## Posts

Showing posts from July, 2022

### 2015 IMO SL #A1

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies$a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}$for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

### 2019 IMO SL #C1

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and$\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k$for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

Let $a$, $b$, $c$ be positive real numbers such that $abc = 1$. Prove that $\frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}.$