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Problems
Contests
National and Regional Contests
Italy Contests
Italy TST
1995 Italy TST
1995 Italy TST
Part of
Italy TST
Subcontests
(4)
4
1
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Locus of circumcentre of PQC as C varies
In a triangle
A
B
C
ABC
A
BC
,
P
P
P
and
Q
Q
Q
are the feet of the altitudes from
B
B
B
and
A
A
A
respectively. Find the locus of the circumcentre of triangle
P
Q
C
PQC
PQC
, when point
C
C
C
varies (with
A
A
A
and
B
B
B
fixed) in such a way that
∠
A
C
B
\angle ACB
∠
A
CB
is equal to
6
0
∘
60^{\circ}
6
0
∘
.
3
1
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Functional inequality condition
A function
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
satisfies the conditions \begin{cases}f(x+24)\le f(x)+24\\ f(x+77)\ge f(x)+77\end{cases} \text{for all}\ x\in\mathbb{R} Prove that
f
(
x
+
1
)
=
f
(
x
)
+
1
f(x+1)=f(x)+1
f
(
x
+
1
)
=
f
(
x
)
+
1
for all real
x
x
x
.
2
1
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21 rectangles on a 8 x 8 chessboard
Twenty-one rectangles of size
3
×
1
3\times 1
3
×
1
are placed on an
8
×
8
8\times 8
8
×
8
chessboard, leaving only one free unit square. What position can the free square lie at?
1
1
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Easy x|yz-1, y|zx-1, z|xy-1
Determine all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of integers greater than
1
1
1
with the property that
x
x
x
divides
y
z
−
1
yz-1
yz
−
1
,
y
y
y
divides
z
x
−
1
zx-1
z
x
−
1
and
z
z
z
divides
x
y
−
1
xy-1
x
y
−
1
.