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National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
2011 Finnish National High School Mathematics Competition
2011 Finnish National High School Mathematics Competition
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
5
1
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Building an arithmetic progression in a game
Two players, the builder and the destroyer, plays the following game. Builder starts and players chooses alternatively different elements from the set
{
0
,
1
,
…
,
10
}
.
\{0,1,\ldots,10\}.
{
0
,
1
,
…
,
10
}
.
Builder wins if some four integer of those six integer he chose forms an arithmetic sequence. Destroyer wins if he can prevent to form such an arithmetic four-tuple. Which one has a winning strategy?
4
1
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Perfect square with sum of digits 2011
Show that there is a perfect square (a number which is a square of an integer) such that sum of its digits is
2011.
2011.
2011.
3
1
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In isosceles ABC, angle BAD < angle DAE
Points
D
D
D
and
E
E
E
divides the base
B
C
BC
BC
of an isosceles triangle
A
B
C
ABC
A
BC
into three equal parts and
D
D
D
is between
B
B
B
and
E
.
E.
E
.
Show that
∠
B
A
D
<
∠
D
A
E
.
\angle BAD<\angle DAE.
∠
B
A
D
<
∠
D
A
E
.
2
1
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Solve $x^4-12x^2+x^2y^2+30\leq 0$ in Z
Find all integers
x
x
x
and
y
y
y
satisfying the inequality
x
4
−
12
x
2
+
x
2
y
2
+
30
≤
0.
x^4-12x^2+x^2y^2+30\leq 0.
x
4
−
12
x
2
+
x
2
y
2
+
30
≤
0.
1
1
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Bisect the area of an equilateral triangle and inscribe a ci
An equilateral triangle has been drawn inside the circle. Split the triangle to two parts with equal area by a line segment parallel to the triangle side. Draw an inscribed circle inside this smaller triangle. What is the ratio of the area of this circle compared to the area of original circle.