Wielkopolska
Liga
VIII W L M
–
PART A
Matematyczna
A1.
Solve the equation
n! + 2 = m3
over the positive integers m, n.
A2.
Let ABC be a triangle. A point D lies on the side BC and a point E lies on the side AD.
Furtermore AC = BC = AD and BD = ED. Prove that
ABE + 2DBE = 90◦ .
A3.
Prove that for all real numbers x, y, z > 21 , if xyz = 1, then
2 2 2
+ + > x + y + z + 3.
x y z
A4.
Suppose that all sides of a triangle ABC have distinct lengths and ACB = 60◦ . Points P 6= A
and Q 6= B lie on the circumcircle of ABC and satisfy AP k BC and BQ k AC. Prove that
AP = BQ.
A5.
For every natural number n > 2 find the greates natural number k with the following property.
There exists a family of k subsets of {1, 2, . . . , n} such that every two distinct sets from this
family have at most one common element.
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 January 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