Problem 3. Suppose that the equation x^3-ax^2+bx-a=0 has three positive real roots (b>0). Find the minimum value of the expression: (b-a)(b^3+3a^3)

Problem 4. Let a,b be positive real numbers and let x,y be positive real numbers less than 1, such that: a/(1-x)+b/(1-y)=1 Prove that: ∛ay+∛bx≤1.

Problem 5. Consider the integer number n>2. Let a_1,a_2,…,a_n and b_1,b_2,…,b_n be two permutations of 0,1,2,…,n-1. Prove that there exist some i≠j such that: n|a_i b_i-a_j b_j Moved to HSO. ~ oVlad

There are $10$ cities in each of the three countries. Each road connects two cities from two different countries (there is at most one road between any two cities.) There are more than $200$ roads between these three countries. Prove that three cities, one city from each country, can be chosen such that there is a road between any two of these cities.

Let $\omega$ be the incircle of $\triangle ABC$ and $D,E,F$ be the tangency points on $BC ,CA, AB$. In $\triangle DEF$ let the altitudes from $E,F$ to $FD,DE$ intersect $AB, AC$ at $X ,Y$. Prove that the second intersection of $(AEX)$ and $(AFY)$ lies on $\omega$