MathDB
Problems
Contests
National and Regional Contests
Nepal Contests
Nepal National Olympiad
2018 Nepal National Olympiad
2018 Nepal National Olympiad
Part of
Nepal National Olympiad
Subcontests
(12)
4c
1
Hide problems
Nepal National Olympiad
Problem Section #4c) A special deck of cards contains
49
49
49
cards, each labeled with a number from
1
1
1
to
7
7
7
and colored with one of seven colors. Each number-color combination appears on exactly one card. John will select a set of eight cards from the deck at random. Given that he gets at least one card of each color and at least one card with each number, the probability that John can discard one of his cards and still have at least one card of each color and at least one card with each number is
p
q
\frac{p}{q}
q
p
, where
p
p
p
and
q
q
q
are relatively prime positive integers. Find
p
+
q
p+q
p
+
q
.
4b
1
Hide problems
Nepal National Olympiad
Problem Section #4b) Let
A
A
A
be a unit square. What is the largest area of a triangle whose vertices lie on the perimeter of
A
A
A
? Justify your answer.
4a
1
Hide problems
Nepal National Olympiad
Problem Section #4a) There is a
6
∗
6
6 * 6
6
∗
6
grid, each square filled with a grasshopper. After the bell rings, each grasshopper jumps to an adjacent square (A square that shares a side). What is the maximum number of empty squares possible?
3c
1
Hide problems
Convex Pentagon
Problem Section #3c) Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a convex pentagon such that
B
C
∥
A
E
,
A
B
=
B
C
+
A
E
BC \parallel AE, AB = BC + AE
BC
∥
A
E
,
A
B
=
BC
+
A
E
, and
∠
A
B
C
=
∠
C
D
E
\angle{ABC} =\angle{CDE}
∠
A
BC
=
∠
C
D
E
. Let
M
M
M
be the midpoint of
C
E
CE
CE
, and let
O
O
O
be the circumcenter of triangle
B
C
D
BCD
BC
D
. Given that
∠
D
M
O
=
9
0
o
\angle{DMO}=90^{o}
∠
D
MO
=
9
0
o
, prove that
2
∠
B
D
A
=
∠
C
D
E
2\angle{BDA} =\angle{CDE}
2∠
B
D
A
=
∠
C
D
E
.
3b
1
Hide problems
Nepal National Olympiad
Problem Section #3NOTE: Neglect that HF and CD.
3a
1
Hide problems
Nepal National Olympiad
Problem Section #3 a) Circles
O
1
O_1
O
1
and
O
2
O_2
O
2
interest at two points
B
B
B
and
C
C
C
, and
B
C
BC
BC
is the diameter of circle
O
1
O_1
O
1
. Construct a tangent line of circle
O
1
O_1
O
1
at
C
C
C
and intersecting circle
O
2
O_2
O
2
at another point
A
A
A
. Join
A
B
AB
A
B
to intersect circle
O
1
O_1
O
1
at point
E
E
E
, then join
C
E
CE
CE
and extend it to intersect circle
O
2
O_2
O
2
at point
F
F
F
. Assume
H
H
H
is an arbitrary point on line segment
A
F
AF
A
F
. Join
H
E
HE
H
E
and extend it to intersect circle
O
1
O_1
O
1
at point
G
G
G
, and then join
B
G
BG
BG
and extend it to intersect the extend of
A
C
AC
A
C
at point
D
D
D
. Prove:
A
H
H
F
=
A
C
C
D
\frac{AH}{HF}=\frac{AC}{CD}
H
F
A
H
=
C
D
A
C
.
2c
1
Hide problems
Nepal National Olympiad
Problem Section #2c). Denote by
Q
+
\mathbb{Q^+}
Q
+
the set of all positive rational numbers. Determine all functions
f
:
Q
+
→
Q
+
f:\mathbb{Q^+}\to\mathbb{Q^+}
f
:
Q
+
→
Q
+
which satisfy the following equation for all
x
,
y
∈
Q
+
:
f
(
f
(
x
)
2
.
y
)
=
x
3
.
f
(
x
y
)
x,y \in \mathbb{Q^+} : f(f(x)^2.y)=x^3.f(xy)
x
,
y
∈
Q
+
:
f
(
f
(
x
)
2
.
y
)
=
x
3
.
f
(
x
y
)
.
2b
1
Hide problems
Nepal National Olympiad
Problem Section #2 b) Find the maximal value of
(
x
3
+
1
)
(
y
3
+
1
)
(x^3+1)(y^3+1)
(
x
3
+
1
)
(
y
3
+
1
)
, where
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
,
x
+
y
=
1
x+y=1
x
+
y
=
1
.
2a
1
Hide problems
Nepal National Olympiad
Problem Section #2 a) If
a
x
+
b
y
=
7
ax+by=7
a
x
+
b
y
=
7
a
x
2
+
b
y
2
=
49
ax^2+by^2=49
a
x
2
+
b
y
2
=
49
a
x
3
+
b
y
3
=
133
ax^3+by^3=133
a
x
3
+
b
y
3
=
133
a
x
4
+
b
y
4
=
406
ax^4+by^4=406
a
x
4
+
b
y
4
=
406
, find the value of
2014
(
x
+
y
−
x
y
)
−
100
(
a
+
b
)
.
2014(x+y-xy)-100(a+b).
2014
(
x
+
y
−
x
y
)
−
100
(
a
+
b
)
.
1c
1
Hide problems
Nepal National Olympiad
Problem Section #1 c) Find all pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
of non-negative integers for which
m
2
+
2.
3
n
=
m
(
2
n
+
1
−
1
)
.
m^2+2.3^n=m(2^{n+1}-1).
m
2
+
2.
3
n
=
m
(
2
n
+
1
−
1
)
.
1b
1
Hide problems
Nepal Regional Olympiad
Problem Section #1 b) Let
a
,
b
a, b
a
,
b
be positive integers such that
b
n
+
n
b^n +n
b
n
+
n
is a multiple of
a
n
+
n
a^n + n
a
n
+
n
for all positive integers
n
n
n
. Prove that
a
=
b
.
a = b.
a
=
b
.
1a
1
Hide problems
Nepal National Olympiad
Problem Section #1 a) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are
189
,
320
,
287
,
264
,
x
189, 320, 287, 264, x
189
,
320
,
287
,
264
,
x
, and y. Find the greatest possible value of:
x
+
y
x + y
x
+
y
.NOTE: There is a high chance that this problems was copied.