2021 Bosnia and Herzegovina Team Selection Test

Part of Bosnia Herzegovina Team Selection Test

Subcontests

(3)

1

maximum no of L-dominoes on a n x n board

An L-shaped figure composed of

4

unit squares (such as shown in the picture) we call L-dominoes. https://cdn.artofproblemsolving.com/attachments/b/2/064b7c7de496f981cd937cbb7392efc1066420.png Determine the maximum number of L-dominoes that can be placed on a board of dimensions

n \times n

n

is natural number, so that no two dominoes overlap and it is possible get from the upper left to the lower right corner of the board by moving only across those squares that are not covered by dominoes. (By moving, we move from someone of the square on it the neighboring square, i.e. the square with which it shares the page).
Note: L-Dominoes can be rotated as well as flipped, giving an symmetrical figure wrt axis compared to the one shown in the picture.

1

1^{k_1}+2^{k_2}+3^{k_3}+...+(p-1)^{k_{p-1}} is divisible by $p

p > 2

be a prime number. Prove that there is a permutation

k_1, k_2, ..., k_{p-1}

1,2,...,p-1

such that the number

1^{k_1}+2^{k_2}+3^{k_3}+...+(p-1)^{k_{p-1}}

is divisible by

p

.
Note: The numbers

k_1, k_2, ..., k_{p-1}

are a permutation of the numbers

1,2,...,p-1

if each of of numbers

1,2,...,p-1

appears exactly once among the numbers

k_1, k_2, ..., k_{p-1}

1

max of y\sqrt{1-x}+z\sqrt{1-y}+x\sqrt{1-z} if 0<=x,y,z<=1

x,y,z

be real numbers from the interval

[0,1]

. Determine the maximum value of expression

W=y\cdot \sqrt{1-x}+z\cdot\sqrt{1-y}+x\cdot\sqrt{1-z}