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Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia JBMO TST
2022 Saudi Arabia JBMO TST
2022 Saudi Arabia JBMO TST
Part of
Saudi Arabia JBMO TST
Subcontests
(4)
4
2
Hide problems
min a such that (a^p -a)/ p =b^2
Determine the smallest positive integer
a
a
a
for which there exist a prime number
p
p
p
and a positive integer
b
≥
2
b \ge 2
b
≥
2
such that
a
p
−
a
p
=
b
2
.
\frac{a^p -a}{p}=b^2.
p
a
p
−
a
=
b
2
.
big birthday cake for m or n persons, not necessarily equal parts
You plan to organize your birthday party, which will be attended either by exactly
m
m
m
persons or by exactly
n
n
n
persons (you are not sure at the moment). You have a big birthday cake and you want to divide it into several parts (not necessarily equal), so that you are able to distribute the whole cake among the people attending the party with everybody getting cake of equal mass (however, one may get one big slice, while others several small slices - the sizes of slices may differ). What is the minimal number of parts you need to divide the cake, so that it is possible, regardless of the number of guests.
1
2
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n + k are composites for all nice numbers n, when n+1, 8n+1 both squares
The positive
n
>
3
n > 3
n
>
3
called ‘nice’ if and only if
n
+
1
n +1
n
+
1
and
8
n
+
1
8n + 1
8
n
+
1
are both perfect squares. How many positive integers
k
≤
15
k \le 15
k
≤
15
such that
4
n
+
k
4n + k
4
n
+
k
are composites for all nice numbers
n
n
n
?
p^5 + p^3 + 2 = q^2 - q
Find all pairs of positive prime numbers
(
p
,
q
)
(p, q)
(
p
,
q
)
such that
p
5
+
p
3
+
2
=
q
2
−
q
.
p^5 + p^3 + 2 = q^2 - q.
p
5
+
p
3
+
2
=
q
2
−
q
.
3
2
Hide problems
2000 consecutive integers on the board
2000
2000
2000
consecutive integers (not necessarily positive) are written on the board. A student takes several turns. On each turn, he partitions the
2000
2000
2000
integers into
1000
1000
1000
pairs, and substitutes each pair by the difference arid the sum of that pair (note that the difference does not need to be positive as the student may choose to subtract the greater number from the smaller one; in addition, all the operations are carried simultaneously). Prove that the student will never again write
2000
2000
2000
consecutive integers on the board.
incenter of PA_0Q lies on altitude of ABC
Let
B
B
1
BB_1
B
B
1
and
C
C
1
CC_1
C
C
1
be the altitudes of acute-angled triangle
A
B
C
ABC
A
BC
, and
A
0
A_0
A
0
is the midpoint of
B
C
BC
BC
. Lines
A
0
B
1
A_0B_1
A
0
B
1
and
A
0
C
1
A_0C_1
A
0
C
1
meet the line passing through
A
A
A
and parallel to
B
C
BC
BC
at points
P
P
P
and
Q
Q
Q
. Prove that the incenter of triangle
P
A
0
Q
PA_0Q
P
A
0
Q
lies on the altitude of triangle
A
B
C
ABC
A
BC
.
2
2
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A, B', P, Q concyclic wanted, 2 altitudes and 2 circles related
Let
B
B
′
BB'
B
B
′
,
C
C
′
CC'
C
C
′
be the altitudes of an acute-angled triangle
A
B
C
ABC
A
BC
. Two circles passing through
A
A
A
and
C
′
C'
C
′
are tangent to
B
C
BC
BC
at points
P
P
P
and
Q
Q
Q
. Prove that
A
,
B
′
,
P
,
Q
A, B', P, Q
A
,
B
′
,
P
,
Q
are concyclic.
max \sqrt{b^2+c^2}/(3-a)+ \sqrt{c^2+a^2}/(3-b)+a+b-2022c if a^2+b^2+c^2=2
Consider non-negative real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfying the condition
a
2
+
b
2
+
c
2
=
2
a^2 + b^2 + c^2 = 2
a
2
+
b
2
+
c
2
=
2
. Find the maximum value of the following expression
P
=
b
2
+
c
2
3
−
a
+
c
2
+
a
2
3
−
b
+
a
+
b
−
2022
c
P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c
P
=
3
−
a
b
2
+
c
2
+
3
−
b
c
2
+
a
2
+
a
+
b
−
2022
c