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National and Regional Contests
Iran Contests
Iran MO (1st Round)
2018 Iran MO (1st Round)
2018 Iran MO (1st Round)
Part of
Iran MO (1st Round)
Subcontests
(25)
25
1
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Dystopian ends for dystopian exams - Iran First Round 2018, P25
Astrophysicists have discovered a minor planet of radius
30
30
30
kilometers whose surface is completely covered in water. A spherical meteor hits this planet and is submerged in the water. This incidence causes an increase of
1
1
1
centimeters to the height of the water on this planet. What is the radius of the meteor in meters?
24
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Sequence involving square roots - Iran First Round 2018, P24
The sequence
{
a
n
}
\{a_n\}
{
a
n
}
is defined as follows: \begin{align*} a_n = \sqrt{1 + \left(1 + \frac 1n \right)^2} + \sqrt{1 + \left(1 - \frac 1n \right)^2}. \end{align*} What is the value of the expression given below? \begin{align*} \frac{4}{a_1} + \frac{4}{a_2} + \dots + \frac{4}{a_{96}}.\end{align*}
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<span class='latex-bold'>(A)</span>\ \sqrt{18241} \qquad<span class='latex-bold'>(B)</span>\ \sqrt{18625} - 1 \qquad<span class='latex-bold'>(C)</span>\ \sqrt{18625} \qquad<span class='latex-bold'>(D)</span>\ \sqrt{19013} - 1\qquad<span class='latex-bold'>(E)</span>\ \sqrt{19013}
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19013
23
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Largest and smallest circles - Iran First Round 2018, P23
Nadia bought a compass and after opening its package realized that the length of the needle leg is
10
10
10
centimeters whereas the length of the pencil leg is
16
16
16
centimeters! Assume that in order to draw a circle with this compass, the angle between the pencil leg and the paper must be at least
30
30
30
degrees but the needle leg could be positioned at any angle with respect to the paper. Let
n
n
n
be the difference between the radii of the largest and the smallest circles that Nadia can draw with this compass in centimeters. Which of the following options is closest to
n
n
n
?
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<span class='latex-bold'>(A)</span>\ 6\qquad<span class='latex-bold'>(B)</span>\ 7\qquad<span class='latex-bold'>(C)</span>\ 10 \qquad<span class='latex-bold'>(D)</span>\ 12\qquad<span class='latex-bold'>(E)</span>\ 20
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1x2 tiles covering a 4x4 area - Iran First Round 2018, P22
There are eight congruent
1
×
2
1\times 2
1
×
2
tiles formed of one blue square and one red square. In how many ways can we cover a
4
×
4
4\times 4
4
×
4
area with these tiles so that each row and each column has two blue squares and two red squares?
21
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Three reflections in an equilateral triangle
The point
P
P
P
is chosen inside or on the equilateral triangle
A
B
C
ABC
A
BC
of side length
1
1
1
. The reflection of
P
P
P
with respect to
A
B
AB
A
B
is
K
K
K
, the reflection of
K
K
K
about
B
C
BC
BC
is
M
M
M
, and the reflection of
M
M
M
with respect to
A
C
AC
A
C
is
N
N
N
. What is the maximum length of
N
P
NP
NP
?
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<span class='latex-bold'>(A)</span>\ 2\sqrt 3\qquad<span class='latex-bold'>(B)</span>\ \sqrt 3\qquad<span class='latex-bold'>(C)</span>\ \frac{\sqrt 3}{2} \qquad<span class='latex-bold'>(D)</span>\ 3\qquad<span class='latex-bold'>(E)</span>\ 1
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Angles in cyclic quadrilateral - Iran First Round 2018, P20
In the convex and cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
, we have
∠
B
=
11
0
∘
\angle B = 110^{\circ}
∠
B
=
11
0
∘
. The intersection of
A
D
AD
A
D
and
B
C
BC
BC
is
E
E
E
and the intersection of
A
B
AB
A
B
and
C
D
CD
C
D
is
F
F
F
. If the perpendicular from
E
E
E
to
A
B
AB
A
B
intersects the perpendicular from
F
F
F
to
B
C
BC
BC
on the circumcircle of the quadrilateral at point
P
P
P
, what is
∠
P
D
F
\angle PDF
∠
P
D
F
in degrees?
19
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Minimum value of x/z - Iran First Round 2018, P19
Let
x
≥
y
≥
z
x \geq y \geq z
x
≥
y
≥
z
be positive real numbers such that \begin{align*}x^2+y^2+z^2 \geq 2xy+2yz+2zx.\end{align*} What is the minimum value of
x
z
\frac{x}{z}
z
x
?
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<span class='latex-bold'>(A)</span>\ 1\qquad<span class='latex-bold'>(B)</span>\ \sqrt 2\qquad<span class='latex-bold'>(C)</span>\ \sqrt 3\qquad<span class='latex-bold'>(D)</span>\ 2\qquad<span class='latex-bold'>(E)</span>\ 4
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18
1
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Rods of lengths 1396, 1439, 2018 - Iran First Round 2017, P18
Three rods of lengths
1396
,
1439
1396, 1439
1396
,
1439
, and
2018
2018
2018
millimeters have been hinged from one tip on the ground. What is the smallest value for the radius of the circle passing through the other three tips of the rods in millimeters?
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Find maximum of m+n if lcm(m,n) = (m-n)^2
Two positive integers
m
m
m
and
n
n
n
are both less than
500
500
500
and
lcm
(
m
,
n
)
=
(
m
−
n
)
2
\text{lcm}(m,n) = (m-n)^2
lcm
(
m
,
n
)
=
(
m
−
n
)
2
. What is the maximum possible value of
m
+
n
m+n
m
+
n
?
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Distinct x, y with (x+y-1)^2 = xy+1 - Iran First Round 2018, P16
A subset of the real numbers has the property that for any two distinct elements of it such as
x
x
x
and
y
y
y
, we have
(
x
+
y
−
1
)
2
=
x
y
+
1
(x+y-1)^2 = xy+1
(
x
+
y
−
1
)
2
=
x
y
+
1
. What is the maximum number of elements in this set?
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<span class='latex-bold'>(A)</span>\ 1\qquad<span class='latex-bold'>(B)</span>\ 2\qquad<span class='latex-bold'>(C)</span>\ 3\qquad<span class='latex-bold'>(D)</span>\ 4\qquad<span class='latex-bold'>(E)</span>\ \text{Infinity}
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Infinity
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Values of a1- a2 + a3 - ... - a20 - Iran First Round 2018, P15
Let
a
1
,
a
2
,
a
3
,
…
,
a
20
a_1, a_2, a_3, \dots, a_{20}
a
1
,
a
2
,
a
3
,
…
,
a
20
be a permutation of the numbers
1
,
2
,
…
,
20
1, 2, \dots, 20
1
,
2
,
…
,
20
. How many different values can the expression
a
1
−
a
2
+
a
3
−
⋯
−
a
20
a_1-a_2+a_3-\dots - a_{20}
a
1
−
a
2
+
a
3
−
⋯
−
a
20
have?
14
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System of three equations - Iran First Round 2018, P14
For how many integers
k
k
k
does the following system of equations has a solution other than
a
=
b
=
c
=
0
a=b=c=0
a
=
b
=
c
=
0
in the set of real numbers? \begin{align*} \begin{cases} a^2+b^2=kc(a+b),\\ b^2+c^2 = ka(b+c),\\ c^2+a^2=kb(c+a).\end{cases}\end{align*}
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1
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Rest in peace, Bahman - Iran First Round 2018, P13
Bahman wants to build an area next to his garden's wall for keeping his poultry. He has three fences each of length
10
10
10
meters. Using the garden's wall, which is straight and long, as well as the three pieces of fence, what is the largest area Bahman can enclose in meters squared?
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<span class='latex-bold'>(A)</span>\ 100 \qquad<span class='latex-bold'>(B)</span>\ 50+25 \sqrt 3\qquad<span class='latex-bold'>(C)</span>\ 50 + 50\sqrt 2\qquad<span class='latex-bold'>(D)</span>\ 75 \sqrt 3 \qquad<span class='latex-bold'>(E)</span>\ 300
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3
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12
1
Hide problems
(a,b,c) such that a, b, c divide a+b+c - Iran First Round 2018, P12
How many triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of positive integers strictly less than
51
51
51
are there such that
a
+
b
+
c
a+b+c
a
+
b
+
c
is divisible by
a
,
b
a, b
a
,
b
, and
c
c
c
?
11
1
Hide problems
No more than 9 stories - Iran First Round 2018, P11
Based on a city's rules, the buildings of a street may not have more than
9
9
9
stories. Moreover, if the number of stories of two buildings is the same, no matter how far they are from each other, there must be a building with a higher number of stories between them. What is the maximum number of buildings that can be built on one side of a street in this city?
10
1
Hide problems
Find the length of CN in triangle ABC - Iran First Round 2018, P10
Consider a triangle
A
B
C
ABC
A
BC
in which
A
B
=
A
C
=
15
AB=AC=15
A
B
=
A
C
=
15
and
B
C
=
18
BC=18
BC
=
18
. Points
D
D
D
and
E
E
E
are chosen on
C
A
CA
C
A
and
C
B
CB
CB
, respectively, such that
C
D
=
5
CD=5
C
D
=
5
and
C
E
=
3
CE=3
CE
=
3
. The point
F
F
F
is chosen on the half-line
D
E
→
\overrightarrow{DE}
D
E
so that
E
F
=
8
EF=8
EF
=
8
. If
M
M
M
is the midpoint of
A
B
AB
A
B
and
N
N
N
is the intersection of
F
M
FM
FM
and
B
C
BC
BC
, what is the length of
C
N
CN
CN
?
9
1
Hide problems
Area of wildfire in the forest park - Iran First Round 2018, P9
A part of a forest park which is located between two roads has caught fire. The fire is spreading at a speed of
10
10
10
kilometers per hour. If the distance between the starting point of the fire and both roads is
10
10
10
kilometers, what is the area of the burned region after two hours in kilometers squared? (Assume that the roads are long, straight parallel lines and the fire does not enter the roads)
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400
π
3
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π
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3
<span class='latex-bold'>(A)</span>\ 200\sqrt 3\qquad<span class='latex-bold'>(B)</span>\ 100 \sqrt 3\qquad<span class='latex-bold'>(C)</span>\ 400\sqrt 3 + 400 \frac{\pi}{3} \qquad<span class='latex-bold'>(D)</span>\ 200\sqrt 3 + 400 \frac{\pi}{3} \qquad<span class='latex-bold'>(E)</span>\ 400\sqrt 3
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3
8
1
Hide problems
Tricky average problem - Iran First Round 2018, P8
The license plate of each automobile in Iran consists of a two-digit and a three-digit number as well as a letter of the Persian alphabet. The digit
0
0
0
is not used in the two numbers. To each license plate, we assign the product of the two numbers on it. For example, if the two numbers are
12
12
12
and
365
365
365
on a license plate, the assigned number would be
12
×
365
=
4380
12 \times 365 = 4380
12
×
365
=
4380
. What is the average of all the assigned numbers to all possible license plates?
7
1
Hide problems
Enclosed area under the graph - Iran First Round 2018, P7
What is the enclosed area between the graph of
y
=
⌊
10
x
⌋
+
1
−
x
2
y=\lfloor 10x \rfloor + \sqrt{1-x^2}
y
=
⌊
10
x
⌋
+
1
−
x
2
in the interval
[
0
,
1
]
[0,1]
[
0
,
1
]
and the
x
x
x
axis?
6
1
Hide problems
Remainder of 18n+17 mod 1920 - Iran First Round 2018, P6
Let
n
n
n
be the smallest positive integer such that the remainder of
3
n
+
45
3n+45
3
n
+
45
, when divided by
1060
1060
1060
, is
16
16
16
. Find the remainder of
18
n
+
17
18n+17
18
n
+
17
upon division by
1920
1920
1920
.
5
1
Hide problems
COVID hits the palaestra - Iran First Round 2018, P5
There are
128
128
128
numbered seats arranged around a circle in a palaestra. The first person to enter the place would sit on seat number
1
1
1
. Since a contagious disease is infecting the people of the city, each person who enters the palaestra would sit on a seat whose distance is the longest to the nearest occupied seat. If there are several such seats, the newly entered person would sit on the seat with the smallest number. What is the number of the seat on which the
39
39
39
th person sits?
4
1
Hide problems
Five points in the plane - Iran First Round 2018, P4
There are
5
5
5
points in the plane no three of which are collinear. We draw all the segments whose vertices are these points. What is the minimum number of new points made by the intersection of the drawn segments?
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5
<span class='latex-bold'>(A)</span>\ 0\qquad<span class='latex-bold'>(B)</span>\ 1\qquad<span class='latex-bold'>(C)</span>\ 2\qquad<span class='latex-bold'>(D)</span>\ 3\qquad<span class='latex-bold'>(E)</span>\ 5
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3
1
Hide problems
Divisibility by 3 in base 4 - Iran First Round 2018, P3
How many
8
8
8
-digit numbers in base
4
4
4
formed of the digits
1
,
2
,
3
1,2, 3
1
,
2
,
3
are divisible by
3
3
3
?
2
1
Hide problems
Least number of plates - Iran First Round 2018, P2
A factory packs its products in cubic boxes. In one store, they put
512
512
512
of these cubic boxes together to make a large
8
×
8
×
8
8\times 8 \times 8
8
×
8
×
8
cube. When the temperature goes higher than a limit in the store, it is necessary to separate the
512
512
512
set of boxes using horizontal and vertical plates so that each box has at least one face which is not touching other boxes. What is the least number of plates needed for this purpose?
1
1
Hide problems
COVID-related probability problem - Iran First Round 2018, P1
In a village with a population of
1000
1000
1000
, two hundred people have been infected by a disease. A diagnostic test can be done to check whether a person is infected, but the result could be erroneous. That is, there is a
5
%
5\%
5%
probability that the test result of an infected person shows that they are not infected and a
5
%
5\%
5%
probability that the test result of a healthy person shows that they are infected. We randomly choose someone from the population of this village and take the diagnostic test from him. What is the probability that the test result declares that person is infected?