Five numbers are chosen from the diagram below, such that no two numbers are chosen from the same row or from the same column. Prove that their sum is always the same. \[\begin{array}{|c|c|c|c|c|}\hline 1&4&7&10&13\\ \hline 16&19&22&25&28\\ \hline 31&34&37&40&43\\ \hline 46&49&52&55&58\\ \hline 61&64&67&70&73\\ \hline \end{array}\]

The function $f_n (x)\ (n=1,2,\cdots)$ is defined as follows. \[f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)\] Evaluate \[\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)\]

Define a [i]modern artwork[/i] to be a nonempty finite set of rectangles in the Cartesian coordinate plane with positive areas, pairwise disjoint interiors, and sides parallel to the coordinate axes. For a modern artwork $S$, define its [i]price[/i] to be the minimum number of colors with which Sean could paint the interiors of rectangles in $S$ such that every rectangle's interior is painted in exactly one color and every two distinct touching rectangles have distinct colors, where two rectangles are [i]touching[/i] if they share infinitely many points. For a positive integer $n$, let $g(n)$ denote the maximum price of any modern artwork with exactly $n$ rectangles. Compute $g(1) + g(2) + \cdots + g(2019).$ [i]Proposed by Yang Liu and Edward Wan[/i]

Let $ ABCD$ be a cyclic quadrilateral. The side lengths of $ ABCD$ are distinct integers less than $ 15$ such that $ BC\cdot CD\equal{}AB\cdot DA$. What is the largest possible value of $ BD$? $ \textbf{(A)}\ \sqrt{\frac{325}{2}} \qquad \textbf{(B)}\ \sqrt{185} \qquad \textbf{(C)}\ \sqrt{\frac{389}{2}} \qquad \textbf{(D)}\ \sqrt{\frac{425}{2}} \qquad \textbf{(E)}\ \sqrt{\frac{533}{2}}$

Let $I$ be the incenter of a triangle $ABC$, $M$ be the midpoint of $AC$, and $W$ be the midpoint of arc $AB$ of the circumcircle not containing $C$. It is known that $\angle AIM = 90^\circ$. Find the ratio $CI:IW$.

What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1,k_2,\ldots,k_n$ for which \[k_1^2+k_2^2+\ldots+k_n^2=2002?\] $\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

Solve the equation, $$\sqrt{x+5}+\sqrt{16-x^2}=x^2-25$$

Solve the system of equations $$\begin{cases} x^2+ [y]=10 \\ y^2+[x]=13 \end{cases}$$ ($[x]$ is the integer part of $x$, $[x]$ is equal to the largest integer not exceeding $x$. For example, $[3,33] = 3$, $[2] = 2$, $[- 3.01] = -4$).

Alexey and Bogdan play a game with two piles of stones. In the beginning, one of the piles contains $2021$ stones, and the second is empty. In one move, each of the guys has to pick up an even number of stones (more than zero) from an arbitrary pile, then transfer half of the stones taken to another pile, and the other half - to remove from the game. Loses the one who cannot make a move. Who will win this game if both strive to win, and Bogdan begins? (Oleksii Masalitin)

For each integer $n\ge2$ prove the inequality $$\log_23+\log_34+\ldots+\log_n(n+1)<n+\ln n-0.9.$$

Call a positive integer [i]emphatic[/i] if it can be written in the form $a^2+b!$, where $a$ and $b$ are positive integers. Prove that there are infinitely many positive integers $n$ such that $n$, $n+1$, and $n+2$ are all [i]emphatic[/i]. [i]Allen Wang[/i]

Prove that if $n$ is a square-free positive integer, there are no coprime positive integers $x$ and $y$ such that $(x + y)^3$ divides $x^n+y^n$

Given a positive integer $n$, we define $\lambda (n)$ as the number of positive integer solutions of $x^2-y^2=n$. We say that $n$ is [i]olympic[/i] if $\lambda (n) = 2021$. Which is the smallest olympic positive integer? Which is the smallest olympic positive odd integer?

Let $M$ be the maximum possible value of $x_1x_2+x_2x_3+\cdots +x_5x_1$ where $x_1, x_2, \cdots x_5$ is a permutation of $(1,2,3,4,5)$ and let $N$ be the number of permutations for which this maximum is attained. Evaluate $M+N$.

For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that $$ \frac {\varphi ( d(n))}{d(\varphi(n))}\le C$$ for all $n\ge 1$ [i]Cyprus[/i]

A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $1/2$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?

Consider the polynomials \[f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;\] \[g(p) = p^3 + 2p + m.\] Find all integral values of $m$ for which $f$ is divisible by $g$.

Let $M = \{(a,b,c,d)|a,b,c,d \in \{1,2,3,4\} \text{ and } abcd > 1\}$. For each $n\in \{1,2,\dots, 254\}$, the sequence $(a_1, b_1, c_1, d_1)$, $(a_2, b_2, c_2, d_2)$, $\dots$, $(a_{255}, b_{255},c_{255},d_{255})$ contains each element of $M$ exactly once and the equality \[|a_{n+1} - a_n|+|b_{n+1} - b_n|+|c_{n+1} - c_n|+|d_{n+1} - d_n| = 1\] holds. If $c_1 = d_1 = 1$, find all possible values of the pair $(a_1,b_1)$.

A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

We are given an acute triangle $ABC$. The angle bisector of $\angle BAC$ cuts $BC$ at $P$. Points $D$ and $E$ lie on segments $AB$ and $AC$, respectively, so that $BC \parallel DE$. Points $K$ and $L$ lie on segments $PD$ and $PE$, respectively, so that points $A$, $D$, $E$, $K$, $L$ are concyclic. Prove that points $B$, $C$, $K$, $L$ are also concyclic. [i]Proposed by Patrik Bak, Slovakia [/i]

$16$ different natural numbers are written on the board, none of which exceeds $30$. Prove that there must be two coprime numbers among the written numbers.

[b]a)[/b] Prove that, for any point in the interior of a triangle, there are two points on the sides of this triangle such that the resultant of the vectors from the interior point those two points is the vector $ 0. $ [b]b)[/b] Prove that, for any point in the interior of a triangle, there are three points on the sides of this triangle such that the resultant of the vectors from the interior point those three points is the vector $ 0. $

Initially, on a board there a positive integer. If board contains the number $x,$ then we may additionally write the numbers $2x+1$ and $\frac{x}{x+2}.$ At some point 2008 is written on the board. Prove, that this number was there from the beginning.

Given is a trihedral angle $ OABC $ with a vertex $ O $ and a point $ P $ in its interior. Let $ V $ be the volume of a parallelepiped with two vertices at points $ O $ and $ P $, whose three edges are contained in the rays $ \overrightarrow{OA} $, $ \overrightarrow{OB} $, $ \overrightarrow{OC} $. Calculate the minimum volume of a tetrahedron whose three faces are contained in the faces of the trihedral angle $OABC$ and the fourth face contains the point $P$.

The fraction $ \frac{a^{\minus{}4}\minus{}b^{\minus{}4}}{a^{\minus{}2}\minus{}b^{\minus{}2}}$ is equal to: $ \textbf{(A)}\ a^{\minus{}6}\minus{}b^{\minus{}6} \qquad \textbf{(B)}\ a^{\minus{}2}\minus{}b^{\minus{}2} \qquad \textbf{(C)}\ a^{\minus{}2}\plus{}b^{\minus{}2} \\ \textbf{(D)}\ a^2\plus{}b^2 \qquad \textbf{(E)}\ a^2\minus{}b^2$