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Problems
Contests
National and Regional Contests
Indonesia Contests
Indonesia Juniors
2007 Indonesia Juniors
2007 Indonesia Juniors
Part of
Indonesia Juniors
Subcontests
(2)
day 2
1
Hide problems
Indonesia Juniors 2007 day 2 OSN SMP
p1. Four kite-shaped shapes as shown below (
a
>
b
a > b
a
>
b
,
a
a
a
and
b
b
b
are natural numbers less than
10
10
10
) arranged in such a way so that it forms a square with holes in the middle. The square hole in the middle has a perimeter of
16
16
16
units of length. What is the possible perimeter of the outermost square formed if it is also known that
a
a
a
and
b
b
b
are numbers coprime? https://cdn.artofproblemsolving.com/attachments/4/1/fa95f5f557aa0ca5afb9584d5cee74743dcb10.png p2. If
a
=
3
p
a = 3^p
a
=
3
p
,
b
=
3
q
b = 3^q
b
=
3
q
,
c
=
3
r
c = 3^r
c
=
3
r
, and
d
=
3
s
d = 3^s
d
=
3
s
and if
p
,
q
,
r
p, q, r
p
,
q
,
r
, and
s
s
s
are natural numbers, what is the smallest value of
p
⋅
q
⋅
r
⋅
s
p\cdot q\cdot r\cdot s
p
⋅
q
⋅
r
⋅
s
that satisfies
a
2
+
b
3
+
c
5
=
d
7
a^2 + b^3 + c^5 = d^7
a
2
+
b
3
+
c
5
=
d
7
3. Ucok intends to compose a key code (password) consisting of 8 numbers and meet the following conditions: i. The numbers used are
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
1, 2, 3, 4, 5, 6, 7, 8
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
, and
9
9
9
. ii. The first number used is at least
1
1
1
, the second number is at least
2
2
2
, third digit-at least
3
3
3
, and so on. iii. The same number can be used multiple times. a) How many different passwords can Ucok compose? b) How many different passwords can Ucok make, if provision (iii) is replaced with: no numbers may be used more than once. p 4. For any integer
a
,
b
a, b
a
,
b
, and
c
c
c
applies
a
×
(
b
+
c
)
=
(
a
×
b
)
+
(
a
×
c
)
a\times (b + c) = (a\times b) + (a\times c)
a
×
(
b
+
c
)
=
(
a
×
b
)
+
(
a
×
c
)
. a) Look for examples that show that
a
+
(
b
×
c
)
≠
(
a
+
b
)
×
(
a
+
c
)
a + (b\times c)\ne (a + b)\times (a + c)
a
+
(
b
×
c
)
=
(
a
+
b
)
×
(
a
+
c
)
. b) Is it always true that
a
+
(
b
×
c
)
=
(
a
+
b
)
×
(
a
+
c
)
a + (b\times c) = (a + b)\times(a + c)
a
+
(
b
×
c
)
=
(
a
+
b
)
×
(
a
+
c
)
? Justify your answer. p5. The results of a survey of
N
N
N
people with the question whether they maintain dogs, birds, or cats at home are as follows:
50
50
50
people keep birds,
61
61
61
people don't have dogs,
13
13
13
people don't keep a cat, and there are at least
74
74
74
people who keep the most a little two kinds of animals in the house. What is the maximum value and minimum of possible value of
N
N
N
?
day 1
1
Hide problems
Indonesia Juniors 2007 day 1 OSN SMP
p1. A set of cards contains
100
100
100
cards, each of which is written with a number from
1
1
1
up to
100
100
100
. On each of the two sides of the card the same number is written, side one is red and the other is green. First of all Leny arranges all the cards with red writing face up. Then Leny did the following three steps: I. Turn over all cards whose numbers are divisible by
2
2
2
II. Turn over all the cards whose numbers are divisible by
3
3
3
III. Turning over all the cards whose numbers are divisible by
5
5
5
, but didn't turn over all cards whose numbers are divisible by
5
5
5
and
2
2
2
. Find the number of Leny cards now numbered in red and face up, p2. Find the area of three intersecting semicircles as shown in the following image. https://cdn.artofproblemsolving.com/attachments/f/b/470c4d2b84435843975a0664fad5fee4a088d5.png p3. It is known that
x
+
1
x
=
7
x+\frac{1}{x}=7
x
+
x
1
=
7
. Determine the value of
A
A
A
so that
A
x
x
4
+
x
2
+
1
=
5
6
\frac{Ax}{x^4+x^2+1}=\frac56
x
4
+
x
2
+
1
A
x
=
6
5
. p4. There are
13
13
13
different gifts that will all be distributed to Ami, Ima, Mai,and Mia. If Ami gets at least
4
4
4
gifts, Ima and Mai respectively got at least
3
3
3
gifts, and Mia got at least
2
2
2
gifts, how many possible gift arrangements are there? p5. A natural number is called a quaprimal number if it satisfies all four following conditions: i. Does not contain zeros. ii. The digits compiling the number are different. iii. The first number and the last number are prime numbers or squares of an integer. iv. Each pair of consecutive numbers forms a prime number or square of an integer.For example, we check the number
971643
971643
971643
. (i)
971643
971643
971643
does not contain zeros. (ii) The digits who compile
971643
971643
971643
are different. (iii) One first number and one last number of
971643
971643
971643
, namely
9
9
9
and
3
3
3
is a prime number or a square of an integer. (iv) Each pair of consecutive numbers, namely
97
,
71
,
16
,
64
97, 71, 16, 64
97
,
71
,
16
,
64
, and
43
43
43
form prime number or square of an integer. So
971643
971643
971643
is a quadratic number.Find the largest
6
6
6
-digit quaprimal number. Find the smallest
6
6
6
-digit quaprimal number. Which digit is never contained in any arbitrary quaprimal number? Explain.