The positive numbers $a, b, c,d,e$ are such that the following identity hold for all real number $x$: $(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3$. Find the smallest value of $d$.

Let $r,s,u,v$ be real numbers. Prove that: $$min\{r-s^2,s-u^2, u-v^2,v-r^2\}\le \frac{1}{4}$$

Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$. What is the minimum possible value of $a + b$?

Given real $a$. Find the least possible area of the rectangle with the sides parallel to the coordinate axes and containing the figure determined by the system of inequalities $$y \le -x^2 \,\,\, and \,\,\, y \ge x^2 - 2x + a$$

Arnaldo and Bernardo play a Super Naval Battle. Each has a board $n \times n$. Arnaldo puts boats on his board (at least one but not known how many). Each boat occupies the $n$ houses of a line or a column and the boats they can not overlap or have a common side. Bernardo marks $m$ houses (representing shots) on your board. After Bernardo marked the houses, Arnaldo says which of them correspond to positions occupied by ships. Bernardo wins, and then discovers the positions of all Arnaldo's boats. Determine the lowest value of $m$ for which Bernardo can guarantee his victory.

A village is constructed in the form of a square, consisting of $9$ blocks , each of side length $\ell$, in a $3 \times 3$ formation . Each block is bounded by a bitumen road . If we commence at a corner of the village, what is the smallest distance we must travel along bitumen roads , if we are to pass along each section of bitumen road at least once and finish at the same corner? (Muscovite folklore)

A triangle $ABC$ and two lines $\ell_1, \ell_2$ are given. Through an arbitrary point $D$ on the side $AB$, a line parallel to $\ell_1$ intersects the $AC$ at point $E$ and a line parallel to $\ell_2$ intersects the $BC$ at point $F$. Construct a point $D$ for which the segment $EF$ has the smallest length.

A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?

Through a point $M$ inside an angle $a$ line is drawn. It cuts off this angle a triangle of the least possible area. Prove that $M$ is the midpoint of the segment on this line that the angle intercepts.

What minimal number of numbers from the set $\{1,2,...,1982\}$ should be deleted to provide the property: [i]none of the remained numbers equals to the product of two other remained numbers[/i]?

Let $n$ be a fixed positive integer and let $x, y$ be positive integers such that $xy = nx+ny$. Determine the minimum and the maximum of $x$ in terms of $n$.

Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]

The real numbers $x, y$ and $z$ are such that $x^2 + y^2 + z^2 = 1$. a) Determine the smallest and the largest possible values of $xy + yz - xz$. b) Prove that there does not exist a triple $(x, y, z)$ of rational numbers, which attains any of the two values in a).

Find all triples of positive real numbers $(a, b, c)$ so that the expression $M = \frac{(a + b)(b + c)(a + b + c)}{abc}$ gets its least value.

Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ?

Prove that there exists a prime number $p$ such that the minimum positive integer $n$ such that $p|2^n -1$ is $3^{2013}$.

At a time $t = 0$, a navy ship is at a point $O$, while an enemy ship is at a point $A$ cruising with speed $v$ perpendicular to $OA = a$. The speed and direction of the enemy ship do not change. The strategy of the navy ship is to travel with constant speed $u$ at a angle $0 < \phi < \pi /2$ to the line $OA$. 1) Let $\phi$ be chosen. What is the minimum distance between the two ships? Under what conditions will the distance vanish? 2) If the distance does not vanish, what is the choice of $\phi$ to minimize the distance? What are directions of the two ships when their distance is minimum?

Determine the smallest positive number $a$ such that the number of all integers belonging to $(a, 2016a]$ is $2016$.

Two players play alternately. The first player is given a pair of positive integers $(x_1, y_1)$. Each player must replace the pair $(x_n, y_n)$ that he is given by a pair of non-negative integers $(x_{n+1}, y_{n+1})$ such that $x_{n+1} = min(x_n, y_n)$ and $y_{n+1} = max(x_n, y_n)- k\cdot x_{n+1}$ for some positive integer $k$. The first player to pass on a pair with $y_{n+1} = 0$ wins. Find for which values of $x_1/y_1$ the first player has a winning strategy.

Consider a rectangular array of single digits $d_{i,j}$ with 10 rows and 7 columns, such that $d_{i+1,j}-d_{i,j}$ is always 1 or -9 for all $1 \leq i \leq 9$ and all $1 \leq j \leq 7$, as in the example below. For $1 \leq i \leq 10$, let $m_i$ be the median of $d_{i,1}$, ..., $d_{i,7}$. Determine the least and greatest possible values of the mean of $m_1$, $m_2$, ..., $m_{10}$. Example: [img]https://cdn.artofproblemsolving.com/attachments/8/a/b77c0c3aeef14f0f48d02dde830f979eca1afb.png[/img]

Let $x, y$ be positive reals such that $x \ne y$. Find the minimum possible value of $(x + y)^2 + \frac{54}{xy(x-y)^2}$ .

A honeybug crawls along the honeycombs with the unite length of their hexagons. He has moved from the node $A$ to the node $B$ along the shortest possible trajectory. Prove that the half of his way he moved in one direction.

Daniel chooses a positive integer $n$ and tells Ana. With this information, Ana chooses a positive integer $k$ and tells Daniel. Daniel draws $n$ circles on a piece of paper and chooses $k$ different points on the condition that each of them belongs to one of the circles he drew. Then he deletes the circles, and only the $k$ points marked are visible. From these points, Ana must reconstruct at least one of the circumferences that Daniel drew. Determine which is the lowest value of $k$ that allows Ana to achieve her goal regardless of how Daniel chose the $n$ circumferences and the $k$ points.

You are traveling in a foreign country whose currency consists of five different-looking kinds of coins. You have several of each coin in your pocket. You remember that the coins are worth $1, 2, 5, 10$, and $20$ florins, but you have no idea which coin is which and you don’t speak the local language. You find a vending machine where a single candy can be bought for $1$ florin: you insert any kind of coin, and receive $1$ candy plus any change owed. You can only buy one candy at a time, but you can buy as many as you want, one after the other. What is the least number of candies that you must buy to ensure that you can determine the values of all the coins? Prove that your answer is correct.

Determine for which $m$ there exist exactly $2^{15}$ subsets $X$ of $\{1,2,...,47\}$ with the following property: $m$ is the smallest element of $X$, and for every $x \in X$, either $x+m \in X$ or $x+m > 47$.