MathDB
Problems
Contests
National and Regional Contests
Pakistan Contests
Pakistan TST
2017 Pakistan TST
2017 Pakistan TST
Part of
Pakistan TST
Subcontests
(3)
Problem 3
1
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Beautiful Disguised Functional!
Find all
f
:
R
+
→
R
+
f:\mathbb{R}^+ \rightarrow \mathbb{R}^+
f
:
R
+
→
R
+
such that for all distinct
x
,
y
,
z
x,y,z
x
,
y
,
z
f
(
x
)
2
−
f
(
y
)
f
(
z
)
=
f
(
x
y
)
f
(
y
)
f
(
z
)
[
f
(
y
z
)
−
f
(
z
x
)
]
f(x)^2-f(y)f(z)=f(x^y)f(y)f(z)[f(y^z)-f(z^x)]
f
(
x
)
2
−
f
(
y
)
f
(
z
)
=
f
(
x
y
)
f
(
y
)
f
(
z
)
[
f
(
y
z
)
−
f
(
z
x
)]
Problem 2
1
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Switching students in a circle, due to height.
There are
n
n
n
students in a circle, one behind the other, all facing clockwise. The students have heights
h
1
<
h
2
<
h
3
<
⋯
<
h
n
h_1 <h_2 < h_3 < \cdots < h_n
h
1
<
h
2
<
h
3
<
⋯
<
h
n
. If a student with height
h
k
h_k
h
k
is standing directly behind a student with height
h
k
−
2
h_{k-2}
h
k
−
2
or lesss, the two students are permitted to switch places Prove that it is not possible to make more than
(
n
3
)
\binom{n}{3}
(
3
n
)
such switches before reaching a position in which no further switches are possible.
Problem 1
1
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Perpendicularity in Cyclic Quad.
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral. The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
meet at
P
P
P
, and
D
A
DA
D
A
and
C
B
CB
CB
meet at
Q
Q
Q
. Suppose
P
Q
PQ
PQ
is perpendicular to
A
C
AC
A
C
. Let
E
E
E
be the midpoint of
A
B
AB
A
B
. Prove that
P
E
PE
PE
is perpendicular to
B
C
BC
BC
.