MathDB

2008 Tuymaada Olympiad

Part of Tuymaada Olympiad

Subcontests

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A loader has a waggon and a little cart

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

Every street in Hamiltonville connects two squares

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
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Portraits of famous scientists hang on a wall

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 54 5\over 4 times greater than Vasya's result. What minimum number of portraits can be on the wall? Author: V. Frank

Several irrational numbers are written on a blackboard

Several irrational numbers are written on a blackboard. It is known that for every two numbers a a and b b on the blackboard, at least one of the numbers a\over b\plus{}1 and b\over a\plus{}1 is rational. What maximum number of irrational numbers can be on the blackboard? Author: Alexander Golovanov
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The interceptions of the A-midline on the incircle

Point I1 I_1 is the reflection of incentre I I of triangle ABC ABC across the side BC BC. The circumcircle of BCI1 BCI_1 intersects the line II1 II_1 again at point P P. It is known that P P lies outside the incircle of the triangle ABC ABC. Two tangents drawn from P P to the latter circle touch it at points X X and Y Y. Prove that the line XY XY contains a medial line of the triangle ABC ABC. Author: L. Emelyanov

Members can be distributed to several rooms

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