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Contests
National and Regional Contests
Iran Contests
Iran MO (3rd Round)
1993 Iran MO (3rd Round)
1993 Iran MO (3rd Round)
Part of
Iran MO (3rd Round)
Subcontests
(5)
6
1
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Prove that number of numbers t in I is less than 1000
Let
x
1
,
x
2
,
…
,
x
12
x_1, x_2, \ldots, x_{12}
x
1
,
x
2
,
…
,
x
12
be twelve real numbers such that for each
1
≤
i
≤
12
1 \leq i \leq 12
1
≤
i
≤
12
, we have
∣
x
i
∣
≥
1
|x_i| \geq 1
∣
x
i
∣
≥
1
. Let
I
=
[
a
,
b
]
I=[a,b]
I
=
[
a
,
b
]
be an interval such that
b
−
a
≤
2
b-a \leq 2
b
−
a
≤
2
. Prove that number of the numbers of the form
t
=
∑
i
=
1
12
r
i
x
i
t= \sum_{i=1}^{12} r_ix_i
t
=
∑
i
=
1
12
r
i
x
i
, where
r
i
=
±
1
r_i=\pm 1
r
i
=
±
1
, which lie inside the interval
I
I
I
, is less than
1000
1000
1000
.
5
1
Hide problems
Show that O1O3 is perpendicular to O2O4
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, diagonals
A
C
AC
A
C
and
B
D
BD
B
D
are equal. We construct four equilateral triangles with centers
O
1
,
O
2
,
O
3
,
O
4
O_1,O_2,O_3,O_4
O
1
,
O
2
,
O
3
,
O
4
on the sides sides
A
B
,
B
C
,
C
D
,
D
A
AB, BC, CD, DA
A
B
,
BC
,
C
D
,
D
A
outside of this quadrilateral, respectively. Show that
O
1
O
3
⊥
O
2
O
4
O_1O_3 \perp O_2O_4
O
1
O
3
⊥
O
2
O
4
.
4
1
Hide problems
Prove that there exists a subset S of positive integers
Prove that there exists a subset
S
S
S
of positive integers such that we can represent each positive integer as difference of two elements of
S
S
S
in exactly one way.
2
1
Hide problems
Find the area of triangle OEF
In the figure below, area of triangles
A
O
D
,
D
O
C
,
AOD, DOC,
A
O
D
,
D
OC
,
and
A
O
B
AOB
A
OB
is given. Find the area of triangle
O
E
F
OEF
OEF
in terms of area of these three triangles.[asy] import graph; size(11.52cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-2.4,xmax=9.12,ymin=-6.6,ymax=5.16; pair A=(0,0), F=(9,0), B=(4,0), C=(3.5,2), D=(1.94,2.59), O=(2.75,1.57); draw(A--(3,4),linewidth(1.2)); draw((3,4)--F,linewidth(1.2)); draw(A--F,linewidth(1.2)); draw((3,4)--B,linewidth(1.2)); draw(A--C,linewidth(1.2)); draw(B--D,linewidth(1.2)); draw((3,4)--O,linewidth(1.2)); draw(C--F,linewidth(1.2)); draw(F--O,linewidth(1.2)); dot(A,ds); label("
A
A
A
",(-0.28,-0.23),NE*lsf); dot(F,ds); label("
F
F
F
",(8.79,-0.4),NE*lsf); dot((3,4),ds); label("
E
E
E
",(3.05,4.08),NE*lsf); dot(B,ds); label("
B
B
B
",(4.05,0.09),NE*lsf); dot(C,ds); label("
C
C
C
",(3.55,2.08),NE*lsf); dot(D,ds); label("
D
D
D
",(1.76,2.71),NE*lsf); dot(O,ds); label("
O
O
O
",(2.57,1.17),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]
1
1
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There exist infinitely many positive integers
Prove that there exist infinitely many positive integers which can't be represented as sum of less than
10
10
10
odd positive integers' perfect squares.