).
|
1+\sqrt{2}
| 0.625 |
In how many ways can we place pawns on a \(4 \times 4\) chessboard such that each row and each column contains exactly two pawns?
|
90
| 0.25 |
How many ways are there to arrange the numbers {1, 2, 3, 4, 5, 6, 7, 8} in a circle so that every two adjacent elements are relatively prime? Consider rotations and reflections of the same arrangement to be indistinguishable.
|
36
| 0.125 |
What is the smallest possible sum of nine consecutive natural numbers if this sum ends in 3040102?
|
83040102
| 0.25 |
The equation \( x^{2}-x+1=0 \) has roots \(\alpha\) and \(\beta\). There is a quadratic function \( f(x) \) that satisfies \( f(\alpha) = \beta \), \( f(\beta) = \alpha \), and \( f(1) = 1 \). What is the expression for \( f(x) \)?
|
f(x) = x^2 - 2x + 2
| 0.875 |
Below are five distinct points on the same line. How many rays originate from one of these five points and do not contain point $B$?
|
4
| 0.75 |
Let \( f(x) = \sin^6\left(\frac{x}{4}\right) + \cos^6\left(\frac{x}{4}\right) \) for all real numbers \( x \). Determine \( f^{(2008)}(0) \) (i.e., \( f \) differentiated 2008 times and then evaluated at \( x = 0 \)).
|
\frac{3}{8}
| 0.875 |
There are 2019 numbers written on the board. One of them occurs more frequently than the others - 10 times. What is the minimum number of different numbers that could be written on the board?
|
225
| 0.875 |
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 |
Tessa has a figure created by adding a semicircle of radius 1 on each side of an equilateral triangle with side length 2, with the semicircles oriented outwards. She then marks two points on the boundary of the figure. What is the greatest possible distance between the two points?
|
3
| 0.25 |
Let \( a \) and \( c \) be positive integers, and let \( b \) be a digit. Determine all triples \( (a, b, c) \) that satisfy the following conditions:
(1) \( (a, b b b \ldots)^{2}=c.777 \ldots \) (infinite decimals);
(2) \( \frac{c+a}{c-a} \) is an integer!
|
(1, 6, 2)
| 0.25 |
The captain's assistant, who had been overseeing the loading of the ship, was smoking one pipe after another from the very start of the process. When $2 / 3$ of the number of loaded containers became equal to $4 / 9$ of the number of unloaded containers, and noon struck, the old sea wolf started smoking his next pipe. When he finished this pipe, the ratio of the number of loaded containers to the number of unloaded containers reversed the ratio that existed before he began smoking this pipe. How many pipes did the second assistant smoke during the loading period (assuming the loading rate and the smoking rate remained constant throughout)?
|
5
| 0.25 |
We can label the squares of an 8 x 8 chessboard from 1 to 64 in 64! different ways. For each way, we find \( D \), the largest difference between the labels of two squares that are adjacent (orthogonally or diagonally). What is the smallest possible \( D \)?
|
9
| 0.625 |
If \(a, b,\) and \(c\) are all multiples of 5, \( a < b < c \), \( c = a + 10 \), then calculate \( \frac{(a-b)(a-c)}{b-c} \).
|
-10
| 0.875 |
A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known that at \( x = 2020 \), all the resulting irreducible polynomials take the value -3. Find \( P(2020) \).
|
243
| 0.875 |
For what value of \( a \) does the equation \( |x-2|=a x-2 \) have an infinite number of solutions?
|
1
| 0.875 |
What is the radius of the smallest circle into which any system of points with a diameter of 1 can be enclosed?
|
\frac{\sqrt{3}}{3}
| 0.25 |
Given $\cos \alpha = \tan \beta$, $\cos \beta = \tan \gamma$, $\cos \gamma = \tan \alpha$, find that $\sin^2 \alpha = \sin^2 \beta = \sin^2 \gamma = \cos^4 \alpha = \cos^4 \beta = \cos^4 \gamma = 4 \sin^2 18^\circ$.
|
4 \sin^2 18^\circ
| 0.625 |
The bases \( AB \) and \( CD \) of trapezoid \( ABCD \) are 101 and 20 respectively, and its diagonals are mutually perpendicular. Find the scalar product of the vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).
|
2020
| 0.625 |
Given a parallelogram \(ABCD\) where \(AB < AC < BC\). Points \(E\) and \(F\) are chosen on the circumcircle \(\omega\) of triangle \(ABC\) such that the tangents to \(\omega\) at these points pass through \(D\). Moreover, segments \(AD\) and \(CE\) intersect. It is found that \(\angle ABF = \angle DCE\). Find the angle \(\angle ABC\).
|
60^\circ
| 0.75 |
The base of a quadrilateral pyramid $S A B C D$ is a parallelogram $A B C D$. 1) Construct a cross-section of the pyramid by a plane passing through the midpoint of edge $A B$ and parallel to plane $S A D$. 2) Find the area of the resulting cross-section if the area of face $S A D$ is 16.
|
12
| 0.125 |
Let $2 \leq r \leq \frac{n}{2}$, and let $\mathscr{A}$ be a family of $r$-element subsets of $Z = \{1, 2, \ldots, n\}$. If $\mathscr{A} = \{A \mid A$ is an $r$-element subset of $Z$ that contains a fixed element $x \in Z\}$, then equality holds.
|
\binom{n-1}{r-1}
| 0.875 |
A trapezoid $ABCD$ is circumscribed around a circle, with side $AB$ perpendicular to the bases, and $M$ is the point of intersection of the diagonals of the trapezoid. The area of triangle $CMD$ is $S$. Find the radius of the circle.
|
R = \sqrt{S}
| 0.75 |
A material point moves in such a way that its velocity is proportional to the distance traveled. At the initial moment, the point was at a distance of 1 meter from the origin, and after 2 seconds, it was at a distance of $e$ meters. Find the equation of motion for the material point.
|
s = e^{\frac{t}{2}}
| 0.5 |
In a grove, there are four types of trees: birches, spruces, pines, and aspens. There are 100 trees in total. It is known that among any 85 trees, there are trees of all four types. What is the smallest number of any trees in this grove that must include trees of at least three types?
|
69
| 0.625 |
The store purchased a batch of greeting cards at the price of 0.21 yuan each and sold them for a total of 14.57 yuan. If each card is sold at the same price and the selling price does not exceed twice the purchase price, how much did the store profit?
|
4.7
| 0.5 |
Let \(\Omega\) be a sphere of radius 4 and \(\Gamma\) be a sphere of radius 2. Suppose that the center of \(\Gamma\) lies on the surface of \(\Omega\). The intersection of the surfaces of \(\Omega\) and \(\Gamma\) is a circle. Compute this circle's circumference.
|
\pi \sqrt{15}
| 0.75 |
Solve the equation \( 2021 \cdot \sqrt[202]{x^{2020}} - 1 = 2020x \) for \( x \geq 0 \). (10 points)
|
x = 1
| 0.375 |
It is known that \(\log_{10}\left(2007^{2006} \times 2006^{2007}\right) = a \times 10^k\), where \(1 \leq a < 10\) and \(k\) is an integer. Find the value of \(k\).
|
4
| 0.625 |
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