Olympiad problems

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Found problems: 28

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2019 Ramnicean Hope, 2

Calculate $ \int_1^4 \frac{\ln x}{(1+x)(4+x)} dx . $ [i]Ovidiu Țâțan[/i]

2006 Victor Vâlcovici, 1

Tags: function , integration , monotone , real analysis , ftc , definite integral , calculus

Let be an even natural number $ n $ and a function $ f:[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ f(x)=\int_0^x \prod_{k=0}^n (s-k) ds. $$ Show that [b]a)[/b] $ f(n)=0. $ [b]b)[/b] $ f $ is globally nonnegative. [i]Gheorghe Grigore[/i]

2019 Ramnicean Hope, 1

Tags: riemann sum , calculus , real analysis , definite integral , substitution

Calculate $ \lim_{n\to\infty }\sum_{t=1}^n\frac{1}{n+t+\sqrt{n^2+nt}} . $ [i]D.M. Bătinețu[/i] and [i]Neculai Stanciu[/i]

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