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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2010 Swedish Mathematical Competition
2010 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
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squares among red squares on an infinite grid
An infinite number of squares on an infinitely square grid paper are painted red. Show that you can draw a number of squares on the paper, with sides along the grid lines, such that:(1) no square in the grid belongs to more than one square (an edge, on the other hand, may belong to more than one square)(2) each red square is located in one of the squares and the number of red squares in such square is at least
1
/
5
1/5
1/5
and at most
4
/
5
4/5
4/5
of the number of squares in the square.[hide=original wording] Ett andligt antal rutor pa ett oandligt rutat papper ar malade roda. Visa att man pa papperet kan rita in ett antal kvadrater, med sidor utefter rutnatets linjer, sadana att : (1) ingen ruta i natet tillhor mer an en kvadrat (en kant kan daremot tillhora mer an en kvadrat), (2) varje rod ruta ligger i nagon av kvadraterna och antalet roda rutor i en sadan kvadratar minst 1/5 och hogst 4/5 av antalet rutor i kvadraten.[url=http://www.mattetavling.se/wp-content/uploads/2011/01/Final10.pdf]sourcePS. I always post the original wording when I doubt about my (using Google) translation.
4
1
Hide problems
a_{100},a_{101},a_{102},... form arithmetic sequence , a_n is divisible by n
We create a sequence by setting
a
1
=
2010
a_1 = 2010
a
1
=
2010
and requiring that
a
n
−
a
n
−
1
≤
n
a_n-a_{n-1}\leq n
a
n
−
a
n
−
1
≤
n
and
a
n
a_n
a
n
is also divisible by
n
n
n
. Show that
a
100
,
a
101
,
a
102
,
…
a_{100},a_{101},a_{102},\dots
a
100
,
a
101
,
a
102
,
…
form an arithmetic sequence.
2
1
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x_1 + x_4 =x_2+x_3, y_1y_4 =y_2y_3 wanted, points on y = mx-k^2
Consider the four lines
y
=
m
x
−
k
2
y = mx-k^2
y
=
m
x
−
k
2
for different integer
k
k
k
. Let
(
x
i
,
y
i
)
(x_i,y_i)
(
x
i
,
y
i
)
,
i
=
1
,
2
,
3
,
4
i = 1,2,3,4
i
=
1
,
2
,
3
,
4
be four different points , such that each belongs to two different lines and on each line pass through just the two of them. Lat
x
1
≤
x
2
≤
x
3
≤
x
4
x_1\leq x_2\leq x_3\leq x_4
x
1
≤
x
2
≤
x
3
≤
x
4
. Show that
x
1
+
x
4
=
x
2
+
x
3
x_1 + x_4 =x_2+x_3
x
1
+
x
4
=
x
2
+
x
3
and
y
1
y
4
=
y
2
y
3
y_1y_4 =y_2y_3
y
1
y
4
=
y
2
y
3
.
3
1
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integer p(x), p (1) = p (2) = 0, p (n) is a prime ,
Find all natural numbers
n
≥
1
n \ge 1
n
≥
1
such that there is a polynomial
p
(
x
)
p(x)
p
(
x
)
with integer coefficients for which
p
(
1
)
=
p
(
2
)
=
0
p (1) = p (2) = 0
p
(
1
)
=
p
(
2
)
=
0
and where
p
(
n
)
p (n)
p
(
n
)
is a prime number .
1
1
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triangle whose three altitudes have lengths 1, 2 ,3
Exists a triangle whose three altitudes have lengths
1
,
2
1, 2
1
,
2
and
3
3
3
respectively?
5
1
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sidelengths (a + b + c) (a + b -c) = 2b^2 and a is the largest possible side
Consider the number of triangles where the side lengths
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy
(
a
+
b
+
c
)
(
a
+
b
−
c
)
=
2
b
2
(a + b + c) (a + b -c) = 2b^2
(
a
+
b
+
c
)
(
a
+
b
−
c
)
=
2
b
2
. Determine the angles in the triangle for which the angle opposite to the side with the length
a
a
a
is as big as possible.