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Problems
Contests
National and Regional Contests
Thailand Contests
Thailand National Olympiad
2010 Thailand Mathematical Olympiad
2010 Thailand Mathematical Olympiad
Part of
Thailand National Olympiad
Subcontests
(10)
10
1
Hide problems
binom (100, p) + 7 is divisible by p
Find all primes
p
p
p
such that
(
100
p
)
+
7
{100 \choose p} + 7
(
p
100
)
+
7
is divisible by
p
p
p
.
9
1
Hide problems
min real root of 2x^5 + 5x^4 + 5x^3 + ax^2 + bx + c = 0, given all roots reals
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be real numbers so that all roots of the equation
2
x
5
+
5
x
4
+
5
x
3
+
a
x
2
+
b
x
+
c
=
0
2x^5 + 5x^4 + 5x^3 + ax^2 + bx + c = 0
2
x
5
+
5
x
4
+
5
x
3
+
a
x
2
+
b
x
+
c
=
0
are real. Find the smallest real root of the equation above.
8
1
Hide problems
modulo 2553 distance d(m, n) <= 36 , |S| >= 70
Define the modulo
2553
2553
2553
distance
d
(
x
,
y
)
d(x, y)
d
(
x
,
y
)
between two integers
x
,
y
x, y
x
,
y
to be the smallest nonnegative integer
d
d
d
equivalent to either
x
−
y
x - y
x
−
y
or
y
−
x
y - x
y
−
x
modulo
2553
2553
2553
. Show that, given a set S of integers such that
∣
S
∣
≥
70
|S| \ge 70
∣
S
∣
≥
70
, there must be
m
,
n
∈
S
m, n \in S
m
,
n
∈
S
with
d
(
m
,
n
)
≤
36
d(m, n) \le 36
d
(
m
,
n
)
≤
36
.
7
1
Hide problems
a^5/bc^2 + b^5/ca^2 + c^5/ab^2 >= a^2 + b^2 + c^2
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive reals. Show that
a
5
b
c
2
+
b
5
c
a
2
+
c
5
a
b
2
≥
a
2
+
b
2
+
c
2
.
\frac{a^5}{bc^2} + \frac{b^5}{ca^2} + \frac{c^5}{ab^2} \ge a^2 + b^2 + c^2.
b
c
2
a
5
+
c
a
2
b
5
+
a
b
2
c
5
≥
a
2
+
b
2
+
c
2
.
4
2
Hide problems
angle wanted, equilateral, AN = BM, 3MB = AB
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an equilateral triangle, and let
M
M
M
and
N
N
N
be points on
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, so that
A
N
=
B
M
AN = BM
A
N
=
BM
and
3
M
B
=
A
B
3MB = AB
3
MB
=
A
B
. Lines
C
M
CM
CM
and
B
N
BN
BN
intersect at
O
O
O
. Find
∠
A
O
B
\angle AOB
∠
A
OB
.
sidelengths and altitudes inequalities , r_3^2 \ge r_1^2 +r_2^2
For
i
=
1
,
2
i = 1, 2
i
=
1
,
2
let
△
A
i
B
i
C
i
\vartriangle A_iB_iC_i
△
A
i
B
i
C
i
be a triangle with side lengths
a
i
,
b
i
,
c
i
a_i, b_i, c_i
a
i
,
b
i
,
c
i
and altitude lengths
p
i
,
q
i
,
r
i
p_i, q_i, r_i
p
i
,
q
i
,
r
i
. Define
a
3
=
a
1
2
+
a
2
2
,
b
3
=
b
1
2
+
b
2
2
a_3 =\sqrt{a_1^2 + a_2^2}, b_3 =\sqrt{b_1^2 + b_2^2}
a
3
=
a
1
2
+
a
2
2
,
b
3
=
b
1
2
+
b
2
2
, and
c
3
=
c
1
2
+
c
2
2
c_3 =\sqrt{c_1^2 + c_2^2}
c
3
=
c
1
2
+
c
2
2
. Prove that
a
3
,
b
3
,
c
3
a_3, b_3, c_3
a
3
,
b
3
,
c
3
are side lengths of a triangle, and if
p
3
,
q
3
,
r
3
p_3, q_3, r_3
p
3
,
q
3
,
r
3
are the lengths of altitudes of this triangle, then
p
3
2
≥
p
1
2
+
p
2
2
p_3^2 \ge p_1^2 +p_2^2
p
3
2
≥
p
1
2
+
p
2
2
,
q
3
2
≥
q
1
2
+
q
2
2
q_3^2 \ge q_1^2 +q_2^2
q
3
2
≥
q
1
2
+
q
2
2
, and
r
3
2
≥
r
1
2
+
r
2
2
r_3^2 \ge r_1^2 +r_2^2
r
3
2
≥
r
1
2
+
r
2
2
3
2
Hide problems
2555...553 divisible by 2553
Show that there are infinitely many positive integers n such that
2
555...55
⏟
n
3
2\underbrace{555...55}_{n}3
2
n
555...55
3
is divisible by
2553
2553
2553
.
(AB+AC)/BC = DE/ MN + 1 , projections on angle bisectors
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a scalene triangle with
A
B
<
B
C
<
C
A
AB < BC < CA
A
B
<
BC
<
C
A
. Let
D
D
D
be the projection of
A
A
A
onto the angle bisector of
∠
A
B
C
\angle ABC
∠
A
BC
, and let
E
E
E
be the projection of
A
A
A
onto the angle bisector of
∠
A
C
B
\angle ACB
∠
A
CB
. The line
D
E
DE
D
E
cuts sides
A
B
AB
A
B
and AC at
M
M
M
and
N
N
N
, respectively. Prove that
A
B
+
A
C
B
C
=
D
E
M
N
+
1
\frac{AB+AC}{BC} =\frac{DE}{MN} + 1
BC
A
B
+
A
C
=
MN
D
E
+
1
5
2
Hide problems
tennis, a^2_1+a^2_2+...+a^2_{2010} =b^2_1 + b^2_2 + ... + b^2_{2010}
In a round-robin table tennis tournament between
2010
2010
2010
athletes, where each match ends with a winner and a loser, let
a
1
,
.
.
.
,
a
2010
a_1,... , a_{2010}
a
1
,
...
,
a
2010
denote the number of wins of each athlete, and let
b
1
,
.
.
,
b
2010
b_1, .., b_{2010}
b
1
,
..
,
b
2010
denote the number of losses of each athlete. Show that
a
1
2
+
a
2
2
+
.
.
.
+
a
2010
2
=
b
1
2
+
b
2
2
+
.
.
.
+
b
2010
2
a^2_1+a^2_2+...+a^2_{2010} =b^2_1 + b^2_2 + ... + b^2_{2010}
a
1
2
+
a
2
2
+
...
+
a
2010
2
=
b
1
2
+
b
2
2
+
...
+
b
2010
2
.
f(x - t, y) + f(x + t, y) + f(x, y - t) + f(x, y + t) = 2010
Determine all functions
f
:
R
×
R
→
R
f : R \times R \to R
f
:
R
×
R
→
R
satisfying the equation
f
(
x
−
t
,
y
)
+
f
(
x
+
t
,
y
)
+
f
(
x
,
y
−
t
)
+
f
(
x
,
y
+
t
)
=
2010
f(x - t, y) + f(x + t, y) + f(x, y - t) + f(x, y + t) = 2010
f
(
x
−
t
,
y
)
+
f
(
x
+
t
,
y
)
+
f
(
x
,
y
−
t
)
+
f
(
x
,
y
+
t
)
=
2010
for all real numbers
x
,
y
x, y
x
,
y
and for all nonzero
t
t
t
6
2
Hide problems
f(-x) = f(x) if f(3x + y) + f(3x-y) = f(x + y) + f(x - y) + 16f(x)
Let
f
:
R
→
R
f : R \to R
f
:
R
→
R
be a function satisfying the functional equation
f
(
3
x
+
y
)
+
f
(
3
x
−
y
)
=
f
(
x
+
y
)
+
f
(
x
−
y
)
+
16
f
(
x
)
f(3x + y) + f(3x-y) = f(x + y) + f(x - y) + 16f(x)
f
(
3
x
+
y
)
+
f
(
3
x
−
y
)
=
f
(
x
+
y
)
+
f
(
x
−
y
)
+
16
f
(
x
)
for all reals
x
,
y
x, y
x
,
y
. Show that
f
f
f
is even, that is,
f
(
−
x
)
=
f
(
x
)
f(-x) = f(x)
f
(
−
x
)
=
f
(
x
)
for all reals
x
x
x
no triples of primes p, q, r satisfy p > r, q > r, and pq | r^p + r^q
Show that no triples of primes
p
,
q
,
r
p, q, r
p
,
q
,
r
satisfy
p
>
r
,
q
>
r
p > r, q > r
p
>
r
,
q
>
r
, and
p
q
∣
r
p
+
r
q
pq | r^p + r^q
pq
∣
r
p
+
r
q
2
2
Hide problems
computational, statring with an isosceles and circle
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an isosceles triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
. A circle passing through
B
B
B
and
C
C
C
intersects sides
A
B
AB
A
B
and
A
C
AC
A
C
at
D
D
D
and
E
E
E
respectively. A point
F
F
F
on this circle is chosen so that
E
F
⊥
B
C
EF\perp BC
EF
⊥
BC
. If
B
C
=
x
BC = x
BC
=
x
,
C
F
=
y
CF = y
CF
=
y
, and
B
F
=
z
BF = z
BF
=
z
, find the length of
D
F
DF
D
F
in terms of
x
,
y
,
z
x, y, z
x
,
y
,
z
.
2010 students from 5 regions in a debate tournament into 3 topics
The Ministry of Education selects
2010
2010
2010
students from
5
5
5
regions of Thailand to participate in a debate tournament, where each pair of students will debate in one of the three topics: politics, economics, and societal problems. Show that there are
3
3
3
students who were born in the same month, come from the same region, are of the same gender , and whose pairwise debates are on the same topic.
1
2
Hide problems
no of ways to distribute 11 balls into 5 boxes with different sizes
Find the number of ways to distribute
11
11
11
balls into
5
5
5
boxes with different sizes, so that each box receives at least one ball, and the total number of balls in the largest and smallest boxes is more than the total number of balls in the remaining boxes.
for every x exists y in {2, 5, 13} such that xy - 1 not a perfect square
Show that, for every positive integer
x
x
x
, there is a positive integer
y
∈
{
2
,
5
,
13
}
y\in \{2, 5, 13\}
y
∈
{
2
,
5
,
13
}
such that
x
y
−
1
xy - 1
x
y
−
1
is not a perfect square.