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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2011 Vietnam Team Selection Test
2011 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(6)
6
1
Hide problems
Exchange of candies amongst n pupils in a round table
Let
n
n
n
be an integer greater than
1.
1.
1.
n
n
n
pupils are seated around a round table, each having a certain number of candies (it is possible that some pupils don't have a candy) such that the sum of all the candies they possess is a multiple of
n
.
n.
n
.
They exchange their candies as follows: For each student's candies at first, there is at least a student who has more candies than the student sitting to his/her right side, in which case, the student on the right side is given a candy by that student. After a round of exchanging, if there is at least a student who has candies greater than the right side student, then he/she will give a candy to the next student sitting to his/her right side. Prove that after the exchange of candies is completed (ie, when it reaches equilibrium), all students have the same number of candies.
5
1
Hide problems
2^{n+2}(2^n-1)-8.3^n+1 is a perfect square
Find all positive integers
n
n
n
such that
A
=
2
n
+
2
(
2
n
−
1
)
−
8
⋅
3
n
+
1
A=2^{n+2}(2^n-1)-8\cdot 3^n +1
A
=
2
n
+
2
(
2
n
−
1
)
−
8
⋅
3
n
+
1
is a perfect square.
3
1
Hide problems
x_i real numbers with sum zero, n>=3
Let
n
n
n
be a positive integer
≥
3.
\geq 3.
≥
3.
There are
n
n
n
real numbers
x
1
,
x
2
,
⋯
x
n
x_1,x_2,\cdots x_n
x
1
,
x
2
,
⋯
x
n
that satisfy:
{
x
1
≥
x
2
≥
⋯
≥
x
n
;
x
1
+
x
2
+
⋯
+
x
n
=
0
;
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
n
(
n
−
1
)
.
\left\{\begin{aligned}&\ x_1\ge x_2\ge\cdots \ge x_n;\\& \ x_1+x_2+\cdots+x_n=0;\\& \ x_1^2+x_2^2+\cdots+x_n^2=n(n-1).\end{aligned}\right.
⎩
⎨
⎧
x
1
≥
x
2
≥
⋯
≥
x
n
;
x
1
+
x
2
+
⋯
+
x
n
=
0
;
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
n
(
n
−
1
)
.
Find the maximum and minimum value of the sum
S
=
x
1
+
x
2
.
S=x_1+x_2.
S
=
x
1
+
x
2
.
4
1
Hide problems
a_n is a sequence with a_0=1, a_1=3, a_{n+2}=...
Let
⟨
a
n
⟩
n
≥
0
\langle a_n\rangle_{n\ge 0}
⟨
a
n
⟩
n
≥
0
be a sequence of integers satisfying
a
0
=
1
,
a
1
=
3
a_0=1, a_1=3
a
0
=
1
,
a
1
=
3
and
a
n
+
2
=
1
+
⌊
a
n
+
1
2
a
n
⌋
∀
n
≥
0.
a_{n+2}=1+\left\lfloor \frac{a_{n+1}^2}{a_n}\right\rfloor \ \ \forall n\ge0.
a
n
+
2
=
1
+
⌊
a
n
a
n
+
1
2
⌋
∀
n
≥
0.
Prove that
a
n
⋅
a
n
+
2
−
a
n
+
1
2
=
2
n
a_n\cdot a_{n+2}-a_{n+1}^2=2^n
a
n
⋅
a
n
+
2
−
a
n
+
1
2
=
2
n
for every natural number
n
.
n.
n
.
2
1
Hide problems
Point A outside a circle (O) with tangents AB,AC
A
A
A
is a point lying outside a circle
(
O
)
(O)
(
O
)
. The tangents from
A
A
A
drawn to
(
O
)
(O)
(
O
)
meet the circle at
B
,
C
.
B,C.
B
,
C
.
Let
P
,
Q
P,Q
P
,
Q
be points on the rays
A
B
,
A
C
AB, AC
A
B
,
A
C
respectively such that
P
Q
PQ
PQ
is tangent to
(
O
)
.
(O).
(
O
)
.
The parallel lines drawn through
P
,
Q
P,Q
P
,
Q
parallel to
C
A
,
B
A
,
CA, BA,
C
A
,
B
A
,
respectively meet
B
C
BC
BC
at
E
,
F
,
E,F,
E
,
F
,
respectively.
(
a
)
(a)
(
a
)
Show that the straight lines
E
Q
EQ
EQ
always pass through a fixed point
M
,
M,
M
,
and
F
P
FP
FP
always pass through a fixed point
N
.
N.
N
.
(
b
)
(b)
(
b
)
Show that
P
M
⋅
Q
N
PM\cdot QN
PM
⋅
QN
is constant.
1
1
Hide problems
Grasshopper jumping from point (1,1) to (m,n) with area =1/2
A grasshopper rests on the point
(
1
,
1
)
(1,1)
(
1
,
1
)
on the plane. Denote by
O
,
O,
O
,
the origin of coordinates. From that point, it jumps to a certain lattice point under the condition that, if it jumps from a point
A
A
A
to
B
,
B,
B
,
then the area of
△
A
O
B
\triangle AOB
△
A
OB
is equal to
1
2
.
\frac 12.
2
1
.
(
a
)
(a)
(
a
)
Find all the positive integral poijnts
(
m
,
n
)
(m,n)
(
m
,
n
)
which can be covered by the grasshopper after a finite number of steps, starting from
(
1
,
1
)
.
(1,1).
(
1
,
1
)
.
(
b
)
(b)
(
b
)
If a point
(
m
,
n
)
(m,n)
(
m
,
n
)
satisfies the above condition, then show that there exists a certain path for the grasshopper to reach
(
m
,
n
)
(m,n)
(
m
,
n
)
from
(
1
,
1
)
(1,1)
(
1
,
1
)
such that the number of jumps does not exceed
∣
m
−
n
∣
.
|m-n|.
∣
m
−
n
∣.