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Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia Pre-TST + Training Tests
2016 Saudi Arabia Pre-TST
2016 Saudi Arabia Pre-TST
Part of
Saudi Arabia Pre-TST + Training Tests
Subcontests
(8)
2.2
2
Hide problems
black and blue vertices of a regular 20 -gon
Ten vertices of a regular
20
20
20
-gon
A
1
A
2
.
.
.
.
A
20
A_1A_2....A_{20}
A
1
A
2
....
A
20
are painted black and the other ten vertices are painted blue. Consider the set consisting of diagonal
A
1
A
4
A_1A_4
A
1
A
4
and all other diagonals of the same length. 1. Prove that in this set, the number of diagonals with two black endpoints is equal to the number of diagonals with two blue endpoints. 2. Find all possible numbers of the diagonals with two black endpoints.
x_1 =|a- b|, y_1 = |b-c|, z_1 = |c-d|, t_1 = |d -a|, sequence
Given four numbers
x
,
y
,
z
,
t
x, y, z, t
x
,
y
,
z
,
t
, let
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
be a permutation of
(
x
,
y
,
z
,
t
)
(x, y, z, t)
(
x
,
y
,
z
,
t
)
and set
x
1
=
∣
a
−
b
∣
x_1 =|a- b|
x
1
=
∣
a
−
b
∣
,
y
1
=
∣
b
−
c
∣
y_1 = |b-c|
y
1
=
∣
b
−
c
∣
,
z
1
=
∣
c
−
d
∣
z_1 = |c-d|
z
1
=
∣
c
−
d
∣
, and
t
1
=
∣
d
−
a
∣
t_1 = |d -a|
t
1
=
∣
d
−
a
∣
. From
x
1
,
y
1
,
z
1
,
t
1
x_1, y_1, z_1, t_1
x
1
,
y
1
,
z
1
,
t
1
, form in the same fashion the numbers
x
2
,
y
2
,
z
2
,
t
2
x_2, y_2, z_2, t_2
x
2
,
y
2
,
z
2
,
t
2
, and so on. It is known that
x
n
=
x
,
y
n
=
y
,
z
n
=
z
,
t
n
=
t
x_n = x, y_n = y, z_n = z, t_n = t
x
n
=
x
,
y
n
=
y
,
z
n
=
z
,
t
n
=
t
for some
n
n
n
. Find all possible values of
(
x
,
y
,
z
,
t
)
(x, y, z, t)
(
x
,
y
,
z
,
t
)
.
2.1
2
Hide problems
x_1 = |x - y|, y_1 = | y -z|, z_1 = |z- x| , sequence
Given three numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
, and set
x
1
=
∣
x
−
y
∣
,
y
1
=
∣
y
−
z
∣
,
z
1
=
∣
z
−
x
∣
x_1 = |x - y|, y_1 = | y -z|, z_1 = |z- x|
x
1
=
∣
x
−
y
∣
,
y
1
=
∣
y
−
z
∣
,
z
1
=
∣
z
−
x
∣
. From
x
1
,
y
1
,
z
1
x_1, y_1, z_1
x
1
,
y
1
,
z
1
, form in the same fashion the numbers
x
2
,
y
2
,
z
2
x_2, y_2, z_2
x
2
,
y
2
,
z
2
, and so on. It is known that
x
n
=
x
,
y
n
=
y
,
z
n
=
z
x_n = x, y_n = y, z_n = z
x
n
=
x
,
y
n
=
y
,
z
n
=
z
for some
n
n
n
. Find all possible values of
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
.
permutaion of 1,2,..., n with any 2 adjacent is 2015 or 2016
1) Prove that there are infinitely many positive integers
n
n
n
such that there exists a permutation of
1
,
2
,
3
,
.
.
.
,
n
1, 2, 3, . . . , n
1
,
2
,
3
,
...
,
n
with the property that the difference between any two adjacent numbers is equal to either
2015
2015
2015
or
2016
2016
2016
. 2) Let
k
k
k
be a positive integer. Is the statement in 1) still true if we replace the numbers
2015
2015
2015
and
2016
2016
2016
by
k
k
k
and
k
+
2016
k + 2016
k
+
2016
, respectively?
1.4
2
Hide problems
F(r) = (p^{rp} - 1) (p - 1)/(p^r - 1) (p^p - 1)
Let
p
p
p
be a given prime. For each prime
r
r
r
, we defind the function as following
F
(
r
)
=
(
p
r
p
−
1
)
(
p
−
1
)
(
p
r
−
1
)
(
p
p
−
1
)
F(r) =\frac{(p^{rp} - 1) (p - 1)}{(p^r - 1) (p^p - 1)}
F
(
r
)
=
(
p
r
−
1
)
(
p
p
−
1
)
(
p
r
p
−
1
)
(
p
−
1
)
. 1. Show that
F
(
r
)
F(r)
F
(
r
)
is a positive integer for any prime
r
≠
p
r \ne p
r
=
p
. 2. Show that
F
(
r
)
F(r)
F
(
r
)
and
F
(
s
)
F(s)
F
(
s
)
are coprime for any primes
r
r
r
and
s
s
s
such that
r
≠
p
,
s
≠
p
r \ne p, s \ne p
r
=
p
,
s
=
p
and
r
≠
s
r \ne s
r
=
s
. 3. Fix a prime
r
≠
p
r \ne p
r
=
p
. Show that there is a prime divisor
q
q
q
of
F
(
r
)
F(r)
F
(
r
)
such that
p
∣
q
−
1
p| q - 1
p
∣
q
−
1
but
p
2
∤
q
−
1
p^2 \nmid q - 1
p
2
∤
q
−
1
.
numbers in a 2015 x 2015 table
The natural numbers
0
,
1
,
2
,
3
,
.
.
.
0, 1, 2, 3, . . .
0
,
1
,
2
,
3
,
...
are written on the square table 2015\times 2015 in a circular order (anti-clockwise) such that
0
0
0
is in the center of the table. The rows and columns are labelled from bottom to top and from left to right respectively. (see figure below) 1. The number
2015
2015
2015
is in which row and which column? 2. We are allowed to perform the following operations: First, we replace the number
0
0
0
in the center by
14
14
14
, after that, each time, we can add
1
1
1
to each of
12
12
12
numbers on
12
12
12
consecutive unit squares in a row, or
12
12
12
consecutive unit squares in a column, or
12
12
12
unit squares in a rectangle
3
×
4
3\times 4
3
×
4
. After a finite number of steps, can we make all numbers on the table are multiples of
2016
2016
2016
? https://cdn.artofproblemsolving.com/attachments/c/d/223b32c0e3f58f62d0d40fa78c09a2cd035ed5.png
1.3
2
Hide problems
a lock has 16 keys arranged in a 4x 4 array
A lock has
16
16
16
keys arranged in a 4\times 4 array, each key oriented either horizontally or vertically. In order to open it, all the keys must be vertically oriented. When a key is switched to another position, all the other keys in the same row and column automatically switch their positions too. Show that no matter what the starting positions are, it is always possible to open this lock. (Only one key at a time can be switched.)
b + 1|a^2 + 1 and a + 1|b^2 + 1
Let
a
,
b
a, b
a
,
b
be two positive integers such that
b
+
1
∣
a
2
+
1
b + 1|a^2 + 1
b
+
1∣
a
2
+
1
,
a
+
1
∣
b
2
+
1
a + 1|b^2 + 1
a
+
1∣
b
2
+
1
. Prove that
a
,
b
a, b
a
,
b
are odd numbers.
2.3
2
Hide problems
u^n - v^n are integers for infinite many n
Let
u
u
u
and
v
v
v
be positive rational numbers with
u
≠
v
u \ne v
u
=
v
. Assume that there are infinitely many positive integers
n
n
n
with the property that
u
n
−
v
n
u^n - v^n
u
n
−
v
n
are integers. Prove that
u
u
u
and
v
v
v
are integers.
circle (I) is inscribed in hexagon with 6 vertices A_b,A_c , B_c , B_a, C_a, C_b
Let
A
B
C
ABC
A
BC
be a non isosceles triangle with circumcircle
(
O
)
(O)
(
O
)
and incircle
(
I
)
(I)
(
I
)
. Denote
(
O
1
)
(O_1)
(
O
1
)
as the circle internal tangent to
(
O
)
(O)
(
O
)
at
A
1
A_1
A
1
and also tangent to segments
A
B
,
A
C
AB,AC
A
B
,
A
C
at
A
b
,
A
c
A_b,A_c
A
b
,
A
c
respectively. Define the circles
(
O
2
)
,
(
O
3
)
(O_2), (O_3)
(
O
2
)
,
(
O
3
)
and the points
B
1
,
C
1
,
B
c
,
B
a
,
C
a
,
C
b
B_1, C_1, B_c , B_a, C_a, C_b
B
1
,
C
1
,
B
c
,
B
a
,
C
a
,
C
b
similarly. 1. Prove that
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 at the point
M
M
M
and
3
3
3
points
I
,
M
,
O
I,M,O
I
,
M
,
O
are collinear. 2. Prove that the circle
(
I
)
(I)
(
I
)
is inscribed in the hexagon with
6
6
6
vertices
A
b
,
A
c
,
B
c
,
B
a
,
C
a
,
C
b
A_b,A_c , B_c , B_a, C_a, C_b
A
b
,
A
c
,
B
c
,
B
a
,
C
a
,
C
b
.
1.1
2
Hide problems
x + y + z + 1/2x yz >= 4 when x^2 +y^2 + z^2 = 2(x y + yz + z x)
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be positive real numbers satisfy the condition
x
2
+
y
2
+
z
2
=
2
(
x
y
+
y
z
+
z
x
)
x^2 +y^2 + z^2 = 2(x y + yz + z x)
x
2
+
y
2
+
z
2
=
2
(
x
y
+
yz
+
z
x
)
. Prove that
x
+
y
+
z
+
1
2
x
y
z
≥
4
x + y + z + \frac{1}{2x yz} \ge 4
x
+
y
+
z
+
2
x
yz
1
≥
4
tangent circle to 3 given circles internally also tangent to a 4th circle
Let
A
B
C
ABC
A
BC
be an acute, non isosceles triangle,
A
X
,
B
Y
,
C
Z
AX, BY, CZ
A
X
,
B
Y
,
CZ
are the altitudes with
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
belong to
B
C
,
C
A
,
A
B
BC, CA,AB
BC
,
C
A
,
A
B
respectively. Respectively denote
(
O
1
)
,
(
O
2
)
,
(
O
3
)
(O_1), (O_2), (O_3)
(
O
1
)
,
(
O
2
)
,
(
O
3
)
as the circumcircles of triangles
A
Y
Z
,
B
Z
X
,
C
X
Y
AY Z, BZX, CX Y
A
Y
Z
,
BZX
,
CX
Y
. Suppose that
(
K
)
(K)
(
K
)
is a circle that internal tangent to
(
O
1
)
,
(
O
2
)
,
(
O
3
)
(O_1), (O_2), (O_3)
(
O
1
)
,
(
O
2
)
,
(
O
3
)
. Prove that
(
K
)
(K)
(
K
)
is tangent to circumcircle of triangle
A
B
C
ABC
A
BC
.
2.4
2
Hide problems
circle tangent to 3 circles wanted, concurrency and collinearity also
Let
A
B
C
ABC
A
BC
be a non isosceles triangle with circumcircle
(
O
)
(O)
(
O
)
and incircle
(
I
)
(I)
(
I
)
. Denote
(
O
1
)
(O_1)
(
O
1
)
as the circle that external tangent to
(
O
)
(O)
(
O
)
at
A
′
A'
A
′
and also tangent to the lines
A
B
,
A
C
AB,AC
A
B
,
A
C
at
A
b
,
A
c
A_b,A_c
A
b
,
A
c
respectively. Define the circles
(
O
2
)
,
(
O
3
)
(O_2), (O_3)
(
O
2
)
,
(
O
3
)
and the points
B
′
,
C
′
,
B
c
,
B
a
,
C
a
,
C
b
B', C', B_c , B_a, C_a, C_b
B
′
,
C
′
,
B
c
,
B
a
,
C
a
,
C
b
similarly. 1. Denote J as the radical center of
(
O
1
)
,
(
O
2
)
,
(
O
3
)
(O_1), (O_2), (O_3)
(
O
1
)
,
(
O
2
)
,
(
O
3
)
and suppose that
J
A
′
JA'
J
A
′
intersects
(
O
1
)
(O_1)
(
O
1
)
at the second point
X
,
J
B
′
X, JB'
X
,
J
B
′
intersects
(
O
2
)
(O_2)
(
O
2
)
at the second point Y , JC' intersects
(
O
3
)
(O_3)
(
O
3
)
at the second point
Z
Z
Z
. Prove that the circle
(
X
Y
Z
)
(X Y Z)
(
X
Y
Z
)
is tangent to
(
O
1
)
,
(
O
2
)
,
(
O
3
)
(O_1), (O_2), (O_3)
(
O
1
)
,
(
O
2
)
,
(
O
3
)
. 2. Prove that
A
A
′
,
B
B
′
,
C
C
′
AA', BB', CC'
A
A
′
,
B
B
′
,
C
C
′
are concurrent at the point
M
M
M
and
3
3
3
points
I
,
M
,
O
I,M,O
I
,
M
,
O
are collinear.
products, (a + i) | b(b + 2016), (a + i) \nmid b, (a + i)\mid (b + 2016)
Let
n
n
n
be a given positive integer. Prove that there are infinitely many pairs of positive integers
(
a
,
b
)
(a, b)
(
a
,
b
)
with
a
,
b
>
n
a, b > n
a
,
b
>
n
such that
∏
i
=
1
2015
(
a
+
i
)
∣
b
(
b
+
2016
)
,
∏
i
=
1
2015
(
a
+
i
)
∤
b
,
∏
i
=
1
2015
(
a
+
i
)
∣
(
b
+
2016
)
\prod_{i=1}^{2015} (a + i) | b(b + 2016), \prod_{i=1}^{2015}(a + i) \nmid b, \prod_{i=1}^{2015} (a + i)\mid (b + 2016)
i
=
1
∏
2015
(
a
+
i
)
∣
b
(
b
+
2016
)
,
i
=
1
∏
2015
(
a
+
i
)
∤
b
,
i
=
1
∏
2015
(
a
+
i
)
∣
(
b
+
2016
)
.
1.2
2
Hide problems
similar triangles nad perpendicularity wanted, inscribed isosceles
Let
A
B
C
ABC
A
BC
be a non isosceles triangle inscribed in a circle
(
O
)
(O)
(
O
)
and
B
E
,
C
F
BE, CF
BE
,
CF
are two angle bisectors intersect at
I
I
I
with
E
E
E
belongs to segment
A
C
AC
A
C
and
F
F
F
belongs to segment
A
B
AB
A
B
. Suppose that
B
E
,
C
F
BE, CF
BE
,
CF
intersect
(
O
)
(O)
(
O
)
at
M
,
N
M,N
M
,
N
respectively. The line
d
1
d_1
d
1
passes through
M
M
M
and perpendicular to
B
M
BM
BM
intersects
(
O
)
(O)
(
O
)
at the second point
P
,
P,
P
,
the line
d
2
d_2
d
2
passes through
N
N
N
and perpendicular to
C
N
CN
CN
intersect
(
O
)
(O)
(
O
)
at the second point
Q
Q
Q
. Denote
H
,
K
H, K
H
,
K
are two midpoints of
M
P
MP
MP
and
N
Q
NQ
NQ
respectively. 1. Prove that triangles
I
E
F
IEF
I
EF
and
O
K
H
OKH
O
KH
are similar. 2. Suppose that S is the intersection of two lines
d
1
d_1
d
1
and
d
2
d_2
d
2
. Prove that
S
O
SO
SO
is perpendicular to
E
F
EF
EF
.
(a + b)/c +(b + c)/a +(c + a)/b >= a + b + c +1/a+1/b+1/c
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive numbers such that
a
2
+
b
2
+
c
2
+
a
b
c
=
4
a^2+b^2+c^2+abc = 4
a
2
+
b
2
+
c
2
+
ab
c
=
4
. Prove that
a
+
b
c
+
b
+
c
a
+
c
+
a
b
≥
a
+
b
+
c
+
1
a
+
1
b
+
1
c
\frac{a + b}{c} +\frac{b + c}{a} +\frac{c + a}{b} \ge a + b + c + \frac{1}{a} + \frac{1}{b} +\frac{1}{c}
c
a
+
b
+
a
b
+
c
+
b
c
+
a
≥
a
+
b
+
c
+
a
1
+
b
1
+
c
1