Christmas Bunny
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### Questions (98)

 25 Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges? 13 Showing that $|\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$ 11 Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''\,dx=\int_0^ty'y''\,dx$. 10 Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$, $f(x)f(yf(x))=f(x+y)$ 10 Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $\sum_{k=1}^n\frac{a_k-b_k}k$

### Reputation (2,005)

 +20 Find $\lim_{n\to\infty}\prod_{k=1}^n\cos\left(\frac{ka}{n\sqrt{n}}\right)$ +10 Prove that $\sum_{k=1}^n \frac{2k+1}{a_1+a_2+…+a_k}<4\sum_{k=1}^n\frac1{a_k}.$ +5 Evaluate $\lim_{n\to\infty}n\int_0^1x^nf(x)dx$ -2 Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''\,dx=\int_0^ty'y''\,dx$.

 3 Proving $n^q$ is algebraic when $n\in \mathbb N$ and $q\in \mathbb Q$. 3 Finding the index such that all partial sums are nonnegative 0 Find the maximum and minimum of $\sum_{i=1}^{n-1}x_ix_{i+1}$ subject to $\sum_{i=1}^nx_i^2=1$. 0 $\sum_{i=1}^nx_i^2+2\sum_{1\le k 3j$, there exists positive integers $s,t$ such that $k > |\sqrt{s}-\sqrt{t}| > j$

### Tags (88)

 3 contest-math × 58 2 real-analysis × 66 3 algorithms × 4 0 analysis × 56 3 integer-programming × 2 0 calculus × 20 3 proof-strategy 0 integration × 17 3 algebraic-number-theory 0 measure-theory × 15

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