Found problems: 5
2014 Bosnia and Herzegovina Junior BMO TST, 4
It is given $5$ numbers $1$, $3$, $5$, $7$, $9$. We get the new $5$ numbers such that we take arbitrary $4$ numbers(out of current $5$ numbers) $a$, $b$, $c$ and $d$ and replace them with $\frac{a+b+c-d}{2}$, $\frac{a+b-c+d}{2}$, $\frac{a-b+c+d}{2}$ and $\frac{-a+b+c+d}{2}$. Can we, with repeated iterations, get numbers:
$a)$ $0$, $2$, $4$, $6$ and $8$
$b)$ $3$, $4$, $5$, $6$ and $7$
2017 Bosnia and Herzegovina EGMO TST, 1
It is given sequence wih length of $2017$ which consists of first $2017$ positive integers in arbitrary order (every number occus exactly once). Let us consider a first term from sequence, let it be $k$. From given sequence we form a new sequence of length 2017, such that first $k$ elements of new sequence are same as first $k$ elements of original sequence, but in reverse order while other elements stay unchanged. Prove that if we continue transforming a sequence, eventually we will have sequence with first element $1$.
2003 Bosnia and Herzegovina Team Selection Test, 1
Board has written numbers: $5$, $7$ and $9$. In every step we do the following: for every pair $(a,b)$, $a>b$ numbers from the board, we also write the number $5a-4b$. Is it possible that after some iterations, $2003$ occurs at the board ?
2000 District Olympiad (Hunedoara), 2
[b]a)[/b] Let $ a,b $ two non-negative integers such that $ a^2>b. $ Show that the equation
$$ \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1 $$
has an infinite number of solutions in the non-negative integers. Here, $ \lfloor\alpha\rfloor $ denotes the floor of $ \alpha. $
[b]b)[/b] Find the floor of $ m=\sqrt{2+\sqrt{2+\underbrace{\cdots}_{\text{n times}}+\sqrt{2}}} , $ where $ n $ is a natural number. Justify.
2019 CMI B.Sc. Entrance Exam, 5
Three positive reals $x , y , z $ satisfy \\
$x^2 + y^2 = 3^2 \\
y^2 + yz + z^2 = 4^2 \\
x^2 + \sqrt{3}xz + z^2 = 5^2 .$ \\
Find the value of $2xy + xz + \sqrt{3}yz$