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National and Regional Contests
France Contests
France Team Selection Test
2000 France Team Selection Test
2000 France Team Selection Test
Part of
France Team Selection Test
Subcontests
(3)
3
2
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France TST 2000 D1Q3
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are positive reals with sum
1
1
1
. Show that
a
2
a
+
b
+
b
2
b
+
c
+
c
2
c
+
d
+
d
2
d
+
a
≥
1
2
\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+d}+\frac{d^2}{d+a} \ge \frac{1}{2}
a
+
b
a
2
+
b
+
c
b
2
+
c
+
d
c
2
+
d
+
a
d
2
≥
2
1
with equality iff
a
=
b
=
c
=
d
=
1
4
a=b=c=d=\frac{1}{4}
a
=
b
=
c
=
d
=
4
1
.
France TST 2000 D2Q3
Find all nonnegative integers
x
,
y
,
z
x,y,z
x
,
y
,
z
such that
(
x
+
1
)
y
+
1
+
1
=
(
x
+
2
)
z
+
1
(x+1)^{y+1} + 1= (x+2)^{z+1}
(
x
+
1
)
y
+
1
+
1
=
(
x
+
2
)
z
+
1
.
2
2
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France TST 2000 D2Q2
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
are points on a circle in that order. Prove that
∣
A
B
−
C
D
∣
+
∣
A
D
−
B
C
∣
≥
2
∣
A
C
−
B
D
∣
|AB-CD|+|AD-BC| \ge 2|AC-BD|
∣
A
B
−
C
D
∣
+
∣
A
D
−
BC
∣
≥
2∣
A
C
−
B
D
∣
.
France TST 2000 D1Q2
A function from the positive integers to the positive integers satisfies these properties 1.
f
(
a
b
)
=
f
(
a
)
f
(
b
)
f(ab)=f(a)f(b)
f
(
ab
)
=
f
(
a
)
f
(
b
)
for any two coprime positive integers
a
,
b
a,b
a
,
b
. 2.
f
(
p
+
q
)
=
f
(
p
)
+
f
(
q
)
f(p+q)=f(p)+f(q)
f
(
p
+
q
)
=
f
(
p
)
+
f
(
q
)
for any two primes
p
,
q
p,q
p
,
q
. Prove that
f
(
2
)
=
2
,
f
(
3
)
=
3
,
f
(
1999
)
=
1999
f(2)=2, f(3)=3, f(1999)=1999
f
(
2
)
=
2
,
f
(
3
)
=
3
,
f
(
1999
)
=
1999
.
1
2
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France TST 2000 D1Q1
Points
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
lie on a circle and
∠
P
S
R
\angle PSR
∠
PSR
is right.
H
,
K
H,K
H
,
K
are the projections of
Q
Q
Q
on lines
P
R
,
P
S
PR,PS
PR
,
PS
. Prove that
H
K
HK
HK
bisects segment
Q
S
QS
QS
.
France TST 2000 D2Q1
Some squares of a
1999
×
1999
1999\times 1999
1999
×
1999
board are occupied with pawns. Find the smallest number of pawns for which it is possible that for each empty square, the total number of pawns in the row or column of that square is at least
1999
1999
1999
.