Olympiad problems

2019 China National Olympiad, 1

Let $a,b,c,d,e\geq -1$ and $a+b+c+d+e=5.$ Find the maximum and minimum value of $S=(a+b)(b+c)(c+d)(d+e)(e+a).$

2018 Peru MO (ONEM), 2

Tags: Peru , national olympiad , algebra , inequalities , High School Olympiads

2) Let $a, b, c$ be real numbers such that $$a+\frac{b}{c}=b+\frac{c}{a}=c+\frac{a}{b}=1$$a) Prove that $ab+bc+ca=0$ and $a+b+c=3$. b) Prove that $|a|+|b|+|c|< 5$

2019 Kosovo National Mathematical Olympiad, 2

Tags: inequalities , High School Olympiads

Show that for any positive real numbers $a,b,c$ the following inequality is true: $$4(a^3+b^3+c^3+3)\geq 3(a+1)(b+1)(c+1)$$ When does equality hold?

1982 Austrian-Polish Competition, 5

Tags: High School Olympiads , algebra

Show that [0,1] cannot be partitioned into two disjoints sets A and B such that B=A+a for some real a.

2003 Junior Balkan MO, 4

Tags: inequalities , High School Olympiads , algebra , JBMO

Let $x, y, z > -1$. Prove that \[ \frac{1+x^2}{1+y+z^2} + \frac{1+y^2}{1+z+x^2} + \frac{1+z^2}{1+x+y^2} \geq 2. \] [i]Laurentiu Panaitopol[/i]