OIFMAT III 2013

Day I - 04 October 2013

1

Find all four-digit perfect squares such that: $\bullet$ All your figures are less than $9$. $\bullet$ By increasing each of its digits by one unit, the resulting number is again a perfect square.

2

We will say that a set $ A $ of points is disastrous if it meets the following conditions: $\bullet$ There are no $ 3 $ collinear points $\bullet$ There is not a trio of mutually equal distances between points. If $ P $ and $ Q $ are points in $ A $, then there are $ M $, $ N $, $ R $ and $ T $ in $ A $ such that: $$ d (P, Q) = \frac {d (M, N) + d (R, T)} {2} $$Show that all disastrous sets are infinite. original wording of second conditionNo existe ni un trĂ­o de distancias entre puntos mutuamente iguales.

3

Legend has it that in a police station in the old west a group of six bandits tried to bribe the Sheriff in charge of the place with six gold coins to free them, the Sheriff was a very honest person so to prevent them from continuing to insist With the idea of bribery, he sat the $6$ bandits around a table and proposed the following: - "Initially the leader will have the six gold coins, in each turn one of you can pass coins to the adjacent companions, but each time you do so you must pass the same amount of coins to each of your neighbors. If at any time they all manage to have the same amount of coins so I will let them go free. " The bandits accepted and began to play. Show that regardless of what moves the bandits make, they cannot win.

4

Show that there exists a set of infinite positive integers such that the sum of an arbitrary finite subset of these is never a perfect square. What happens if we change the condition from not being a perfect square to not being a perfect power?

5

In an acute triangle $ ABC $ with circumcircle $ \Omega $ and circumcenter $ O $, the circle $ \Gamma $ is drawn, passing through the points $ A $, $ O $ and $ C $ together with its diameter $ OQ $, then the points $ M $ and $ N $ are chosen on the lines $ AQ $ and $ AC $, respectively, in such a way that the quadrilateral $ AMBN $ is a parallelogram. Prove that the point of intersection of the lines $ MN $ and $ BQ $ lies on the circle $ \Gamma $.

Day II - 19 October 2013

6

The acute triangle $ABC$ is inscribed in a circle with center $O$. Let $D$ be the intersection of the bisector of angle $BAC$ with segment $BC$ and $ P$ the intersection point of $AB$ with the perpendicular on $OA$ passing through $D$. Show that $AC = AP$.

7

Define $ a \circledast b = a + b-2ab $. Calculate the value of $$A=\left( ...\left(\left(\frac{1}{2014}\circledast \frac{2}{2014}\right)\circledast\frac{3}{2014}\right)...\right)\circledast\frac{2013}{2014}$$

8

$ABCD$ is a trapezoid with $AB$ parallel to $CD$. The external bisectors of the angles at $ B$ and $C$ intersect at $ P$. The external bisectors of the angles at $ A$ and $D$ intersect at $Q$. Show that the length of $PQ$ is equal to half the perimeter of the trapezoid $ABCD$.

9

Let $ a, b \in Z $, prove that if the expression $ a \cdot 2013^n + b $ is a perfect square for all natural $n$, then $ a $ is zero.

10

Prove that the sequence defined by: $$ y_ {n + 1} = \frac {1} {2} (3y_ {n} + \sqrt {5y_ {n} ^ {2} -4}) , \,\, \forall n \ge 0$$with $ y_ {0} = 1$ consists only of integers.