MathDB
Problems
Contests
National and Regional Contests
Italy Contests
ITAMO
2024 ITAMO
2024 ITAMO
Part of
ITAMO
Subcontests
(6)
6
1
Hide problems
Maximizing a sum of integer fractions
For each integer
n
n
n
, determine the smallest real number
M
n
M_n
M
n
such that
1
a
1
+
a
1
a
2
+
a
2
a
3
+
⋯
+
a
n
−
1
a
n
≤
M
n
\frac{1}{a_1}+\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots+\frac{a_{n-1}}{a_n} \le M_n
a
1
1
+
a
2
a
1
+
a
3
a
2
+
⋯
+
a
n
a
n
−
1
≤
M
n
for any
n
n
n
-tuple
(
a
1
,
a
2
,
…
,
a
n
)
(a_1,a_2,\dots,a_n)
(
a
1
,
a
2
,
…
,
a
n
)
of integers such that
1
<
a
1
<
a
2
<
⋯
<
a
n
1<a_1<a_2<\dots<a_n
1
<
a
1
<
a
2
<
⋯
<
a
n
.
5
1
Hide problems
Placing few guards in a fortress to oversee everything
A fortress is a finite collection of cells in an infinite square grid with the property that one can pass from any cell of the fortress to any other by a sequence of moves to a cell with a common boundary line (but it can have "holes"). The walls of a fortress are the unit segments between cells belonging to the fortress and cells not belonging to the fortress. The area
A
A
A
of a fortress is the number of cells it consists of. The perimeter
P
P
P
is the total length of its walls. Each cell of the fortress can contain a guard which can oversee the cells to the top, the bottom, the right and the left of this cell, up until the next wall (it also oversees its own cell).(a) Determine the smallest integer
k
k
k
such that
k
k
k
guards suffice to oversee all cells of any fortress of perimeter
P
≤
2024
P \le 2024
P
≤
2024
. (b) Determine the smallest integer
k
k
k
such that
k
k
k
guards suffice to oversee all cells of any fortress of area
A
≤
2024
A \le 2024
A
≤
2024
.
4
1
Hide problems
Segments of equal length in a circumscribed rectangle
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle with
A
B
<
B
C
AB<BC
A
B
<
BC
and circumcircle
Γ
\Gamma
Γ
. Let
P
P
P
be a point on the arc
B
C
BC
BC
(not containing
A
A
A
) and let
Q
Q
Q
be a point on the arc
C
D
CD
C
D
(not containing
A
A
A
) such that
B
P
=
C
Q
BP=CQ
BP
=
CQ
.The circle with diameter
A
Q
AQ
A
Q
intersects
A
P
AP
A
P
again in
S
S
S
. The perpendicular to
A
Q
AQ
A
Q
through
B
B
B
intersects
A
P
AP
A
P
in
X
X
X
. (a) Show that
X
S
=
P
S
XS=PS
XS
=
PS
. (b) Show that
A
X
=
D
Q
AX=DQ
A
X
=
D
Q
.
3
1
Hide problems
Is 72^71 egyptian?
A positive integer
n
n
n
is called egyptian if there exists a strictly increasing sequence
0
<
a
1
<
a
2
<
⋯
<
a
k
=
n
0<a_1<a_2<\dots<a_k=n
0
<
a
1
<
a
2
<
⋯
<
a
k
=
n
of integers with last term
n
n
n
such that
1
a
1
+
1
a
2
+
⋯
+
1
a
k
=
1.
\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_k}=1.
a
1
1
+
a
2
1
+
⋯
+
a
k
1
=
1.
(a) Determine if
n
=
72
n=72
n
=
72
is egyptian. (b) Determine if
n
=
71
n=71
n
=
71
is egyptian. (c) Determine if
n
=
7
2
71
n=72^{71}
n
=
7
2
71
is egyptian.
2
1
Hide problems
The locus of midpoints of unit segments in a square
We are given a unit square in the plane. A point
M
M
M
in the plane is called median if there exists points
P
P
P
and
Q
Q
Q
on the boundary of the square such that
P
Q
PQ
PQ
has length one and
M
M
M
is the midpoint of
P
Q
PQ
PQ
. Determine the geometric locus of all median points.
1
1
Hide problems
Taking the difference with pi becomes periodic
Let
x
0
=
202
4
2024
x_0=2024^{2024}
x
0
=
202
4
2024
and
x
n
+
1
=
∣
x
n
−
π
∣
x_{n+1}=|x_n-\pi|
x
n
+
1
=
∣
x
n
−
π
∣
for
n
≥
0
n \ge 0
n
≥
0
. Show that there exists a value of
n
n
n
such that
x
n
+
2
=
x
n
x_{n+2}=x_n
x
n
+
2
=
x
n
.