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Problems
Contests
International Contests
IberoAmerican
1989 IberoAmerican
1989 IberoAmerican
Part of
IberoAmerican
Subcontests
(3)
3
2
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Side lengths of a triangle inequality
Let
a
,
b
a,b
a
,
b
and
c
c
c
be the side lengths of a triangle. Prove that:
a
−
b
a
+
b
+
b
−
c
b
+
c
+
c
−
a
c
+
a
<
1
16
\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}<\frac{1}{16}
a
+
b
a
−
b
+
b
+
c
b
−
c
+
c
+
a
c
−
a
<
16
1
Infinitely many solutions to diophantine equation
Show that the equation
2
x
2
−
3
x
=
3
y
2
2x^2-3x=3y^2
2
x
2
−
3
x
=
3
y
2
has infinitely many solutions in positive integers.
2
2
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Trigonometric inequality for non-obtuse angles
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be real numbers such that
0
≤
x
,
y
,
z
≤
π
2
0\le x,y,z\le\frac{\pi}{2}
0
≤
x
,
y
,
z
≤
2
π
. Prove the inequality
π
2
+
2
sin
x
cos
y
+
2
sin
y
cos
z
≥
sin
2
x
+
sin
2
y
+
sin
2
z
.
\frac{\pi}{2}+2\sin x\cos y+2\sin y\cos z\ge\sin 2x+\sin 2y+\sin 2z.
2
π
+
2
sin
x
cos
y
+
2
sin
y
cos
z
≥
sin
2
x
+
sin
2
y
+
sin
2
z
.
Function defined on N with conditions
Let the function
f
f
f
be defined on the set
N
\mathbb{N}
N
such that\text{(i)}\ \ f(1)=1 \text{(ii)}\ f(2n+1)=f(2n)+1 \text{(iii)} f(2n)=3f(n)Determine the set of values taken
f
f
f
.
1
2
Hide problems
x,y,z system of equations
Determine all triples of real numbers that satisfy the following system of equations:
x
+
y
−
z
=
−
1
x
2
−
y
2
+
z
2
=
1
−
x
3
+
y
3
+
z
3
=
−
1
x+y-z=-1\\ x^2-y^2+z^2=1\\ -x^3+y^3+z^3=-1
x
+
y
−
z
=
−
1
x
2
−
y
2
+
z
2
=
1
−
x
3
+
y
3
+
z
3
=
−
1
Prove M P · OA = BC · OQ
The incircle of the triangle
A
B
C
ABC
A
BC
is tangent to sides
A
C
AC
A
C
and
B
C
BC
BC
at
M
M
M
and
N
N
N
, respectively. The bisectors of the angles at
A
A
A
and
B
B
B
intersect
M
N
MN
MN
at points
P
P
P
and
Q
Q
Q
, respectively. Let
O
O
O
be the incentre of
△
A
B
C
\triangle ABC
△
A
BC
. Prove that
M
P
⋅
O
A
=
B
C
⋅
O
Q
MP\cdot OA=BC\cdot OQ
MP
⋅
O
A
=
BC
⋅
OQ
.