Wielkopolska
Liga
VIII W L M
–
PART C
Matematyczna
C1.
Let P be a polynomial with real coefficients. Suppose that every positive integer appears in
the sequence P (1), P (2), P (3), . . . at least once. Prove that the degree of P is 1.
C2.
Find all primes p with the following property. In the decimal expansion of the fraction
have 0 at the p-th place.
C3.
A square of size n × n is divided into n2 unit squares. We colour every side of every unit square
with one of two colours: black or white, so that every unite square has two black sides and two
white sides. Prove that the number of white unit segments at the boarder of the square n × n
is even.
C4.
The diagonals of a quadrilateral ABCD inscribed into a circle with center O intersect at the
point P . The circles circumscribed at the triangles ABP and CDP intersect at the point
E 6= P , and the circles circumscribed at the triangles BCP and DAP intersect at the point
F 6= P . Prove that all the points E, F, O, P lie at the same circle.
C5.
A real-valued function f is defined for every positive real argument. Suppose that for every
x > 0 we have
f (x)2 = 1 + (x − 1)f (x + 1).
1
p
we
Prove that if f (x) > 0 for every x > 1, then f (x) > x for every x > 1.
Please send the solutions by the registered letter to:
Wielkopolska Liga Matematyczna
(dr Bartlomiej Bzdȩga)
Collegium Mathematicum
ul.Umultowska 87
61-614 Poznań
The deadline is
31st March 2017.
(The post stamp date decides.)
Every solution should be written separately, one-sided, at
A4 format. You can write in English or Polish. Please put
you name, your school name and the class in the upper left
corner of every sheet. Your email address would be welcome.
Before sending your solutions please read the
WLM Rules available at the www page of WLM (only
Polish version is available).
All information on Wielkopolska Liga Matematyczna and
ratings after every part one can find at
wlm.wmi.amu.edu.pl