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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
1985 Iran MO (2nd round)
1985 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(7)
7
1
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(a+c)/(b+c) > a/b - [Iran Second Round 1985]
Let
a
,
b
a,b
a
,
b
and
c
c
c
be real numbers with
b
,
c
>
0.
b,c >0.
b
,
c
>
0.
Prove that if
a
<
b
(
a
>
b
)
,
a<b \ ( a>b),
a
<
b
(
a
>
b
)
,
then
a
+
c
b
+
c
>
a
b
(
a
+
c
b
+
c
<
a
b
)
\frac{a+c}{b+c} > \frac ab \qquad ( \frac{a+c}{b+c} < \frac ab)
b
+
c
a
+
c
>
b
a
(
b
+
c
a
+
c
<
b
a
)
And then prove that
a
+
c
b
+
c
\frac{a+c}{b+c}
b
+
c
a
+
c
is between
1
1
1
and
a
b
.
\frac ab.
b
a
.
6
1
Hide problems
Ring - [Iran Second Round 1985]
In The ring
R
\mathbf R
R
, we have
∀
x
∈
R
:
x
2
=
x
\forall x \in \mathbf R : x^2=x
∀
x
∈
R
:
x
2
=
x
. Prove that in this ringi) Every element is equals to its additive inverse.ii) This ring has commutative property.
5
2
Hide problems
Find the derivative of f - [Iran Second Round 1985]
Let
f
:
R
→
R
f: \mathbb R \to \mathbb R
f
:
R
→
R
and
g
:
R
→
R
g: \mathbb R \to \mathbb R
g
:
R
→
R
be two functions satisfying \forall x,y \in \mathbb R: \begin{cases} f(x+y)=f(x)f(y),\\ f(x)= x g(x)+1\end{cases} \text{and} \lim_{x \to 0} g(x)=1. Find the derivative of
f
f
f
in an arbitrary point
x
.
x.
x
.
Archery - [Iran Second Round 1985]
In the Archery with an especial gun, the probability of goal is
90
%
.
90 \%.
90%.
If we continue our work until we goal.i) What is the probability which exactly
3
3
3
balls consumed.ii) What is the probability which at least
3
3
3
balls consumed.
3
2
Hide problems
Inequality of Functions - [Iran Second Round 1985]
Let
f
:
R
→
R
,
g
:
R
→
R
f: \mathbb R \to \mathbb R,g: \mathbb R \to \mathbb R
f
:
R
→
R
,
g
:
R
→
R
and
φ
:
R
→
R
\varphi: \mathbb R \to \mathbb R
φ
:
R
→
R
be three ascendant functions such that
f
(
x
)
≤
g
(
x
)
≤
φ
(
x
)
∀
x
∈
R
.
f(x) \leq g(x) \leq \varphi(x) \qquad \forall x \in \mathbb R.
f
(
x
)
≤
g
(
x
)
≤
φ
(
x
)
∀
x
∈
R
.
Prove that
f
(
f
(
x
)
)
≤
g
(
g
(
x
)
)
≤
φ
(
φ
(
x
)
)
∀
x
∈
R
.
f(f(x)) \leq g(g(x)) \leq \varphi(\varphi(x)) \qquad \forall x \in \mathbb R.
f
(
f
(
x
))
≤
g
(
g
(
x
))
≤
φ
(
φ
(
x
))
∀
x
∈
R
.
Note. The function is
k
(
x
)
k(x)
k
(
x
)
ascendant if for every
x
,
y
∈
D
k
,
x
≤
y
x,y \in D_k, x \leq {y}
x
,
y
∈
D
k
,
x
≤
y
we have
g
(
x
)
≤
g
(
y
)
g(x)\leq{g(y)}
g
(
x
)
≤
g
(
y
)
.
Find the angle - [Iran Second Round 1985]
Find the angle between two common sections of the page
2
x
+
y
−
z
=
0
2x+y-z=0
2
x
+
y
−
z
=
0
and the cone
4
x
2
−
y
2
+
3
z
2
=
0.
4x^2-y^2+3z^2=0.
4
x
2
−
y
2
+
3
z
2
=
0.
4
2
Hide problems
At least one of x or y is integer - [Iran Second Round 1985]
Let
x
x
x
and
y
y
y
be two real numbers. Prove that the equations \lfloor x \rfloor + \lfloor y \rfloor =\lfloor x +y \rfloor , \lfloor -x \rfloor + \lfloor -y \rfloor =\lfloor -x-y \rfloor Holds if and only if at least one of
x
x
x
or
y
y
y
be integer.
G_a is a subgroup of G - [Iran Second Round 1985]
Let
G
G
G
be a group and let
a
a
a
be a constant member of it. Prove that
G
a
=
{
x
∣
∃
n
∈
Z
,
x
=
a
n
}
G_a = \{x | \exists n \in \mathbb Z , x=a^n\}
G
a
=
{
x
∣∃
n
∈
Z
,
x
=
a
n
}
Is a subgroup of
G
.
G.
G
.
2
2
Hide problems
Plot the triangle - [Iran Second Round 1985]
In the triangle
A
B
C
ABC
A
BC
the length of side
A
B
AB
A
B
, and height
A
H
AH
A
H
are known. also we know that
∠
B
=
2
∠
C
.
\angle B = 2 \angle C.
∠
B
=
2∠
C
.
Plot this triangle.
Find the value od the fraction - [Iran Second Round 1985]
Let
x
,
y
x, y
x
,
y
and
z
z
z
be three positive real numbers for which
x
2
+
y
2
+
z
2
=
x
y
+
y
z
+
z
x
.
x^2+y^2+z^2=xy+yz+zx.
x
2
+
y
2
+
z
2
=
x
y
+
yz
+
z
x
.
Find the value of
x
x
+
y
+
z
.
\frac{\sqrt x}{\sqrt x + \sqrt y+ \sqrt z}.
x
+
y
+
z
x
.
1
2
Hide problems
The number is an integer (Cosine) - [Iran Second Round 1985]
Let
α
\alpha
α
be an angle such that
cos
α
=
p
q
\cos \alpha = \frac pq
cos
α
=
q
p
, where
p
p
p
and
q
q
q
are two integers. Prove that the number
q
n
cos
n
α
q^n \cos n \alpha
q
n
cos
n
α
is an integer.
Inscribe a triangle - [Iran Second Round 1985]
Inscribe in the triangle
A
B
C
ABC
A
BC
a triangle with minimum perimeter.