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On a blackboard, there are 1989 consecutive natural numbers written from $1, 2, 3, \ldots, 1989$. A transformation is performed as follows: erase any two numbers on the blackboard and replace them with the remainder when the sum of the two erased numbers is divided by 19. After performing this transformation multiple times, there are two numbers left on the blackboard: one is 89 and the other is a single-digit number. What is this single-digit number? | 2 | 0.75 |
What is the minimum number of different numbers that must be chosen from $1, 2, 3, \ldots, 1000$ to ensure that among the chosen numbers, there exist 3 different numbers that can form the side lengths of a triangle? | 16 | 0.625 |
2011 is a four-digit number such that the sum of its digits is 4. How many four-digit numbers have a digit sum of 4? | 20 | 0.875 |
Based on the definition of the derivative, find \( f^{\prime}(0) \):
\[
f(x) = \begin{cases}
\frac{\cos x - \cos 3x}{x}, & x \neq 0 \\
0, & x = 0
\end{cases}
\] | 4 | 0.875 |
Find the sum of the digits of all numbers in the sequence \(1, 2, 3, \ldots, 99, 100\). | 901 | 0.5 |
The owner of an apartment insured it for 3,750,000 rubles (the actual cost of the apartment is 7,500,000 rubles). The actual damage amounted to 2,750,000 rubles. The insurance compensation under the proportional liability system was 1,350,000 rubles. Determine the amount of the deductible provided for in the contract. | 50000 | 0.875 |
Find the limits of the following functions:
1) \( f(x) = 2x - 3 - \frac{1}{x} \) as \( x \rightarrow 1 \):
2) \( y = \frac{x^3 - 3x^2 + 2x - 5}{x^2 + 2} \) as \( x \rightarrow -1 \):
3) \( y = x \sin \frac{1}{x} \) as \( x \rightarrow 0 \). | 0 | 0.375 |
Let \( n \) be a positive integer, and \( d \) be a digit in decimal notation. If \(\frac{n}{810} = \overline{0.d25d25d25\cdots}\), find \( n \). | 750 | 0.625 |
The recruits stood in a row one behind the other, all facing the same direction. Among them were three brothers: Peter, Nikolai, and Denis. There were 50 people ahead of Peter, 100 people ahead of Nikolai, and 170 people ahead of Denis. On the command "About-face!" everyone turned in the opposite direction. In doing so, it turned out that in front of one of the brothers now stood four times as many people as in front of another brother. How many recruits might there be in total, including the brothers? List all possible options. | 211 | 0.75 |
Let \( m > 1 \). Under the constraints
\[
\begin{cases}
y \geqslant x, \\
y \leqslant mx, \\
x + y \leqslant 1
\end{cases}
\]
the maximum value of the objective function \( z = x + my \) is less than 2. Determine the range of \( m \). | (1, 1 + \sqrt{2}) | 0.875 |
Does there exist a polyhedron where all the faces are triangles except one which is a pentagon, and such that all vertices have an even degree? | \text{No} | 0.625 |
The number 100 is represented as a sum of several two-digit numbers, and in each addend, the digits are swapped. What is the largest possible number that could be obtained in the new sum? | 406 | 0.25 |
In triangle \(ABC\), the median \(BK\), the angle bisector \(BE\), and the altitude \(AD\) are given.
Find the side \(AC\), if it is known that the lines \(BK\) and \(BE\) divide the segment \(AD\) into three equal parts, and \(AB=4\). | \sqrt{13} | 0.125 |
Given \( S_{n}=1+2+3+\cdots+n \) for \( n \in \mathbf{N} \), find the maximum value of the function \( f(n)=\frac{S_{n}}{(n+32) S_{n+1}} \). | \frac{1}{50} | 0.875 |
Show that the line
$$
\left(m^{2}+6m+3\right)x - \left(2m^{2}+18m+2\right)y - 3m + 2 = 0
$$
passes through a fixed point, regardless of the value assigned to \( m \). | (-1, -\frac{1}{2}) | 0.75 |
Given a triangle $ABC$, an inscribed circle touches the sides $AB, AC, BC$ at points $C_1, B_1, A_1$ respectively. Find the radius of the excircle $w$, which touches the side $AB$ at point $D$, the extension of side $BC$ at point $E$, and the extension of side $AC$ at point $G$. It is known that $CE = 6$, the radius of the inscribed circle is $1$, and $CB_1 = 1$. | 6 | 0.25 |
A parallelepiped is composed of white and black unit cubes in a checkerboard pattern. It is known that there are $1 \frac{12}{13} \%$ more black cubes than white cubes. Find the surface area of the parallelepiped, given that each side of the parallelepiped is greater than 1. | 142 | 0.625 |
As shown in the figure, an "L"-shaped paper with a perimeter of 52 cm can be split along the dashed line into two identical rectangles. If the longest side length is 16 cm, what is the area of the "L"-shaped paper in square centimeters? | 120 | 0.125 |
Given the graph of the function \(y = x + \frac{x}{x}\) and the point \((1, 0)\), how many lines can be drawn through this point such that they do not intersect the given graph? | 2 | 0.625 |
A shooter makes three attempts. Success (hitting the target) and failure (missing the target) of each attempt are independent of the outcomes of the other attempts, and the probability of successfully completing each attempt is constant and equal to \( p \). Find the probability of successfully completing two out of three attempts. | 3p^2(1-p) | 0.625 |
The triangle \(PQR\) is isosceles with \(PR = QR\). Angle \(PRQ = 90^\circ\) and length \(PQ = 2 \text{ cm}\). Two arcs of radius \(1 \text{ cm}\) are drawn inside triangle \(PQR\). One arc has its center at \(P\) and intersects \(PR\) and \(PQ\). The other arc has its center at \(Q\) and intersects \(QR\) and \(PQ\). What is the area of the shaded region, in \(\text{cm}^2\)? | 1 - \frac{\pi}{4} | 0.375 |
Using the greedy algorithm, propose a coloring of the first graph involved in exercise 6 (or rather its more readable planar version available in the solution to this exercise). Deduce its chromatic number. | 3 | 0.375 |
As shown in the figure, in the right triangle $ABC$, point $F$ is on $AB$ such that $AF = 2FB$. The quadrilateral $EBCD$ is a parallelogram. What is the ratio $FD: EF$? | 2:1 | 0.125 |
Through a point located inside a triangle, lines are drawn parallel to the sides of the triangle.
These lines divide the triangle into three smaller triangles and three quadrilaterals. Let \( a \), \( b \), and \( c \) be the heights of the three smaller triangles parallel to the sides of the original triangle. Find the height of the original triangle parallel to these. | a + b + c | 0.375 |
It is known that the numbers \(x, y, z \) form an arithmetic progression in that order with a common difference \(\alpha = \arcsin \frac{\sqrt{7}}{4}\), and the numbers \(\frac{1}{\sin x}, \frac{4}{\sin y}, \frac{1}{\sin z}\) also form an arithmetic progression in that order. Find \(\sin ^{2} y\). | \frac{7}{13} | 0.875 |
Let the function \( f(x) = ax^2 + bx + c \) where \( a \neq 0 \) satisfy \( |f(0)| \leq 2 \), \( |f(2)| \leq 2 \), and \( |f(-2)| \leq 2 \). Determine the maximum value of \( y = |f(x)| \) for \( x \in [-2, 2] \). | \frac{5}{2} | 0.25 |
Fill the five numbers $2015, 2016, 2017, 2018, 2019$ into the five boxes labeled " $D, O, G, C, W$ " such that $D+O+G=C+O+W$. How many different ways can this be done? | 24 | 0.25 |
In a convex 13-gon, all the diagonals are drawn. They divide it into polygons. Consider a polygon among them with the largest number of sides. What is the maximum number of sides it can have? | 13 | 0.125 |
Let the real numbers \( x_{1}, x_{2}, \cdots, x_{1997} \) satisfy the following two conditions:
1. \( -\frac{1}{\sqrt{3}} \leq x_{i} \leq \sqrt{3} \) for \( i = 1, 2, \cdots, 1997 \);
2. \( x_{1} + x_{2} + \cdots + x_{1997} = -318 \sqrt{3} \).
Find the maximum value of \( x_{1}^{12} + x_{2}^{12} + \cdots + x_{1997}^{12} \) and justify your answer. | 189548 | 0.375 |
Leon has cards with digits from 1 to 7. How many ways are there to glue two three-digit numbers (one card will not be used) so that their product is divisible by 81, and their sum is divisible by 9? | 36 | 0.125 |
The base of a right prism is an isosceles trapezoid \(ABCD\) with \(AB = CD = 13\), \(BC = 11\), and \(AD = 21\). The area of the diagonal cross-section of the prism is 180. Find the total surface area of the prism. | 906 | 0.5 |
Given a regular 15-sided polygon with a side length of \(2a\). What is the area of the annulus bounded by the circles inscribed in and circumscribed around the 15-sided polygon? | \pi a^2 | 0.25 |
It is known that 999973 has exactly three distinct prime factors. Find the sum of these prime factors. | 171 | 0.5 |
Calculate the limit of the function:
$$\lim _{x \rightarrow 0} \frac{e^{4 x}-e^{-2 x}}{2 \operatorname{arctg} x-\sin x}$$ | 6 | 0.875 |
Given that \( x \) is a four-digit number and the sum of its digits is \( y \). When the value of \( \frac{x}{y} \) is minimized, \( x = \) _______ | 1099 | 0.75 |
In the expression \( S=\sqrt{x_{1}-x_{2}+x_{3}-x_{4}} \), where \( x_{1}, x_{2}, x_{3}, x_{4} \) is a permutation of the numbers \( 1, 2, 3, 4 \), find the number of different permutations that make \( S \) a real number. | 16 | 0.875 |
There are four inequalities: $\sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2}}}}<2$, $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}<2$, $\sqrt{3 \sqrt{3 \sqrt{3 \sqrt{3}}}}<3$, $\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3}}}}<3$. The number of incorrect inequalities is ______. | 0 | 0.875 |
The sequence \( a_{1}, a_{2}, a_{3}, \ldots \) is an arithmetic sequence with common difference 3 and \( a_{1}=1 \). The sequence \( b_{1}, b_{2}, b_{3}, \ldots \) is an arithmetic sequence with common difference 10 and \( b_{1}=2 \). What is the smallest integer larger than 2023 that appears in both sequences? | 2032 | 0.875 |
In the diagram below, $P$ is a point on the semi-circle with the diameter $AB$. The point $L$ is the foot of the perpendicular from $P$ onto $AB$, and $K$ is the midpoint of $PB$. The tangents to the semicircle at $A$ and at $P$ meet at the point $Q$. It is given that $PL$ intersects $QB$ at the point $M$, and $KL$ intersects $QB$ at the point $N$. Suppose $\frac{AQ}{AB}=\frac{5}{12}$, $QM=25$ cm and $MN = x$ cm. Find the value of $x$. | 12 | 0.25 |
Given the complex numbers \( z_{1}=1+a i \) and \( z_{2}=2^{\frac{3}{4}}\left(\cos \frac{3 \pi}{8}+i \sin \frac{3 \pi}{8}\right) \), find all real values of \( a \) for which \( z_{1}^{3}=z_{2}^{2} \). | 1 | 0.625 |
In a class, there are 15 boys and 15 girls. On Women's Day, some boys called some girls to congratulate them (no boy called the same girl more than once). It turned out that the children can be uniquely divided into 15 pairs, such that each pair consists of a boy and a girl whom he called. What is the maximum number of calls that could have been made? | 120 | 0.125 |
Professor Antônio discovered an interesting property related to the integer $x$ that represents his age. He told his students that $x^{2}=\overline{a b a c}$ and that $x=\overline{a b}+\overline{a c}$. What is the professor's age?
Note: We are using a bar to distinguish the decimal representation of the four-digit number $\overline{a b c d}$ from the product $a \cdot b \cdot c \cdot d$. For example, if $\overline{a b c d}=1267$, then $a=1, b=2, c=6,$ and $d=7$. The notation is the same for numbers with other quantities of digits. | 45 | 0.5 |
The diagram shows a triangle \(ABC\) with an area of \(12 \, \text{cm}^2\). The sides of the triangle are extended to points \(P, Q, R, S, T\) and \(U\) so that \(PA = AB = BS\), \(QA = AC = CT\), and \(RB = BC = CU\). What is the area (in \(\text{cm}^2\)) of hexagon \(PQRSTU\)? | 156 | 0.625 |
Let $s(n)$ denote the sum of all odd digits of the number $n$. For example, $s(4)=0$, $s(173)=11$, and $s(1623)=4$. Calculate the value of the sum $s(1)+s(2)+s(3)+\ldots+s(321)$. | 1727 | 0.375 |
A pot contains $3 \pi$ liters of water taken at a temperature of $t=0{ }^{\circ} C$ and brought to a boil in 12 minutes. After this, without removing the pot from the stove, ice at a temperature of $t=0{ }^{\circ} C$ is added. The next time the water begins to boil is after 15 minutes. Determine the mass of the added ice. The specific heat capacity of water is $c_{B}=4200 \, \text{J} / \text{kg} \cdot { }^{\circ} \mathrm{C}$, the specific heat of fusion of ice is $\lambda=3.3 \times 10^{5} \, \text{J} / \text{kg}$, the density of water is $\rho = 1000 \, \text{kg} / \text{m}^{3}$.
Answer: 2.1 kg | 2.1 \, \text{kg} | 0.25 |
Let \( m \) and \( n \) be relatively prime positive integers. Determine all possible values of
\[ \operatorname{gcd}\left(2^{m}-2^{n}, 2^{m^{2}+mn+n^{2}}-1\right). \] | 1 \text{ and } 7 | 0.875 |
Given a trapezoid \(A B C D\) with right angles at \(A\) and \(D\). Let \(A B = a\), \(C D = b\), and \(A D = h\).
1) What relationship must exist between \(a\), \(b\), and \(h\) for the diagonals \(A C\) and \(B D\) to be perpendicular?
2) Let \(M\) be the midpoint of \([A D]\). What relationship must exist between \(a\), \(b\), and \(h\) for the triangle \(B M C\) to be a right triangle at \(M\)? | h^2 = 4ab | 0.625 |
In space, there are 4 pairwise skew lines \( l_{1}, l_{2}, l_{3}, l_{4} \), such that no three of them are parallel to the same plane. Draw a plane \( P \) such that the points \( A_{1}, A_{2}, A_{3}, A_{4} \) at the intersections of these lines with \( P \) form a parallelogram. How many lines do the centers of such parallelograms trace out? | 3 | 0.25 |
A sequence \(\left(a_{n}\right)\) satisfies the conditions:
$$
a_{1}=3, \quad a_{2}=9, \quad a_{n+1}=4 a_{n}-3 a_{n-1} \quad (n > 1)
$$
Find the formula for the general term of the sequence. | a_n = 3^n | 0.875 |
Find the real solution(s) to the equation \((x+y)^{2} = (x+1)(y-1)\). | (-1,1) | 0.875 |
a) What is the radius of the smallest circle that can contain $n$ points ($n=2,3,4,...,10,11$), one of which coincides with the center of the circle, so that the distance between any two points is at least 1?
b)* How many points can be placed inside a circle of radius 2 so that one of the points coincides with the center of the circle and the distance between any two points is at least 1? | 19 | 0.125 |
Given an integer \( n \geqslant 2 \). Let \( a_{1}, a_{2}, \cdots, a_{n} \) and \( b_{1}, b_{2}, \cdots, b_{n} \) be positive numbers that satisfy
\[ a_{1} + a_{2} + \cdots + a_{n} = b_{1} + b_{2} + \cdots + b_{n}, \]
and for any \( i, j \) ( \( 1 \leqslant i < j \leqslant n \)), it holds that \( a_{i}a_{j} \geqslant b_{i} + b_{j} \). Find the minimum value of \( a_{1} + a_{2} + \cdots + a_{n} \). | 2n | 0.875 |
In how many ways can a $2 \times n$ rectangle be tiled with $2 \times 1$ dominoes? | F_{n+1} | 0.875 |
In the sequence $\{a_n\}$, $a_4 = 1$, $a_{11} = 9$, and the sum of any three consecutive terms is 15. What is $a_{2016}$? | 5 | 0.875 |
On each edge of a regular tetrahedron of side 1, there is a sphere with that edge as diameter. S is the intersection of the spheres (so it is all points whose distance from the midpoint of every edge is at most 1/2). Show that the distance between any two points of S is at most 1/√6. | \frac{1}{\sqrt{6}} | 0.625 |
Find the sine of the angle at the vertex of an isosceles triangle, given that the perimeter of any inscribed rectangle, with two vertices lying on the base, is a constant value. | \frac{4}{5} | 0.75 |
The number zero is written on a board. Peter is allowed to perform the following operations:
- Apply to one of the numbers written on the board a trigonometric (sin, $\cos$, $\operatorname{tg}$, or ctg) or inverse trigonometric (arcsin, arccos, $\operatorname{arctg}$, or arcctg) function and write the result on the board;
- Write on the board the quotient or product of two already written numbers.
Help Peter write $\sqrt{3}$ on the board. | \sqrt{3} | 0.875 |
The point \( P \) on the curve \( y = \frac{x+1}{x-1} \) (\( x \in \mathbf{R}, x \neq 1 \)) is at a minimum distance \( d \) from the origin \( O \). Find \( d \). | 2 - \sqrt{2} | 0.75 |
If the function \( f(x) = \frac{(\sqrt{1008} x + \sqrt{1009})^{2} + \sin 2018 x}{2016 x^{2} + 2018} \) has a maximum value of \( M \) and a minimum value of \( m \), then \( M + m \) equals \_\_\_\_. | 1 | 0.875 |
In triangle \(ABC\), \(AB = (b^2 - 1)\) cm, \(BC = a^2\) cm, and \(AC = 2a\) cm, where \(a\) and \(b\) are positive integers greater than 1. Find the value of \(a - b\). | 0 | 0.875 |
For which values of \( n \) is the polynomial \((x+1)^{n} - x^{n} - 1\) divisible by:
a) \( x^{2} + x + 1 \)
b) \( \left(x^{2} + x + 1\right)^{2} \)
c) \( \left(x^{2} + x + 1\right)^{3} \)? | n = 1 | 0.25 |
Two players, \(A\) and \(B\), play rock-paper-scissors continuously until player \(A\) wins 2 consecutive games. Suppose each player is equally likely to use each hand sign in every game. What is the expected number of games they will play? | 12 | 0.375 |
Point $P$ is located on side $AB$ of square $ABCD$ such that $AP : PB = 1 : 4$. Point $Q$ lies on side $BC$ of the square and divides it in the ratio $BQ : QC = 5$. Lines $DP$ and $AQ$ intersect at point $E$. Find the ratio of the lengths $AE : EQ$. | 6:29 | 0.625 |
An ant starts from vertex \( A \) of rectangular prism \( ABCD-A_1B_1C_1D_1 \) and travels along the surface to reach vertex \( C_1 \) with the shortest distance being 6. What is the maximum volume of the rectangular prism? | 12\sqrt{3} | 0.875 |
In the pyramid $ABCD$, the area of the face $ABC$ is four times the area of the face $ABD$. Point $M$ is taken on the edge $CD$ such that $CM: MD = 2$. Through point $M$, planes parallel to the faces $ABC$ and $ABD$ are drawn. Find the ratio of the areas of the resulting sections. | 1 | 0.125 |
Let \( A \) be a set of integers where the smallest element is 1 and the largest element is 100. Each element, except 1, is equal to the sum of two elements from \( A \) (it can be twice the same element). Find the minimum number of elements in set \( A \). | 9 | 0.5 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c = 10$ and $\frac{\cos A}{\cos B} = \frac{b}{a} = \frac{4}{3}$, point $P$ is a point on the incircle of $\triangle ABC$. Let $d$ be the sum of the squares of the distances from $P$ to vertices $A$, $B$, and $C$. Find $d_{\min} + d_{\max}$. | 160 | 0.625 |
Let \( n \) be an integer. Determine the remainder \( b \) of \( n^{a} - n \) divided by 30. | 0 | 0.75 |
Henry starts to read a 290-page book on a Sunday. He reads four pages every day except on Sundays when he reads 25 pages. How many days does it take him to finish the book? | 41 | 0.75 |
Let \(ABC\) be a triangle with \(AB = 3\), \(BC = 4\), and \(CA = 5\). What is the distance from \(B\) to line \(AC\)? | \frac{12}{5} | 0.25 |
Calculate the circulation of the vector field given in spherical coordinates: \( \mathbf{F} = (2r) \mathbf{e}_{r} + (R + r) \sin \theta \mathbf{e}_{\varphi} \) along the circle \( L: \{ r = R, \theta = \frac{\pi}{2} \} \) in the positive direction of the angle \(\varphi\), directly and using Stokes' theorem. | 4\pi R^2 | 0.375 |
For real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right]=1994\). | n = 312 | 0.375 |
Two brothers sold a flock of sheep that belonged to both of them, receiving as many rubles for each sheep as there were sheep in the flock. The brothers divided the money received as follows: first the elder brother took ten rubles from the total amount, then the second brother took ten rubles, after which the first brother took another ten rubles, and so on. The younger brother lacked ten rubles in the end, so he took all the remaining small change, and in addition, to make the division fair, the elder brother gave the younger brother his pocket knife. What was the value of the pocket knife? | 2 | 0.5 |
If the real number \( x \) satisfies \( \log _{2} x = 1 + \cos \theta \) where \( \theta \in \left[ -\frac{\pi}{2}, 0 \right] \), then the maximum value of the function \( f(x) = |x-1| + 2|x-3| \) is ____________. | 5 | 0.75 |
There are 4 numbers written on a board. Vasya multiplied the first of these numbers by \(\sin \alpha\), the second by \(\cos \alpha\), the third by \(\operatorname{tg} \alpha\), and the fourth by \(\operatorname{ctg} \alpha\) (for some angle \(\alpha\)) and obtained a set of the same 4 numbers (possibly in a different order). What is the maximum number of distinct numbers that could have been written on the board? | 3 | 0.375 |
A natural number is called interesting if all its digits are different, and the sum of any two adjacent digits is the square of a natural number. Find the largest interesting number. | 6310972 | 0.125 |
A sequence of real numbers \( a_{0}, a_{1}, \ldots, a_{9} \) with \( a_{0}=0 \), \( a_{1}=1 \), and \( a_{2}>0 \) satisfies
\[ a_{n+2} a_{n} a_{n-1} = a_{n+2} + a_{n} + a_{n-1} \]
for all \( 1 \leq n \leq 7 \), but cannot be extended to \( a_{10} \). In other words, no values of \( a_{10} \in \mathbb{R} \) satisfy
\[ a_{10} a_{8} a_{7} = a_{10} + a_{8} + a_{7} .\]
Compute the smallest possible value of \( a_{2} \). | \sqrt{2}-1 | 0.625 |
Ten Cs are written in a row. Some Cs are upper-case and some are lower-case, and each is written in one of two colors, green and yellow. It is given that there is at least one lower-case C, at least one green C, and at least one C that is both upper-case and yellow. Furthermore, no lower-case C can be followed by an upper-case C, and no yellow C can be followed by a green C. In how many ways can the Cs be written? | 36 | 0.25 |
Let \( \triangle ABC \) be an equilateral triangle with side length 1. Points \( A_1 \) and \( A_2 \) are chosen on side \( BC \), points \( B_1 \) and \( B_2 \) are chosen on side \( CA \), and points \( C_1 \) and \( C_2 \) are chosen on side \( AB \) such that \( BA_1 < BA_2 \), \( CB_1 < CB_2 \), and \( AC_1 < AC_2 \).
Suppose that the three line segments \( B_1C_2 \), \( C_1A_2 \), and \( A_1B_2 \) are concurrent, and the perimeters of triangles \( AB_2C_1 \), \( BC_2A_1 \), and \( CA_2B_1 \) are all equal. Find all possible values of this common perimeter. | 1 | 0.25 |
How many solutions does the equation
$$
15x + 6y + 10z = 1973
$$
have in integers that satisfy the following inequalities:
$$
x \geq 13, \quad y \geq -4, \quad z > -6
$$ | 1953 | 0.375 |
Calculate the limit of the function:
$$\lim_{x \rightarrow \pi} (x + \sin x)^{\sin x + x}$$ | \pi^\pi | 0.5 |
Find the largest positive integer \( n \) such that \( n! \) ends with exactly 100 zeros. | 409 | 0.25 |
Let \( n \) be a positive integer. If the sum \( 1 + 2 + \cdots + n \) is exactly equal to a three-digit number, and that three-digit number has all identical digits, what are all possible values of \( n \)? | 36 | 0.5 |
In the first quadrant of the Cartesian coordinate plane, the points with integer coordinates are numbered as follows:
- (0,0) is point number 1
- (1,0) is point number 2
- (1,1) is point number 3
- (0,1) is point number 4
- (0,2) is point number 5
- (1,2) is point number 6
- (2,2) is point number 7
- (2,1) is point number 8
- (2,0) is point number 9
Following the sequence indicated by the arrows, what are the coordinates of the 2000th point? | (44, 25) | 0.25 |
The height of the triangular pyramid \(ABCD\) dropped from vertex \(D\) passes through the point of intersection of the heights of triangle \(ABC\). Additionally, it is known that \(DB = 3\), \(DC = 2\), and \(\angle BDC = 90^{\circ}\). Find the ratio of the area of face \(ADB\) to the area of face \(ADC\). | \frac{3}{2} | 0.75 |
When passengers boarded an empty tram, half of them took seats. How many passengers boarded initially if, after the first stop, their number increased by exactly $8 \%$ and it is known that the tram accommodates no more than 70 people? | 50 | 0.625 |
The school table tennis championship was held in an Olympic system format. The winner won six matches. How many participants in the tournament won more games than they lost? (In an Olympic system tournament, participants are paired up. Those who lose a game in the first round are eliminated. Those who win in the first round are paired again. Those who lose in the second round are eliminated, and so on. In each round, a pair was found for every participant.) | 16 | 0.875 |
Given vectors \(\vec{a} = (1, \sin \theta)\) and \(\vec{b} = (\cos \theta, \sqrt{3})\) where \(\theta \in \mathbf{R}\), find the range of the magnitude \(|\vec{a} - \vec{b}|\). | [1, 3] | 0.625 |
A rectangular chessboard of size \( m \times n \) is composed of unit squares (where \( m \) and \( n \) are positive integers not exceeding 10). A piece is placed on the unit square in the lower-left corner. Players A and B take turns moving the piece. The rules are as follows: either move the piece any number of squares upward, or any number of squares to the right, but you cannot move off the board or stay in the same position. The player who cannot make a move loses (i.e., the player who first moves the piece to the upper-right corner wins). How many pairs of integers \( (m, n) \) are there such that the first player A has a winning strategy? | 90 | 0.875 |
A sports team's members have unique numbers taken from the integers 1 to 100. If no member's number is the sum of the numbers of any two other members, nor is it twice the number of any other member, what is the maximum number of people this team can have? | 50 | 0.75 |
Given a large cube of dimensions \( 4 \times 4 \times 4 \) composed of 64 unit cubes, select 16 of the unit cubes to be painted red, such that in every \( 1 \times 1 \times 4 \) rectangular prism within the large cube that is composed of 4 unit cubes, exactly 1 unit cube is red. How many different ways are there to select these 16 red unit cubes? Explain why. | 576 | 0.25 |
If the odd function \( y=f(x) \) defined on \( \mathbf{R} \) is symmetrical about the line \( x=1 \), and when \( 0 < x \leqslant 1 \), \( f(x)=\log_{3}x \), find the sum of all real roots of the equation \( f(x)=-\frac{1}{3}+f(0) \) in the interval \( (0,10) \). | 30 | 0.125 |
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that
\[ f(x+y) + y \leq f(f(f(x))) \]
holds for all \( x, y \in \mathbb{R} \). | f(x) = c - x | 0.125 |
If for a natural number \( n (\geqslant 2) \), there exist integers \( a_{1}, a_{2}, \cdots, a_{n} \) such that
\[
a_{1}+a_{2}+\cdots+a_{n}=a_{1} \cdot a_{2} \cdot \cdots \cdot a_{n}=1990,
\]
find the minimum value of \( n \). | 5 | 0.375 |
If the real numbers \( x \) and \( y \) satisfy \( x^{2} + 2 \cos y = 1 \), what is the range of values that \( x - \cos y \) can take? | [-1, \sqrt{3} + 1] | 0.125 |
Inside the cube $ABCDEFG A_{1} B_{1} C_{1} D_{1}$, there is the center $O$ of a sphere with radius 10. The sphere intersects the face $A A_{1} D_{1} D$ along a circle of radius 1, the face $A_{1} B_{1} C_{1} D_{1}$ along a circle of radius 1, and the face $C D D_{1} C_{1}$ along a circle of radius 3. Find the length of the segment $OD_{1}$. | 17 | 0.5 |
a) What is the maximum number of knights that can be placed on a chessboard of 64 squares so that no two knights threaten each other?
b) Determine the number of different arrangements of the maximum possible number of knights where no two knights threaten each other. | 2 | 0.5 |
A printing house determines the cost of printing a book by adding the cost of the cover to the cost of each page and rounding the result up to the nearest whole ruble (for example, if the result is 202 rubles and 1 kopek, it is rounded up to 203 rubles). It is known that the cost of a book with 104 pages is 134 rubles, and a book with 192 pages costs 181 rubles. What is the cost of printing the cover, given that it is a whole number of rubles and the cost of a single page is a whole number of kopeks? | 77 | 0.875 |
A natural number that can be expressed as the sum of two consecutive non-zero natural numbers and can also be expressed as the sum of three consecutive non-zero natural numbers is called a "good number". What is the largest "good number" less than or equal to 2011? | 2007 | 0.75 |
A $72 \mathrm{db}$ $4,5 \mathrm{~V}$ $2 \Omega$ internal resistance batteries are connected together. How should we connect them so that the current through an external resistance of $R=36 \Omega$ is
a) maximum,
b) minimum?
What are the values of these currents? (Only consider configurations consisting of series-connected groups of equal numbers of batteries connected in parallel.) | 0.125 \text{ A} | 0.375 |
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