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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1977 Swedish Mathematical Competition
1977 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
Hide problems
a^2 + b^2 + c^2 > 2, a^3 + b^3 + c^3 <2, a^4 + b^4 + c^4 > 2
Show that there are positive reals
a
a
a
,
b
b
b
,
c
c
c
such that
{
a
2
+
b
2
+
c
2
>
2
a
3
+
b
3
+
c
3
<
2
a
4
+
b
4
+
c
4
>
2
\left\{ \begin{array}{l} a^2 + b^2 + c^2 > 2 \\ a^3 + b^3 + c^3 <2 \\ a^4 + b^4 + c^4 > 2 \\ \end{array} \right.
⎩
⎨
⎧
a
2
+
b
2
+
c
2
>
2
a
3
+
b
3
+
c
3
<
2
a
4
+
b
4
+
c
4
>
2
5
1
Hide problems
at least four 2x2 squares have sum greater than 100 , in a 8x8 board
The numbers
1
,
2
,
3
,
.
.
.
,
64
1, 2, 3, ... , 64
1
,
2
,
3
,
...
,
64
are written in the cells of an
8
×
8
8 \times 8
8
×
8
board (in some order, one per cell). Show that at least four
2
×
2
2 \times 2
2
×
2
squares have sum greater than
100
100
100
.
4
1
Hide problems
cos^3 y / cos x + sin^3 y/ sin x =1 if cos y / cos x + sin y/ sin x =-1
Show that if
cos
x
cos
y
+
sin
x
sin
y
=
−
1
\frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1
cos
y
cos
x
+
sin
y
sin
x
=
−
1
then
cos
3
y
cos
x
+
sin
3
y
sin
x
=
1
\frac{\cos^3 y}{\cos x}+\frac{\sin^3 y}{\sin x}=1
cos
x
cos
3
y
+
sin
x
sin
3
y
=
1
2
1
Hide problems
3, 4, 5 distances of an interior point of equilateral from vertices
There is a point inside an equilateral triangle side
d
d
d
whose distance from the vertices is
3
,
4
,
5
3, 4, 5
3
,
4
,
5
. Find
d
d
d
.
3
1
Hide problems
xy + yz + zx = 3n^2 - 1, x + y + z = 3n , when x >=y>= z
Show that the only integral solution to
{
x
y
+
y
z
+
z
x
=
3
n
2
−
1
x
+
y
+
z
=
3
n
\left\{ \begin{array}{l} xy + yz + zx = 3n^2 - 1\\ x + y + z = 3n \\ \end{array} \right.
{
x
y
+
yz
+
z
x
=
3
n
2
−
1
x
+
y
+
z
=
3
n
with
x
≥
y
≥
z
x \geq y \geq z
x
≥
y
≥
z
is
x
=
n
+
1
x=n+1
x
=
n
+
1
,
y
=
n
y=n
y
=
n
,
z
=
n
−
1
z=n-1
z
=
n
−
1
.
1
1
Hide problems
largest integer d such that p^d divides p^4!
p
p
p
is a prime. Find the largest integer
d
d
d
such that
p
d
p^d
p
d
divides
p
4
!
p^4!
p
4
!
.