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Problems
Contests
National and Regional Contests
Ecuador Contests
Ecuador Juniors
2018 Ecuador Juniors
2018 Ecuador Juniors
Part of
Ecuador Juniors
Subcontests
(6)
4
1
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spiral by similar triangles
Given a positive integer
n
>
1
n > 1
n
>
1
and an angle
α
<
9
0
o
\alpha < 90^o
α
<
9
0
o
, Jaime draws a spiral
O
P
0
P
1
.
.
.
P
n
OP_0P_1...P_n
O
P
0
P
1
...
P
n
of the following form (the figure shows the first steps):
∙
\bullet
∙
First draw a triangle
O
P
0
P
1
OP_0P_1
O
P
0
P
1
with
O
P
0
=
1
OP_0 = 1
O
P
0
=
1
,
∠
P
1
O
P
0
=
α
\angle P_1OP_0 = \alpha
∠
P
1
O
P
0
=
α
and
P
1
P
0
O
=
9
0
o
P_1P_0O = 90^o
P
1
P
0
O
=
9
0
o
∙
\bullet
∙
then for every integer
1
≤
i
≤
n
1 \le i \le n
1
≤
i
≤
n
draw the point
P
i
+
1
P_{i+1}
P
i
+
1
so that
∠
P
i
+
1
O
P
i
=
α
\angle P_{i+1}OP_i = \alpha
∠
P
i
+
1
O
P
i
=
α
,
∠
P
i
+
1
P
i
O
=
9
0
o
\angle P_{i+1}P_iO = 90^o
∠
P
i
+
1
P
i
O
=
9
0
o
and
P
i
−
1
P_{i-1}
P
i
−
1
and
P
i
+
1
P_{i+1}
P
i
+
1
are in different half-planes with respect to the line
O
P
i
OP_i
O
P
i
https://cdn.artofproblemsolving.com/attachments/f/2/aa3913989dac1cf04f2b42b5d630b2e096dcb6.png a) If
n
=
6
n = 6
n
=
6
and
α
=
3
0
o
\alpha = 30^o
α
=
3
0
o
, find the length of
P
0
P
n
P_0P_n
P
0
P
n
. b) If
n
=
2018
n = 2018
n
=
2018
and
α
=
4
5
o
\alpha= 45^o
α
=
4
5
o
, find the length of
P
0
P
n
P_0P_n
P
0
P
n
.
5
1
Hide problems
interesting numbers, remainder 2 under division by 4, in reverse order also
We call a positive integer interesting if the number and the number with its digits written in reverse order both leave remainder
2
2
2
in division by
4
4
4
. a) Determine if
2018
2018
2018
is an interesting number. b) For every positive integer
n
n
n
, find how many interesting
n
n
n
-digit numbers there are.
2
1
Hide problems
bw cells in 30x30 board
Danielle divides a
30
×
30
30 \times30
30
×
30
board into
100
100
100
regions that are
3
×
3
3 \times 3
3
×
3
squares squares each and then paint some squares black and the rest white. Then to each region assigns it the color that has the most squares painted with that color. a) If there are more black regions than white, what is the minimum number
N
N
N
of cells that Danielle can paint black? b) In how many ways can Danielle paint the board if there are more black regions than white and she uses the minimum number
N
N
N
of black squares?
1
1
Hide problems
z^4 - z^3 - 2z^2 - 3z - 1= 0 - 2018 Ecuador Juniors (OMEC) L2 p1
Find all reals
z
z
z
such that
z
4
−
z
3
−
2
z
2
−
3
z
−
1
=
0
z^4 - z^3 - 2z^2 - 3z - 1= 0
z
4
−
z
3
−
2
z
2
−
3
z
−
1
=
0
.
3
1
Hide problems
(PQRS)<= 1/2 (ABCD), quadr. inside a square
Let
A
B
C
D
ABCD
A
BC
D
be a square. Point
P
,
Q
,
R
,
S
P, Q, R, S
P
,
Q
,
R
,
S
are chosen on the sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
, respectively, such that
A
P
+
C
R
≥
A
B
≥
B
Q
+
D
S
AP + CR \ge AB \ge BQ + DS
A
P
+
CR
≥
A
B
≥
BQ
+
D
S
. Prove that
a
r
e
a
(
P
Q
R
S
)
≤
1
2
a
r
e
a
(
A
B
C
D
)
area \,\, (PQRS) \le \frac12 \,\, area \,\, (ABCD)
a
re
a
(
PQRS
)
≤
2
1
a
re
a
(
A
BC
D
)
and determine all cases when equality holds.
6
1
Hide problems
max even pos. integer not sum of composite odd numbers
What is the largest even positive integer that cannot be expressed as the sum of two composite odd numbers?