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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1992 Swedish Mathematical Competition
1992 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
3
1
Hide problems
25x25 inequality system 2x_{25} - 5x_1 + 3x_2 >= 0
Solve:
{
2
x
1
−
5
x
2
+
3
x
3
≥
0
2
x
2
−
5
x
3
+
3
x
4
≥
0
.
.
.
2
x
23
−
5
x
24
+
3
x
25
≥
0
2
x
24
−
5
x
25
+
3
x
1
≥
0
2
x
25
−
5
x
1
+
3
x
2
≥
0
\begin{cases} 2x_1 - 5x_2 + 3x_3 \ge 0 \\ 2x_2 - 5x_3 + 3x4 \ge 0 \\ ...\\ 2x_{23} - 5x_{24} + 3x_{25} \ge 0\\ 2x_{24} - 5x_{25} + 3x_1 \ge 0\\ 2x_{25} - 5x_1 + 3x_2 \ge 0 \end{cases}
⎩
⎨
⎧
2
x
1
−
5
x
2
+
3
x
3
≥
0
2
x
2
−
5
x
3
+
3
x
4
≥
0
...
2
x
23
−
5
x
24
+
3
x
25
≥
0
2
x
24
−
5
x
25
+
3
x
1
≥
0
2
x
25
−
5
x
1
+
3
x
2
≥
0
4
1
Hide problems
diophantine inequalities a < b, a < 4c, b c^3 <= a c^3 + b
Find all positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
such that
a
<
b
a < b
a
<
b
,
a
<
4
c
a < 4c
a
<
4
c
, and
b
c
3
≤
a
c
3
+
b
b c^3 \le a c^3 + b
b
c
3
≤
a
c
3
+
b
.
6
1
Hide problems
x_1/y_1 + x_2/y_2 + x_3/y_3 = 0 if 3 collinear point lie on y^2 = x^3
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
,
(
x
3
,
y
3
)
(x_1, y_1), (x_2, y_2), (x_3, y_3)
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
,
(
x
3
,
y
3
)
lie on a straight line and on the curve
y
2
=
x
3
y^2 = x^3
y
2
=
x
3
. Show that
x
1
y
1
+
x
2
y
2
+
x
3
y
3
=
0
\frac{x_1}{y_1} + \frac{x_2}{y_2}+\frac{x_3}{y_3} = 0
y
1
x
1
+
y
2
x
2
+
y
3
x
3
=
0
.
5
1
Hide problems
right triangle if a^2 + b^2 = 2cR
A triangle has sides
a
,
b
,
c
a, b, c
a
,
b
,
c
with longest side
c
c
c
, and circumradius
R
R
R
. Show that if
a
2
+
b
2
=
2
c
R
a^2 + b^2 = 2cR
a
2
+
b
2
=
2
c
R
, then the triangle is right-angled.
2
1
Hide problems
11 to 99 in square grid 9x9
The squares in a
9
×
9
9\times 9
9
×
9
grid are numbered from
11
11
11
to
99
99
99
, where the first digit is the row and the second the column. Each square is colored black or white. Squares
44
44
44
and
49
49
49
are black. Every black square shares an edge with at most one other black square, and each white square shares an edge with at most one other white square. What color is square
99
99
99
?
1
1
Hide problems
is (19^{92} - 91^{29})/90 an integer?
Is
1
9
92
−
9
1
29
90
\frac{19^{92} - 91^{29}}{90}
90
1
9
92
−
9
1
29
an integer?