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National and Regional Contests
The Philippines Contests
Philippine MO
2012 Philippine MO
2012 Philippine MO
Part of
Philippine MO
Subcontests
(5)
5
1
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14th Philippine Mathematical Olympiad 2011-National Stage #5
There are exactly
120
120
120
Twitter subscribers from National Science High School. Statistics show that each of
10
10
10
given celebrities has at least
85
85
85
followers from National Science High School. Prove that there must be two students such that each of the
10
10
10
celebrities is being followed in Twitter by at least one of these students.
4
1
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14th Philippine Mathematical Olympiad 2011-National Stage #4
Let
⋆
\star
⋆
be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers
x
x
x
and
y
y
y
, (i)
(
x
+
1
)
⋆
0
=
(
0
⋆
x
)
+
1
(x + 1)\star 0 = (0\star x) + 1
(
x
+
1
)
⋆
0
=
(
0
⋆
x
)
+
1
(ii)
0
⋆
(
y
+
1
)
=
(
y
⋆
0
)
+
1
0\star (y + 1) = (y\star 0) + 1
0
⋆
(
y
+
1
)
=
(
y
⋆
0
)
+
1
(iii)
(
x
+
1
)
⋆
(
y
+
1
)
=
(
x
⋆
y
)
+
1
(x + 1)\star (y + 1) = (x\star y) + 1
(
x
+
1
)
⋆
(
y
+
1
)
=
(
x
⋆
y
)
+
1
. If
123
⋆
456
=
789
123\star 456 = 789
123
⋆
456
=
789
, find
246
⋆
135
246\star 135
246
⋆
135
.
3
1
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14th Philippine Mathematical Olympiad 2011-National Stage #3
If
a
b
>
0
ab>0
ab
>
0
and
0
<
x
<
π
2
\displaystyle 0<x<\frac{\pi}{2}
0
<
x
<
2
π
, prove that
(
1
+
a
2
sin
x
)
(
1
+
b
2
cos
x
)
≥
(
1
+
2
a
b
)
2
sin
2
x
2
.
\left ( 1+\frac{a^2}{\sin x} \right ) \left ( 1+\frac{b^2}{\cos x} \right ) \geq \frac{(1+\sqrt{2}ab)^2 \sin 2x}{2}.
(
1
+
sin
x
a
2
)
(
1
+
cos
x
b
2
)
≥
2
(
1
+
2
ab
)
2
sin
2
x
.
2
1
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14th Philippine Mathematical Olympiad 2011-National Stage #2
Let
f
f
f
be a polynomial function with integer coefficients and
p
p
p
be a prime number. Suppose there are at least four distinct integers satisfying
f
(
x
)
=
p
f(x) = p
f
(
x
)
=
p
. Show that
f
f
f
does not have integer zeros.
1
1
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14th Philippine Mathematical Olympiad 2011-National Stage #1
A computer generates even integers half of the time and another computer generates even integers a third of the time. If
a
i
a_i
a
i
and
b
i
b_i
b
i
are the integers generated by the computers, respectively, at time
i
i
i
, what is the probability that
a
1
b
1
+
a
2
b
2
+
⋯
+
a
k
b
k
a_1b_1 +a_2b_2 +\cdots + a_kb_k
a
1
b
1
+
a
2
b
2
+
⋯
+
a
k
b
k
is an even integer.