MathDB
Problems
Contests
National and Regional Contests
Italy Contests
ITAMO
2004 ITAMO
2004 ITAMO
Part of
ITAMO
Subcontests
(5)
6
1
Hide problems
prove that xyz=x+y+z+2
Let
P
P
P
be a point inside a triangle
A
B
C
ABC
A
BC
. Lines
A
P
,
B
P
,
C
P
AP,BP,CP
A
P
,
BP
,
CP
meet the opposite sides of the triangle at points
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
respectively. Denote
x
=
A
P
P
A
′
,
y
=
B
P
P
B
′
x =\frac{AP}{PA'}, y = \frac{BP}{PB'}
x
=
P
A
′
A
P
,
y
=
P
B
′
BP
and
z
=
C
P
P
C
′
z = \frac{CP}{PC'}
z
=
P
C
′
CP
. Prove that
x
y
z
=
x
+
y
+
z
+
2
xyz = x+y+z+2
x
yz
=
x
+
y
+
z
+
2
.
5
1
Hide problems
sequence expressed as sum of 2 sequences
Decide if the following statement is true or false: For every sequence
{
x
n
}
n
∈
N
\{x_n\}_{n\in \mathbb{N}}
{
x
n
}
n
∈
N
of non-negative real numbers, there exist sequences
{
a
n
}
n
∈
N
\{a_n\}_{n\in\mathbb{N}}
{
a
n
}
n
∈
N
and
{
b
n
}
n
∈
N
\{b_n\}_{n\in\mathbb{N}}
{
b
n
}
n
∈
N
of non-negative real numbers such that: (a)
x
n
=
a
n
+
b
n
x_n = a_n + b_n
x
n
=
a
n
+
b
n
for all
n
n
n
; (b)
a
1
+
⋯
+
a
n
≤
n
a_1 + \cdots + a_n \le n
a
1
+
⋯
+
a
n
≤
n
for infinitely many values of
n
n
n
; (c)
b
1
+
⋯
+
b
n
≤
n
b_1 + \cdots + b_n \le n
b
1
+
⋯
+
b
n
≤
n
for infinitely many values of
n
n
n
.
4
1
Hide problems
describe the winning strategy
Antonio and Bernardo play the following game. They are given two piles of chips, one with
m
m
m
and the other with
n
n
n
chips. Antonio starts, and thereafter they make in turn one of the following moves: (i) take a chip from one pile; (ii) take a chip from each of the piles; (ii) remove a chip from one of the piles and put it onto the other. Who cannot make any more moves, loses. Decide, as a function of
m
m
m
and
n
n
n
if one of the players has a winning strategy, and in the case of the affirmative answer describe that strategy.
1
1
Hide problems
low temperature in christmas
Observing the temperatures recorded in Cesenatico during the December and January, Stefano noticed an interesting coincidence: in each day of this period, the low temperature is equal to the sum of the low temperatures the preceeding day and the succeeding day. Given that the low temperatures in December
3
3
3
and January
31
31
31
were
5
∘
C
5^\circ \text C
5
∘
C
and
2
∘
C
2^\circ \text C
2
∘
C
respectively, find the low temperature in December
25
25
25
.
3
1
Hide problems
can it be xpressed as sum of two squares
(a) Is
200
5
2004
2005^{2004}
200
5
2004
the sum of two perfect squares? (b) Is
200
4
2005
2004^{2005}
200
4
2005
the sum of two perfect squares?