2018 Bosnia and Herzegovina Team Selection
Test
Bosnia and Herzegovina Team Selection Test 2018
Day 1 April 21st
1 In acute triangle ABC (AB < AC) let D, E and F be foots of perpedicular
from A, B and C to BC, CA and AB, respectively. Let P and Q be points on
line EF such that DP ⊥ EF and BQ = CQ. Prove that ∠ADP = ∠P BQ
2 Let a1 , a2 , . . . an , k, and M be positive integers such that
1 1 1
+ + ··· + =k and a1 a2 · · · an = M.
a1 a2 an
If M > 1, prove that the polynomial
P (x) = M (x + 1)k − (x + a1 )(x + a2 ) · · · (x + an )
has no positive roots.
3 Find all values of positive integers a and b such that it is possible to put a
ones and b zeros in every of vertices in polygon with a + b sides so it is possible
to rotate numbers in those vertices with respect to primary position and after
rotation one neighboring 0 and 1 switch places and in every other vertices other
than those two numbers remain the same.
Day 2 April 22nd
4 Every square of 1000 × 1000 board is colored black or white. It is known that
exists one square 10 × 10 such that all squares inside it are black and one square
10 × 10 such that all squares inside are white. For every square K 10 × 10 we
define its power m(K) as an absolute value of difference between number of
white and black squares 1 × 1 in square K. Let T be a square 10 × 10 which has
minimum power among all squares 10 × 10 in this board. Determine maximal
possible value of m(T )
5 Let p ≥ 2 be a prime number. Eduardo and Fernando play the following game
making moves alternately: in each move, the current player chooses an index i
in the set {1, 2, . . . , p−1} that was not chosen before by either of the two players
and then chooses an element ai from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Eduardo
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2018 Bosnia and Herzegovina Team Selection
Test
has the first move. The game ends after all the indices have been chosen .Then
the following number is computed:
X
p−1
p−1
2
M = a0 + a1 10 + a2 10 + · · · + ap−1 10 = ai .10i
i=0
.
The goal of Eduardo is to make M divisible by p, and the goal of Fernando is
to prevent this.
Prove that Eduardo has a winning strategy.
Proposed by Amine Natik, Morocco
6 Let O be the circumcenter of an acute triangle ABC. Line OA intersects the
altitudes of ABC through B and C at P and Q, respectively. The altitudes
meet at H. Prove that the circumcenter of triangle P QH lies on a median of
triangle ABC.
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Contributors: gobathegreat, math90, Muradjl, fastlikearabbit
