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Problems
Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
2017 Finnish National High School Mathematics Comp
2017 Finnish National High School Mathematics Comp
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
1
1
Hide problems
euclidean division problem for starters
By dividing the integer
m
m
m
by the integer
n
,
22
n, 22
n
,
22
is the quotient and
5
5
5
the remainder. As the division of the remainder with
n
n
n
continues, the new quotient is
0.4
0.4
0.4
and the new remainder is
0.2
0.2
0.2
. Find
m
m
m
and
n
n
n
.
4
1
Hide problems
game with integers, winning strategy the one who plays first
Let
m
m
m
be a positive integer. Two players, Axel and Elina play the game HAUKKU (
m
m
m
) proceeds as follows: Axel starts and the players choose integers alternately. Initially, the set of integers is the set of positive divisors of a positive integer
m
m
m
.The player in turn chooses one of the remaining numbers, and removes that number and all of its multiples from the list of selectable numbers. A player who has to choose number
1
1
1
, loses. Show that the beginner player, Axel, has a winning strategy in the HAUKKU (
m
m
m
) game for all
m
∈
Z
+
m \in Z_{+}
m
∈
Z
+
.PS. As member Loppukilpailija noted, it should be written
m
>
1
m>1
m
>
1
, as the statement does not hold for
m
=
1
m = 1
m
=
1
.
5
1
Hide problems
A, B, and Q lie on a circle with center T
Let
A
A
A
and
B
B
B
be two arbitrary points on the circumference of the circle such that
A
B
AB
A
B
is not the diameter of the circle. The tangents to the circle drawn at points
A
A
A
and
B
B
B
meet at
T
T
T
. Next, choose the diameter
X
Y
XY
X
Y
so that the segments
A
X
AX
A
X
and
B
Y
BY
B
Y
intersect. Let this be the intersection of
Q
Q
Q
. Prove that the points
A
,
B
A, B
A
,
B
, and
Q
Q
Q
lie on a circle with center
T
T
T
.
3
1
Hide problems
22 220 038^m-22 220 038^n has 8 zeros at the end
Consider positive integers
m
m
m
and
n
n
n
for which
m
>
n
m> n
m
>
n
and the number
2222003
8
m
−
2222003
8
n
22 220 038^m-22 220 038^n
2222003
8
m
−
2222003
8
n
has are eight zeros at the end. Show that
n
>
7
n> 7
n
>
7
.
2
1
Hide problems
calculate x^2+y^2 and x^4+y^4 when x^3+y^3=2 and x+y=1
Determine
x
2
+
y
2
x^2+y^2
x
2
+
y
2
and
x
4
+
y
4
x^4+y^4
x
4
+
y
4
, when
x
3
+
y
3
=
2
x^3+y^3=2
x
3
+
y
3
=
2
and
x
+
y
=
1
x+y=1
x
+
y
=
1