Минко Балкански 19.05.2007

[frostinlove]

хайде някои които днес е ходил на това състезание да поства условията и ако има отговори че ме мързи да пиша

[Реклама]

[sd_pld]

Аз бях,ама нямам отговори или решения естествено

[frostinlove]

a znaesh li adresa koito napisaha na dyskata na koito da tyrsim rezultati?

[sd_pld]

[martosss]

Дайте задачките де, не ни карайте да се мъчим

[M_Velinova]

Eто сайта, но резултати и задачите не видях.
http://www.iheb.org/

[frostinlove]

айде някой да даде класиране моля! или линк поне


Exercise 1. Let a, b et c be the lengths of the sides BC, AC et AB of a triangle ABC, and let ma,mband mcbe the lengths of the medians out of A, B and C.a) Prove that34≤(ma+mb+mc)/(a+b+c)≤ 1. Can one of these two inequalities become an equality?b) Can one improve the lower or the upper bound in these inequalities?Exercise
2. Let a, b, c and d be four real numbers such that ac − bd = 1. Prove that a2+ b^2+ c^2+ d^2+ ad + bc ≥√3.Exercise
3. A casino offers the following game : An ordinary dice with 6 sides, numbered from 1 to6, is thrown and you are offered to win the amount it shows in euros; if you not satisfied by this amount,you can refuse it; in that case the dice is thrown a second time, and you can still take the amount ineuros or decline it; if you refuse it again then the dice is thrown one last time; this time you can notrefuse anymore, and you have to take the amount in euros shown on the dice.What is the maximum amount M that a player would be willing to pay in order to play this gameand win on average? In other terms what is minimal price M at which the casino can propose thisgame to its clients, without loosing money on average. (For example, if M = 0.99 then the casinolooses money at each game, because the player pays 0.99 and earns an amount between 1 and 6. On thecontrary if M > 6 no one will be willing to play.)
Exercise 4. Let I be the center of the inscribed circle in a non equilateral triangle ABC and letG be its barycenter (intersection point of the medians). Prove that GI is parallel to one of the sides ofthe triangle if, and only if, the lengths of the sides of the triangle form an arithmetic progression.
Exercise 5. Let p be the biggest prime divisor of 2007. Prove that there do not exist naturalnumbers x and k such as p divides x^2k+ 2008^2007.
Exercise 6. Let n ≥ 1 be a natural number. Find the number Mnof different ways to put correctlyn opening parentheses “(” and n closing parentheses “)”. Correctly means that if you read it form theleft to the right, there are never more closing parentheses than opening parentheses. For example, for n= 2 the correct combinations are (()) and ()().Hint : one could notice that Mnis also the number of different mountains that can be built out ofn ascending segments like / and n descending segments like \ which never go under the ground. Thefigure here below shows the possible mountains for n = 2