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Problems
Contests
National and Regional Contests
Indonesia Contests
Indonesia Juniors
2003 Indonesia Juniors
2003 Indonesia Juniors
Part of
Indonesia Juniors
Subcontests
(2)
day 2
1
Hide problems
Indonesia Juniors 2003 day 2 OSN SMP
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
.
day 1
1
Hide problems
Indonesia Juniors 2003 day 1 OSN SMP
p1. The pattern
A
B
C
C
C
D
D
D
D
A
B
B
C
C
C
D
D
D
D
A
B
B
C
C
C
D
D
D
D
.
.
.
ABCCCDDDDABBCCCDDDDABBCCCDDDD...
A
BCCC
DDDD
A
BBCCC
DDDD
A
BBCCC
DDDD
...
repeats to infinity. Which letter ranks in place
2533
2533
2533
? p2. Prove that if
a
>
2
a > 2
a
>
2
and
b
>
3
b > 3
b
>
3
then
a
b
+
6
>
3
a
+
2
b
ab + 6 > 3a + 2b
ab
+
6
>
3
a
+
2
b
. p3. Given a rectangle
A
B
C
D
ABCD
A
BC
D
with size
16
16
16
cm
×
25
\times 25
×
25
cm,
E
B
F
G
EBFG
EBFG
is kite, and the length of
A
E
=
5
AE = 5
A
E
=
5
cm. Determine the length of
E
F
EF
EF
. https://cdn.artofproblemsolving.com/attachments/2/e/885af838bcf1392eb02e2764f31ae83cb84b78.pngp4. Consider the following series of statements. It is known that
x
=
1
x = 1
x
=
1
. Since
x
=
1
x = 1
x
=
1
then
x
2
=
1
x^2 = 1
x
2
=
1
. So
x
2
=
x
x^2 = x
x
2
=
x
. As a result,
x
2
−
1
=
x
−
1
x^2 - 1 = x- 1
x
2
−
1
=
x
−
1
(
x
−
1
)
(
x
+
1
)
=
(
x
−
1
)
⋅
1
(x -1) (x + 1) = (x - 1) \cdot 1
(
x
−
1
)
(
x
+
1
)
=
(
x
−
1
)
⋅
1
Using the rule out, we get
x
+
1
=
1
x + 1 = 1
x
+
1
=
1
1
+
1
=
1
1 + 1 = 1
1
+
1
=
1
2
=
1
2 = 1
2
=
1
The question. a. If
2
=
1
2 = 1
2
=
1
, then every natural number must be equal to
1
1
1
. Prove it. b. The result of
2
=
1
2 = 1
2
=
1
is something that is impossible. Of course there's something wrong in the argument above? Where is the fault? Why is that you think wrong? p5. To calculate
(
1998
)
(
1996
)
(
1994
)
(
1992
)
+
16
\sqrt{(1998)(1996)(1994)(1992)+16}
(
1998
)
(
1996
)
(
1994
)
(
1992
)
+
16
. someone does it in a simple way as follows:
200
0
2
−
2
×
5
×
2000
+
5
2
−
5
2000^2-2 \times 5\times 2000 + 5^2 - 5
200
0
2
−
2
×
5
×
2000
+
5
2
−
5
? Is the way that person can justified? Why? p6. To attract customers, a fast food restaurant give gift coupons to everyone who buys food at the restaurant with a value of more than
25
,
000
25,000
25
,
000
Rp.. Behind every coupon is written one of the following numbers:
9
9
9
,
12
12
12
,
42
42
42
,
57
57
57
,
69
69
69
,
21
21
21
, 15,
75
75
75
,
24
24
24
and
81
81
81
. Successful shoppers collect coupons with the sum of the numbers behind the coupon is equal to 100 will be rewarded in the form of TV
2
1
′
′
21''
2
1
′′
. If the restaurant owner provides as much as
10
10
10
2
1
′
′
21''
2
1
′′
TV pieces, how many should be handed over to the the customer? p7. Given is the shape of the image below. https://cdn.artofproblemsolving.com/attachments/4/6/5511d3fb67c039ca83f7987a0c90c652b94107.png The centers of circles
B
B
B
,
C
C
C
,
D
D
D
, and
E
E
E
are placed on the diameter of circle
A
A
A
and the diameter of circle
B
B
B
is the same as the radius of circle
A
A
A
. Circles
C
C
C
,
D
D
D
, and
E
E
E
are equal and the pairs are tangent externally such that the sum of the lengths of the diameters of the three circles is the same with the radius of the circle
A
A
A
. What is the ratio of the circumference of the circle
A
A
A
with the sum of the circumferences of circles
B
B
B
,
C
C
C
,
D
D
D
, and
E
E
E
? p8. It is known that
a
+
b
+
c
=
0
a + b + c = 0
a
+
b
+
c
=
0
. Prove that
a
3
+
b
3
+
c
3
=
3
a
b
c
a^3 + b^3 + c^3 = 3abc
a
3
+
b
3
+
c
3
=
3
ab
c
.