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Found problems: 2

1986 Traian Lălescu, 2.3
Tags: function , inequalities , calculus , derivative , ftc , rolle theorem , integral
Let $ f:[0,2]\longrightarrow \mathbb{R} $ a differentiable function having a continuous derivative and satisfying $ f(0)=f(2)=1 $ and $ |f’|\le 1. $ Show that $$ \left| \int_0^2 f(t) dt\right| >1. $$

2005 IberoAmerican Olympiad For University Students, 7
Tags: polynomial , algebra , rolle theorem , real analysis
Prove that for any integers $n,p$, $0<n\leq p$, all the roots of the polynomial below are real: \[P_{n,p}(x)=\sum_{j=0}^n {p\choose j}{p\choose {n-j}}x^j\]