On March $6$, $2002$, the celebrations of the $500$th anniversary of the birth of by mathematician Pedro Nunes. That morning, only ten people entered the Viva bookstore for science. Each of these people bought exactly $3$ different books. Furthermore, any two people bought at least one copy of the same book. The Adventures of Mathematics by Pedro Nunes was one of the books that achieved the highest number of sales in this morning. What is the smallest value this number could have taken?

At a party every girl danced with at least one boy, but not with all of them. Similarly, every boy danced with at least one girl, but not with all of them. Show that there were two girls $G$ and $G'$ and two boys $B$ and $B'$, such that each of $B$ and $G$ danced, $B'$ and $G'$ danced, but $B$ and $G'$ did not dance, and $B'$ and $G$ did not dance.

Let $k \le n$ be positive integers and $x$ be a real number with $0 \le x < 1/n$. Prove that $${n \choose 0} - {n \choose 1} x +{n \choose 2} x^2 - ... + (-1)^k {n \choose k} x^k > 0$$

Let $\mathbb{Q}$ be the set of rational numbers. A function $f: \mathbb{Q} \to \mathbb{Q}$ is called aquaesulian if the following property holds: for every $x,y \in \mathbb{Q}$, \[ f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y). \] Show that there exists an integer $c$ such that for any aquaesulian function $f$ there are at most $c$ different rational numbers of the form $f(r) + f(-r)$ for some rational number $r$, and find the smallest possible value of $c$.

Find all functions $f,g,h: \mathbb{R} \mapsto \mathbb{R}$ such that $f(x) - g(y) = (x-y) \cdot h(x+y)$ for $x,y \in \mathbb{R}.$

For any real number$m$, the equation $$x^2+(m-2)x- (m+3)=0$$ has two solutions, denoted $x_1 $and $ x_2$. Determine $m$ such that $x_1^2+x_2^2$ is the minimum possible.

Let $S = \{1, 2, \ldots, 2016\}$, and let $f$ be a randomly chosen bijection from $S$ to itself. Let $n$ be the smallest positive integer such that $f^{(n)}(1) = 1$, where $f^{(i)}(x) = f(f^{(i-1)}(x))$. What is the expected value of $n$?

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfying \[ f(xy)(f(x) - f(y)) = (x-y)f(x)f(y) \] for all $x,y$.

Let $ABC$ be a triangle in which no angle is $90^{\circ}$. For any point $P$ in the plane of the triangle, let $A_1, B_1, C_1$ denote the reflections of $P$ in the sides $BC,CA,AB$ respectively. Prove that (i) If $P$ is the incenter or an excentre of $ABC$, then $P$ is the circumenter of $A_1B_1C_1$; (ii) If $P$ is the circumcentre of $ABC$, then $P$ is the orthocentre of $A_1B_1C_1$; (iii) If $P$ is the orthocentre of $ABC$, then $P$ is either the incentre or an excentre of $A_1B_1C_1$.

In a triangle ABC, D is midpoint of BC . If $\angle ADB = 45 ^{\circ}$ and $\angle ACD = 30^{\circ}$, determine $\angle BAD.$

Given a positive integer $k$, let $||k||$ denote the absolute difference between $k$ and the nearest perfect square. For example, $||13||=3$ since the nearest perfect square to $13$ is $16$. Compute the smallest positive integer $n$ such that $\frac{||1|| + ||2|| + ...+ ||n||}{n}=100$.

Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$

The polynomial $-400x^5+2660x^4-3602x^3+1510x^2+18x-90$ has five rational roots. Suppose you guess a rational number which could possibly be a root (according to the rational root theorem). What is the probability that it actually is a root?

There are $100$ stones in a pile. A partition of the heap in $k $ piles is called [i]special [/i] if it meets that the number of stones in each pile is different and also for any partition of any of the piles into two new piles it turns out that between the $k + 1$ piles there are two that have the same number of stones (each pile contains at least one stone). a) Find the maximum number $k$, such that there is a special partition of the $100$ stones into $k $ piles. b) Find the minimum number $k $, such that there is a special partition of the $100$ stones in $k $ piles.

Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that \[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]

The product of three positive numbers is $1$ and their sum is greater than the sum of their inverses. Prove that one of these numbers is greater than $1$, while the other two are smaller than $1$.

Consider quadrilateral $ABCD $ inscribed in circle $\omega $. $P\equiv AC\cap BD$. $E$, $F$ lie on sides $AB$, $CD$ respectively such that $\hat {APE}=\hat {DPF} $. Circles $\omega_1$, $\omega_2$ are tangent to $\omega$ at $X $, $Y $ respectively and also both tangent to the circumcircle of $\triangle PEF $ at $P $. Prove that: $$\frac {EX}{EY}=\frac {FX}{FY} $$ [i]Proposed by Ali Zamani [/i]

We are given an infinite row of cells extending infinitely in both directions. Some cells contain one or more stones. The total number of stones is finite. At each move, the player performs one of the following three operations: [b]1. [/b]Take three stones from some cell, and add one stone to the cells located one cell to the left and one cell to the right, each skipping one cell in between. [b]2. [/b]Take two stones from some cell, and add one stone to the cell one cell to the left, skipping one cell and one stone to the adjacent cell to the right. [b]3.[/b] Take one stone from each of two adjacent cells, and add one stone to the cell to the right of these two cells. The process ends when no moves are possible. Prove that the process always terminates and the final distribution of stones does not depend on the choices of moves made by the player. [img]https://i.imgur.com/IjcIDOa.png[/img] [i]Proposed by Luka Tsulaia, Georgia[/i]

Suppose $a_1, \dots, a_n$ are integers whose greatest common divisor is 1. Let $S$ be a set of integers with the following properties: (a) For $i=1, \dots, n$, $a_i \in S$. (b) For $i,j = 1, \dots, n$ (not necessarily distinct), $a_i - a_j \in S$. (c) For any integers $x,y \in S$, if $x+y \in S$, then $x-y \in S$. Prove that $S$ must be equal to the set of all integers.

Consider an $ABCD$ parallelogram of area $1$. Let $E$ be the center of gravity of the triangle $ABC, F$ the center of gravity of the triangle $BCD, G$ the center of gravity of the triangle $CDA$ and $H$ the center of gravity of the triangle $DAB$. Calculate the area of quadrilateral $EFGH$.

Let $c$ be a fixed positive integer, and $\{x_k\}^{\inf}_{k=1}$ be a sequence such that $x_1=c$ and $x_n=x_{n-1}+\lfloor \frac{2x_{n-1}-2}{n} \rfloor$ for $n \ge 2$. Determine the explicit formula of $x_n$ in terms of $n$ and $c$. (Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

Let $S$ be the set of all positive integers $n$ such that the sum of all factors of $n$, including $1$ and $n$, is $120$. Compute the sum of all numbers in $S$. [i]Proposed by Evin Liang[/i]

Theo Travel, who has $5$ children, has already visited $8$ countries of the eurozone. From every country, he brought $5$ not necessarily distinct coins home. Moreover, among these $40$ coins there are exactly $5$ of every value ($1,2,5,10,20,$ and $50$ ct, $1$ and $2$ euro). He wants to give each child $8$ coins such that they are from different countries and that each child gets the same amount of money. Is this always possible?

For each $k$ , find the least $n$ in terms of $k$ st the following holds: There exists $n$ real numbers $a_1 , a_2 ,\cdot \cdot \cdot , a_n$ st for each $i$ : $$0 < a_{i+1} - a_{i} < a_i - a_{i-1}$$ And , there exists $k$ pairs $(i,j)$ st $a_i - a_j = 1$.

$ABCD$ is any convex quadrilateral. Squares center $E, F, G, H$ are constructed on the outside of the edges $AB, BC, CD$ and $DA$ respectively. Show that $EG$ and $FH$ are equal and perpendicular.