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In the plane Cartesian coordinate system, the ellipse $\Gamma$: $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, point $P$ is inside the ellipse $\Gamma$ and moves on the line $y=x$. Points $K$ and $L$ are on $\Gamma$, such that the directions of $\overrightarrow{P K}$ and $\overrightarrow{P L}$ are along the positive direction of the $x$-axis and $y$-axis respectively. Point $Q$ is such that $P K Q L$ forms a rectangle. Does there exist two points $A$ and $B$ in the plane such that when the rectangle $P K Q L$ changes, $|Q A|-|Q B|$ is a non-zero constant? If such points exist, find all possible lengths of line segment $A B$; if not, explain why. | \frac{5\sqrt{3}}{3} | 0.875 |
Let $[a]$ represent the greatest integer less than or equal to $a$. Determine the largest positive integer solution to the equation $\left[\frac{x}{7}\right]=\left[\frac{x}{8}\right]+1$. | 104 | 0.875 |
Given a triplet of real numbers $(a, b, c)$, we can perform the following action: choose $x$ and $y$ from the triplet and replace them with $(x-y) / \sqrt{2}$ and $(x+y) / \sqrt{2}$. Can we transform the initial triplet $(2, \sqrt{2}, 1 / \sqrt{2})$ into the final triplet $(1, \sqrt{2}, 1+\sqrt{2})$? | \text{No} | 0.625 |
What is the largest number of white and black chips that can be placed on a chessboard so that on each horizontal and each vertical, the number of white chips is exactly twice the number of black chips? | 48 \text{ chips} | 0.625 |
Let \( ABCD \) be a parallelogram. Let \( E \) be the midpoint of \( AB \) and \( F \) be the midpoint of \( CD \). Points \( P \) and \( Q \) are on segments \( EF \) and \( CF \), respectively, such that \( A, P \), and \( Q \) are collinear. Given that \( EP = 5 \), \( PF = 3 \), and \( QF = 12 \), find \( CQ \). | 8 | 0.375 |
For any $n \geq 3$ points $A_{1}$, $A_{2}, \ldots, A_{n}$ in the plane, none of which are collinear, let $\alpha$ be the smallest angle $\angle A_{i} A_{j} A_{k}$ formed by any triple of distinct points $A_{i}, A_{j}, A_{k}$. For each value of $n$, find the maximum value of $\alpha$. Determine the configurations of the points for which this value is attained. | \frac{180^\circ}{n} | 0.875 |
A collection of \( n \) squares on the plane is called tri-connected if the following criteria are satisfied:
(i) All the squares are congruent.
(ii) If two squares have a point \( P \) in common, then \( P \) is a vertex of each of the squares.
(iii) Each square touches exactly three other squares.
How many positive integers \( n \) are there with \( 2018 \leq n \leq 3018 \), such that there exists a collection of \( n \) squares that is tri-connected? | 501 | 0.5 |
Find the smallest three-digit number ABC that is divisible by the two-digit numbers AB and BC (the digit A cannot be 0, but the digit B can be; different letters do not necessarily represent different digits). | 110 | 0.5 |
Assume we have a box of matches, each match being 1 unit in length. We are also given a cardboard square with side length \( n \) units. Divide the square into \( n^2 \) smaller square units with straight lines. Arrange the matches on the cardboard subject to the following conditions:
1. Each match must cover the side of one of the small squares.
2. Exactly two sides of each small square (including edge squares) must be covered by matches.
3. Matches cannot be placed on the edge of the cardboard.
For which values of \( n \) does this problem have a solution? | n \text{ is even} | 0.375 |
A regular hexagonal truncated pyramid is inscribed in a sphere of radius \( R \). The plane of the lower base passes through the center of the sphere, and the lateral edge forms an angle of \( 60^{\circ} \) with the plane of the base. Determine the volume of the pyramid. | \frac{21 R^3}{16} | 0.125 |
Boys were collecting apples. Each boy collected either 10 apples or 10% of the total number of apples collected, and there were both types of boys. What is the minimum number of boys that could have been? | 6 | 0.625 |
How many pairs of integers \( x, y \), between 1 and 1000, exist such that \( x^{2} + y^{2} \) is divisible by 7? | 20164 | 0.5 |
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 13. Find \( x \). | 728 | 0.375 |
Given that \( a \) and \( b \) are integers, and \( a + b \) is a root of the equation
\[
x^{2} + ax + b = 0
\]
find the maximum possible value of \( b \). | 9 | 0.75 |
Given a regular triangular pyramid \( P-ABC \) with a volume of \( 9 \sqrt{3} \), the dihedral angle between the face \( PAB \) and the base \( ABC \) is \( 60^\circ \). Point \( D \) is on the line segment \( AB \) such that \( AD = \frac{1}{6} AB \), point \( E \) is on the line segment \( AC \) such that \( AE = \frac{1}{6} AC \), and point \( F \) is the midpoint of \( PC \). The plane \( DEF \) intersects the line segment \( PB \) at point \( G \). Find the area of the quadrilateral \( DEFG \). | \sqrt{57} | 0.5 |
The probability of randomly selecting 5 different numbers from the set {1, 2, ..., 20} such that at least two of them are consecutive. | \frac{232}{323} | 0.75 |
What is the value of the following infinite product?
$$
z=9 \cdot \sqrt[3]{9} \cdot \sqrt[9]{9} \ldots \sqrt[3^{n}]{9} \ldots
$$ | 27 | 0.75 |
A square can be divided into four congruent figures. If each of the congruent figures has an area of 1, what is the area of the square? | 4 | 0.875 |
In triangle \( \triangle ABC \), \( AB = AC \), \( AD \) and \( BE \) are the angle bisectors of \( \angle A \) and \( \angle B \) respectively, and \( BE = 2 AD \). What is the measure of \( \angle BAC \)? | 108^\circ | 0.125 |
Given \( z \in \mathbf{C} \). If the equation with respect to \( x \):
$$
4 x^{2}-8 z x+4 i+3=0
$$
has real roots, then the minimum value of \( |z| \) is _______. | 1 | 0.125 |
In February of a non-leap year, Kirill and Vova decided to eat ice cream according to the following rules:
1. If the day of the month was even and the day of the week was Wednesday or Thursday, they would each eat seven servings of ice cream.
2. If the day of the week was Monday or Tuesday and the day of the month was odd, they would each eat three servings of ice cream.
3. If the day of the week was Friday, the number of servings each of them ate would be equal to the day of the month.
On all other days and under other conditions, eating ice cream was prohibited. What is the maximum number of servings of ice cream that Vova could eat in February under these conditions? | 110 | 0.25 |
On the island of knights and liars, each inhabitant was asked about each of the others: whether the person is a liar or a knight. A total of 42 answers of "knight" and 48 answers of "liar" were received. What is the maximum number of knights that could have been on the island? Justify your answer. (It is known that knights always tell the truth, and liars always lie.) | 6 | 0.875 |
There are 20 rooms, some with lights on and some with lights off. The occupants of these rooms prefer to match the majority of the rooms. Starting from room one, if the majority of the remaining 19 rooms have their lights on, the occupant will turn the light on; otherwise, they will turn the light off. Initially, there are 10 rooms with lights on and 10 rooms with lights off, and the light in the first room is on. After everyone in these 20 rooms has had a turn, how many rooms will have their lights off?
A. 0
B. 10
C. 11
D. 20 | 20 | 0.75 |
For every natural number $n$, let $\mathbf{S}(n)$ denote the sum of the digits of $n$. Calculate $\mathbf{S}^{5}(2018^{2018^{2018}})$. | 7 | 0.625 |
Let \( m \) and \( n \) be two positive integers that satisfy
$$
\frac{m}{n}=\frac{1}{10 \times 12}+\frac{1}{12 \times 14}+\frac{1}{14 \times 16}+\cdots+\frac{1}{2012 \times 2014}
$$
Find the smallest possible value of \( m+n \). | 10571 | 0.5 |
Each of the 10 people is either a knight, who always tells the truth, or a liar, who always lies. Each of them picked an integer. Then the first person said, "My number is greater than 1," the second said, "My number is greater than 2," and so on, with the tenth person saying, "My number is greater than 10." After that, all ten, speaking in some order, said one of the following statements: "My number is less than 1," "My number is less than 2," ..., "My number is less than 10" (each person said exactly one of these ten phrases). What is the maximum number of knights that could be among these 10 people? | 8 | 0.75 |
Gavrila got on the train with a fully charged smartphone, and by the end of the trip, his smartphone was fully discharged. He spent half of the total time playing Tetris and the other half watching cartoons. It is known that the smartphone completely discharges in 3 hours of video playback or 5 hours of playing Tetris. What distance did Gavrila travel if the train moved at an average speed of 80 km/h for half the way and at an average speed of 60 km/h for the other half? Give the answer in kilometers, rounded to the nearest whole number if necessary. | 257 | 0.875 |
Points \( M, N, \) and \( K \) are located on the lateral edges \( A A_{1}, B B_{1}, \) and \( C C_{1} \) of the triangular prism \( A B C A_{1} B_{1} C_{1} \) such that \( A M : A A_{1} = 1 : 2, B N : B B_{1} = 1 : 3, \) and \( C K : C C_{1} = 1 : 4 \). Point \( P \) belongs to the prism. Find the maximum possible volume of the pyramid \( M N K P \) if the volume of the prism is 16. | 4 | 0.125 |
Rearrange the four digits of 2016 to form a four-digit perfect square; then this four-digit perfect square is $\qquad$ . | 2601 | 0.75 |
Through a point in the interior of a triangle \(ABC\), three lines are drawn, one parallel to each side. These lines divide the sides of the triangle into three regions each. Let \(a, b\), and \(c\) be the lengths of the sides opposite \(\angle A, \angle B\), and \(\angle C\), respectively, and let \(a', b'\), and \(c'\) be the lengths of the middle regions of the sides opposite \(\angle A, \angle B\), and \(\angle C\), respectively. Find the numerical value of \(a' / a + b' / b + c' / c\). | 1 | 0.875 |
Let \(A \cup B \cup C = \{1, 2, 3, 4, 5, 6\}\), and \(A \cap B = \{1, 2\}\). Additionally, \(\{1, 2, 3, 4\} \subseteq B \cup C\). Determine the number of distinct groups \((A, B, C)\) that meet these conditions. (Note: The order of \(A\), \(B\), and \(C\) matters, so different orderings are considered different groups.) | 1600 | 0.375 |
At each vertex of a tetrahedron, there is an ant. At a given moment, each ant randomly chooses an edge and moves along it to the neighboring vertex. What is the probability that two ants will meet either midway on the edge or at the end of their journey? | \frac{25}{27} | 0.125 |
What is the smallest natural number ending in 4 such that placing the last digit at the front of the number gives a number that is four times the original number? | 102564 | 0.625 |
Find all functions \( f: \mathbb{R} \longrightarrow \mathbb{R} \) such that \( f \) is monotone and there exists an \( n \in \mathbb{N} \) such that for all \( x \):
\[ f^{n}(x) = -x \] | f(x) = -x | 0.875 |
Given a triangle \( \triangle ABC \) with \( \angle ABC = 80^\circ \), \( \angle ACB = 70^\circ \), and \( BC = 2 \). A perpendicular line is drawn from \( A \) to \( BC \), and another perpendicular line is drawn from \( B \) to \( AC \). The two perpendicular lines intersect at \( H \). Find the length of \( AH \). | 2 \sqrt{3} | 0.75 |
Calculate \(\sec \frac{2 \pi}{9} + \sec \frac{4 \pi}{9} + \sec \frac{6 \pi}{9} + \sec \frac{8 \pi}{9}\). | 4 | 0.75 |
There are 306 different cards with numbers \(3, 19, 3^{2}, 19^{2}, \ldots, 3^{153}, 19^{153}\) (each card has exactly one number, and each number appears exactly once). How many ways can you choose 2 cards such that the product of the numbers on the selected cards is a perfect square? | 17328 | 0.5 |
In an ornithological park, there are birds of several species, with a total of 2021 individuals. The birds are seated in a row, and it turns out that between any two birds of the same species, there is an even number of birds. What is the smallest number of bird species that could be? | 1011 | 0.125 |
An inscribed circle is drawn inside isosceles trapezoid \(ABCD\) with \(AB = CD\). Let \(M\) be the point where the circle touches side \(CD\), \(K\) be the intersection point of the circle with segment \(AM\), and \(L\) be the intersection point of the circle with segment \(BM\). Calculate the value of \(\frac{AM}{AK} + \frac{BM}{BL}\). | 10 | 0.25 |
Calculate the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty} \frac{(n+1)^{3}-(n-1)^{3}}{(n+1)^{2}+(n-1)^{2}}
\] | 3 | 0.875 |
We will call a date diverse if its representation in the format DD/MM/YY (day-month-year) includes all digits from 0 to 5. How many diverse dates are there in the year 2013? | 2 | 0.5 |
Find \( g(2021) \) if for any real numbers \( x, y \) the following equation holds:
\[ g(x-y) = g(x) + g(y) - 2022(x + y) \] | 4086462 | 0.5 |
Plot the graph of the function \( \mathrm{y} = \sqrt{4 \sin ^{4} x - 2 \cos 2x + 3} + \sqrt{4 \cos ^{4} x + 2 \cos 2x + 3} \). | 4 | 0.75 |
Find the curvature of the curve:
1) \( x=t^{2}, y=2 t^{3} \) at the point where \( t=1 \);
2) \( y=\cos 2 x \) at the point where \( x=\frac{\pi}{2} \). | 4 | 0.875 |
For how many natural numbers \( n \) not exceeding 600 are the triples of numbers
\[ \left\lfloor \frac{n}{2} \right\rfloor, \left\lfloor \frac{n}{3} \right\rfloor, \left\lfloor \frac{n}{5} \right\rfloor \quad\text{and}\quad \left\lfloor \frac{n+1}{2} \right\rfloor, \left\lfloor \frac{n+1}{3} \right\rfloor, \left\lfloor \frac{n+1}{5} \right\rfloor \]
distinct? As always, \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \). | 440 | 0.375 |
If a zero is appended to the right of a given number and the result is subtracted from 143, the resulting value is three times the given number. What is the given number? | 11 | 0.875 |
Find the smallest positive integer \( n \) such that in any 2-coloring of \( K_n \), there exist two monochromatic triangles of the same color that do not share any edge.
(1991 China National Training Team Problem) | 8 | 0.25 |
In a mathematics competition consisting of three problems, A, B, and C, among the 39 participants, each person solved at least one problem. Among those who solved problem A, there are 5 more people who only solved A than those who solved A and any other problems. Among those who did not solve problem A, the number of people who solved problem B is twice the number of people who solved problem C. Additionally, the number of people who only solved problem A is equal to the combined number of people who only solved problem B and those who only solved problem C. What is the maximum number of people who solved problem A? | 23 | 0.125 |
a) Consider a prime $p$ that divides $10^{n} + 1$ for some positive integer $n$. For example, $p = 7$ divides $10^{3} + 1$. By analyzing the main period of the decimal representation of $\frac{1}{p}$, verify that the number of times the digit $i$ appears is equal to the number of times the digit $9 - i$ appears for each $i \in \{0, 1, 2, \ldots, 9\}$.
b) Consider a prime number $p$ that does not divide 10 and suppose that the period of the decimal representation of $\frac{1}{p}$ is $2k$. Is it always possible to break the period into two numbers that sum up to $10^k - 1$? For example, the period of $\frac{1}{7}$ has a length of $6 = 2k$, as it is equal to 142857. See that $142 + 857 = 999 = 10^3 - 1 = 10^k - 1$.
c) Given
$$
x = \frac{1}{1998} + \frac{1}{19998} + \frac{1}{199998} + \ldots
$$
when we write $2x$ as a decimal number, what is the $59^{\text{th}}$ digit after the decimal point? | 1 | 0.5 |
The sequence \(\left|a_{n}\right|\) satisfies \(a_{1} = 19\), \(a_{2} = 98\), and \(a_{n+2} = a_{n} - \frac{2}{a_{n+11}}\). When \(a_{n} = 0\), what is \(m\)? | 933 | 0.25 |
Given \(0 \leq x \leq \pi\), and
\[
3 \sin \frac{x}{2} = \sqrt{1 + \sin x} - \sqrt{1 - \sin x}
\]
Find \(\tan x\). | 0 | 0.875 |
A square is divided into rectangles whose sides are parallel to the sides of the square. For each of these rectangles, the ratio of its shorter side to its longer side is calculated. Show that the sum of these ratios is at least 1. | 1 | 0.75 |
On a board, the digit 1 is written $n$ times. An operation consists of choosing two numbers $a$ and $b$ written on the board, erasing them, and writing $\frac{a+b}{4}$ in their place. Show that the number written on the board after $n-1$ steps is greater than or equal to $\frac{1}{n}$. | \frac{1}{n} | 0.25 |
In rectangle \(ABCD\), points \(E\) and \(F\) lie on sides \(AB\) and \(CD\) respectively such that both \(AF\) and \(CE\) are perpendicular to diagonal \(BD\). Given that \(BF\) and \(DE\) separate \(ABCD\) into three polygons with equal area, and that \(EF = 1\), find the length of \(BD\). | \sqrt{3} | 0.375 |
A cup is filled with a salt solution at a concentration of 15%. There are three iron balls in large, medium, and small sizes, with their volume ratio being 10:5:3. First, the small ball is placed into the cup of salt solution, causing 10% of the salt solution to overflow. After removing the small ball, the medium ball is placed into the cup and then removed. Next, the large ball is placed into the cup and then removed. Finally, pure water is added to the cup to fill it to the top. What is the final concentration of the salt solution in the cup? | 10 \% | 0.125 |
Find the total number of positive integers \( n \) not more than 2013 such that \( n^4 + 5n^2 + 9 \) is divisible by 5.
| 1611 | 0.625 |
Find the exact value of \(\tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{1}{5}\right) + \tan^{-1}\left(\frac{1}{8}\right)\). | \frac{\pi}{4} | 0.875 |
Denote the circles around $A$ and $B$ in problem 1448 by $k_{1}$ and $k_{2}$, respectively, with radii $r_{1}$ and $r_{2}$. Let $k$ be a circle that is tangent to $k_{1}$ and $k_{2}$ from the inside, with radius $r_{3}$. What does the ratio $r_{3} / r_{1}$ approach when, fixing $A$ and $B$, the point $C$ approaches $A$, i.e., as $r_{1}$ approaches 0? | \frac{1}{2} | 0.25 |
Let's calculate the mass of sodium carbonate:
$$
m\left(\mathrm{Na}_{2} \mathrm{CO}_{3}\right)=n \cdot M=0.125 \cdot 106=13.25 \text{ g}
$$
Next, let's determine the mass fraction of sodium carbonate in the initial solution:
$$
\omega\left(\mathrm{Na}_{2} \mathrm{CO}_{3}\right)=\frac{m\left(\mathrm{Na}_{2} \mathrm{CO}_{3}\right) \cdot 100\%}{m_{\mathrm{p}-\mathrm{pa}}\left(\mathrm{Na}_{2} \mathrm{CO}_{3}\right)}=\frac{13.25 \cdot 100\%}{132.5}=10\%
$$ | 10\% | 0.5 |
Using 6 squares with a side length of 1 can form a shape with a side length of 1. To form a regular hexagon with a side length of 5, how many equilateral triangles with a side length of 1 are needed? | 150 | 0.5 |
A regular tessellation is formed by identical regular \( m \)-polygons for some fixed \( m \). Find the sum of all possible values of \( m \). | 13 | 0.875 |
Given the quadratic polynomial \( f(x) = a x^{2} - a x + 1 \), it is known that \( |f(x)| \leq 1 \) for \( 0 \leq x \leq 1 \). Find the maximum possible value of \( a \). | 8 | 0.75 |
If a natural number \( n \) can be uniquely represented as the sum \( a_1 + a_2 + \cdots + a_k \) and the product \( a_1 a_2 \cdots a_k \) of \( k \) natural numbers (\( k \geq 2 \)), then this natural number is called a "good" number. For example, 10 is a "good" number because \( 10 = 5 + 2 + 1 + 1 + 1 = 5 \times 2 \times 1 \times 1 \times 1 \), and this representation is unique. Use the notion of prime numbers to determine which natural numbers are "good" numbers. | n | 0.125 |
Solve the equation \(3^{x} + 4^{y} = 5^{z}\) for positive integers \(x, y,\) and \(z\). | x = 2, y = 2, z = 2 | 0.125 |
Jirka drew a square grid with 25 squares. He then wanted to color each square so that no two squares of the same color share a common vertex.
What is the minimum number of colors Jirka had to use? | 4 | 0.25 |
Given two parallel lines at a distance of 15 units from each other, there is a point \( M \) that is 3 units away from one of them. A circle is drawn through the point \( M \) and tangent to both lines. Find the distance between the projections of the center and the point \( M \) onto one of the given lines. | 6 | 0.875 |
Given that the interior angles \(A, B, C\) of triangle \(\triangle ABC\) are opposite to the sides \(a, b, c\) respectively, and that \(A - C = \frac{\pi}{2}\), and \(a, b, c\) form an arithmetic sequence, find the value of \(\cos B\). | \frac{3}{4} | 0.375 |
Evaluate the limit as \( x \) approaches 1 of \( \frac{\sin 3 \pi x}{\sin 2 \pi x} \). | -\frac{3}{2} | 0.875 |
Given that \( p \) and \( q \) are real numbers with \( pq = b \) and \( p^{2}q + q^{2}p + p + q = 70 \). If \( c = p^{2} + q^{2} \), find the value of \( c \). | 31 | 0.25 |
What is the common ratio of the geometric sequence \( a + \log_{2} 3, a + \log_{4} 3, a + \log_{8} 3 \)? | \frac{1}{3} | 0.875 |
At the first site, higher-class equipment was used, while at the second site, first-class equipment was used, with higher-class being less than first-class. Initially, 30% of the equipment from the first site was transferred to the second site. Then, 10% of the equipment at the second site was transferred to the first site, with half of the transferred equipment being first-class. After this, the amount of higher-class equipment at the first site exceeded that at the second site by 6 units, and the total amount of equipment at the second site increased by more than 2% compared to the initial amount. Find the total amount of first-class equipment. | 17 | 0.625 |
Find the number of subsets $\{a, b, c\}$ of $\{1,2,3,4, \ldots, 20\}$ such that $a<b-1<c-3$. | 680 | 0.375 |
Two circles touch each other internally at point \( C \). A tangent is drawn at point \( P \) on the smaller circle, which intersects the larger circle at points \( A \) and \( B \). Show that \( CP \) intersects the arc \( AB \) of the larger circle at its midpoint \( Q \). | Q | 0.5 |
Given the vectors $\boldsymbol{a}=(0,1)$, $\boldsymbol{b}=\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$, and $\boldsymbol{c}=\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$. Real numbers $x, y, z$ satisfy $x \boldsymbol{a}+y \boldsymbol{b}+z \boldsymbol{c}=(1,2)$. Find the minimum value of $x^{2}+y^{2}+z^{2}$. | \frac{10}{3} | 0.375 |
In a circle there are 101 numbers written. It is known that among any five consecutive numbers, there are at least two positive numbers. What is the minimum number of positive numbers that can be among these 101 written numbers? | 41 | 0.5 |
The diagonal $BD$ of parallelogram $ABCD$ forms angles of $45^{\circ}$ with side $BC$ and with the altitude from vertex $D$ to side $AB$.
Find the angle $ACD$. | 45^\circ | 0.875 |
Let \( n > 0 \) be an integer. Each face of a regular tetrahedron is painted in one of \( n \) colors (the faces are not necessarily painted different colors.) Suppose there are \( n^3 \) possible colorings, where rotations, but not reflections, of the same coloring are considered the same. Find all possible values of \( n \). | 1 \text{ and } 11 | 0.875 |
Find the positive real solution \((x, y, z)\) of the system of equations:
$$
\left\{\begin{array}{l}
2 x^{3}=2 y\left(x^{2}+1\right)-\left(z^{2}+1\right), \\
2 y^{4}=3 z\left(y^{2}+1\right)-2\left(x^{2}+1\right), \\
2 z^{5}=4 x\left(z^{2}+1\right)-3\left(y^{2}+1\right)
\end{array}\right.
$$ | (1, 1, 1) | 0.875 |
A flea named Kuzya can make a jump exactly 15 mm in any direction on a plane. Its task is to get from point $A$ to point $B$ on the plane, with the distance between them being 2020 cm. What is the minimum number of jumps it must make to accomplish this? | 1347 | 0.625 |
Anya calls a date beautiful if all 6 digits of its record are different. For example, 19.04.23 is a beautiful date, while 19.02.23 and 01.06.23 are not. How many beautiful dates are there in the year 2023? | 30 | 0.25 |
Let the function \( f(x) = \frac{1}{2} + \log_{2} \frac{x}{1-x} \). Define \( S_n = \sum_{i=1}^{n-1} f\left(\frac{i}{n}\right) \), where \( n \in \mathbf{N}^{*} \) and \( n \geqslant 2 \). Determine \( S_n \). | \frac{n-1}{2} | 0.625 |
Given the numbers \(a, b, c, d\) such that \(a^{2} + b^{2} = 1\), \(c^{2} + d^{2} = 1\), and \(ac + bd = 0\). Calculate \(ab + cd\). | 0 | 0.875 |
All positive integer solutions $(a, b)$ of the equation $a^{3}-a^{2} b+a^{2}+2 a+2 b+1=0$ are $\qquad$ . | (5, 7) | 0.25 |
What is the minimum force required to press a cube with a volume of $10 \, \text{cm}^{3}$, floating in water, so that it is completely submerged? The density of the material of the cube is $400 \, \text{kg/m}^{3}$, and the density of water is $1000 \, \text{kg/m}^{3}$. Give the answer in SI units. Assume the acceleration due to gravity is $10 \, \text{m/s}^{2}$. | 0.06 \, \text{N} | 0.75 |
The sum of the non-negative real numbers \( s_{1}, s_{2}, \ldots, s_{2004} \) is 2. Additionally, we know that
\[ s_{1} s_{2} + s_{2} s_{3} + \cdots + s_{2003} s_{2004} + s_{2004} s_{1} = 1 \]
Determine the minimum and maximum possible values of the expression
\[ S = s_{1}^2 + s_{2}^2 + \cdots + s_{2004}^2 \]
under these conditions. | 2 | 0.25 |
Let the odd function \( f(x) \) have a domain of \( [-2,2] \), and be decreasing in the interval \( [-2,0] \), satisfying
$$
f(1-m)+f\left(1-m^{2}\right)<0.
$$
Determine the range of values for the real number \( m \). | [-1, 1) | 0.75 |
Given a triangle \(ABC\). Point \(P\) is the center of the inscribed circle. Find the angle \(B\), given that \(R_{ABC} = R_{APC}\), where \(R_{ABC}\) and \(R_{APC}\) are the circumradii of triangles \(ABC\) and \(APC\) respectively. | 60^\circ | 0.875 |
In the interval $[0, \pi]$, how many solutions does the trigonometric equation $\cos 7x = \cos 5x$ have? | 7 | 0.25 |
Roma thought of a natural number for which the sum of its digits is divisible by 8. He then added 2 to the number and again obtained a number whose sum of digits is divisible by 8. Find the smallest number that Roma could have thought of. | 699 | 0.625 |
As shown in the figure, two ants, A and B, move on the following circumferences. \( AC \) is the diameter of the larger circle. Point \( B \) is on \( AC \), with \( AB \) and \( BC \) being the diameters of two smaller circles respectively. Ant A moves clockwise on the larger circle, while Ant B moves in a "8" pattern along the two smaller circles in the direction indicated by the arrow (A → B → C → B → A). Both ants start from point \( A \) at the same time and keep moving continuously. The speed ratio of Ant A to Ant B is \( v_{\text{A}} : v_{\text{B}} = 3 : 2 \). After \( T_1 \) minutes, the two ants meet. Then, Ant A increases its speed by \(\frac{1}{3}\) while Ant B's speed remains unchanged, and they continue to move along their original paths. After \( T_2 \) minutes, the two ants meet again. Given that \( T_1 + T_2 = 100^3 - 99^3 + 98^3 - 97^3 + \dots + 2^3 - 1^3 \), determine how many minutes it takes for Ant A to complete one round of the larger circle at its original speed (express the answer as an improper fraction). | \frac{1015000}{9} | 0.375 |
Even natural numbers \( a \) and \( b \) are such that the greatest common divisor (GCD) of \( a \) and \( b \) plus the least common multiple (LCM) of \( a \) and \( b \) equals \( 2^{23} \). How many distinct values can the LCM of \( a \) and \( b \) take? | 22 | 0.25 |
After expanding and combining like terms in the expression \((x+y+z)^{2028} + (x-y-z)^{2028}\), how many monomials of the form \(x^{a} y^{b} z^{c}\) have a non-zero coefficient? | 1030225 | 0.125 |
How many four-digit numbers satisfy the following two conditions:
(1) The sum of any two adjacent digits is not greater than 2;
(2) The sum of any three adjacent digits is not less than 3. | 1 | 0.5 |
Let \( ABCDEF \) be a regular hexagon. A frog starts at vertex \( A \) and can jump to one of the two adjacent vertices randomly on each jump. If the frog reaches point \( D \) within 5 jumps, it stops jumping. If the frog cannot reach \( D \) within 5 jumps, it stops jumping after completing 5 jumps. Determine the total number of different jumping sequences possible from the start until it stops. | 26 | 0.375 |
The Minions need to make jam within the specified time. Kevin can finish the job 4 days earlier if he works alone, while Dave would finish 6 days late if he works alone. If Kevin and Dave work together for 4 days and then Dave completes the remaining work alone, the job is completed exactly on time. How many days would it take for Kevin and Dave to complete the job if they work together? | 12 | 0.875 |
If digits in different positions are considered different, then in a base-$d$ numbering system, $nd$ digits allow writing $d^n$ numbers (from 0 to $d^n-1$). Which numbering system is the most economical in this respect, i.e., allows writing the most significant number of numbers using a given number of digits? (When comparing numbering systems with bases $d_1$ and $d_2$, we only consider sets of $m$ digits, where $m$ is divisible by both $d_1$ and $d_2$.) | 3 | 0.625 |
Let \( n \) be a natural number with the following property: from the numbers \( 1, 2, 3, \cdots, n \), any selection of 51 different numbers must include two numbers whose sum is equal to 101. What is the maximum value of \( n \) with this property? | 100 | 0.25 |
In parallelogram \(ABCD\), \(AB = 1\), \(BC = 4\), and \(\angle ABC = 60^\circ\). Suppose that \(AC\) is extended from \(A\) to a point \(E\) beyond \(C\) so that triangle \(ADE\) has the same area as the parallelogram. Find the length of \(DE\). | 2\sqrt{3} | 0.625 |
Given a fixed point \( A(2,0) \) on a plane and a moving point \( P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right) \), determine the area of the figure swept by line segment \( AP \) as \( t \) changes from \( 15^{\circ} \) to \( 45^{\circ} \). | \frac{\pi}{6} | 0.625 |
Find the value of the expression:
\[ \frac{\sin (2 \alpha + 2 \pi) + 2 \sin (4 \alpha - \pi) + \sin (6 \alpha + 4 \pi)}{\cos (6 \pi - 2 \alpha) + 2 \cos (4 \alpha - \pi) + \cos (6 \alpha - 4 \pi)}. \] | \tan 4\alpha | 0.625 |
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