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Problems
Contests
International Contests
IMO Longlists
1983 IMO Longlists
1983 IMO Longlists
Part of
IMO Longlists
Subcontests
(51)
75
1
Hide problems
Find the sum of the fiftieth powers of sides and diagonals
Find the sum of the fiftieth powers of all sides and diagonals of a regular
100
100
100
-gon inscribed in a circle of radius
R
.
R.
R
.
74
1
Hide problems
Find the locus of points M in the plane
In a plane we are given two distinct points
A
,
B
A,B
A
,
B
and two lines
a
,
b
a, b
a
,
b
passing through
B
B
B
and
A
A
A
respectively
(
a
∋
B
,
b
∋
A
)
(a \ni B, b \ni A)
(
a
∋
B
,
b
∋
A
)
such that the line
A
B
AB
A
B
is equally inclined to a and b. Find the locus of points
M
M
M
in the plane such that the product of distances from
M
M
M
to
A
A
A
and a equals the product of distances from
M
M
M
to
B
B
B
and
b
b
b
(i.e.,
M
A
⋅
M
A
′
=
M
B
⋅
M
B
′
MA \cdot MA' = MB \cdot MB'
M
A
⋅
M
A
′
=
MB
⋅
M
B
′
, where
A
′
A'
A
′
and
B
′
B'
B
′
are the feet of the perpendiculars from
M
M
M
to
a
a
a
and
b
b
b
respectively).
73
1
Hide problems
There exist two points P and Q in the plane of the triangle
Let
A
B
C
ABC
A
BC
be a nonequilateral triangle. Prove that there exist two points
P
P
P
and
Q
Q
Q
in the plane of the triangle, one in the interior and one in the exterior of the circumcircle of
A
B
C
ABC
A
BC
, such that the orthogonal projections of any of these two points on the sides of the triangle are vertices of an equilateral triangle.
72
1
Hide problems
Cosines inequality on n variables - ILL 1983
Prove that for all
x
1
,
x
2
,
…
,
x
n
∈
R
x_1, x_2,\ldots , x_n \in \mathbb R
x
1
,
x
2
,
…
,
x
n
∈
R
the following inequality holds:
∑
n
≥
i
>
j
≥
1
cos
2
(
x
i
−
x
j
)
≥
n
(
n
−
2
)
4
\sum_{n \geq i >j \geq 1} \cos^2(x_i - x_j ) \geq \frac{n(n-2)}{4}
n
≥
i
>
j
≥
1
∑
cos
2
(
x
i
−
x
j
)
≥
4
n
(
n
−
2
)
68
1
Hide problems
Find the fourth root of the equation
Three of the roots of the equation
x
4
−
p
x
3
+
q
x
2
−
r
x
+
s
=
0
x^4 -px^3 +qx^2 -rx+s = 0
x
4
−
p
x
3
+
q
x
2
−
r
x
+
s
=
0
are
tan
A
,
tan
B
\tan A, \tan B
tan
A
,
tan
B
, and
tan
C
\tan C
tan
C
, where
A
,
B
A, B
A
,
B
, and
C
C
C
are angles of a triangle. Determine the fourth root as a function only of
p
,
q
,
r
p, q, r
p
,
q
,
r
, and
s
.
s.
s
.
67
1
Hide problems
Prove that altitudes of the tetrahedron are concurrent
The altitude from a vertex of a given tetrahedron intersects the opposite face in its orthocenter. Prove that all four altitudes of the tetrahedron are concurrent.
65
1
Hide problems
Prove the fraction on cotangents
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral whose diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect in a point
P
P
P
. Prove that
A
P
P
C
=
cot
∠
B
A
C
+
cot
∠
D
A
C
cot
∠
B
C
A
+
cot
∠
D
C
A
\frac{AP}{PC}=\frac{\cot \angle BAC + \cot \angle DAC}{\cot \angle BCA + \cot \angle DCA}
PC
A
P
=
cot
∠
BC
A
+
cot
∠
D
C
A
cot
∠
B
A
C
+
cot
∠
D
A
C
64
1
Hide problems
Find the sum of all of the face angles of the polyhedron
The sum of all the face angles about all of the vertices except one of a given polyhedron is
5160
5160
5160
. Find the sum of all of the face angles of the polyhedron.
62
1
Hide problems
Prove that there is a point E on AB
A
A
A
circle
γ
\gamma
γ
is drawn and let
A
B
AB
A
B
be a diameter. The point
C
C
C
on
γ
\gamma
γ
is the midpoint of the line segment
B
D
BD
B
D
. The line segments
A
C
AC
A
C
and
D
O
DO
D
O
, where
O
O
O
is the center of
γ
\gamma
γ
, intersect at
P
P
P
. Prove that there is a point
E
E
E
on
A
B
AB
A
B
such that
P
P
P
is on the circle with diameter
A
E
.
AE.
A
E
.
61
1
Hide problems
Is it possible to find integers p and q ?
Let
a
a
a
and
b
b
b
be integers. Is it possible to find integers
p
p
p
and
q
q
q
such that the integers
p
+
n
a
p+na
p
+
na
and
q
+
n
b
q +nb
q
+
nb
have no common prime factor no matter how the integer
n
n
n
is chosen ?
59
1
Hide problems
Solve the equation tan^2(2x) + 2 tan(2x) · tan(3x) −1 = 0
Solve the equation
tan
2
(
2
x
)
+
2
tan
(
2
x
)
⋅
tan
(
3
x
)
−
1
=
0.
\tan^2(2x) + 2 \tan(2x) \cdot \tan(3x) -1 = 0.
tan
2
(
2
x
)
+
2
tan
(
2
x
)
⋅
tan
(
3
x
)
−
1
=
0.
57
1
Hide problems
Find a number n in the system of base n^2 + 1
In the system of base
n
2
+
1
n^2 + 1
n
2
+
1
find a number
N
N
N
with
n
n
n
different digits such that:(i)
N
N
N
is a multiple of
n
n
n
. Let
N
=
n
N
′
.
N = nN'.
N
=
n
N
′
.
(ii) The number
N
N
N
and
N
′
N'
N
′
have the same number
n
n
n
of different digits in base
n
2
+
1
n^2 + 1
n
2
+
1
, none of them being zero.(iii) If
s
(
C
)
s(C)
s
(
C
)
denotes the number in base
n
2
+
1
n^2 + 1
n
2
+
1
obtained by applying the permutation
s
s
s
to the
n
n
n
digits of the number
C
C
C
, then for each permutation
s
,
s
(
N
)
=
n
s
(
N
′
)
.
s, s(N) = ns(N').
s
,
s
(
N
)
=
n
s
(
N
′
)
.
56
1
Hide problems
Determine the greatest common divisor of the coefficients
Consider the expansion
(
1
+
x
+
x
2
+
x
3
+
x
4
)
496
=
a
0
+
a
1
x
+
⋯
+
a
1984
x
1984
.
(1 + x + x^2 + x^3 + x^4)^{496} = a_0 + a_1x + \cdots + a_{1984}x^{1984}.
(
1
+
x
+
x
2
+
x
3
+
x
4
)
496
=
a
0
+
a
1
x
+
⋯
+
a
1984
x
1984
.
(a) Determine the greatest common divisor of the coefficients
a
3
,
a
8
,
a
13
,
…
,
a
1983
.
a_3, a_8, a_{13}, \ldots , a_{1983}.
a
3
,
a
8
,
a
13
,
…
,
a
1983
.
(b) Prove that
1
0
340
<
a
992
<
1
0
347
.
10^{340 }< a_{992} < 10^{347}.
1
0
340
<
a
992
<
1
0
347
.
55
1
Hide problems
Find maximum of M(a) for a≤1983
For every
a
∈
N
a \in \mathbb N
a
∈
N
denote by
M
(
a
)
M(a)
M
(
a
)
the number of elements of the set
{
b
∈
N
∣
a
+
b
is a divisor of
a
b
}
.
\{ b \in \mathbb N | a + b \text{ is a divisor of } ab \}.
{
b
∈
N
∣
a
+
b
is a divisor of
ab
}
.
Find
max
a
≤
1983
M
(
a
)
.
\max_{a\leq 1983} M(a).
max
a
≤
1983
M
(
a
)
.
53
1
Hide problems
Prove that a ∈ {0, 1, . . ., n} and z_k ∈ {1, i}
Let
a
∈
R
a \in \mathbb R
a
∈
R
and let
z
1
,
z
2
,
…
,
z
n
z_1, z_2, \ldots, z_n
z
1
,
z
2
,
…
,
z
n
be complex numbers of modulus
1
1
1
satisfying the relation
∑
k
=
1
n
z
k
3
=
4
(
a
+
(
a
−
n
)
i
)
−
3
∑
k
=
1
n
z
k
‾
\sum_{k=1}^n z_k^3=4(a+(a-n)i)- 3 \sum_{k=1}^n \overline{z_k}
k
=
1
∑
n
z
k
3
=
4
(
a
+
(
a
−
n
)
i
)
−
3
k
=
1
∑
n
z
k
Prove that
a
∈
{
0
,
1
,
…
,
n
}
a \in \{0, 1,\ldots, n \}
a
∈
{
0
,
1
,
…
,
n
}
and
z
k
∈
{
1
,
i
}
z_k \in \{1, i \}
z
k
∈
{
1
,
i
}
for all
k
.
k.
k
.
49
1
Hide problems
Inequality on k variables
Given positive integers
k
,
m
,
n
k,m, n
k
,
m
,
n
with
k
m
≤
n
km \leq n
km
≤
n
and non-negative real numbers
x
1
,
…
,
x
k
x_1, \ldots , x_k
x
1
,
…
,
x
k
, prove that
n
(
∏
i
=
1
k
x
i
m
−
1
)
≤
m
∑
i
=
1
k
(
x
i
n
−
1
)
.
n \left( \prod_{i=1}^k x_i^m -1 \right) \leq m \sum_{i=1}^k (x_i^n-1).
n
(
i
=
1
∏
k
x
i
m
−
1
)
≤
m
i
=
1
∑
k
(
x
i
n
−
1
)
.
48
1
Hide problems
Inequality in parallelepiped
Prove that in any parallelepiped the sum of the lengths of the edges is less than or equal to twice the sum of the lengths of the four diagonals.
47
1
Hide problems
If angles are equal to pi/3 then lines are concurent
In a plane, three pairwise intersecting circles
C
1
,
C
2
,
C
3
C_1, C_2, C_3
C
1
,
C
2
,
C
3
with centers
M
1
,
M
2
,
M
3
M_1,M_2,M_3
M
1
,
M
2
,
M
3
are given. For
i
=
1
,
2
,
3
i = 1, 2, 3
i
=
1
,
2
,
3
, let
A
i
A_i
A
i
be one of the points of intersection of
C
j
C_j
C
j
and
C
k
(
{
i
,
j
,
k
}
=
{
1
,
2
,
3
}
)
C_k \ (\{i, j, k \} = \{1, 2, 3 \})
C
k
({
i
,
j
,
k
}
=
{
1
,
2
,
3
})
. Prove that if
∠
M
3
A
1
M
2
=
∠
M
1
A
2
M
3
=
∠
M
2
A
3
M
1
=
π
3
\angle M_3A_1M_2 = \angle M_1A_2M_3 = \angle M_2A_3M_1 = \frac{\pi}{3}
∠
M
3
A
1
M
2
=
∠
M
1
A
2
M
3
=
∠
M
2
A
3
M
1
=
3
π
(directed angles), then
M
1
A
1
,
M
2
A
2
M_1A_1, M_2A_2
M
1
A
1
,
M
2
A
2
, and
M
3
A
3
M_3A_3
M
3
A
3
are concurrent.
46
1
Hide problems
Prove that u_n \to \infty [IMO Longlist 1983]
Let
f
f
f
be a real-valued function defined on
I
=
(
0
,
+
∞
)
I = (0,+\infty)
I
=
(
0
,
+
∞
)
and having no zeros on
I
I
I
. Suppose that
lim
x
→
+
∞
f
′
(
x
)
f
(
x
)
=
+
∞
.
\lim_{x \to +\infty} \frac{f'(x)}{f(x)}=+\infty.
x
→
+
∞
lim
f
(
x
)
f
′
(
x
)
=
+
∞.
For the sequence
u
n
=
ln
∣
f
(
n
+
1
)
f
(
n
)
∣
u_n = \ln \left| \frac{f(n+1)}{f(n)} \right|
u
n
=
ln
f
(
n
)
f
(
n
+
1
)
, prove that
u
n
→
+
∞
u_n \to +\infty
u
n
→
+
∞
as
n
→
+
∞
.
n \to +\infty.
n
→
+
∞.
45
1
Hide problems
What happens after the operation is repeated n times ?
Let two glasses, numbered
1
1
1
and
2
2
2
, contain an equal quantity of liquid, milk in glass
1
1
1
and coffee in glass
2
2
2
. One does the following: Take one spoon of mixture from glass
1
1
1
and pour it into glass
2
2
2
, and then take the same spoon of the new mixture from glass
2
2
2
and pour it back into the first glass. What happens after this operation is repeated
n
n
n
times, and what as
n
n
n
tends to infinity?
44
1
Hide problems
Determine the different coin with three weighings
We are given twelve coins, one of which is a fake with a different mass from the other eleven. Determine that coin with three weighings and whether it is heavier or lighter than the others.
43
1
Hide problems
Determine the positions of the points P, Q, R, S
Given a square
A
B
C
D
ABCD
A
BC
D
, let
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
, and
S
S
S
be four variable points on the sides
A
B
,
B
C
,
C
D
AB, BC, CD
A
B
,
BC
,
C
D
, and
D
A
DA
D
A
, respectively. Determine the positions of the points
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
, and
S
S
S
for which the quadrilateral
P
Q
R
S
PQRS
PQRS
is a parallelogram, a rectangle, a square, or a trapezoid.
42
1
Hide problems
Find the ratio of the area of the octagon
Consider the square
A
B
C
D
ABCD
A
BC
D
in which a segment is drawn between each vertex and the midpoints of both opposite sides. Find the ratio of the area of the octagon determined by these segments and the area of the square
A
B
C
D
.
ABCD.
A
BC
D
.
40
1
Hide problems
What is the ratio c/a if R = a ?
Four faces of tetrahedron
A
B
C
D
ABCD
A
BC
D
are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are
a
<
b
<
c
a < b < c
a
<
b
<
c
, determine the radius of the sphere circumscribed about the tetrahedron as a function on
a
,
b
a, b
a
,
b
, and
c
c
c
. What is the ratio
c
/
a
c/a
c
/
a
if
R
=
a
?
R = a \ ?
R
=
a
?
39
1
Hide problems
Alpha is the root of the equation E(x) = x^3 - 5x -50 = 0
If
α
\alpha
α
is the real root of the equation
E
(
x
)
=
x
3
−
5
x
−
50
=
0
E(x) = x^3 - 5x -50 = 0
E
(
x
)
=
x
3
−
5
x
−
50
=
0
such that
x
n
+
1
=
(
5
x
n
+
50
)
1
/
3
x_{n+1} = (5x_n + 50)^{1/3}
x
n
+
1
=
(
5
x
n
+
50
)
1/3
and
x
1
=
5
x_1 = 5
x
1
=
5
, where
n
n
n
is a positive integer, prove that:(a)
x
n
+
1
3
−
α
3
=
5
(
x
n
−
α
)
x_{n+1}^3 - \alpha^3 = 5(x_n - \alpha)
x
n
+
1
3
−
α
3
=
5
(
x
n
−
α
)
(b)
α
<
x
n
+
1
<
x
n
.
\alpha < x_{n+1} < x_n.
α
<
x
n
+
1
<
x
n
.
38
1
Hide problems
Find the explicit form of u_n
Let
{
u
n
}
\{u_n \}
{
u
n
}
be the sequence defined by its first two terms
u
0
,
u
1
u_0, u_1
u
0
,
u
1
and the recursion formula
u
n
+
2
=
u
n
−
u
n
+
1
.
u_{n+2 }= u_n - u_{n+1}.
u
n
+
2
=
u
n
−
u
n
+
1
.
(a) Show that
u
n
u_n
u
n
can be written in the form
u
n
=
α
a
n
+
β
b
n
u_n = \alpha a^n + \beta b^n
u
n
=
α
a
n
+
β
b
n
, where
a
,
b
,
α
,
β
a, b, \alpha, \beta
a
,
b
,
α
,
β
are constants independent of
n
n
n
that have to be determined.(b) If
S
n
=
u
0
+
u
1
+
⋯
+
u
n
S_n = u_0 + u_1 + \cdots + u_n
S
n
=
u
0
+
u
1
+
⋯
+
u
n
, prove that
S
n
+
u
n
−
1
S_n + u_{n-1}
S
n
+
u
n
−
1
is a constant independent of
n
.
n.
n
.
Determine this constant.
37
1
Hide problems
1983 points on a circle
The points
A
1
,
A
2
,
…
,
A
1983
A_1,A_2, \ldots , A_{1983}
A
1
,
A
2
,
…
,
A
1983
are set on the circumference of a circle and each is given one of the values
±
1
\pm 1
±
1
. Show that if the number of points with the value
+
1
+1
+
1
is greater than
1789
1789
1789
, then at least
1207
1207
1207
of the points will have the property that the partial sums that can be formed by taking the numbers from them to any other point, in either direction, are strictly positive.
36
1
Hide problems
What is the largest possible value of k ?
The set
X
X
X
has
1983
1983
1983
members. There exists a family of subsets
{
S
1
,
S
2
,
…
,
S
k
}
\{S_1, S_2, \ldots , S_k \}
{
S
1
,
S
2
,
…
,
S
k
}
such that: (i) the union of any three of these subsets is the entire set
X
X
X
, while (ii) the union of any two of them contains at most
1979
1979
1979
members. What is the largest possible value of
k
?
k ?
k
?
34
1
Hide problems
Sum on distances
In a plane are given n points
P
i
(
i
=
1
,
2
,
…
,
n
)
P_i \ (i = 1, 2, \ldots , n)
P
i
(
i
=
1
,
2
,
…
,
n
)
and two angles
α
\alpha
α
and
β
\beta
β
. Over each of the segments
P
i
P
i
+
1
(
P
n
+
1
=
P
1
)
P_iP_{i+1} \ (P_{n+1} = P_1)
P
i
P
i
+
1
(
P
n
+
1
=
P
1
)
a point
Q
i
Q_i
Q
i
is constructed such that for all
i
i
i
: (i) upon moving from
P
i
P_i
P
i
to
P
i
+
1
,
Q
i
P_{i+1}, Q_i
P
i
+
1
,
Q
i
is seen on the same side of
P
i
P
i
+
1
P_iP_{i+1}
P
i
P
i
+
1
, (ii)
∠
P
i
+
1
P
i
Q
i
=
α
,
\angle P_{i+1}P_iQ_i = \alpha,
∠
P
i
+
1
P
i
Q
i
=
α
,
(iii)
∠
P
i
P
i
+
1
Q
i
=
β
.
\angle P_iP_{i+1}Q_i = \beta.
∠
P
i
P
i
+
1
Q
i
=
β
.
Furthermore, let
g
g
g
be a line in the same plane with the property that all the points
P
i
,
Q
i
P_i,Q_i
P
i
,
Q
i
lie on the same side of
g
g
g
. Prove that
∑
i
=
1
n
d
(
P
i
,
g
)
=
∑
i
=
1
n
d
(
Q
i
,
g
)
.
\sum_{i=1}^n d(P_i, g)= \sum_{i=1}^n d(Q_i, g).
i
=
1
∑
n
d
(
P
i
,
g
)
=
i
=
1
∑
n
d
(
Q
i
,
g
)
.
where
d
(
M
,
g
)
d(M,g)
d
(
M
,
g
)
denotes the distance from point
M
M
M
to line
g
.
g.
g
.
32
1
Hide problems
Function and inequality
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers and let
[
x
]
[x]
[
x
]
denote the greatest integer that does not exceed the real number
x
x
x
. Suppose that
f
f
f
is a function defined on the set of non-negative integers
n
n
n
and taking real values such that
f
(
0
)
=
0
f(0) = 0
f
(
0
)
=
0
and
f
(
n
)
≤
a
n
+
f
(
[
b
n
]
)
+
f
(
[
c
n
]
)
,
for all
n
≥
1.
f(n) \leq an + f([bn]) + f([cn]), \qquad \text{ for all } n \geq 1.
f
(
n
)
≤
an
+
f
([
bn
])
+
f
([
c
n
])
,
for all
n
≥
1.
Prove that if
b
+
c
<
1
b + c < 1
b
+
c
<
1
, there is a real number
k
k
k
such that
f
(
n
)
≤
k
n
for all
n
(
1
)
f(n) \leq kn \qquad \text{ for all } n \qquad (1)
f
(
n
)
≤
kn
for all
n
(
1
)
while if
b
+
c
=
1
b + c = 1
b
+
c
=
1
, there is a real number
K
K
K
such that
f
(
n
)
≤
K
n
log
2
n
f(n) \leq K n \log_2 n
f
(
n
)
≤
K
n
lo
g
2
n
for all
n
≥
2
n \geq 2
n
≥
2
. Show that if
b
+
c
=
1
b + c = 1
b
+
c
=
1
, there may not be a real number
k
k
k
that satisfies
(
1
)
.
(1).
(
1
)
.
30
1
Hide problems
Prove that there exist a unique sequence
Prove the existence of a unique sequence
{
u
n
}
(
n
=
0
,
1
,
2
…
)
\{u_n\} \ (n = 0, 1, 2 \ldots )
{
u
n
}
(
n
=
0
,
1
,
2
…
)
of positive integers such that
u
n
2
=
∑
r
=
0
n
(
n
+
r
r
)
u
n
−
r
for all
n
≥
0
u_n^2 = \sum_{r=0}^n \binom{n+r}{r} u_{n-r} \qquad \text{for all } n \geq 0
u
n
2
=
r
=
0
∑
n
(
r
n
+
r
)
u
n
−
r
for all
n
≥
0
29
1
Hide problems
Find cos theta
Let
O
O
O
be a point outside a given circle. Two lines
O
A
B
,
O
C
D
OAB, OCD
O
A
B
,
OC
D
through
O
O
O
meet the circle at
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
, where
A
,
C
A,C
A
,
C
are the midpoints of
O
B
,
O
D
OB,OD
OB
,
O
D
, respectively. Additionally, the acute angle
θ
\theta
θ
between the lines is equal to the acute angle at which each line cuts the circle. Find
cos
θ
\cos \theta
cos
θ
and show that the tangents at
A
,
D
A,D
A
,
D
to the circle meet on the line
B
C
.
BC.
BC
.
28
1
Hide problems
Show that the triangle is equilateral
Show that if the sides
a
,
b
,
c
a, b, c
a
,
b
,
c
of a triangle satisfy the equation
2
(
a
b
2
+
b
c
2
+
c
a
2
)
=
a
2
b
+
b
2
c
+
c
2
a
+
3
a
b
c
,
2(ab^2 + bc^2 + ca^2) = a^2b + b^2c + c^2a + 3abc,
2
(
a
b
2
+
b
c
2
+
c
a
2
)
=
a
2
b
+
b
2
c
+
c
2
a
+
3
ab
c
,
then the triangle is equilateral. Show also that the equation can be satisfied by positive real numbers that are not the sides of a triangle.
26
1
Hide problems
Show that 2abc-ab-bc-ca cannot be represented in such form
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive integers satisfying
gcd
(
a
,
b
)
=
gcd
(
b
,
c
)
=
gcd
(
c
,
a
)
=
1
\gcd (a, b) = \gcd (b, c) = \gcd (c, a) = 1
g
cd
(
a
,
b
)
=
g
cd
(
b
,
c
)
=
g
cd
(
c
,
a
)
=
1
. Show that
2
a
b
c
−
a
b
−
b
c
−
c
a
2abc-ab-bc-ca
2
ab
c
−
ab
−
b
c
−
c
a
cannot be represented as
b
c
x
+
c
a
y
+
a
b
z
bcx+cay +abz
b
c
x
+
c
a
y
+
ab
z
with nonnegative integers
x
,
y
,
z
.
x, y, z.
x
,
y
,
z
.
25
1
Hide problems
How many permutations are sorted ?
How many permutations
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
of
{
1
,
2
,
.
.
.
,
n
}
\{1, 2, . . ., n \}
{
1
,
2
,
...
,
n
}
are sorted into increasing order by at most three repetitions of the following operation: Move from left to right and interchange
a
i
a_i
a
i
and
a
i
+
1
a_{i+1}
a
i
+
1
whenever
a
i
>
a
i
+
1
a_i > a_{i+1}
a
i
>
a
i
+
1
for
i
i
i
running from
1
1
1
up to
n
−
1
?
n - 1 \ ?
n
−
1
?
24
1
Hide problems
show that 0 < f(x) -x < c for every 0 < x < 1
Every
x
,
0
≤
x
≤
1
x, 0 \leq x \leq 1
x
,
0
≤
x
≤
1
, admits a unique representation
x
=
∑
j
=
0
∞
a
j
2
−
j
x = \sum_{j=0}^{\infty} a_j 2^{-j}
x
=
∑
j
=
0
∞
a
j
2
−
j
, where all the
a
j
a_j
a
j
belong to
{
0
,
1
}
\{0, 1\}
{
0
,
1
}
and infinitely many of them are
0
0
0
. If
b
(
0
)
=
1
+
c
2
+
c
,
b
(
1
)
=
1
2
+
c
,
c
>
0
b(0) = \frac{1+c}{2+c}, b(1) =\frac{1}{2+c},c > 0
b
(
0
)
=
2
+
c
1
+
c
,
b
(
1
)
=
2
+
c
1
,
c
>
0
, and
f
(
x
)
=
a
0
+
∑
j
=
0
∞
b
(
a
0
)
⋯
b
(
a
j
)
a
j
+
1
f(x)=a_0 + \sum_{j=0}^{\infty}b(a_0) \cdots b(a_j) a_{j+1}
f
(
x
)
=
a
0
+
j
=
0
∑
∞
b
(
a
0
)
⋯
b
(
a
j
)
a
j
+
1
show that
0
<
f
(
x
)
−
x
<
c
0 < f(x) -x < c
0
<
f
(
x
)
−
x
<
c
for every
x
,
0
<
x
<
1.
x, 0 < x < 1.
x
,
0
<
x
<
1.
22
1
Hide problems
Does there exist an infinite number of sets
Does there exist an infinite number of sets
C
C
C
consisting of
1983
1983
1983
consecutive natural numbers such that each of the numbers is divisible by some number of the form
a
1983
a^{1983}
a
1983
, with
a
∈
N
,
a
≠
1
?
a \in \mathbb N, a \neq 1?
a
∈
N
,
a
=
1
?
21
1
Hide problems
Prove that there are infinitely many positive integers n
Prove that there are infinitely many positive integers
n
n
n
for which it is possible for a knight, starting at one of the squares of an
n
×
n
n \times n
n
×
n
chessboard, to go through each of the squares exactly once.
20
1
Hide problems
Functional n-th root
Let
f
f
f
and
g
g
g
be functions from the set
A
A
A
to the same set
A
A
A
. We define
f
f
f
to be a functional
n
n
n
-th root of
g
g
g
(
n
n
n
is a positive integer) if
f
n
(
x
)
=
g
(
x
)
f^n(x) = g(x)
f
n
(
x
)
=
g
(
x
)
, where
f
n
(
x
)
=
f
n
−
1
(
f
(
x
)
)
.
f^n(x) = f^{n-1}(f(x)).
f
n
(
x
)
=
f
n
−
1
(
f
(
x
))
.
(a) Prove that the function
g
:
R
→
R
,
g
(
x
)
=
1
/
x
g : \mathbb R \to \mathbb R, g(x) = 1/x
g
:
R
→
R
,
g
(
x
)
=
1/
x
has an infinite number of
n
n
n
-th functional roots for each positive integer
n
.
n.
n
.
(b) Prove that there is a bijection from
R
\mathbb R
R
onto
R
\mathbb R
R
that has no nth functional root for each positive integer
n
.
n.
n
.
18
1
Hide problems
Four parts question
Let
b
≥
2
b \geq 2
b
≥
2
be a positive integer.(a) Show that for an integer
N
N
N
, written in base
b
b
b
, to be equal to the sum of the squares of its digits, it is necessary either that
N
=
1
N = 1
N
=
1
or that
N
N
N
have only two digits.(b) Give a complete list of all integers not exceeding
50
50
50
that, relative to some base
b
b
b
, are equal to the sum of the squares of their digits.(c) Show that for any base b the number of two-digit integers that are equal to the sum of the squares of their digits is even.(d) Show that for any odd base
b
b
b
there is an integer other than
1
1
1
that is equal to the sum of the squares of its digits.
17
1
Hide problems
In how many ways can the numbers be arranged in an array ?
In how many ways can
1
,
2
,
…
,
2
n
1, 2,\ldots, 2n
1
,
2
,
…
,
2
n
be arranged in a
2
×
n
2 \times n
2
×
n
rectangular array
(
a
1
a
2
⋯
a
n
b
1
b
2
⋯
b
n
)
\left(\begin{array}{cccc}a_1& a_2 & \cdots & a_n\\b_1& b_2 & \cdots & b_n\end{array}\right)
(
a
1
b
1
a
2
b
2
⋯
⋯
a
n
b
n
)
for which: (i)
a
1
<
a
2
<
⋯
<
a
n
,
a_1 < a_2 < \cdots < a_n,
a
1
<
a
2
<
⋯
<
a
n
,
(ii)
b
1
<
b
2
<
⋯
<
b
n
,
b_1 < b_2 <\cdots < b_n,
b
1
<
b
2
<
⋯
<
b
n
,
(iii)
a
1
<
b
1
,
a
2
<
b
2
,
…
,
a
n
<
b
n
?
a_1 < b_1, a_2 < b_2, \ldots, a_n < b_n \ ?
a
1
<
b
1
,
a
2
<
b
2
,
…
,
a
n
<
b
n
?
15
1
Hide problems
Find all possible finite sequences {n_0,n_1,...,n_k}
Find all possible finite sequences
{
n
0
,
n
1
,
n
2
,
…
,
n
k
}
\{n_0, n_1, n_2, \ldots, n_k \}
{
n
0
,
n
1
,
n
2
,
…
,
n
k
}
of integers such that for each
i
,
i
i, i
i
,
i
appears in the sequence
n
i
n_i
n
i
times
(
0
≤
i
≤
k
)
.
(0 \leq i \leq k).
(
0
≤
i
≤
k
)
.
14
1
Hide problems
Find the set of all such points M
Let
ℓ
\ell
ℓ
be tangent to the circle
k
k
k
at
B
B
B
. Let
A
A
A
be a point on
k
k
k
and
P
P
P
the foot of perpendicular from
A
A
A
to
ℓ
\ell
ℓ
. Let
M
M
M
be symmetric to
P
P
P
with respect to
A
B
AB
A
B
. Find the set of all such points
M
.
M.
M
.
13
1
Hide problems
Prove that a_i and a_j exist for every prime p
Let
p
p
p
be a prime number and
a
1
,
a
2
,
…
,
a
(
p
+
1
)
/
2
a_1, a_2, \ldots, a_{(p+1)/2}
a
1
,
a
2
,
…
,
a
(
p
+
1
)
/2
different natural numbers less than or equal to
p
.
p.
p
.
Prove that for each natural number
r
r
r
less than or equal to
p
p
p
, there exist two numbers (perhaps equal)
a
i
a_i
a
i
and
a
j
a_j
a
j
such that
p
≡
a
i
a
j
(
m
o
d
r
)
.
p \equiv a_i a_j \pmod r.
p
≡
a
i
a
j
(
mod
r
)
.
12
1
Hide problems
In how many different ways it can be done ?
The number
0
0
0
or
1
1
1
is to be assigned to each of the
n
n
n
vertices of a regular polygon. In how many different ways can this be done (if we consider two assignments that can be obtained one from the other through rotation in the plane of the polygon to be identical)?
11
1
Hide problems
Find the point C
A boy at point
A
A
A
wants to get water at a circular lake and carry it to point
B
B
B
. Find the point
C
C
C
on the lake such that the distance walked by the boy is the shortest possible given that the line
A
B
AB
A
B
and the lake are exterior to each other.
10
1
Hide problems
Which number between 1,2,...,1983 has the most divisors ?
Which of the numbers
1
,
2
,
…
,
1983
1, 2, \ldots , 1983
1
,
2
,
…
,
1983
has the largest number of divisors?
7
1
Hide problems
Find x such that x^4+x^3+x^2+x+1 is perfect square (old)
Find all numbers
x
∈
Z
x \in \mathbb Z
x
∈
Z
for which the number
x
4
+
x
3
+
x
2
+
x
+
1
x^4 + x^3 + x^2 + x + 1
x
4
+
x
3
+
x
2
+
x
+
1
is a perfect square.
5
1
Hide problems
The set Q^2 of points in R^2
Consider the set
Q
2
\mathbb Q^2
Q
2
of points in
R
2
\mathbb R^2
R
2
, both of whose coordinates are rational.(a) Prove that the union of segments with vertices from
Q
2
\mathbb Q^2
Q
2
is the entire set
R
2
\mathbb R^2
R
2
.(b) Is the convex hull of
Q
2
\mathbb Q^2
Q
2
(i.e., the smallest convex set in
R
2
\mathbb R^2
R
2
that contains
Q
2
\mathbb Q^2
Q
2
) equal to
R
2
\mathbb R^2
R
2
?
3
1
Hide problems
On tetrahedron ABCD and its altitudes
(a) Given a tetrahedron
A
B
C
D
ABCD
A
BC
D
and its four altitudes (i.e., lines through each vertex, perpendicular to the opposite face), assume that the altitude dropped from
D
D
D
passes through the orthocenter
H
4
H_4
H
4
of
△
A
B
C
\triangle ABC
△
A
BC
. Prove that this altitude
D
H
4
DH_4
D
H
4
intersects all the other three altitudes.(b) If we further know that a second altitude, say the one from vertex A to the face
B
C
D
BCD
BC
D
, also passes through the orthocenter
H
1
H_1
H
1
of
△
B
C
D
\triangle BCD
△
BC
D
, then prove that all four altitudes are concurrent and each one passes through the orthocenter of the respective triangle.
2
1
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At least one of the airlines can offer a round trip
Seventeen cities are served by four airlines. It is noted that there is direct service (without stops) between any two cities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings.