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Problems
Contests
National and Regional Contests
Nigeria Contests
Nigerian Senior Mathematics Olympiad Round 4
2019 Nigerian Senior MO Round 4
2019 Nigerian Senior MO Round 4
Part of
Nigerian Senior Mathematics Olympiad Round 4
Subcontests
(4)
3
1
Hide problems
ant moves on the cartesian plane, broken line rational and integer
An ant is moving on the cooridnate plane, starting form point
(
0
,
−
1
)
(0,-1)
(
0
,
−
1
)
along a straight line until it reaches the
x
x
x
- axis at point
(
x
,
0
)
(x,0)
(
x
,
0
)
where
x
x
x
is a real number. After it turns
9
0
o
90^o
9
0
o
to the left and moves again along a straight line until it reaches the
y
y
y
-axis . Then it again turns left and moves along a straight line until it reaches the
x
x
x
-axis, where it once more turns left by
9
0
o
90^o
9
0
o
and moves along a straight line until it finally reached the
y
y
y
-axis. Can both the length of the ant's journey and distance between it's initial and final point be: (a) rational numbers ? (b) integers? Justify your answersPS. Collected [url=https://artofproblemsolving.com/community/c949609_2019_nigerian_senior_mo_round_4]here
4
1
Hide problems
x_{n+2}=3x_{n+1}-2 x_n , y_n=x^2_n+2^{n+2}, y_n is odd's perfect square
We consider the real sequence (
x
n
x_n
x
n
) defined by
x
0
=
0
,
x
1
=
1
x_0=0, x_1=1
x
0
=
0
,
x
1
=
1
and
x
n
+
2
=
3
x
n
+
1
−
2
x
n
x_{n+2}=3x_{n+1}-2 x_{n}
x
n
+
2
=
3
x
n
+
1
−
2
x
n
for
n
=
0
,
1
,
2
,
.
.
.
n=0,1,2,...
n
=
0
,
1
,
2
,
...
We define the sequence (
y
n
y_n
y
n
) by
y
n
=
x
n
2
+
2
n
+
2
y_n=x^2_n+2^{n+2}
y
n
=
x
n
2
+
2
n
+
2
for every nonnegative integer
n
n
n
. Prove that for every
n
>
0
,
y
n
n>0, y_n
n
>
0
,
y
n
is the square of an odd integer.
2
1
Hide problems
LX and BC meet at given circle, perpendularity wanted
Let
K
,
L
,
M
K,L, M
K
,
L
,
M
be the midpoints of
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
repectively on a given triangle
A
B
C
ABC
A
BC
. Let
Γ
\Gamma
Γ
be a circle passing through
B
B
B
and tangent to the circumcircle of
K
L
M
KLM
K
L
M
, say at
X
X
X
. Suppose that
L
X
LX
L
X
and
B
C
BC
BC
meet at
Γ
\Gamma
Γ
. Show that
C
X
CX
CX
is perpendicular to
A
B
AB
A
B
.
1
1
Hide problems
1 <= f(x)-x<= 2019, f(f(x))= x mod 2019 , f^k(x)=x+2019 k
Let
f
:
N
→
N
f: N \to N
f
:
N
→
N
be a function satisfying (a)
1
≤
f
(
x
)
−
x
≤
2019
1\le f(x)-x \le 2019
1
≤
f
(
x
)
−
x
≤
2019
∀
x
∈
N
\forall x \in N
∀
x
∈
N
(b)
f
(
f
(
x
)
)
≡
x
f(f(x))\equiv x
f
(
f
(
x
))
≡
x
(mod
2019
2019
2019
)
∀
x
∈
N
\forall x \in N
∀
x
∈
N
Show that
∃
x
∈
N
\exists x \in N
∃
x
∈
N
such that
f
k
(
x
)
=
x
+
2019
k
,
∀
k
∈
N
f^k(x)=x+2019 k, \forall k \in N
f
k
(
x
)
=
x
+
2019
k
,
∀
k
∈
N