2016 Austria Beginners' Competition

Part of Austrian MO Beginners' Competition

Subcontests

(4)

1

Problem 4 -- Pentagon Proof

ABCDE

be a convex pentagon with ﬁve equal sides and right angles at

C

D

P

denote the intersection point of the diagonals

AC

BD

. Prove that the segments

PA

PD

have the same length.
(Gottfried Perz)

1

Problem 3 -- Nine Number Bubbles

We consider the following ﬁgure:
[See attachment]
We are looking for labellings of the nine ﬁelds with the numbers 1, 2, ..., 9. Each of these numbers has to be used exactly once. Moreover, the six sums of three resp. four numbers along the drawn lines have to be be equal. Give one such labelling. Show that all such labellings have the same number in the top ﬁeld. How many such labellings do there exist? (Two labellings are considered diﬀerent, if they disagree in at least one ﬁeld.)
(Walther Janous)

1

Problem 2 -- Square Dancing Inequality

Prove that all real numbers

x \ne -1

y \ne -1

xy = 1

satisfy the following inequality:

\left(\frac{2+x}{1+x}\right)^2 + \left(\frac{2+y}{1+y}\right)^2 \ge \frac92

1

Problem 1 -- Distance to n/3

Determine all nonnegative integers

n

having two distinct positive divisors with the same distance from

\tfrac{n}{3}

.
(Richard Henner)