IOM 2016

Find all positive integers $n$ such that there exist $n$ consecutive positive integers whose sum is a perfect square.

Let $a_1, . . . , a_n$ be positive integers satisfying the inequality $\sum_{i=1}^{n}\frac{1}{a_n}\le \frac{1}{2}$ . Every year, the government of Optimistica publishes its Annual Report with n economic indicators. For each $i = 1, . . . , n$ ,the possible values of the $i-th$ indicator are $1, 2, . . . , a_i$ . The Annual Report is said to be optimistic if at least $n - 1$ indicators have higher values than in the previous report. Prove that the government can publish optimistic Annual Reports in an infinitely long sequence.

Let $A_1A_2 . . . A_n$ be a cyclic convex polygon whose circumcenter is strictly in its interior. Let $B_1, B_2, ..., B_n$ be arbitrary points on the sides $A_1A_2, A_2A_3, ..., A_nA_1$ , respectively, other than the vertices. Prove that $\frac{B_1B_2}{A_1A_3}+ \frac{B_2B_3}{A_2A_4}+...+\frac{B_nB_1}{A_nA_2}>1$ .

A convex quadrilateral $ABCD$ has right angles at $A$ and $C$ . A point $E$ lies on the extension of the side $AD$ beyond $D$ so that $\angle ABE =\angle ADC$ . The point $K$ is symmetric to the point $C$ with respect to point $A$ . Prove that $\angle ADB =\angle AKE$ .

Let $r(x)$ be a polynomial of odd degree with real coefficients. Prove that there exist only finitely many (or none at all) pairs of polynomials $p(x)$ and $q(x)$ with real coefficients satisfying the equation $(p(x))^3 + q(x^2) = r(x)$ .

In a country with $n$ cities, some pairs of cities are connected by one-way flights operated by one of two companies $A$ and $B$ . Two cities can be connected by more than one flight in either direction. An $AB$ -word $w$ is called implementable if there is a sequence of connected flights whose companies' names form the word $w$ . Given that every $AB$ -word of length $2^n$ is implementable, prove that every finite $AB$ -word is implementable. (An $AB$ -word of length $k$ is an arbitrary sequence of $k$ letters $A$ or $B$ ; e.g. $AABA$ is a word of length $4$ .)