In a fish shop with 28 kinds of fish, there are 28 fish sellers. In every seller, there exists only one type of each fish kind, depending on where it comes, Mediterranean or Black Sea. Each of the $k$ people gets exactly one fish from each seller and exactly one fish of each kind. For any two people, there exists a fish kind which they have different types of it (one Mediterranean, one Black Sea). What is the maximum possible number of $k$?

Let $S$ be a set of $\displaystyle { 2n \choose n } + 1$ real numbers, where $n$ is an positive integer. Prove that there exists a monotone sequence $\{a_i\}_{1\leq i \leq n+2} \subset S$ such that \[ |x_{i+1} - x_1 | \geq 2 | x_i - x_1 | , \] for all $i=2,3,\ldots, n+1$.

Let $D_{1}, D_{2}, . . . , D_{n}$ be rectangular parallelepipeds in space, with edges parallel to the $x, y, z$ axes. For each $D_{i}$, let $x_{i} , y_{i} , z_{i}$ be the lengths of its projections onto the $x, y, z$ axes, respectively. Suppose that for all pairs $D_{i}$ , $D_{j}$, if at least one of $x_{i} < x_{j}$ , $y_{i} < y_{j}$, $z_{i} < z_{j}$ holds, then $x_{i} \leq x_{j}$ , $y_{i} \leq y_{j}$, and $z_{i} < z_{j}$ . If the volume of the region $\bigcup^{n}_{i=1}{D_{i}}$ equals 1997, prove that there is a subset $\{D_{i_{1}}, D_{i_{2}}, . . . , D_{i_{m}}\}$ of the set $\{D_{1}, . . . , D_{n}\}$ such that $(i)$ if $k \not= l $ then $D_{i_{k}} \cap D_{i_{l}} = \emptyset $, and $(ii)$ the volume of $\bigcup^{m}_{k=1}{D_{i_{k}}}$ is at least 73.

Let $ n\ge 3 $ be a natural number and $ A_0A_1...A_{n-1} $ a regular polygon. Consider $ B_0 $ on the segment $ A_0A_1 $ such that $ A_0B_0<\frac{1}{2}A_0A_1; B_1 $ on $ A_1A_2 $ so that $ A_1B_1<\frac{1}{2} A_1A_2; $ etc.; $ B_{n-2} $ on $ A_{n-2}A_{n-1} $ so that $ A_{n-2}B_{n-2} <\frac{1}{2} A_{n-2}A_{n-1} , $ and $ B_{n-1} $ on $ A_{n-1}A_0 $ with $ A_{n-1}B_{n-1} <\frac{1}{2} A_{n-1}A_{0} . $ Show that the perimeter of any ploygon that has its vertices on the segments $ A_1B_1,A_2B_2,...,A_{n-1}B_{n-1}, $ is equal or greater than the perimeter of $ B_0B_1...B_{n-1} . $

Find the sum of all integers $n$ with this property: both $n$ and $n + 2021$ are perfect squares.

There are four points $A, B, C, D$ on a circle. Circles are drawn through each pair of neighboring points. Denote the intersection points of neighboring circles by $A_1, B_1, C_1, D_1$. (Some of these points may coincide with previously given ones.) Prove that points $A_1, B_1, C_1, D_1$ lie on one circle.

Each square of a $7 \times 8$ board is painted black or white, in such a way that each $3 \times 3$ subboard has at least two black squares that are neighboring. What is the least number of black squares that can be on the entire board? Clarification: Two squares are [i]neighbors [/i] if they have a common side.

Given a positive integer $N$. Prove that: there are infinitely elements of the [i]Fibonacci[/i] sequence that are divisible by $N$.

Given real numbers $b_0,b_1,\ldots, b_{2019}$ with $b_{2019}\neq 0$, let $z_1,z_2,\ldots, z_{2019}$ be the roots in the complex plane of the polynomial \[ P(z) = \sum_{k=0}^{2019}b_kz^k. \] Let $\mu = (|z_1|+ \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\ldots, z_{2019}$ to the origin.  Determine the largest constant $M$ such that $\mu\geq M$ for all choices of $b_0,b_1,\ldots, b_{2019}$ that satisfy \[ 1\leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019. \]

Find the sum of the solutions to $\dfrac{1}{1+\dfrac{1}{|x-25|}}=\frac{49}{50}$.

The figure below shows a smaller square within a larger square. Both squares have integer side lengths. The region inside the larger square but outside the smaller square has area $52$. Find the area of the larger square. [img]https://cdn.artofproblemsolving.com/attachments/a/f/2cb8c70109196bf30f88aef0c53bbac07d6cc3.png[/img]

Given a convex quadrilateral $ABCD$. The points $P$ and $Q$ are the midpoints of the diagonals $AC$ and $BD$ respectively. The line $PQ$ intersects the lines $AB$ and $CD$ at $N$ and $M$ respectively. Prove that the circumcircles of triangles $NAP$, $NBQ$, $MQD$, and $MPC$ have a common point.

The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught $n$ fish for various values of $n$. \[ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\ \hline \text{number of contestants who caught} \ n \ \text{fish} & 9 & 5 & 7 & 23 & \dots & 5 & 2 & 1 \\ \hline \end{array} \] In the newspaper story covering the event, it was reported that (a) the winner caught 15 fish; (b) those who caught 3 or more fish averaged 6 fish each; (c) those who caught 12 or fewer fish averaged 5 fish each. What was the total number of fish caught during the festival?

Let $a_1, a_2, \ldots, a_8$ be real numbers. Prove that $$\sum_{i=1}^{8} \left( a_i^2 + a_i a_{i+2} \right) \geq \sum_{i=1}^{8} \left( a_i a_{i+1} + a_i a_{i+3} \right),$$ where the indices are taken modulo 8, i.e., $a_9 = a_1$, $a_{10} = a_2$, and $a_{11} = a_3$. In which cases does equality hold? [i]Proposed by Vukašin Pantelić and Andrija Živadinović[/i]

For which positive integers $n \ge 4$ one can find n points in the plane, no three collinear, such that for each triangle formed with three of the $n$ points which are on the convex hull, exactly one of the $n - 3$ remaining points belongs to its interior.

Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write \[\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots , \] where $r=r(p)$; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by \[ f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}\] a) Prove that $S$ is infinite. b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$

The numbers from $1$ to $2n$ are arranged in a certain order in the cell of the strip $1 \times 2n$. Let's call a [i]flip[/i] an operation that takes one cell from the left half of the strip and one cell from the right half of the strip, after which it swaps the numbers written in them, but only if the larger of these numbers is located to the left of the smaller one. Prove that if all $n^2$ possible flips are performed in any order, then all numbers from $1$ to $n$ will be written in the left, and all numbers from $n + 1$ up to $2n$ — in the right half of the strip.

Find all possible pairs of integers $ a$ and $ b$ such that $ab = 160 + 90 (a,b)$, where $(a, b)$ is the greatest common divisor of $ a$ and $ b$.

Point $D$ lies on side $AB$ of triangle $ABC$, and $$\frac{AD}{DC} = \frac{AB}{BC}.$$ Prove that angle $C$ is obtuse. (Sergey Berlov)

I have a set $A$ containing $n$ distinct integers. This set has the property that if $a,b \in A$, then $12 \nmid |a+b|$ and $12 \nmid |a-b|$. What is the largest possible value of $n$?

Let $a, b, c \in\mathbb{R}^+$. Prove that $$\sum_{cyc}ab\left(\frac{1}{2a+c}+\frac{1}{2b+c}\right)<\sum_{cyc}\frac{a^3+b^3}{c^2+ab}.$$

Find all polynomials $P$ with integer coefficients such that wherever $a, b \in N$ and $a+b$ is a square we have $P(a) + P(b)$ is also a square.

Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of integers defined recursively as $ x_{n+2}=5x_{n+1}-x_n. $ Prove that $ \left( x_n\right)_{n\ge 1} $ has a subsequence whose terms are multiples of $ 22 $ if $ \left( x_n\right)_{n\ge 1} $ has a term that is multiple of $ 22. $

The bisectors of the angles $B,C$ of a triangle $ABC$ intersect the opposite sides in $B', C'$ respectively. Prove that the straight line $B'C'$ intersects the inscribed circle in two different points.

In the coordinate plane a parabola passes through the points $(7,6)$, $(7,12)$, $(18,19)$, and $(18,48)$. The axis of symmetry of the parabola is a line with slope $\tfrac{r}{s}$ where r and s are relatively prime positive integers. Find $r + s$.