A car starts from point \( O \) and moves along a straight road with a constant speed \( v \). At the same time, a person on a bicycle, located at a distance \( a \) from point \( O \) and \( b \) from the road, begins to ride with the intention of delivering a letter to the car driver. What is the minimum speed the cyclist must maintain to achieve this goal?

Solution: \boxed{\frac{v b}{a}}


Given that \(a\), \(b\), \(c\), and \(d\) are positive integers, and
\[
\log_{a} b=\frac{3}{2}, \quad \log_{c} d=\frac{5}{4}, \quad \text{and} \quad a-c=9,
\]
find the value of \(a + b + c + d\).


In triangle \(ABC\), point \(K\) on side \(AB\) and point \(M\) on side \(AC\) are positioned such that \(AK:KB = 3:2\) and \(AM:MC = 4:5\). Determine the ratio in which the line through point \(K\) parallel to side \(BC\) divides segment \(BM\).


One hundred friends, including Petya and Vasya, live in several cities. Petya found out the distance from his city to the city of each of the remaining 99 friends and summed these 99 numbers. Vasya did the same. Petya obtained 1000 km. What is the largest number Vasya might have obtained? (Consider the cities as points on a plane; if two people live in the same city, the distance between their cities is considered to be zero.)

Solution: \boxed{99000 \text{ km}}


A ruler of a certain country, for purely military reasons, wanted there to be more boys than girls among his subjects. Therefore, he decreed that no family should have more than one girl. Thus, every woman in this country who had children would have a girl as the last and only last child because no woman, having given birth to a girl, would dare to have more children. What proportion of boys made up the total mass of children in this country?


As shown in the figure, three circles intersect to create seven regions. Fill these regions with the integers from $1$ to $7$, ensuring that the sum of the four numbers in each circle is equal. What is the maximum possible sum for each circle?


The diagonal of a regular 2006-gon \(P\) is called good if its ends divide the boundary of \(P\) into two parts, each containing an odd number of sides. The sides of \(P\) are also called good. Let \(P\) be divided into triangles by 2003 diagonals, none of which have common points inside \(P\). What is the maximum number of isosceles triangles, each of which has two good sides, that such a division can have?


Given the complex number sequence $\left\{z_{n}\right\}$ which satisfies $z_{1}=1$ and $z_{n+1}=\overline{z_{n}}+1+n i$ for $n=1,2, \cdots$, find $z_{2015}$.

Solution: \boxed{2015 + 1007i}


\( z_{1}, z_{2}, z_{3} \) are the three roots of the polynomial
\[ P(z) = z^{3} + a z + b \]
and satisfy the condition
\[ \left|z_{1}\right|^{2} + \left|z_{2}\right|^{2} + \left|z_{3}\right|^{2} = 250 \]
Moreover, the three points \( z_{1}, z_{2}, z_{3} \) in the complex plane form a right triangle. Find the length of the hypotenuse of this right triangle.


Anya arranges pebbles on the sand. First, she placed one pebble, then added pebbles to form a pentagon, then made a larger outer pentagon from pebbles, and then another outer pentagon, and so on, as shown in the figure. The number of pebbles arranged in the first four pictures are: 1, 5, 12, and 22. If she continues to create these pictures, how many pebbles will be in the 10th picture?


Let the function \( f: \mathbf{R} \rightarrow \mathbf{R} \) satisfy \( f(0) = 1 \) and for any \( x, y \in \mathbf{R} \), \( f(x y + 1) = f(x) f(y) - f(y) - x + 2 \). Find \( f(x) \).

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


Let \( c \) be a positive real number. If \( x^{2} + 2 \sqrt{c} x + b = 0 \) has one real root only, find the value of \( c \).

Solution: \boxed{c = b}


The Fibonacci numbers are defined by \( F_{1} = F_{2} = 1 \), and \( F_{n} = F_{n-1} + F_{n-2} \) for \( n \geq 3 \). If the number
\[ \frac{F_{2003}}{F_{2002}} - \frac{F_{2004}}{F_{2003}} \]
is written as a fraction in lowest terms, what is the numerator?


It is known about the numbers \( x_1 \) and \( x_2 \) that \( x_1 + x_2 = 2 \sqrt{1703} \) and \( \left| x_1 - x_2 \right| = 90 \). Find \( x_1 \cdot x_2 \).


Given 100 numbers. Each number is increased by 2. The sum of the squares of the numbers remains unchanged. Each resulting number is then increased by 2 again. How has the sum of the squares changed now?


Find the equation of the tangent line to the given curve at the point with the abscissa \( x_{0} = 1 \).

\[ y = \frac{x^{29} + 6}{x^{4} + 1}, \quad x_{0} = 1 \]

Solution: \boxed{y = 7.5x - 4}


The parabola $\Pi_{1}$ with upward-facing branches passes through the points with coordinates $(10,0)$ and $(13,0)$. The parabola $\Pi_{2}$ with upward-facing branches also passes through the point with coordinates $(13,0)$. It is also known that the vertex of $\Pi_{1}$ bisects the segment connecting the origin and the vertex of $\Pi_{2}$. At what abscissa does the parabola $\Pi_{2}$ intersect the $x$-axis again?


Given the sequence \(\left\{a_{n}\right\}\) with the general term
\[ a_{n} = n^{4} + 6n^{3} + 11n^{2} + 6n, \]
find the sum of the first 12 terms \( S_{12} \).


On a plane, a regular hexagon with side length \( a \) is drawn. For any \( n \in \mathbf{N} \), greater than 1, construct a segment of length \( a / n \) using only a straightedge.

Solution: \boxed{\frac{a}{n}}


In $\triangle ABC$, point $E$ is on $AB$, point $F$ is on $AC$, and $BF$ intersects $CE$ at point $P$. If the areas of quadrilateral $AEPF$ and triangles $BEP$ and $CFP$ are all equal to 4, what is the area of $\triangle BPC$?


Exactly half of the population of the island Nevezennya are hares, and all the others are rabbits. If an inhabitant of the island Nevezennya states something, they always sincerely believe in what they are saying. However, hares earnestly make mistakes on average in one out of every four cases, while rabbits earnestly make mistakes on average in one out of every three cases. One day, a creature came to the center of the island and shouted: "I am not a hare!". It thought for a moment and sadly said: "I am not a rabbit." What is the probability that it is indeed a hare?


Nine points are drawn on a sheet of paper, as shown in the following figure:

a) In how many ways is it possible to choose three collinear points?

b) In how many ways is it possible to choose four points such that three of them are collinear?


In a geometric sequence $\left\{a_{n}\right\}$ where all terms are positive, given that $2 a_{4}+a_{3}-2 a_{2}-a_{1}=8$, find the minimum value of $2 a_{8}+a_{7}$.


Pasha and Sasha made three identical toy cars. Sasha did one-fifth of the total work. After that, they sold the cars and divided the proceeds proportionally to the work done. Pasha noticed that if he gave Sasha 400 rubles and Sasha made and sold another similar car, they would have equal amounts of money. How much does one toy car cost?

Solution: \boxed{1000 \text{ rubles}}


Find the number of triples of natural numbers \((a, b, c)\) that satisfy the system of equations

$$
\left\{\begin{array}{l}
\gcd(a, b, c)=15 \\
\text{lcm}(a, b, c)=3^{15} \cdot 5^{18}
\end{array}\right.
$$


Let \( p \) be a prime number. Find all possible values of the remainder when \( p^{2} - 1 \) is divided by 12.

Solution: \boxed{0, 3, 8}


In the triangle \( ABC \), on the longest side \( BC \) with length \( b \), point \( M \) is chosen. Find the minimum distance between the centers of the circles circumscribed around triangles \( BAM \) and \( ACM \).


Compute the limit as \( x \) approaches 0 of the expression \( \frac{e^{x \cos x} - 1 - x}{\sin(x^2)} \).


Given that the sequence \( a_1, a_2, \cdots, a_n, \cdots \) satisfies \( a_1 = a_2 = 1 \) and \( a_3 = 2 \), and for any \( n \in \mathbf{N}^{*} \), it holds that \( a_n \cdot a_{n+1} \cdot a_{n+2} \cdot a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3} \). Find the value of \( \sum_{i=1}^{2023} a_i \).


What is the minimum number of points that can be chosen on a circle with a circumference of 1956 so that for each of these points there is exactly one chosen point at a distance of 1 and exactly one at a distance of 2 (distances are measured along the circle)?


In rectangle \(ABCD\), side \(AB\) is 6 and side \(BC\) is 11. From vertices \(B\) and \(C\), angle bisectors are drawn intersecting side \(AD\) at points \(X\) and \(Y\) respectively. Find the length of segment \(XY\).


On Earth, the Autobots discovered a new energy source, "energy crystals," but it was seized by the Decepticons. The Decepticons manufactured cubic containers to transport the energy crystals back to Cybertron. Each energy crystal is a rectangular prism measuring 30 cm in length, 25 cm in width, and 5 cm in height. The container was fully loaded with energy crystals. At least how many energy crystals are there in one container?


For a positive integer \( n \), let \( S_{n} \) be the minimum value of \( \sum_{k=1}^{n} \sqrt{(2k-1)^{2} + a_{k}^{2}} \), where \( a_{1}, a_{2}, \cdots, a_{n} \) are positive real numbers whose sum is 17. There exists a unique \( n \) such that \( S_{n} \) is also an integer. Find \( n \).


The café "Buratino" operates 6 days a week with Mondays off. Kolya said that from April 1 to April 20, the café was open for 17 days, and from April 10 to April 30, it was open for 18 days. It is known that he made a mistake once. What was the date of the last Tuesday in April?


Let \([x]\) denote the greatest integer less than or equal to the real number \(x\). For example, \([3]=3\), \([2.7]=2\), and \([-2.2]=-3\). Find the last two digits of \(\left[\frac{10^{93}}{10^{31}+3}\right]\).


Five girls ran a race. Fiona started first, followed by Gertrude, then Hannah, then India, and lastly Janice. Whenever a girl overtook another girl, she was awarded a point. India was first to finish, followed by Gertrude, Fiona, Janice, and lastly Hannah. What is the lowest total number of points that could have been awarded?
A) 9
B) 8
C) 7
D) 6
E) 5


The number \(123456789(10)(11)(12)(13)(14)\) is written in the base-15 number system, which means the number is equal to:

\[
(14) + (13) \cdot 15 + (12) \cdot 15^{2} + (11) \cdot 15^{3} + \ldots + 2 \cdot 15^{12} + 15^{13}
\]

What remainder does this number give when divided by 7?


In trapezoid \(ABCD\), the side \(AB\) is perpendicular to the base \(BC\). A circle passes through points \(C\) and \(D\) and is tangent to line \(AB\) at point \(E\).

Find the distance from point \(E\) to line \(CD\), if \(AD = 4\) and \(BC = 3\).

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


2 apples and 3 oranges cost 6 dollars.
4 apples and 7 oranges cost 13 dollars.
16 apples and 23 oranges cost $C$ dollars. Find $C$.

If $K=\frac{6 \cos \theta+5 \sin \theta}{2 \cos \theta+3 \sin \theta}$ and $\tan \theta=2$, find $K$.


It is given that \(\log \frac{x}{2}=0.5\) and \(\log \frac{y}{5}=0.1\). If \(\log xy=c\), find \(c\).


The fourth-degree polynomial \(x^{4}-18 x^{3}+k x^{2}+200 x-1984\) has four roots, and the product of two of these roots is \(-32\). Find the real number \(k\).


It is known that the constant term \( a_0 \) of the polynomial \( P(x) \) with integer coefficients is less than 100 in absolute value, and \( P(20) = P(16) = 2016 \). Find \( a_0 \).


Several equally sized teams of guards slept an equal number of nights. Each guard slept more nights than the number of guards in a team but fewer nights than the number of teams. How many guards are there in a team if all the guards together slept a total of 1001 person-nights?


Vanya thought of a two-digit number, then swapped its digits and multiplied the resulting number by itself. The result turned out to be four times the original number. What number did Vanya think of?


How many nonempty subsets of {1, 2, ..., 10} have the property that the sum of their largest element and smallest element is 11?


A barcode is composed of alternate strips of black and white, where the leftmost and rightmost strips are always black. Each strip (of either color) has a width of 1 or 2. The total width of the barcode is 12. The barcodes are always read from left to right. How many distinct barcodes are possible?


In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse." How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.


Solve the equation \( x^2 + y^2 + z^2 = 2xyz \) in integers.

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


If the numbers 826 and 4373 are divided by the same natural number, the remainders are 7 and 8, respectively. Find all possible values of the divisor.


Let \( n \) be a fixed integer, \( n \geq 2 \).

1. Determine the smallest constant \( c \) such that the inequality \[
\sum_{1 \leq i < j \leq n} x_i x_j (x_i^2 + x_j^2) \leq c \left( \sum_{i=1}^n x_i \right)^4
\] holds for all non-negative real numbers \( x_1, x_2, \cdots, x_n \).

2. For this constant \( c \), determine the necessary and sufficient conditions for equality to hold.


The school organized an outing for 1511 people and rented 42-seater and 25-seater buses. If each person must have exactly one seat and each seat is occupied by one person, how many different bus rental arrangements are possible?


A square \(ABCD\) has a side-length of 2, and \(M\) is the midpoint of \(BC\). The circle \(S\) inside the quadrilateral \(AMCD\) touches the three sides \(AM\), \(CD\), and \(DA\). What is its radius?

Solution: \boxed{3 - \sqrt{5}}


From the set \(\{1,2,3, \cdots, 14\}\), pick \(a_{1}, a_{2}, a_{3}\) in ascending order, such that \(a_{2} - a_{1} \geqslant 3\) and \(a_{3} - a_{2} \geqslant 3\). How many different ways are there to pick such \(a_{1}, a_{2}, a_{3}\)?


A full container holds 150 watermelons and melons with a total value of 24,000 rubles. The total value of all watermelons is equal to the total value of all melons. How much does one watermelon cost in rubles, given that the container can hold 120 melons (without watermelons) or 160 watermelons (without melons)?


There are $n$ points inside a convex quadrilateral, along with the 4 vertices of the quadrilateral, for a total of $n+4$ points. No three points lie on the same line. By using these points as vertices to cut the quadrilateral into triangles, the maximum number of triangles that can be formed is 1992. Find the value of $n$.


The function \( f \) is defined on the set of positive integers and satisfies \( f(1) = 2002 \) and \( f(1) + f(2) + \cdots + f(n) = n^{2} f(n) \) for \( n > 1 \). What is the value of \( f(2002) \)?


The numerators and denominators of the fractions \(\frac{2018}{2011}\) and \(\frac{2054}{2019}\) are each reduced by the same positive integer \(a\), resulting in the new fractions being equal. What is the positive integer \(a\)?


Given constants $\lambda \neq 0$, $n$, and $m$ as positive integers with $m > n$ and $m \neq 2n$, the function $f(x)$ defined by the equation
$$
f(x+\lambda)+f(x-\lambda)=2 \cos \frac{2n\pi}{m} f(x)
$$
is a periodic function with period $m\lambda$.

Solution: \boxed{m\lambda}


A square with a side length of 100 was cut into two equal rectangles. They were then placed next to each other as shown in the picture. Find the perimeter of the resulting figure.


Consider a triangle \(ABC\), where \(AB = 20\), \(BC = 25\), and \(CA = 17\). \(P\) is a point on the plane. What is the minimum value of \(2 \times PA + 3 \times PB + 5 \times PC\)?


In a movie theater, five friends took seats from 1 to 5 (the leftmost seat is number 1). During the movie, Anya went to get popcorn. When she returned, she found that Varya had moved one seat to the right, Galya had moved two seats to the left, and Diana and Ella had swapped seats, leaving the end seat for Anya. In which seat was Anya sitting before she got up?


On a board, there are 48 ones written. Every minute, Carlson erases two arbitrary numbers and writes their sum on the board, and then eats an amount of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 48 minutes?


Is it possible to cut off a corner from a cube with a 20 cm edge so that the cut has the shape of a triangle with sides of 7, 8, and 11 cm?

Solution: \boxed{\text{No}}


We consider all natural numbers less than \( p^3 \), where \( p \) is a prime number. How many of these numbers are coprime with \( p^3 \)?

Solution: \boxed{p^3 - p^2}


Five people are standing in a line, each wearing a different hat numbered $1, 2, 3, 4, 5$. Each person can only see the hats of the people in front of them. Xiao Wang cannot see any hats; Xiao Kong can only see hat number 4; Xiao Tian cannot see hat number 3 but can see hat number 1; Xiao Yan sees three hats but does not see hat number 3; Xiao Wei sees hats numbered 3 and 2. What number hat is Xiao Tian wearing?


In \(\triangle ABC\), \(AB = 9\), \(BC = 8\), and \(AC = 7\). The bisector of \(\angle A\) meets \(BC\) at \(D\). The circle passing through \(A\) and touching \(BC\) at \(D\) cuts \(AB\) and \(AC\) at \(M\) and \(N\) respectively. Find \(MN\).


In unit square \(ABCD\), points \(E, F, G\) are chosen on sides \(BC, CD,\) and \(DA\) respectively such that \(AE\) is perpendicular to \(EF\) and \(EF\) is perpendicular to \(FG\). Given that \(GA = \frac{404}{1331}\), find all possible values of the length of \(BE\).


Four teams, including Quixajuba, are competing in a volleyball tournament where:

- Each team plays against every other team exactly once;
- Any match ends with one team winning;
- In any match, the teams have an equal probability of winning;
- At the end of the tournament, the teams are ranked by the number of victories.

a) Is it possible that, at the end of the tournament, all teams have the same number of victories? Why?

b) What is the probability that the tournament ends with Quixajuba alone in first place?

c) What is the probability that the tournament ends with three teams tied for first place?


Calculate the area of the region bounded by the parabolas \( y = x^{2} \) and \( y = 8 - x^{2} \).


Vitya and Masha were born in the same year in June. Find the probability that Vitya is at least one day older than Masha.


Find all functions \( f \) from \(\mathbb{R}\) to \(\mathbb{R}\) such that for all pairs \((x, y)\) of real numbers,
\[
f(x+y) = x + f(f(y))
\]

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


What type of curve is defined by the equation \( y^{2} - 4x - 2y + 1 = 0 \)? If the line \( y = kx + 2 \) is tangent to this curve, what should be the value of \( k \)?


70 numbers are arranged in a row. Except for the first and last number, three times each number exactly equals the sum of its two adjacent numbers. The sequence starts with the following numbers on the far left: \(0, 1, 3, 8, 21, \cdots\). What is the remainder when the rightmost number is divided by 6?


(Theorem of Helly) We consider four convex sets in the plane such that the intersection of any three of them is always non-empty.

- a) Show that the intersection of all four convex sets is non-empty.
- b) Does the theorem hold true if we replace 4 with $n \geqslant 4$?

Solution: \boxed{\text{Yes}}


Find the number of ordered triples of positive integers \((a, b, c)\) such that
$$
6a + 10b + 15c = 3000.
$$


In the representation of three two-digit numbers, there are no zeros, and in each of them, both digits are different. Their sum is 41. What could their sum be if the digits in them are swapped?


In the right triangle \(ABC\), the leg \(AB\) is 21, and the leg \(BC\) is 28. A circle with center \(O\) on the hypotenuse \(AC\) is tangent to both legs.

Find the radius of the circle.


Let \( XYZ \) be an acute-angled triangle. Let \( s \) be the side length of the square which has two adjacent vertices on side \( YZ \), one vertex on side \( XY \), and one vertex on side \( XZ \). Let \( h \) be the distance from \( X \) to the side \( YZ \) and \( b \) be the distance from \( Y \) to \( Z \).

(a) If the vertices have coordinates \( X=(2,4), Y=(0,0) \), and \( Z=(4,0) \), find \( b, h \), and \( s \).

(b) Given the height \( h=3 \) and \( s=2 \), find the base \( b \).

(c) If the area of the square is 2017, determine the minimum area of triangle \( XYZ \).


David goes to a store that sells a bottle of orange juice for $2.80 and a pack of six bottles for $15.00. He needs to buy 22 bottles for his birthday. What is the minimum amount he will spend?

Solution: \boxed{56.20 \text{ reais}}


Let \( A_{1} A_{2} \cdots A_{21} \) be a regular 21-sided polygon inscribed in a circle. Select \( n \) different vertices from \( A_{1}, A_{2}, \cdots, A_{21} \) and color them red such that the distance between any two red points is different. Find the maximum value of the positive integer \( n \).

(Sichuan Province Middle School Mathematics Competition, 2014)


Suppose \( a, b, c, d \) are real numbers such that
\[ |a-b| + |c-d| = 99 \]
\[ |a-c| + |b-d| = 1 \]

Determine all possible values of \( |a-d| + |b-c| \).


Given the function \(\mathrm{f}(\mathrm{x}) = \mathrm{x} - \ln(\mathrm{ax} + 2 \mathrm{a} + 1) + 2\), if \(\mathrm{f}(\mathrm{x}) \geq 0\) holds for any \(\mathrm{x} \geq -2\), find the range of the real number \(\mathrm{a}\).

Solution: \boxed{[0, 1]}


What number has the property that if it is added to the numbers 100 and 164 separately, it results in perfect squares each time?


In a box, there are white and blue balls, with the number of white balls being 8 times greater than the number of blue balls. It is known that if you take out 100 balls, there will definitely be at least one blue ball among them. How many balls are there in the box in total?


The length of the curve given by the parametric equations \(\left\{\begin{array}{l}x=2 \cos ^{2} \theta \\ y=3 \sin ^{2} \theta\end{array}\right.\) (where \(\theta\) is the parameter) is?


A boat travels 16 kilometers downstream and 8 kilometers upstream in a total of 4 hours. It also travels 12 kilometers downstream and 10 kilometers upstream in the same amount of time. How many hours does it take for the boat to travel 24 kilometers downstream and then return?


Ctibor marked a square land plot on a map with a scale of 1:50000 and calculated that its side corresponds to $1 \mathrm{~km}$ in reality. He then resized the map on a copier such that the marked square had an area $1.44 \mathrm{~cm}^{2}$ smaller than the original.

What was the scale of the resized map?

Hint: What were the dimensions of the marked plot on the original map?


Solve the system of equations:
\[ \begin{cases} 9y^2 - 4x^2 = 144 - 48x, \\ 9y^2 + 4x^2 = 144 + 18xy. \end{cases} \]

After obtaining the solutions \(\left(x_1, y_1\right), \left(x_2, y_2\right), \ldots, \left(x_n, y_n\right)\), write the sum of the squares of the solutions:
\[ x_1^2 + x_2^2 + \ldots + x_n^2 + y_1^2 + y_2^2 + \ldots + y_n^2. \]


Find the minimum value of the function \( f(x) = x^2 - 4x - \frac{8}{x} + \frac{4}{x^2} + 5 \) on the ray \( x < 0 \).

Solution: \boxed{9 + 8 \sqrt{2}}


Find the maximum value of the function \( f(x) = 8 \sin x + 15 \cos x \).


Determine whether the number

$$
\frac{1}{2 \sqrt{1}+1 \sqrt{2}}+\frac{1}{3 \sqrt{2}+2 \sqrt{3}}+\frac{1}{4 \sqrt{3}+3 \sqrt{4}}+\cdots+\frac{1}{100 \sqrt{99}+99 \sqrt{100}}
$$

is rational or irrational. Explain your answer.


Four schools each send 3 representatives to form \( n \) groups for social practice activities (each representative can participate in multiple groups). The conditions are: (1) Representatives from the same school are not in the same group; (2) Any two representatives from different schools participate in exactly one group together. What is the minimum value of \( n \)?


Find the volume of the body bounded by two cylinders \( x^{2} + y^{2} = R^{2} \) and \( x^{2} + z^{2} = R^{2} \).

Solution: \boxed{\frac{16}{3} R^3}


The figure shows a semicircle, where \( B \) is a point on its diameter \( AC \) such that \( AB = 6 \) and \( BC = 12 \). The line perpendicular to this diameter meets the semicircle at \( D \). The semicircles with diameters \( AB \) and \( BC \), drawn as shown, meet \( AD \) and \( CD \) at \( E \) and \( F \), respectively. Find the distance between \( E \) and \( F \).


Four people, A, B, C, and D, are playing a table tennis tournament (with no draws). Each person plays a match against every other person. After all matches, the results are as follows: A won 3 matches, B won 1 match, and D did not win any match. How many matches did C win?


Xiaoming sequentially adds even numbers $2, 4, 6, \cdots$ up to a certain number. However, he accidentally omitted adding one even number and obtained an incorrect total of 2014. What is the omitted even number?


\( AD \) and \( BC \) are both perpendicular to \( AB \), and \( CD \) is perpendicular to \( AC \). If \( AB = 4 \) and \( BC = 3 \), find \( CD \).


The right triangles $\triangle M D C$ and $\triangle A D K$ share a common right angle $\angle D$. Point $K$ is on $C D$ and divides it in the ratio $2: 3$ counting from point $C$. Point $M$ is the midpoint of side $A D$. Find the sum of $\angle A K D$ and $\angle M C D$, if $A D: C D=2: 5$.


I5.1 In the figure, find \( a \).

I5.2 If the lines \( a x + b y = 1 \) and \( 10 x - 34 y = 3 \) are perpendicular to each other, find \( b \).

I5.3 If the \( b^{\text {th }} \) day of May in a year is Friday and the \( c^{\text {th }} \) day of May in the same year is Tuesday, where \( 16 < c < 24 \), find \( c \).

I5.4 \( c \) is the \( d^{\text {th }} \) prime number. Find \( d \).

Solution: \boxed{d = 9}


In $\triangle ABC$, $\angle C = 90^\circ$, $\angle B = 30^\circ$, and $AC = 1$. Point $M$ is the midpoint of $AB$. The triangle $\triangle ACM$ is folded along $CM$ so that the distance between points $A$ and $B$ is $\sqrt{2}$. Find the distance from point $A$ to the plane $BCM$.


For which real values of $x$ does the function $f(x)=x^{2}-x+1+\sqrt{2 x^{4}-18 x^{2}+12 x+68}$ have a minimum value? What is the minimum value?


Solution: \boxed{x = \frac{4}{3}}


Given the set \( S = \{1, 2, 3, \ldots, 2000, 2001\} \), if a subset \( T \) of \( S \) has the property that for any three elements \( x, y, z \) in \( T \), \( x + y \neq z \), what is the maximum number of elements that \( T \) can have?


There is a target on the wall consisting of five zones: a central circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the number of points awarded for hitting each zone is inversely proportional to the probability of hitting that zone, and the bullseye is worth 315 points. How many points is hitting the blue (penultimate) zone worth?


Given a natural number \( x = 9^n - 1 \), where \( n \) is an odd natural number. It is known that \( x \) has exactly three distinct prime factors, one of which is 61. Find \( x \).


A running competition on an unpredictable distance is conducted as follows. On a round track, two points \( A \) and \( B \) are chosen randomly (with the help of a spinning arrow), after which the athletes run from \( A \) to \( B \) along the shorter arc. A spectator bought a ticket to the stadium and wants the athletes to run past their seat (so they can take a good photograph). What is the probability that this will happen?


Given an arithmetic sequence \(a_{1}, a_{2}, \ldots, a_{n}\) with a non-zero common difference, and the terms \(a_{3}, a_{4}, a_{7}, a_{n}\) form a geometric sequence, determine the number of terms in the sequence.


In the following two equations, the same Chinese character represents the same digit, and different Chinese characters represent different digits:
数字花园 + 探秘 = 2015,
探秘 + 1 + 2 + 3 + ... + 10 = 花园
So the four-digit 数字花园 = ______


The weight of a caught fish follows a normal distribution with parameters $a = 375$ g and $\sigma = 25$ g. Find the probability that the weight of one fish will be: a) between 300 and 425 g; b) not more than 450 g; c) more than 300 g.


A circle is circumscribed around a square with side length \( a \). Another square is inscribed in one of the resulting segments. Determine the area of this inscribed square.


Elisa has 24 science books and also books on math and literature. If Elisa had one more math book, then $\frac{1}{9}$ of her books would be math books and one-fourth would be literature books. If Elisa has fewer than 100 books, how many math books does she have?


In the regular tetrahedron \(ABCD\), points \(E\) and \(F\) are on edges \(AB\) and \(AC\) respectively, such that \(BE = 3\) and \(EF = 4\), and \(EF\) is parallel to face \(BCD\). What is the area of \(\triangle DEF\)?


1. If the circumcenter of triangle \( \triangle ABO \) is on the ellipse, find the value of the real number \( p \).
2. If the circumcircle of triangle \( \triangle ABO \) passes through the point \( N\left(0, \frac{13}{2}\right) \), find the value of the real number \( p \).

Ellipse \( C_1: \frac{x^{2}}{4} + y^{2} = 1 \) intersects the parabola \( C_2: x^{2} = 2py \) (where \( p > 0 \)) at points \( A \) and \( B \). \( O \) is the origin.


You are given an \( m \times n \) chocolate bar divided into \( 1 \times 1 \) squares. You can break a piece of chocolate by splitting it into two pieces along a straight line that does not cut through any of the \( 1 \times 1 \) squares. What is the minimum number of times you have to break the bar in order to separate all the \( 1 \times 1 \) squares?

Solution: \boxed{m \times n - 1}


One million bucks (i.e., one million male deer) are in different cells of a \( 1000 \times 1000 \) grid. The left and right edges of the grid are then glued together, and the top and bottom edges of the grid are glued together, so that the grid forms a doughnut-shaped torus. Furthermore, some of the bucks are honest bucks, who always tell the truth, and the remaining bucks are dishonest bucks, who never tell the truth. Each of the million bucks claims that "at most one of my neighboring bucks is an honest buck." A pair of neighboring bucks is said to be a buckaroo pair if exactly one of them is an honest buck. What is the minimum possible number of buckaroo pairs in the grid?

Note: Two bucks are considered to be neighboring if their cells \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) satisfy either: \(x_{1} = x_{2}\) and \(y_{1} - y_{2} \equiv \pm 1 \pmod{1000}\), or \(x_{1} - x_{2} \equiv \pm 1 \pmod{1000}\) and \(y_{1} = y_{2}\).


It is known that \( f(x) \) is a continuous monotonically increasing function. It is also known that \( f(0) = 0 \) and \( f(1) = 1 \). Find the area of the figure bounded by the graphs of the functions \( f(x / 4) \), \( 4f(x) \), and the line \( x + y = 5 \).


A smooth sphere with a radius of 1 cm is dipped in red paint and released between two absolutely smooth concentric spheres with radii of 4 cm and 6 cm respectively (this sphere is outside the smaller sphere but inside the larger one). Upon contact with both spheres, the sphere leaves a red mark. During its movement, the sphere followed a closed path, resulting in a red-bordered area on the smaller sphere with an area of 37 square cm. Find the area of the region bordered by the red contour on the larger sphere. Give your answer in square centimeters, rounded to the nearest hundredth if necessary.


The equation \(x^3 - 9x^2 + 8x + 2 = 0\) has 3 real roots \(p, q, r\). Find the value of \(\frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2}\).


The base of a pyramid is a triangle with sides \(3 \ \mathrm{cm}\), \(p \ \mathrm{cm}\), and \(5 \ \mathrm{cm}\). If the height and volume of the pyramid are \(q \ \mathrm{cm}\) and \(12 \ \mathrm{cm}^{3}\) respectively, find \(q\).


Consider a 4x4 square grid of 16 cells. In the top right cell, I place a minus sign (-), and in all other cells I place a plus sign (+). I repeatedly perform an operation that consists of changing all the signs in a single row, a single column, or a single diagonal. Can we, after a certain number of operations, have only plus signs (+) in the grid?

Solution: \boxed{\text{No}}


The largest divisor of a natural number \( N \), smaller than \( N \), was added to \( N \), producing a power of ten. Find all such \( N \).


In a computer program written in Turbo Pascal, a function $\operatorname{Random}(x)$ is used, generating random integers from 1 to $x$. What is the probability that a number divisible by 5 will appear when this function is executed if $x=100$?


In the Cartesian coordinate system, two points $A(0, a)$ and $B(0, b)$ are given on the positive $y$-axis, where $a > b > 0$. Point $C$ is on the positive $x$-axis and is such that $\angle ACB$ is maximized. What are the coordinates of point $C$?

Solution: \boxed{(\sqrt{ab}, 0)}


Given a right-angled triangle \(15 ABC\) with \(\angle BAC = 90^\circ\), squares \(ABDE\) and \(BCFG\) are constructed on sides \(AB\) and \(BC\) respectively. The area of square \(ABDE\) is \(8 \ \text{cm}^2\) and the area of square \(BCFG\) is \(26 \ \text{cm}^2\). Find the area of triangle \(DBG\) in \(\text{cm}^2\).


As shown in the figure, $M$ and $N$ are points that divide the diagonals $AC$ and $CE$ of regular hexagon $ABCDEF$ internally, with $AM : AC = CN : CE = r$. If points $B$, $M$, and $N$ are collinear, find the value of $r$.


A $7 \times 7$ chessboard has 2 squares painted yellow, and the remaining squares painted green. If one coloring can be obtained from another through rotation in the plane of the chessboard, then the two colorings are considered the same. How many different colorings are possible?


Kola is twice as old as Ola was when Kola was as old as Ola is now. When Ola is as old as Kola is now, their combined age will be 36 years. How old is Kola now?


Petra has three different dictionaries and two different novels on a shelf. How many ways are there to arrange the books if she wants to keep the dictionaries together and the novels together?


Xiaoming saw a tractor pulling a rope slowly on the road and decided to measure the length of the rope. If Xiaoming walks in the direction the tractor is moving, it takes him a total of 140 steps to walk from one end of the rope to the other. If Xiaoming walks in the opposite direction to the tractor, it takes him 20 steps to walk from one end of the rope to the other. The speeds of both the tractor and Xiaoming remain constant, and Xiaoming covers 1 meter with each step. What is the length of the rope in meters?


A sequence consists of the digits \(122333444455555 \ldots\) such that each positive integer \(n\) is repeated \(n\) times, in increasing order. Find the sum of the 4501st and 4052nd digits of this sequence.


Find the probability that a randomly chosen five-digit natural number with non-repeating digits, composed of the digits 1, 2, 3, 4, 5, 6, 7, 8, is divisible by 8 without a remainder.


Calculate the limit of the function:

\[
\lim _{x \rightarrow 0}\left(2-e^{\sin x}\right)^{\operatorname{ctg} \pi x}
\]


During the summer vacation, Xiao Ming went to an amusement park and met four classmates: A, B, C, and D. Xiao Ming and the four classmates all shook hands. A shook hands with 3 people, B shook hands with 2 people, and C shook hands with 1 person. With how many people did D shake hands?


Let line \( l: y = kx + m \) (where \( k \) and \( m \) are integers) intersect the ellipse \( \frac{x^2}{16} + \frac{y^2}{12} = 1 \) at two distinct points \( A \) and \( B \), and intersect the hyperbola \( \frac{x^2}{4} - \frac{y^2}{12} = 1 \) at two distinct points \( C \) and \( D \). The vector sum \( \overrightarrow{AC} + \overrightarrow{BD} = \overrightarrow{0} \). The total number of such lines is _____.


In an apartment building, each entrance has the same number of floors, and each floor has the same number of apartments. The number of floors in the building is greater than the number of apartments on each floor, the number of apartments on each floor is greater than the number of entrances, and the number of entrances is more than one. How many floors are in the building, if there are a total of 105 apartments?


Given a trapezoid \(ABCD\) and a point \(M\) on the side \(AB\) such that \(DM \perp AB\). It is found that \(MC = CD\). Find the length of the upper base \(BC\), if \(AD = d\).


Given the function \( f(x) = a + x - b^x \) has a zero \( x_0 \in (n, n+1) \) (\(n \in \mathbf{Z}\)), where the constants \( a \) and \( b \) satisfy the conditions \( 2019^a = 2020 \) and \( 2020^b = 2019 \). Determine the value of \( n \).


Given the ellipse \( C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \( a > b > 0 \), let \( F \) be the right focal point of the ellipse \( C \). A line \( l \) is drawn through the intersection of the right directrix \( x = 2a \) and the \( x \)-axis, and intersects the ellipse \( C \) at points \( A \) and \( B \). The arithmetic mean of \( \frac{1}{|AF|} \) and \( \frac{1}{|BF|} \) is \( \frac{1}{24} \). Find the maximum area of triangle \( \triangle ABF \).

Solution: \boxed{192 \sqrt{3}}


It is known that $\sin \alpha+\sin \beta=2 \sin (\alpha+\beta)$ and $\alpha+\beta \neq 2 \pi n (n \in \mathbb{Z})$. Find $\operatorname{tg} \frac{\alpha}{2} \operatorname{tg} \frac{\beta}{2}$.


Color each vertex of a quadrangular pyramid so that the endpoints of each edge are of different colors. If only 4 colors are available, find the total number of different coloring methods.


There are people with masses of 150, 60, 70, 71, 72, 100, 101, 102, and 103 kg standing in front of an elevator. The elevator has a load capacity of 200 kg. What is the minimum number of trips required for all the people to get upstairs?


In triangle \(ABC\), angle \(C\) is a right angle, and \(AC: AB = 4: 5\). A circle with its center on leg \(AC\) is tangent to the hypotenuse \(AB\) and intersects leg \(BC\) at point \(P\), such that \(BP: PC = 2: 3\). Find the ratio of the radius of the circle to leg \(BC\).


Let \( f: \mathbf{R} \rightarrow \mathbf{R} \) satisfy the functional equation
$$
f\left(x^{3}\right)+f\left(y^{3}\right)=(x+y) f\left(x^{2}\right)+f\left(y^{2}\right)-f(x y).
$$

Find the explicit form of the function \( f(x) \).
(2012, Albania Mathematical Olympiad)

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


There are mittens in a bag: right and left. A total of 12 pairs: 10 red and 2 blue. How many mittens do you need to pull out to surely get a pair of mittens of the same color?


The sequence \( a_{0}, a_{1}, \dots \) is defined as follows:

\[ a_{0} = 1995, \]
\[ a_{n} = n a_{n-1} + 1, \quad \text{for } n \geq 1. \]

What is the remainder when \( a_{2000} \) is divided by 9?


Find the greatest integer value of \(a\) for which the equation

\[
(x-a)(x-7) + 3 = 0
\]

has at least one integer root.


Find the number of natural numbers not exceeding 2022 and not belonging to either the arithmetic progression \(1, 3, 5, \ldots\) or the arithmetic progression \(1, 4, 7, \ldots\).


A novice economist-cryptographer received a cryptogram from the ruler, which contained the next secret decree about introducing a per unit tax in a certain market. The cryptogram specified the amount of tax revenue that needed to be collected and emphasized that it would be impossible to collect a higher amount of tax revenue in this market. Unfortunately, the economist-cryptographer deciphered the cryptogram incorrectly — the digits in the tax revenue amount were determined in the wrong order. Based on these erroneous data, a decision was made to introduce a per unit tax on the consumer at a rate of 30 monetary units per unit of goods. The market supply is given by $Qs=6P-312$, and the market demand is linear. Additionally, it is known that when the price changes by one unit, the change in demand is 1.5 times less than the change in supply. After the tax was introduced, the consumer price rose to 118 monetary units.

1) Restore the market demand function.
2) Determine the amount of tax revenue collected at the chosen rate.
3) Determine the rate of the per unit tax that would achieve the ruler's decree.
4) What is the amount of tax revenue the ruler indicated to collect?

Solution: \boxed{T_{\max} = 8640}


The sum of the non-negative numbers \(a\), \(b\), and \(c\) is equal to 3. Find the maximum value of the expression \(ab + bc + 2ca\).


Given complex numbers \( z_{1}, z_{2}, z_{3} \) such that \( \left|z_{1}\right| \leq 1 \), \( \left|z_{2}\right| \leq 1 \), and \( \left|2 z_{3}-\left(z_{1}+z_{2}\right)\right| \leq \left|z_{1}-z_{2}\right| \). What is the maximum value of \( \left|z_{3}\right| \)?


What is the maximum number of non-empty subsets that can be chosen from a set of 100 elements such that any two chosen subsets are either disjoint or one contains the other?


Let \( N \) denote the number of subsets of \(\{1,2,3, \ldots, 100\}\) that contain more prime numbers than multiples of 4. Compute the largest integer \( k \) such that \( 2^{k} \) divides \( N \).


Let \(a\), \(b\), and \(c\) be three positive real numbers such that \(abc \geq 1\). Show that:

\[
\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c} \leq 1
\]

Determine the case of equality.

Solution: \boxed{a = b = c = 1}


Samson writes down the number 123456789 on a piece of paper. He can insert multiplication signs between any two adjacent digits, any number of times at different places, or none at all. By reading the digits between the multiplication signs as individual numbers, he creates an expression made up of the products of these numbers. For example, 1234$\cdot$56$\cdot$789. What is the maximum possible value of the resulting number?


Two circles have centers that are \( d \) units apart, and each has a diameter \( \sqrt{d} \). For any \( d \), let \( A(d) \) be the area of the smallest circle that contains both of these circles. Find \( \lim _{d \rightarrow \infty} \frac{A(d)}{d^{2}} \).


Five friends - Kristina, Nadya, Marina, Liza, and Galya - gather in the park every day after buying ice cream from the shop around the corner. One day, they had a conversation.

Kristina: There were five people in front of me.

Marina: I was the first in line!

Liza: There was no one after me.

Nadya: I was standing next to Marina.

Galya: There was only one person after me.

The girls are friends, so they do not lie to each other. How many people were between Kristina and Nadya?


There are 60 people sitting at a large round table; each person is either a knight or a liar. Each person made the statement: "Of the five people sitting consecutively to my right, at least two are liars." How many knights can be sitting at this table?


a) In how many ways can 9 people arrange themselves I) on a bench II) around a circular table?
b) In how many ways can 5 men and 4 women arrange themselves on a bench such that I) no two people of the same gender sit next to each other? II) the men and women sit in separate groups (only 1 man and 1 woman sit next to each other)?


There are three batches of parts, each containing 20 parts. The number of standard parts in the first, second, and third batches is 20, 15, and 10 respectively. A part is randomly selected from one of these batches and is found to be standard. After returning the part to the batch, another part is randomly selected from the same batch and is also found to be standard. Find the probability that the parts were selected from the third batch.


a) The line segments meeting at the vertex form a number of angles, one between each adjacent pair of segments. Suppose we color half the angles white and half the angles black such that no two angles next to each other are the same color. Show that the sum of the black angles is $180^{\circ}$.

b) Show that the number of mountain folds and the number of valley folds meeting at the vertex differ by exactly 2.


Given \( n+1 \) distinct values of \( x \) such that the polynomials
$$
\begin{array}{l}
f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} \\
g(x)=b_{n} x^{n}+b_{n-1} x^{n-1}+\cdots+b_{1} x+b_{0}
\end{array}
$$
have equal values, then \( f(x) = g(x) \).

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


On a circle, 15 white points and 1 black point are marked. Consider all possible (convex) polygons with their vertices at these points.

We will classify them into two types:

- Type 1: those with only white vertices.
- Type 2: those with the black point as one of the vertices.

Are there more polygons of Type 1 or Type 2? How many more are there?


In $\triangle ABC$, $AB > AC$, $\angle BAC = 45^\circ$. Point $E$ is the intersection of the external angle bisector of $\angle BAC$ with the circumcircle of $\triangle ABC$. Point $F$ is on $AB$ such that $EF \perp AB$. Given $AF = 1$ and $BF = 5$, find the area of $\triangle ABC$.

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


In an acute triangle \( \mathrm{ABC} \), the distance between the centers of the circumcircles of triangles \( C_2HA_1 \) and \( A_1HA_2 \) is equal to \( AC \).

Solution: \boxed{AC}


The side of an equilateral triangle is $a$. Find the radius of the escribed circle.


In triangle $ABC$, the median $CM$ and the angle bisector $BL$ were drawn. Then, all segments and points except for the points $A(2, 8)$, $M(4, 11)$, and $L(6, 6)$ were erased from the diagram. What are the coordinates of point $C$?

Solution: \boxed{(14, 2)}


Consider all 7-digit numbers formed by all possible permutations of the digits in the number 1234567. How many of these numbers leave a remainder of 5 when divided by 7? The answer is \(6!\).


In triangle ABC, AC=1, AB=2, and O is the point of intersection of the angle bisectors. A segment passing through point O parallel to side BC intersects sides AC and AB at points K and M, respectively. Find the perimeter of triangle AKM.


Determine the largest positive integer $N$ such that there exists a $6 \times N$ table $T$ that satisfies:
(1) Each column is a permutation of $1, 2, \cdots, 6$.
(2) For any two columns $i \neq j$, there exists a row $r (r \in \{1,2, \cdots, 6\})$ such that $t_{ri}=t_{rj}$.
(3) For any two columns $i \neq j$, there exists a row $s (s \in \{1,2, \cdots, 6\})$ such that $t_{si} \neq t_{sj}$.


A store has two types of toys, Big White and Little Yellow, with a total of 60 toys. It is known that the price ratio of Big White to Little Yellow is 6:5 (both prices are in integer yuan). Selling all of them results in a total of 2016 yuan. How many Big Whites are there?


A function \( f \) satisfies the following conditions for all nonnegative integers \( x \) and \( y \):

- \( f(0, x) = f(x, 0) = x \)
- If \( x \geq y \geq 0 \), \( f(x, y) = f(x - y, y) + 1 \)
- If \( y \geq x \geq 0 \), \( f(x, y) = f(x, y - x) + 1 \)

Find the maximum value of \( f \) over \( 0 \leq x, y \leq 100 \).


A positive integer is said to be a "palindrome" if it reads the same from left to right as from right to left. For example, 2002 is a palindrome. Find the sum of all 4-digit palindromes.


Find the largest interval over which \( f(x) = \sqrt{x - 1} + \sqrt{x + 24 - 10\sqrt{x - 1}} \) is real and constant.

Solution: \boxed{[1, 26]}


Calculate the limit of the function:

$$
\lim _{x \rightarrow 1}\left(1+e^{x}\right)^{\frac{\sin \pi x}{1-x}}
$$

Solution: \boxed{(1 + e)^\pi}


Given quadratic polynomials \( f_{1}(x), f_{2}(x), \ldots, f_{100}(x) \) with identical coefficients for \( x^{2} \) and \( x \), but differing constant terms; each polynomial has two roots. For each polynomial \( f_{i}(x) \), one root is chosen and denoted by \( x_{i} \). What values can the sum \( f_{2}\left(x_{1}\right) + f_{3}\left(x_{2}\right) + \ldots + f_{100}\left(x_{99}\right) + f_{1}\left(x_{100}\right) \) take?


Let $R$ be the set of all real numbers. Find all functions $f: R \rightarrow R$ such that for all $x, y \in R$, the following equation holds:
$$
f\left(x^{2}+f(y)\right)=y+(f(x))^{2}.
$$

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


Find the remainder when \( 30! - 1 \) is divided by 930.


There are 36 criminal gangs operating in Chicago, some of which are hostile to each other. Each gangster belongs to several gangs, and any two gangsters are members of different sets of gangs. It is known that no gangster belongs to two gangs that are hostile to each other. Furthermore, if a gangster does not belong to a particular gang, that gang is hostile to one of the gangs the gangster is a member of. What is the maximum number of gangsters that can be in Chicago?


An ordered pair of non-negative integers $(m, n)$ is called "simple" if, when performing the addition $m+n$, no carrying is required. The sum $m+n$ is called the sum of the ordered pair $(m, n)$. Find the number of "simple" ordered pairs of non-negative integers whose sum is 1492.


Given that $\alpha, \beta, \gamma$ are all acute angles and $\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1$, find the minimum value of $\tan \alpha \cdot \tan \beta \cdot \tan \gamma$.


An archipelago consists of \( N \geq 7 \) islands. Any two islands are connected by at most one bridge. It is known that there are no more than 5 bridges leading from each island, and among any 7 islands, there are always two islands connected by a bridge. What is the maximum possible value of \( N \)?


Let the lengths of the three sides of a triangle be integers \( l, m, n \) 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.


The graphs of the functions \( y = ax^2 \), \( y = bx \), and \( y = c \) intersect at a point located above the x-axis. Determine how many roots the equation \( ax^2 + bx + c = 0 \) can have.


Given some triangles with side lengths \(a \,\text{cm}, 2 \,\text{cm}\) and \(b \,\text{cm}\), where \(a\) and \(b\) are integers and \(a \leq 2 \leq b\). If there are \(q\) non-congruent classes of triangles satisfying the above conditions, find the value of \(q\).


Solve the equation for real numbers:

$$
\frac{1}{\log _{\frac{1}{2}} x}+\frac{1}{\log _{\frac{2}{3}} x}+\cdots+\frac{1}{\log _{\frac{9}{10}} x}=1
$$


Let \(\log_{14} 16\) be equal to \(a\). Express \(\log_{8} 14\) in terms of \(a\).


The edges of \( K_{2017} \) are each labeled with 1, 2, or 3 such that any triangle has a sum of labels of at least 5. Determine the minimum possible average of all \( \binom{2017}{2} \) labels.

Solution: \boxed{2 - \frac{1}{2017}}


Given point \( A \) is a point inside a unit circle centered at \( O \), satisfying \( |\overrightarrow{OA}| = \frac{1}{2} \). Points \( B \) and \( C \) are any two points on the unit circle \( O \). Determine the range of values for \( \overrightarrow{AC} \cdot \overrightarrow{BC} \).

Solution: \boxed{[-\frac{1}{8}, 3]}


There are four different passwords, $A$, $B$, $C$, and $D$, used by an intelligence station. Each week, one of the passwords is used, and each week it is randomly chosen with equal probability from the three passwords not used in the previous week. Given that the password used in the first week is $A$, find the probability that the password used in the seventh week is also $A$ (expressed as a simplified fraction).


Given a natural number \( n \geq 5 \), find:
1. The minimum number of distinct values produced by \( a_i + a_j \) for \( \{a_1, a_2, \cdots, a_n\} \) where \( 1 \leq i < j \leq n \).
2. Identify all \( n \)-element sets that achieve this minimum value.


Solve the equation \(\frac{15}{x\left(\sqrt[3]{35-8 x^{3}}\right)}=2x+\sqrt[3]{35-8 x^{3}}\). Write the sum of all obtained solutions as the answer.


Show that the sum of the cubes of three consecutive numbers is always divisible by 9.


Let \(a, b, c > 0\) such that \(abc = 1\). Show that
\[ \sum_{cyc} \frac{ab}{ab + a^5 + b^5} \leq 1. \]


Let the set \( A = \left\{\frac{1}{2}, \frac{1}{7}, \frac{1}{11}, \frac{1}{13}, \frac{1}{15}, \frac{1}{32}\right\} \) have non-empty subsets \( A_{1}, A_{2}, \cdots, A_{63} \). Denote the product of all elements in the subset \( A_{i} \) (where \( i=1,2,\cdots,63 \)) as \( p_{i} \) (the product of elements in a single-element subset is the element itself). Then,
\[
p_{1} + p_{2} + \cdots + p_{63} =
\]


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

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


A parallelepiped $A B C D A_1 B_1 C_1 D_1$ is given. Points $M, L$, and $K$ are taken on the edges $A D, A_1 D_1$, and $B_1 C_1$ respectively, such that $B_1 K = \frac{1}{3} A_1 L$ and $A M = \frac{1}{2} A_1 L$. It is known that $K L = 2$. Find the length of the segment by which the plane $K L M$ intersects the parallelogram $A B C D$.


Nine integers from 1 to 5 are written on a board. It is known that seven of them are at least 2, six are greater than 2, three are at least 4, and one is at least 5. Find the sum of all the numbers.


A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of a square is 24, and the perimeter of a small rectangle is 16. What is the perimeter of the large rectangle?

The perimeter of a shape is the sum of its side lengths.


Show that \( x^2 + y^2 + 1 > x(y + 1) \) for all reals \( x, y \). Find the largest \( k \) such that \( x^2 + y^2 + 1 \geq kx(y + 1) \) for all reals \( x, y \). Find the largest \( k \) such that \( m^2 + n^2 + 1 \geq km(n + 1) \) for all integers \( m, n \).


Solve the inequality

$$
8 \cdot \frac{|x+1|-|x-7|}{|2x-3|-|2x-9|} + 3 \cdot \frac{|x+1|+|x-7|}{|2x-3|+|2x-9|} \leq 8
$$

Record the sum of its integer solutions that satisfy the condition $|x| < 120$.


To the eight-digit number 20222023, append one digit to the left and one digit to the right so that the resulting ten-digit number is divisible by 72. Determine all possible solutions.


Determine the value of \(a\) such that in the equation

$$
x^{2}+a x+2 a=0
$$

one root is four times the other root.


Let function \( f(x) = 1 - |1 - 2x| \) and \( g(x) = x^2 - 2x + 1 \) for \( x \in [0,1] \), and define

\[
F(x) =
\begin{cases}
f(x) & \text{if } f(x) \geq g(x), \\
g(x) & \text{if } f(x) < g(x).
\end{cases}
 \]

Determine the number of real roots of the equation \( F(x) \cdot 2^x = 1 \).


The function \( f(x) \) is defined on the set of real numbers, and satisfies the equations \( f(2+x) = f(2-x) \) and \( f(7+x) = f(7-x) \) for all real numbers \( x \). Let \( x = 0 \) be a root of \( f(x) = 0 \). Denote the number of roots of \( f(x) = 0 \) in the interval \(-1000 \leq x \leq 1000 \) by \( N \). Find the minimum value of \( N \).


Given a geometric sequence $\left\{a_{n}\right\}$ with a common ratio $q \in (1,2)$, and that $a_{n}$ is a positive integer for $1 \leq n \leq 6$, find the minimum value of $a_{6}$.


Calculate the limit of the function:

\[ \lim_{x \to 10} \frac{\lg x - 1}{\sqrt{x - 9} - 1} \]

Solution: \boxed{\frac{1}{5 \ln 10}}


From any \( n \)-digit \((n>1)\) number \( a \), we can obtain a \( 2n \)-digit number \( b \) by writing two copies of \( a \) one after the other. If \( \frac{b}{a^{2}} \) is an integer, find the value of this integer.


In a game, \( N \) people are in a room. Each of them simultaneously writes down an integer between 0 and 100 inclusive. A person wins the game if their number is exactly two-thirds of the average of all the numbers written down. There can be multiple winners or no winners in this game. Let \( m \) be the maximum possible number such that it is possible to win the game by writing down \( m \). Find the smallest possible value of \( N \) for which it is possible to win the game by writing down \( m \) in a room of \( N \) people.


It is given that \(a, b, c\) are three real numbers such that the roots of the equation \(x^{2} + 3x - 1 = 0\) also satisfy the equation \(x^{4} + a x^{2} + b x + c = 0\). Find the value of \(a + b + 4c + 100\).


The parabola with equation \( y = ax^2 + bx + c \) passes through the points \( (4, 0) \), \( \left(\frac{t}{3}, 0\right) \), and \( (0, 60) \). What is the value of \( a \)?


Over the course of three years, Marina did not invest the funds in her Individual Investment Account (IIA) into financial instruments and therefore did not receive any income from it. However, she gained the right to an investment tax deduction for depositing her own money into the IIA.

The tax deduction is provided for the amount of money deposited into the IIA during the tax period, but not more than 400,000 rubles in total per year.

Marina is entitled to receive 13% of the amount deposited into the IIA as a refund of the Personal Income Tax (PIT) that she paid on her income.

The PIT amount deducted from Marina's annual salary is equal to \( 30,000 \text{ rubles} \times 12 \text{ months} \times 0.13 = 46,800 \text{ rubles} \).

The tax deduction for the first year is \( 100,000 \text{ rubles} \times 0.13 = 13,000 \text{ rubles} \). This amount does not exceed the PIT deducted from Marina's annual salary.

The tax deduction for the second year is \( 400,000 \text{ rubles} \times 0.13 = 52,000 \text{ rubles} \). This amount exceeds the PIT deducted from Marina's annual salary. Therefore, the tax deduction will be limited to the PIT paid for the second year, or 46,800 rubles.

The tax deduction for the third year is \( 400,000 \text{ rubles} \times 0.13 = 52,000 \text{ rubles} \). This amount also exceeds the PIT deducted from Marina's annual salary. Therefore, the tax deduction will be limited to the PIT paid for the third year, or 46,800 rubles.

The total amount of tax deduction for 3 years is \( 13,000 + 46,800 + 46,800 = 106,600 \text{ rubles} \).

The return on Marina's transactions over 3 years is \( \frac{106,600}{1,000,000} \times 100 \% = 10.66 \% \).

The annual return on Marina's transactions is \( \frac{10.66 \%}{3} = 3.55 \% \).


Let \( a \) and \( b \) be positive integers where \( a \) is less than \( b \). Calculate the sum of the irreducible fractions between \( a \) and \( b \) with denominator 7.

Solution: \boxed{3(b^2 - a^2)}


In a certain kingdom, the workforce consists only of a clan of dwarves and a clan of elves. Historically, in this kingdom, dwarves and elves have always worked separately, and no enterprise has ever allowed itself to hire both at the same time. The aggregate labor supply of dwarves is given by the function \( w_{\text{dwarves}}^{S} = 1 + \frac{L}{3} \), and the aggregate labor supply of elves is \( w_{\text{elves}}^{S} = 3 + L \). The inverse function of aggregate labor demand for dwarves is \( w_{\text{dwarves}}^{D} = 10 - \frac{2L}{3} \), and the inverse function of aggregate labor demand for elves is \( w_{\text{elves}}^{D} = 18 - 2L \). Recently, a newly enthroned king became very concerned that the wage rates of his subjects are different, so he issued a law stating that the wages of elves and dwarves must be equal and that workers should not be discriminated against based on their clan affiliation. The king believes that regulatory intervention in wage rates will negatively impact the kingdom's economy overall and mandates that all his subjects behave in a perfectly competitive manner. By how many times will the wage of the group of workers whose wage was lower before the king's intervention increase if the firms in the kingdom are indifferent to hiring elves or dwarves?

Solution: \boxed{1.25 \text{ times}}


There are 10 cards, each with two different numbers from the set {1, 2, 3, 4, 5}, and no two cards have the exact same pair of numbers. These 10 cards are placed in five boxes labeled 1, 2, 3, 4, and 5. A card labeled with numbers i and j can only be placed in box i or box j. A placement is called "good" if and only if the number of cards in box 1 is greater than the number of cards in each of the other boxes. Find the total number of good placements.


Does there exist a natural number that, when divided by the sum of its digits, gives 2014 as both the quotient and the remainder? If there is more than one such number, write their sum as the answer. If no such number exists, write 0 as the answer.


Suppose that for the positive numbers \( x \), \( y \), and \( z \),

\[ x^2 + xy + y^2 = 9, \quad y^2 + yz + z^2 = 16, \quad z^2 + zx + x^2 = 25. \]

Determine the value of \( xy + yz + zx \).

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


A square is inscribed in an isosceles right triangle such that one vertex of the square is located on the hypotenuse, the opposite vertex coincides with the right angle vertex of the triangle, and the other vertices lie on the legs. Find the side of the square if the leg of the triangle is $a$.


Two squares, $ABCD$ and $BEFG$, with side lengths of $8 \text{ cm}$ and $6 \text{ cm}$ respectively, are placed next to each other as shown in the diagram. Line $DE$ intersects $BG$ at point $P$. What is the area of the shaded region $APEG$ in the diagram?

Solution: \boxed{18 \text{ cm}^2}


There are 31 ones written on a board. Each minute, Karlson erases any two numbers and writes their sum on the board, then eats an amount of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 31 minutes?


Given the equation \(x^{2} - 2ax + 64 =0\):
1. If the equation has real roots, find the range of the real number \(a\).
2. If \(a\) is a positive integer and the equation has integer roots, find the maximum value of the positive integer \(a\).


Calculate the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.

$$
y=2+\operatorname{ch} x, 0 \leq x \leq 1
$$


Solution: \boxed{a^4 + b^4 = 2}


Given a positive integer \( n \), let \( P(x) \) and \( Q(x) \) be polynomials with real coefficients of degree no greater than \( n \). They satisfy the equation \( x^{n+1} P(x) + (x+1)^{n+1} Q(x) = 1 \). Find \( Q(x) \), and determine the value of \( Q\left(-\frac{1}{2}\right) \).


In the coordinate plane, a point is called an integer point if both its x-coordinate and y-coordinate are integers. For any natural number \( n \), the point \( O \) (the origin) is connected to the point \( A_n(n, n+3) \). Let \( f(n) \) denote the number of integer points on the line segment \( OA_n \) excluding the endpoints. Find the value of \( f(1) + f(2) + \cdots + f(1990) \).


The bases \( AB \) and \( CD \) of the trapezoid \( ABCD \) are 155 and 13 respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors \( \overrightarrow{AD} \) and \( \overrightarrow{BC} \).


At 1:00 PM, two identical recreational boats set off in opposite directions from a pier on a river. At the same time, a raft also departed from the pier. An hour later, one of the boats turned around and started moving back. The other boat did the same at 3:00 PM. What is the speed of the current if, at the moment the boats met, the raft had drifted 7.5 km from the pier?

Solution: \boxed{2.5 \text{ km/h}}


In a regular polygon with 67 sides, all segments joining two vertices, including the sides of the polygon, are drawn. We choose $n$ of these segments and assign each one a color from 10 possible colors. Find the minimum value of $n$ that guarantees, regardless of which $n$ segments are chosen and how the colors are assigned, that there will always be a vertex of the polygon that belongs to 7 segments of the same color.


Find the smallest positive period of the function \( f(x) = \cos(\sqrt{2} x) + \sin\left(\frac{3}{8} \sqrt{2} x\right) \).

Solution: \boxed{8 \sqrt{2} \pi}


Each cell of a \(50 \times 50\) square contains a number equal to the count of \(1 \times 16\) rectangles (both vertical and horizontal) for which this cell is an endpoint. How many cells contain numbers that are greater than or equal to 3?


As shown in Figure 1.4.23, in the isosceles triangle \( \triangle ABC \), \( AB = AC \) and \( \angle A = 120^\circ \). Point \( D \) is on side \( BC \), and \( BD = 1 \), \( DC = 2 \). Find the length of \( AD \).


Let \( f(x) \) be a function defined on \( \mathbf{R} \). If there exist two distinct real numbers \( x_{1}, x_{2} \in \mathbf{R} \) such that \( f\left(\frac{x_{1}+x_{2}}{2}\right) = \frac{f\left(x_{1}\right) + f\left(x_{2}\right)}{2} \), then the function \( f(x) \) is said to have property \( \mathrm{P} \). Determine which of the following functions does not have property \( \mathrm{P} \):

\[
1. \quad f(x) =
\begin{cases}
\frac{1}{x} & (x \neq 0) \\
0 & (x = 0)
\end{cases}
\]
\[
2. \quad f(x) = x^2
\]
\[
3. \quad f(x) = \left| x^2 - 1 \right|
\]
\[
4. \quad f(x) = x^3
\]

A: 1

B: 2

C: 3

D: 4

Solution: \boxed{ B }


Find \( p \) if each of the numbers \( p \), \( 2p + 1 \), and \( 4p + 1 \) is prime.


Let \( k, \alpha \) and \( 10k - \alpha \) be positive integers. What is the remainder when the following number is divided by 11?

\[
8^{10k + \alpha} + 6^{10k - \alpha} - 7^{10k - \alpha} - 2^{10k + \alpha}
\]


Given 6 different thin rods with lengths $a, b, c, d, e, f$, any three of which can form a triangle. How many distinct tetrahedral edge frameworks can be assembled from the rods, where the frameworks are not equivalent by rotation or reflection?


How many times does 24 divide into 100 factorial (100!)?


Among the positive integers less than 10,000, if we exchange the digit in the highest place with the digit in the lowest place, we obtain a new number that is 1.2 times the original number. What is the sum of all numbers that satisfy this condition?


On an island, there live three tribes: knights, who always tell the truth; liars, who always lie; and tricksters, who sometimes tell the truth and sometimes lie. At a round table sit 100 representatives of these tribes.

Each person at the table said two sentences: 1) "To my left sits a liar"; 2) "To my right sits a trickster". How many knights and liars are at the table if half of those present are tricksters?


As shown in the figure, the square $DEOF$ is inscribed in a quarter circle. If the radius of the circle is 1 centimeter, then the area of the shaded region is $\qquad$ square centimeters. (Use $\pi = 3.14$.)


The product of all natural numbers from 1 to \( n \) is denoted as \( n! \) (read as "n-factorial"). Which number is greater, \( 200! \) or \( 100^{200} \)?


Find the lateral surface area of a regular triangular pyramid if its height is 4 and its slant height is 8.


Xiaoming is riding a bicycle, while Xiaoming's father is walking. They start from locations $A$ and $B$ respectively, moving towards each other. After meeting, Xiaoming continues for another 18 minutes to reach $B$. It is known that Xiaoming's cycling speed is 4 times that of his father's walking speed, and it takes Xiaoming's father a certain number of minutes to walk from the meeting point to $A$. How long does Xiaoming's father need to walk from the meeting point to $A$?

Solution: \boxed{288 \text{ minutes}}


The number 2019 is expressed as a sum of different odd natural numbers. What is the maximum possible number of terms in this sum?


Three volleyballs with a radius of 18 lie on a horizontal floor, each pair touching one another. A tennis ball with a radius of 6 is placed on top of them, touching all three volleyballs. Find the distance from the top of the tennis ball to the floor. (All balls are spherical in shape.)


Given that the polynomial \( f \) satisfies the equation \( f(x^{2} + 1) - f(x^{2} - 1) = 4x^{2} + 6 \), determine the polynomial \( f(x^{2} + 1) - f(x^{2}) \).

Solution: \boxed{2x^2 + 4}


Determine all real values of \(a\) for which there exist five non-negative real numbers \(x_{1}, \ldots, x_{5}\) satisfying the following equations:

\[
\sum_{k=1}^{5} k x_{k} = a
\]

\[
\sum_{k=1}^{5} k^{3} x_{k} = a^{2}
\]

\[
\sum_{k=1}^{5} k^{5} x_{k} = a^{3}
\]

Solution: \boxed{0, 1, 4, 9, 16, 25}


In a math competition, the possible scores for each problem for each participating team are 0 points, 3 points, or 5 points. By the end of the competition, the sum of the total scores of three teams is 32 points. If the total score of any single team can reach 32 points, how many different possible combinations are there for the total scores of these three teams?


Pensioners on one of the planets in Alpha Centauri enjoy painting the cells of $2016 \times 2016$ boards with gold and silver paints. One day, it turned out that for all the boards painted that day, each $3 \times 3$ square contained exactly $A$ gold cells, and each $2 \times 4$ or $4 \times 2$ rectangle contained exactly $Z$ gold cells. For which values of $A$ and $Z$ is this possible?

Solution: \boxed{A = 9, Z = 8}


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?


Anton, Borya, Vasya, and Grisha met. It is known that each of them is either from the tribe of knights (who always tell the truth) or from the tribe of liars (who always lie). Anton said that he and Grisha are from different tribes. Borya and Vasya called each other liars. Grisha claimed that among the four of them, there are at least two knights. How many knights are there actually?


Find the number of four-digit numbers, composed of the digits 1, 2, 3, 4, 5, 6, 7 (each digit can be used no more than once), that are divisible by 15.


A ball thrown vertically upwards has its height above the ground expressed as a quadratic function with respect to its time of motion. Xiaohong throws two balls vertically upwards one after the other, with a 1-second interval between them. Assume the initial height above the ground for both balls is the same, and each reaches the same maximum height 1.1 seconds after being thrown. If the first ball's height matches the second ball's height at $t$ seconds after the first ball is thrown, determine $t = \qquad$ .


Gru and the Minions plan to make money through cryptocurrency mining. They chose Ethereum as one of the most stable and promising currencies. They bought a system unit for 9499 rubles and two graphics cards for 31431 rubles each. The power consumption of the system unit is 120 W, and for each graphics card, it is 125 W. The mining speed for one graphics card is 32 million hashes per second, allowing it to earn 0.00877 Ethereum per day. 1 Ethereum equals 27790.37 rubles. How many days will it take for the team's investment to pay off, considering electricity costs of 5.38 rubles per kWh? (20 points)


Three sportsmen called Primus, Secundus, and Tertius take part in a race every day. Primus wears the number '1' on his shirt, Secundus wears '2', and Tertius wears '3'.

On Saturday, Primus wins, Secundus is second, and Tertius is third. Using their shirt numbers this result is recorded as '123'.

On Sunday, Primus starts the race in the lead with Secundus in second. During Sunday's race:
- Primus and Secundus change places exactly 9 times,
- Secundus and Tertius change places exactly 10 times,
- Primus and Tertius change places exactly 11 times.

How will Sunday's result be recorded?


Two congruent parallelograms have equal sides parallel to each other. In both parallelograms, we mark the vertices, the midpoints of the sides, and the intersection point of the diagonals. We choose one point from the marked points of the first parallelogram and another point from the marked points of the second parallelogram in every possible way, and construct the midpoint of the segment connecting the two chosen points.

How many different midpoints do we get in this way?


The base of the quadrilateral pyramid \( S A B C D \) is a square \( A B C D \), and \( S A \) is the height of the pyramid. Let \( M \) and \( N \) be the midpoints of the edges \( S C \) and \( A D \), respectively. What is the maximum possible area of the triangle \( B S A \) if \( M N = 3 \)?


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)?


Given the sequence \( a_{n} = 1 + n^{3} \) where the sequence is \(\{2, 9, 28, 65, \ldots\} \) and \( \delta_{n} = \operatorname{gcd}(a_{n+1}, a_{n}) \), find the maximum value that \(\delta_{n}\) can take.


Let $\mathbb{R}_{\geq 0}$ denote the set of nonnegative real numbers. Find all functions $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ such that, for all $x, y \in \mathbb{R}_{\geq 0}$,

$$
f\left(\frac{x+f(x)}{2}+y\right)=2 x-f(x)+f(f(y))
$$

and

$$
(f(x)-f(y))(x-y) \geq 0
$$

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


The dividend is six times larger than the divisor, and the divisor is four times larger than the quotient. Find the dividend.


Find the integer solutions of the equation

$$
x^{4}+y^{4}=3 x^{3} y
$$


Suppose the three sides of a triangular field are all integers, and its area is equal to the perimeter. What is the largest possible area of the field?


Let \( N_{0} \) be the set of non-negative integers, and \( f: N_{0} \rightarrow N_{0} \) be a function such that \( f(0)=0 \) and for any \( n \in N_{0} \), \( [f(2n+1)]^{2} - [f(2n)]^{2} = 6f(n) + 1 \) and \( f(2n) > f(n) \). Determine how many elements in \( f(N_{0}) \) are less than 2004.


In triangle \(ABC\), angle \(B\) is \(120^\circ\) and \(AB = 2BC\). The perpendicular bisector of side \(AB\) intersects \(AC\) at point \(D\). Find the ratio \(AD : DC\).


Write the five-digit number "12345" 403 times to form a 2015-digit number "$123451234512345 \cdots$". Starting from the left, sequentially delete all digits located in the odd positions of this number to form a new number. Repeat this process of deleting all digits in the odd positions of the newly formed number until only one digit remains. What is the last remaining digit?


Petya ascended a moving escalator, counting 75 steps, and then descended the same escalator (i.e., moving against the direction of the escalator), counting 150 steps. While descending, Petya walked three times faster than when ascending. How many steps does the escalator have when it is stopped?


Calculate: $(10 \times 19 \times 20 \times 53 \times 100 + 601) \div 13 = \ ?$


Anton wrote three natural numbers \(a\), \(b\), and \(c\) on the board. Ira drew three rectangles with dimensions \(a \times b\), \(a \times c\), and \(b \times c\) on the board. It turned out that the difference in areas of some pair of rectangles is 1, and the difference in areas of another pair of rectangles is 49. What can \(a + b + c\) equal? List all possible options.


Given the polynomial \( P(x) = x^5 - x^2 + 1 \) with roots \( r_1, r_2, \ldots, r_5 \), and another polynomial \( Q(x) = x^2 + 1 \), find \( Q(r_1) Q(r_2) Q(r_3) Q(r_4) Q(r_5) \).


Petya glued a paper cube and wrote numbers from 1 to 6 on its faces such that the sums of the numbers on any two opposite faces are equal. Vasya wants to cut this cube to obtain the net shown in the figure. He tries to make the sums of the numbers in the horizontal and vertical directions on this net as close as possible. What is the smallest positive difference he can get, regardless of how Petya arranged the numbers?


On 2016 cards, the numbers from 1 to 2016 were written (each number exactly once). Then \( k \) cards were taken. What is the smallest \( k \) such that among them there will be two cards with numbers whose square root difference is less than 1?


Given that $F$ is the right focus of the ellipse $\frac{x^{2}}{1+a^{2}} + y^{2} = 1$ for $a > 0$, $M(m, 0)$ and $N(0, n)$ are points on the $x$-axis and $y$-axis respectively, and they satisfy $\overrightarrow{M N} \cdot \overrightarrow{N F} = 0$. Let point $P$ satisfy $\overrightarrow{O M} = 2 \overrightarrow{O N} + \overrightarrow{P O}$.

(1) Find the locus $C$ of point $P$.

(2) For any line passing through point $F$ that intersects the locus $C$ at points $A$ and $B$, let lines $O A$ and $O B$ intersect the line $x = -a$ at points $S$ and $T$ respectively (with $O$ being the origin). Determine whether $\overrightarrow{F S} \cdot \overrightarrow{F T}$ is a constant value. If it is, find this constant; if not, provide an explanation.


The purchase of a book cost 1 ruble plus an additional one-third of the price of the book. What is the price of the book?

Solution: \boxed{1.5 \text{ rubles}}


Given \( n \) points on a plane where any three points form the vertices of a right triangle, determine the maximum value of \( n \).


A certain conference is attended by \(12k\) people (\(k\) is a positive integer), and each person knows exactly \(3k+6\) other attendees. Assume that for any two people, the number of people who know both of them is the same. Determine the number of people attending this conference.
(IMC 36th Preliminary Problem)


Find the largest four-digit number in which all the digits are different and which is divisible by any of its digits (don't forget to explain why it is the largest).


The pages in a book are numbered as follows: the first sheet contains two pages (numbered 1 and 2), the second sheet contains the next two pages (numbered 3 and 4), and so on. A mischievous boy named Petya tore out several consecutive sheets: the first torn-out page has the number 185, and the number of the last torn-out page consists of the same digits but in a different order. How many sheets did Petya tear out?


We assembled a large cube from \( k^{3} \) congruent small cubes. What are those values of \( k \) for which it is possible to color the small cubes with two colors in such a way that each small cube has exactly two neighbors of the same color? (Two small cubes are considered neighbors if they share a face.)

Solution: \boxed{k \text{ is even}}


In the number $2016^{* * * *} 02 * *$, each of the six asterisks must be replaced with any of the digits $0, 2, 4, 5, 7, 9$ (digits may be repeated) so that the resulting 12-digit number is divisible by 15. How many ways can this be done?


The domain of the function $f$ is the set of real numbers. For each number $a$, it holds that if $a<x<a+100$, then $a \leq f(x) \leq a+100$. Determine the function $f$.

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


What is the smallest positive integer \( n \) which cannot be written in any of the following forms?
- \( n = 1 + 2 + \cdots + k \) for a positive integer \( k \).
- \( n = p^k \) for a prime number \( p \) and integer \( k \).
- \( n = p + 1 \) for a prime number \( p \).


Two dice are thrown. Find the probability that the sum of the points on the rolled faces is even and at least one of the dice shows a six.


Oleg writes a real number above each column of a blank $50 \times 50$ grid (outside the grid) and a real number to the left of each row (outside the grid). It is known that all 100 real numbers he writes are distinct, and there are exactly 50 rational numbers and 50 irrational numbers among them. Then, Oleg writes the sum of the number to the left of the row and the number above the column in each cell of the grid (creating an addition table). What is the maximum number of rational numbers that can appear in this grid?


The worm consists of a white head and several segments, as shown in the figure.

When the worm is born, it has a head and one white segment. Each day, a new segment is added to the worm in one of the following ways:

- Either a white segment splits into a white and a gray segment.
- Or a gray segment splits into a gray and a white segment.

(On the fourth day, the worm reaches maturity and stops growing - its body consists of a head and four segments.)

How many different color variants of mature worms of this species can exist?

Hint: What could a two-day-old worm look like?


Given that \( a \) and \( b \) are integers, and \(\frac{127}{a}-\frac{16}{b}=1\). What is the maximum value of \( b \)?


Let \( n \geq 2 \) be a fixed integer. Find the least constant \( C \) such that the inequality
\[ \sum_{i<j} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq C\left(\sum_{i} x_{i}\right)^{4} \]
holds for every \( x_{1}, \ldots, x_{n} \geq 0 \) (the sum on the left consists of \(\binom{n}{2}\) summands). For this constant \( \bar{C} \), characterize the instances of equality.


The sum
$$
\frac{1}{1 \times 2 \times 3}+\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\cdots+\frac{1}{100 \times 101 \times 102}
$$
can be expressed as $\frac{a}{b}$, a fraction in its simplest form. Find $a+b$.


On a $6 \times 6$ grid, place Go pieces in all squares, with one piece in each square. The number of white pieces in each row must be different from one another, and the number of white pieces in each column must be the same. How many black Go pieces are there in total on the $6 \times 6$ grid?
A. 18
B. 14
C. 12
D. 10


Let \( n \geqslant 4, \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} ; \beta_{1}, \beta_{2}, \cdots, \beta_{n} \) be two sets of real numbers, satisfying \( \sum_{j=1}^{n} \alpha_{j}^{2}<1 \) and \( \sum_{j=1}^{n} \beta_{j}^{2}<1 \). Define
\[ A^{2}=1-\sum_{j=1}^{n} \alpha_{j}^{2}, \, B^{2}=1-\sum_{j=1}^{n} \beta_{j}^{2}, \, W=\frac{1}{2}\left(1-\sum_{j=1}^{n} \alpha_{j} \beta_{j}\right)^{2} . \]

Find all real numbers \(\lambda\) such that the equation \( x^{n}+\lambda\left(x^{n-1}+\cdots+x^{3}+W x^{2}+A B x+1\right)=0 \) has only real roots.


For which \( n \) can we find a cyclic shift \( a_1, a_2, \ldots, a_n \) of \( 1, 2, 3, \ldots, n \) (i.e., \( i, i+1, i+2, \ldots, n, 1, 2, \ldots, i-1 \) for some \( i \)) and a permutation \( b_1, b_2, \ldots, b_n \) of \( 1, 2, 3, \ldots, n \) such that \( 1 + a_1 + b_1 = 2 + a_2 + b_2 = \ldots = n + a_n + b_n \)?

Solution: \boxed{n \text{ is odd}}


Solution: \boxed{-2 \ln 2}


The non-negative reals \(a, b, c, d\) satisfy \(a^5 + b^5 \leq 1\) and \(c^5 + d^5 \leq 1\). Show that \(a^2c^3 + b^2d^3 \leq 1\).


Place 6 points inside a rectangle of dimensions \( 4 \times 3 \). Show that there are at least two points whose distance is less than or equal to \( \sqrt{5} \).


When I saw Eleonora, I found her very pretty. After a brief trivial conversation, I told her my age and asked how old she was. She answered:

- When you were as old as I am now, you were three times older than me. And when I will be three times older than I am now, together our ages will sum up to exactly a century.

How old is this capricious lady?


Find all integers \( n \) for which \( n^2 + 20n + 11 \) is a perfect square.


Piercarlo chooses \( n \) integers from 1 to 1000 inclusive. None of his integers is prime, and no two of them share a factor greater than 1. What is the greatest possible value of \( n \)?


The picture shows several circles connected by segments. Tanya chooses a natural number \( n \) and places different natural numbers in the circles so that the following property holds for all these numbers:

If numbers \( a \) and \( b \) are not connected by a segment, then the sum \( a^2 + b^2 \) must be coprime with \( n \). If they are connected, then the numbers \( a^2 + b^2 \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?


As shown in the figure, a truck travels from $A$ to $C$ at a speed of 50 km/h, while at the same time, delivery person A and delivery person B set out from $B$ to travel to $A$ and $C$ respectively, with both traveling at the same speed. If A travels 3 km before meeting the truck, and 12 minutes later, the truck and B arrive at $C$ simultaneously, then what is the distance between cities $A$ and $C$ in kilometers?

Solution: \boxed{17.5 \text{ km}}


In the city of liars and knights, there are 366 inhabitants, each born on a different day of a leap year. All inhabitants of the city answered two questions. To the question "Were you born in February?" 100 people answered affirmatively, and to the question "Were you born on the 30th day?" 60 people answered affirmatively. How many knights were born in February?


Gavrila saw a squirrel in the window, sitting on a tree branch directly opposite him at a distance of 3 meters 75 centimeters. He decided to feed the creature and threw a nut horizontally at a speed of 5 m/s directly towards the squirrel. Will the squirrel be able to catch the nut if it can jump with great speed in any direction for a distance of 1 meter 70 centimeters? Assume that the acceleration due to gravity is $g = 10 \mathrm{~m} / \mathrm{s}^{2}$ and neglect air resistance.

Solution: \boxed{\text{No}}


Given the sets
$$
\begin{array}{l}
A=\{2,0,1,7\}, \\
B=\{x \mid x=a b, a, b \in A\} .
\end{array}
$$
determine the number of elements in set $B$.


Mr. Canada chooses a positive real \( a \) uniformly at random from \((0,1]\), chooses a positive real \( b \) uniformly at random from \((0,1]\), and then sets \( c = \frac{a}{a+b} \). What is the probability that \( c \) lies between \(\frac{1}{4}\) and \(\frac{3}{4}\)?


Calculate the value of
$$
I_{m}=\cot ^{2} \frac{\pi}{2 m+1}+\cdots+\cot ^{2} \frac{m \pi}{2 m+1}.
$$


From 24 identical wooden cubes, a "pipe" was glued - a cube $3 \times 3 \times 3$ with the "core" of three cubes removed. Can a diagonal be drawn in each square on the surface of the "pipe" to form a closed path that does not pass through any vertex more than once?

Solution: \boxed{\text{No}}


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


In the given set of numbers, one number is equal to the average of all the numbers, the largest number is 7 greater than the average, the smallest number is 7 less than the average, and most numbers in the set have below-average values.

What is the smallest number of numbers that can be in the set?


Given a triangle \( \triangle ABC \) with area 1, and side length \( a \) opposite to angle \( A \), find the minimum value of \( a^2 + \frac{1}{\sin A} \).


On a line, the points $A, B, C, D$ are marked in that order. Point $M$ is the midpoint of segment $AC$, and point $N$ is the midpoint of segment $BD$. Find the length of segment $MN$, given that $AD=68$ and $BC=26$.


A covered rectangular football field with a length of 90 m and a width of 60 m is being designed to be illuminated by four floodlights, each hanging from some point on the ceiling. Each floodlight illuminates a circle, with a radius equal to the height at which the floodlight is hanging. Determine the minimally possible height of the ceiling, such that the following conditions are met: every point on the football field is illuminated by at least one floodlight, and the height of the ceiling must be a multiple of 0.1 m (for example, 19.2 m, 26 m, 31.9 m, etc.).

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


Given positive real numbers \(a_{1}, a_{2}, \cdots, a_{n}\) and non-negative real numbers \(b_{1}, b_{2}, \cdots, b_{n}\) satisfying:
1. \(a_{1} + a_{2} + \cdots + a_{n} + b_{1} + b_{2} + \cdots + b_{n} = n\)
2. \(a_{1} a_{2} \cdots a_{n} + b_{1} b_{2} \cdots b_{n} = \frac{1}{2}\)

Find the maximum value of \(a_{1} a_{2} \cdots a_{n} \left( \frac{b_{1}}{a_{1}} + \frac{b_{2}}{a_{2}} + \cdots + \frac{b_{n}}{a_{n}} \right)\).


Given that \( \tan \left(\frac{\pi}{4}+\alpha\right)=2 \), find the value of \( \frac{1}{2 \sin \alpha \cos \alpha+\cos ^{2} \alpha} \).


We build a $4 \times 4 \times 4$ cube out of sugar cubes. How many different rectangular parallelepipeds can the sugar cubes determine, if the rectangular parallelepipeds differ in at least one sugar cube?


What is the smallest number of distinct integers needed so that among them one can select both a geometric progression and an arithmetic progression of length 5?


Famous skater Tony Hawk rides a skateboard (segment $AB$) in a ramp, which is a semicircle with a diameter $PQ$. Point $M$ is the midpoint of the skateboard, and $C$ is the foot of the perpendicular dropped from point $A$ to the diameter $PQ$. What values can the angle $\angle ACM$ take if the angular measure of arc $AB$ is $24^{\circ}$?


Given that \( p \) is a prime number and \( r \) is the remainder when \( p \) is divided by 210, if \( r \) is a composite number that can be expressed as the sum of two perfect squares, find \( r \).

Solution: \boxed{r = 169}


a) What is the maximum number of queens that can be placed on a chessboard with 64 squares such that no two queens threaten each other?
b) The same question for a chessboard with \( n^2 \) squares.

Solution: \boxed{n}


A semicircle with radius 2021 has diameter \( AB \) and center \( O \). Points \( C \) and \( D \) lie on the semicircle such that \(\angle AOC < \angle AOD = 90^{\circ} \). A circle of radius \( r \) is inscribed in the sector bounded by \( OA \) and \( OC \) and is tangent to the semicircle at \( E \). If \( CD = CE \), compute \(\lfloor r \rfloor\).


Can a circle be circumscribed around the quadrilateral \( A B C D \) if \( \angle A D C=30^{\circ} \), \( A B=3 \), \( B C=4 \), and \( A C=6 \)?

Solution: \boxed{\text{No}}


In the 100-digit number 12345678901234...7890, all digits in odd positions were crossed out; in the resulting 50-digit number, all digits in odd positions were crossed out again, and so on. The crossing out continued as long as there were digits to cross out. Which digit was crossed out last?


Given a right triangular prism \(ABC-A_{1}B_{1}C_{1}\) with the base being a right triangle, \(\angle ACB = 90^{\circ}\), \(AC = 6\), \(BC = CC_{1} = \sqrt{2}\), and \(P\) is a moving point on \(BC_{1}\), find the minimum value of \(CP + PA_{1}\).


Let the function \( f(z) \) (where \( z \) is a complex number) satisfy \( f(f(z)) = (z \bar{z} - z - \bar{z})^{2} \). If \( f(1) = 0 \), then what is \( |f(\mathrm{i}) - 1| \)?


Find the sum of natural numbers from 1 to 3000 inclusive, which have common divisors with the number 3000 that are greater than 1.


A large rectangle is divided into four identical smaller rectangles by slicing parallel to one of its sides. The perimeter of the large rectangle is 18 metres more than the perimeter of each of the smaller rectangles. The area of the large rectangle is 18 square metres more than the area of each of the smaller rectangles. What is the perimeter in metres of the large rectangle?


Let \( x \) and \( y \) be positive numbers whose sum is equal to 2. Find the maximum value of the expression \( x^2 y^2 (x^2 + y^2) \).


On the edge \(AD\) and the diagonal \(A_1C\) of the parallelepiped \(ABCDA_1B_1C_1D_1\), points \(M\) and \(N\) are taken respectively, such that the line \(MN\) is parallel to the plane \(BDC_1\) and \(AM:AD = 1:5\). Find the ratio \(CN:CA_1\).


Points \( M \), \( N \), and \( K \) are located on the lateral edges \( AA_1 \), \( BB_1 \), and \( CC_1 \) of the triangular prism \( ABC A_1 B_1 C_1 \) such that \( \frac{AM}{AA_1} = \frac{2}{3} \), \( \frac{BN}{BB_1} = \frac{3}{5} \), \( \frac{CK}{CC_1} = \frac{4}{7} \). Point \( P \) belongs to the prism. Find the maximum possible volume of the pyramid \( MNKP \), if the volume of the prism is 27.


Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.


Is it possible to place natural numbers from 1 to 42 (each exactly once) in a $6 \times 7$ (6 rows and 7 columns) rectangular table such that the sum of numbers in each $1 \times 2$ vertical rectangle is even?

Solution: \boxed{\text{No}}


In the following addition problem, eight different letters each represent one of the digits from 2 to 9. The same letters represent the same digits, and different letters represent different digits. Find $\overline{\mathrm{NINE}} = \quad$
$$
\begin{array}{r}
O N E \\
T W O \\
+\quad S I X \\
\hline N I N E
\end{array}
$$


All 25 students of the 7th grade "A" class participated in a quiz consisting of three rounds. In each round, each participant scored a certain number of points. It is known that in each round, as well as in the sum of all three rounds, all participants scored different amounts of points.

Kolia from 7th grade "A" finished third in the first round, fourth in the second, and fifth in the third round of the quiz. What is the lowest possible overall position Kolia could achieve among all his classmates based on the total points from all three rounds?


A point $M$ is taken on the side $AB$ of the trapezoid $ABCD$ such that $AM: BM = 2: 3$. A point $N$ is taken on the opposite side $CD$ such that the segment $MN$ divides the trapezoid into parts, one of which has an area three times greater than the other. Find the ratio $CN: DN$ if $BC: AD = 1: 2$.


A number is said to be TOP if it has 5 digits and when the product of the 1st and 5th digits is equal to the sum of the 2nd, 3rd, and 4th digits. For example, 12,338 is TOP because it has 5 digits and $1 \cdot 8 = 2 + 3 + 3$.

a) What is the value of $a$ such that $23,4a8$ is TOP?

b) How many TOP numbers end with 2 and start with 1?

c) How many TOP numbers start with 9?


Given the sequence \(\left\{a_{n}\right\}\) where \(a_{1}=7\) and \(\frac{a_{n+1}}{a_{n}}=a_{n}+2\) for \(n=1,2,3,\cdots\), find the smallest positive integer \(n\) such that \(a_{n}>4^{2018}\).


Given that \( x \) and \( y \) are real numbers such that \( |x| + x + y = 10 \) and \( |y| + x - y = 10 \), if \( P = x + y \), find the value of \( P \).


Find all three-digit numbers \(\overline{abc}\) that satisfy \(\overline{abc} = (a + b + c)^3\).


If the edge length of the cube $A_{1} A_{2} A_{3} A_{4}-B_{1} B_{2} B_{3} B_{4}$ is 1, then the number of elements in the set $\left\{x \mid x=\overrightarrow{A_{1} B_{1}} \cdot \overrightarrow{A_{i} B_{j}}, i \in\{1,2,3,4\}, j \in\{1,2,3,4\}\right\}$ is $\qquad$.


On the island of Friends and Foes, every citizen is either a Friend (who always tells the truth) or a Foe (who always lies). Seven citizens are sitting in a circle. Each declares "I am sitting between two Foes". How many Friends are there in the circle?


In a table containing $A$ columns and 100 rows, natural numbers from 1 to $100 \cdot A$ are written by rows in ascending order, starting from the first row. The number 31 is in the fifth row. In which row is the number 100?


Let the function \( f(x) \) be a differentiable function defined on the interval \( (-\infty, 0) \), with its derivative denoted as \( f'(x) \), and satisfying the inequality \( 2 f(x) + x f'(x) > x^2 \). Then, find the solution set of the inequality:
\[ (x+2017)^{2} f(x+2017) - f(-1) > 0. \]

Solution: \boxed{(-\infty, -2018)}


At a mathematics competition, three problems were given: $A$, $B$, and $C$. There were 25 students who each solved at least one problem. Among the students who did not solve problem $A$, twice as many solved $B$ as solved $C$. One more student solved only problem $A$ than the number of those who also solved problem $A$. Half of the students who solved only one problem did not solve $A$. How many students solved only problem $B$?


For the set \( S = \left\{ \left( a_1, a_2, \cdots, a_5 \right) \mid a_i = 0 \text{ or } 1, i = 1, 2, \cdots, 5 \right\} \), the distance between any two elements \( A = \left( a_1, a_2, \cdots, a_5 \right) \) and \( B = \left( b_1, b_2, \cdots, b_5 \right) \) is defined as:
\[ d(A, B) = \left| a_1 - b_1 \right| + \cdots + \left| a_5 - b_5 \right| \]
Find a subset \( T \) of \( S \) such that the distance between any two elements in \( T \) is greater than 2. What is the maximum number of elements that subset \( T \) can contain? Provide a proof for your conclusion.


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?


Twenty phones are connected by wires in such a way that each wire connects two phones, each pair of phones is connected by at most one wire, and no more than two wires extend from any one phone. The wires need to be colored (each wire entirely with one color) such that the wires extending from each phone are of different colors. What is the minimum number of colors needed for such a coloring?


Leon has cards with digits from 1 to 7. How many ways are there to combine these cards into two three-digit numbers (one card will not be used) so that each of them is divisible by 9?


In a circle, an inscribed hexagon has three consecutive sides each of length 3, and the other three sides each of length 5. A chord of the circle splits the hexagon into two quadrilaterals: one quadrilateral has three sides each of length 3, and the other quadrilateral has three sides each of length 5. If the length of the chord is $\frac{m}{n}$, where $m$ and $n$ are coprime positive integers, find the value of $m+n$.

(A) 309
(B) 349
(C) 369
(D) 389
(E) 409


Find the value of \( a > 1 \) for which the equation \( a^x = \log_a x \) has a unique solution.

Solution: \boxed{a = e^{1/e}}


In square ABCD, diagonals AC and BD meet at point E. Point F is on CD such that ∠CAF = ∠FAD. If AF intersects ED at point G, and EG = 24 cm, find the length of CF.

Solution: \boxed{48 \text{ cm}}


Numbers $1,2,3,4,5,6,7,$ and $8$ are placed at the vertices of a cube such that the sum of any three numbers belonging to any face of the cube is not less than 10. Find the minimum possible sum of four numbers belonging to one face.


Find all pairs of positive integers \(x, y\) which satisfy the equation

\[
2x^2 + 5y^2 = 11(xy - 11).
\]

Solution: \boxed{(14, 27)}


The diagram shows a circle and a trapezium \(ABCD\) in which \(AD\) is parallel to \(BC\) and \(AB = DC\). All four sides of \(ABCD\) are tangents to the circle. The circle has a radius of 4, and the area of \(ABCD\) is 72. What is the length of \(AB\)?


There are 9 cards, each with a digit from 1 to 9 written on it (one digit per card). Three cards are randomly chosen and placed in a sequence. What is the probability that the resulting three-digit number is divisible by 3?


In an ornithological park, there are birds of several species, with a total of 2021 individuals. The birds are seated in a row, and it turns out that between any two birds of the same species, there is an even number of birds. What is the smallest number of bird species that could be?


In an $8 \times 8$ frame with a width of 2 cells (as shown in the figure) there are a total of 48 cells. How many cells are there in a $254 \times 254$ frame with a width of 2 cells?


Let $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$. Let $C$ be a point on $C_{1}$ and $D$ a point on $C_{2}$ such that $(A B)$ bisects $\widehat{C A D}$. Let $E$ be the intersection of $[A C]$ and $C_{2}$, and $F$ the intersection of $[A D]$ and $C_{1}$.

Show that $C E = D F$.

Solution: \boxed{CE = DF}


Given that positive integers \( a, b, c \) (\( a < b < c \)) form a geometric sequence, and
\[ \log_{2016} a + \log_{2016} b + \log_{2016} c = 3, \]
find the maximum value of \( a + b + c \).


It is given that \( x \) and \( y \) are two real numbers such that
\[ (x+y)^{4} + (x-y)^{4} = 4112 \]
and \( x^{2} - y^{2} = 16 \).

Find the value of \( x^{2} + y^{2} \).


Let a right-angled triangle have side lengths \( a > b > c \). Show that for \( n > 2 \), \( a^n > b^n + c^n \).

Solution: \boxed{a^n > b^n + c^n}


Find all prime numbers \( p \) and \( q \) such that \( p^{q+1} + q^{p+1} \) is a square of a natural number.

Solution: \boxed{p =  q = 2}


Let $O$ be the vertex of a parabola, $F$ be the focus, and $PQ$ be a chord passing through $F$. Given that $|OF|=a$ and $|PQ|=b$, find the area of $\triangle OPQ$.

Solution: \boxed{a \sqrt{a b}}