MathDB
Problems
Contests
National and Regional Contests
Indonesia Contests
Indonesia Juniors
2005 Indonesia Juniors
2005 Indonesia Juniors
Part of
Indonesia Juniors
Subcontests
(2)
day 2
1
Hide problems
Indonesia Juniors 2005 day 2 OSN SMP
p1. Among the numbers
1
5
\frac15
5
1
and
1
4
\frac14
4
1
there are infinitely many fractional numbers. Find
999
999
999
decimal numbers between
1
5
\frac15
5
1
and
1
4
\frac14
4
1
so that the difference between the next fractional number with the previous fraction constant. (i.e. If
x
1
,
x
2
,
x
3
,
x
4
,
.
.
.
,
x
999
x_1, x_2, x_3, x_4,..., x_{999}
x
1
,
x
2
,
x
3
,
x
4
,
...
,
x
999
is a fraction that meant, then
x
2
−
x
1
=
x
3
−
x
3
=
.
.
.
=
x
n
−
x
n
−
1
=
.
.
.
=
x
999
−
x
998
x_2 - x_1= x_3 - x_3= ...= x_n - x_{n-1}=...=x_{999}-x_{998}
x
2
−
x
1
=
x
3
−
x
3
=
...
=
x
n
−
x
n
−
1
=
...
=
x
999
−
x
998
) p2. The pattern in the image below is: "Next image obtained by adding an image of a black equilateral triangle connecting midpoints of the sides of each white triangle that is left in the previous image." The pattern is continuous to infinity. https://cdn.artofproblemsolving.com/attachments/e/f/81a6b4d20607c7508169c00391541248b8f31e.png It is known that the area of the triangle in Figure
1
1
1
is
1
1
1
unit area. Find the total area of the area formed by the black triangles in figure
5
5
5
. Also find the total area of the area formed by the black triangles in the
20
20
20
th figure. p3. For each pair of natural numbers
a
a
a
and
b
b
b
, we define
a
∗
b
=
a
b
+
a
−
b
a*b = ab + a - b
a
∗
b
=
ab
+
a
−
b
. The natural number
x
x
x
is said to be the constituent of the natural number
n
n
n
if there is a natural number
y
y
y
that satisfies
x
∗
y
=
n
x*y = n
x
∗
y
=
n
. For example,
2
2
2
is a constituent of
6
6
6
because there is a natural number 4 so that
2
∗
4
=
2
⋅
4
+
2
−
4
=
8
+
2
−
4
=
6
2*4 = 2\cdot 4 + 2 - 4 = 8 + 2 - 4 = 6
2
∗
4
=
2
⋅
4
+
2
−
4
=
8
+
2
−
4
=
6
. Find all constituent of
2005
2005
2005
. p4. Three people want to eat at a restaurant. To find who pays them to make a game. Each tossing one coin at a time. If the result is all heads or all tails, then they toss again. If not, then "odd person" (i.e. the person whose coin appears different from the two other's coins) who pay. Determine the number of all possible outcomes, if the game ends in tossing: a. First. b. Second. c. Third. d. Tenth. p5. Given the equation
x
2
+
3
y
2
=
n
x^2 + 3y^2 = n
x
2
+
3
y
2
=
n
, where
x
x
x
and
y
y
y
are integers. If
n
<
20
n < 20
n
<
20
what number is
n
n
n
, and which is the respective pair
(
x
,
y
)
(x,y)
(
x
,
y
)
? Show that it is impossible to solve
x
2
+
3
y
2
=
8
x^2 + 3y^2 = 8
x
2
+
3
y
2
=
8
in integers.
day 1
1
Hide problems
Indonesia Juniors 2005 day 1 OSN SMP
p1.
A
A
A
is a set of numbers. The set
A
A
A
is closed to subtraction, meaning that the result of subtracting two numbers in
A
A
A
will be returns a number in
A
A
A
as well. If it is known that two members of
A
A
A
are
4
4
4
and
9
9
9
, show that: a.
0
∈
A
0\in A
0
∈
A
b.
13
∈
A
13 \in A
13
∈
A
c.
74
∈
A
74 \in A
74
∈
A
d. Next, list all the members of the set
A
A
A
. p2.
(
2
,
0
,
4
,
1
)
(2, 0, 4, 1)
(
2
,
0
,
4
,
1
)
is one of the solutions/answers of
x
1
+
x
2
+
x
3
+
x
4
=
7
x_1+x_2+x_3+x_4=7
x
1
+
x
2
+
x
3
+
x
4
=
7
. If all solutions belong on the set of not negative integers , specify as many possible solutions/answers from
x
1
+
x
2
+
x
3
+
x
4
=
7
x_1+x_2+x_3+x_4=7
x
1
+
x
2
+
x
3
+
x
4
=
7
p3. Adi is an employee at a textile company on duty save data. One time Adi was asked by the company leadership to prepare data on production increases over five periods. After searched by Adi only found four data on the increase, namely
4
%
4\%
4%
,
9
%
9\%
9%
,
7
%
7\%
7%
, and
5
%
5\%
5%
. One more data, namely the
5
5
5
th data, was not found. Investigate increase of 5th data production, if Adi only remembers that the arithmetic mean and median of the five data are the same. p4. Find all pairs of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
that satisfy the system of the following equations:
{
x
(
y
+
1
)
=
y
2
−
1
y
(
x
+
1
)
=
x
2
−
1
\left\{\begin{array}{l} x(y+1)=y^2-1 \\ y(x+1)=x^2-1 \end{array} \right.
{
x
(
y
+
1
)
=
y
2
−
1
y
(
x
+
1
)
=
x
2
−
1
p5. Given the following image.
A
B
C
D
ABCD
A
BC
D
is square, and
E
E
E
is any point outside the square
A
B
C
D
ABCD
A
BC
D
. Investigate whether the relationship
A
E
2
+
C
E
2
=
B
E
2
+
D
E
2
AE^2 + CE^2 = BE^2 +DE^2
A
E
2
+
C
E
2
=
B
E
2
+
D
E
2
holds in the picture below. https://cdn.artofproblemsolving.com/attachments/2/5/a339b0e4df8407f97a4df9d7e1aa47283553c1.png