# Solution: 2013-06 Inequality on the unit interval

Let $$f : [0, 1] \to \mathbb{R}$$ be a continuously differentiable function with $$f(0) = 0$$ and $$0 < f'(x) \leq 1$$. Prove that $\left( \int_0^1 f(x) dx \right)^2 \geq \int_0^1 [f(x)]^3 dx.$

The best solution was submitted by 박훈민, 13학번. Congratulations!

Other solutions were submitted by 라준현(08학번, +3), 김호진(09학번, +3), 서기원(09학번, +3), 김범수(10학번, +3), 황성호(13학번, +3), 홍혁표(13학번, +3), 김준(13학번, +3), 전한솔(고려대 13학번, +3), 이시우(POSTECH 13학번, +3), 한대진(신현여중 교사, +3). Thank you for your participation.

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# Midterm break

The problem of the week will take a break during the midterm period and return on May 3, Friday. Good luck on your midterm exams!

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# 2013-06 Inequality on the unit interval

Let $$f : [0, 1] \to \mathbb{R}$$ be a continuously differentiable function with $$f(0) = 0$$ and $$0 < f'(x) \leq 1$$. Prove that $\left( \int_0^1 f(x) dx \right)^2 \geq \int_0^1 [f(x)]^3 dx.$

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# Solution: 2013-05 Zeros of a cosine series

Let $F(x) = \sum_{n=1}^{1000} \cos (n^{3/2} x).$
Prove that $$F$$ has at least $$80$$ zeros in the interval $$(0, 2013)$$.

The best solution was submitted by 황성호, 13학번. Congratulations!

Other solutions were submitted by 라준현(08학번, +3), 김호진(09학번, +2). Thank you for your participation. Sincere apology for the error in the first version last Friday.

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# 2013-05 Zeros of a cosine series

Let $F(x) = \sum_{n=1}^{1000} \cos (n^{3/2} x).$
Prove that $$F$$ has at least $$80$$ zeros in the interval $$(0, 2013)$$.

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Let $$H$$ be an $$N \times N$$ real symmetric matrix. Suppose that $$|H_{kk}| < 1$$ for $$1 \leq k \leq N$$. Prove that, if $$|H_{ij}| > 4$$ for some $$i, j$$, then the largest eigenvalue of $$H$$ is larger than $$3$$.