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Problems
Contests
National and Regional Contests
Mongolia Contests
Mongolian Mathematical Olympiad
2000 Mongolian Mathematical Olympiad
2000 Mongolian Mathematical Olympiad
Part of
Mongolian Mathematical Olympiad
Subcontests
(6)
Problem 6
2
Hide problems
side ratio in triangle, angle bisectors form cyclic quadrilateral
In a triangle
A
B
C
ABC
A
BC
, the angle bisector at
A
,
B
,
C
A,B,C
A
,
B
,
C
meet the opposite sides at
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
, respectively. Prove that if the quadrilateral
B
A
1
B
1
C
1
BA_1B_1C_1
B
A
1
B
1
C
1
is cyclic, then
A
C
A
B
+
B
C
=
A
B
A
C
+
C
B
+
B
C
B
A
+
A
C
.
\frac{AC}{AB+BC}=\frac{AB}{AC+CB}+\frac{BC}{BA+AC}.
A
B
+
BC
A
C
=
A
C
+
CB
A
B
+
B
A
+
A
C
BC
.
# of naturals not exceeding n divisible by exactly one of a set of primes
Given distinct prime numbers
p
1
,
…
,
p
s
p_1,\ldots,p_s
p
1
,
…
,
p
s
and a positive integer
n
n
n
, find the number of positive integers not exceeding
n
n
n
that are divisible by exactly one of the
p
i
p_i
p
i
.
Problem 5
2
Hide problems
system of inequalities, number of integer solutions
Given a natural number
n
n
n
, find the number of quadruples
(
x
,
y
,
u
,
v
)
(x,y,u,v)
(
x
,
y
,
u
,
v
)
of integers with
1
≤
x
,
y
,
y
,
v
≤
n
1\le x,y,y,v\le n
1
≤
x
,
y
,
y
,
v
≤
n
satisfy the following inequalities: \begin{align*} &1\le v+x-y\le n,\\ &1\le x+y-u\le n,\\ &1\le u+v-y\le n,\\ &1\le v+x-u\le n. \end{align*}
inequality in positive integers
Let
m
,
n
,
k
m,n,k
m
,
n
,
k
be positive integers with
m
≥
2
m\ge2
m
≥
2
and
k
≥
log
2
(
m
−
1
)
k\ge\log_2(m-1)
k
≥
lo
g
2
(
m
−
1
)
. Prove that
∏
s
=
1
n
m
s
−
1
m
s
<
1
2
n
+
1
2
k
+
1
.
\prod_{s=1}^n\frac{ms-1}{ms}<\sqrt[2^{k+1}]{\frac1{2n+1}}.
s
=
1
∏
n
m
s
m
s
−
1
<
2
k
+
1
2
n
+
1
1
.
Problem 4
2
Hide problems
functional inequality over R
Suppose that a function
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
satisfies the following conditions:(i)
∣
f
(
a
)
−
f
(
b
)
∣
≤
∣
a
−
b
∣
\left|f(a)-f(b)\right|\le|a-b|
∣
f
(
a
)
−
f
(
b
)
∣
≤
∣
a
−
b
∣
for all
a
,
b
∈
R
a,b\in\mathbb R
a
,
b
∈
R
; (ii)
f
(
f
(
f
(
0
)
)
)
=
0
f(f(f(0)))=0
f
(
f
(
f
(
0
)))
=
0
.Prove that
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
.
distance between towns in country
In a country with
n
n
n
towns, the distance between the towns numbered
i
i
i
and
j
j
j
is denoted by
x
i
j
x_{ij}
x
ij
. Suppose that the total length of every cyclic route which passes through every town exactly once is the same. Prove that there exist numbers
a
i
,
b
i
a_i,b_i
a
i
,
b
i
(
i
=
1
,
…
,
n
i=1,\ldots,n
i
=
1
,
…
,
n
) such that
x
i
j
=
a
i
+
b
j
x_{ij}=a_i+b_j
x
ij
=
a
i
+
b
j
for all distinct
i
,
j
i,j
i
,
j
.
Problem 3
2
Hide problems
cutting up cube into unit cubes
A cube of side
n
n
n
is cut into
n
3
n^3
n
3
unit cubes, and m of these cubes are marked so that the centers of any three marked cubes do not form a right-angled triangle with legs parallel to sides of the cube. Find the maximum possible value of
m
m
m
.
points moving around circles
Two points
A
A
A
and
B
B
B
move around two different circles in the plane with the same angular velocity. Suppose that there is a point
C
C
C
which is equidistant from
A
A
A
and
B
B
B
at every moment. Prove that, at some moment,
A
A
A
and
B
B
B
will coincide.
Problem 2
2
Hide problems
externally tangent circles, prove collinearity of intersection points
Circles
ω
1
,
ω
2
,
ω
3
\omega_1,\omega_2,\omega_3
ω
1
,
ω
2
,
ω
3
with centers
O
1
,
O
2
,
O
3
O_1,O_2,O_3
O
1
,
O
2
,
O
3
, respectively, are externally tangent to each other. The circle
ω
1
\omega_1
ω
1
touches
ω
2
\omega_2
ω
2
at
P
1
P_1
P
1
and
ω
3
\omega_3
ω
3
at
P
2
P_2
P
2
. For any point
A
A
A
on
ω
1
\omega_1
ω
1
,
A
1
A_1
A
1
denotes the point symmetric to
A
A
A
with respect to
O
1
O_1
O
1
. Show that the intersection points of
A
P
2
AP_2
A
P
2
with
ω
3
\omega_3
ω
3
,
A
1
P
3
A_1P_3
A
1
P
3
with
ω
2
\omega_2
ω
2
, and
A
P
3
AP_3
A
P
3
with
A
1
P
2
A_1P_2
A
1
P
2
lie on a line.
inequality in vectors
Let
n
≥
2
n\ge2
n
≥
2
. For any two
n
n
n
-vectors
x
⃗
=
(
x
1
,
…
,
x
n
)
\vec x=(x_1,\ldots,x_n)
x
=
(
x
1
,
…
,
x
n
)
and
y
⃗
=
(
y
1
,
…
,
y
n
)
\vec y=(y_1,\ldots,y_n)
y
=
(
y
1
,
…
,
y
n
)
, we define
f
(
x
⃗
,
y
⃗
)
=
x
1
y
1
‾
−
∑
i
=
2
n
x
i
y
i
‾
.
f\left(\vec x,\vec y\right)=x_1\overline{y_1}-\sum_{i=2}^nx_i\overline{y_i}.
f
(
x
,
y
)
=
x
1
y
1
−
i
=
2
∑
n
x
i
y
i
.
Prove that if
f
(
x
⃗
,
x
⃗
)
≥
0
f\left(\vec x,\vec x\right)\ge0
f
(
x
,
x
)
≥
0
, and
f
(
y
⃗
,
y
⃗
)
≥
0
f\left(\vec y,\vec y\right)\ge0
f
(
y
,
y
)
≥
0
, then
∣
f
(
x
⃗
,
y
⃗
)
∣
2
≥
f
(
x
⃗
,
x
⃗
)
f
(
y
⃗
,
y
⃗
)
\left|f\left(\vec x,\vec y\right)\right|^2\ge f\left(\vec x,\vec x\right)f\left(\vec y,\vec y\right)
∣
f
(
x
,
y
)
∣
2
≥
f
(
x
,
x
)
f
(
y
,
y
)
.
Problem 1
2
Hide problems
sequence of radical (nt)
Let
rad
(
k
)
\operatorname{rad}(k)
rad
(
k
)
denote the product of prime divisors of a natural number
k
k
k
(define
rad
(
1
)
=
1
\operatorname{rad}(1)=1
rad
(
1
)
=
1
). A sequence
(
a
n
)
(a_n)
(
a
n
)
is defined by setting
a
1
a_1
a
1
arbitrarily, and
a
n
+
1
=
a
n
+
rad
(
a
n
)
a_{n+1}=a_n+\operatorname{rad}(a_n)
a
n
+
1
=
a
n
+
rad
(
a
n
)
for
n
≥
1
n\ge1
n
≥
1
. Prove that the sequence
(
a
n
)
(a_n)
(
a
n
)
contains arithmetic progressions of arbitrary length.
range of (x+y+z)^2/xyz, x,y,z in N
Find all integers that can be written in the form
(
x
+
y
+
z
)
2
x
y
z
\frac{(x+y+z)^2}{xyz}
x
yz
(
x
+
y
+
z
)
2
, where
x
,
y
,
z
x,y,z
x
,
y
,
z
are positive integers.