Given the function \( f(x) = \log_{2}(x + 1) \), the graph of \( y = f(x) \) is shifted 1 unit to the left and then the y-coordinates of all points on the graph are stretched to twice their original values (the x-coordinates remain unchanged), resulting in the graph of the function \( y = g(x) \). What is the maximum value of the function \( F(x) = f(x) - g(x) \)?


For a positive integer \( n \), let \( x_n \) be the real root of the equation \( n x^{3} + 2 x - n = 0 \). Define \( a_n = \left[ (n+1) x_n \right] \) (where \( [x] \) denotes the greatest integer less than or equal to \( x \)) for \( n = 2, 3, \ldots \). Then find \( \frac{1}{1005} \left( a_2 + a_3 + a_4 + \cdots + a_{2011} \right) \).


A circle with radius 1 is circumscribed around triangle \( A P K \). The extension of side \( A P \) beyond vertex \( P \) cuts off a segment \( B K \) from the tangent to the circle through vertex \( K \), where \( B K \) is equal to 7. Find the area of triangle \( A P K \), given that angle \( A B K \) is equal to \( \arctan \frac{2}{7} \).


In how many ways can all natural numbers from 1 to 200 be painted in red and blue so that no sum of two different numbers of the same color equals a power of two?


Natural numbers \(a\) and \(b\) are such that \(a^{a}\) is divisible by \(b^{b}\), but \(a\) is not divisible by \(b\). Find the smallest possible value of \(a + b\), given that \(b\) is coprime with 210.


A provincial TV station has $n$ ad inserts in one day, broadcasting a total of $m$ ads. The first time, they broadcast one ad and $\frac{1}{8}$ of the remaining $(m-1)$ ads. The second time, they broadcast two ads and $\frac{1}{8}$ of the remaining ads. Following this pattern, during the $n$-th insert, they broadcast the remaining $n$ ads (with $n > 1$). How many times did they broadcast ads that day? Find the total number of ads $m$.

Solution: \boxed{n = 7, m = 49}


For \(x, y, z \geq 1\), find the minimum value of the expression

$$
A = \frac{\sqrt{3 x^{4} + y} + \sqrt{3 y^{4} + z} + \sqrt{3 z^{4} + x} - 3}{x y + y z + z x}
$$


Solve the following system of equations:

$$
\begin{aligned}
& \frac{6}{3x + 4y} + \frac{4}{5x - 4z} = \frac{7}{12} \\
& \frac{9}{4y + 3z} - \frac{4}{3x + 4y} = \frac{1}{3} \\
& \frac{2}{5x - 4z} + \frac{6}{4y + 3z} = \frac{1}{2}
\end{aligned}
$$

Solution: \boxed{x = 4, y = 3, z = 2}


A sequence of integers \( a_1, a_2, a_3, \ldots \) is defined by
\[
\begin{array}{c}
a_1 = k, \\
a_{n+1} = a_n + 8n \text{ for all integers } n \geq 1.
\end{array}
\]

Find all values of \( k \) such that every term in the sequence is a square.


Given that the integer part of the real number \(\frac{2+\sqrt{2}}{2-\sqrt{2}}\) is \(a\), and the decimal part is \(1 - b\), find the value of \(\frac{(b-1) \cdot (5-b)}{\sqrt{a^{2}-3^{2}}}\).


How many natural numbers \( \mathrm{N} \) greater than 300 exist such that exactly two out of the numbers \( 4 \mathrm{~N}, \mathrm{~N} - 300, \mathrm{N} + 45, 2\mathrm{N} \) are four-digit numbers?


Calculate the double integral

$$
\iint_{D} \frac{x}{y^{5}} \, dx \, dy
$$

where the region $D$ is defined by the inequalities

$$
1 \leq \frac{x^{2}}{16}+y^{2} \leq 3, \quad y \geq \frac{x}{4}, \quad x \geq 0
$$


For a natural number \( N \), if at least seven out of the nine natural numbers from 1 to 9 are factors of \( N \), \( N \) is called a "seven-star number." What is the smallest "seven-star number" greater than 2000?


In triangle \( \triangle ABC \), it is known that \(AB = 2\), \(BC = 4\), \(BD = \sqrt{6}\) is the angle bisector of \(\angle B\). Find the length of the median \(BE\) on side \(AC\).


Let \( ABC \) be an isosceles triangle with \( AB = AC \). Let \( D \) and \( E \) be the midpoints of segments \( AB \) and \( AC \), respectively. Suppose that there exists a point \( F \) on ray \( \overrightarrow{DE} \) outside of \( ABC \) such that triangle \( BFA \) is similar to triangle \( ABC \). Compute \( \frac{AB}{BC} \).


Given the complex numbers \( z_{1}, z_{2}, z_{3} \) satisfying:
\[
\begin{array}{l}
\left|z_{1}\right| \leq 1, \left|z_{2}\right| \leq 2, \\
\left|2z_{3} - z_{1} - z_{2}\right| \leq \left|z_{1} - z_{2}\right|.
\end{array}
\]
What is the maximum value of \( \left|z_{3}\right| \)?


Count the number of functions \( f: \mathbb{Z} \rightarrow \{\text{'green','blue'}\} \) such that \( f(x) = f(x+22) \) for all integers \( x \) and there does not exist an integer \( y \) with \( f(y) = f(y+2) = \text{'green'} \).


What is the measure, in degrees, of the smallest positive angle \( x \) for which \( 4^{\sin ^{2} x} \cdot 2^{\cos ^{2} x} = 2 \sqrt[4]{8} \)?


How many different ways (up to rotation) are there of labeling the faces of a cube with the numbers 1, 2, ..., 6?


Can the number, formed by writing out the integers from 1 to \( n \) (where \( n > 1 \)) in a row, be the same when read from left to right and from right to left?

Solution: \boxed{\text{No}}


Given that point \( P \) is inside \( \triangle ABC \), and satisfies
$$
\overrightarrow{AP}=\frac{1}{3} \overrightarrow{AB}+\frac{1}{4} \overrightarrow{AC} ,
$$
and let the areas of \( \triangle PBC \), \( \triangle PCA \), and \( \triangle PAB \) be \( S_{1}, S_{2}, S_{3} \) respectively. Determine the ratio \( S_{1}: S_{2}: S_{3} \).

Solution: \boxed{5: 4: 3}


Find the number of triples of natural numbers \(a, b,\) and \(c\) not exceeding 2017 such that the polynomial \(x^{11} + a x^7 + b x^3 + c\) has a rational root.


In a paper, a $4 \times 6$ grid was drawn, and then the diagonal from $A$ to $B$ was traced.

Observe that the diagonal $AB$ intersects the grid at 9 points.

If the grid were of size $12 \times 17$, at how many points would the diagonal $AB$ intersect the grid?


Given that the positive rational numbers \(a\) and \(b\) satisfy:
\[ a+b=a^{2}+b^{2}=s \]

If \(s\) is not an integer, then denote:
\[ s=\frac{m}{n} \text{ where } (m, n) \text{ are positive integers with } \gcd(m, n)=1. \]

Let \(p\) be the smallest prime factor of \(n\). Find the smallest possible value of \(p\).


Write in a row five numbers such that the sum of each pair of adjacent numbers is negative, and the sum of all the numbers is positive.

Solution: \boxed{3, -4, 3, -4, 3}


13 children sat at a round table and agreed that boys would lie to girls and tell the truth to each other, while girls would lie to boys and tell the truth to each other. One of the children said to his right neighbor: "Most of us are boys." That neighbor said to his right neighbor: "Most of us are girls," and he said to his right neighbor: "Most of us are boys," and so on, until the last child said to the first: "Most of us are boys." How many boys were at the table?


Find the coefficient of \( x^{6} \) in the expansion of
$$
(x+1)^{6} \cdot \sum_{i=0}^{6} x^{i}
$$


Masha wrote a three-digit number on the board, and Vera wrote the same number next to it but swapped the last two digits. After that, Polina added the resulting numbers and got a four-digit sum, the first three digits of which are 195. What is the last digit of this sum? (The answer needs to be justified.)


As shown in Figure 2, in the quadrilateral pyramid $P-ABCD$, $PA \perp$ the base $ABCD$, $BC=CD=2$, $AC=4$, $\angle ACB=\angle ACD=\frac{\pi}{3}$, and $F$ is the midpoint of $PC$. Given that $AF \perp PB$, find:

1. The length of $PA$;
2. The sine value of the dihedral angle $B-AF-D$.


Anya did not tell Misha how old she is but mentioned that on each of her birthdays, her mother puts as many coins into a piggy bank as Anya's age. Misha estimated that there are no less than 110 and no more than 130 coins in the piggy bank. How old is Anya?


A roulette can land on any number from 0 to 2007 with equal probability. The roulette is spun repeatedly. Let $P_{k}$ be the probability that at some point the sum of the numbers that have appeared in all spins equals $k$. Which number is greater: $P_{2007}$ or $P_{2008}$?


Let the set \( A = \{0, 1, 2, \ldots, 9\} \). The collection \( \{B_1, B_2, \ldots, B_k\} \) is a family of non-empty subsets of \( A \). When \( i \neq j \), the intersection \( B_i \cap B_j \) has at most two elements. Find the maximum value of \( k \).


If \( y \) is the greatest value of \(\frac{14}{5+3 \sin \theta}\), find \( y \).

In the figure, \(100 \cos \alpha = k\). Find \( k \).

When \(3 x^{2}+4 x + a\) is divided by \(x+2\), the remainder is 5. Find \(a\).

The solution for \(3 t^{2}-5 t-2<0\) is \(-\frac{1}{3}<t<m\). Find \(m\).


Find the sum of the squares of the natural divisors of the number 1800. (For example, the sum of the squares of the natural divisors of the number 4 is \(1^{2} + 2^{2} + 4^{2} = 21\)).


If the domains of the functions \( f(x) \) and \( g(x) \) are both the set of non-negative real numbers, and for any \( x \geq 0 \), \( f(x) \cdot g(x) = \min \{ f(x), g(x) \} \), given \( f(x) = 3 - x \) and \( g(x) = \sqrt{2x + 5} \), then the maximum value of \( f(x) \cdot g(x) \) is ______ .

Solution: \boxed{2 \sqrt{3} - 1}


A locomotive approaching with a speed of $20 \mathrm{~m/s}$ sounded its whistle, and it was heard by a person standing at the railway crossing 4 seconds before the train arrived. How far was the locomotive when it started whistling? (The speed of sound is $340 \mathrm{~m/s}$.)

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


$[a]$ denotes the greatest integer less than or equal to $a$. Given that $\left(\left[\frac{1}{7}\right]+1\right) \times\left(\left[\frac{2}{7}\right]+1\right) \times\left(\left[\frac{3}{7}\right]+1\right) \times \cdots \times$ $\left(\left[\frac{\mathrm{k}}{7}\right]+1\right)$ leaves a remainder of 7 when divided by 13, find the largest positive integer $k$ not exceeding 48.


Find the natural values of \( x \) that satisfy the system of inequalities

\[
\begin{cases}
\log _{\sqrt{2}}(x-1) < 4 \\
\frac{x}{x-3} + \frac{x-5}{x} < \frac{2x}{3-x}
\end{cases}
\]

Solution: \boxed{x = 2}


From the numbers $1, 2, 3, \cdots, 2003$, select $k$ numbers such that among the selected $k$ numbers, it is always possible to find three numbers that can form the side lengths of an acute-angled triangle. Find the minimum value of $k$ that satisfies the above condition.


In triangle \( \triangle ABC \), the sides opposite to angles \( \angle A \), \( \angle B \), and \( \angle C \) are \( a \), \( b \), and \( c \) respectively. Given that:
\[ a^2 + 2(b^2 + c^2) = 2\sqrt{2} \]
find the maximum value of the area of triangle \( \triangle ABC \).


Find the smallest natural decimal number \(n\) whose square starts with the digits 19 and ends with the digits 89.


Let $[x]$ denote the greatest integer less than or equal to $x$. When $0 \leqslant x \leqslant 10$, find the number of distinct integers represented by the function $f(x) = [x] + [2x] + [3x] + [4x]$.


Seryozha and Lena have several chocolate bars, each weighing no more than 100 grams. No matter how they divide these chocolate bars, the total weight of the chocolate bars for one of them will not exceed 100 grams. What is the maximum possible total weight of all the chocolate bars?

Solution: \boxed{300 \text{ grams}}


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


The function \( f \) is such that \( f(2x - 3y) - f(x + y) = -2x + 8y \) for all \( x \) and \( y \). Find all possible values of the expression \(\frac{f(5t) - f(t)}{f(4t) - f(3t)}\).


Find the largest value of \( x \) such that \(\sqrt[3]{x} + \sqrt[3]{10 - x} = 1\).

Solution: \boxed{5 + 2\sqrt{13}}


Given the function \( f(x) = x^2 \cos \frac{\pi x}{2} \), and the sequence \(\left\{a_n\right\}\) in which \( a_n = f(n) + f(n+1) \) where \( n \in \mathbf{Z}_{+} \). Find the sum of the first 100 terms of the sequence \(\left\{a_n\right\}\), denoted as \( S_{100} \).


Point \( C \) is located 12 km downstream from point \( B \). A fisherman set out in a rowboat from point \( A \), which is upstream from point \( B \), and reached \( C \) in 4 hours. For the return trip, he took 6 hours. On another occasion, the fisherman used a motorized boat, tripling his relative speed in the water, and traveled from \( A \) to \( B \) in 45 minutes. Determine the speed of the current, assuming it is constant.

Solution: \boxed{1 \text{ km/h}}


Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow x}\left(\sin \sqrt{n^{2}+1} \cdot \operatorname{arctg} \frac{n}{n^{2}+1}\right)
$$


For a set \( S \), let \( |S| \) represent the number of elements in \( S \), and let \( n(S) \) represent the number of subsets of \( S \), including the empty set and \( S \) itself. If sets \( A \), \( B \), and \( C \) satisfy \( n(A) + n(B) + n(C) = n(A \cup B \cup C) \) and \( |A| = |B| = 100 \), find the minimum possible value of \( |A \cap B \cap C| \).


Seven dwarfs stood at the corners of their garden, each at one corner, and stretched a rope around the entire garden. Snow White started from Doc and walked along the rope. First, she walked four meters to the east where she met Prof. From there, she continued two meters north before reaching Grumpy. From Grumpy, she walked west and after two meters met Bashful. Continuing three meters north, she reached Happy. She then walked west and after four meters met Sneezy, from where she had three meters south to Sleepy. Finally, she followed the rope by the shortest path back to Doc, thus walking around the entire garden.

How many square meters is the entire garden?

Hint: Draw the shape of the garden, preferably on graph paper.


A fly and $\mathrm{k}$ spiders are moving on a $2019 \times 2019$ grid. On its turn, the fly can move by 1 square and the $k$ spiders can each move by 1 square. What is the minimal $k$ for which the spiders are guaranteed to catch the fly?


When the number "POTOP" was added together 99,999 times, the resulting number had the last three digits of 285. What number is represented by the word "POTOP"? (Identical letters represent identical digits.)


In the rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the lengths of the edges are given as $A B=54, A D=90, A A_{1}=60$. A point $E$ is marked at the midpoint of edge $A_{1} B_{1}$, and a point $F$ is marked at the midpoint of edge $B_{1} C_{1}$. Find the distance between lines $A E$ and $B F$.


a) The difference between two consecutive but not adjacent angles is $100^{\circ}$. Determine the angle formed by their bisectors.

b) In the drawing below, $D A$ is the bisector of angle $\angle C A B$. Determine the value of angle $\angle D A E$ knowing that $\angle C A B + \angle E A B = 120^{\circ}$ and $\angle C A B - \angle E A B = 80^{\circ}$.


In parallelogram \(ABCD\), points \(A_{1}, A_{2}, A_{3}, A_{4}\) and \(C_{1}, C_{2}, C_{3}, C_{4}\) are respectively the quintisection points of \(AB\) and \(CD\). Points \(B_{1}, B_{2}\) and \(D_{1}, D_{2}\) are respectively the trisection points of \(BC\) and \(DA\). Given that the area of quadrilateral \(A_{4} B_{2} C_{4} D_{2}\) is 1, find the area of parallelogram \(ABCD\).


Let \( f(n) \) be the last non-zero digit in the decimal representation of \( n! \). Show that for distinct integers \( a_i \geq 0 \), \( f(5^{a_1} + 5^{a_2} + \ldots + 5^{a_r}) \) depends only on \( a_1 + \ldots + a_r = a \). Write the value as \( g(a) \). Find the smallest period for \( g \), or show that it is not periodic.


Solve the equation for all values of the parameter \( a \):

$$
3 x^{2}+2 a x-a^{2}=\ln \frac{x-a}{2 x}
$$

Solution: \boxed{x = -a}


Among 30 people with different ages, select two groups: the first group with 12 people and the second group with 15 people, such that the oldest person in the first group is younger than the youngest person in the second group. Determine the number of ways to make this selection.


What is the largest integer divisible by all positive integers less than its cube root?


A cube with a side length of 2 is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?


A case contains 20 cassette tapes with disco music and 10 with techno music. A DJ randomly picks two tapes one after the other. What is the probability that
1) The first tape contains disco music;
2) The second tape also contains disco music.

Consider two scenarios:
a) The DJ returns the first tape to the case before picking the second tape;
b) The DJ does not return the first tape to the case.


Let the function \( f(x) = \cos x \cos (x - \theta) - \frac{1}{2} \cos \theta \), where \( x \in \mathbb{R} \) and \( 0 < \theta < \pi \). It is known that \( f(x) \) attains its maximum value at \( x = \frac{\pi}{3} \).

(1) Find the value of \( \theta \).

(2) Let \( g(x) = 2 f\left(\frac{3}{2} x\right) \). Determine the minimum value of the function \( g(x) \) on the interval \(\left[0, \frac{\pi}{3}\right] \).


\( f(n) \) is defined on the set of positive integers, and satisfies the following conditions:
1. For any positive integer \( n \), \( f[f(n)] = 4n + 9 \).
2. For any non-negative integer \( k \), \( f(2^k) = 2^{k+1} + 3 \).

Determine \( f(1789) \).


In triangle \( \triangle ABC \), \( AC > AB \). Point \( P \) is the intersection of the perpendicular bisector of \( BC \) and the internal angle bisector of \( \angle A \). Draw \( PX \perp AB \), intersecting the extension of \( AB \) at point \( X \). Draw \( PY \perp AC \), intersecting \( AC \) at point \( Y \). Let \( Z \) be the intersection of \( XY \) and \( BC \). Find the value of \( \frac{BZ}{ZC} \).


The base of the pyramid is an equilateral triangle with a side length of 6. One of the lateral edges is perpendicular to the plane of the base and has a length of 4. Find the radius of the sphere circumscribed around the pyramid.


Let the quadratic function \( f(x) = ax^2 + bx + c \) (where \( a, b, c \in \mathbf{R} \) and \( a \neq 0 \)) satisfy the following conditions:
1. For \( x \in \mathbf{R} \), \( f(x-4) = f(2-x) \);
2. For \( x \in (0, 2) \), \( f(x) \leq \left( \frac{x+1}{2} \right)^2 \);
3. The minimum value of \( f(x) \) on \( \mathbf{R} \) is 0.

Find the maximum value of \( m \) (where \( m > 1 \)) such that there exists \( t \in \mathbf{R} \) for which \( f(x+t) \leq x \) for all \( x \in [1, m] \).


Find the number of different arrangements of 8 rooks on different white squares of an 8x8 chessboard such that no rook can attack another. A chess rook attacks all squares on the row and column intersecting its position.


Ezekiel has a rectangular piece of paper with an area of 40. The width of the paper is more than twice the height. He folds the bottom left and top right corners at $45^{\circ}$ and creates a parallelogram with an area of 24. What is the perimeter of the original rectangle?


A clock's minute hand has length 4 and its hour hand length 3. What is the distance between the tips at the moment when it is increasing most rapidly?


At what angle to the x-axis is the tangent to the graph of the function \( g(x) = x^2 \ln x \) inclined at the point \( x_0 = 1 \)?


Anya is arranging pebbles in the sand. She first placed one pebble, then added pebbles to form a pentagon. Next, she created a larger external pentagon with pebbles, and after that another external pentagon, and so on, as shown in the picture. The numbers of pebbles she arranged in the first four pictures are: 1, 5, 12, and 22. If she continues to make such pictures, how many pebbles will there be in the 10th picture?


In a tournament, each participant plays a match against every other participant. The winner of a match earns 1 point, the loser 0 points, and if the match is a draw, both players earn half a point. At the end of the tournament, participants are ranked according to their score (if several participants have the same score, their order is chosen randomly). Each participant has earned half of their points in their matches against the last ten participants (in the ranking). How many people participated in this tournament?


On a drying rack, $n$ socks are hanging in a random order (as they were taken out of the washing machine). Among them are two of Absent-Minded Scientist's favorite socks. The socks are hidden by a drying sheet, so the Scientist cannot see them and picks one sock at a time by feel. Find the expected number of socks the Scientist will have taken by the time he has both of his favorite socks.


Given \( n > 1 \), find the maximum value of \( \sin^2 x_1 + \sin^2 x_2 + \ldots + \sin^2 x_n \), where \( x_i \) are non-negative and have sum \( \pi \).


We repeatedly toss a coin until we get either three consecutive heads ($HHH$) or the sequence $HTH$ (where $H$ represents heads and $T$ represents tails). What is the probability that $HHH$ occurs before $HTH$?


Write the equation of a plane passing through the point \(A\) perpendicular to the vector \(\overrightarrow{BC}\).

Given:
\[ A(5, -1, 2) \]
\[ B(2, -4, 3) \]
\[ C(4, -1, 3) \]

Solution: \boxed{2x + 3y - 7 = 0}


Define the lengths of intervals $(m, n)$, $[m, n)$, $(m, n]$, and $[m, n]$ to be $n - m$ ($n, m \in \mathbf{R}$ and $n > m$). Find the sum of the lengths of the intervals for real numbers $x$ that satisfy the inequality
\[
\frac{1}{x-20}+\frac{1}{x-17} \geqslant \frac{1}{512}
\]


Is a triangle necessarily isosceles if the center of its inscribed circle is equidistant from the midpoints of two of its sides?

Solution: \boxed{\text{No}}


Calculate the limit of the function:

$\lim _{x \rightarrow 0}\left(\operatorname{tg}\left(\frac{\pi}{4}-x\right)\right)^{\left(e^{x}-1\right) / x}$


Find all pairs of positive integers \((x, y)\) such that \(x^2 = y^2 + 7y + 6\).

Solution: \boxed{(6, 3)}


Find the circulation of the vector field

$$
\vec{a}=\frac{y}{3} \vec{i} 3-3 x \vec{j}+x \vec{k}
$$

along the closed contour $\Gamma$

$$
\left\{\begin{array}{l}
x=2 \cos t \\
y=2 \sin t \\
z=1-2 \cos t - 2 \sin t
\end{array} \quad t \in [0,2\pi]
\right.
$$

Solution: \boxed{-\frac{52 \pi}{3}}


The cost of 60 copies of the first volume and 75 copies of the second volume is 2700 rubles. In reality, the total payment for all these books was only 2370 rubles because a discount was applied: 15% off the first volume and 10% off the second volume. Find the original price of these books.


Let $A = \left\{a_1, a_2, a_3\right\}, B = \{-1, 0, 1\}$.
(1) How many distinct mappings are there from $A$ to $B$?
(2) Determine the number of mappings $f: A \rightarrow B$ that satisfy $f(a_1) > f(a_2) \geqslant f(a_3)$.


Given the function \( f(x) = -\frac{1}{2} x^{2} + \frac{13}{2} \) on the interval \([a, b]\), if the minimum value on this interval is \(2a\) and the maximum value is \(2b\), find the interval \([a, b]\).

Solution: \boxed{[1, 3]}


Find the maximum value of the positive real number \( k \) such that for any positive real numbers \( a \) and \( b \), the following inequality holds:
$$
\sqrt{a^{2}+k b^{2}}+\sqrt{b^{2}+k a^{2}} \geq a+b+(k-1) \sqrt{a b}
$$


Solution: \boxed{p}


Vera and Anya attend a mathematics club in which more than \(91\%\) of the members are boys. Find the smallest possible number of participants in the club.


On the base \(AC\) of an isosceles triangle \(ABC (AB = BC)\), point \(M\) is marked. It is known that \(AM = 7\), \(MB = 3\), \(\angle BMC = 60^\circ\). Find the length of segment \(AC\).


In the plane Cartesian coordinate system \( xOy \), the focus of the parabola \( \Gamma: y^2 = 2px \) (with \( p > 0 \)) is \( F \). A tangent is drawn to \( \Gamma \) at a point \( P \) different from \( O \), and this tangent intersects the y-axis at point \( Q \). Given that \( |FP| = 2 \) and \( |FQ| = 1 \), find the dot product of the vectors \( \overrightarrow{OP} \) and \( \overrightarrow{OQ} \).


Let \( S = \{1, 2, 3, 4\} \), and consider a sequence \( a_1, a_2, \cdots, a_n \) of \( n \) terms where \( a_i \in S \) such that for any non-empty subset \( B \) of \( S \), there exists a subsequence of \( n \) adjacent terms that forms the set \( B \). Find the minimum value of \( n \).


Let \(PABC\) be a tetrahedron such that \(\angle APB = \angle APC = \angle BPC = 90^\circ\), \(\angle ABC = 30^\circ\), and \(AP^2\) equals the area of triangle \(ABC\). Compute \(\tan \angle ACB\).

Solution: \boxed{8 + 5\sqrt{3}}


To move a chess piece from the point $(0,0)$ to the point $(m, n)$, where $m$ and $n$ are natural numbers, and the movement must be along the coordinate axis directions, only changing direction at points with integer coordinates. How many shortest paths are there to move the chess piece?

Solution: \boxed{\binom{m+n}{m}}


In the quadrilateral \(ABCD\), the lengths of the sides \(BC\) and \(CD\) are 2 and 6, respectively. The points of intersection of the medians of triangles \(ABC\), \(BCD\), and \(ACD\) form an equilateral triangle. What is the maximum possible area of quadrilateral \(ABCD\)? If necessary, round the answer to the nearest 0.01.


The bisectors of the exterior angles $B$ and $C$ of triangle $ABC$ intersect at point $M$.

a) Can angle $BMC$ be obtuse?

b) Find angle $BAC$ given that $\angle BMC = \frac{\angle BAM}{2}$.


Given the function \( f(x) = x^2 - ax + \frac{a}{2} \) where \( a > 0 \) and \( x \in [0,1] \), find the minimum value \( g(a) \) of \( f(x) \), and then find the maximum value of \( g(a) \).


If integer \( x \) satisfies \( x \geq 3+\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3}}}}} \), find the minimum value of \( x \).


Let $k > 0$ be an integer, and let $p = p(n)$ be a function of $n$ such that for large $n$, we have $p \geqslant (6k \ln n) n^{-1}$. Show that
$$
\lim_{n \rightarrow \infty} P\left[\alpha \geqslant \frac{1}{2} \frac{n}{k}\right] = 0.
$$


In triangle \( ABC \), let \( J \) be the center of the excircle tangent to side \( BC \) at \( A_1 \) and to the extensions of sides \( AC \) and \( AB \) at \( B_1 \) and \( C_1 \), respectively. Suppose that the lines \( A_1B_1 \) and \( AB \) are perpendicular and intersect at \( D \). Let \( E \) be the foot of the perpendicular from \( C_1 \) to line \( DJ \). Determine the angles \(\angle B E A_1\) and \(\angle A E B_1\).


Given that \(a\), \(b\), and \(c\) are three distinct real numbers, and the three quadratic equations
$$
\begin{array}{l}
x^{2}+a x+b=0, \\
x^{2}+b x+c=0, \\
x^{2}+c x+a=0
\end{array}
$$
each pair of equations has exactly one common root. Find the value of \(a^{2}+b^{2}+c^{2}\).


What is the value of \((7 - 6 \times (-5)) - 4 \times (-3) \div (-2)\) ?


Let \(ABCD\) be a rectangle with \(AB=20\) and \(AD=23\). Let \(M\) be the midpoint of \(CD\), and let \(X\) be the reflection of \(M\) across point \(A\). Compute the area of triangle \(XBD\).


There are five types of gifts priced at 2 yuan, 5 yuan, 8 yuan, 11 yuan, and 14 yuan, and five types of packaging boxes priced at 3 yuan, 5 yuan, 7 yuan, 9 yuan, and 11 yuan. Each gift is paired with one packaging box. How many different total prices are there?


The sum of the first thirteen terms of an arithmetic progression is $50\%$ of the sum of the last thirteen terms of this progression. The sum of all terms of this progression, excluding the first three terms, is to the sum of all terms excluding the last three terms in the ratio $5:4$. Find the number of terms in this progression.


A circle is drawn inside a regular hexagon so that it touches all six sides of the hexagon. The area of the circle is \(\pi \times 64 \sqrt{3}\). What is the area of the hexagon?


What prime numbers less than 17 divide the number \( 2002^{2002} - 1 \)?


Find the smallest positive real number $\lambda$ such that for any three complex numbers $\mathrm{z}_{1}$, $\mathrm{z}_{2}$, $\mathrm{z}_{3} \in \{\mathrm{z} \in \mathbb{C} \mid |\mathrm{z}| < 1\}$, if $\mathrm{z}_{1} + \mathrm{z}_{2} + \mathrm{z}_{3} = 0$, then $\left| \mathrm{z}_{1} \mathrm{z}_{2} + \mathrm{z}_{2} \mathrm{z}_{3} + \mathrm{z}_{3} \mathrm{z}_{1} \right|^{2} + \left| \mathrm{z}_{1} \mathrm{z}_{2} \mathrm{z}_{3} \right|^{2} < \lambda$.


Four people are sitting at four sides of a table, and they are dividing a 32-card Hungarian deck equally among themselves. If one selected player does not receive any aces, what is the probability that the player sitting opposite them also has no aces among their 8 cards?


Given that $\sum_{i=1}^{n} a_{i} x_{i}=p$ and $\sum_{i=1}^{n} a_{i}=q$, with $a_{i} > 0 \ (i=1,2,\ldots,n)$, where $p$ and $q$ are constants, find the minimum value of $\sum_{i=1}^{n} a_{i} x_{i}^{2}$.


If the 3-digit decimal number \( n = \overline{abc} \) satisfies that \( a \), \( b \), and \( c \) form an arithmetic sequence, then what is the maximum possible value of a prime factor of \( n \)?


Five soccer teams play a match where each team plays every other team exactly once. Each match awards 3 points to the winner, 0 points to the loser, and 1 point to each team in the event of a draw. After all matches have been played, the total points of the five teams are found to be five consecutive natural numbers. Let the teams ranked 1st, 2nd, 3rd, 4th, and 5th have drawn $A$, $B$, $C$, $D$, and $E$ matches respectively. Determine the five-digit number $\overline{\mathrm{ABCDE}}$.


Inside a right triangle \(ABC\) with hypotenuse \(AC\), a point \(M\) is chosen such that the areas of triangles \(ABM\) and \(BCM\) are one-third and one-quarter of the area of triangle \(ABC\) respectively. Find \(BM\) if \(AM = 60\) and \(CM = 70\). If the answer is not an integer, round it to the nearest whole number.


Find all positive integers \( x \) for which \( p(x) = x^2 - 10x - 22 \), where \( p(x) \) denotes the product of the digits of \( x \).


As shown in the figure, a large rectangle is divided into 9 smaller rectangles. The areas of the three small rectangles located at the corners are 9, 15, and 12, respectively. Find the area of the small rectangle located at the fourth corner.


As shown in the figure, the graph \(\Gamma\) of the function \(y = \cos x + 1\) for \(0 \leq x \leq 2\pi\) is drawn inside the rectangle \(OABC\). When \(AB\) coincides with \(OC\), forming a cylinder, find the eccentricity of the curve \(\Gamma\) on the surface of the cylinder.


Calculate the value of the product \(\left(1-\frac{1}{4}\right) \cdot\left(1-\frac{1}{9}\right) \cdot\left(1-\frac{1}{16}\right) \cdot \ldots \cdot\left(1-\frac{1}{2021^{2}}\right)\).


Let \( A, B, C, \) and \( D \) be points randomly selected independently and uniformly within the unit square. What is the probability that the six lines \( \overline{AB}, \overline{AC}, \overline{AD}, \overline{BC}, \overline{BD}, \) and \( \overline{CD} \) all have positive slopes?


The arithmetic mean of three two-digit natural numbers \( x, y, z \) is 60. What is the maximum value that the expression \( \frac{x + y}{z} \) can take?


What is the minimum number of cells that need to be marked in a $7 \times 7$ grid so that in each vertical or horizontal $1 \times 4$ strip there is at least one marked cell?


For normal operation, the bus depot must have at least eight buses on the line, and currently, it has ten. The probability that each bus does not go on the line is 0.1. Find the probability of normal operation of the bus depot for the coming day.


Given a natural number \( x = 6^n + 1 \), where \( n \) is an odd natural number. It is known that \( x \) has exactly three distinct prime divisors, one of which is 11. Find \( x \).


The cells of a $50 \times 50$ table are colored in $n$ colors such that for any cell, the union of its row and column contains cells of all $n$ colors. Find the maximum possible number of blue cells if:

(a) $n=2$;

(b) $n=25$.


Determine the amount of released gas substance based on the chemical reaction equation:
$$
\mathrm{n}\left(\mathrm{CO}_{2}\right)=\mathrm{n}\left(\mathrm{CaCO}_{3}\right)=2.4 \text{ mol}
$$

Calculate the volume of the released gas:
$$
V\left(\mathrm{CO}_{2}\right)=n \cdot V_m=2.4 \cdot 22.4=53.76 \text{ L}
$$

Solution: \boxed{53.76 \text{ L}}


How many three-digit natural numbers \( n \) are there for which the number \( n^3 - n^2 \) is a perfect square?


Find all solutions to the following equation:

$$
\cos ^{2} x - 2 \cos x \cos y \cos (x+y) + \cos ^{2}(x+y) = a
$$

What is the condition for the equation to be solvable?

Solution: \boxed{0 \leq a \leq 1}


Nine digits: \(1, 2, 3, \ldots, 9\) are written in a certain order (forming a nine-digit number). Consider all consecutive triples of digits, and find the sum of the resulting seven three-digit numbers. What is the largest possible value of this sum?


A square fits snugly between a horizontal line and two touching circles with a radius of 1000. The line is tangent to the circles. What is the side length of the square?


Into how many maximum parts can the surface of a sphere (sphere) be divided by 3, 4, and generally $n$ circles?

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


Calculate the double integral \(\iint_{D}(x + 2y) \, dx \, dy\), where the region \(D\) is bounded by the parabolas \(y = x - x^{2}\), \(y = 1 - x^{2}\), and the \(Oy\)-axis.


Given the numbers \( x, y, z \), and \( k \) such that

\[ \frac{7}{x+y} = \frac{k}{x+z} = \frac{11}{z-y} \]

determine the value of \( k \).


A large circular table has 60 chairs around it. What is the largest number of people who can sit around the table so that each person is only sitting next to exactly one other person?


What is the maximum number of parts into which 5 segments can divide a plane?


Find the distance from the point $M_0$ to the plane that passes through the three points $M_1$, $M_2$, and $M_3$.

$M_1(1, 0, 2)$

$M_2(1, 2, -1)$

$M_3(2, -2, 1)$

$M_0(-5, -9, 1)$


Let $p$ be a prime number. Show that $\binom{2p}{p} \equiv 2 \pmod{p}$.


Find the smallest natural number \( k \) such that for some natural number \( a \), greater than 500,000, and some natural number \( b \), the equation \(\frac{1}{a} + \frac{1}{a+k} = \frac{1}{b}\) holds.


How many five-digit numbers divisible by 3 are there that include the digit 6?


An acute angle equals to \(60^{\circ}\) and contains two circles that are tangential to each other externally. The radius of the smaller circle is \(r\). Find the radius of the larger circle.


Formulate the equation of the normal to the given curve at the point with abscissa \( x_0 \).

\[ y = \frac{1 + \sqrt{x}}{1 - \sqrt{x}}, \quad x_{0} = 4 \]

Solution: \boxed{y = -2x + 5}


On a plane, a semicircle with a diameter \( AB = 36 \) cm was constructed; inside it, a semicircle with a diameter \( OB = 18 \) cm was constructed (with \( O \) being the center of the larger semicircle). Then, a circle was constructed, touching both semicircles and segment \( AO \). Find the radius of this circle. Justify your answer.

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


Calculate the limit of the function:

$$\lim _{x \rightarrow 0}\left(\frac{\arcsin ^{2} x}{\arcsin ^{2} 4 x}\right)^{2 x+1}$$


The denominators of two irreducible fractions are 600 and 700. What is the smallest possible value of the denominator of their sum when written as an irreducible fraction?

Note: We say that the fraction \( p / q \) is irreducible if the integers \( p \) and \( q \) do not have common prime factors in their factorizations. For example, \( \frac{5}{7} \) is an irreducible fraction.


Find the equation of the line \( L \) such that the graph of the function

\[ y = x^4 - 4x^3 - 26x^2 \]

lies on one side of this line and has two points in common with it.

Solution: \boxed{y = -60x - 225}


Point \( M \) divides the side \( BC \) of the parallelogram \( ABCD \) in the ratio \( BM:MC = 1:2 \). The line \( AM \) intersects the diagonal \( BD \) at point \( K \). Find the area of the quadrilateral \( CMKD \) if the area of the parallelogram \( ABCD \) is 1.


Two groups have an equal number of students. Each student studies at least one language: English or French. It is known that 5 people in the first group and 5 in the second group study both languages. The number of students studying French in the first group is three times less than in the second group. The number of students studying English in the second group is four times less than in the first group. What is the minimum possible number of students in one group?


Given the function \( f(x) = \frac{2x^2 + \sqrt{2} \sin \left(x + \frac{\pi}{4}\right)}{2x^2 + \cos x} \), with maximum and minimum values \( a \) and \( b \) respectively, find the value of \( a + b \).


Through points \(A(0 ; 14)\) and \(B(0 ; 4)\), two parallel lines are drawn. The first line, passing through point \(A\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(K\) and \(L\). The second line, passing through point \(B\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(M\) and \(N\).

What is the value of \(\frac{A L - A K}{B N - B M}?\)


Calculate the volume of the solid formed by rotating around the $O Y$ axis the curvilinear trapezoid which is bounded by the hyperbola $x y=2$ and the lines $y_{1}=1, y_{2}=4$, and $y_{3}=0$.


Find all functions \( f : [1, +\infty) \rightarrow [1, +\infty) \) that satisfy the following conditions:
1. \( f(x) \leq 2(x + 1) \);
2. \( f(x + 1) = \frac{1}{x} \left[ (f(x))^2 - 1 \right] \).

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


In triangle \(ABC\), angle \(A\) is \(40^\circ\). The triangle is randomly thrown onto a table. Find the probability that vertex \(A\) ends up east of the other two vertices.


Find the maximum value of the parameter \(a\) for which the equation \((|x-2|+2a)^{2}-3(|x-2|+2a)+4a(3-4a)=0\) has three solutions. Specify the largest value in your answer.


Is there a figure that has exactly two axes of symmetry but does not have a center of symmetry?

Solution: \boxed{\text{No}}


Point \( M \) divides the side \( BC \) of the parallelogram \( ABCD \) in the ratio \( BM : MC = 2 \). Line \( AM \) intersects the diagonal \( BD \) at point \( K \). Find the area of the quadrilateral \( CMKD \) if the area of the parallelogram \( ABCD \) is 1.


2016 students are lined up in a row and count off from left to right according to $1,2 \cdots, n$ $(n \geqslant 2)$. If the 2016th student counts to $n$, all students who count to $n$ in this round will receive a New Year's gift. How many students will definitely not receive a New Year's gift, regardless of what $n$ is?


There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.

All the students in this class sat at a round table.

- Several students said: "There is exactly one troublemaker next to me."
- All other students said: "There are exactly two troublemakers next to me."

What is the minimum number of troublemakers that can be in the class?


The Small and the Big islands have a rectangular shape and are divided into rectangular counties. Each county has a road laid along one of the diagonals. On each island, these roads form a closed path that does not pass through any point more than once. Here is how the Small island is structured, which has only six counties (see the illustration).

Draw how the Big island can be structured if it has an odd number of counties. How many counties did you come up with?

Solution: \boxed{9 \text{ fiefs}}


For how many positive numbers less than 1000 is it true that among the numbers $2, 3, 4, 5, 6, 7, 8,$ and $9$, there is exactly one that is not its divisor?


Reflect the parabola \( y = x^2 \) over the point \( (1, 1) \). What is the equation of the reflected image?

Solution: \boxed{y = -x^2 + 4x - 2}


In triangle \( ABC \), a point \( K \) is taken on side \( AB \), such that \( AK: BK = 1:2 \). Another point \( L \) is taken on side \( BC \), such that \( CL: BL = 2:1 \). Point \( Q \) is the intersection of lines \( AL \) and \( CK \). Find the area of triangle \( ABC \) if it is known that \( S_{BQC} = 1 \).


Solve the following system of equations for positive numbers:

$$
\begin{cases}
x^{y} = z \\
y^{z} = x \\
z^{x} = y
\end{cases}
$$

Solution: \boxed{(1, 1, 1)}


Let $S$ be a set of $n$ different real numbers, and $A_{s}$ be the set of all distinct averages of any two distinct elements from $S$. For a given $n \geq 2$, what is the minimum possible number of elements in $A_{s}$?


There are three saline solutions with concentrations of 5%, 8%, and 9%, labeled A, B, and C, weighing 60g, 60g, and 47g respectively. We need to prepare 100g of a saline solution with a concentration of 7%. What is the maximum and minimum amount of solution A (5% concentration) that can be used? Please write down the sum of these two numbers as the answer.


How many natural five-digit numbers have the product of their digits equal to 2000?


Given that \( z_{1} \) and \( z_{2} \) are complex numbers such that \( \left|z_{1}\right|=3 \), \( \left|z_{2}\right|=5 \), and \( \left|z_{1}+z_{2}\right|=7 \), find the value of \( \arg \left(\frac{z_{2}}{z_{1}}\right)^{3} \).

Solution: \boxed{\pi}


Let \( n \) be a natural number less than 50. Find the sum of all possible values of \( n \) such that \( 4n + 5 \) and \( 7n + 6 \) have a common divisor greater than 1.


Solve the inequality \(4 + x^2 + 2x \sqrt{2 - x^2} < 8 \sqrt{2 - x^2} + 5x\). (20 points)

Solution: \boxed{(-1, \sqrt{2}]}


In the right triangle \( \triangle ABC \),

\[
\angle A = 90^\circ, \, AB = AC
\]

\( M \) and \( N \) are the midpoints of \( AB \) and \( AC \) respectively. \( D \) is an arbitrary point on the segment \( MN \) (excluding points \( M \) and \( N \)). The extensions of \( BD \) and \( CD \) intersect \( AC \) and \( AB \) at points \( F \) and \( E \) respectively. If

\[
\frac{1}{BE} + \frac{1}{CF} = \frac{3}{4}
\]

then find the length of \( BC \).

Solution: \boxed{BC = 4\sqrt{2}}


Given the family of curves \(2(2 \sin \theta - \cos \theta + 3)x^2 - (8 \sin \theta + \cos \theta + 1)y = 0\), where \(\theta\) is a parameter, find the maximum length of the chord intercepted by these curves on the line \(y = 2x\).


We start with the numbers \( a, b, c, d \). We then replace them with \( a' = a - b \), \( b' = b - c \), \( c' = c - d \), \( d' = d - a \). We carry out this process 1996 times. Is it possible to end up with numbers \( A, B, C, D \) such that \( |BC - AD| \), \( |AC - BD| \), \( |AB - CD| \) are all primes?

Solution: \boxed{\text{No}}


In triangle \( \triangle ABC \), let \( BC = a \), \( CA = b \), and \( AB = c \). If \( 9a^2 + 9b^2 - 19c^2 = 0 \), then find the value of \( \frac{\cot C}{\cot A + \cot B} \).


Roma decided to create his own multiplication table. The rows correspond to the numbers \( 12, 13, 14, \ldots, 60 \) and the columns to the numbers \( 15, 16, 17, \ldots, 40 \). In the cells of the table, he wrote the products of the pairs of row and column numbers. How many of these products will be even numbers?


If the number $\mathrm{a} 2016 \mathrm{~b} 2017$ is divisible by 33, how many such numbers are there?


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


Is there a triangular pyramid whose heights are 1, 2, 3, and 6?

Solution: \boxed{\text{No}}


Let complex number \( z = \cos \frac{4\pi}{7} + i \sin \frac{4\pi}{7} \). Determine the value of \( \left| \frac{z}{1+z^2} + \frac{z^2}{1+z^4} + \frac{z^3}{1+z^6} \right| \).


There are two circles: one centered at point \(A\) with a radius of 5, and another centered at point \(B\) with a radius of 15. Their common internal tangent touches the circles at points \(C\) and \(D\) respectively. The lines \(AB\) and \(CD\) intersect at point \(E\). Find \(CD\) if \(BE = 39\).


Find all prime numbers \( p \) such that the numbers \( p+7 \) and \( p^{2}+7 \) are double the squares of natural numbers.


Given a sequence of positive integers \(\{y_{n}\}\) satisfying
\[
y_{n+2} = \frac{y_{n} + 2013}{1 + y_{n+1}} \quad (n \geq 1),
\]
find the minimum value of \(y_{1} + y_{2}\).


A cuckoo clock chimes "cuckoo" as many times as the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock at 9:05 and started turning the minute hand until the clock advanced by 7 hours. How many times did the clock chime "cuckoo" during this period?


Let \( N = 2019^2 - 1 \). How many positive factors of \( N^2 \) do not divide \( N \)?


Inside a large rectangle of size $A \times B$ there is a smaller rectangle of size $a \times b$ (see the figure).

Find the difference between the total area of the yellow quadrilaterals and the total area of the green quadrilaterals, given that $A=20$, $B=30$, $a=4$, $b=7$.


Given the complex numbers \( z_{1} = \sin \alpha + 2i \) and \( z_{2} = 1 + i \cos \alpha \) (where \( \alpha \) is a real number and \( i \) is the imaginary unit), what is the minimum value of \( \frac{13 - \left| z_{1} + i z_{2} \right|^{2}}{\left| z_{1} - i z_{2} \right|} \)?


The clock runs 3 minutes fast every 24 hours. The clock was set accurately. After what minimum time will the clock show the correct time again?

Solution: \boxed{240 \text{ days}}


In Figure 1, \(\triangle ABC\) is an isosceles triangle with \(AB = BC = 20 \text{ cm}\) and \(\tan \angle BAC = \frac{4}{3}\). If the radius of the inscribed circle of \(\triangle ABC\) is \(r\) cm, find the value of \(r\).


From the set \(\{1,2,3, \cdots, 1000\}\), three numbers \(a_{1}, a_{2}, a_{3}\) are randomly selected without replacement. From the remaining 997 numbers, another three numbers \(b_{1}, b_{2}, b_{3}\) are randomly selected without replacement. Let \(p\) be the probability of the following event: A rectangular brick of dimensions \(a_{1} \times a_{2} \times a_{3}\) can be placed into a rectangular box of dimensions \(b_{1} \times b_{2} \times b_{3}\) after appropriate rotation, with the edges of the brick remaining parallel to the edges of the box. If \(p\) is expressed as a reduced fraction, what is the sum of the numerator and the denominator?


Find all triplets of three strictly positive integers such that:

$$
x+\frac{1}{y+\frac{1}{z}}=\frac{10}{7}
$$

Solution: \boxed{(1, 2, 3)}


In a rectangular parallelepiped $\mathrm{ABCDA}_{1} \mathrm{~B}_{1} \mathrm{C}_{1} \mathrm{D}_{1}$, with $\mathrm{AB}=1$ cm, $\mathrm{AD}=2$, and $\mathrm{AA}_{1}=1$, find the minimum area of triangle $\mathrm{PA}_{1} \mathrm{C}$, where vertex $\mathrm{P}$ lies on the line $\mathrm{AB}_{1}$.


In $\triangle ABC$, point $P$ is an arbitrary point on side $BC$. Lines $PE$ and $PF$ are parallel to $BA$ and $CA$, respectively. If $S_{\triangle ABC} = 1$, show that at least one of the areas $S_{\triangle BPF}$, $S_{\triangle PCE}$, or $S_{\square PEAF}$ is not less than $4/9$. (Here, $S$ denotes the area of the figure.)


Each cell in a \(6 \times 6\) table is filled with either +1 or -1 such that the product of all numbers in any row and any column is positive. How many ways can this be accomplished?


Let \(ABC\) be a triangle with \(AB=7\), \(BC=9\), and \(CA=4\). Let \(D\) be the point such that \(AB \parallel CD\) and \(CA \parallel BD\). Let \(R\) be a point within triangle \(BCD\). Lines \(\ell\) and \(m\) going through \(R\) are parallel to \(CA\) and \(AB\) respectively. Line \(\ell\) meets \(AB\) and \(BC\) at \(P\) and \(P'\) respectively, and \(m\) meets \(CA\) and \(BC\) at \(Q\) and \(Q'\) respectively. If \(S\) denotes the largest possible sum of the areas of triangles \(BPP'\), \(RP'Q'\), and \(CQQ'\), determine the value of \(S^2\).


In a regular \(n\)-gon, each side and each diagonal must be painted such that any two of these segments that share a common point are painted differently. What is the minimum number of colors needed for this?

Solution: \boxed{n}


A circle passes through vertices \(A\), \(B\), and \(C\) of a parallelogram \(ABCD\) with sides \(AB = 5\) and \(BC = 2\) and intersects the line containing the diagonal \(BD\) at point \(E\). It is known that \(BE = 8\). Find \(BD\).


How many ways are there to tile a $2 \times 12$ rectangle with non-overlapping $1 \times 2$ dominoes?


Given a linear function \( f(x) \), it is known that the distance between the intersection points of the graphs \( y = x^{2} \) and \( y = f(x) \) is \( \sqrt{10} \), and the distance between the intersection points of the graphs \( y = x^{2} - 1 \) and \( y = f(x) + 1 \) is \( \sqrt{42} \). Find the distance between the intersection points of the graphs \( y = x^{2} + 1 \) and \( y = f(x) + 2 \).


Thirty students are going on an excursion to St. Petersburg along with their parents, some of whom will be driving cars. Each car can accommodate 5 people, including the driver. What is the minimum number of parents needed to be invited on the excursion?


Given a quadrilateral \(A B C D\), see the image. Point \(T_{1}\) is the centroid of triangle \(B C D\), point \(T_{2}\) is the centroid of triangle \(A B D\), and points \(T_{1}\) and \(T_{2}\) lie on segment \(A C\). The length of segment \(T_{1} T_{2}\) is \(3 \text{ cm}\) and point \(D\) has a distance of \(3 \text{ cm}\) from segment \(A C\).

Determine the area of quadrilateral \(A B C D\).

(E. Patáková)

Solution: \boxed{27 \text{ cm}^2}


David has a very unique calculator that performs only two operations, the usual addition $(+)$ and another operation denoted by $*$, which satisfies:

(i) $a * a = a$,

(ii) $a * 0 = 2a$ and

(iii) $(a * b) + (c * d) = (a + c) * (b + d)$,

for any integers $a$ and $b$. What are the results of the operations $(2 * 3) + (0 * 3)$ and $1024 * 48$?


If we pick uniformly a random square of area 1 with sides parallel to the $x$ and $y$ axes that lies entirely within the 5-by-5 square bounded by the lines $x=0$, $x=5$, $y=0$, and $y=5$ (the corners of the square need not have integer coordinates), what is the probability that the point $(x, y)=(4.5, 0.5)$ lies within the square of area 1?


Given the circumcenter \(O\) of \(\triangle ABC\), and \(3 \overrightarrow{OA} + 4 \overrightarrow{OB} + 5 \overrightarrow{OC} = \mathbf{0}\), find the value of \(\cos \angle ABC\).


A square \( KLMN \) with side length 1 is inscribed in triangle \( ABC \): points \( K \) and \( L \) lie on side \( AC \), and points \( M \) and \( N \) lie on sides \( AB \) and \( BC \) respectively. The area of the square is half the area of the triangle. Find the length of the altitude \( BH \) of triangle \(ABC\).


Given that \(a, b, c, d\) are non-zero real numbers and \(f(x)=\frac{a x+b}{c x+d}\), where \(x \in \mathbf{R}\), and it is known that \(f(19) = 19\) and \(f(97) = 97\). If for any real number \(x\) (where \(x \neq -\frac{d}{c}\)), \(f[f(x)] = x\) holds true, find the only number outside the range of \(f(x)\).


A pair of vertices of a simple polygon is said to be "visible" to each other if and only if they are adjacent or the line segment connecting them lies entirely inside the polygon (excluding the endpoints on the boundary). Given a simple polygon with \( n \) vertices, where each vertex is exactly visible to four other vertices, find all possible values of the positive integer \( n \).

Note: A simple polygon is a polygon without holes and does not intersect itself.


Let \( n \) be a positive integer. Define \( S_n = \left\{\left(a_1, a_2, \cdots, a_{2^n}\right) \mid a_i = 0 \text{ or } 1, 1 \leq i \leq 2^n\right\} \). For any two elements \( a = \left(a_1, a_2, \cdots, a_{2^n}\right) \) and \( b = \left(b_1, b_2, \cdots, b_{2^n}\right) \) in \( S_n \), define \( d(a, b) = \sum_{i=1}^{2^n} \left|a_i - b_i\right| \).

If \( A \subseteq S_n \) such that for any two distinct elements \( a, b \in A \), \( d(a, b) \geq 2^{n-1} \), then \( A \) is called a good subset. Find the maximum number of elements in a good subset of \( S_n \).


Petya is trying to arrange identical cardboard rectangles of size \(14 \times 10\) on a table to form a square. Can he achieve this? Suggest your own way of constructing such a square. What is the minimum number of rectangles needed for this?


Given \(a = \frac{11 \times 66 + 12 \times 67 + 13 \times 68 + 14 \times 69 + 15 \times 70}{11 \times 65 + 12 \times 66 + 13 \times 67 + 14 \times 68 + 15 \times 69} \times 100\), what is the integer part of \(a\)?


A piece of iron wire is 78 centimeters long, and a red dot is painted on it every 3 centimeters. The wire is bent at the red dots to form a rectangle. What is the maximum area of the resulting rectangle in square centimeters?


What is the maximum number of sides that a convex polygon can have if its interior angles form an arithmetic sequence with a common difference of $d=1^{\circ}$?


There are $n$ circular paper pieces, and every two pieces have a common point. It is possible to hook all $n$ pieces with 7 hooks.


A seven-digit number, where the sum of every two adjacent digits from left to right is 9, 7, 9, 2, 8, 11. What is this seven-digit number?


Provide the smallest interval in which the product of the sum of three positive numbers and the sum of their reciprocals can fall.

Solution: \boxed{[9, \infty)}


Let vectors $\overrightarrow{a_{1}}=(1,5)$, $\overrightarrow{a_{2}}=(4,-1)$, $\overrightarrow{a_{3}}=(2,1)$, and let $\lambda_{1}, \lambda_{2}, \lambda_{3}$ be non-negative real numbers such that $\lambda_{1}+\frac{\lambda_{2}}{2}+\frac{\lambda_{3}}{3}=1$. Find the minimum value of $\left|\lambda_{1} \overrightarrow{a_{1}}+\lambda_{2} \overrightarrow{a_{2}}+\lambda_{3} \overrightarrow{a_{3}}\right|$.


Find the largest natural number that cannot be represented as the sum of two composite numbers. (Recall that a natural number is called composite if it is divisible by some natural number other than itself and one.)


In the number \(2016 * * * * 02 *\), each of the 5 asterisks needs to be replaced by any of the digits \(0, 2, 4, 7, 8, 9\) (digits can repeat) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done?


Given that \(x = 2a^5 = 3b^2\) where \(a\) and \(b\) are positive integers, find the least possible value of \(x\).


Find all polynomials \( f(x) \) such that \( f(2x) = f'(x) f''(x) \).


The parabola \( y = x^2 \) intersects with the line \( y = 25 \). A circle is constructed on the segment between the points of intersection of the parabola and the line as its diameter. Find the area of the convex polygon whose vertices are the points of intersection of the given circle and the parabola. In your answer, indicate the nearest integer to this area.


Find all natural numbers \( n \) for which \( (n+1)! \) is divisible by the sum \( 1! + 2! + \ldots + n! \).

Solution: \boxed{1 \text{ and } 2}


A five-digit number $\overline{ABCDE}$ is a multiple of 2014, and $\overline{CDE}$ has exactly 16 factors. What is the smallest value of $\overline{ABCDE}$?


In an aquarium, there are three types of fish: gold, silver, and red fish. If a cat eats all the gold fish, the number of fish becomes 1 less than $\frac{2}{3}$ of the original number. If a cat eats all the red fish, the number of fish becomes 4 more than $\frac{2}{3}$ of the original number. Which type of fish—gold or silver—are there more of, and by how many?


Given the numbers \( x, y, z \in \left[0, \frac{\pi}{2}\right] \), find the maximum value of the expression

\[ A = \sin(x-y) + \sin(y-z) + \sin(z-x). \]

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


Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \), \( f(1) = 1 \), and \( \left| f^{\prime}(x) \right| \leq 2 \) for all real numbers \( x \). If \( a \) and \( b \) are real numbers such that the set of possible values of \( \int_{0}^{1} f(x) \, dx \) is the open interval \( (a, b) \), determine \( b - a \).


How many fractions between \(\frac{1}{6}\) and \(\frac{1}{3}\) inclusive can be written with a denominator of 15?


A pair of natural numbers ($a, p$) is called good if the number $a^{3}+p^{3}$ is divisible by $a^{2}-p^{2}$, and $a > p$.

(a) Provide any possible value of $a$ for which the pair $(a, 19)$ is good.

(b) Find the number of good pairs for which $p$ is a prime number less than 24.


In the diagram, the circle has radius \(\sqrt{5}\). Rectangle \(ABCD\) has \(C\) and \(D\) on the circle, \(A\) and \(B\) outside the circle, and \(AB\) tangent to the circle. What is the area of \(ABCD\) if \(AB = 4AD\)?


In the triangle $A B C$ with an area of one unit, let $D$ and $E$ be the trisection points on side $A B$, and let $F$ be the midpoint of side $A C$. The line $F E$ intersects line $C B$ at $G$, and the line $F D$ intersects line $C B$ at $H$. Determine the area of triangle $F G H$.


The area of trapezoid \(ABCD\) is 30. Point \(P\) is the midpoint of the lateral side \(AB\). Point \(R\) on side \(CD\) is chosen such that \(2CD = 3RD\). Lines \(AR\) and \(PD\) intersect at point \(Q\). Find the area of triangle \(APQ\) if \(AD = 2BC\).


A robot is located in one of the cells of an infinite grid paper and can be given the following commands:
- move up (the robot moves to the adjacent cell above);
- move down (the robot moves to the adjacent cell below);
- move left (the robot moves to the adjacent cell to the left);
- move right (the robot moves to the adjacent cell to the right).

For example, if the robot executes a sequence of four commands (up, right, down, left), it will obviously return to its initial position, which means it will be in the same cell from which it started moving. How many different sequences of 8 commands return the robot to its initial position?


Let \( p \) and \( q \) be positive integers such that \( \frac{5}{8} < \frac{p}{q} < \frac{7}{8} \). What is the smallest value of \( p \) so that \( p + q = 2005 \)?


How many 4-digit positive multiples of 4 can be formed from the digits 0, 1, 2, 3, 4, 5, 6 such that each digit appears without repetition?


Determine the largest possible natural number \( n \) for which there are \( n \) different subsets of \( S = \{1,2, \ldots, 2017\} \) such that the union of no two of these subsets is equal to \( S \).


We know that the numbers 2013, 2014, and 2015 have the same number of divisors. What is the smallest value of \( n \) such that \( n \), \( n+1 \), and \( n+2 \) are consecutive natural numbers with the same number of divisors?


What is the minimum length of the second longest side of a triangle with an area of one unit?


How many natural numbers \( x \), less than 10000, are there for which \( 2^x - x^2 \) is divisible by 7?


Given positive real numbers \( a \) and \( b \) (where \( a > b \)), what is the product of the distances from the points \(\left(\sqrt{a^{2}-b^{2}}, 0\right)\) and \(\left(-\sqrt{a^{2}-b^{2}}, 0\right)\) to the line
\[
\frac{x \cos \theta}{a}+\frac{y \sin \theta}{b}=1
\]


Given the number $200 \ldots 002$ (100 zeros). We need to replace two of the zeros with non-zero digits such that the resulting number is divisible by 66. How many ways can this be done?


Bully Vasya loves to run on the escalator in the subway, and he runs down twice as fast as up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is moving down, it takes Vasya 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down on an escalator that is moving up? (The escalator always moves at a constant speed.)


How many 5-digit numbers \(\overline{abcde}\) exist such that digits \(b\) and \(d\) are each the sum of the digits to their immediate left and right? (That is, \(b = a + c\) and \(d = c + e\).)


How many 3-digit positive integers have exactly one even digit?


Given a $3 \times 3$ grid (like a Tic-Tac-Toe board), four randomly selected cells have been randomly placed with four tokens.

Find the probability that among these four tokens, three are aligned in a row either vertically, horizontally, or diagonally.


If natural numbers \( a \), \( b \), and \( c \) each leave a remainder of 5 when divided by 14, what is the remainder when \( a + b + c \) is divided by 14?


Three fractions \(A\), \(B\), and \(C\) have numerators and denominators that are natural numbers. The ratio of their numerators is \(3:2:1\), and the ratio of their denominators is \(2:3:4\). The sum of these three fractions is \(\frac{29}{60}\). Find \(A - B - C\).


In triangle \(ABC\), a point \(D\) is marked on side \(BC\) such that \(BD: DC = 1:3\). Additionally, points \(E\) and \(K\) are marked on side \(AC\) such that \(E\) lies between \(A\) and \(K\). Segment \(AD\) intersects segments \(BE\) and \(BK\) at points \(M\) and \(N\) respectively, and it is given that \(BM: ME = 7:5\) and \(BN: NK = 2:3\). Find the ratio \(MN: AD\).


By the properties of absolute values (\(|x| = |-x|\)), replacing \(x\) with \(-x\) does not change the given inequality.

Therefore, the region defined by this inequality will be symmetric with respect to the y-axis, and it is sufficient to find the area of the figure when \(x \geq 0\). In the region \(x \geq 0\), the inequality can be rewritten as: \(\left| x - \frac{y^2}{2} \right| + x + \frac{y^2}{2} \leq 2 - y\).

Expanding the absolute value sign, we get two regions:

I: \(\left\{ \begin{array}{c} x \geq \frac{y^2}{2} \\ x - \frac{y^2}{2} + x + \frac{y^2}{2} \leq 2 - y \end{array} \leftrightarrow \left\{ \begin{array}{c} x \geq \frac{y^2}{2} \\ y + 2x \leq 2 \end{array} \right. \right. \)

II: \(\left\{ \begin{array}{c} x \leq \frac{y^2}{2} \\ - x + \frac{y^2}{2} + x + \frac{y^2}{2} \leq 2 - y \end{array} \leftrightarrow \left\{ \begin{array}{c} x \leq \frac{y^2}{2} \\ y^2 + y - 2 \leq 0 \end{array} \leftrightarrow \left\{ \begin{array}{c} x \leq \frac{y^2}{2} \\ -2 \leq y \leq 1 \end{array} \right. \right. \right. \)

Combining regions I and II for \(x \geq 0\) and considering that \(x = \frac{y^2}{2}\) represents the graph of a parabola symmetric about the Ox axis with its vertex at the origin. The sign of the modular expression is determined by substituting the coordinate of an arbitrary point. Clearly, the figure obtained in the region \(x \geq 0\) is a trapezoid \(ABCD\). The area of this trapezoid \(S\) is equal to: \(\frac{1}{2} \times 3(2 + \frac{1}{2}) = \frac{15}{4}\). Therefore, the area of the entire figure will be \(\frac{15}{2}\). Answer: \(\frac{15}{2}\).

By writing the equation in the form: \(\sqrt{(x + a)^2 + a - a^2} + \sqrt{(x - a)^2 + a - a^2} = 2\sqrt{b}\), we get that if \(x_0\) is a root, then \(-x_0\) is also a root of this equation. Unique solution implies \(x_0 = 0\). Substituting \(x = 0\) into the equation, we find that \(2\sqrt{a} = 2\sqrt{b} \Rightarrow a = b\) and \(a \geq 0\).

Therefore, we get the equation \(\sqrt{x^2 + 2bx + b} + \sqrt{x^2 - 2bx + b} = 2\sqrt{b}\). Squaring both sides, we get: \(\sqrt{(x^2 + b)^2 - 4b^2x^2} = b - x^2\). Squaring again, we get: \(x^4 + b^2 + 2x^2b - 4b^2x^2 = b^2 - 2bx^2 + x^4 \Rightarrow b(b - 1)x^2 = 0\).

If \(b \neq 0\), \(b \neq 1\), then \(x = 0\) is the only solution. If \(b = 0\), then \(x = 0\), meaning \(x = 0\) is the only solution. Finally, substituting \(b = 1\), we have:

\(\sqrt{x^2 + 2x + 1} + \sqrt{x^2 - 2x + 1} = 2 \Rightarrow |x + 1| + |x - 1| = 2\).
The solution to this equation is the interval \(x \in [-1, 1]\), so in this case, uniqueness does not hold.


On an island, there are two tribes: knights and liars. Knights always tell the truth, and liars always lie. One day, 80 people sat at a round table, and each of them declared: "Among the 11 people sitting immediately after me in a clockwise direction, there are at least 9 liars." How many knights are sitting at the round table? Indicate all possible options.


Let $k$ be a given positive integer. Find the smallest positive integer $n$ such that among any $n$ integers, there always exist two positive integers whose sum or difference is divisible by $2k$.

Solution: \boxed{k + 2}


How many natural numbers $n$ exist for which $4^n - 15$ is a perfect square of an integer?


All the positive integers which are co-prime to 2012 are grouped in increasing order in such a way that the \( n^{\text{th}} \) group has \( 2n - 1 \) numbers. For example, the first three groups are (1), (3, 5, 7), (9, 11, 13, 15, 17). It is known that 2013 belongs to the \( k^{\text{th}} \) group. Find the value of \( k \). (Note: Two integers are said to be co-prime if their greatest common divisor is 1.)


If a positive divisor is chosen at random from \(10^{99}\), what is the probability that it is also a multiple of \(10^{88}\)? Express the probability as \(\frac{m}{n}\) where \(m\) and \(n\) are coprime, and find \(m+n\).


A snail crawls from one tree to another. In half a day, it covered \( l_{1}=5 \) meters. Then, it got tired of this and turned back, crawling \( l_{2}=4 \) meters. It got tired and fell asleep. The next day, the same process repeats. The distance between the trees is \( s=30 \) meters. On which day of its journey will the snail reach the tree? (10 points)


Given that \( m, n, k \) are positive integers, if there exists a pair of positive integers \( (a, b) \) such that
\[
(1+a) n^{2}-4(m+a) n+4 m^{2}+4 a+b(k-1)^{2}<3,
\]
then the number of possible values of \( m+n+k \) is \(\quad\) .


In a test where the full score is 100 points, and all students' scores are non-negative integers, let the scores of \( n \) students be \( a_{1}, a_{2}, \cdots, a_{n} \), and the number of students who scored at least \( k \) points be \( b_{k} \) ( \(1 \leq k \leq 100\)). Let \( M=\sum_{i=1}^{n} a_{i} \) and \( N=\sum_{i=1}^{100} b_{i} \). Determine the relationship between \( M \) and \( N \).

Solution: \boxed{M=N}


Two differentiable real functions \( f(x) \) and \( g(x) \) satisfy
\[ \frac{f^{\prime}(x)}{g^{\prime}(x)} = e^{f(x) - g(x)} \]
for all \( x \), and \( f(0) = g(2003) = 1 \). Find the largest constant \( c \) such that \( f(2003) > c \) for all such functions \( f, g \).

Solution: \boxed{1 - \ln 2}


What is the smallest \( n \) for which there is a solution to the following system of equations:
\[
\sin x_1 + \sin x_2 + \cdots + \sin x_n = 0,
\]
\[
\sin x_1 + 2 \sin x_2 + \cdots + n \sin x_n = 100?
\]


In a hexagon described around a circle, five consecutive sides are given as $a, b, c, d, e$. Find the sixth side.

Solution: \boxed{a - b + c - d + e}


Xixi and Shanshan each have a certain number of point cards.
Xixi says to Shanshan: "If you give me 3 cards, the number of my cards will be 3 times yours."
Shanshan says to Xixi: "If you give me 4 cards, the number of my cards will be 4 times yours."
Xixi says to Shanshan: "If you give me 5 cards, the number of my cards will be 5 times yours."
It is known that exactly one of the above statements is incorrect. How many point cards did Xixi originally have?


Find the number of positive integers \(x\) that satisfy the inequality \(\left|3+\log _{x} \frac{1}{3}\right|<\frac{8}{3}\).


Determine all integers \( n \geq 1 \) such that \( n \) divides \( 2^{n} - 1 \).

Solution: \boxed{n = 1}


Let the real numbers \(a_1, a_2, \cdots, a_{100}\) satisfy the following conditions: (i) \(a_1 \geq a_2 \geq \cdots \geq a_{100} \geq 0\); (ii) \(a_1 + a_2 \leq 100\); (iii) \(a_3 + a_4 + \cdots + a_{100} \leq 100\). Find the maximum value of \(a_1^2 + a_2^2 + \cdots + a_{100}^2\) and the values of \(a_1, a_2, \cdots, a_{100}\) that achieve this maximum.


Ana, Bia, Cátia, Diana, and Elaine work as street vendors selling sandwiches. Daily, they visit Mr. Manoel's snack bar and take an identical number of sandwiches to sell. One day, Mr. Manoel was sick and left a note explaining his absence and asking each vendor to take $\frac{1}{5}$ of the sandwiches. Ana arrived first, followed the note's instructions, and left to sell her sandwiches. Bia arrived next but thought she was the first and took $\frac{1}{5}$ of what was left. Cátia, Diana, and Elaine arrived together and equally divided the remaining sandwiches since Cátia knew Ana and Bia had already picked up their shares.

a) What fraction of the total sandwiches did Bia take?

b) Who received the smallest quantity of sandwiches? Who received the largest quantity?

c) By the end of the division, none of the vendors had more than 20 sandwiches. How many sandwiches did Mr. Manoel leave for them?


Which three-digit numbers are equal to the sum of the factorials of their digits?


If an odd square not divisible by 9 is decreased by one, the result is divisible by 24.


Find the number of natural numbers \( k \), not exceeding 291000, such that \( k^2 - 1 \) is divisible by 291.


In triangle \(ABC\), \(AB = 32\), \(AC = 35\), and \(BC = x\). What is the smallest positive integer \(x\) such that \(1 + \cos^2 A\), \(\cos^2 B\), and \(\cos^2 C\) form the sides of a non-degenerate triangle?


Given a cube $ABCD-A_{1}B_{1}C_{1}D_{1}$ with edge length 1, find the length of the path traced by a moving point $P$ on the surface of the cube such that the volume of the tetrahedron $P-BDD_{1}B_{1}$ is $\frac{1}{3}$.


Given that the sequences $\left\{a_{n}\right\}$, $\left\{b_{n}\right\}$, and $\left\{c_{n}\right\}$ are all arithmetic sequences, and

$$
a_{1}+b_{1}+c_{1}=0, \quad a_{2}+b_{2}+c_{2}=1,
$$

find $a_{2015}+b_{2015}+c_{2015}$.


We say that a set $E$ of natural numbers is special when, for any two distinct elements $a, b \in E$, $(a-b)^{2}$ divides the product $ab$.

(a) Find a special set composed of three elements.

(b) Does there exist a special set composed of four natural numbers that are in arithmetic progression?

Solution: \boxed{\text{No}}


Find the maximum value of \(3 \sin \left(x+\frac{\pi}{9}\right)+5 \sin \left(x+\frac{4 \pi}{9}\right)\), where \(x\) ranges over all real numbers.


Given a positive integer $n \geqslant 2$, find the maximum value of the constant $C(n)$ such that for all real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying $x_{i} \in (0, 1)$ for $i = 1, 2, \cdots, n$ and
$$
\left(1 - x_{i}\right)\left(1 - x_{j}\right) \geqslant \frac{1}{4} \quad \text{for} \quad 1 \leqslant i < j \leqslant n,
$$
the following inequality holds
$$
\sum_{i=1}^{n} x_{i} \geqslant C(n) \sum_{1 \leqslant i < j \leqslant n}\left(2x_{i}x_{j} + \sqrt{x_{i}x_{j}}\right).
$$


For what values of the parameter \( a \) will the minimum value of the function

$$
f(x)=|7x - 3a + 8| + |5x + 4a - 6| + |x - a - 8| - 24
$$

be the smallest?

Solution: \boxed{a = \frac{82}{43}}


Let \( p \) and \( q \) represent two consecutive prime numbers. For some fixed integer \( n \), the set \( \{ n - 1, 3n - 19, 38 - 5n, 7n - 45 \} \) represents \( \{ p, 2p, q, 2q \} \), but not necessarily in that order. Find the value of \( n \).


Given the functions \( f(x) = x^2 + 4x + 3 \) and \( g(x) = x^2 + 2x - 1 \), find all integer solutions to the equation \( f(g(f(x))) = g(f(g(x))) \).

Solution: \boxed{x = -2}


The first term of the geometric sequence \( \{a_n\} \) is \( a_1 = 1536 \), and the common ratio is \( q = -\frac{1}{2} \). Let \( f(n) \) denote the product of the first \( n \) terms. For which \( n \) is \( f(n) \) maximized?

Solution: \boxed{n = 12}


On a board, the numbers $1, 2, 3, \ldots, 235$ were written. Petya erased several of them. It turned out that among the remaining numbers, no number is divisible by the difference of any two others. What is the maximum number of numbers that could remain on the board?


In the number system with base $A+1$,

$$
(A A A A)^{2}=A A A C B B B C,
$$

where different letters represent different digits. Determine the values of $A$, $B$, and $C$.

Solution: \boxed{A=2, B=0, C=1}


Solution: \boxed{x = 1}


A pack of \( n \) cards, including three aces, is well shuffled. Cards are turned over in turn. Show that the expected number of cards that must be turned over to reach the second ace is \( \frac{n+1}{2} \).


We know that the number of factors of 2013, 2014, and 2015 are the same. What is the smallest value of \( n \) among three consecutive natural numbers \( n, n+1, \) and \( n+2 \) that have the same number of factors?


Denote by \(\langle x\rangle\) the fractional part of the real number \(x\) (for instance, \(\langle 3.2\rangle = 0.2\)). A positive integer \(N\) is selected randomly from the set \(\{1, 2, 3, \ldots, M\}\), with each integer having the same probability of being picked, and \(\left\langle\frac{87}{303} N\right\rangle\) is calculated. This procedure is repeated \(M\) times and the average value \(A(M)\) is obtained. What is \(\lim_{M \rightarrow \infty} A(M)\)?


Four small animals change seats. Initially, the mouse sits in seat 1, the monkey in seat 2, the rabbit in seat 3, and the cat in seat 4. They continuously exchange seats. The first time, the top and bottom rows swap. The second time, the left and right columns swap after the first exchange. The third time, the top and bottom rows swap again. The fourth time, the left and right columns swap again, and so on. After the tenth exchange, in which seat is the rabbit sitting?


Let \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\), and satisfy the equation \(\sin^2 \alpha + \sin^2 \beta - \frac{\sqrt{6}}{2} \sin \alpha - \frac{\sqrt{10}}{2} \sin \beta + 1 = 0\). Find \(\alpha + \beta\).


In an aquarium, there are three types of fish: gold, silver, and red fish. If a cat eats all the gold fish, the number of fish becomes 1 less than $\frac{2}{3}$ of the original number. If a cat eats all the red fish, the number of fish becomes 4 more than $\frac{2}{3}$ of the original number. Which type of fish—gold or silver—are there more of, and by how many?


Adva van egy egyenlő oldalú háromszög egyik oldala \(a\). Szerkesszünk e háromszög fölé egy merőleges hasábot és egy merőleges gúlát. Mekkora e testek köbtartalma és felülete, ha mindegyiknek a magassága \(m\)? Határozzuk meg \(m\)-et úgy, hogy a hasáb oldalfelülete egyenlő legyen a gúla oldalfelületével.

Solution: \boxed{m = \frac{a}{6}}


How many values of the parameter \( a \) exist such that the equation

$$
4a^{2} + 3x \log x + 3 \log^{2} x = 13a \log x + ax
$$

has a unique solution?


If the acute angle $\theta$ satisfies $\sin (\pi \cos \theta) = \cos (\pi \sin \theta)$, then what is $\sin 2\theta$?


Vasya cut a triangle out of cardboard and numbered its vertices with the numbers 1, 2, and 3. It turns out that if Vasya rotates his triangle 15 times clockwise around the vertex numbered 1 by an angle equal to the angle at this vertex, the triangle returns to its original position. If Vasya rotates his triangle 6 times clockwise around the vertex numbered 2 by an angle equal to the angle at this vertex, the triangle returns to its original position. Vasya claims that if he rotates his triangle \( n \) times around the vertex numbered 3 by an angle equal to the angle at this vertex, the triangle will return to its original position. What is the minimal \( n \) that Vasya could name so that his statement is true for at least some cardboard triangle?


Given that point \( P \) lies on the hyperbola \( \frac{x^2}{16} - \frac{y^2}{9} = 1 \), and the distance from \( P \) to the right directrix of this hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of the hyperbola, find the x-coordinate of \( P \).


Consider a rectangle \( ABCD \) where the side lengths are \( \overline{AB}=4 \) and \( \overline{BC}=8 \). Points \( M \) and \( N \) are fixed on sides \( BC \) and \( AD \), respectively, such that the quadrilateral \( BMDN \) is a rhombus. Calculate the area of this rhombus.


In the diagram below, \(ABCD\) is a square. The points \(A, B\), and \(G\) are collinear. The line segments \(AC\) and \(DG\) intersect at \(E\), and the line segments \(DG\) and \(BC\) intersect at \(F\). Suppose that \(DE = 15 \text{ cm}\), \(EF = 9 \text{ cm}\), and \(FG = x \text{ cm}\). Find the value of \(x\).


For which natural numbers \( n \) is the integer \( n^2 + n + 1 \) a perfect square?


Given the following numbers: 20172017, 20172018, 20172019, 20172020, and 20172021. Is there a number among them that is relatively prime to all the others? If so, which one?


A rectangle can be divided into \( n \) equal squares. The same rectangle can also be divided into \( n + 76 \) equal squares. Find all possible values of \( n \).


Find all positive integers \( n \) and \( m \) that satisfy the equation
\[
n^{5} + n^{4} = 7^{m} - 1.
\]

Solution: \boxed{(2, 2)}


Find the sum of all four-digit numbers, in which only the digits $1, 2, 3, 4, 5$ are used, and each digit occurs no more than once.


Vasya has \( n \) candies of several types, where \( n \geq 145 \). It is known that if any group of at least 145 candies is chosen from these \( n \) candies (in particular, one can choose the group of all \( n \) candies), there exists a type of candy such that the chosen group contains exactly 10 candies of that type. Find the largest possible value of \( n \).


The teacher wrote the following numbers on the board:

$$
1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43
$$

Two neighboring numbers always differ by the same value, in this case by 3. Then, she erased all the numbers except for 1, 19, and 43. Next, she added several integers between these three numbers such that each pair of neighboring numbers again differ by the same value and no number is written more than once.

In how many ways could the teacher have added the numbers?

(Note: Can the number 5 be included among the added numbers?)


There is one three-digit number and two two-digit numbers written on the board. The sum of the numbers containing the digit seven is 208. The sum of the numbers containing the digit three is 76. Find the sum of all three numbers.


In triangle \(ABC\), angle \(A\) is \(40^\circ\). The triangle is randomly thrown onto a table. Find the probability that vertex \(A\) ends up east of the other two vertices.


In how many ways can an \( n \times n \) grid be filled with zeros and ones such that each row and each column contains an even number of ones? Each cell of the grid must contain either a zero or a one.


Arrange the elements of the set \(\left\{2^{x}+2^{y} \mid x, y \in \mathbf{N}, x < y\right\}\) in ascending order. What is the 60th number? (Answer with a number).


Seryozha and Misha, while walking in the park, stumbled upon a meadow surrounded by linden trees. Seryozha walked around the meadow, counting the trees. Misha did the same, but started at a different tree (although he walked in the same direction). The tree that was the 20th for Seryozha was the 7th for Misha, and the tree that was the 7th for Seryozha was the 94th for Misha. How many trees were growing around the meadow?


Find all pairs of integers $(m, n)$ such that
$$
m^{3}-n^{3}=2 m n+8
$$

Solution: \boxed{(0, -2), (2, 0)}


Let $a=\sqrt{4+\sqrt{5-a}}, b=\sqrt{4+\sqrt{5+b}}, c=\sqrt{4-\sqrt{5-c}},$ and $d=\sqrt{4-\sqrt{5+d}}.$

Calculate $a b c d$.


Calculate the contour integrals:
1) \(\oint_{-i} 2 x \, dx - (x + 2 y) \, dy\)
2) \(\oint_{+i} y \cos x \, dx + \sin x \, dy\)

along the perimeter of the triangle with vertices \(A(-1, 0)\), \(B(0, 2)\), and \(C(2, 0)\).


A non-empty finite set is called a trivial set if the sum of the squares of all its elements is an odd number. Given the set \( A = \{1, 2, 3, \ldots, 2016, 2017\} \), find the number of trivial subsets of \( A \) that are proper subsets (i.e., not equal to \( A \) itself). (You may express the answer using exponents.)

Solution: \boxed{2^{2016} - 1}


Maria travels to school by a combination of walking and skateboarding. She can get there in 38 minutes if she walks for 25 minutes and skateboards for 13 minutes, or in 31 minutes if she walks for 11 minutes and skateboards for 20 minutes. How long (in minutes) would it take her to walk to school?


The function \( y = f(t) \) is such that the sum of the roots of the equation \( f(\sin x) = 0 \) in the interval \([3 \pi / 2, 2 \pi]\) is \( 33 \pi \), and the sum of the roots of the equation \( f(\cos x) = 0 \) in the interval \([\pi, 3 \pi / 2]\) is \( 23 \pi \). What is the sum of the roots of the second equation in the interval \([\pi / 2, \pi]?\)

Solution: \boxed{17 \pi}


Let $[x]$ denote the greatest integer not exceeding the real number $x$. If
\[ A = \left[\frac{7}{8}\right] + \left[\frac{7^2}{8}\right] + \cdots + \left[\frac{7^{2019}}{8}\right] + \left[\frac{7^{2020}}{8}\right], \]
what is the remainder when $A$ is divided by 50?


How many roots does \( \arctan x = x^{2} - 1.6 \) have, where the arctan function is defined in the range \( -\frac{\pi}{2} < \arctan x < \frac{\pi}{2} \)?


Six pirates - a captain and five crew members - are sitting around a campfire facing the center. They need to divide a treasure of 180 gold coins. The captain proposes a way to divide the coins (i.e., how many coins each pirate should receive: each pirate must receive a whole non-negative number of coins; different pirates may receive different amounts of coins). After this, the other five pirates vote on the captain's proposal. A pirate will vote "yes" only if he receives more coins than each of his two neighbors. The proposal is accepted if at least three out of the five crew members vote "yes".

What is the maximum number of coins the captain can receive under these rules?


A set of five volumes of an encyclopedia is arranged in ascending order on a shelf, i.e., from left to right, volumes 1 through 5 are lined up. We want to rearrange them in descending order, i.e., from left to right, volumes 5 through 1, but each time we are only allowed to swap the positions of two adjacent volumes. What is the minimum number of such swaps required to achieve the desired arrangement?


In the Cartesian coordinate system $xOy$, the graph of the parabola $y=ax^2 - 3x + 3 \ (a \neq 0)$ is symmetric with the graph of the parabola $y^2 = 2px \ (p > 0)$ with respect to the line $y = x + m$. Find the product of the real numbers $a$, $p$, and $m$.


Solution: \boxed{8x^3 - 27}


Given that $a$ and $b$ are positive integers, and $a - b \sqrt{3} = (2 - \sqrt{3})^{100}$, find the unit digit of $a \cdot b$.


Find all functions \( f: \mathbf{Z}^{*} \rightarrow \mathbf{R} \) (where \(\mathbf{Z}^{*}\) is the set of non-negative integers) that satisfy \( f(n+m) + f(n-m) = f(3n) \) for all \( m, n \in \mathbf{Z}^{*} \) with \( n \geqslant m \).

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


Given $\triangle ABC$ with $AB = 2$, $AC = 1$, and $\angle BAC = 120^\circ$, $O$ is the circumcenter of $\triangle ABC$. If $\overrightarrow{AO} = \lambda \overrightarrow{AB} + \mu \overrightarrow{AC}$, find $\lambda + \mu$.


Solve the following system of equations:

$$
\begin{aligned}
& x + y = a \\
& \frac{1}{x} + \frac{1}{y} = \frac{1}{b}
\end{aligned}
$$

Determine the values of \(a\) and \(b\) for which the roots are real. What relationship must exist between \(a\) and \(b\) for one root to be three times the other?

Solution: \boxed{b = \frac{3a}{16}}


Numbers from 1 to 6 are placed on the faces of a cube. The cube is rolled twice. The first time, the sum of the numbers on the four lateral faces was 12, and the second time it was 15. What number is written on the face opposite the one where the number 3 is written?


One-and-a-half-liter mineral water bottles are narrowed in the shape of an "hourglass" to make them easier to grip. The normal circumference of the bottle is $27.5 \mathrm{~cm}$, while at the waist - which is a $1 \mathrm{~cm}$ tall cylindrical section - it is only $21.6 \mathrm{~cm}$. The cylindrical sections with different circumferences are connected above and below the waist by truncated cone sections each $2 \mathrm{~cm}$ high. How much taller are such bottles compared to their regular counterparts with the same volume and normal circumference, but without the narrowing?

Solution: \boxed{1.18 \text{ cm}}


Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all real numbers \( x \) and \( y \):

\[
f\left(x^{2} + f(y)\right) = y + f(x)^{2}
\]

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


On a plate, there are different candies of three types: 2 lollipops, 3 chocolate candies, and 5 jelly candies. Sveta ate all of them one by one, choosing each next candy at random. Find the probability that the first and last candies she ate were of the same type.


In $\triangle ABC$, $\angle ABC = \angle ACB = 40^\circ$, and $P$ is a point inside the triangle such that $\angle PAC = 20^\circ$ and $\angle PCB = 30^\circ$. Find the measure of $\angle PBC$.


Five identical balls are moving in one direction in a straight line at some distance from each other, while five other identical balls are moving towards them. The speeds of all the balls are the same. Upon collision, any two balls bounce off in opposite directions with the same speed they had before the collision. How many total collisions will occur between the balls?


A positive unknown number less than 2022 was written on the board next to the number 2022. Then, one of the numbers on the board was replaced by their arithmetic mean. This replacement was done 9 more times, and the arithmetic mean was always an integer. Find the smaller of the numbers that were initially written on the board.


Points \( P \) and \( Q \) are located on side \( BC \) of triangle \( ABC \), with \( BP: PQ: QC = 1: 2: 3 \). Point \( R \) divides side \( AC \) of this triangle such that \( AR: RC = 1: 2 \). What is the ratio of the area of quadrilateral \( PQST \) to the area of triangle \( ABC \), if \( S \) and \( T \) are the intersection points of line \( BR \) with lines \( AQ \) and \( AP \), respectively?


Solve the following inequalities

1. \(3x + 1 \geq -2\)
2. \(y \geq 1\) and \(-2y \geq -2\)
3. \(y^2(x^2 + 1) - 1 \leq x^2\)

Solution: \boxed{-1 \leq y \leq 1}


Let the side lengths of a triangle be integers \( l \), \( m \), and \( n \), with \( l > m > n \). It is given that \( \left\{\frac{3^{l}}{10^{4}}\right\} = \left\{\frac{3^{m}}{10^{4}}\right\} = \left\{\frac{3^{n}}{10^{4}}\right\} \), where \(\{x\} = x - [x]\) and \([x]\) denotes the greatest integer less than or equal to \( x \). Find the minimum perimeter of such a triangle.


Let triangle \(ABC\) have \(AB=5\), \(BC=6\), and \(AC=7\), with circumcenter \(O\). Extend ray \(AB\) to point \(D\) such that \(BD=5\), and extend ray \(BC\) to point \(E\) such that \(OD=OE\). Find \(CE\).

Solution: \boxed{\sqrt{59} - 3}


From the 11 positive integers $1, 2, 3, \cdots, 11$, if you randomly choose 3 different positive integers $a, b, c$, what is the probability that their product $abc$ is divisible by 4?


Several people were seated around a round table such that the distances between neighboring people were equal. One of them was given a card with the number 1, and the rest were given cards with numbers 2, 3, and so on, in a clockwise direction.

The person with the card numbered 31 noticed that the distance from him to the person with the card numbered 7 is the same as the distance to the person with the card numbered 14. How many people are seated at the table in total?


The base of an isosceles triangle is $4 \sqrt{2} \text{ cm}$, and the median of one of the equal sides is $5 \text{ cm}$. Find the lengths of the equal sides.

Solution: \boxed{6 \text{ cm}}


Given $M$ as the set of all rational numbers within the open interval $(0,1)$. Does there exist a subset $A$ of $M$ such that every element in $M$ can be uniquely represented as the sum of a finite number of distinct elements from $A$?

Solution: \boxed{\text{No}}


Calculate the limit of the function:

\[
\lim _{x \rightarrow \frac{\pi}{3}} \frac{1-2 \cos x}{\pi-3 x}
\]


Choose 3 different numbers from 1 to 300 such that their sum is divisible by 3. How many such combinations are there?


\( P(x) \) is a fourth-degree polynomial with integer coefficients, with a leading coefficient that is positive. It is known that \( P(\sqrt{3}) = P(\sqrt{5}) \). Find the value(s) of \( x \) at which \( P(x) \) takes its minimum value.

Solution: \boxed{\pm 2}


Randomly select a permutation \(\sigma = \left(x_{1}, x_{2}, \ldots, x_{n}\right)\) of \(1, 2, \ldots, n\). If \(x_{i} = i\), then \(i\) is called a fixed point of \(\sigma\). Let the number of fixed points in the permutation be denoted as the random variable \(X_{n}\). Find the expectation \(E\left(X_{n}\right)\) of \(X_{n}\).


In the diagram below, \( \triangle ABC \) is a triangle with \( AB = 39 \text{ cm}, BC = 45 \text{ cm}, \) and \( CA = 42 \text{ cm} \). The tangents at \( A \) and \( B \) to the circumcircle of \( \triangle ABC \) meet at the point \( P \). The point \( D \) lies on \( BC \) such that \( PD \) is parallel to \( AC \). It is given that the area of \( \triangle ABD \) is \( x \text{ cm}^2 \). Find the value of \( x \).


An equilateral triangle is subdivided into 4 smaller equilateral triangles. Using red and yellow to paint the vertices of the triangles, each vertex must be colored and only one color can be used per vertex. If two colorings are considered the same when they can be made identical by rotation, how many different colorings are possible?


A torpedo boat is anchored 9 km from the nearest point on the shore. A messenger needs to be sent from the boat to a camp located 15 km along the shore from the nearest point of the boat. If the messenger travels on foot at a speed of 5 km/h and rows at a speed of 4 km/h, at which point on the shore should he land to reach the camp in the shortest possible time?


Two cars, Car A and Car B, start from point A and point B, respectively, and move towards each other simultaneously. They meet after 3 hours, after which Car A turns back towards point A and Car B continues moving forward. Once Car A reaches point A, it turns towards point B again and meets Car B half an hour later. How many minutes does Car B take to travel from point A to point B?

Solution: \boxed{432 \text{ minutes}}


Given \(\sin x + \sin y = 0.6\) and \(\cos x + \cos y = 0.8\), find \(\cos x \cdot \cos y\).


Given \( a, b \in \mathbb{Z} \), and \( a + b \) is a root of the equation \( x^{2} + ax + b = 0 \), what is the maximum possible value of \( b \)?


Buying an album. Masha was short of 2 kopecks, Kolya was short of 34 kopecks, and Fedya was short of 35 kopecks. When they pooled their money, it was still not enough to buy the album. How much does the album cost?


There are two coal mines, Mine A and Mine B. Each gram of coal from Mine A releases 4 calories of heat when burned, and each gram of coal from Mine B releases 6 calories of heat when burned. The price per ton of coal at the production site is 20 yuan for Mine A and 24 yuan for Mine B. It is known that the transportation cost for a ton of coal from Mine A to city $N$ is 8 yuan. What should the transportation cost per ton of coal from Mine B to city $N$ be for it to be more economical than transporting coal from Mine A?

Solution: \boxed{18 \text{ Yuan}}


Let the function \( f(x) \) be defined on \([0,1]\), satisfying \( f(0) = f(1) \) and for any \( x, y \in [0,1] \) having \( |f(x) - f(y)| < |x - y| \). Find the smallest real number \( m \) such that for any function \( f(x) \) meeting the above conditions and for any \( x, y \in [0,1] \), we have \( |f(x) - f(y)| < m \).


What is "Romney"? If

$$
\frac{N}{O}=. \text{Romney Romney Romney} \ldots
$$

is the decimal representation of a certain proper fraction, where each letter represents some decimal digit, find the value of the word Romney (the letters $N$ and $n$ represent the same digit; the same applies to $O$ and $o$).


A semicircle with a radius of 1 is drawn inside a semicircle with a radius of 2. A circle is drawn such that it touches both semicircles and their common diameter. What is the radius of this circle?


Given the sequence elements \( a_{n} \) such that \( a_{1}=1337 \) and \( a_{2n+1}=a_{2n}=n-a_{n} \) for all positive integers \( n \). Determine the value of \( a_{2004} \).


Calculate the volumes of the solids formed by the rotation of the figures bounded by the graphs of the functions. Axis of rotation is \(O x\).

$$
y=1-x^{2}, x=0, x=\sqrt{y-2}, x=1
$$

Solution: \boxed{5 \pi}


Chester traveled from Hualien to Lukang in Changhua to participate in the Hua Luogeng Gold Cup Math Competition. Before leaving, his father checked the car’s odometer, which displayed a palindromic number of 69,696 kilometers (a palindromic number reads the same forward and backward). After driving for 5 hours, they arrived at the destination with the odometer showing another palindromic number. During the journey, the father's driving speed never exceeded 85 kilometers per hour. What is the maximum possible average speed (in kilometers per hour) that Chester's father could have driven?


The edges of a rectangular parallelepiped are in the ratio $3: 4: 12$. A diagonal cross-section is made through the largest edge. Find the sine of the angle between the plane of this cross-section and the diagonal of the parallelepiped that does not lie in that plane.


In triangle \(ABC\), \(\angle A = 2 \angle C\). Suppose that \(AC = 6\), \(BC = 8\), and \(AB = \sqrt{a} - b\), where \(a\) and \(b\) are positive integers. Compute \(100a + b\).


Compute the area of the figure bounded by one arch of the cycloid given by the parametric equations $x = 2(t - \sin t)$ and $y = 2(1 - \cos t)$.


Defined on $\mathbf{R}$, the function $f$ satisfies
$$
f(1+x)=f(9-x)=f(9+x).
$$

Given $f(0)=0$, and $f(x)=0$ has $n$ roots in the interval $[-4020, 4020]$, find the minimum value of $n$.


Two spheres with equal mass and charge are initially at a distance of \( l \) from each other when they are released without an initial velocity. After \( t \) seconds, the distance between them doubles. How much time will it take for the distance between the spheres to double if they are initially released from a distance of \( 3l \)? Air resistance can be neglected.

Solution: \boxed{3 \sqrt{3} t}


Subset \( S \subseteq \{1, 2, 3, \ldots, 1000\} \) is such that if \( m \) and \( n \) are distinct elements of \( S \), then \( m + n \) does not belong to \( S \). What is the largest possible number of elements in \( S \)?


Given a regular 2017-sided polygon \(A_{1} A_{2} \cdots A_{2017}\) inscribed in a unit circle \(\odot O\), choose any two different vertices \(A_{i}, A_{j}\). Find the probability that \(\overrightarrow{O A_{i}} \cdot \overrightarrow{O A_{j}} > \frac{1}{2}\).


Given the parabola \( y^2 = 4p(x + p) \) (where \( p > 0 \)), two mutually perpendicular chords \( AB \) and \( CD \) pass through the origin \( O \). Find the minimum value of \( |AB| + |CD| \).


Let \(\mathbb{Q}^{+}\) denote the set of positive rational numbers. Find, with proof, all functions \(f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}\) such that, for all positive rational numbers \(x\) and \(y\), we have
\[
f(x) = f(x + y) + f\left(x + x^2 f(y)\right).
\]

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


There are $N$ prime numbers written on a board (not necessarily distinct). It turns out that the sum of any three numbers on the board is also a prime number. What is the largest possible value of $N$?


The positive integers \( x \) and \( y \) satisfy the equation \( y x^{2} + x y^{2} = 70 \). What is the value of \( x^{4} + y^{4} \)?


Let \( n \) be a natural number. For any real numbers \( x, y, z \), the inequality \(\left(x^{2}+y^{2}+z^{2}\right) \leq n\left(x^{4}+y^{4}+z^{4}\right)\) always holds. Find the minimum value of \( n \).


The three-digit number ends with the digit 2. If this digit is moved to the beginning of the number, the resulting number will be 18 greater than the original number. Find this number.


At the namesake festival, 45 Alexanders, 122 Borises, 27 Vasily, and several Gennady attended. At the beginning of the festival, all of them lined up so that no two people with the same name stood next to each other. What is the minimum number of Gennadys that could have attended the festival?


One ball is taken at random from a bag containing $(b-4)$ white balls and $(b+46)$ red balls. If $\frac{c}{6}$ is the probability that the ball is white, find $c$.


In a certain book, there were 100 statements written as follows:
1) "In this book, there is exactly one false statement."
2) "In this book, there are exactly two false statements."
...
3) "In this book, there are exactly one hundred false statements."

Which of these statements is true?


A mouse is gnawing on a cube of cheese with an edge of 3, divided into 27 unit cubes. When the mouse eats a cube, it moves to another cube that shares a face with the previous one. Can the mouse eat the entire cube except for the central unit cube?

Solution: \boxed{\text{No}}


The side of an equilateral triangle inscribed in a circle is $a$. Calculate the area of the square inscribed in the same circle.

Solution: \boxed{\frac{2 a^2}{3}}


The product of the midline of a trapezoid and the segment connecting the midpoints of its diagonals equals 25. Find the area of the trapezoid if its height is three times the difference of its bases.


A $3 \times 3$ table is initially filled with zeros. In one move, any $2 \times 2$ square in the table is chosen, and all zeros in it are replaced with crosses, and all crosses with zeros. Let's call a "pattern" any arrangement of crosses and zeros in the table. How many different patterns can be obtained as a result of such moves? Patterns that can be transformed into each other by a $90^\circ$ or $180^\circ$ rotation are considered different.


In a single round-robin table tennis tournament, a) 10 players, b) 11 players competed. What is the minimum number of matches the champion must have won, given that there was no tie for first place?


There exists \( x_{0} < 0 \) such that \( x^{2} + |x - a| - 2 < 0 \) (where \( a \in \mathbb{Z} \)) is always true. Find the sum of all values of \( a \) that satisfy this condition.


As shown in the figure, point $D$ is an interior point of $\triangle ABC$. $BD$ intersects $AC$ at $E$, and $CD$ intersects $AB$ at $F$, with $AF = BF = CD$ and $CE = DE$. Find the measure of $\angle BFC$.


For \(x, y, z \geq 1\), find the minimum value of the expression

$$
A = \frac{\sqrt{3 x^{4} + y} + \sqrt{3 y^{4} + z} + \sqrt{3 z^{4} + x} - 3}{x y + y z + z x}
$$


A town has \( c \) celebrities and \( m \) weekly magazines. One week each celebrity was mentioned in an odd number of magazines, and each magazine mentioned an odd number of celebrities. Show that \( m \) and \( c \) are both even or both odd. How many ways are there for each magazine to mention an odd number of celebrities so that each celebrity is mentioned an odd number of times?


Find the rank of the \( (2n+1) \times (2n+1) \) skew-symmetric matrix with entries given by:
- \( a_{ij} = 1 \) for \( i - j = -2n, -(2n-1), \ldots, -(n+1) \)
- \( a_{ij} = -1 \) for \( i - j = -n, -(n-1), \ldots, -1 \)
- \( a_{ij} = 1 \) for \( i - j = 1, 2, \ldots, n \)
- \( a_{ij} = -1 \) for \( i - j = n+1, n+2, \ldots, 2n+1 \)

In other words, the main diagonal is zeros, the diagonals immediately below the main diagonal (up to \( n \) diagonals) are \( 1s \), the diagonals below those are \( -1s \), the diagonals immediately above the main diagonal (up to \( n \) diagonals) are \( -1s \), and the diagonals above those are \( 1s \).


Is it possible to place 8 different numbers, none of which are divisible by 13, from the range of 1 to 245 at the vertices of a cube such that the numbers in adjacent vertices have a common divisor greater than 1, while the numbers in non-adjacent vertices do not share any common divisor greater than 1?

Solution: \boxed{\text{No}}