Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly $ 5\over 4$ times greater than Vasya's result. What minimum number of portraits can be on the wall? Author: V. Frank

Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? Author: L. Emelyanov Tuymaada 2008, Junior League, First Day, Problem 2.Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime.

100 unit squares of an infinite squared plane form a $ 10\times 10$ square. Unit segments forming these squares are coloured in several colours. It is known that the border of every square with sides on grid lines contains segments of at most two colours. (Such square is not necessarily contained in the original $ 10\times 10$ square.) What maximum number of colours may appear in this colouring? Author: S. Berlov

Point $ I_1$ is the reflection of incentre $ I$ of triangle $ ABC$ across the side $ BC$. The circumcircle of $ BCI_1$ intersects the line $ II_1$ again at point $ P$. It is known that $ P$ lies outside the incircle of the triangle $ ABC$. Two tangents drawn from $ P$ to the latter circle touch it at points $ X$ and $ Y$. Prove that the line $ XY$ contains a medial line of the triangle $ ABC$. Author: L. Emelyanov

Click for solution Let $ D$, $ E$, $ F$ be the tangency points of the incircle $ \rho(I)$ with the sides $ BC$, $ CA$ and $ AB$, respectively and let $ M$ be the midpoint of small arc $ BC$, not containing the vertex $ A$. Since $ MB = MI = MC$, the reflection $ M_{1}$ of $ M$ into the sideline $ BC$ is the circumcenter of the $ BI_{1}CP$. See that $ PD$ is the $ P$-altitude of triangle $ BPC$ and the reflection $ I_{1}$ of $ I$ into $ D$ lies on the circumcircle of $ BPC$. Thus, $ I$ is the orthocenter of $ BPC$ and since $ XY$ is the polar of $ P$ wrt. $ \rho$ and $ DE$ is the polar of $ B$, it follows that the lines $ XY$, $ DF$ and $ CI$ are concurrent. On the other hand, since $ XY$ is the polar of $ P$, we have that $ PI \perp XY$ and so $ XY \| BC$. In this case, the conclusion follows from http://www.mathlinks.ro/viewtopic.php?t=132338.

A loader has a waggon and a little cart. The waggon can carry up to 1000 kg, and the cart can carry only up to 1 kg. A finite number of sacks with sand lie in a storehouse. It is known that their total weight is more than 1001 kg, while each sack weighs not more than 1 kg. What maximum weight of sand can the loader carry in the waggon and the cart, regardless of particular weights of sacks? Author: M.Ivanov, D.Rostovsky, V.Frank

Let $ ABCD$ be an isosceles trapezoid with $ AD \parallel BC$. Its diagonals $ AC$ and $ BD$ intersect at point $ M$. Points $ X$ and $ Y$ on the segment $ AB$ are such that $ AX = AM$, $ BY = BM$. Let $ Z$ be the midpoint of $ XY$ and $ N$ is the point of intersection of the segments $ XD$ and $ YC$. Prove that the line $ ZN$ is parallel to the bases of the trapezoid. Author: A. Akopyan, A. Myakishev

A set $ X$ of positive integers is called nice if for each pair $ a$, $ b\in X$ exactly one of the numbers $ a + b$ and $ |a - b|$ belongs to $ X$ (the numbers $ a$ and $ b$ may be equal). Determine the number of nice sets containing the number 2008. Author: Fedor Petrov

250 numbers are chosen among positive integers not exceeding 501. Prove that for every integer $ t$ there are four chosen numbers $ a_1$, $ a_2$, $ a_3$, $ a_4$, such that $ a_1 + a_2 + a_3 + a_4 - t$ is divisible by 23. Author: K. Kokhas

Several irrational numbers are written on a blackboard. It is known that for every two numbers $ a$ and $ b$ on the blackboard, at least one of the numbers $ a\over b+1$ and $ b\over a+1$ is rational. What maximum number of irrational numbers can be on the blackboard? Author: Alexander Golovanov

Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? Author: L. Emelyanov Tuymaada 2008, Junior League, First Day, Problem 2.Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime.

Point $ I_1$ is the reflection of incentre $ I$ of triangle $ ABC$ across the side $ BC$. The circumcircle of $ BCI_1$ intersects the line $ II_1$ again at point $ P$. It is known that $ P$ lies outside the incircle of the triangle $ ABC$. Two tangents drawn from $ P$ to the latter circle touch it at points $ X$ and $ Y$. Prove that the line $ XY$ contains a medial line of the triangle $ ABC$. Author: L. Emelyanov

Click for solution Let $ D$, $ E$, $ F$ be the tangency points of the incircle $ \rho(I)$ with the sides $ BC$, $ CA$ and $ AB$, respectively and let $ M$ be the midpoint of small arc $ BC$, not containing the vertex $ A$. Since $ MB = MI = MC$, the reflection $ M_{1}$ of $ M$ into the sideline $ BC$ is the circumcenter of the $ BI_{1}CP$. See that $ PD$ is the $ P$-altitude of triangle $ BPC$ and the reflection $ I_{1}$ of $ I$ into $ D$ lies on the circumcircle of $ BPC$. Thus, $ I$ is the orthocenter of $ BPC$ and since $ XY$ is the polar of $ P$ wrt. $ \rho$ and $ DE$ is the polar of $ B$, it follows that the lines $ XY$, $ DF$ and $ CI$ are concurrent. On the other hand, since $ XY$ is the polar of $ P$, we have that $ PI \perp XY$ and so $ XY \| BC$. In this case, the conclusion follows from http://www.mathlinks.ro/viewtopic.php?t=132338.

A group of persons is called good if its members can be distributed to several rooms so that nobody is acquainted with any person in the same room but it is possible to choose a person from each room so that all the chosen persons are acquainted with each other. A group is called perfect if it is good and every set of its members is also good. A perfect group planned a party. However one of its members, Alice, brought here acquaintance Bob, who was not originally expected, and introduced him to all her other acquaintances. Prove that the new group is also perfect. Author: C. Berge

Every street in the city of Hamiltonville connects two squares, and every square may be reached by streets from every other. The governor discovered that if he closed all squares of any route not passing any square more than once, every remained square would be reachable from each other. Prove that there exists a circular route passing every square of the city exactly once. Author: S. Berlov

A set $ X$ of positive integers is called nice if for each pair $ a$, $ b\in X$ exactly one of the numbers $ a + b$ and $ |a - b|$ belongs to $ X$ (the numbers $ a$ and $ b$ may be equal). Determine the number of nice sets containing the number 2008. Author: Fedor Petrov

A loader has two carts. One of them can carry up to 8 kg, and another can carry up to 9 kg. A finite number of sacks with sand lie in a storehouse. It is known that their total weight is more than 17 kg, while each sack weighs not more than 1 kg. What maximum weight of sand can the loader carry on his two carts, regardless of particular weights of sacks? Author: M.Ivanov, D.Rostovsky, V.Frank

A convex hexagon is given. Let $ s$ be the sum of the lengths of the three segments connecting the midpoints of its opposite sides. Prove that there is a point in the hexagon such that the sum of its distances to the lines containing the sides of the hexagon does not exceed $ s.$ Author: N. Sedrakyan