Wielkopolska
Liga
VIII W L M
–
PART B
Matematyczna
B1.
A cuboid P of size a × b × c consists of a · b · c cubes of size 1 × 1 × 1. Two players play the
following game with no draw possible. They take turns and in every round a player sticks a
needle in P , parallelly to a chosen side of P , which means that he drills a, b or c cubes. No
cube can be drilled twice. A player loses the game when he has no legal move. Find all triples
(a, b, c) such that the first player has a winning strategy.
B2.
Distinct natural numbers a, b, d satisfy the following conditions:
d = gcd(a, b)
Prove that d <
and
d + 1 = gcd(a + 1, b + 1).
p
|a − b|.
B3.
Let ABC be a trinagle and let D, E, F be the points such that C, A, B are the midpoints of
the segments BD, CE, AF , respectively. Prove that the triangles ABC and DEF have the
same centroid.
B4.
There are n > 1 positive integers at the blackboard. They are less than 2n and not necessarily
distinct. For several rounds, as long as possible, we make the following
moves. We choose two
ab
numbers a, b > 1 at the blackboard, erase them and write a+b at the blackboard. Prove that
at least one number 1 will remain at the blackboard.
(By bxc we mean the greatest integer not exeeding x.)
B5.
Let p1 , p2 , . . . , pk be distinct primes, less than n. Assume that the division of n by p1 , p2 , . . . , pk
gives the nonzero remainders r1 , r2 , . . . , rk , respectively. Let p = min{p1 , p2 , . . . , pk } and r =
max{r1 , r2 , . . . , rk }. Prove that nr > pk .
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
28th February 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