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National and Regional Contests
Indonesia Contests
Indonesia Juniors
2003 Indonesia Juniors
day 2
day 2
Part of
2003 Indonesia Juniors
Problems
(1)
Indonesia Juniors 2003 day 2 OSN SMP
Source:
10/30/2021
p1. It is known that
a
1
=
2
a_1=2
a
1
=
2
,
a
2
=
3
a_2=3
a
2
=
3
. For
k
>
2
k > 2
k
>
2
, define
a
k
=
1
2
a
k
−
2
+
1
3
a
k
−
1
a_k=\frac{1}{2}a_{k-2}+\frac{1}{3}a_{k-1}
a
k
=
2
1
a
k
−
2
+
3
1
a
k
−
1
. Find the infinite sum of of
a
1
+
a
2
+
a
3
+
.
.
.
a_1+a_2+a_3+...
a
1
+
a
2
+
a
3
+
...
p2. The multiplied number is a natural number in two-digit form followed by the result time. For example,
7
×
8
=
56
7\times 8 = 56
7
×
8
=
56
, then
7856
7856
7856
and
8756
8756
8756
are multiplied numbers .
2
×
3
=
6
2\times 3 = 6
2
×
3
=
6
, then
236
236
236
and
326
326
326
are multiplied.
2
×
0
=
0
2\times 0 = 0
2
×
0
=
0
, then
200
200
200
is the multiplied. For the record, the first digit of the number times can't be
0
0
0
. a. What is the difference between the largest and the smallest multiplied number? b. Find all the multiplied numbers that consist of three digits and each digit is square number. c. Given the following "boxes" that must be filled with multiple numbers. https://cdn.artofproblemsolving.com/attachments/b/6/ac086a3d1a0549fae909c072224605430daf1d.png Determine the contents of the shaded box. Is this content the only one? d. Complete all the empty boxes above with multiplied numbers. p3. Look at the picture of the arrangement of three squares below. https://cdn.artofproblemsolving.com/attachments/1/3/c0200abae77cc73260b083117bf4bafc707eea.pngProve that
∠
B
A
X
+
∠
C
A
X
=
4
5
o
\angle BAX + \angle CAX = 45^o
∠
B
A
X
+
∠
C
A
X
=
4
5
o
p4. Prove that
(
n
−
1
)
n
(
n
3
+
1
)
(n-1)n (n^3 + 1)
(
n
−
1
)
n
(
n
3
+
1
)
is always divisible by
6
6
6
for all natural number
n
n
n
.