MathDB
Problems
Contests
National and Regional Contests
Malaysia Contests
JOM
JOM 2015
JOM 2015
Part of
JOM
Subcontests
(5)
4
1
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Strictly Increasing Sequences
Given a natural number
n
≥
3
n\ge 3
n
≥
3
, determine all strictly increasing sequences
a
1
<
a
2
<
⋯
<
a
n
a_1<a_2<\cdots<a_n
a
1
<
a
2
<
⋯
<
a
n
such that
gcd
(
a
1
,
a
2
)
=
1
\text{gcd}(a_1,a_2)=1
gcd
(
a
1
,
a
2
)
=
1
and for any pair of natural numbers
(
k
,
m
)
(k,m)
(
k
,
m
)
satisfy
n
≥
m
≥
3
n\ge m\ge 3
n
≥
m
≥
3
,
m
≥
k
m\ge k
m
≥
k
,
a
1
+
a
2
+
⋯
+
a
m
a
k
\frac{a_1+a_2+\cdots +a_m}{a_k}
a
k
a
1
+
a
2
+
⋯
+
a
m
is a positive integer.
2
1
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AB + CD = BC
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral. Let angle bisectors of
∠
B
\angle B
∠
B
and
∠
C
\angle C
∠
C
intersect at
E
E
E
. Let
A
B
AB
A
B
intersect
C
D
CD
C
D
at
F
F
F
. Prove that if
A
B
+
C
D
=
B
C
AB+CD=BC
A
B
+
C
D
=
BC
, then
A
,
D
,
E
,
F
A,D,E,F
A
,
D
,
E
,
F
is cyclic.
5
1
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Navi and Ozna
Navi and Ozna are playing a game where Ozna starts first and the two take turn making moves. A positive integer is written on the waord. A move is to (i) subtract any positive integer at most 2015 from it or (ii) given that the integer on the board is divisible by
2014
2014
2014
, divide by
2014
2014
2014
. The first person to make the integer
0
0
0
wins. To make Navi's condition worse, Ozna gets to pick integers
a
a
a
and
b
b
b
,
a
≥
2015
a\ge 2015
a
≥
2015
such that all numbers of the form
a
n
+
b
an+b
an
+
b
will not be the starting integer, where
n
n
n
is any positive integer.Find the minimum number of starting integer where Navi wins.
1
1
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Game on Graph
Baron and Peter are playing a game. They are given a simple finite graph
G
G
G
with
n
≥
3
n\ge 3
n
≥
3
vertex and
k
k
k
edges that connects the vertices. First Peter labels two vertices A and B, and places a counter at A. Baron starts first. A move for Baron is move the counter along an edge. Peter's move is to remove an edge from the graph. Baron wins if he reaches
B
B
B
, otherwise Peter wins.Given the value of
n
n
n
, what is the largest
k
k
k
so that Peter can always win?
3
1
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Weird Inequality
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers greater or equal to
3
3
3
. Prove that
3
(
a
b
c
+
b
+
2
c
)
≥
2
(
a
b
+
2
a
c
+
3
b
c
)
3(abc+b+2c)\ge 2(ab+2ac+3bc)
3
(
ab
c
+
b
+
2
c
)
≥
2
(
ab
+
2
a
c
+
3
b
c
)
and determine all equality cases.