Undergraduate contests Jozsef Wildt International Math Competition 2019 Jozsef Wildt International Math Competition W. 60 Prove these two inequalities on heights, inradius and exradius of a tetrahedon Problem Statement In all tetrahedron A B C D ABCD A BC D ( n ( n + 2 ) ) 1 n ∑ c y c ( ( h a − r ) 2 ( h a n − r n ) ( h a n + 2 − r n + 2 ) ) 1 n ≤ 1 r 2 (n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(h_a-r)^2}{(h_a^n-r^n)(h_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2} ( n ( n + 2 ) ) n 1 cyc ∑ ( ( h a n − r n ) ( h a n + 2 − r n + 2 ) ( h a − r ) 2 ) n 1 ≤ r 2 1 ( n ( n + 2 ) ) 1 n ∑ c y c ( ( r a − r ) 2 ( r a n − r n ) ( r a n + 2 − r n + 2 ) ) 1 n ≤ 1 r 2 (n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(r_a-r)^2}{(r_a^n-r^n)(r_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2} ( n ( n + 2 ) ) n 1 cyc ∑ ( ( r a n − r n ) ( r a n + 2 − r n + 2 ) ( r a − r ) 2 ) n 1 ≤ r 2 1 n ∈ N ∗ n\in \mathbb{N}^* n ∈ N ∗