MathDB
Problems
Contests
National and Regional Contests
Romania Contests
Romania EGMO Team Selection Test
2023 Romania EGMO TST
2023 Romania EGMO TST
Part of
Romania EGMO Team Selection Test
Subcontests
(3)
P4
1
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Inequality with absolute value
Let
n
⩾
3
n\geqslant 3
n
⩾
3
be an integer and
a
1
,
…
,
a
n
a_1,\ldots,a_n
a
1
,
…
,
a
n
be nonzero real numbers, with sum
S
S{}
S
. Prove that
∑
i
=
1
n
∣
S
−
a
i
a
i
∣
⩾
n
−
1
n
−
2
.
\sum_{i=1}^n\left|\frac{S-a_i}{a_i}\right|\geqslant\frac{n-1}{n-2}.
i
=
1
∑
n
a
i
S
−
a
i
⩾
n
−
2
n
−
1
.
P3
1
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Geometrical inequality
Let
D
D{}
D
be a point inside the triangle
A
B
C
ABC
A
BC
. Let
E
E{}
E
and
F
F{}
F
be the projections of
D
D{}
D
onto
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. The lines
B
D
BD
B
D
and
C
D
CD
C
D
intersect the circumcircle of
A
B
C
ABC
A
BC
the second time at
M
M{}
M
and
N
N{}
N
, respectively. Prove that
E
F
M
N
⩾
r
R
,
\frac{EF}{MN}\geqslant \frac{r}{R},
MN
EF
⩾
R
r
,
where
r
r{}
r
and
R
R{}
R
are the inradius and circumradius of
A
B
C
ABC
A
BC
, respectively.
P1
1
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St Petersburg 2008 #7
A square with side
2008
2008
2008
is broken into regions that are all squares with side
1
1
1
. In every region, either
0
0
0
or
1
1
1
is written, and the number of
1
1
1
's and
0
0
0
's is the same. The border between two of the regions is removed, and the numbers in each of them are also removed, while in the new region, their arithmetic mean is recorded. After several of those operations, there is only one square left, which is the big square itself. Prove that it is possible to perform these operations in such a way, that the final number in the big square is less than
1
2
1
0
6
\frac{1}{2^{10^6}}
2
1
0
6
1
.