JBMO 2013

Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers.

Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.

Show that $$\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16$$

for all positive real numbers $a$ and $b$ such that $ab\geq 1$.

Let $n$ be a positive integer. Two players, Alice and Bob, are playing the following game:

• Alice chooses $n$ real numbers; not necessarily distinct.
• Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are $\frac{n(n-1)}{2}$ such sums; not necessarily distinct.)
• Bob wins if he finds correctly the initial $n$ numbers chosen by Alice with only one guess. Can Bob be sure to win for the following cases?

a. $n=5$ b. $n=6$ c. $n=8$

Justify your answer(s).

[For example, when $n=4$, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]