MathDB
Problems
Contests
International Contests
Tuymaada Olympiad
2010 Tuymaada Olympiad
2010 Tuymaada Olympiad
Part of
Tuymaada Olympiad
Subcontests
(4)
3
4
Show problems
2
4
Show problems
4
4
Show problems
1
3
Hide problems
Tuymaada 2010, Junior League, Problem 1
Misha and Sahsa play a game on a
100
×
100
100\times 100
100
×
100
chessboard. First, Sasha places
50
50
50
kings on the board, and Misha places a rook, and then they move in turns, as following (Sasha begins):At his move, Sasha moves each of the kings one square in any direction, and Misha can move the rook on the horizontal or vertical any number of squares. The kings cannot be captured or stepped over. Sasha's purpose is to capture the rook, and Misha's is to avoid capture.Is there a winning strategy available for Sasha?
Tuymaada 2010, Junior League, Problem 5
We have a set
M
M
M
of real numbers with
∣
M
∣
>
1
|M|>1
∣
M
∣
>
1
such that for any
x
∈
M
x\in M
x
∈
M
we have either
3
x
−
2
∈
M
3x-2\in M
3
x
−
2
∈
M
or
−
4
x
+
5
∈
M
-4x+5\in M
−
4
x
+
5
∈
M
. Show that
M
M
M
is infinite.
Three quadratic trinomials all with an integer root
Baron Münchausen boasts that he knows a remarkable quadratic triniomial with positive coefficients. The trinomial has an integral root; if all of its coefficients are increased by
1
1
1
, the resulting trinomial also has an integral root; and if all of its coefficients are also increased by
1
1
1
, the new trinomial, too, has an integral root. Can this be true?