MathDB
Problems
Contests
National and Regional Contests
Thailand Contests
Thailand National Olympiad
2005 Thailand Mathematical Olympiad
2005 Thailand Mathematical Olympiad
Part of
Thailand National Olympiad
Subcontests
(21)
18
1
Hide problems
sum 1/ (1+ tan^{2548} (k\pi /2548))
Compute the sum
∑
k
=
0
1273
1
1
+
t
a
n
2548
(
k
π
2548
)
\sum_{k=0}^{1273}\frac{1}{1 + tan^{2548}\left(\frac{k\pi}{2548}\right)}
k
=
0
∑
1273
1
+
t
a
n
2548
(
2548
kπ
)
1
17
1
Hide problems
given a * b = (a+b+1)/(ab+12)
For
a
,
b
≥
0
a, b \ge 0
a
,
b
≥
0
we define
a
∗
b
=
a
+
b
+
1
a
b
+
12
a * b = \frac{a+b+1}{ab+12}
a
∗
b
=
ab
+
12
a
+
b
+
1
. Compute
0
∗
(
1
∗
(
2
∗
(
.
.
.
(
2003
∗
(
2004
∗
2005
)
)
.
.
.
)
)
)
0*(1*(2*(... (2003*(2004*2005))...)))
0
∗
(
1
∗
(
2
∗
(
...
(
2003
∗
(
2004
∗
2005
))
...
)))
.
21
1
Hide problems
min of cos(a-b) + cos(b-c) + cos(c-a)
Compute the minimum value of
c
o
s
(
a
−
b
)
+
c
o
s
(
b
−
c
)
+
c
o
s
(
c
−
a
)
cos(a-b) + cos(b-c) + cos(c-a)
cos
(
a
−
b
)
+
cos
(
b
−
c
)
+
cos
(
c
−
a
)
as
a
,
b
,
c
a,b,c
a
,
b
,
c
ranges over the real numbers.
20
1
Hide problems
max of ab^{1/2}c^{1/3}d^{1/4} if 36a + 4b + 4c + 3d = 25, for a, b, c, d > 0
Let
a
,
b
,
c
,
d
>
0
a, b, c, d > 0
a
,
b
,
c
,
d
>
0
satisfy
36
a
+
4
b
+
4
c
+
3
d
=
25
36a + 4b + 4c + 3d = 25
36
a
+
4
b
+
4
c
+
3
d
=
25
. What is the maximum possible value of
a
b
1
/
2
c
1
/
3
d
1
/
4
ab^{1/2}c^{1/3}d^{1/4}
a
b
1/2
c
1/3
d
1/4
?
19
1
Hide problems
P(4) + P(0) for monic P(x) degree 4 when ...
Let
P
(
x
)
P(x)
P
(
x
)
be a monic polynomial of degree
4
4
4
such that for
k
=
1
,
2
,
3
k = 1, 2, 3
k
=
1
,
2
,
3
, the remainder when
P
(
x
)
P(x)
P
(
x
)
is divided by
x
−
k
x - k
x
−
k
is equal to
k
k
k
. Find the value of
P
(
4
)
+
P
(
0
)
P(4) + P(0)
P
(
4
)
+
P
(
0
)
.
16
1
Hide problems
sum of roots of (2 - x)^{2005} + x^{2005} = 0.
Compute the sum of roots of
(
2
−
x
)
2005
+
x
2005
=
0
(2 - x)^{2005} + x^{2005} = 0
(
2
−
x
)
2005
+
x
2005
=
0
.
15
1
Hide problems
f(2548) =? f(x + 2y) + 2f(y - 2x) = 3x -4y + 6
A function
f
:
R
→
R
f : R \to R
f
:
R
→
R
satisfy the functional equation
f
(
x
+
2
y
)
+
2
f
(
y
−
2
x
)
=
3
x
−
4
y
+
6
f(x + 2y) + 2f(y - 2x) = 3x -4y + 6
f
(
x
+
2
y
)
+
2
f
(
y
−
2
x
)
=
3
x
−
4
y
+
6
for all reals
x
,
y
x, y
x
,
y
. Compute
f
(
2548
)
f(2548)
f
(
2548
)
.
14
1
Hide problems
f(2) =? , f(m + n) = f(m) + f(n) + 2mn - 2548, f(2548) = -2548
A function
f
:
Z
→
Z
f : Z \to Z
f
:
Z
→
Z
is given so that
f
(
m
+
n
)
=
f
(
m
)
+
f
(
n
)
+
2
m
n
−
2548
f(m + n) = f(m) + f(n) + 2mn - 2548
f
(
m
+
n
)
=
f
(
m
)
+
f
(
n
)
+
2
mn
−
2548
for all positive integers
m
,
n
m, n
m
,
n
. Given that
f
(
2548
)
=
−
2548
f(2548) = -2548
f
(
2548
)
=
−
2548
, find the value of
f
(
2
)
f(2)
f
(
2
)
.
13
1
Hide problems
odd k, exists m, k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372
Find all odd integers
k
k
k
for which there exists a positive integer
m
m
m
satisfying the equation
k
+
(
k
+
5
)
+
(
k
+
10
)
+
.
.
.
+
(
k
+
5
(
m
−
1
)
)
=
1372
k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372
k
+
(
k
+
5
)
+
(
k
+
10
)
+
...
+
(
k
+
5
(
m
−
1
))
=
1372
.
12
1
Hide problems
no of even n, 0 <=n <=100 such that 5 | n^2 x 2^{{2n}^2}+ 1
Find the number of even integers n such that
0
≤
n
≤
100
0 \le n \le 100
0
≤
n
≤
100
and
5
∣
n
2
⋅
2
2
n
2
+
1
5 | n^2 \cdot 2^{{2n}^2}+ 1
5∣
n
2
⋅
2
2
n
2
+
1
.
8
1
Hide problems
sum of alternates signed sums of a subset
For each subset
T
T
T
of
S
=
{
1
,
2
,
.
.
.
,
7
}
S = \{1, 2, ... , 7\}
S
=
{
1
,
2
,
...
,
7
}
, the result
r
(
T
)
r(T)
r
(
T
)
of T is computed as follows: the elements of
T
T
T
are written, largest to smallest, and alternating signs
(
+
,
−
)
(+, -)
(
+
,
−
)
starting with
+
+
+
are put in front of each number. The value of the resulting expression is
r
(
T
)
r(T)
r
(
T
)
. (For example, for
T
=
{
2
,
4
,
7
}
T =\{2, 4, 7\}
T
=
{
2
,
4
,
7
}
, we have
r
(
T
)
=
+
7
−
4
+
2
=
5
r(T) = +7 - 4 + 2 = 5
r
(
T
)
=
+
7
−
4
+
2
=
5
.) Compute the sum of
r
(
T
)
r(T)
r
(
T
)
as
T
T
T
ranges over all subsets of
S
S
S
.
10
1
Hide problems
remainder \sum k^{2005\cdot 2^{2005}} : 2^{2005}
What is the remainder when
∑
k
=
1
2005
k
2005
⋅
2
2005
\sum_{k=1}^{2005}k^{2005\cdot 2^{2005}}
∑
k
=
1
2005
k
2005
⋅
2
2005
is divided by
2
2005
2^{2005}
2
2005
?
9
1
Hide problems
gcd ( (135^{90}-45^{90}) / 90^2 , 90^2 )
Compute gcd
(
13
5
90
−
4
5
90
9
0
2
,
9
0
2
)
\left( \frac{135^{90}-45^{90}}{90^2} , 90^2 \right)
(
9
0
2
13
5
90
−
4
5
90
,
9
0
2
)
11
1
Hide problems
min x, 2^{254} divides x^{2005} + 1
Find the smallest positive integer
x
x
x
such that
2
254
2^{254}
2
254
divides
x
2005
+
1
x^{2005} + 1
x
2005
+
1
.
7
1
Hide problems
ways of 2548 as a sum of at least two positive integers
How many ways are there to express
2548
2548
2548
as a sum of at least two positive integers, where two sums that differ in order are considered different?
5
1
Hide problems
how many ways are there 6 rolls of a die to sum to 21
A die is thrown six times. How many ways are there for the six rolls to sum to
21
21
21
?
6
2
Hide problems
no of pos. integer (x_1+x_2+x_3)^2 (y_1+y_2) = 2548
Find the number of positive integer solutions to the equation
(
x
1
+
x
2
+
x
3
)
2
(
y
1
+
y
2
)
=
2548
(x_1+x_2+x_3)^2(y_1+y_2) = 2548
(
x
1
+
x
2
+
x
3
)
2
(
y
1
+
y
2
)
=
2548
.
sum [(2a - b )(a-b)]^2 >=5
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be distinct real numbers. Prove that
(
2
a
−
b
a
−
b
)
2
+
(
2
b
−
c
b
−
c
)
2
+
(
2
c
−
a
c
−
a
)
2
≥
5
\left(\frac{2a - b}{a -b} \right)^2+\left(\frac{2b - c}{b - c} \right)^2+\left(\frac{2c - a}{c - a} \right)^2 \ge 5
(
a
−
b
2
a
−
b
)
2
+
(
b
−
c
2
b
−
c
)
2
+
(
c
−
a
2
c
−
a
)
2
≥
5
4
2
Hide problems
BO x OC wanted, O incenter of ABC , inscribed in circle with diameter BC
Triangle
△
A
B
C
\vartriangle ABC
△
A
BC
is inscribed in the circle with diameter
B
C
BC
BC
. If
A
B
=
3
AB = 3
A
B
=
3
,
A
C
=
4
AC = 4
A
C
=
4
, and
O
O
O
is the incenter of
△
A
B
C
\vartriangle ABC
△
A
BC
, then find
B
O
⋅
O
C
BO \cdot OC
BO
⋅
OC
.
P is circumcenter of a circles tangent to all 1 circle and 2 semicircles
Let
O
1
O_1
O
1
be the center of a semicircle
ω
1
\omega_1
ω
1
with diameter
A
B
AB
A
B
and let
O
2
O_2
O
2
be the center of a circle
ω
2
\omega_2
ω
2
inscribed in
ω
1
\omega_1
ω
1
and which is tangent to
A
B
AB
A
B
at
O
1
O_1
O
1
. Let
O
3
O_3
O
3
be a point on
A
B
AB
A
B
that is the center of a semicircle
ω
3
\omega_3
ω
3
which is tangent to both
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
. Let
P
P
P
be the intersection of the line through
O
3
O_3
O
3
perpendicular to
A
B
AB
A
B
and the line through
O
2
O_2
O
2
parallel to
A
B
AB
A
B
. Show that
P
P
P
is the center of a circle
Γ
\Gamma
Γ
tangent to all of
ω
1
,
ω
2
\omega_1, \omega_2
ω
1
,
ω
2
and
ω
3
\omega_3
ω
3
.
3
2
Hide problems
ratio AB/BC , isosceles , <ABC = 2<BAC
Triangle
△
A
B
C
\vartriangle ABC
△
A
BC
is isosceles with
A
B
=
A
C
AB = AC
A
B
=
A
C
and
∠
A
B
C
=
2
∠
B
A
C
\angle ABC = 2\angle BAC
∠
A
BC
=
2∠
B
A
C
. Compute
A
B
B
C
\frac{AB}{BC}
BC
A
B
.
f(f(n)) = 2n for all positive integers n
Does there exist a function
f
:
Z
+
→
Z
+
f : Z^+ \to Z^+
f
:
Z
+
→
Z
+
such that
f
(
f
(
n
)
)
=
2
n
f(f(n)) = 2n
f
(
f
(
n
))
=
2
n
for all positive integers
n
n
n
? Justify your answer, and if the answer is yes, give an explicit construction.
2
2
Hide problems
ratio wanted, 2 feet of altitudes, BH=R
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an acute triangle, and let
A
′
A'
A
′
and
B
′
B'
B
′
be the feet of altitudes from
A
A
A
to
B
C
BC
BC
and from
B
B
B
to
C
A
CA
C
A
, respectively; the altitudes intersect at
H
H
H
. If
B
H
BH
B
H
is equal to the circumradius of
△
A
B
C
\vartriangle ABC
△
A
BC
, find
A
′
B
A
B
\frac{A'B}{AB}
A
B
A
′
B
.
a, b \in S such that a \ne b and 10 | a^3b - ab^3
Let
S
S
S
be a set of three distinct integers. Show that there are
a
,
b
∈
S
a, b \in S
a
,
b
∈
S
such that
a
≠
b
a \ne b
a
=
b
and
10
∣
a
3
b
−
a
b
3
10 | a^3b - ab^3
10∣
a
3
b
−
a
b
3
.
1
2
Hide problems
computational, incscribed trapezoid ABCD, DC = 4AD, diameter AB
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid inscribed in a unit circle with diameter
A
B
AB
A
B
. If
D
C
=
4
A
D
DC = 4AD
D
C
=
4
A
D
, compute
A
D
AD
A
D
.
AH x FH = GH^2 wanted, circle with diameter BC related
A point
A
A
A
is chosen outside a circle with diameter
B
C
BC
BC
so that
△
A
B
C
\vartriangle ABC
△
A
BC
is acute. Segments
A
B
AB
A
B
and
A
C
AC
A
C
intersect the circle at
D
D
D
and
E
E
E
, respectively, and
C
D
CD
C
D
intersects
B
E
BE
BE
at
F
F
F
. Line
A
F
AF
A
F
intersects the circle again at
G
G
G
and intersects
B
C
BC
BC
at
H
H
H
. Prove that
A
H
⋅
F
H
=
G
H
2
AH \cdot F H = GH^2
A
H
⋅
F
H
=
G
H
2
. .