MathDB
Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia IMO TST
2019 Saudi Arabia IMO TST
2019 Saudi Arabia IMO TST
Part of
Saudi Arabia IMO TST
Subcontests
(3)
3
2
Hide problems
regular hexagon divided into 6n^2 regular triangles , 2n coins
Let regular hexagon is divided into
6
n
2
6n^2
6
n
2
regular triangles. Let
2
n
2n
2
n
coins are put in different triangles such, that no any two coins lie on the same layer (layer is area between two consecutive parallel lines). Let also triangles are painted like on the chess board. Prove that exactly
n
n
n
coins lie on black triangles. https://cdn.artofproblemsolving.com/attachments/0/4/96503a10351b0dc38b611c6ee6eb945b5ed1d9.png
concurrency and tangent line to circle wanted, incircle ,line passing incenter
Let
A
B
C
ABC
A
BC
be an acute nonisosceles triangle with incenter
I
I
I
and
(
d
)
(d)
(
d
)
is an arbitrary line tangent to
(
I
)
(I)
(
I
)
at
K
K
K
. The lines passes through
I
I
I
, perpendicular to
I
A
,
I
B
,
I
C
IA, IB, IC
I
A
,
I
B
,
I
C
cut
(
d
)
(d)
(
d
)
at
A
1
,
B
1
,
C
1
A_1, B_1,C_1
A
1
,
B
1
,
C
1
respectively. Suppose that
(
d
)
(d)
(
d
)
cuts
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at
M
,
N
,
P
M,N, P
M
,
N
,
P
respectively. The lines through
M
,
N
,
P
M,N,P
M
,
N
,
P
and respectively parallel to the internal bisectors of
A
,
B
,
C
A, B, C
A
,
B
,
C
in triangle
A
B
C
ABC
A
BC
meet each other to define a triange
X
Y
Z
XYZ
X
Y
Z
. Prove that three lines
A
A
1
,
B
B
1
,
C
C
1
AA_1, BB_1, CC_1
A
A
1
,
B
B
1
,
C
C
1
are concurrent and
I
K
IK
I
K
is tangent to the circle
(
X
Y
Z
)
(XY Z)
(
X
Y
Z
)
2
2
Hide problems
product or number of all divisor of $sm, sn are equal
Find all pair of integers
(
m
,
n
)
(m,n)
(
m
,
n
)
and
m
≥
n
m \ge n
m
≥
n
such that there exist a positive integer
s
s
s
and a) Product of all divisor of
s
m
,
s
n
sm, sn
s
m
,
s
n
are equal. b) Number of divisors of
s
m
,
s
n
sm,sn
s
m
,
s
n
are equal.
f(n + 1)f(n + 2)... f(n + k) / f(1)f(2) ... f(k) \in Z
Let non-constant polynomial
f
(
x
)
f(x)
f
(
x
)
with real coefficients is given with the following property: for any positive integer
n
n
n
and
k
k
k
, the value of expression
f
(
n
+
1
)
f
(
n
+
2
)
.
.
.
f
(
n
+
k
)
f
(
1
)
f
(
2
)
.
.
.
f
(
k
)
∈
Z
\frac{f(n + 1)f(n + 2)... f(n + k)}{ f(1)f(2) ... f(k)} \in Z
f
(
1
)
f
(
2
)
...
f
(
k
)
f
(
n
+
1
)
f
(
n
+
2
)
...
f
(
n
+
k
)
∈
Z
Prove that
f
(
x
)
f(x)
f
(
x
)
is divisible by
x
x
x
1
2
Hide problems
n^3 - n^2 <= f(n) (f(f(n)))^2 <= n^3 + n^2
Find all functions
f
:
Z
+
→
Z
+
f : Z^+ \to Z^+
f
:
Z
+
→
Z
+
such that
n
3
−
n
2
≤
f
(
n
)
⋅
(
f
(
f
(
n
)
)
)
2
≤
n
3
+
n
2
n^3 - n^2 \le f(n) \cdot (f(f(n)))^2 \le n^3 + n^2
n
3
−
n
2
≤
f
(
n
)
⋅
(
f
(
f
(
n
))
)
2
≤
n
3
+
n
2
for every
n
n
n
in positive integers
a_n is smallest pos integer such a_0 + a_1 +... + a_n is divisible by n
Let
a
0
a_0
a
0
be an arbitrary positive integer. Let
(
a
n
)
(a_n)
(
a
n
)
be infinite sequence of positive integers such that for every positive integer
n
n
n
, the term
a
n
a_n
a
n
is the smallest positive integer such that
a
0
+
a
1
+
.
.
.
+
a
n
a_0 + a_1 +... + a_n
a
0
+
a
1
+
...
+
a
n
is divisible by
n
n
n
. Prove that there exist
N
N
N
such that
a
n
+
1
=
a
n
a_{n+1} = a_n
a
n
+
1
=
a
n
for all
n
≥
N
n \ge N
n
≥
N