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Let \( P(x) \) be the polynomial of minimal degree such that \( P(k) = \frac{720k}{k^2 - 1} \) for \( k \in \{2, 3, 4, 5\} \). Find the value of \( P(6) \). | 48 | 0.5 |
Points \( A, B, C, D \) are on the same circle, and \( BC = CD = 4 \). Let \( E \) be the intersection of \( AC \) and \( BD \), with \( AE = 6 \). If the lengths of segments \( BE \) and \( DE \) are both integers, what is the length of \( BD \)? | 7 | 0.75 |
Calculate the limit of the function:
\[
\lim _{x \rightarrow a}\left(2-\frac{x}{a}\right)^{\tan\left(\frac{\pi x}{2a}\right)}
\] | e^{\frac{2}{\pi}} | 0.875 |
Find the value of $\frac{1^{4}+2009^{4}+2010^{4}}{1^{2}+2009^{2}+2010^{2}}$. | 4038091 | 0.375 |
Two prime numbers that differ by exactly 2 are called twin primes. For example, 3 and 5 are twin primes, as are 29 and 31. In number theory research, twin primes are one of the most popular topics. If a pair of twin primes both do not exceed 200, what is the maximum sum of these two prime numbers? | 396 | 0.875 |
Given three points \( A, B, C \) forming a triangle with angles \( 30^{\circ}, 45^{\circ}, 105^{\circ} \). Two of these points are selected, and the perpendicular bisector of the segment connecting them is drawn, after which the third point is reflected relative to this perpendicular bisector, resulting in a fourth point \( D \). This procedure continues with the set of the four resulting points: two points are chosen, a perpendicular bisector is drawn, and all points are reflected relative to it. What is the maximum number of distinct points that can be obtained as a result of repeatedly applying this procedure? | 12 | 0.125 |
In a Cartesian coordinate system, \( A(1,2) \), \( B(3,0) \), and \( P \) are points on the circle \( (x-3)^{2}+(y-2)^{2}=1 \). Suppose
$$
\overrightarrow{O P}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B} \quad (\lambda, \mu \in \mathbf{R}).
$$
Find the minimum value of \( 11\lambda + 9\mu \). | 12 | 0.625 |
Admiral Ackbar needs to send a 5-character message through hyperspace to the Rebels. Each character is a lowercase letter, and the same letter may appear more than once in a message. When the message is beamed through hyperspace, the characters come out in a random order. Ackbar chooses his message so that the Rebels have at least a \(\frac{1}{2}\) chance of getting the same message he sent. How many distinct messages could he send? | 26 | 0.5 |
Given the function \( f(x)=\frac{m-2 \sin x}{\cos x} \) is monotonically decreasing on the interval \(\left(0, \frac{\pi}{2}\right)\), find the range of the real number \( m \). | (-\infty, 2] | 0.25 |
Teacher Kadrnožková bought tickets at the zoo ticket office for her students and herself. An adult ticket was more expensive than a student ticket but not more than twice the price. Teacher Kadrnožková paid a total of 994 CZK. Teacher Hnízdo had three more students than his colleague, and therefore, he paid 1120 CZK for his students and himself.
1. How many students did teacher Hnízdo have with him?
2. How much did an adult ticket cost? | 70 | 0.125 |
At vertex $A$ of a unit square $ABCD$, an ant begins its journey. It needs to reach point $C$, where the entrance to an anthill is located. Points $A$ and $C$ are separated by a vertical wall in the shape of an isosceles right triangle with hypotenuse $BD$. Find the length of the shortest path the ant must take to reach the anthill. | 2 | 0.625 |
Given that \( y = f(x) \) is an increasing function on \(\mathbb{R}\) and \( f(x) = f^{-1}(x) \), show that \( f(x) = x \). | f(x) = x | 0.875 |
Two ants crawled along their own closed routes on a $7 \times 7$ board. Each ant crawled only along the sides of the cells of the board and visited each of the 64 vertices of the cells exactly once. What is the minimum possible number of such sides that both the first and the second ant crawled along? | 16 | 0.75 |
Along the shore of a circular lake, there are apple trees. Petya and Vasya start walking from point $A$ on the shore in opposite directions along the shore, counting all the apple trees they encounter and all the apples growing on the trees. When they meet at some point $B$, they compare their results. It turns out that Petya counted twice as many apple trees as Vasya, and seven times more apples than Vasya. Surprised by this result, they decided to repeat the experiment. They started from point $B$ in the same directions and met again at point $C$. It turned out that on the way from $B$ to $C$, Petya again counted twice as many apple trees as Vasya, and seven times more apples than Vasya. Their surprise grew, and they decided to repeat the experiment once more. Setting out from $C$ in the same directions, they met at point $D$. It turned out that Petya again counted twice as many apple trees as Vasya. Who counted more apples on the way from $C$ to $D$, and by how many times?
Answer: On the way from $C$ to $D$, Vasya counted 3 times more apples than Petya. | 3 | 0.75 |
The side length of the base of a regular triangular pyramid is \(a\). The lateral edge forms an angle of \(60^{\circ}\) with the plane of the base. Find the radius of the sphere circumscribed around the pyramid. | \frac{2a}{3} | 0.625 |
Solve the equation \(x^{3} - \lfloor x \rfloor = 3\), where \(\lfloor x \rfloor\) represents the greatest integer less than or equal to \(x\). | \sqrt[3]{4} | 0.875 |
In triangle \(ABC\), the angle bisectors \(AD\) and \(BE\) are drawn. Find the measure of angle \(C\), given that \(AD \cdot BC = BE \cdot AC\) and \(AC \neq BC\). | 60^{\circ} | 0.875 |
\(A\) and \(B\) are positive integers less than 10 such that \(21A104 \times 11 = 2B8016 \times 9\).
Find \(A\).
Find \(B\). | 5 | 0.875 |
Calculate the definite integral:
$$
\int_{\pi / 4}^{\arccos (1 / \sqrt{26})} \frac{36 \, dx}{(6 - \tan x) \sin 2x}
$$ | 6 \ln 5 | 0.875 |
Let \( \triangle DEF \) be a triangle and \( H \) the foot of the altitude from \( D \) to \( EF \). If \( DE = 60 \), \( DF = 35 \), and \( DH = 21 \), what is the difference between the minimum and the maximum possible values for the area of \( \triangle DEF \)? | 588 | 0.875 |
Petya places "+" and "-" signs in all possible ways into the expression $1 * 2 * 3 * 4 * 5 * 6$ at the positions of the asterisks. For each arrangement of signs, he calculates the resulting value and writes it on the board. Some numbers may appear on the board multiple times. Petya then sums all the numbers on the board. What is the sum that Petya obtains? | 32 | 0.875 |
There are 10 different balls that need to be placed into 8 different empty boxes, with each box containing at least one ball. How many ways are there to do this? | 30240000 | 0.125 |
On an island, there are only knights, who always tell the truth, and liars, who always lie. There are at least two knights and at least two liars. One day, each islander pointed to each of the others in turn and said either "You are a knight!" or "You are a liar!". The phrase "You are a liar!" was said exactly 230 times. How many times was the phrase "You are a knight!" said? | 526 | 0.75 |
In the isosceles triangle \(ABC\) (\(AB = BC\)), a point \(D\) is taken on the side \(BC\) such that \(BD : DC = 1 : 4\). In what ratio does the line \(AD\) divide the altitude \(BE\) of triangle \(ABC\), counting from vertex \(B\)? | 1:2 | 0.875 |
At the school reunion, 45 people attended. It turned out that any two of them who have the same number of acquaintances among the attendees are not acquainted with each other. What is the maximum number of pairs of acquaintances that could be among the attendees? | 870 | 0.125 |
For positive integers \( n \) and \( k \), let \( \mho(n, k) \) be the number of distinct prime divisors of \( n \) that are at least \( k \). For example, \( \mho(90,3) = 2 \), since the only prime factors of 90 that are at least 3 are 3 and 5. Find the closest integer to
\[
\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} \frac{\mho(n, k)}{3^{n+k-7}}
\] | 167 | 0.625 |
Find the smallest positive integer \( n \) that satisfies the following condition: For any \( n \) points \( A_1, A_2, \dots, A_n \) chosen on the circumference of a circle \( O \), among the \(\binom{n}{2}\) angles \(\angle A_i O A_j \) (where \( 1 \leq i < j \leq n \)), there are at least 2007 angles that do not exceed \( 120^\circ \). | 91 | 0.25 |
A digit was crossed out from a six-digit number, resulting in a five-digit number. When this five-digit number was subtracted from the original six-digit number, the result was 654321. Find the original six-digit number. | 727023 | 0.75 |
In an equilateral triangular prism \( S-ABC \), the lateral edges are equal in length to the edges of the base. If \( E \) and \( F \) are the midpoints of \( SC \) and \( AB \) respectively, what is the angle between the skew lines \( EF \) and \( SA \)? | 45^{\circ} | 0.375 |
In triangle \( \triangle ABC \), \(\cos A\), \(\sin A\), and \(\tan B\) form a geometric sequence with a common ratio of \(\frac{3}{4}\). What is \(\cot C\)? | -\frac{53}{96} | 0.75 |
Rectangle \( R_{0} \) has sides of lengths 3 and 4. Rectangles \( R_{1}, R_{2} \), and \( R_{3} \) are formed such that:
- all four rectangles share a common vertex \( P \),
- for each \( n=1,2,3 \), one side of \( R_{n} \) is a diagonal of \( R_{n-1} \),
- for each \( n=1,2,3 \), the opposite side of \( R_{n} \) passes through a vertex of \( R_{n-1} \) such that the center of \( R_{n} \) is located counterclockwise of the center of \( R_{n-1} \) with respect to \( P \).
Compute the total area covered by the union of the four rectangles. | 30 | 0.125 |
A cube is dissected into 6 pyramids by connecting a given point in the interior of the cube with each vertex of the cube, so that each face of the cube forms the base of a pyramid. The volumes of five of these pyramids are 200, 500, 1000, 1100, and 1400. What is the volume of the sixth pyramid? | 600 | 0.5 |
ABC is an acute-angled triangle with area 1. A rectangle R has its vertices on the sides of the triangle, with two vertices on BC, one on AC, and one on AB. Another rectangle S has its vertices on the sides of the triangle formed by the points A, R3, and R4, with two vertices on R3R4 and one on each of the other two sides. What is the maximum total area of rectangles R and S over all possible choices of triangle and rectangles? | \frac{2}{3} | 0.625 |
Horizontal parallel segments \( AB = 10 \) and \( CD = 15 \) are the bases of trapezoid \( ABCD \). Circle \(\gamma\) of radius 6 has its center within the trapezoid and is tangent to sides \( AB \), \( BC \), and \( DA \). If side \( CD \) cuts out an arc of \(\gamma\) measuring \(120^\circ\), find the area of \(ABCD\). | \frac{225}{2} | 0.125 |
Calculate the limit of the function:
\[
\lim _{x \rightarrow -1} \frac{x^{3}-2 x-1}{x^{4}+2 x+1}
\] | -\frac{1}{2} | 0.875 |
Let $x$ and $y$ be non-zero real numbers such that $\frac{x \sin \frac{\pi}{5} + y \cos \frac{\pi}{5}}{x \cos \frac{\pi}{5} - y \sin \frac{\pi}{5}} = \tan \frac{9\pi}{20}$.
(1) Find the value of $\frac{y}{x}$;
(2) In triangle $ABC$, if $\tan C=\frac{y}{x}$, find the maximum value of $\sin 2A + 2 \cos B$. | \frac{3}{2} | 0.75 |
\[
\frac{\cos 68^{\circ} \cos 8^{\circ} - \cos 82^{\circ} \cos 22^{\circ}}{\cos 53^{\circ} \cos 23^{\circ} - \cos 67^{\circ} \cos 37^{\circ}}
\] | 1 | 0.625 |
A coin is tossed 10 times. Find the probability that two heads do not appear consecutively at any time. | \frac{9}{64} | 0.5 |
Given that \(\theta_{1}, \theta_{2}, \cdots, \theta_{n}\) are all non-negative real numbers and satisfy the equation
\[
\theta_{1} + \theta_{2} + \cdots + \theta_{n} = \pi.
\]
Find the maximum value of \(\sin^2 \theta_{1} + \sin^2 \theta_{2} + \cdots + \sin^2 \theta_{n}\). | \frac{9}{4} | 0.25 |
Let \( x \) and \( y \) be two non-zero numbers such that \( x^{2} + xy + y^{2} = 0 \) ( \( x \) and \( y \) are complex numbers, but that is not very important). Find the value of
$$
\left(\frac{x}{x+y}\right)^{2013}+\left(\frac{y}{x+y}\right)^{2013}
$$ | -2 | 0.75 |
Given the sequence $\left\{a_{n}\right\}$ which satisfies $a_{n} = a_{n-1} - a_{n-2}$ for $n \geq 3$. If the sum of its first 1492 terms is 1985 and the sum of its first 1985 terms is 1492, find the sum of its first 2001 terms. | 986 | 0.375 |
The numbers \(a, b, c, d\) belong to the interval \([-6, 6]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 156 | 0.625 |
The integers from 1 to \( n \) are written in increasing order from left to right on a blackboard. David and Goliath play the following game: starting with David, the two players alternate erasing any two consecutive numbers and replacing them with their sum or product. Play continues until only one number on the board remains. If it is odd, David wins, but if it is even, Goliath wins. Find the 2011th smallest positive integer greater than 1 for which David can guarantee victory. | 4022 | 0.625 |
In the figure below, triangle \(ABC\) and rectangle \(PQRS\) have the same area and the same height of 1. For each value of \(x\) between 0 and 1, trapezoid \(ABED\) of height \(x\) is drawn, and then rectangle \(PQNM\) of the same area as the trapezoid is drawn, as shown in the figure. Let \(f\) be the function that assigns to each \(x\) the height of rectangle \(PQNM\).
a) What is the ratio between \(AB\) and \(PQ\)?
b) What is the value of \(f\left(\frac{1}{2}\right)\)?
c) Find the expression of \(f(x)\) and draw the graph of \(f\). | 2x - x^2 | 0.375 |
How many children did my mother have?
If you asked me this question, I would only tell you that my mother dreamed of having at least 19 children, but she couldn't make this dream come true; however, I had three times more sisters than cousins, and twice as many brothers as sisters. How many children did my mother have? | 10 | 0.75 |
A part of a book has fallen out. The number of the first fallen page is 387, and the number of the last page consists of the same digits but in a different order. How many sheets fell out of the book? | 176 | 0.25 |
Each of the 12 knights sitting around a round table has chosen a number, and all the numbers are different. Each knight claims that the number they have chosen is greater than the numbers chosen by their neighbors on the right and left. What is the maximum number of these claims that can be true? | 6 | 0.75 |
If \( a, b, c, \) and \( d \) satisfy \( \frac{a}{b} = \frac{2}{3} \) and \( \frac{c}{b} = \frac{1}{5} \) and \( \frac{c}{d} = \frac{7}{15} \), what is the value of \( \frac{a b}{c d} \)? | \frac{70}{9} | 0.875 |
Is there a positive integer \( m \) such that the equation
\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{abc} = \frac{m}{a+b+c} \]
has infinitely many solutions in positive integers \( a, b, c \)? | 12 | 0.375 |
Let $a$ be an integer such that $x^2 - x + a$ divides $x^{13} + x + 90$. Find the value of $a$. | 2 | 0.75 |
What is the smallest number of regular hexagons with a side length of 1 needed to completely cover a disc with a radius of 1? | 3 | 0.5 |
It is known that all 'krakozyabrs' have either horns or wings (or both). From the results of a global census of 'krakozyabrs,' it was found that 20% of the 'krakozyabrs' with horns also have wings, and 25% of the 'krakozyabrs' with wings also have horns. How many 'krakozyabrs' are left in the world if it is known that their number is more than 25 but less than 35? | 32 | 0.875 |
Given an isosceles trapezoid $\mathrm{ABCE}$ with bases $\mathrm{BC}$ and $\mathrm{AE}$, where $\mathrm{BC}$ is smaller than $\mathrm{AE}$ with lengths 3 and 4 respectively. The smaller lateral side $\mathrm{AB}$ is equal to $\mathrm{BC}$. Point $\mathrm{D}$ lies on $\mathrm{AE}$ such that $\mathrm{AD}$ : $\mathrm{DE}=3:1$. Point $\mathrm{F}$ lies on $\mathrm{AD}$ such that $\mathrm{AF}$ : $\mathrm{FD}=2:1$. Point $\mathrm{G}$ lies on $\mathrm{BD}$ such that $\mathrm{BG}$ : $\mathrm{GD}=1:2$. Determine the angle measure $\angle \mathrm{CFG}$. | 45 | 0.25 |
We need to calculate the expression with three significant digits' accuracy. To simplify the division, find an expression of the form (2) - with coefficients $A, B, C$ such that when multiplying it by (1), we obtain a rational denominator (a formula table can be used).
$$
\begin{gathered}
529 \\
\hline 12 \sqrt[3]{9}+52 \sqrt[3]{3}+49 \\
A \sqrt[3]{9}+B \sqrt[3]{3}+C
\end{gathered}
$$ | 3.55 | 0.625 |
120 identical spheres are tightly packed in the shape of a regular triangular pyramid. How many spheres are in the base? | 36 | 0.875 |
If \( n \in \mathbf{N}^{*} \), then \( \lim_{n \rightarrow \infty} \sin^{2}\left(\pi \sqrt{n^{2}+n}\right) = \) ? | 1 | 0.625 |
A semicircle with a radius of $3 \text{ cm}$ contains a $1 \text{ cm}$ radius semicircle and a $2 \text{ cm}$ radius semicircle within it, as shown in the figure. Calculate the radius of the circle that is tangent to all three semicircular arcs. | \frac{6}{7} | 0.25 |
Usually, I go up the escalator in the subway. I have calculated that when I walk up the moving escalator, I ascend 20 steps, and it takes me exactly 60 seconds. My wife walks up the stairs more slowly and only ascends 16 steps; therefore, her total time to ascend the escalator is longer - it is 72 seconds.
How many steps would I have to climb if the escalator suddenly breaks down? | 40 | 0.75 |
In triangle \(ABC\), angle \(A\) is \(40^\circ\). The triangle is randomly thrown onto a table. Find the probability that vertex \(A\) ends up east of the other two vertices. | \frac{7}{18} | 0.375 |
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{2}} \frac{1-\sin ^{3} x}{\cos ^{2} x}$ | \frac{3}{2} | 0.875 |
Find the distance from the point \( M_{0} \) to the plane passing through the three points \( M_{1}, M_{2}, M_{3} \).
$$
\begin{aligned}
& M_{1}(3, 10, -1) \\
& M_{2}(-2, 3, -5) \\
& M_{3}(-6, 0, -3) \\
& M_{0}(-6, 7, -10)
\end{aligned}
$$ | 7 | 0.875 |
Given 1997 points on a plane such that from any four points, three can be chosen to be collinear (on the same line). Show that at least 1996 of the points are on the same line. | 1996 | 0.625 |
Draw a rectangle. Connect the midpoints of the opposite sides to get 4 congruent rectangles. Connect the midpoints of the lower right rectangle for a total of 7 rectangles. Repeat this process infinitely. Let \( n \) be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing an edge have the same color and \( m \) be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing a corner have the same color. Find the ordered pair \((n, m)\). | (3, 4) | 0.25 |
Let \([x]\) be the largest integer not greater than \(x\), for example, \([2.5] = 2\). If \(a = 1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \cdots + \frac{1}{2004^{2}}\) and \(S = [a]\), find the value of \(a\). | 1 | 0.875 |
The equation \( x^{4} - 7x - 3 = 0 \) has exactly two real roots \( a \) and \( b \), where \( a > b \). Find the value of the expression \( \frac{a - b}{a^{4} - b^{4}} \). | \frac{1}{7} | 0.625 |
The sum of the digits of the birth years of Jean and Jack are equal, and the age of each of them starts with the same digit. Can you determine the difference in their ages? | 9 | 0.625 |
Four students from Harvard, one of them named Jack, and five students from MIT, one of them named Jill, are going to see a Boston Celtics game. However, they found out that only 5 tickets remain, so 4 of them must go back. Suppose that at least one student from each school must go see the game, and at least one of Jack and Jill must go see the game. How many ways are there of choosing which 5 people can see the game? | 104 | 0.75 |
A rectangle has the same perimeter and area as a rhombus, one of whose angles measures $30^{\circ}$. What is the ratio of the sides of the rectangle? | 3 + 2\sqrt{2} | 0.375 |
A small village has \( n \) people. During their yearly elections, groups of three people come up to a stage and vote for someone in the village to be the new leader. After every possible group of three people has voted for someone, the person with the most votes wins. This year, it turned out that everyone in the village had the exact same number of votes! If \( 10 \leq n \leq 100 \), what is the number of possible values of \( n \)? | 61 | 0.625 |
In the figure, \( AB \) is tangent to the circle at point \( A \), \( BC \) passes through the center of the circle, and \( CD \) is a chord of the circle that is parallel to \( AB \). If \( AB = 6 \) and \( BC = 12 \), what is the length of \( CD \)? | 7.2 \; \text{units} | 0.25 |
Describe all positive integer solutions $(m, n)$ of the equation $8m - 7 = n^2$ and provide the first value of $m$ (if it exists) greater than 1959. | 2017 | 0.375 |
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{\sqrt{3 n-1}-\sqrt[3]{125 n^{3}+n}}{\sqrt[3]{n}-n}
$$ | 5 | 0.75 |
A pentagon is inscribed around a circle, with the lengths of its sides being whole numbers, and the lengths of the first and third sides equal to 1. Into what segments does the point of tangency divide the second side? | \frac{1}{2} | 0.375 |
Given the function \( f(x) = \frac{m - 2 \sin x}{\cos x} \) is monotonically decreasing on the interval \(\left(0, \frac{\pi}{2} \right)\), find the range of the real number \( m \). | (-\infty, 2] | 0.75 |
Calculate the area of a pentagon whose vertices have the coordinates: \((-1, 2)\), \((1, 1)\), \((2, 3)\), \((3, -5)\), and \((-2, -4)\). | 25.5 | 0.875 |
The sum of the positive numbers \(a, b, c,\) and \(d\) is 4. Find the minimum value of the expression
$$
\frac{a^{8}}{\left(a^{2}+b\right)\left(a^{2}+c\right)\left(a^{2}+d\right)}+\frac{b^{8}}{\left(b^{2}+c\right)\left(b^{2}+d\right)\left(b^{2}+a\right)}+\frac{c^{8}}{\left(c^{2}+d\right)\left(c^{2}+a\right)\left(c^{2}+b\right)}+\frac{d^{8}}{\left(d^{2}+a\right)\left(d^{2}+b\right)\left(d^{2}+c\right)}
$$ | \frac{1}{2} | 0.875 |
A positive number \( x \) was increased by 69%. By what percentage did the number \( \sqrt{\frac{x}{5}} \) increase? | 30\% | 0.875 |
Circles \(\omega_{1}\) and \(\omega_{2}\) intersect at points \(A\) and \(B\), and a circle centered at point \(O\) encompasses circles \(\omega_{1}\) and \(\omega_{2}\), touching them at points \(C\) and \(D\) respectively. It turns out that points \(A, C\), and \(D\) are collinear. Find the angle \(\angle ABO\). | 90^\circ | 0.625 |
Find the minimum number of planes that divide a cube into at least 300 parts. | 13 | 0.75 |
For any positive integer \( n \), if
$$
1^{n} + 2^{n} + \cdots + (n-1)^{n} < C n^{n},
$$
then the minimum value of \( C \) is ______. | \frac{1}{e-1} | 0.75 |
Find all odd natural numbers greater than 500 but less than 1000, for which the sum of the last digits of all divisors (including 1 and the number itself) is equal to 33. | 729 | 0.375 |
In an eight-digit number, each digit (except the last one) is greater than the following digit. How many such numbers are there? | 45 | 0.625 |
Given positive integers \( m \), \( n \), and \( r \), where \( 1 \leq r \leq m \leq n \). An \( m \times n \) grid has a set of \( m \) squares that form a generalized diagonal (or simply diagonal) if each pair of squares in the set are in different rows and different columns.
Consider coloring some squares red on an \( m \times n \) grid such that each row and each column contains at most \( r \) red squares. Determine the minimum value of the positive integer \( a \) such that for any coloring scheme, it is possible to find \( a \) diagonals on the grid such that all the red squares are on these diagonals. | r | 0.25 |
What is the smallest natural number $n$ for which there exist natural numbers $x$ and $y$ satisfying the equation a) $x \cdot (x+n) = y^{2}$; b) $x \cdot (x+n) = y^{3}$? | n = 2 | 0.125 |
In the bottom-left corner of a chessboard, there is a checker piece. It can be moved one square up, one square to the right, or one square diagonally down-left. Is it possible to move the checker in such a way that it visits all the squares on the board exactly once? (7 points) | \text{No} | 0.375 |
Point \( M \) is the midpoint of side \( BC \) of the triangle \( ABC \), where \( AB = 17 \), \( AC = 30 \), and \( BC = 19 \). A circle is constructed with diameter \( AB \). A point \( X \) is chosen arbitrarily on this circle. What is the minimum possible length of the segment \( MX \)? | 6.5 | 0.5 |
The number $\overline{x y z t}$ is a perfect square such that the number $\overline{t z y x}$ is also a perfect square, and the quotient of the numbers $\overline{x y z t}$ and $\overline{t z y x}$ is also a perfect square. Determine the number $\overline{x y z t}$. (The overline indicates that the number is written in the decimal system.) | 9801 | 0.5 |
Given points $A(a-1, a)$ and $B(a, a+1)$, and a moving point $P$ such that its distance to the point $M(1,0)$ is greater by 1 than its distance to the $y$-axis. The locus of $P$ is a curve $C$. The line segment $AB$ intersects curve $C$ at exactly one point. Determine the range of the value of $a$. | [-1, 0] \cup [1, 2] | 0.875 |
The complex numbers \( z_1, z_2, \cdots, z_{100} \) satisfy: \( z_1 = 3 + 2i \), \( z_{n+1} = \overline{z_n} \cdot i^n \) for \( n = 1, 2, \cdots, 99 \) (where \( i \) is the imaginary unit). Find the value of \( z_{99} + z_{100} \). | -5 + 5i | 0.875 |
Is there a natural number \( n \) such that the number \( 6n^2 + 5n \) is a power of 2? | \text{No} | 0.375 |
Compute the positive integer less than 1000 which has exactly 29 positive proper divisors. (Here we refer to positive integer divisors other than the number itself.) | 720 | 0.625 |
Pedro wants to paint a cube-shaped box in such a way that faces sharing a common edge are painted in different colors. Calculate the minimum number of colors needed to paint the box in this manner. | 3 | 0.375 |
As shown in the figure, $\angle 1$ is equal to 100 degrees, $\angle 2$ is equal to 60 degrees, and $\angle 3$ is equal to 90 degrees. Find $\angle 4$. | 110^\circ | 0.875 |
Find the sum of the squares of the distances from a point \( M \), taken on the diameter of some circle, to the ends of any chord parallel to this diameter, if the radius of the circle is \( R \) and the distance from point \( M \) to the center of the circle is \( a \). | 2(a^2 + R^2) | 0.125 |
In quadrilateral \( \square ABCD \), point \( M \) lies on diagonal \( BD \) with \( MD = 3BM \). Let \( AM \) intersect \( BC \) at point \( N \). Find the value of \( \frac{S_{\triangle MND}}{S_{\square ABCD}} \). | \frac{1}{8} | 0.125 |
There are 1993 boxes containing balls arranged in a line from left to right. If the leftmost box contains 7 balls, and every four consecutive boxes contain a total of 30 balls, then how many balls are in the rightmost box? | 7 | 0.125 |
Two circles with radii 2 and 3 touch each other externally at point $A$. Their common tangent passing through point $A$ intersects their other two common tangents at points $B$ and $C$. Find $BC$. | 2 \sqrt{6} | 0.375 |
Three frogs on a swamp jumped in sequence. Each one landed exactly in the middle of the segment between the other two. The jump length of the second frog is 60 cm. Find the jump length of the third frog. | 30 \text{ cm} | 0.75 |
Given: Three vertices \( A, B, \) and \( C \) of a square are on the parabola \( y = x^2 \).
Find: The minimum possible area of such a square. | 2 | 0.375 |
Calculate the double integrals over the regions bounded by the specified lines:
a) \(\iint x^{2} y \, dx \, dy\); \(y=0\); \(y=\sqrt{2ax - x^{2}}\);
b) \(\iint_{\Omega} \sin(x+y) \, dx \, dy\); \(y=0\); \(y=x\); \(x+y=\frac{\pi}{2}\);
c) \(\iint_{\Omega} x^{2}(y-x) \, dx \, dy\); \(x=y^{2}\); \(y=x^{2}\). | -\frac{1}{504} | 0.75 |
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