MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
1985 Vietnam Team Selection Test
1985 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
2
2
Hide problems
The locus of the incenter of triangle SBC
Let
A
B
C
ABC
A
BC
be a triangle with AB \equal{} AC. A ray
A
x
Ax
A
x
is constructed in space such that the three planar angles of the trihedral angle
A
B
C
x
ABCx
A
BC
x
at its vertex
A
A
A
are equal. If a point
S
S
S
moves on
A
x
Ax
A
x
, find the locus of the incenter of triangle
S
B
C
SBC
SBC
.
The equation has an odd number of solutions
Find all real values of a for which the equation (a \minus{} 3x^2 \plus{} \cos \frac {9\pi x}{2})\sqrt {3 \minus{} ax} \equal{} 0 has an odd number of solutions in the interval [ \minus{} 1,5]
3
2
Hide problems
Existence of triangle satisfies trigonometric conditions
Does there exist a triangle
A
B
C
ABC
A
BC
satisfying the following two conditions: (a)
sin
2
A
+
sin
2
B
+
sin
2
C
=
cot
A
+
cot
B
+
cot
C
{ \sin^2A + \sin^2B + \sin^2C = \cot A + \cot B + \cot C}
sin
2
A
+
sin
2
B
+
sin
2
C
=
cot
A
+
cot
B
+
cot
C
(b)
S
≥
a
2
−
(
b
−
c
)
2
S\ge a^2 - (b - c)^2
S
≥
a
2
−
(
b
−
c
)
2
where
S
S
S
is the area of the triangle
A
B
C
ABC
A
BC
.
infinitely many points of discontinuity
Suppose a function
f
:
R
→
R
f: \mathbb R\to \mathbb R
f
:
R
→
R
satisfies f(f(x)) \equal{} \minus{} x for all
x
∈
R
x\in \mathbb R
x
∈
R
. Prove that
f
f
f
has infinitely many points of discontinuity.
1
2
Hide problems
Real number between two subsequences
The sequence
(
x
n
)
(x_n)
(
x
n
)
of real numbers is defined by x_1\equal{}\frac{29}{10} and x_{n\plus{}1}\equal{}\frac{x_n}{\sqrt{x_n^2\minus{}1}}\plus{}\sqrt{3} for all
n
≥
1
n\ge 1
n
≥
1
. Find a real number
a
a
a
(if exists) such that x_{2k\minus{}1}>a>x_{2k}.
Inequality with sum of inradii in polygon
A convex polygon
A
1
,
A
2
,
⋯
,
A
n
A_1,A_2,\cdots ,A_n
A
1
,
A
2
,
⋯
,
A
n
is inscribed in a circle with center
O
O
O
and radius
R
R
R
so that
O
O
O
lies inside the polygon. Let the inradii of the triangles A_1A_2A_3, A_1A_3A_4, \cdots , A_1A_{n \minus{} 1}A_n be denoted by r_1,r_2,\cdots ,r_{n \minus{} 2}. Prove that r_1 \plus{} r_2 \plus{} ... \plus{} r_{n \minus{} 2}\leq R(n\cos \frac {\pi}{n} \minus{} n \plus{} 2).