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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1982 Swedish Mathematical Competition
1982 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
Hide problems
(2a-1) sin x + (1-a) sin(1-a)x >= 0 for 0 <=a <= 1$ and 0 <= x <= \pi
Show that
(
2
a
−
1
)
sin
x
+
(
1
−
a
)
sin
(
1
−
a
)
x
≥
0
(2a-1) \sin x + (1-a) \sin(1-a)x \geq 0
(
2
a
−
1
)
sin
x
+
(
1
−
a
)
sin
(
1
−
a
)
x
≥
0
for
0
≤
a
≤
1
0 \leq a \leq 1
0
≤
a
≤
1
and
0
≤
x
≤
π
0 \leq x \leq \pi
0
≤
x
≤
π
.
5
1
Hide problems
4 points of same color forming a rectangle in a 12x12 array, in 3 colours
Each point in a
12
×
12
12 \times 12
12
×
12
array is colored red, white or blue. Show that it is always possible to find
4
4
4
points of the same color forming a rectangle with sides parallel to the sides of the array.
4
1
Hide problems
AD = DE = EC = n, AB = 33, AC = 21,BC = m where m,n integers
A
B
C
ABC
A
BC
is a triangle with
A
B
=
33
AB = 33
A
B
=
33
,
A
C
=
21
AC = 21
A
C
=
21
and
B
C
=
m
BC = m
BC
=
m
, an integer. There are points
D
D
D
,
E
E
E
on the sides
A
B
AB
A
B
,
A
C
AC
A
C
respectively such that
A
D
=
D
E
=
E
C
=
n
AD = DE = EC = n
A
D
=
D
E
=
EC
=
n
, an integer. Find
m
m
m
.
3
1
Hide problems
point P in ABCD such that (PAB)=(PBC)=(PCD)=(PDA)
Show that there is a point
P
P
P
inside the quadrilateral
A
B
C
D
ABCD
A
BC
D
such that the triangles
P
A
B
PAB
P
A
B
,
P
B
C
PBC
PBC
,
P
C
D
PCD
PC
D
,
P
D
A
PDA
P
D
A
have equal area. Show that
P
P
P
must lie on one of the diagonals.
2
1
Hide problems
abc >= (a+b-c)(b+c-a)(c+a-b) for a,b,c>0
Show that
a
b
c
≥
(
a
+
b
−
c
)
(
b
+
c
−
a
)
(
c
+
a
−
b
)
abc \geq (a+b-c)(b+c-a)(c+a-b)
ab
c
≥
(
a
+
b
−
c
)
(
b
+
c
−
a
)
(
c
+
a
−
b
)
for positive reals
a
a
a
,
b
b
b
,
c
c
c
.
1
1
Hide problems
no of solutions of x^2 - [x^2] = \left(x - [x]\right)^2 if 1 <= x <=n
How many solutions does
x
2
−
[
x
2
]
=
(
x
−
[
x
]
)
2
x^2 - [x^2] = \left(x - [x]\right)^2
x
2
−
[
x
2
]
=
(
x
−
[
x
]
)
2
have satisfying
1
≤
x
≤
n
1 \leq x \leq n
1
≤
x
≤
n
?