MathDB
Problems
Contests
National and Regional Contests
China Contests
XES Mathematics Olympiad
the 14th XMO
the 14th XMO
Part of
XES Mathematics Olympiad
Subcontests
(4)
P4
1
Hide problems
123 = 246
In an
n
n
n
by
n
n
n
grid, each cell is filled with an integer between
1
1
1
and
6
6
6
. The outmost cells all contain the number
1
1
1
, and any two cells that share a vertex has difference not equal to
3
3
3
. For any vertex
P
P
P
inside the grid (not including the boundary), there are
4
4
4
cells that have
P
P
P
has a vertex. If these four cells have exactly three distinct numbers
i
i
i
,
j
j
j
,
k
k
k
(two cells have the same number), and the two cells with the same number have a common side, we call
P
P
P
an
i
j
k
ijk
ijk
-type vertex. Let there be
A
i
j
k
A_{ijk}
A
ijk
vertices that are
i
j
k
ijk
ijk
-type. Prove that
A
123
≡
A
246
(
m
o
d
2
)
A_{123}\equiv A_{246} \pmod 2
A
123
≡
A
246
(
mod
2
)
.
P3
1
Hide problems
PG // EF
In quadrilateral
A
B
C
D
ABCD
A
BC
D
,
E
E
E
and
F
F
F
are midpoints of
A
B
AB
A
B
and
C
D
CD
C
D
, and
G
G
G
is the intersection of
A
D
AD
A
D
with
B
C
BC
BC
.
P
P
P
is a point within the quadrilateral, such that
P
A
=
P
B
PA=PB
P
A
=
PB
,
P
C
=
P
D
PC=PD
PC
=
P
D
, and
∠
A
P
B
+
∠
C
P
D
=
18
0
∘
\angle APB+\angle CPD=180^{\circ}
∠
A
PB
+
∠
CP
D
=
18
0
∘
. Prove that
P
G
PG
PG
and
E
F
EF
EF
are parallel.
P2
1
Hide problems
Fractional parts
Let
p
p
p
be a prime. Define
f
n
(
k
)
f_n(k)
f
n
(
k
)
to be the number of positive integers
1
≤
x
≤
p
−
1
1\leq x\leq p-1
1
≤
x
≤
p
−
1
such that
(
{
x
p
}
−
{
k
p
}
)
(
{
n
x
p
}
−
{
k
p
}
)
<
0.
\left(\left\{\frac{x}{p}\right\}-\left\{\frac{k}{p}\right\}\right)\left(\left\{\frac{nx}{p}\right\}-\left\{\frac{k}{p}\right\}\right)<0.
(
{
p
x
}
−
{
p
k
}
)
(
{
p
n
x
}
−
{
p
k
}
)
<
0.
Let
a
n
=
f
n
(
1
2
)
+
f
n
(
3
2
)
+
⋯
+
f
n
(
2
p
−
1
2
)
a_n=f_n\left(\frac 12\right)+f_n\left(\frac 32\right)+\dots+f_n\left(\frac{2p-1}{2}\right)
a
n
=
f
n
(
2
1
)
+
f
n
(
2
3
)
+
⋯
+
f
n
(
2
2
p
−
1
)
, find
min
{
a
2
,
a
3
,
…
,
a
p
−
1
}
\min\{a_2, a_3, \dots, a_{p-1}\}
min
{
a
2
,
a
3
,
…
,
a
p
−
1
}
.
P1
1
Hide problems
Rounding distances
Nonnegative reals
x
1
x_1
x
1
,
x
2
x_2
x
2
,
…
\dots
…
,
x
n
x_n
x
n
satisfies
x
1
+
x
2
+
⋯
+
x
n
=
n
x_1+x_2+\dots+x_n=n
x
1
+
x
2
+
⋯
+
x
n
=
n
. Let
∣
∣
x
∣
∣
||x||
∣∣
x
∣∣
be the distance from
x
x
x
to the nearest integer of
x
x
x
(e.g.
∣
∣
3.8
∣
∣
=
0.2
||3.8||=0.2
∣∣3.8∣∣
=
0.2
,
∣
∣
4.3
∣
∣
=
0.3
||4.3||=0.3
∣∣4.3∣∣
=
0.3
). Let
y
i
=
x
i
∣
∣
x
i
∣
∣
y_i = x_i ||x_i||
y
i
=
x
i
∣∣
x
i
∣∣
. Find the maximum value of
∑
i
=
1
n
y
i
2
\sum_{i=1}^n y_i^2
∑
i
=
1
n
y
i
2
.