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Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
2015 Finnish National High School Mathematics Comp
2015 Finnish National High School Mathematics Comp
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
5
1
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6 out of 10 questions correct, probability p and neg. how many for 7 points?
Mikko takes a multiple choice test with ten questions. His only goal is to pass the test, and this requires seven points. A correct answer is worth one point, and answering wrong results in the deduction of one point. Mikko knows for sure that he knows the correct answer in the six first questions. For the rest, he estimates that he can give the correct answer to each problem with probability
p
,
0
<
p
<
1
p, 0 < p < 1
p
,
0
<
p
<
1
. How many questions Mikko should try?
4
1
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bw squares in nxn grid, coloring with limits on 2x2 square coloring
Let
n
n
n
be a positive integer. Every square in a
n
×
n
n \times n
n
×
n
-square grid is either white or black. How many such colourings exist, if every
2
×
2
2 \times 2
2
×
2
-square consists of exactly two white and two black squares? The squares in the grid are identified as e.g. in a chessboard, so in general colourings obtained from each other by rotation are different.
3
1
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largest k so that 12^k is a factor of 120!
Determine the largest integer
k
k
k
for which
1
2
k
12^k
1
2
k
is a factor of
120
!
120!
120
!
2
1
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volume of a lateral right square pyramid
The lateral edges of a right square pyramid are of length
a
a
a
. Let
A
B
C
D
ABCD
A
BC
D
be the base of the pyramid,
E
E
E
its top vertex and
F
F
F
the midpoint of
C
E
CE
CE
. Assuming that
B
D
F
BDF
B
D
F
is an equilateral triangle, compute the volume of the pyramid.
1
1
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\sqrt{1+ \sqrt {1+x}}=\sqrt[3]{x}
Solve the equation
1
+
1
+
x
=
x
3
\sqrt{1+\sqrt {1+x}}=\sqrt[3]{x}
1
+
1
+
x
=
3
x
for
x
≥
0
x \ge 0
x
≥
0
.