For positive integers \( n \), let \( f(n) \) be the product of the digits of \( n \). Find the largest positive integer \( m \) such that
\[
\sum_{n=1}^{\infty} \frac{f(n)}{m^{\left\lfloor \log_{10} n \right\rfloor}}
\]
is an integer.


Find all pairs of natural numbers \(a\) and \(b\) such that \(3^{a} + 4^{b}\) is a perfect square.

Solution: \boxed{(2, 2)}


The height of a cone is equal to the diameter of its base. Find the ratio of the area of its base to its lateral surface area.


A natural number greater than 1 is called "good" if it is equal to the product of its distinct proper divisors (excluding 1 and the number itself). Find the sum of the first ten "good" natural numbers.


Consider a four-digit natural number with the following property: if we swap its first two digits with the second two digits, we get a four-digit number that is 99 less.

How many such numbers are there in total, and how many of them are divisible by 9?


What is the minimum number of sides of a regular polygon that approximates the area of its circumscribed circle with an error of less than 1 per thousand (0.1%)?


A convex quadrilateral \(EFGH\) has vertices \(E, F, G, H\) lying respectively on the sides \(AB, BC, CD,\) and \(DA\) of another quadrilateral \(ABCD\). It satisfies the equation \(\frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA} = 1\). Given that points \(E, F, G,\) and \(H\) lie on the sides of quadrilateral \(E_1F_1G_1H_1\) respectively with \(\frac{E_1A}{AH_1} = \lambda\), find the value of \(\frac{F_1C}{CG_1}\).

Solution: \boxed{\lambda}


Given the function \( f(x) = \frac{\sin \left(x + 45^\circ \right)}{\sin \left(x + 60^\circ \right)}, \) for \( x \in \left[ 0^\circ, 90^\circ \right] \), find the product of the maximum and minimum values of \( f(x) \).


The first, second, and third terms of a geometric sequence are distinct and equal to the second, fourth, and seventh terms of an arithmetic sequence, respectively. The product of these three numbers is 64. Find the first term of the geometric sequence.


Find the smallest positive real number \( r \) with the following property: For every choice of 2023 unit vectors \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_{2023} \in \mathbb{R}^2 \), a point \( \mathbf{p} \) can be found in the plane such that for each subset \( S \) of \(\{1, 2, \ldots, 2023\}\), the sum
\[ \sum_{i \in S} \mathbf{v}_i \]
lies inside the disc \(\left\{ \mathbf{x} \in \mathbb{R}^2 : \| \mathbf{x} - \mathbf{p} \| \leq r \right\}\).


The numbers \(a, b, c, d\) belong to the interval \([-4.5, 4.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\).


On the radius \( AO \) of a circle with center \( O \), point \( M \) is chosen. On one side of \( AO \) on the circle, points \( B \) and \( C \) are chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \( BC \), given that the radius of the circle is 15 and \(\sin \alpha = \frac{\sqrt{21}}{5}\)?


How many sets of integers \((a, b, c)\) satisfy \(2 \leq a \leq b \leq c\) and \(a b c = 2013 \times 2014\)?


Max has 2015 cans, numbered from 1 to 2015, and an unlimited supply of coins.

Consider the following initial configurations:

(a) All cans are empty.

(b) Can number 1 contains 1 coin, can number 2 contains 2 coins, and so on, up to can number 2015, which contains 2015 coins.

(c) Can number 1 contains 2015 coins, can number 2 contains 2014 coins, and so on, up to can number 2015, which contains 1 coin.

Max chooses a number \( n \) in each step, with \( 1 \leq n \leq 2015 \), and adds \( n \) coins to every can except can number \( n \).

Determine for each of the three initial configurations (a), (b), and (c) whether Max can achieve a state where all cans contain the same number of coins after a finite number of steps, with at least one step performed.

Solution: \boxed{Yes}


Let \(ABCD\) be a parallelogram. Let \(M\) be the midpoint of segment \([BC]\) and \(N\) be the midpoint of segment \([CD]\). The lines \((AN)\) and \((BD)\) intersect at \(Q\), and the lines \((AM)\) and \((BD)\) intersect at \(P\). Show that \(BP = PQ = QD\).

Solution: \boxed{PQ = QD}


Real numbers \(a, b, c, d\) are such that \(a + b = \frac{9}{c - d}\) and \(c + d = \frac{25}{a - b}\). What is the minimum value that the quantity \(a^2 + b^2 + c^2 + d^2\) can take?


In the dihedral angle $\alpha - E F - \beta $, $AE \subset \alpha, BF \subset \beta$, and $AE \perp EF, BF \perp EF$. Given $EF = 1$, $AE = 2$, and $AB = \sqrt{2}$, find the maximum volume of the tetrahedron $ABEF$.


Morning discovered that they have a total of 20 coins consisting of 10-cent and 50-cent coins, and the total amount of money is 8 yuan. How many 10-cent coins are there?


Given 6000 cards, each with a unique natural number from 1 to 6000 written on it. It is required to choose two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done?


The diagram shows a segment of a circle such that \( CD \) is the perpendicular bisector of the chord \( AB \). Given that \( AB = 16 \) and \( CD = 4 \), find the diameter of the circle.


If the positive numbers \( a \), \( b \), and \( c \) satisfy \( a + 2b + 3c = abc \), then the minimum value of \( abc \) is \(\qquad\).


Find the area of the triangle intercepted by the line \( y = 3x + 1 \) from the region defined by the inequality \( |x - 1| + |y - 2| \leq 2 \).


Let \( s(n) \) denote the sum of the digits of the number \( n \). For example, \( s(2345) = 2 + 3 + 4 + 5 = 14 \). Observe that:

\[ 40 - s(40) = 36 = 9 \times 4 ;\]
\[ 500 - s(500) = 495 = 9 \times 55 ;\]
\[ 2345 - s(2345) = 2331 = 9 \times 259. \]

(a) What can be said about the number \( n - s(n) \)?

(b) Using the previous item, calculate \( s(s(s(2^{2009}))) \).

Hint: Show that the desired number is less than 9.


Around a circle, 130 trees are planted: birches and lindens (both types are present). Each tree has a sign that reads: "Next to it, two different types of trees are growing." It is known that among all the trees, this statement is false for all lindens and exactly one birch. How many birches could there be? List all possible options.


A merchant bought several sacks of salt in Tver and sold them in Moscow with a profit of 100 rubles. With all the money he gained, he bought salt again in Tver (at the Tver price) and sold it in Moscow (at the Moscow price). This time, the profit amounted to 120 rubles. How much money did he spend on the first purchase?


In the shown diagram, fill the numbers \( 1, 2, \ldots, 10 \) into the ten circles such that the sum of the numbers in any five consecutive adjacent circles does not exceed a certain integer \( M \). Find the minimum value of \( M \) and complete your diagram.


On the bisector of angle \( B A C \) of triangle \( A B C \), there is a point \( M \), and on the extension of side \( A B \) beyond point \( A \), there is a point \( N \) such that \( A C = A M = 1 \) and \( \angle A N M = \angle C N M \). Find the radius of the circumcircle of triangle \( C N M \).


If a number leaves a remainder of 3 when divided by 4 and a remainder of 5 when divided by 9, what is the remainder when it is divided by 36?


Given the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\) with the left and right foci \(F_{1}\) and \(F_{2}\) respectively, a line \(l\) passes through the right focus and intersects the ellipse at points \(P\) and \(Q\). Find the maximum area of the inscribed circle of \(\triangle F_{1}PQ\).


Expanding the expression \((1+\sqrt{5})^{213}\) using the Binomial Theorem, we obtain terms of the form \(C_{213}^{k}(\sqrt{5})^{k}\). Find the value of \(k\) for which this term takes the maximum value.


What is the minimum number of factors that need to be crossed out from the number \( 99! = 1 \cdot 2 \cdot \ldots \cdot 99 \) so that the product of the remaining factors ends in \( 2 \)?


Find all natural numbers $n$ such that $n^{5}-2 n^{4}-7 n^{2}-7 n+3=0$.


The graphs of the functions \( y = ax^{2} + bx + 1 \) and \( y = x^{2} + cx + d \) intersect at the point with coordinates \( (2, 4) \). What is the value of the expression \( 4a + d \), given that \( b + c = 1 \)?


A quadrilateral \(ABCD\) is given, where \(AD \parallel BC\). The angle bisector of \(\angle A\) intersects side \(CD\) at point \(X\) and the extension of side \(BC\) beyond point \(C\) at point \(Y\). It is found that \(\angle AXC = 90^\circ\). Find the length of segment \(AB\), given that \(AD = 19\) and \(CY = 16\).


If the number 6 is placed at the right end of a two-digit number, the resulting three-digit number is greater than the original two-digit number by 474. What is the original two-digit number? ___


There is a box containing 7 blue and 5 red balls. What is the minimum number of balls that need to be drawn to ensure there are at least 2 blue and 1 red balls among them?

Solution: \boxed{8 \text{ balls}}


Find the square root of \( 25 \cdot 26 \cdot 27 \cdot 28 + 1 \).


The cells of a $2 \times 2019$ table must be filled with numbers (exactly one number in each cell) according to the following rules. The top row must contain 2019 real numbers, none of which are equal to each other, and the bottom row must contain the same 2019 numbers, but in a different order. Each of the 2019 columns must contain two different numbers, and the sum of these two numbers must be a rational number. What is the maximum number of irrational numbers that could be in the first row of the table?

(S. Kudrya)


Given that \( r, s, t \) are integers, and the set \( \{a \mid a = 2^r + 2^s + 2^t, 0 \leq t < s < r\} \) forms a sequence \(\{a_n\} \) from smallest to largest as \(7, 11, 13, 14, \cdots\), find \( a_{36} \).


Let $A B C D$ be a rhombus. Let $F$ be a point on the segment $[A D]$ and $E$ a point on the segment $[A B]$. The lines $(F C)$ and $(B D)$ intersect at $L$, the lines $(E C)$ and $(B D)$ intersect at $K$. The lines $(F K)$ and $(B C)$ intersect at $Q$ and the lines $(E L)$ and $(D C)$ intersect at $P$. Show that $C P = C Q$.

Solution: \boxed{CP = CQ}


From 4 male students and 2 female students, 3 students are chosen at random to participate in a speech contest. Let the random variable $\xi$ denote the number of female students among the 3 chosen students.
(I) Find the probability distribution of $\xi$.
(II) Calculate the expected value of $\xi$.
(III) Find the probability that the number of female students among the 3 chosen is less than or equal to 1.


What is the maximum number of cells on an $8 \times 8$ chessboard that can be cut by a single straight line?


If \( a + \log_{3} 2017 \), \( a + \log_{9} 2017 \), \( a + \log_{27} 2017 \) (\( a \in \mathbb{R} \)) form a geometric sequence, what is the common ratio?


If \(\sin ^{3} x - \cos ^{3} x = -1\), then what is \(\sin x - \cos x\)?


Let \(ABC\) be a triangle with \(AB = 13\), \(BC = 14\), and \(CA = 15\). Let \(D\) be the foot of the altitude from \(A\) to \(BC\). The inscribed circles of triangles \(ABD\) and \(ACD\) are tangent to \(AD\) at \(P\) and \(Q\), respectively, and are tangent to \(BC\) at \(X\) and \(Y\), respectively. Let \(PX\) and \(QY\) meet at \(Z\). Determine the area of triangle \(XYZ\).


Find any solution to the following rebus:

$$
\begin{array}{r}
\mathrm{ABA} \\
+\mathrm{ABC} \\
\mathrm{ACC} \\
\hline
1416
\end{array}
$$

where \(A\), \(B\), and \(C\) are three different non-zero digits.

Enter the values of the digits \(A\), \(B\), and \(C\).

Solution: \boxed{A = 4, B = 7, C = 6}


In how many different ways can 22 be written as the sum of 3 different prime numbers? Determine the number of triples \((a, b, c)\) of prime numbers with \(1 < a < b < c\) and \(a + b + c = 22\).


Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Find the number of elements in the set $\left\{n \left|\, n=\left[\frac{k^{2}}{2005}\right], 1 \leqslant k \leqslant 2004, k \in \mathbf{N}\right\}$.


For each positive integer \(1 \leqq k \leqq 100\), let \(a_{k}\) denote the sum \(\frac{1}{k}+\frac{1}{k+1}+\ldots+\frac{1}{100}\). Calculate the value of

\[ a_{1} + a_{1}^{2} + a_{2}^{2} + \ldots + a_{100}^{2}. \]


Two identical circles touch each other from the inside of a third circle and touch each other. Connecting the three centers forms a triangle with a perimeter of 18. Find the radius of the larger circle.


Let \( P(x) \) be a polynomial of degree 2010. Suppose \( P(n) = \frac{n}{1+n} \) for all \( n = 0, 1, 2, \ldots, 2010 \). Find \( P(2012) \).


Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{1+2+3+\ldots+n}{\sqrt{9 n^{4}+1}}
$$


Given that the orthocenter of triangle \( \triangle OAB \) is exactly the focus of the parabola \( y^2 = 4x \), where \( O \) is the origin and points \( A \) and \( B \) lie on the parabola, find the area \( S \) of triangle \( \triangle OAB \).


Steph Curry is playing the following game and he wins if he has exactly 5 points at some time. Flip a fair coin. If heads, shoot a 3-point shot which is worth 3 points. If tails, shoot a free throw which is worth 1 point. He makes \(\frac{1}{2}\) of his 3-point shots and all of his free throws. Find the probability he will win the game. (Note he keeps flipping the coin until he has exactly 5 or goes over 5 points).


Given that the element \((1,2) \in A \cap B\), where \(A = \left\{ (x, y) \mid a x - y^2 + b = 0 \right\}\) and \(B = \left\{ (x, y) \mid x^2 - a y - b = 0 \right\}\), find the values of \(a\) and \(b\).

Solution: \boxed{a = -3, b = 7}


A circle with a radius of 1 is randomly placed inside a $15 \times 36$ rectangle $ABCD$. Find the probability that this circle does not intersect the diagonal $AC$.


Given two circles with radii \( R \) and \( r \). Their common internal tangents are perpendicular to each other. Find the area of the triangle formed by these tangents and the common external tangent to the given circles.

Solution: \boxed{S = Rr}


First, select $n$ numbers from $1,2, \cdots, 2020$, then choose any two numbers $a$ and $b$ from these $n$ numbers, ensuring $a \neq 2b$. Find the maximum value of $n$.


Let \( a_{1}, a_{2}, \cdots, a_{20} \in \{1,2, \cdots, 5\} \) and \( b_{1}, b_{2}, \cdots, b_{20} \in \{1,2, \cdots, 10\} \). Define the set \( X = \left\{(i, j) \mid 1 \leq i < j \leq 20, \left(a_{i} - a_{j}\right)\left(b_{i} - b_{j}\right)<0\right\} \). Find the maximum number of elements in \( X \).


At an exchange point, there are two types of transactions:
1) Give 2 euros - receive 3 dollars and a candy as a gift.
2) Give 5 dollars - receive 3 euros and a candy as a gift.

When the wealthy Buratino came to the exchange point, he only had dollars. When he left, he had fewer dollars, he did not get any euros, but he received 50 candies. How many dollars did Buratino spend for such a "gift"?


Find the maximum value of the expression

$$
\frac{a}{x} + \frac{a+b}{x+y} + \frac{a+b+c}{x+y+z}
$$

where \( a, b, c \in [2,3] \), and the triplet of numbers \( x, y, z \) is some permutation of the triplet \( a, b, c \).


The function \( f(x) \) defined on \(\mathbf{R}\) is symmetric about the point \(\left(-\frac{3}{4}, 0\right)\), and satisfies \( f(x) = -f\left(x + \frac{3}{2}\right) \), \( f(-1) = 1 \), and \( f(0) = -2 \). Find the value of \( f(1) + f(2) + \cdots + f(2008) \).


In a table tennis tournament, each participant played against every other participant once. Each match was officiated by one referee. All referees officiated a different number of matches. Player Ivanov claims that all his matches were officiated by different referees. Players Petrov and Sidorov claim the same about their matches. Is it possible that none of them is mistaken?

Solution: \boxed{\text{No}}


We are given a fixed point on a circle of radius 1. Starting from this point and moving along the circumference in the positive direction for curved distances $0,1,2,\ldots$ from it, we obtain points with abscissas $n=0,1,2,\ldots$ respectively. How many points among them should we take to ensure that some two of them are less than the distance $1/5$ apart?


Anton writes down all positive integers that are divisible by 2. Berta writes down all positive integers that are divisible by 3. Clara writes down all positive integers that are divisible by 4. The orderly Dora notes the numbers written by the others. She arranges these numbers in ascending order and does not write any number more than once. What is the 2017th number in her list?


Let \( G, A_{1}, A_{2}, A_{3}, A_{4}, B_{1}, B_{2}, B_{3}, B_{4}, B_{5} \) be ten points on a circle such that \( G A_{1} A_{2} A_{3} A_{4} \) is a regular pentagon and \( G B_{1} B_{2} B_{3} B_{4} B_{5} \) is a regular hexagon, and \( B_{1} \) lies on the minor arc \( G A_{1} \). Let \( B_{5} B_{3} \) intersect \( B_{1} A_{2} \) at \( G_{1} \), and let \( B_{5} A_{3} \) intersect \( G B_{3} \) at \( G_{2} \). Determine the degree measure of \( \angle G G_{2} G_{1} \).


The sequence of real numbers \( a_1, a_2, \cdots, a_n, \cdots \) is defined by the following equation: \( a_{n+1} = 2^n - 3a_n \) for \( n = 0, 1, 2, \cdots \).
1. Find an expression for \( a_n \) in terms of \( a_0 \) and \( n \).
2. Find \( a_0 \) such that \( a_{n+1} > a_n \) for any positive integer \( n \).


In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, it is given that $A B = B C = 4$ and $A A_{1} = 2$. Point $P$ lies on the plane $A_{1} B C$, and it holds that $\overrightarrow{D P} \cdot \overrightarrow{P B} = 0$. Find the area of the plane region formed by all such points $P$ satisfying the given condition.


After the latest work season, the electrified part of the Middle Earth Tundra doubled. At the same time, its non-electrified part decreased by 25%. What fraction of the entire Tundra was its non-electrified part at the beginning of the work season?


Let $ABC$ be a triangle, and let $D$ be the foot of the angle bisector from $A$. The circumcircle of $ABD$ intersects $(AC)$ at $E$. The circumcircle of $ADC$ intersects $(AB)$ at $F$. Show that $BF = CE$.

Solution: \boxed{BF = CE}


Let \( p(x) = 2x^3 - 3x^2 + 1 \). How many squares of integers are there among the numbers \( p(1), p(2), \ldots, p(2016) \)?


On an $8 \times 8$ chessboard, 64 checkers numbered from 1 to 64 are placed. 64 students, one by one, approach the board and flip only those checkers whose numbers are divisible by the ordinal number of the current student. A "king" (or "crowned" checker) is a checker that has been flipped an odd number of times. How many "kings" will there be on the board after the last student steps away?


Given the parabola \( E: y^{2} = 2x \) with focus \( F \), two lines \( l_{1} \) and \( l_{2} \) are drawn through \( F \) with their slopes' product equal to \( -\frac{1}{2} \). Line \( l_{1} \) intersects \( E \) at points \( A \) and \( C \), and line \( l_{2} \) intersects \( E \) at points \( B \) and \( D \). Find the minimum value of \( |A C| + 2|B D| \).

Solution: \boxed{8 \sqrt{2} + 6}


Let \( a \) and \( b \) be distinct real numbers. If the quadratic function \( f(x) = x^2 + ax + b \) satisfies \( f(a) = f(b) \), then the value of \( f(2) \) is ______.


Find the ratio of the volumes of a regular tetrahedron and an octahedron that have equal total surface areas.


Group the positive integers $1, 2, \cdots, n, \cdots$ in such a way that the $k$-th group contains $k+1$ numbers: $(1, 2), (3, 4, 5), (6, 7, 8, 9), \cdots$. Determine which group contains the number 2016.


Find the least prime factor of the number represented by \(1 \underbrace{0000 \cdots 00}_{2010 \text{-many}} 1\).


The numbers \( a \) and \( b \) are such that the polynomial \( x^{4} + x^{3} + 2x^{2} + ax + b \) is the square of some other polynomial. Find \( b \).


A store received apples of the first grade worth 228 rubles and apples of the second grade worth 180 rubles. During unloading, the apples got mixed up. Calculations showed that if all the apples are now sold at a price 90 kopeks lower than the price per kilogram of first grade apples, the planned revenue will be achieved. How many kilograms of apples were delivered, given that there were 5 kg more second grade apples than first grade apples?

Solution: \boxed{85 \text{ kg}}


A geometric progression \( b_{1}, b_{2}, \ldots \) is such that \( b_{25} = 2 \tan \alpha \) and \( b_{31} = 2 \sin \alpha \) for some acute angle \( \alpha \). Find the term number \( n \) for which \( b_{n} = \sin 2\alpha \).


The distance between the centers of two circles with radii \( r \) and \( R \) is \( a \), where \( a > r + R \). Find the smallest distance between points, one of which lies on the first circle and the other on the second circle (the distance between the circles).

Solution: \boxed{a - r - R}


Determine the integers $a$ and $b$ such that $a^{4}+(a+b)^{4}+b^{4}$ is a perfect square.

Solution: \boxed{a = 0, b = 0}


Miss Spelling has enough sheets of paper to give each pupil in her class 3 sheets and have 31 sheets left over. Alternatively, she could give each pupil 4 sheets and have 8 sheets left over. How many sheets of paper does she have?


Given the real numbers \( x \) and \( y \) that satisfy the equation
\[ 2x = \ln(x + y - 1) + \ln(x - y - 1) + 4, \]
find the value of \( 2015 x^2 + 2016 y^3 \).


A positive integer \( n \) cannot be divided by 2 or 3, and there do not exist non-negative integers \( a \) and \( b \) such that \( \left|2^a - 3^b\right| = n \). Find the minimum value of \( n \).


We will say that a number has the form $\overline{a b a}$ if its first and third digits are the same; the second digit does not have to be different. For example, 101 and 222 have this form, while 220 and 123 do not. Similarly, let's define the form $\overline{\overline{b a b c}}$. How many numbers of the form $\overline{a b a b c}$ are divisible by 5?


Masha wrote on the board, in ascending order, all the natural divisors of a certain number \( N \) (the first divisor is 1, and the largest divisor is the number \( N \) itself). It turned out that the third-to-last divisor is 21 times larger than the second divisor. What is the largest possible value of \( N \)?


Find all prime numbers that can be represented both as the sum and as the difference of two prime numbers.


A school math club is designing a desert tree planting scheme on graph paper as follows: The \(k\)-th tree is planted at point \(P_{k}(x_{k}, y_{k})\), where \(x_{1}=1\) and \(y_{1}=1\). For \(k \geq 2\), the coordinates are given by the equations:
\[ \left\{\begin{array}{l} x_{k} = x_{k-1} + 1 - 5\left[\frac{k-1}{5}\right] + 5\left[\frac{k-2}{5}\right], \\ y_{k} = y_{k-1} + \left[\frac{k-1}{5}\right] - \left[\frac{k-2}{5}\right]. \end{array}\right. \]

where \([a]\) denotes the integer part of the real number \(a\) (e.g. \([2.6] = 2\) and \([0.6] = 0\)). According to this scheme, what are the coordinates of the 2008th tree planting point?

Solution: \boxed{(3, 402)}


Solve the inequality \(\sqrt{x^{2}-3 x-54}-\sqrt{x^{2}-27 x+162}<8 \sqrt{\frac{x+6}{x-9}}\) and find the sum of its integer solutions within the interval \([-25, 25]\).


In a game, Wang Liang drew the cards $7, 7, 7, 3$ and found that $7 + 7 + 7 + 3 = 24$. If a combination of 4 cards that adds up to 24 is called a "friendly hand," how many different friendly hands have a largest number of 7?

Solution: \boxed{4 \text{ groups}}


The graphs of the equations
\[
y = -x + 8
\]
\[
173y = -289x + 2021
\]
on the Cartesian plane intersect at \((a, b)\). Find \(a + b\).


In the coordinate plane, points $A(1, 9)$, $C(5, 8)$, $D(8, 2)$, and $E(2, 2)$ are given. Find the area of the pentagon $ABCDE$, where $B$ is the intersection point of lines $EC$ and $AD$.


Let \(a_{1}, a_{2}, \cdots, a_{17}\) be a permutation of \(1, 2, \cdots, 17\), satisfying
$$
\left(a_{1}-a_{2}\right)\left(a_{2}-a_{3}\right) \cdots\left(a_{16}-a_{17}\right)\left(a_{17}-a_{1}\right) = 2^{n}.
$$

Find the maximum value of the positive integer \(n\).


In the cells of a $5 \times 5$ square, natural numbers from 1 to 5 are placed so that in each column, each row, and each of the two main diagonals, all numbers are different. Can the sum of the numbers in the shaded cells in the drawing be equal to 19?

Solution: \boxed{\text{No}}


How many diagonals does a regular undecagon (11-sided polygon) have?


Three natural numbers are such that the last digit of the sum of any two of them is the last digit of the third number. The product of these three numbers was written on the board, and then everything except the last three digits of this product was erased. What three digits could remain on the board? Find all possible answers.


A circle with a radius of 2 passes through the midpoints of three sides of triangle \(ABC\), where the angles at vertices \(A\) and \(B\) are \(30^{\circ}\) and \(45^{\circ}\), respectively.

Find the height drawn from vertex \(A\).

Solution: \boxed{2 + 2\sqrt{3}}


$6 \cdot 35$ Given that $\lg 3=0.47712$, then the incorrect logarithmic value among the following four options is
(A) $\lg 1.5=0.17609$.
(B) $\lg 5=0.69897$.
(C) $\lg 2=0.30103$.
(D) $\lg 7=0.84519$.

Solution: \boxed{(D)}


Larry can swim from Harvard to MIT (with the current of the Charles River) in 40 minutes, or back (against the current) in 45 minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in 15 minutes? (Assume that the speed of the current and Larry's swimming and rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss.


How many different positions can appear on a chessboard if both players, starting from the initial position, make just one move each?


The sum of the first three terms of an increasing arithmetic sequence is 15. If 1 is subtracted from each of the first two terms, and 1 is added to the third term, the three resulting numbers form a geometric sequence. Find the sum of the first 10 terms of the arithmetic sequence.


Given the set \( A = \{1, 2, 3, \ldots, 1002\} \), Petya and Vasya play a game. Petya names a number \( n \), and Vasya then selects a subset of \( A \) consisting of \( n \) elements. Vasya wins if there are no two coprime numbers in the subset he selects; otherwise, Petya wins. What is the minimum \( n \) that Petya must name to guarantee a win?


Given that \( \cos A + \cos B + \cos C = \sin A + \sin B + \sin C = 0 \), find the value of \( \cos^4 A + \cos^4 B + \cos^4 C \).


Find the number of integer solutions of the inequality \(\sqrt{3 \cos \frac{\pi x}{2} - \cos \frac{\pi x}{4} + 1} - \sqrt{6} \cdot \cos \frac{\pi x}{4} \geq 0\), belonging to the interval [1991, 2013].


One day in October, Nikita looked at the wall clock and saw that four flies had fallen asleep exactly on the numbers 12, 3, 6, and 9.

Find the probability that exactly 20 minutes after Nikita looked at the clock, two of the flies had already been swept away by the minute hand.


Consider all possible 100-digit natural numbers, in which only the digits $1, 2, 3$ are used. How many of them are divisible by 3 exactly?


A square grid of 16 dots contains the corners of nine \( 1 \times 1 \) squares, four \( 2 \times 2 \) squares, and one \( 3 \times 3 \) square, for a total of 14 squares whose sides are parallel to the sides of the grid. What is the smallest possible number of dots you can remove so that, after removing those dots, each of the 14 squares is missing at least one corner? Justify your answer by showing both that the number of dots you claim is sufficient and by explaining why no smaller number of dots will work.


Find all primes \( p \) for which the numbers \( p+1 \) and \( p^2+1 \) are twice the squares of natural numbers.

Solution: \boxed{p = 7}


Find all functions \( f: \mathbf{Q} \rightarrow \mathbf{Q} \) (where \( \mathbf{Q} \) denotes the set of rational numbers) such that:

1. \( f(1) = 2 \)
2. For any \( x, y \in \mathbf{Q} \),
\[ f(x y) = f(x) f(y) - f(x+y) + 1 \]

Solution: \boxed{f(x) = x + 1}


In a quadrilateral \(ABCD\), where \(AB = BC\), \(\angle A = \angle B = 20^\circ\), and \(\angle C = 30^\circ\). The extension of side \(AD\) intersects \(BC\) at point \(M\), and the extension of side \(CD\) intersects \(AB\) at point \(N\). Find the angle \(\angle AMN\).


In the figure, many identical matchsticks form a rectangle. The length of this rectangle is composed of 60 matchsticks, and the width is composed of 10 matchsticks. Finally, these matchsticks are distributed equally among more than 100 children, with each child receiving the same number of matchsticks and no matchsticks left over. How many children are there in total?


There are a total of 10040 numbers written on a blackboard, including 2006 of the number 1, 2007 of the number 2, 2008 of the number 3, 2009 of the number 4, and 2010 of the number 5. Each operation consists of erasing 4 different numbers and writing the 5th number (for example, erasing one 1, one 2, one 3, and one 4, and writing one 5; or erasing one 2, one 3, one 4, and one 5, and writing one 1, etc.). If after a finite number of operations, there are exactly two numbers left on the blackboard, what is the product of these two numbers?


Positive integers \(a_{1}, a_{2}, \cdots, a_{18}\) satisfy the following two conditions:
1. \(a_{1} < a_{2} < a_{3} < \cdots < a_{18}\)
2. \(a_{1} + a_{2} + \cdots + a_{18} = 2001\)

What is the maximum value of \(a_{9}\)?


Three tired cowboys walked into a saloon and hung their hats on a bison horn at the entrance. When they left late at night, they were unable to distinguish one hat from another and took three hats at random. Find the probability that none of them took their own hat.


Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{e^{1 / n}+\sin \left(\frac{n}{n^{2}+1}\right) \cdot \cos n}{1+\cos \left(\frac{1}{n}\right)}$$

Solution: \boxed{ \frac{1}{2} }


In the decimal system, the place value for each digit corresponds to powers of 10. For instance, \( 24_{10} = 2 \times 10 + 4 \times 1 \) and \( 365_{10} = 3 \times 10^2 + 6 \times 10 + 5 \times 1 \). Computers use the binary system, where the place value corresponds to powers of 2. For example, \( 7_{10} = 1 \times 2^2 + 1 \times 2 + 1 \times 1 = 111_{2} \) and \( 12_{10} = 1 \times 2^3 + 1 \times 2^2 + 0 \times 2 + 0 \times 1 = 1100_{2} \).

If a natural number can be expressed in base \( m \) as \( 45_m \) and in base \( n \) as \( 54_n \), find the smallest values of \( m \) and \( n \).

Minimum values of \( m = \) \_\_\_\_\_ , \( n = \) \_\_\_\_\_ .

Solution: \boxed{m = 11, \quad n = 9}


The area of a right triangle is 10 units. The radius of the inscribed circle is 1 unit. What is the radius of the circumscribed circle of the triangle?

Solution: \boxed{4.5 \text{ units}}


There is a triple \( k, m, n \) of positive integers without common factors such that \( k \log_{400} 5 + m \log_{400} 2 = n \). Find the sum \( k + m + n \).


Calculate the area of the parallelogram formed by the vectors $a$ and $b$.

$a = 3p + 2q$

$b = 2p - q$

$|p| = 4$

$|q| = 3$

$\angle (p, q) = \frac{3\pi}{4}$


Given that the positive real numbers \(a, b, c\) satisfy the equation \(a^{2}+4 b^{2}+9 c^{2}=4 b+12 c-2\), find the minimum value of \(\frac{1}{a}+\frac{2}{b}+\frac{3}{c}\).


Given the conditions in the diagram, what is the ratio of the angles \(\frac{\alpha}{\beta}\)?


Koschei is counting his gold coins. When he counts them by tens, there are 7 coins left over, and 3 coins are missing to make a whole dozen. Koschei's total wealth is estimated to be between $300 and $400 coins. How many coins does Koschei have?


Given a circle on a plane with center \( O \) and a point \( P \) outside the circle. Construct a regular \( n \)-sided polygon inscribed in the circle, and denote its vertices as \( A_{1}, A_{2}, \ldots, A_{n} \). Show that

$$
\lim _{n \rightarrow \infty} \sqrt[n]{\overline{P A}_{1} \cdot P A_{2} \cdot \ldots \cdot \overline{P A}_{n}}=\overline{P O}.
$$

Solution: \boxed{\overline{P O}}


Define: \(a \Delta b = a \times 1 \underbrace{00 \cdots 00}_{b \uparrow 0} + b, a \square b = a \times 10 + b\) (where \(a, b\) are natural numbers), compute \(2018 \square (123 \Delta 4)\).


Find all surjective functions \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) such that for all \( m, n \in \mathbb{N}^{*} \), the numbers \( f(m+n) \) and \( f(m)+f(n) \) have exactly the same prime divisors.

Solution: \boxed{f(n) = n}


Two circles touch each other externally at point \( C \). A line is tangent to the first circle at point \( A \) and to the second circle at point \( B \). The line \( AC \) intersects the second circle at point \( D \), distinct from \( C \). Find \( BC \), given that \( AC = 9 \) and \( CD = 4 \).


A line with a slope of 2 and a line with a slope of -4 each have a $y$-intercept of 6. The distance between the $x$-intercepts of these lines is:
(A) 2
(B) 6
(C) $\frac{3}{2}$
(D) $\frac{5}{2}$
(E) $\frac{9}{2}$


What is the maximum number of parts into which a sphere can be divided by $n$ circles?

Solution: \boxed{n^2 - n + 2}


A sequence of natural numbers $\left\{x_{n}\right\}$ is constructed according to the following rules:
$$
x_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, \text{ for } n \geq 1.
$$

It is known that some term in the sequence is 1000. What is the smallest possible value of $a+b$?


Xiaoming constructed a sequence using the four digits $2, 0, 1, 6$ by continuously appending these digits in order: 2, 20, 201, 2016, 20162, 201620, 2016201, 20162016, 201620162, … In this sequence, how many prime numbers are there?


Given a sequence of positive real numbers \( x_0, x_1, \cdots, x_{1995} \) that satisfies the following conditions:
(1) \( x_0 = x_{1995} \);
(2) \( x_{i-1} + \frac{2}{x_{i-1}} = 2 x_i + \frac{1}{x_i}, \) for \( i = 1, 2, \cdots, 1995 \).
Find the maximum value of \( x_0 \) among all the sequences that satisfy these conditions.


On a sheet of graph paper, two rectangles are drawn. For the first rectangle, the vertical side is shorter than the horizontal side, and for the second rectangle, it's the opposite. Find the maximum possible area of their intersection if each rectangle contains more than 2010 but less than 2020 cells.


Two spheres are inscribed in a dihedral angle so that they touch each other. The radius of one sphere is twice the radius of the other, and the line connecting the centers of the spheres forms an angle of $30^{\circ}$ with the edge of the dihedral angle. Find the measure of the dihedral angle. In your answer, provide the cosine of this angle, rounded to two decimal places if necessary.


Let \( x, y \in \mathbf{R}^{+} \), and \(\frac{19}{x}+\frac{98}{y}=1\). Find the minimum value of \( x + y \).

Solution: \boxed{117 + 14 \sqrt{38}}


Find the smallest \( n > 2016 \) such that \( 1^n + 2^n + 3^n + 4^n \) is not divisible by 10.


The distance between two cells on an infinite chessboard is defined as the minimum number of moves a king needs to travel between these cells. Three cells on the board are marked, and the pairwise distances between them are all equal to 100. How many cells exist such that the distance from these cells to each of the three marked cells is 50?


In triangle \(A B C\), side \(B C\) equals 4, and the median drawn to this side equals 3. Find the length of the common chord of two circles, each of which passes through point \(A\) and is tangent to \(B C\), with one tangent at point \(B\) and the other at point \(C\).


Consider the geometric sequence \( \left(a+\log _{2} 3\right),\left(a+\log _{4} 3\right),\left(a+\log _{8} 3\right) \). What is the common ratio of this sequence?


ABC is an equilateral triangle with area 1. A' is the point on the side BC such that BA' = 2·A'C. Define B' and C' similarly. Show that the lines AA', BB' and CC' enclose a triangle with area 1/7.


a) There are 128 coins of two different weights, with an equal number of coins of each weight. How can you guarantee finding two coins of different weights using a balance scale without weights in no more than seven weighings?

b) There are eight coins of two different weights, with an equal number of coins of each weight. How can you guarantee finding two coins of different weights using a balance scale without weights in exactly two weighings?


Let \( a, b, c, d \) be real numbers defined by

$$
a=\sqrt{4-\sqrt{5-a}}, \quad b=\sqrt{4+\sqrt{5-b}}, \quad c=\sqrt{4-\sqrt{5+c}}, \quad d=\sqrt{4+\sqrt{5+d}}
$$

Calculate their product.


Given a strictly increasing sequence made up of powers of 3 or the sum of distinct powers of 3: $1,3,4,9,10,12,13, \cdots$, find the 100th term of this sequence.


A bag contains \( d \) balls of which \( x \) are black, \( x+1 \) are red, and \( x+2 \) are white. If the probability of drawing a black ball randomly from the bag is less than \(\frac{1}{6}\), find the value of \( d \).


The cost of 3 hamburgers, 5 milkshakes, and 1 order of fries at a certain fast food restaurant is $23.50. At the same restaurant, the cost of 5 hamburgers, 9 milkshakes, and 1 order of fries is $39.50. What is the cost of 2 hamburgers, 2 milkshakes, and 2 orders of fries at this restaurant?


The integers from 1 to \( k \) are concatenated to form the integer \( N = 123456789101112 \ldots \). Determine the smallest integer value of \( k > 2019 \) such that \( N \) is divisible by 9.


Paul and Caroline arranged to meet under the Arc de Triomphe between 11 AM and noon. Each of them will arrive at a random time within this interval. According to their agreement, if the other person does not appear within fifteen minutes, the first one to arrive will not wait any longer.

What is the probability that Paul and Caroline will meet?


Find all values of the parameter \( b \) for which there exists a value of \( a \) such that the system

$$
\left\{\begin{array}{l}
x^{2}+y^{2}+2 b(b-x+y)=4 \\
y=\frac{9}{(x+a)^{2}+1}
\end{array}\right.
$$

has at least one solution \((x, y)\).

Solution: \boxed{b \in [-11, 2)}


On a certain day, there were some nuts in a bag. On the next day, the same number of nuts was added to the bag, but eight nuts were taken out. On the third day, the same thing happened: the same number of nuts as were already in the bag was added, but eight were taken out. The same process occurred on the fourth day, and after this, there were no nuts left in the bag. How many nuts were in the bag at the very beginning?


On the island of misfortune, there live truth-tellers, who always tell the truth, and liars, who always lie. One day, 2023 natives, among which there are $N$ liars, stood in a circle, and each said, "Both of my neighbors are liars." How many different values can $N$ take?


The area of a triangle is $6 \sqrt{6}$, its perimeter is 18, and the distance from the center of the inscribed circle to one of the vertices is $\frac{2 \sqrt{42}}{3}$. Find the shortest side of the triangle.


Two arithmetic progressions are given. The first and fifth terms of the first progression are 7 and -5, respectively. The first term of the second progression is 0, and the last term is $7 / 2$. Find the sum of the terms of the second progression, given that the third terms of both progressions are equal.


Let's call a word any finite sequence of letters from the Russian alphabet. How many distinct five-letter words can be formed from the letters of the word "САМСА"? How many from the letters of the word "ПАСТА"? Provide the sum of the numbers found.


A granite pedestal. When constructing a square foundation and a cubic pedestal for a monument, granite cubic blocks of size \(1 \times 1\) meter were used. The pedestal used exactly as many blocks as the square foundation upon which it stood. All the blocks were used whole and uncut.

Look at the picture and try to determine the total number of blocks used. The foundation has a thickness of one block.


Convert from kilometers to miles. In the problem 3.125, the Fibonacci numeral system was introduced as being useful when converting distances from kilometers to miles or vice versa.

Suppose we want to find out how many miles are in 30 kilometers. For this, we represent the number 30 in the Fibonacci numeral system:

$$
30=21+8+1=F_{8}+F_{6}+F_{2}=(1010001)_{\mathrm{F}}
$$

Now we need to shift each number one position to the right, obtaining

$$
F_{7}+F_{5}+F_{1}=13+5+1=19=(101001)_{\mathrm{F}}
$$

So, the estimated result is 19 miles. (The correct result is approximately 18.46 miles.) Similarly, conversions from miles to kilometers are done.

Explain why this algorithm works. Verify that it gives a rounded number of miles in $n$ kilometers for all $n \leqslant 100$, differing from the correct answer by less than $2 / 3$ miles.


Find the volume of an inclined triangular prism, where the area of one of its lateral faces is \( S \), and the distance from the plane of this face to the opposite edge is \( d \).

Solution: \boxed{\frac{1}{2} S d}


Out of 100 tourists going on an overseas trip, 10 people do not know either German or French, 76 people know German, and 83 know French. How many tourists know both languages?


On a blackboard, there are the numbers 1, 2, 3, 4, 5, and 6. In each move, João can replace two numbers \(a\) and \(b\) with \(a \cdot b + a + b\). Find all the possibilities for the last number on the blackboard.


How many integer roots of the equation

$$
\cos 2 \pi x + \cos \pi x = \sin 3 \pi x + \sin \pi x
$$

lie between the roots of the equation \( x^2 + 10x - 17 = 0 \)?


Solution: \boxed{A > B}


In an isosceles triangle with each of the equal sides being \( b \), angle bisectors are drawn from the base angles. The segment of the line between the points of intersection of the angle bisectors with the equal sides is equal to \( t \). Find the base of the triangle.

Solution: \boxed{ \frac{bt}{b-t} }


Let \( S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \). How many (potentially empty) subsets \( T \) of \( S \) are there such that, for all \( x \), if \( x \) is in \( T \) and \( 2x \) is in \( S \), then \( 2x \) is also in \( T \)?


In a certain city, vehicle license plates are numbered consecutively from "10000" to "99999". How many license plates out of these 90,000 have the digit 9 appearing at least once and where the sum of the digits is a multiple of 9?


In a 1000 meter race, it is known that when runner A reaches the finish line, runner B is 50 meters away from the finish line. When runner B reaches the finish line, runner C is 100 meters away from the finish line. How many meters away is runner C from the finish line when runner A reaches the finish line?

Solution: \boxed{145 \text{ meters}}


A supermarket purchases 500 kg of sugar oranges at a cost price of 4.80 yuan per kilogram, with an expected weight loss of 10%. If the supermarket intends to achieve a profit of 20% on the sale of these sugar oranges, what should be the retail price per kilogram?

Solution: \boxed{6.4 \, \text{yuan}}


The sum of the ages of four people, A, B, C, and D, is 100 years. A is 32 years old, and the sum of the ages of A and B is three times the sum of the ages of the other two people. C is 3 years older than D. What is the sum of the age of the oldest person and the youngest person among these four people?


In the figure, \(ABCD\) is a trapezoid, and \(AB \parallel CD\). The areas of triangles \(ABO\) and \(OCD\) are 16 and 4, respectively. Find \(\frac{DC}{AB}\).


From all the numbers with a digit sum of 25, find the one whose product of digits is maximum. If there are several such numbers, provide the smallest one among them.


Grandma baked 19 pancakes. The grandchildren came home from school and started eating them. While the younger grandchild eats 1 pancake, the older grandchild eats 3 pancakes, and during this time, grandma manages to bake 2 more pancakes. When they finished, there were 11 pancakes left on the plate. How many pancakes did the older grandchild eat?


Members of the mathematical circle "Seeker" noticed that the number 289 can be obtained by squaring the sum of the digits 8 and 9, i.e., \( 289 = (8 + 9)^{2} \). They decided to find such three-digit numbers with a hundreds digit less than four, each of which equals the sum of its tens and units digits raised to the power of the hundreds digit. Help them solve this problem.


Given a quadrilateral \( A B C D \) with the properties: \( A B = B D \), \( \angle A B D = \angle D B C \), \( \angle B C D = 90^\circ \). Point \( E \) is marked on segment \( B C \) such that \( A D = D E \). What is the length of segment \( B D \), given that \( B E = 7 \) and \( E C = 5 \)?


Li Gang delivered honeycomb briquettes to Grandma Wang, a dependent of a military personnel. The first time, he delivered $\frac{3}{8}$ of the total. The second time, he delivered 50 pieces. At this point, the amount delivered was exactly $\frac{5}{7}$ of the amount not yet delivered. How many pieces of honeycomb briquettes have not been delivered yet?


A collection of 2011 circles divides the plane into \( N \) regions in such a way that any pair of circles intersects at two points and no point lies on three circles. Find the last four digits of \( N \).


A herd of 183 elephants could drink the lake in 1 day, and a herd of 37 elephants could do it in 5 days.

In how many days will one elephant drink the lake?

Solution: \boxed{365 \text{ days}}


A 4x4 chessboard composed of 16 small squares has 8 of these squares painted black such that each row and each column contains exactly 2 black squares. How many different ways are there to paint the squares?


Katya placed a square with a perimeter of 40 cm next to a square with a perimeter of 100 cm as shown in the picture. What is the perimeter of the resulting figure in centimeters?

Solution: \boxed{120 \text{ cm}}


Solve the integral equation

$$
\int_{0}^{\infty} \frac{e^{-x t /(1-x)}}{1-x} e^{-t} \varphi(t) d t=1-x, \quad |x|<1
$$

Solution: \boxed{\varphi(t) = t}


A natural number \( A \) is written twice in a row (for example, writing 12 as 1212) to form a new number \( B \). If \( B \) is a multiple of 2016, what is the smallest value of \( A \)?


On Monday, Misha has five classes, and on Tuesday, he has six. To determine in which of these classes unpredictable teachers will ask him for his homework, Misha flips a coin 11 times – once for each possible occasion of being asked. If heads, Misha believes he will be asked about his homework in that class; if tails, he won't be asked. After his classes on Tuesday, Misha noted that he guessed correctly 7 times. Find the probability that exactly 3 of his correct predictions were for Monday.


Dina has a calculating machine, labelled \( f \), that takes one number as input and calculates an output. The machine \( f \) calculates its output by multiplying its input by 2 and then subtracting 3. For example, if Dina inputs 2.16 into \( f \), the output is 1.32. If Dina inputs a number \( x \) into \( f \), she gets a first output which she then inputs back into \( f \) to obtain a second output, which is -35. What is the value of \( x \)?


Given the two-variable function
$$
f(a, b)=\max _{x \in[-1,1]}\left\{\left|x^{2}-a x-b\right|\right\},
$$
find the minimum value of \( f(a, b) \).


A function \( f \) is called periodic if it takes at least two different values, and there exists \( p > 0 \) such that \( f(x + p) = f(x) \) for any \( x \). Each of these numbers \( p \) is called a period of the function \( f \).

Is it possible to construct functions \( g \) and \( h \) with periods 1 and \( \pi \) respectively, such that \( g + h \) is also a periodic function?

Solution: \boxed{\text{No}}


Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6. The distance between the centers of the spheres is 13. If a plane is tangent to these two spheres and intersects the surface of the cylinder forming an ellipse, then the sum of the lengths of the major axis and the minor axis of this ellipse is ___.


Find the true absolute error of the number \( a_{0} = 245.2 \) if \( a = 246 \).


At an Antarctic station, there are $n$ polar explorers, all of different ages. With probability $p$, a friendship develops between any two polar explorers, independent of other friendships or dislikes. When the winter ends and it's time to go home, in each pair of friends, the older one gives friendly advice to the younger one. Find the expected number of those who did not receive any friendly advice.


What is the maximum number of regions into which \( n \) circles can divide the plane?

Solution: \boxed{n(n-1) + 2}


Upon entering the atmosphere, an asteroid heated up significantly and exploded as it approached the Earth's surface, breaking into a large number of fragments. Scientists collected all the fragments and divided them into groups based on their size. It was found that one-fifth of all the fragments had a diameter of 1 to 3 meters, another 26 were very large (more than 3 meters in diameter), and the rest were divided into several groups, each consisting of 1/7 of the total number of fragments. What is the total number of fragments collected by the scientists?


As shown in the figure, a rectangular table has 8 columns. The numbers $1, 2, \cdots$ are filled into the table in a specific order (from left to right, moving to the next row and continuing from left to right once a row is filled). A student first paints the cell containing the number 1 black, then skips 1 cell and paints the cell containing the number 3 black; next, he skips 2 cells and paints the cell containing the number 6 black; then he skips 3 cells and paints the cell containing the number 10 black. This process continues until every column contains at least one black cell (then he stops). What is the number in the last cell he paints black?


One degree on the Celsius scale is equal to 1.8 degrees on the Fahrenheit scale, and $0^{\circ}$ Celsius corresponds to $32^{\circ}$ Fahrenheit.

Can the temperature be expressed as the same number of degrees in both Celsius and Fahrenheit?


Find the derivative.

\[ y = \frac{1}{4} \cdot \ln \frac{x-1}{x+1} - \frac{1}{2} \cdot \operatorname{arctg} \, x \]

Solution: \boxed{\frac{1}{x^4 - 1}}


There are three flasks containing concentrated acid: the first with 10 g, the second with 20 g, and the third with 30 g. There is also a fourth flask with water. If some amount of water from the fourth flask is added to the first flask, and the remaining water is poured into the second flask, the concentration of acid in the first flask will be 5%, and in the second flask will be $23 \frac{1}{3} \%$. What will be the concentration of acid in the third flask if all the water from the fourth flask is poured into it?


The sequence of numbers \( a_1, a_2, a_3, \ldots, a_n, \ldots \) satisfies the relationship \( a_n = a_{n-1} \cdot a_{n-3} \) for \( n = 4, 5, 6, \ldots \). Find \( a_{2019} \), given that \( a_1 = 1 \), \( a_2 = 1 \), \( a_3 = -1 \).


The sum of \(\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\frac{1}{4 \times 5 \times 6}+\ldots+\frac{1}{13 \times 14 \times 15}+\frac{1}{14 \times 15 \times 16}\) is \(\frac{m}{n}\) in its lowest terms. Find the value of \(m+n\).


Among the first hundred elements of the arithmetic progression \(3, 7, 11, \ldots\), find those that are also elements of the arithmetic progression \(2, 9, 16, \ldots\). Provide the sum of the found numbers in your answer.


In the year $3000-, the World Hockey Championship will follow new rules: 12 points will be awarded for a win, 5 points will be deducted for a loss, and no points will be awarded for a draw. If the Brazilian team plays 38 matches, scores 60 points, and loses at least once, how many wins can they achieve? List all possible scenarios and justify why there cannot be any others.


Given a parallelepiped \( ABCD A_1 B_1 C_1 D_1 \), point \( X \) is chosen on edge \( A_1 D_1 \) and point \( Y \) is chosen on edge \( BC \). It is known that \( A_1 X = 5 \), \( B Y = 3 \), and \( B_1 C_1 = 14 \). The plane \( C_1 X Y \) intersects the ray \( DA \) at point \( Z \). Find \( DZ \).


If \( w \) is a positive integer with \( w^{2} - 5w = 0 \), what is the value of \( w \)?


Two cars leave a station one after another and head towards a dock. The speed of the first car is 90 km/h. At 8:32 AM, the distance between the first car and the station is 1.2 times the distance of the second car from the station. At 9:20 AM, the distance between the first car and the station is twice the distance of the second car from the station. If the first car left the station at 8:08 AM, what is the speed of the second car in kilometers per hour?


In cube \( ABCD-A_{1}B_{1}C_{1}D_{1} \) with \( AA_{1} = 1 \), \( E \) and \( F \) being the midpoints of edges \( CC_{1} \) and \( DD_{1} \) respectively, what is the area of the cross-section of the circumscribed sphere of the cube cut by plane \( AEF \)?


A finite increasing sequence $a_{1}, a_{2}, \ldots, a_{n}$ ($n \geq 3$) of natural numbers is given, such that for all $k \leq n-2$, the equality $a_{k+2}=3 a_{k+1}-2 a_{k}-1$ holds. The sequence must contain the term $a_{k}=2021$. Determine the maximum number of three-digit numbers divisible by 25 that this sequence can contain.


Find the number of terms in a finite geometric progression where the first, second, and last terms are 3, 12, and 3072, respectively.


Flat Albert and his buddy Mike are watching the game on Sunday afternoon. Albert is drinking lemonade from a two-dimensional cup which is an isosceles triangle whose height and base measure 9 cm and 6 cm; the opening of the cup corresponds to the base, which points upwards. Every minute after the game begins, the following takes place: if \( n \) minutes have elapsed, Albert stirs his drink vigorously and takes a sip of height \(\frac{1}{n^{2}} \) cm. Shortly afterwards, while Albert is busy watching the game, Mike adds cranberry juice to the cup until it's once again full in an attempt to create Mike's cranberry lemonade. Albert takes sips precisely every minute, and his first sip is exactly one minute after the game begins. After an infinite amount of time, let \( A \) denote the amount of cranberry juice that has been poured (in square centimeters). Find the integer nearest \(\frac{27}{\pi^{2}} A\).


Three "mryak" are 10 rubles more expensive than five "bryak". Six "mryak" are 31 rubles more expensive than eight "bryak". By how many rubles are seven "mryak" more expensive than nine "bryak"?


A body is thrown vertically upward with a velocity that changes according to the law \( v = (29.4 - 9.8t) \ \mathrm{m/s} \). Find the maximum height attained.

Solution: \boxed{44.1 \text{ meters}}


A ceiling is covered by an $8 \times 8$ grid where each cell contains a lightbulb. On the wall, there are switches that can toggle the state of each bulb in a row or a column. Initially, all the lightbulbs are on. Is it possible to achieve a configuration where only one lightbulb is on?

Solution: \boxed{\text{No}}


A covered rectangular soccer field of length 90 meters and width 60 meters is being designed. It must be illuminated by four floodlights, each hung at some point on the ceiling. Each floodlight illuminates a circle with a radius equal to the height at which it is hung. Determine the minimum possible height of the ceiling such that the following conditions are satisfied: every point on the soccer field is illuminated by at least one floodlight; the height of the ceiling must be a multiple of 0.1 meters (e.g., 19.2 meters, 26 meters, 31.9 meters).

Solution: \boxed{27.1 \text{ meters}}


Chickens and rabbits are kept in the same cage. There are 20 fewer chickens than rabbits. The number of rabbit legs is 10 more than three times the number of chicken legs. How many chickens are there?


13 children sat around a circular table and agreed that boys would lie to girls but tell the truth to each other. Conversely, girls would lie to boys but tell the truth to each other. One of the children told their right-hand neighbor, "The majority of us are boys." That child then told their right-hand neighbor, "The majority of us are girls," and the third child told their right-hand neighbor, "The majority of us are boys," and so on, continuing in this pattern until the last child told the first child, "The majority of us are boys." How many boys were at the table?


Calculate the limit of the numerical sequence:

\[ \lim _{n \rightarrow \infty} \frac{\sqrt[3]{n^{2}+2}-5 n^{2}}{n-\sqrt{n^{4}-n+1}} \]


Given a set \( \mathrm{S} = \{1, 2, 3, \ldots, 199, 200\} \), a subset \(\mathrm{A} = \{a, b, c\}\) of \(\mathrm{S}\) is said to be "nice" if \( a + c = 2b \). How many "nice" subsets does \(\mathrm{S}\) have? (Note: The order of the elements inside the set does not matter. For example, we consider \(\{a, b, c\}\) or \(\{a, c, b\}\) or \(\{c, b, a\}\) to be the same set.)


If \( a < b < c < d < e \) are consecutive positive integers, \( b + c + d \) is a perfect square, and \( a + b + c + d + e \) is a perfect cube, what is the minimum value of \( c \)?


Let \( ABCD \) be a trapezoid with \( AD \parallel BC \), such that side \( CD \) is tangent to the circle with diameter \( AB \). If \( G \) is the midpoint of \( CD \) and \( CD = 8 \text{ cm} \), determine the measure of the height \( GF \).

Solution: \boxed{4 \text{ cm}}


Find the lateral surface area of a right triangular prism with height \( h \), given that a line passing through the center of the top base and the midpoint of a side of the bottom base is inclined at an angle of \( 60^{\circ} \) to the plane of the base.


Two milk processing plants must process an equal amount of a certain volume of milk together. The second plant started working $a$ working days later but processed daily $m$ liters more than the first one. After $5a/9$ working days since the start of their joint work, $1/3$ of the entire task remained incomplete. How many working days were required to complete the task if the work was finished simultaneously and each plant processed half of the specified volume of milk?


Find all functions \( f: \mathbb{Z}_{+} \rightarrow \mathbb{Z}_{+} \) such that for any positive integer \( n \), the following equation holds:
\[ f(f(f(n))) + f(f(n)) + f(n) = 3n. \]

Solution: \boxed{f(n) = n}


The equations of the sides of a quadrilateral are:

$$
y = -x + 7, \quad y = \frac{x}{2} + 1, \quad y = -\frac{3}{2} x + 21, \quad y = \frac{7}{4} x + \frac{3}{2}
$$

Determine the coordinates of the vertices of the quadrilateral and its area.


Circles with centers \( O_1 \) and \( O_2 \) and equal radii are inscribed in the angles at \( B \) and \( C \) of triangle \( ABC \) respectively. Point \( O \) is the center of the incircle of triangle \( ABC \). These circles touch side \( BC \) at points \( K_1, K_2, \) and \( K \) respectively, with \( BK_1 = 4, CK_2 = 8, \) and \( BC = 18 \).

a) Find the length of segment \( CK \).

b) Suppose the circle with center \( O_1 \) touches side \( AB \) at point \( K_3 \). Find the angle \( ABC \), given that point \( O_1 \) is the center of the circumscribed circle around triangle \( OK_1K_3 \).


If the lines \( y = x + d \) and \( x = -y + d \) intersect at the point \( (d-1, d) \), find the value of \( d \).


What is the maximum number of checkers that can be placed on an $8 \times 8$ board such that each one is under attack?


In a class, the percentage of girls is more than $25 \%$ but less than $30 \%$. What is the smallest number of students that such a class can have?


Let \( s(n) \) denote the sum of the digits (in base ten) of a positive integer \( n \). Compute the number of positive integers \( n \) of at most \( 10^4 \) that satisfy
\[ s(11n) = 2s(n). \]


Find the degree measure of the angle

$$
\delta=\arccos \left(\left(\sin 2541^{\circ}+\sin 2542^{\circ}+\cdots+\sin 6141^{\circ}\right)^{\cos 2520^{\circ}}+\cos 2521^{\circ}+\cdots+\cos 6120^{\circ}\right)
$$


In the right trapezoid \( ABCD \), it is known that \(\angle A = \angle D = 90^\circ\), \(DE \perp AC\) at point \(E\), \(\angle ACD = \angle EBC = 30^\circ\), and \(AD = \sqrt{3}\). Find \(BC\).


In $\triangle ABC$, let $A$, $B$, and $C$ be the three interior angles, and $a$, $b$, and $c$ be the sides opposite these angles, respectively. Given that:
\[2 \sqrt{2}\left(\sin ^{2} A-\sin ^{2} C\right) = (a-b) \sin B,\]
and the circumradius of $\triangle ABC$ is $\sqrt{2}$.
(1) Find angle $C$.
(2) Find the maximum value of the area $S$ of $\triangle ABC$.


Find two coprime numbers \(a\) and \(b\) that satisfy the equation:

$$
\frac{5a + 7b}{6a + 5b} = \frac{29}{28}
$$

Solution: \boxed{a = 3, \quad b = 2}


Given the equation $8 \cdot 21$, find the integer $k$ such that $36+k$, $300+k$, and $596+k$ are the squares of three consecutive terms in an arithmetic sequence.

Solution: \boxed{k = 925}


Given any natural numbers \(a\) and \(b\), we define: \(a \oplus b = a + b + 1\) and \(a \otimes b = a \times b - 1\). What is the value of \((5 \oplus 7) \oplus (2 \otimes 4)\)?


Boris was given a Connect Four game set for his birthday, but his color-blindness makes it hard to play the game. Still, he enjoys the shapes he can make by dropping checkers into the set. If the number of shapes possible modulo (horizontal) flips about the vertical axis of symmetry is expressed as \( 9(1+2+\cdots+n) \), find \( n \). (Note: the board is a vertical grid with seven columns and eight rows. A checker is placed into the grid by dropping it from the top of a column, and it falls until it hits either the bottom of the grid or another checker already in that column. Also, \( 9(1+2+\cdots+n) \) is the number of shapes possible, with two shapes that are horizontal flips of each other counted as one. In other words, the shape that consists solely of 3 checkers in the rightmost row and the shape that consists solely of 3 checkers in the leftmost row are to be considered the same shape.)


Alice has six magical pies in her pocket - two that increase in size (eat - grow), and the remaining are decreasing (eat - shrink). When Alice met Mary Ann, she blindly took three pies from her pocket and gave them to Mary. Find the probability that one of the girls does not have a single size-increasing pie.


Given that $\theta$ is an acute angle and $\sin \frac{\theta}{2}=\sqrt{\frac{x-1}{2x}}$, find the value of $\operatorname{tg} \theta$ (where $\operatorname{tg}$ represents the tangent function).

(A) $x$

(B) $\frac{1}{x}$

(C) $\frac{\sqrt{x-1}}{x+1}$

(D) $\frac{\sqrt{x^{2}-1}}{x}$

(E) $\sqrt{x^{2}-1}$

(Source: The 24th Annual High School Mathematics Examination, 1973)

Solution: \boxed{\sqrt{x^2 - 1}}


Given \( 2014 + y = 2015 + x = 2016 + z \) and \( y \times x \times z = 504 \), find \( y \times x + x \times z \).


When Gulliver arrived in Lilliput, he found that everything there was exactly 12 times shorter than in his homeland. Can you determine how many Lilliputian matchboxes can fit into Gulliver's matchbox?


The volume of a rectangular cuboid is 8 cubic centimeters, and the total surface area is 32 square centimeters. If its length, width, and height form a geometric progression, then the sum of all its edges is $\qquad$ .


A natural number is called interesting if all its digits are distinct, and the sum of any two adjacent digits is a perfect square. Find the largest interesting number.


From four classes, 28% of the students received a grade of "5" on the final math exam, 35% received a "4", 25% received a "3", and 12% received a "2". How many students took the exam if there are no more than 30 students in each class?


In the city of Trocalândia, 20% of the cats think they are dogs, and 25% of the dogs think they are cats. One day, a veterinary psychologist decided to test all the cats and dogs in Trocalândia and found that 30% of the total thought they were cats. What proportion of the tested animals were dogs?


Numbers \(a\) and \(b\) are such that the polynomial \(x^{4} + x^{3} + 2x^{2} + ax + b\) is the square of some other polynomial. Find \(b\).


Given that \((m-2)\) is a positive integer and it is also a factor of \(3m^2 - 2m + 10\), find the sum of all such values of \(m\).


The supply of a certain good in a competitive market can be described by the function $Q^{S}(p) = 2 + 8p$ (where $Q$ is the quantity sold, and $p$ is the price per unit). It is known that at a price of 2 monetary units, buyers are willing to purchase no more than 8 units of the product, and at a price of 3 monetary units, no more than 6 units of the product, with the demand curve being linear.

a) What will be the equilibrium price and quantity of the product sold?

b) To increase consumption of this product, the government has introduced a per unit subsidy for its producers amounting to $s = 1$ for each unit produced. By how many units will the quantity sold increase? (14 points).


Given that \( M \) is the midpoint of the height \( D D_{1} \) of a regular tetrahedron \( ABCD \), find the dihedral angle \( A-M B-C \) in radians.


Given that \( a_{k} \) is the number of integer terms in \( \log_{2} k, \log_{3} k, \cdots, \log_{2018} k \). Calculate \( \sum_{k=1}^{2018} a_{k} \).


A regular 2022-gon has a perimeter of 6.28. To the nearest positive integer, compute the area of the 2022-gon.


Consider three collinear points \(B\), \(C\), and \(D\) such that \(C\) is between \(B\) and \(D\). Let \(A\) be a point not on the line \(BD\) such that \(AB = AC = CD\).

(a) If \(\angle BAC = 36^\circ\), then verify that
\[ \frac{1}{CD} - \frac{1}{BD} = \frac{1}{CD + BD} \]

(b) Now, suppose that
\[ \frac{1}{CD} - \frac{1}{BD} = \frac{1}{CD + BD} \]

Verify that \(\angle BAC = 36^\circ\).


(a) In the diagram, $\triangle PQR$ is right-angled at $R$. If $PQ = 25$ and $RQ = 24$, determine the perimeter and area of $\triangle PQR$.

(b) In the diagram, $\triangle ABC$ is right-angled at $C$ with $AB = c$, $AC = b$, and $BC = a$. Also, $\triangle ABC$ has a perimeter of 144 and an area of 504. Determine all possible values of $c$.


The base of an inclined parallelepiped is a rhombus with a side length of 60. A diagonal section plane passing through the longer diagonal of the base is perpendicular to the base's plane. The area of this section is 7200. Find the shorter diagonal of the base if the lateral edge is 80 and forms an angle of $60^\circ$ with the base plane.


The vertices of a quadrilateral are \( A(0, 0) \), \( B(0, -1) \), \( C(1, 0) \), \( D\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \). Let \( K \) be the intersection point of the lines \( AB \) and \( CD \). Calculate the distance \( AK \).

Solution: \boxed{\sqrt{2} + 1}


The medians of one triangle are equal to the sides of another triangle. Find the ratio of the areas of these triangles.


A square has side length \( s \) and a diagonal of length \( s+1 \). Write the area of the square in the form \( a+b \sqrt{c} \) where \( a \), \( b \), and \( c \) are positive integers.

Solution: \boxed{3 + 2\sqrt{2}}


The sum of the first two digits of a four-digit number is equal to the sum of the last two digits. The sum of the first and the last digits equals the third digit. The sum of the second and fourth digits is twice the sum of the first and third digits. What is this four-digit number?


Given positive numbers \(a, b, c, d\), find the minimum value of the expression

$$
A = \left( \frac{a+b}{c} \right)^4 + \left( \frac{b+c}{d} \right)^4 + \left( \frac{c+d}{a} \right)^4 + \left( \frac{d+a}{b} \right)^4
$$


If \( a \) and \( b \) are positive integers such that \( a^{2} - b^{4} = 2009 \), find \( a + b \).


The legs \( AC \) and \( CB \) of the right triangle \( ABC \) are 15 and 8, respectively. A circular arc with radius \( CB \) is drawn from center \( C \), cutting off a part \( BD \) from the hypotenuse. Find \( BD \).


An art exhibition opens at 8:00, but people have already been queuing to enter before that. From the time the first visitor arrives, the number of visitors arriving per minute is consistent. If 3 entry points are opened, the queue ends at 8:09. If 5 entry points are opened, the queue ends at 8:05. What time did the first visitor arrive?


Given a rectangle of dimensions \(100 \times 101\), divided by grid lines into unit squares, find the number of segments into which these grid lines divide its diagonal.


If \( x, y \) and \( z \) are positive real numbers such that \(\frac{x+y-z}{z}=\frac{x-y+z}{y}=\frac{-x+y+z}{x}\) and \( a=\frac{(x+y) \cdot(y+z) \cdot(z+x)}{x y z} \), find the value of \( a \).

Let \( u \) and \( t \) be positive integers such that \( u+t+u t=4 a+2 \). If \( b=u+t \), find the value of \( b \).


Determine \( p \) so that the expression

$$
x^{2}-4 b x+4 a b+p^{2}-2 p x
$$

is a perfect square.

Solution: \boxed{a - b}


A and B arrange to meet at a certain place between 0:00 and 1:00. The person who arrives first should wait for 20 minutes before leaving. If the times of their arrival are independent and equally likely at any moment between 0:00 and 1:00, what is the probability that they will meet?


There are knights who always tell the truth and liars who always lie living on an island. One day, 30 residents of this island sat around a round table. Each of them said one of the following two phrases: "My neighbor on the left is a liar" or "My neighbor on the right is a liar." What is the minimum number of knights that can be seated at the table?


Let \( T \) be the set of all positive divisors of \( 60^{100} \). \( S \) is a subset of \( T \) such that no number in \( S \) is a multiple of another number in \( S \). Find the maximum value of \( |S| \).


Let $X$ be a subset of $\{1,2,3, \ldots, 10000\}$ with the following property: If $a, b \in X, a \neq b$, then $a \cdot b \notin X$. What is the maximal number of elements in $X$?


Let \(\lfloor x \rfloor\) denote the greatest integer less than or equal to \(x\). How many positive integers less than 2005 can be expressed in the form \(\lfloor x \lfloor x \rfloor \rfloor\) for some positive real \(x\)?


Given \(\frac{\cos \alpha \cos \frac{\beta}{2}}{\cos \left(\alpha - \frac{\beta}{2}\right)} + \frac{\cos \beta \cos \frac{\alpha}{2}}{\cos \left(\beta - \frac{\alpha}{2}\right)} = 1\), find the value of \(\cos \alpha + \cos \beta\).


Real numbers \( x \) and \( y \) satisfy \( \tan x = x \) and \( \tan y = y \), and \( |x| \neq |y| \). Find the value of \(\frac{\sin (x+y)}{x+y} - \frac{\sin (x-y)}{x-y} \).


What real values of \( x \) satisfy the equations:
(a) \(\sqrt{x+\sqrt{2x-1}} + \sqrt{x-\sqrt{2x-1}} = \sqrt{2}\),
(b) \(\sqrt{x+\sqrt{2x-1}} + \sqrt{x-\sqrt{2x-1}} = 1\),
(c) \(\sqrt{x+\sqrt{2x-1}} + \sqrt{x-\sqrt{2x-1}} = 2\)?

(Here, the square roots are non-negative.)


In triangle \(ABC\), let the parabola with focus \(A\) and directrix \(BC\) intersect sides \(AB\) and \(AC\) at \(A_1\) and \(A_2\), respectively. Similarly, let the parabola with focus \(B\) and directrix \(CA\) intersect sides \(BC\) and \(BA\) at \(B_1\) and \(B_2\), respectively. Finally, let the parabola with focus \(C\) and directrix \(AB\) intersect sides \(CA\) and \(CB\) at \(C_1\) and \(C_2\), respectively. If triangle \(ABC\) has sides of length 5, 12, and 13, find the area of the triangle determined by lines \(A_1C_2\), \(B_1A_2\), and \(C_1B_2\).


Five numbers form a decreasing arithmetic progression. The sum of their cubes is zero, and the sum of their fourth powers is 136. Find the smallest of these numbers.

Solution: \boxed{-2 \sqrt{2}}


Given four positive integers \(a, b, c,\) and \(d\) satisfying the equations \(a^2 = c(d + 20)\) and \(b^2 = c(d - 18)\). Find the value of \(d\).


In a rectangular table, the letters of the word "olympiada" are arranged in a specific order.

| O | Л | И | M | П | И | A | Д | A |
| - | - | - | - | - | - | - | - | - |
| Л | И | M | П | И | A | Д | A | O |
| И | M | П | И | A | Д | A | O | Л |
| M | П | И | A | Д | A | O | Л | И |

Starting from the letter "O" located in the top left corner of the table, one must read the word "olympiada", ending with the letter "A". The transition between letters is allowed between adjacent cells in the table. Find the number of different ways to read the word "olympiada" according to the given table.


Given a triangular pyramid $S-ABC$ whose base is an equilateral triangle, the projection of point $A$ on the face $SBC$ is the orthocenter $H$ of triangle $\triangle SBC$. The dihedral angle between the planes $H-AB-C$ is $30^{\circ}$, and $SA = 2\sqrt{3}$. Find the volume of the triangular pyramid $S-ABC$.


Solve the system of equations. Provide the largest value of $y$ in the solution.

$$
\left\{\begin{array}{l}
x^{2}+3 x y-y^{2}=27 \\
3 x^{2}-x y+y^{2}=27
\end{array}\right.
$$


The number \( x \) satisfies \( x^{2}-4x+2=0 \). What is the value of \( x+\frac{2}{x} \)?


For a Spring Festival event for supporting military families, the First Residents' Committee bought 9 kg of oranges and 10 kg of apples, spending a total of 73.8 yuan; the Second Residents' Committee bought 17 kg of pears and 6 kg of bananas, spending a total of 69.8 yuan. If the unit price of oranges and pears is the same, and the unit price of apples and bananas is also the same, how much is the cost per kilogram of oranges and the cost per kilogram of bananas?

Solution: \boxed{2.2, \ 5.4}


Find the largest five-digit number whose digits' product equals 120.


Determine the real number \( a \) having the property that \( f(a) = a \) is a relative minimum of \( f(x) = x^4 - x^3 - x^2 + ax + 1 \).


Given that \(\sin \alpha + \sin \beta = 2 \sin (\alpha + \beta)\) with \(\alpha + \beta \neq 2 \pi n (n \in \mathbb{Z})\). Find \(\tan \frac{\alpha}{2} \tan \frac{\beta}{2}\).


Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfying for all \( x, y \in \mathbb{R} \):

\[ f(x+y) = f(x) - f(y) \]

Solution: \boxed{f(x) = 0}


N is the set of positive integers. Find all functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that \( f(f(n)) + f(n) = 2n + 2001 \) or \( 2n + 2002 \).

Solution: \boxed{f(n) = n + 667}


Without using any tables, find the exact value of the product:
\[ P = \cos \frac{\pi}{15} \cos \frac{2\pi}{15} \cos \frac{3\pi}{15} \cos \frac{4\pi}{15} \cos \frac{5\pi}{15} \cos \frac{6\pi}{15} \cos \frac{7\pi}{15}. \]


Two circular tracks $\alpha$ and $\beta$ of the same radius are tangent to each other. A car $A$ travels clockwise on track $\alpha$ and a car $B$ travels counterclockwise on track $\beta$. At the start, cars $A$ and $B$ are on the same line with the center of track $\alpha$, and this line is tangent to track $\beta$. After the start, the cars begin to approach the point of tangency of the tracks. Each car completes one full lap on its track in one hour (and never switches to the other track). For how much time during this hour will the distance between the cars be at least the diameter of each track?


Petya is playing a computer game called "Pile of Stones." Initially, there are 16 stones in the pile. Players take turns to take 1, 2, 3, or 4 stones from the pile. The player who takes the last stone wins. Petya is playing for the first time and therefore takes a random number of stones each time, while still following the rules of the game. The computer plays according to the following algorithm: on each turn, it takes as many stones as necessary to put itself in the most advantageous position. Petya always starts the game. What is the probability that Petya will win?


Given that \( AM^2 = CM \cdot MN \) or \( \frac{AM}{MN} = \frac{CM}{AM} \), and since \( AM = MK \), we have \( \frac{AM}{MN} = \frac{CM}{MK} \).

Consider triangles \( \triangle AMN \) and \( \triangle CMK \). With \( \angle AMN = \angle CMK \) and \( \frac{AM}{MN} = \frac{CM}{MK} \), the triangles are similar by the Angle-Angle-Angle (AAA) criterion. From this similarity, it follows that \( \angle ANC = \angle CKA \).

Now, consider triangles \( \triangle BNC \) and \( \triangle BKA \). With \( \angle B \) being common and \( \angle BNC = \angle BKA \) (since they are adjacent to equal angles), the triangles \( \triangle ABK \) and \( \triangle CBN \) are similar by the Side-Angle-Side (SAS) criterion. From this similarity, it follows that \( \frac{BK}{NB} = \frac{AB}{BC} \).

Next, consider triangles \( \triangle BNK \) and \( \triangle BCA \). Given \( \frac{BK}{NB} = \frac{AB}{BC} \) and \( \angle B \) being common, the triangles \( \triangle BNK \) and \( \triangle CBN \) are similar by the Side-Angle-Side (SAS) criterion. From this similarity, it follows that \( \angle BKN = \angle BAC \).

Thus,
\[ \angle BAC = 180^\circ - (\angle ABC + \angle BCA) = 180^\circ - (47^\circ + 64^\circ) = 69^\circ = \angle BKN \]


It takes 18 doodads to make 5 widgets. It takes 11 widgets to make 4 thingamabobs. How many doodads does it take to make 80 thingamabobs?


In the central cell of a $21 \times 21$ board, there is a piece. In one move, the piece can be moved to an adjacent cell sharing a side. Alina made 10 moves. How many different cells can the piece end up in?


Find the maximum and minimum values of the function \( y = x^3 - 3x^2 + 5 \) on the interval \([-1, 1]\).


According to a survey conducted in the 7th "E" grade class, it was found that $20\%$ of the students who are interested in mathematics are also interested in physics, and $25\%$ of the students who are interested in physics are also interested in mathematics. Only Peter and Vasya are not interested in either of these subjects. How many students are there in the 7th "E" grade class, given that there are more than 20 but less than 30 of them?


The "Buratino" cafe is open 6 days a week with a day off on Mondays. Kolya made two statements: "From April 1 to April 20, the cafe worked 18 days," and "From April 10 to April 30, the cafe also worked 18 days." It is known that he made a mistake once. How many days did the cafe work from April 1 to April 13?


On the sides \( AB \) and \( BC \) of an equilateral triangle \( ABC \), points \( L \) and \( K \) are marked respectively. \( M \) is the point of intersection of segments \( AK \) and \( CL \). It is known that the area of triangle \( AMC \) is equal to the area of quadrilateral \( LBKM \). Find the angle \( AMC \).


Little Pang, Little Dingding, Little Ya, and Little Qiao's four families, totaling 8 parents and 4 children, went to the amusement park together. The ticket prices are as follows: adult tickets are 100 yuan per person; children's tickets are 50 yuan per person; if there are 10 or more people, they can buy group tickets, which are 70 yuan per person. What is the minimum amount they need to spend to buy the tickets?


Given the ellipse \(\frac{x^{2}}{5}+\frac{y^{2}}{4}=1\) with its right focus as \(F\) and its upper vertex as \(A\). Point \(P\) is a moving point on the ellipse located in the first quadrant. Find the maximum area of triangle \(\triangle APF\).

Solution: \boxed{\sqrt{6} - 1}


There are fewer than 50 students in a class. $\frac{1}{7}$ of the students live on Gorky Street, $\frac{1}{3}$ live on Pushkin Street, $\frac{1}{2}$ live on Gaidar Street, and the rest live on Korabelnaya Street. How many students live on Korabelnaya Street?


In the quadrilateral \(ABCD\), \(AB = 2\), \(BC = 4\), \(CD = 5\). Find its area given that it is both circumscribed and inscribed.


$$\sin \left(2 \alpha - \frac{3}{2} \pi\right) + \cos \left(2 \alpha - \frac{8}{3} \pi\right) + \cos \left(\frac{2}{3} \pi + 2 \alpha\right)$$


In a regular decagon, all diagonals were drawn. How many pairwise non-similar triangles are there in this figure?


In triangle \( A B C \), the angles \(\angle B = 30^\circ\) and \(\angle A = 90^\circ\) are known. Point \(K\) is marked on side \( A C \), and points \(L\) and \( M \) are marked on side \( B C \) such that \( K L = K M \) (point \( L \) lies on segment \( B M\)).

Find the length of segment \( L M \), given that \( A K = 4 \), \( B L = 31 \), and \( M C = 3 \).


If \( 2 \times 3^{8} \) can be expressed as the sum of \( k \) consecutive positive integers, what is the maximum value of \( k \)?


There are $n$ marbles, and they are divided into two piles in any manner. Determine the product of the numbers of marbles in the two piles. Then, take one of the piles and divide it into two piles in any manner. Determine the product of the numbers of marbles in these two new piles. Repeat this process: each time take any pile and divide it into two piles, determine the product of the numbers of marbles in the two piles, and continue until no pile can be further divided. Denote the sum of all these products as $S$. Determine the possible values of $S$.


Given the functions \( f(x) = \frac{x^2 + 4x + 3}{x^2 + 7x + 14} \) and \( g(x) = \frac{x^2 - 5x + 10}{x^2 + 5x + 20} \):
1. Find the maximum value of the function \( f(x) \).
2. Find the maximum value of the function \( g(x)^{f(x)} \).


Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in 3 days, and Anne and Carl can do it in 5 days. How many days does it take for all of them working together, if Carl gets injured at the end of the first day and can't come back? Express your answer as a fraction in lowest terms.


Calculate the result of the expression \((11 \times 24 - 23 \times 9) \div 3 + 3\).


Find the number of divisors of the number \( a = 2^3 \cdot 3^2 \cdot 5^2 \) that are divisible by 3. Find the sum of such divisors.


The line \( y = kx - 2 \) intersects the parabola \( y^2 = 8x \) at points \( A \) and \( B \). If the x-coordinate of the midpoint of the line segment \( AB \) is 2, then the length of the line segment \( AB \) is \( |AB| \).

Solution: \boxed{2 \sqrt{15}}


Determine the function \( f: \mathbb{R}^+ \rightarrow \mathbb{R}^+ \) such that for all \( x, y \in \mathbb{R}^+ \), the equation \( f(x f(y)) = f(xy) + x \) holds.

Solution: \boxed{f(x) = x + 1}


In the triangle \(ABC\), we draw the altitude \(AA_1\) from vertex \(A\). What is the length of \(BA_1\), if \(AB = 8 \text{ cm}\), \(AC = 10 \text{ cm}\), and \(BC = 12 \text{ cm}\)?

Solution: \boxed{4.5 \text{ cm}}


To write down the sequence of numbers from 1 to \( n \), 2893 digits were needed. What is the value of \( n \)?


The non-negative numbers \( x_{1}, x_{2}, \ldots, x_{n} \) have a sum of 1.

How large can the value of
\[ x_{1} x_{2} + x_{2} x_{3} + \ldots + x_{n-2} x_{n-1} + x_{n-1} x_{n} \]
be?


Calculate the length of the arc of the curve given by the equation in a rectangular coordinate system.

$$
y=\ln \cos x+2,0 \leq x \leq \frac{\pi}{6}
$$


Find the maximum and minimum values of \( \cos x + \cos y + \cos z \), where \( x, y, z \) are non-negative reals with sum \( \frac{4\pi}{3} \).


Given that $\triangle ABC$ is an isosceles right triangle with $\angle A=90^{\circ}$, and $\overrightarrow{AB}=\boldsymbol{a}+\boldsymbol{b}$, $\overrightarrow{AC}=a-b$. If $a=(\cos \theta, \sin \theta)$ where $\theta \in \mathbf{R}$, then the area of $\triangle ABC$ is equal to $\qquad$.


Find all pairs \((x, y)\) of positive integers such that \(x^3 - y^3 = xy + 41\).

Solution: \boxed{(5, 4)}


A freight train departed from Moscow at $x$ hours $y$ minutes and arrived in Saratov at $y$ hours $z$ minutes. The travel time was $z$ hours $x$ minutes.

Find all possible values of $x$.

Solution: \boxed{0 \text{ and } 12}


The numbers 100 and 90 were divided by the same number. In the first case, the remainder was 4, and in the second case, the remainder was 18. What number were they divided by?


Let \( f(x) = x^3 + ax + b \), with \( a \neq b \), and suppose the tangent lines to the graph of \( f \) at \( x = a \) and \( x = b \) are parallel. Find \( f(1) \).


Let $a \in \mathbb{R}$. Consider the following functional equation:

$$
f(0)=0, \quad f(1)=1, \quad \text{if } x \leq y, \quad f\left(\frac{x+y}{2}\right)=(1-a)f(x)+a f(y)
$$

Find the values of $a$ for which the equation has at least one solution from $\mathbb{R}$ to $\mathbb{R}$. The same question applies when we require a continuous solution.


To celebrate the anniversary of the St. Petersburg Mathematical Olympiads, the mint produced three commemorative coins. One coin was correct, the second coin had two heads, and the third coin had two tails. The director of the mint randomly selected one of these three coins without looking and flipped it. It landed on heads. What is the probability that the second side of this coin is also heads?


An isosceles triangle has a base \(a\), and its vertex angle is less than \(120^{\circ}\). The height from the base to the opposite vertex is \(m\). Reflect each vertex of the triangle over the line containing the opposite side. The three new points form another isosceles triangle with base \(a^{\prime}\) and corresponding height \(m^{\prime}\). Show that

\[ \frac{a^{\prime}}{a} + \frac{m^{\prime}}{m} = 4 \]


Show that

$$
n^{12}-n^{8}-n^{4}+1
$$

is divisible by 512 when $n$ is an arbitrary odd integer.


The four sides of quadrilateral \(ABCD\) are equal in length. Determine the perimeter of \(ABCD\) given that it has an area of 120 and \(AC = 10\).


Divers extracted a certain number of pearls, not exceeding 1000. The distribution of the pearls happens as follows: each diver in turn approaches the heap of pearls and takes either exactly half or exactly one-third of the remaining pearls. After all divers have taken their share, the remainder of the pearls is offered to the sea god. What is the maximum number of divers that could have participated in the pearl extraction?


A quadrilateral \(ABCD\) is inscribed in a circle. The angle bisectors of angles \(B\) and \(C\) intersect at a point that lies on segment \(AD\).

Find \(AD\), if \(AB=5\) and \(CD=3\).

Solution: \boxed{AD = 8}


A wagon was loaded with several kilograms of apple jam, among which there was $20\%$ good and $80\%$ bad jam. Every day, half of the bad jam spoiled and was discarded. After a few days, it turned out that the wagon contained $20\%$ bad and $80\%$ good jam. How many days have passed since the loading?


Given the sequence $\left\{a_{n}\right\}$ where $a_{1}=1, a_{2}=4, a_{3}=10$, and for any $n \geq 2$, the relation $a_{n+1}^{2} - 2 a_{n}^{2} = a_{n} a_{n+2} - 2 a_{n-1} a_{n+1}$ holds, determine the units digit of $a_{21}$.


Solution: \boxed{a_{100} = -3}


The figure shows a board consisting of nine squares. A counter originally on square $X$ can be moved either upwards or to the right one square at a time. By how many different routes may the counter be moved from $X$ to $Y$?


Calculate the arc lengths of the curves given by the parametric equations.

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=4(\cos t+t \sin t) \\
y=4(\sin t-t \cos t)
\end{array}\right. \\
& 0 \leq t \leq 2 \pi
\end{aligned}
$$


If \( X=\sqrt{(100)(102)(103)(105)+(Q-3)} \) is an integer and \( R \) is the units digit of \( X \), find the value of \( R \).

If \( S \) is the sum of the last 3 digits (hundreds, tens, units) of the product of the positive roots of \( \sqrt{2012} \cdot x^{\log_{2012} x}=x^{R} \), find the value of \( S \).


On a board, there are written four non-zero numbers such that the sum of any three of them is less than the fourth number. What is the minimum number of negative numbers that can be written on the board? Justify your answer.


How many five-digit numbers can be formed

a) with all identical digits,

b) with two different digits,

c) with three different digits,

d) with four different digits,

e) with five different digits?


A group of tourists started their hike from a campsite. Fifteen minutes later, a tourist named Ivan remembered that he had forgotten a flashlight at the campsite and went back to get it, walking at a speed faster than that of the main group. After retrieving the flashlight, he started catching up with the group at the same increased speed and managed to reach them only 2.5 hours after he initially left to get the flashlight. Assuming the speeds of the group and Ivan (when not with the group) are constant, find how many times Ivan's speed is greater than the group's speed. Express the answer as a whole number or a decimal.


Let $A B C D$ be a tetrahedron with volume $V$. Point $E$ is the midpoint of edge $A D$, and point $F$ is on the extension of $A B$ such that $B F = A B$. The plane passing through points $C$, $E$, and $F$ intersects $B D$ at point $G$. Determine the volume of the tetrahedron $C D G E$.


In the triangular pyramid \(P-ABC\), \(\angle APC = \angle CPB = \angle BPA = \frac{\pi}{2}\), and \(PA = PB = 3\), \(PC = 4\). Also, \(M\) is a point within the base \(ABC\). What is the minimum value of the sum of the squares of the distances from \(M\) to the three lateral faces of the triangular pyramid?


The circles $\omega_{1}$ and $\omega_{2}$ with centers $O_{1}$ and $O_{2}$ respectively intersect at point $A$. The segment $O_{2}A$ intersects the circle $\omega_{1}$ again at point $K$, and the segment $O_{1}A$ intersects the circle $\omega_{2}$ again at point $L$. A line passing through point $A$ parallel to $KL$ intersects the circles $\omega_{1}$ and $\omega_{2}$ again at points $C$ and $D$ respectively. The segments $CK$ and $DL$ intersect at point $N$. Find the angle between the lines $O_{1}A$ and $O_{2}N$.


The numbers from 1 to 8 are arranged in a circle. A number is called big if it is greater than its neighbors, and small if it is smaller than its neighbors. In the arrangement, each number is either big or small. What is the maximum possible sum of the small numbers?


Let \(\mathrm{p} > 3\) be a prime number, and define the set \(M = \{1, 2, \ldots, p-1\}\). Consider the set
\[ T = \left\{\left(n, x_{n}\right) : p \mid (n x_{n} - 1) \text{, and } n, x_{n} \in M\right\}. \]
Find the smallest non-negative residue modulo \(\mathrm{p}\) of
\[ \sum_{\left(\mathrm{n}, x_{n}\right) \in \mathrm{T}} \mathrm{n}\left\lfloor \frac{\mathrm{nx}}{\mathrm{p}} \right\rfloor. \]


In a triangle, the sum of the squares of two sides is equal to the square of the diameter of the circumscribed circle. Is it true that the triangle is a right triangle?

Solution: \boxed{\text{No}}


Suppose that the function

$$
y = ax^3 + bx^2 + cx + d
$$

has both extrema. What is the relationship among the coefficients if the straight line connecting the points of the curve corresponding to the extrema passes through the origin?

Solution: \boxed{9ad = bc}


If \(\log _{4}(x+2 y) + \log _{4}(x-2 y) = 1\), what is the minimum value of \(|x| - |y|\)?


There are more than 20 but less than 30 students in the class, and everyone has a different birthday. Petya said, "The number of students older than me is twice the number of students younger than me." Katya said, "The number of students older than me is three times less than the number of students younger than me." How many students are there in the class if both Petya and Katya are correct?


In the diagram, each street segment is 500 meters long. Lele starts from point $A$ and chooses the shortest path to point $B$. Dongdong starts from point $B$ and chooses the shortest path to point $A$. Lele walks 18 meters per minute, and Dongdong walks 24 meters per minute. The probability that they meet on their way is ________.


A four-digit number whose last digit is not 0, if the first two digits can divide 2014, and the product of the first two digits and the last two digits can be divided by 2014, then what is the largest four-digit number?


Find the polynomials with integer coefficients \( P \) such that:

$$
\forall n,\; n \text{ divides } P\left(2^{n}\right)
$$

Solution: \boxed{P(x) = 0}


\( f \) is a mapping from the set of natural numbers \( N \) to the set \( A \). If for \( x, y \in \mathbf{N} \), \( x - y \) is a prime number, it always holds that \( f(x) \neq f(y) \). How many elements must \( A \) have at least?


Given that the base of the triangular pyramid \( S-ABC \) is an isosceles right triangle with \( AB \) as the hypotenuse, and \( SA = SB = SC = 2 \), and \( AB = 2 \), if points \( S \), \( A \), \( B \), and \( C \) are all on the surface of a sphere centered at \( O \), then the distance from point \( O \) to the plane \( ABC \) is \(\qquad\).


Polina went to the cafeteria and saw that 2 puff pastries cost 3 times more than 2 pies. Polina didn't have enough money for 2 puff pastries, but she had enough for 1 pie and 1 puff pastry. She then wondered how many times more money she spent buying 1 puff pastry and 1 pie instead of 2 pies. Help Polina answer this question.


Point \( O \) is the center of the circumcircle of triangle \( ABC \). Points \( Q \) and \( R \) are chosen on sides \( AB \) and \( BC \), respectively. Line \( QR \) intersects the circumcircle of triangle \( ABR \) again at point \( P \) and intersects the circumcircle of triangle \( BCQ \) again at point \( S \). Lines \( AP \) and \( CS \) intersect at point \( K \). Find the angle between lines \( KO \) and \( QR \).


Find the smallest positive integer \( n \) that satisfies the following condition: If the set \( A = \{1, 2, 3, \ldots, n\} \) is arbitrarily divided into 63 pairwise disjoint subsets \( A_1, A_2, A_3, \ldots, A_{63} \) (which are non-empty and their union is the set \( A \)), then there always exist two positive integers \( x \) and \( y \) that belong to the same subset \( A_i \) (\( 1 \leq i \leq 63 \)) such that \( x > y \) and \( 31x \leq 32y \).


Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{2^{n+1}+3^{n+1}}{2^{n}+3^{n}}$$


Find the sum of the first 10 elements that appear in both the arithmetic progression $\{5, 8, 11, 13, \ldots\}$ and the geometric progression $\{20, 40, 80, 160, \ldots\}$. (10 points)


Let $a \triangle b$ and $a \nabla b$ represent the minimum and maximum of two numbers $a$ and $b$, respectively. For example, $3 \triangle 4 = 3$ and $3 \nabla 4 = 4$. Determine the total number of distinct values for the expression $5 \nabla [4 \nabla (x \triangle 4)]$ for different numbers $x$.


Given the equation $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$, find the product of the real roots.


Six persons \(A, B, C, D, E\), and \(F\) are to rotate for night shifts in alphabetical order with \(A\) serving on the first Sunday, \(B\) on the first Monday and so on. In the fiftieth week, which day does \(A\) serve on? (Represent Sunday by 0, Monday by 1, ..., Saturday by 6 in your answer.)


Let \( a \) and \( b \) be two real numbers. Show that \( a^{2} + b^{2} \geq 2ab \).

Solution: \boxed{a^2 + b^2 \geq 2ab}


What percentage error do we make if we approximate the side of a regular heptagon by taking half of the chord corresponding to the $120^\circ$ central angle?


The volume of tetrahedron \(ABCD\) is \(\frac{1}{6}\), and it satisfies \(\angle ACB = 45^\circ\), \(AD + BC + \frac{AC}{\sqrt{2}} = 3\). Find \(CD\).


In response to the call of the commander, 55 soldiers came: archers and swordsmen. All of them were dressed either in golden or black armor. It is known that swordsmen tell the truth when wearing black armor and lie when wearing golden armor, while archers do the opposite.

- To the question "Are you wearing golden armor?" 44 people responded affirmatively.
- To the question "Are you an archer?" 33 people responded affirmatively.
- To the question "Is today Monday?" 22 people responded affirmatively.

How many archers in golden armor came in response to the commander's call?


Find the number of natural numbers \( k \) not exceeding 242400, such that \( k^2 + 2k \) is divisible by 303.


Several even natural numbers are written in a row. Each number, except for the leftmost one, is obtained by subtracting the largest digit of its left neighbor from its left neighbor. Can the row contain more than 12 numbers?

Solution: \boxed{Yes}


On a farm, there are pigs and horses. What is the minimum number of pigs on the farm such that they make up between $54\%$ and $57\%$ of the total number of animals?


If real numbers \( x \) and \( y \) satisfy \( 4x^2 - 4xy + 2y^2 = 1 \), then the sum of the maximum and minimum values of \( 3x^2 + xy + y^2 \) is \(\qquad\)


A box contains one hundred multicolored balls: 28 red, 20 green, 13 yellow, 19 blue, 11 white, and 9 black. What is the minimum number of balls that must be drawn from the box, without looking, to ensure that at least 15 balls of one color are among them?


On the side \( DE \) of a regular hexagon \( ABCDEF \), point \( K \) is chosen so that the line \( AK \) divides the area of the hexagon in the ratio \( 3:1 \). In what ratio does point \( K \) divide the side \( DE \) ? Answer: \( 3:1 \).


Vitya collects toy cars from the "Retro" series. The problem is that the total number of different models in the series is unknown — it's a big commercial secret. However, it is known that different cars are produced in equal quantities, so it can be assumed that all models are evenly and randomly distributed across different online stores.

On different websites, Vitya found several offers, but upon closer inspection, it turned out that among the offered cars, there were only 12 different ones. Vitya is almost convinced that there are only 12 cars in total and that further search is pointless. But who knows?

How many more offers from other stores does Vitya need to review to be convinced that there are indeed 12 models in the series? Vitya considers himself convinced of something if the probability of this event is greater than 0.99.


Find the maximum value that the product of natural numbers, whose sum is equal to 2020, can take.

Solution: \boxed{2^2 \cdot 3^{672}}


In the trapezoid \(KLMN\), the bases \(KN\) and \(LM\) are 12 and 3, respectively. From point \(Q\), which lies on side \(MN\), a perpendicular \(QP\) is dropped to side \(KL\). It is known that \(P\) is the midpoint of side \(KL\), \(PM = 4\), and the area of quadrilateral \(PLMQ\) is four times smaller than the area of quadrilateral \(PKNQ\).

Find the length of segment \(PN\).


Circle \(O\) has chord \(AB\). A smaller circle is tangent to circle \(O\) at \(T\) and tangent to \(AB\) at \(X\) such that \(AX = 2XB\). What is \(\frac{AT}{BT}\)?


A circle with a radius of 1 is randomly placed inside a $15 \times 36$ rectangle $ABCD$. Find the probability that this circle does not intersect with the diagonal $AC$.


Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty}\left(\frac{4 n^{2}+4 n-1}{4 n^{2}+2 n+3}\right)^{1-2 n}
$$


Shawn's password to unlock his phone is four digits long, made up of two 5s and two 3s. How many different possibilities are there for Shawn's password?


Calculate the limit of the function:

$$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\operatorname{tg} 3 x}{\operatorname{tg} x}$$


Natasha and Inna bought identical boxes of tea bags. It is known that one tea bag is enough for either two or three cups of tea. Natasha's box lasted for 41 cups of tea, while Inna's box lasted for 58 cups. How many tea bags were in the box?


In triangle \( \triangle ABC \), \(\angle ABC = 40^{\circ}\), \(\angle ACB = 20^{\circ}\), point \( N \) is inside \( \triangle ABC \), \(\angle NBC = 30^{\circ}\), and \(\angle NAB = 20^{\circ}\). Find the measure of \(\angle NCB\).