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National and Regional Contests
Italy Contests
Italy TST
2000 Italy TST
2000 Italy TST
Part of
Italy TST
Subcontests
(4)
1
1
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Fraction diophantine equation
Determine all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of positive integers such that
13
x
2
+
1996
y
2
=
z
1997
\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997}
x
2
13
+
y
2
1996
=
1997
z
2
1
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angle bisection
Let
A
B
C
ABC
A
BC
be an isosceles right triangle and
M
M
M
be the midpoint of its hypotenuse
A
B
AB
A
B
. Points
D
D
D
and
E
E
E
are taken on the legs
A
C
AC
A
C
and
B
C
BC
BC
respectively such that
A
D
=
2
D
C
AD=2DC
A
D
=
2
D
C
and
B
E
=
2
E
C
BE=2EC
BE
=
2
EC
. Lines
A
E
AE
A
E
and
D
M
DM
D
M
intersect at
F
F
F
. Show that
F
C
FC
FC
bisects the
∠
D
F
E
\angle DFE
∠
D
FE
.
3
1
Hide problems
sequence bounding
Given positive numbers
a
1
a_1
a
1
and
b
1
b_1
b
1
, consider the sequences defined by a_{n+1}=a_n+\frac{1}{b_n}, b_{n+1}=b_n+\frac{1}{a_n} (n \ge 1) Prove that
a
25
+
b
25
≥
10
2
a_{25}+b_{25} \geq 10\sqrt{2}
a
25
+
b
25
≥
10
2
.
4
1
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problems in competition
On a mathematical competition
n
n
n
problems were given. The final results showed that: (i) on each problem, exactly three contestants scored
7
7
7
points; (ii) for each pair of problems, exactly one contestant scored
7
7
7
points on both problems. Prove that if
n
≥
8
n \geq 8
n
≥
8
, then there is a contestant who got
7
7
7
points on each problem. Is this statement necessarily true if n \equal{} 7?