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Problems
Contests
International Contests
IMO Longlists
1986 IMO Longlists
1986 IMO Longlists
Part of
IMO Longlists
Subcontests
(59)
80
1
Hide problems
Find the angle between planes
Let
A
B
C
D
ABCD
A
BC
D
be a tetrahedron and
O
O
O
its incenter, and let the line
O
D
OD
O
D
be perpendicular to
A
D
AD
A
D
. Find the angle between the planes
D
O
B
DOB
D
OB
and
D
O
C
.
DOC.
D
OC
.
79
1
Hide problems
If two angles are equal, then other angle bisects another !
Let
A
A
1
,
B
B
1
,
C
C
1
AA_1,BB_1, CC_1
A
A
1
,
B
B
1
,
C
C
1
be the altitudes in an acute-angled triangle
A
B
C
ABC
A
BC
,
K
K
K
and
M
M
M
are points on the line segments
A
1
C
1
A_1C_1
A
1
C
1
and
B
1
C
1
B_1C_1
B
1
C
1
respectively. Prove that if the angles
M
A
K
MAK
M
A
K
and
C
A
A
1
CAA_1
C
A
A
1
are equal, then the angle
C
1
K
M
C_1KM
C
1
K
M
is bisected by
A
K
.
AK.
A
K
.
78
1
Hide problems
Geometric inequality with sin and cos
If
T
T
T
and
T
1
T_1
T
1
are two triangles with angles
x
,
y
,
z
x, y, z
x
,
y
,
z
and
x
1
,
y
1
,
z
1
x_1, y_1, z_1
x
1
,
y
1
,
z
1
, respectively, prove the inequality
cos
x
1
sin
x
+
cos
y
1
sin
y
+
cos
z
1
sin
z
≤
cot
x
+
cot
y
+
cot
z
.
\frac{\cos x_1}{\sin x}+\frac{\cos y_1}{\sin y}+\frac{\cos z_1}{\sin z} \leq \cot x+\cot y+\cot z.
sin
x
cos
x
1
+
sin
y
cos
y
1
+
sin
z
cos
z
1
≤
cot
x
+
cot
y
+
cot
z
.
75
1
Hide problems
Determine the maximum possible area of the triangle
The incenter of a triangle is the midpoint of the line segment of length
4
4
4
joining the centroid and the orthocenter of the triangle. Determine the maximum possible area of the triangle.
73
1
Hide problems
There are infinitely many pairs (i,j) for each k
Let
(
a
i
)
i
∈
N
(a_i)_{i\in \mathbb N}
(
a
i
)
i
∈
N
be a strictly increasing sequence of positive real numbers such that
lim
i
→
∞
a
i
=
+
∞
\lim_{i \to \infty} a_i = +\infty
lim
i
→
∞
a
i
=
+
∞
and
a
i
+
1
/
a
i
≤
10
a_{i+1}/a_i \leq 10
a
i
+
1
/
a
i
≤
10
for each
i
i
i
. Prove that for every positive integer
k
k
k
there are infinitely many pairs
(
i
,
j
)
(i, j)
(
i
,
j
)
with
1
0
k
≤
a
i
/
a
j
≤
1
0
k
+
1
.
10^k \leq a_i/a_j \leq 10^{k+1}.
1
0
k
≤
a
i
/
a
j
≤
1
0
k
+
1
.
72
1
Hide problems
Determine a and b in a game
A one-person game with two possible outcomes is played as follows: After each play, the player receives either
a
a
a
or
b
b
b
points, where
a
a
a
and
b
b
b
are integers with
0
<
b
<
a
<
1986
0 < b < a < 1986
0
<
b
<
a
<
1986
. The game is played as many times as one wishes and the total score of the game is defined as the sum of points received after successive plays. It is observed that every integer
x
≥
1986
x \geq 1986
x
≥
1986
can be obtained as the total score whereas
1985
1985
1985
and
663
663
663
cannot. Determine
a
a
a
and
b
.
b.
b
.
71
1
Hide problems
Two lines intersect in a point on the nine-point circle
Two straight lines perpendicular to each other meet each side of a triangle in points symmetric with respect to the midpoint of that side. Prove that these two lines intersect in a point on the nine-point circle.
68
1
Hide problems
On the equation x^4 + ax^3 + bx^2 + ax + 1 = 0
Consider the equation
x
4
+
a
x
3
+
b
x
2
+
a
x
+
1
=
0
x^4 + ax^3 + bx^2 + ax + 1 = 0
x
4
+
a
x
3
+
b
x
2
+
a
x
+
1
=
0
with real coefficients
a
,
b
a, b
a
,
b
. Determine the number of distinct real roots and their multiplicities for various values of
a
a
a
and
b
b
b
. Display your result graphically in the
(
a
,
b
)
(a, b)
(
a
,
b
)
plane.
66
1
Hide problems
One hundred red points and one hundred blue points
One hundred red points and one hundred blue points are chosen in the plane, no three of them lying on a line. Show that these points can be connected pairwise, red ones with blue ones, by disjoint line segments.
65
1
Hide problems
Prove that there is a point M on the circle C
Let
A
1
A
2
A
3
A
4
A_1A_2A_3A_4
A
1
A
2
A
3
A
4
be a quadrilateral inscribed in a circle
C
C
C
. Show that there is a point
M
M
M
on
C
C
C
such that
M
A
1
−
M
A
2
+
M
A
3
−
M
A
4
=
0.
MA_1 -MA_2 +MA_3 -MA_4 = 0.
M
A
1
−
M
A
2
+
M
A
3
−
M
A
4
=
0.
64
1
Hide problems
a_n is perfect square for every n
Let
(
a
n
)
n
∈
N
(a_n)_{n\in \mathbb N}
(
a
n
)
n
∈
N
be the sequence of integers defined recursively by
a
1
=
a
2
=
1
,
a
n
+
2
=
7
a
n
+
1
−
a
n
−
2
a_1 = a_2 = 1, a_{n+2} = 7a_{n+1} - a_n - 2
a
1
=
a
2
=
1
,
a
n
+
2
=
7
a
n
+
1
−
a
n
−
2
for
n
≥
1
n \geq 1
n
≥
1
. Prove that
a
n
a_n
a
n
is a perfect square for every
n
.
n.
n
.
63
1
Hide problems
A'B', B'C', C'A intersects the incircle in two points
Let
A
A
′
,
B
B
′
,
C
C
′
AA',BB', CC'
A
A
′
,
B
B
′
,
C
C
′
be the bisectors of the angles of a triangle
A
B
C
(
A
′
∈
B
C
,
B
′
∈
C
A
,
C
′
∈
A
B
)
ABC \ (A' \in BC, B' \in CA, C' \in AB)
A
BC
(
A
′
∈
BC
,
B
′
∈
C
A
,
C
′
∈
A
B
)
. Prove that each of the lines
A
′
B
′
,
B
′
C
′
,
C
′
A
′
A'B', B'C', C'A'
A
′
B
′
,
B
′
C
′
,
C
′
A
′
intersects the incircle in two points.
62
1
Hide problems
Determine all pairs (x,y) such that p^x − y^3 = 1
Determine all pairs of positive integers
(
x
,
y
)
(x, y)
(
x
,
y
)
satisfying the equation
p
x
−
y
3
=
1
p^x - y^3 = 1
p
x
−
y
3
=
1
, where
p
p
p
is a given prime number.
61
1
Hide problems
Find the greatest integer p
Given a positive integer
n
n
n
, find the greatest integer
p
p
p
with the property that for any function
f
:
P
(
X
)
→
C
f : \mathbb P(X) \to C
f
:
P
(
X
)
→
C
, where
X
X
X
and
C
C
C
are sets of cardinality
n
n
n
and
p
p
p
, respectively, there exist two distinct sets
A
,
B
∈
P
(
X
)
A,B \in \mathbb P(X)
A
,
B
∈
P
(
X
)
such that
f
(
A
)
=
f
(
B
)
=
f
(
A
∪
B
)
f(A) = f(B) = f(A \cup B)
f
(
A
)
=
f
(
B
)
=
f
(
A
∪
B
)
. (
P
(
X
)
\mathbb P(X)
P
(
X
)
is the family of all subsets of
X
X
X
.)
60
1
Hide problems
Three variables inequality
Prove the inequality
(
−
a
+
b
+
c
)
2
(
a
−
b
+
c
)
2
(
a
+
b
−
c
)
2
≥
(
−
a
2
+
b
2
+
c
2
)
(
a
2
−
b
2
+
c
2
)
(
a
2
+
b
2
−
c
2
)
(-a+b+c)^2(a-b+c)^2(a+b-c)^2 \geq (-a^2+b^2+c^2)(a^2-b^2+c^2)(a^2+b^2-c^2)
(
−
a
+
b
+
c
)
2
(
a
−
b
+
c
)
2
(
a
+
b
−
c
)
2
≥
(
−
a
2
+
b
2
+
c
2
)
(
a
2
−
b
2
+
c
2
)
(
a
2
+
b
2
−
c
2
)
for all real numbers
a
,
b
,
c
.
a, b, c.
a
,
b
,
c
.
57
1
Hide problems
So many points
In a triangle
A
B
C
ABC
A
BC
, the incircle touches the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
in the points
A
′
,
B
′
,
C
′
A',B', C'
A
′
,
B
′
,
C
′
, respectively; the excircle in the angle
A
A
A
touches the lines containing these sides in
A
1
,
B
1
,
C
1
A_1,B_1, C_1
A
1
,
B
1
,
C
1
, and similarly, the excircles in the angles
B
B
B
and
C
C
C
touch these lines in
A
2
,
B
2
,
C
2
A_2,B_2, C_2
A
2
,
B
2
,
C
2
and
A
3
,
B
3
,
C
3
A_3,B_3, C_3
A
3
,
B
3
,
C
3
. Prove that the triangle
A
B
C
ABC
A
BC
is right-angled if and only if one of the point triples
(
A
′
,
B
3
,
C
′
)
,
(A',B_3, C'),
(
A
′
,
B
3
,
C
′
)
,
(
A
3
,
B
′
,
C
3
)
,
(
A
′
,
B
′
,
C
2
)
,
(
A
2
,
B
2
,
C
′
)
,
(
A
2
,
B
1
,
C
2
)
,
(
A
3
,
B
3
,
C
1
)
,
(A_3,B', C_3), (A',B', C_2), (A_2,B_2, C'), (A_2,B_1, C_2), (A_3,B_3, C_1),
(
A
3
,
B
′
,
C
3
)
,
(
A
′
,
B
′
,
C
2
)
,
(
A
2
,
B
2
,
C
′
)
,
(
A
2
,
B
1
,
C
2
)
,
(
A
3
,
B
3
,
C
1
)
,
(
A
1
,
B
2
,
C
1
)
,
(
A
1
,
B
1
,
C
3
)
(A_1,B_2, C_1), (A_1,B_1, C_3)
(
A
1
,
B
2
,
C
1
)
,
(
A
1
,
B
1
,
C
3
)
is collinear.
56
1
Hide problems
Construct M such that the sum is maximal
Let
A
1
A
2
A
3
A
4
A
5
A
6
A_1A_2A_3A_4A_5A_6
A
1
A
2
A
3
A
4
A
5
A
6
be a hexagon inscribed into a circle with center
O
O
O
. Consider the circular arc with endpoints
A
1
,
A
6
A_1,A_6
A
1
,
A
6
not containing
A
2
A_2
A
2
. For any point
M
M
M
of that arc denote by
h
i
h_i
h
i
the distance from
M
M
M
to the line
A
i
A
i
+
1
(
1
≤
i
≤
5
)
A_iA_{i+1} \ (1 \leq i \leq 5)
A
i
A
i
+
1
(
1
≤
i
≤
5
)
. Construct
M
M
M
such that the sum
h
1
+
⋯
+
h
5
h_1 + \cdots + h_5
h
1
+
⋯
+
h
5
is maximal.
55
1
Hide problems
Determine all n-digit numbers
Given an integer
n
≥
2
n \geq 2
n
≥
2
, determine all
n
n
n
-digit numbers
M
0
=
a
1
a
2
⋯
a
n
‾
(
a
i
≠
0
,
i
=
1
,
2
,
.
.
.
,
n
)
M_0 = \overline{a_1a_2 \cdots a_n} \ (a_i \neq 0, i = 1, 2, . . ., n)
M
0
=
a
1
a
2
⋯
a
n
(
a
i
=
0
,
i
=
1
,
2
,
...
,
n
)
divisible by the numbers
M
1
=
a
2
a
3
⋯
a
n
a
1
‾
M_1 = \overline{a_2a_3 \cdots a_na_1}
M
1
=
a
2
a
3
⋯
a
n
a
1
,
M
2
=
a
3
a
4
⋯
a
n
a
1
a
2
‾
M_2 = \overline{a_3a_4 \cdots a_na_1 a_2}
M
2
=
a
3
a
4
⋯
a
n
a
1
a
2
,
⋯
\cdots
⋯
,
M
n
−
1
=
a
n
a
1
a
2
.
.
.
a
n
−
1
‾
.
M_{n-1} = \overline{a_na_1a_2 . . .a_{n-1}}.
M
n
−
1
=
a
n
a
1
a
2
...
a
n
−
1
.
54
1
Hide problems
Find the least integer n
Find the least integer
n
n
n
with the following property: For any set
V
V
V
of
8
8
8
points in the plane, no three lying on a line, and for any set
E
E
E
of n line segments with endpoints in
V
V
V
, one can find a straight line intersecting at least
4
4
4
segments in
E
E
E
in interior points.
53
1
Hide problems
IMO Long List 1986 - Find S(r, v, n)
For given positive integers
r
,
v
,
n
r, v, n
r
,
v
,
n
let
S
(
r
,
v
,
n
)
S(r, v, n)
S
(
r
,
v
,
n
)
denote the number of
n
n
n
-tuples of non-negative integers
(
x
1
,
⋯
,
x
n
)
(x_1, \cdots, x_n)
(
x
1
,
⋯
,
x
n
)
satisfying the equation
x
1
+
⋯
+
x
n
=
r
x_1 +\cdots+ x_n = r
x
1
+
⋯
+
x
n
=
r
and such that
x
i
≤
v
x_i \leq v
x
i
≤
v
for
i
=
1
,
⋯
,
n
i = 1, \cdots , n
i
=
1
,
⋯
,
n
. Prove that
S
(
r
,
v
,
n
)
=
∑
k
=
0
m
(
−
1
)
k
(
n
k
)
(
r
−
(
v
+
1
)
k
+
n
−
1
n
−
1
)
S(r, v, n)=\sum_{k=0}^{m} (-1)^k \binom nk \binom{r - (v + 1)k + n - 1}{n-1}
S
(
r
,
v
,
n
)
=
k
=
0
∑
m
(
−
1
)
k
(
k
n
)
(
n
−
1
r
−
(
v
+
1
)
k
+
n
−
1
)
Where
m
=
{
n
,
[
r
v
+
1
]
}
.
m=\left\{n,\left[\frac{r}{v+1}\right]\right\}.
m
=
{
n
,
[
v
+
1
r
]
}
.
52
1
Hide problems
IMO Long List 1986 System of equations
Solve the system of equations
tan
x
1
+
cot
x
1
=
3
tan
x
2
,
\tan x_1 +\cot x_1=3 \tan x_2,
tan
x
1
+
cot
x
1
=
3
tan
x
2
,
tan
x
2
+
cot
x
2
=
3
tan
x
3
,
\tan x_2 +\cot x_2=3 \tan x_3,
tan
x
2
+
cot
x
2
=
3
tan
x
3
,
⋮
\vdots
⋮
tan
x
n
+
cot
x
n
=
3
tan
x
1
\tan x_n +\cot x_n=3 \tan x_1
tan
x
n
+
cot
x
n
=
3
tan
x
1
51
1
Hide problems
IMO Long List 1986 geometric inequality
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be the lengths of the sides of a quadrilateral circumscribed about a circle and let
S
S
S
be its area. Prove that
S
≤
a
b
c
d
S \leq \sqrt{abcd}
S
≤
ab
c
d
and find conditions for equality.
50
1
Hide problems
IMO Long List 1986 Geometry problem 50
Let
D
D
D
be the point on the side
B
C
BC
BC
of the triangle
A
B
C
ABC
A
BC
such that
A
D
AD
A
D
is the bisector of
∠
C
A
B
\angle CAB
∠
C
A
B
. Let
I
I
I
be the incenter of
A
B
C
.
ABC.
A
BC
.
(a) Construct the points
P
P
P
and
Q
Q
Q
on the sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, such that
P
Q
PQ
PQ
is parallel to
B
C
BC
BC
and the perimeter of the triangle
A
P
Q
APQ
A
PQ
is equal to
k
⋅
B
C
k \cdot BC
k
⋅
BC
, where
k
k
k
is a given rational number.(b) Let
R
R
R
be the intersection point of
P
Q
PQ
PQ
and
A
D
AD
A
D
. For what value of
k
k
k
does the equality
A
R
=
R
I
AR = RI
A
R
=
R
I
hold?(c) In which case do the equalities
A
R
=
R
I
=
I
D
AR = RI = ID
A
R
=
R
I
=
I
D
hold?
49
1
Hide problems
The radius of each circle is the reciprocal of an integer
Let
C
1
,
C
2
C_1, C_2
C
1
,
C
2
be circles of radius
1
/
2
1/2
1/2
tangent to each other and both tangent internally to a circle
C
C
C
of radius
1
1
1
. The circles
C
1
C_1
C
1
and
C
2
C_2
C
2
are the first two terms of an infinite sequence of distinct circles
C
n
C_n
C
n
defined as follows:
C
n
+
2
C_{n+2}
C
n
+
2
is tangent externally to
C
n
C_n
C
n
and
C
n
+
1
C_{n+1}
C
n
+
1
and internally to
C
C
C
. Show that the radius of each
C
n
C_n
C
n
is the reciprocal of an integer.
48
1
Hide problems
The points Qn are on one of the sides of P containing A or D
Let
P
P
P
be a convex
1986
1986
1986
-gon in the plane. Let
A
,
D
A,D
A
,
D
be interior points of two distinct sides of P and let
B
,
C
B,C
B
,
C
be two distinct interior points of the line segment
A
D
AD
A
D
. Starting with an arbitrary point
Q
1
Q_1
Q
1
on the boundary of
P
P
P
, define recursively a sequence of points
Q
n
Q_n
Q
n
as follows: given
Q
n
Q_n
Q
n
extend the directed line segment
Q
n
B
Q_nB
Q
n
B
to meet the boundary of
P
P
P
in a point
R
n
R_n
R
n
and then extend
R
n
C
R_nC
R
n
C
to meet the boundary of
P
P
P
again in a point, which is defined to be
Q
n
+
1
Q_{n+1}
Q
n
+
1
. Prove that for all
n
n
n
large enough the points
Q
n
Q_n
Q
n
are on one of the sides of
P
P
P
containing
A
A
A
or
D
D
D
.
46
1
Hide problems
IMO Long List 1986 matrix problem-Find values of k
We wish to construct a matrix with
19
19
19
rows and
86
86
86
columns, with entries
x
i
j
∈
{
0
,
1
,
2
}
(
1
≤
i
≤
19
,
1
≤
j
≤
86
)
x_{ij} \in \{0, 1, 2\} \ (1 \leq i \leq 19, 1 \leq j \leq 86)
x
ij
∈
{
0
,
1
,
2
}
(
1
≤
i
≤
19
,
1
≤
j
≤
86
)
, such that:(i) in each column there are exactly
k
k
k
terms equal to
0
0
0
;(ii) for any distinct
j
,
k
∈
{
1
,
.
.
.
,
86
}
j, k \in \{1, . . . , 86\}
j
,
k
∈
{
1
,
...
,
86
}
there is
i
∈
{
1
,
.
.
.
,
19
}
i \in \{1, . . . , 19\}
i
∈
{
1
,
...
,
19
}
with
x
i
j
+
x
i
k
=
3.
x_{ij} + x_{ik} = 3.
x
ij
+
x
ik
=
3.
For what values of
k
k
k
is this possible?
45
1
Hide problems
IMO Long List 1986 with n variables
Given
n
n
n
real numbers
a
1
≤
a
2
≤
⋯
≤
a
n
a_1 \leq a_2 \leq \cdots \leq a_n
a
1
≤
a
2
≤
⋯
≤
a
n
, define M_1=\frac 1n \sum_{i=1}^{n} a_i , M_2=\frac{2}{n(n-1)} \sum_{1 \leq i
a
1
≤
M
1
−
Q
≤
M
1
+
Q
≤
a
n
a_1 \leq M_1 - Q \leq M_1 + Q \leq a_n
a
1
≤
M
1
−
Q
≤
M
1
+
Q
≤
a
n
and that equality holds if and only if
a
1
=
a
2
=
⋯
=
a
n
.
a_1 = a_2 = \cdots = a_n.
a
1
=
a
2
=
⋯
=
a
n
.
42
1
Hide problems
Total number of placements
The integers
1
,
2
,
⋯
,
n
2
1, 2, \cdots, n^2
1
,
2
,
⋯
,
n
2
are placed on the fields of an
n
×
n
n \times n
n
×
n
chessboard
(
n
>
2
)
(n > 2)
(
n
>
2
)
in such a way that any two fields that have a common edge or a vertex are assigned numbers differing by at most
n
+
1
n + 1
n
+
1
. What is the total number of such placements?
41
1
Hide problems
If A'B'C' is equilateral then so is ABC
Let
M
,
N
,
P
M,N,P
M
,
N
,
P
be the midpoints of the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
of a triangle
A
B
C
ABC
A
BC
. The lines
A
M
,
B
N
,
C
P
AM, BN, CP
A
M
,
BN
,
CP
intersect the circumcircle of
A
B
C
ABC
A
BC
at points
A
′
,
B
′
,
C
′
A',B', C'
A
′
,
B
′
,
C
′
, respectively. Show that if
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
is an equilateral triangle, then so is
A
B
C
.
ABC.
A
BC
.
40
1
Hide problems
Max. value of 2m+7n
Find the maximum value that the quantity
2
m
+
7
n
2m+7n
2
m
+
7
n
can have such that there exist distinct positive integers
x
i
(
1
≤
i
≤
m
)
,
y
j
(
1
≤
j
≤
n
)
x_i \ (1 \leq i \leq m), y_j \ (1 \leq j \leq n)
x
i
(
1
≤
i
≤
m
)
,
y
j
(
1
≤
j
≤
n
)
such that the
x
i
x_i
x
i
's are even, the
y
j
y_j
y
j
's are odd, and
∑
i
=
1
m
x
i
+
∑
j
=
1
n
y
j
=
1986.
\sum_{i=1}^{m} x_i +\sum_{j=1}^{n} y_j=1986.
∑
i
=
1
m
x
i
+
∑
j
=
1
n
y
j
=
1986.
39
1
Hide problems
Number of mappings
Let
S
S
S
be a
k
k
k
-element set.(a) Find the number of mappings
f
:
S
→
S
f : S \to S
f
:
S
→
S
such that \text{(i) } f(x) \neq x \text{ for } x \in S, \text{(ii) } f(f(x)) = x \text{ for }x \in S.(b) The same with the condition
(i)
\text{(i)}
(i)
left out.
37
1
Hide problems
The set can be partitioned into 27 sets
Prove that the set
{
1
,
2
,
.
.
.
,
1986
}
\{1, 2, . . . , 1986\}
{
1
,
2
,
...
,
1986
}
can be partitioned into
27
27
27
disjoint sets so that no one of these sets contains an arithmetic triple (i.e., three distinct numbers in an arithmetic progression).
35
1
Hide problems
Maximum and minimum value of |a| + |b| + |c|
Establish the maximum and minimum values that the sum
∣
a
∣
+
∣
b
∣
+
∣
c
∣
|a| + |b| + |c|
∣
a
∣
+
∣
b
∣
+
∣
c
∣
can have if
a
,
b
,
c
a, b, c
a
,
b
,
c
are real numbers such that the maximum value of
∣
a
x
2
+
b
x
+
c
∣
|ax^2 + bx + c|
∣
a
x
2
+
b
x
+
c
∣
is
1
1
1
for
−
1
≤
x
≤
1.
-1 \leq x \leq 1.
−
1
≤
x
≤
1.
34
1
Hide problems
Polynomial identity
For each non-negative integer
n
n
n
,
F
n
(
x
)
F_n(x)
F
n
(
x
)
is a polynomial in
x
x
x
of degree
n
n
n
. Prove that if the identity
F
n
(
2
x
)
=
∑
r
=
0
n
(
−
1
)
n
−
r
(
n
r
)
2
r
F
r
(
x
)
F_n(2x)=\sum_{r=0}^{n} (-1)^{n-r} \binom nr 2^r F_r(x)
F
n
(
2
x
)
=
r
=
0
∑
n
(
−
1
)
n
−
r
(
r
n
)
2
r
F
r
(
x
)
holds for each n, then
F
n
(
t
x
)
=
∑
r
=
0
n
(
n
r
)
t
r
(
1
−
t
)
n
−
r
F
r
(
x
)
F_n(tx)=\sum_{r=0}^{n} \binom nr t^r (1-t)^{n-r} F_r(x)
F
n
(
t
x
)
=
r
=
0
∑
n
(
r
n
)
t
r
(
1
−
t
)
n
−
r
F
r
(
x
)
32
1
Hide problems
Solve in positive integers
Find, with proof, all solutions of the equation
1
x
+
2
y
−
3
z
=
1
\frac 1x +\frac 2y- \frac 3z = 1
x
1
+
y
2
−
z
3
=
1
in positive integers
x
,
y
,
z
.
x, y, z.
x
,
y
,
z
.
31
1
Hide problems
PQ passes through the circumcenter of ABC
Let
P
P
P
and
Q
Q
Q
be distinct points in the plane of a triangle
A
B
C
ABC
A
BC
such that
A
P
:
A
Q
=
B
P
:
B
Q
=
C
P
:
C
Q
AP : AQ = BP : BQ = CP : CQ
A
P
:
A
Q
=
BP
:
BQ
=
CP
:
CQ
. Prove that the line
P
Q
PQ
PQ
passes through the circumcenter of the triangle.
30
1
Hide problems
Convex polyhedron
Prove that a convex polyhedron all of whose faces are equilateral triangles has at most
30
30
30
edges.
29
1
Hide problems
Binary operation *star*
We define a binary operation
⋆
\star
⋆
in the plane as follows: Given two points
A
A
A
and
B
B
B
in the plane,
C
=
A
⋆
B
C = A \star B
C
=
A
⋆
B
is the third vertex of the equilateral triangle ABC oriented positively. What is the relative position of three points
I
,
M
,
O
I, M, O
I
,
M
,
O
in the plane if
I
⋆
(
M
⋆
O
)
=
(
O
⋆
I
)
⋆
M
I \star (M \star O) = (O \star I)\star M
I
⋆
(
M
⋆
O
)
=
(
O
⋆
I
)
⋆
M
holds?
27
1
Hide problems
Only one local maximum
In an urn there are n balls numbered
1
,
2
,
⋯
,
n
1, 2, \cdots, n
1
,
2
,
⋯
,
n
. They are drawn at random one by one without replacement and the numbers are recorded. What is the probability that the resulting random permutation has only one local maximum? A term in a sequence is a local maximum if it is greater than all its neighbors.
24
1
Hide problems
There exist not fewer than 1986 distinct squares
Two families of parallel lines are given in the plane, consisting of
15
15
15
and
11
11
11
lines, respectively. In each family, any two neighboring lines are at a unit distance from one another; the lines of the first family are perpendicular to the lines of the second family. Let
V
V
V
be the set of
165
165
165
intersection points of the lines under consideration. Show that there exist not fewer than
1986
1986
1986
distinct squares with vertices in the set
V
.
V .
V
.
23
1
Hide problems
Rectangle
Let
I
I
I
and
J
J
J
be the centers of the incircle and the excircle in the angle
B
A
C
BAC
B
A
C
of the triangle
A
B
C
ABC
A
BC
. For any point
M
M
M
in the plane of the triangle, not on the line
B
C
BC
BC
, denote by
I
M
I_M
I
M
and
J
M
J_M
J
M
the centers of the incircle and the excircle (touching
B
C
BC
BC
) of the triangle
B
C
M
BCM
BCM
. Find the locus of points
M
M
M
for which
I
I
M
J
J
M
II_MJJ_M
I
I
M
J
J
M
is a rectangle.
22
1
Hide problems
Common divisors of two terms of the sequence
Let
(
a
n
)
n
≥
0
(a_n)_{n \geq 0}
(
a
n
)
n
≥
0
be the sequence of integers defined recursively by
a
0
=
0
,
a
1
=
1
,
a
n
+
2
=
4
a
n
+
1
+
a
n
a_0 = 0, a_1 = 1, a_{n+2} = 4a_{n+1} + a_n
a
0
=
0
,
a
1
=
1
,
a
n
+
2
=
4
a
n
+
1
+
a
n
for
n
≥
0.
n \geq 0.
n
≥
0.
Find the common divisors of
a
1986
a_{1986}
a
1986
and
a
6891
.
a_{6891}.
a
6891
.
21
1
Hide problems
Find the maximum value of the product
Let
A
B
AB
A
B
be a segment of unit length and let
C
,
D
C, D
C
,
D
be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set
{
A
,
B
,
C
,
D
}
.
\{A,B,C,D\}.
{
A
,
B
,
C
,
D
}
.
20
1
Hide problems
Alpha set
For any angle α with
0
<
α
<
18
0
∘
0 < \alpha < 180^{\circ}
0
<
α
<
18
0
∘
, we call a closed convex planar set an
α
\alpha
α
-set if it is bounded by two circular arcs (or an arc and a line segment) whose angle of intersection is
α
\alpha
α
. Given a (closed) triangle
T
T
T
, find the greatest
α
\alpha
α
such that any two points in
T
T
T
are contained in an
α
\alpha
α
-set
S
⊂
T
.
S \subset T .
S
⊂
T
.
19
1
Hide problems
f(x)=x forall x ∈ [0, 1]
Let
f
:
[
0
,
1
]
→
[
0
,
1
]
f : [0, 1] \to [0, 1]
f
:
[
0
,
1
]
→
[
0
,
1
]
satisfy
f
(
0
)
=
0
,
f
(
1
)
=
1
f(0) = 0, f(1) = 1
f
(
0
)
=
0
,
f
(
1
)
=
1
and
f
(
x
+
y
)
−
f
(
x
)
=
f
(
x
)
−
f
(
x
−
y
)
f(x + y) - f(x) = f(x) - f(x - y)
f
(
x
+
y
)
−
f
(
x
)
=
f
(
x
)
−
f
(
x
−
y
)
for all
x
,
y
≥
0
x, y \geq 0
x
,
y
≥
0
with
x
−
y
,
x
+
y
∈
[
0
,
1
]
.
x - y, x + y \in [0, 1].
x
−
y
,
x
+
y
∈
[
0
,
1
]
.
Prove that
f
(
x
)
=
x
f(x) = x
f
(
x
)
=
x
for all
x
∈
[
0
,
1
]
.
x \in [0, 1].
x
∈
[
0
,
1
]
.
17
1
Hide problems
Right-faced tetrahedron
We call a tetrahedron right-faced if each of its faces is a right-angled triangle.(a) Prove that every orthogonal parallelepiped can be partitioned into six right-faced tetrahedra.(b) Prove that a tetrahedron with vertices
A
1
,
A
2
,
A
3
,
A
4
A_1,A_2,A_3,A_4
A
1
,
A
2
,
A
3
,
A
4
is right-faced if and only if there exist four distinct real numbers
c
1
,
c
2
,
c
3
c_1, c_2, c_3
c
1
,
c
2
,
c
3
, and
c
4
c_4
c
4
such that the edges
A
j
A
k
A_jA_k
A
j
A
k
have lengths
A
j
A
k
=
∣
c
j
−
c
k
∣
A_jA_k=\sqrt{|c_j-c_k|}
A
j
A
k
=
∣
c
j
−
c
k
∣
for
1
≤
j
<
k
≤
4.
1\leq j < k \leq 4.
1
≤
j
<
k
≤
4.
16
1
Hide problems
Find the least n_k
Given a positive integer
k
k
k
, find the least integer
n
k
n_k
n
k
for which there exist five sets
S
1
,
S
2
,
S
3
,
S
4
,
S
5
S_1, S_2, S_3, S_4, S_5
S
1
,
S
2
,
S
3
,
S
4
,
S
5
with the following properties: |S_j|=k \text{ for } j=1, \cdots , 5 , |\bigcup_{j=1}^{5} S_j | = n_k ; |S_i \cap S_{i+1}| = 0 = |S_5 \cap S_1|, \text{for } i=1,\cdots ,4
15
1
Hide problems
Prove that a subset of N exist
Let
N
=
B
1
∪
⋯
∪
B
q
\mathbb N = B_1\cup\cdots \cup B_q
N
=
B
1
∪
⋯
∪
B
q
be a partition of the set
N
\mathbb N
N
of all positive integers and let an integer
l
∈
N
l \in \mathbb N
l
∈
N
be given. Prove that there exist a set
X
⊂
N
X \subset \mathbb N
X
⊂
N
of cardinality
l
l
l
, an infinite set
T
⊂
N
T \subset \mathbb N
T
⊂
N
, and an integer
k
k
k
with
1
≤
k
≤
q
1 \leq k \leq q
1
≤
k
≤
q
such that for any
t
∈
T
t \in T
t
∈
T
and any finite set
Y
⊂
X
Y \subset X
Y
⊂
X
, the sum
t
+
∑
y
∈
Y
y
t+ \sum_{y \in Y} y
t
+
∑
y
∈
Y
y
belongs to
B
k
.
B_k.
B
k
.
13
1
Hide problems
f_ij for a set{1,2,3,...,n}
Let
N
=
{
1
,
2
,
…
,
n
}
N = \{1, 2, \ldots, n\}
N
=
{
1
,
2
,
…
,
n
}
,
n
≥
3
n \geq 3
n
≥
3
. To each pair
i
≠
j
i \neq j
i
=
j
of elements of
N
N
N
there is assigned a number
f
i
j
∈
{
0
,
1
}
f_{ij} \in \{0, 1\}
f
ij
∈
{
0
,
1
}
such that
f
i
j
+
f
j
i
=
1
f_{ij} + f_{ji} = 1
f
ij
+
f
ji
=
1
. Let
r
(
i
)
=
∑
i
≠
j
f
i
j
r(i)=\sum_{i \neq j} f_{ij}
r
(
i
)
=
∑
i
=
j
f
ij
, and write
M
=
max
i
∈
N
r
(
i
)
M = \max_{i\in N} r(i)
M
=
max
i
∈
N
r
(
i
)
,
m
=
min
i
∈
N
r
(
i
)
m = \min_{i\in N} r(i)
m
=
min
i
∈
N
r
(
i
)
. Prove that for any
w
∈
N
w \in N
w
∈
N
with
r
(
w
)
=
m
r(w) = m
r
(
w
)
=
m
there exist
u
,
v
∈
N
u, v \in N
u
,
v
∈
N
such that
r
(
u
)
=
M
r(u) = M
r
(
u
)
=
M
and
f
u
v
f
v
w
=
1
f_{uv}f_{vw} = 1
f
uv
f
v
w
=
1
.
12
1
Hide problems
A1A2A3A4 is a regular tetrahedron
Let
O
O
O
be an interior point of a tetrahedron
A
1
A
2
A
3
A
4
A_1A_2A_3A_4
A
1
A
2
A
3
A
4
. Let
S
1
,
S
2
,
S
3
,
S
4
S_1, S_2, S_3, S_4
S
1
,
S
2
,
S
3
,
S
4
be spheres with centers
A
1
,
A
2
,
A
3
,
A
4
A_1,A_2,A_3,A_4
A
1
,
A
2
,
A
3
,
A
4
, respectively, and let
U
,
V
U, V
U
,
V
be spheres with centers at
O
O
O
. Suppose that for
i
,
j
=
1
,
2
,
3
,
4
,
i
≠
j
i, j = 1, 2, 3, 4, i \neq j
i
,
j
=
1
,
2
,
3
,
4
,
i
=
j
, the spheres
S
i
S_i
S
i
and
S
j
S_j
S
j
are tangent to each other at a point
B
i
j
B_{ij}
B
ij
lying on
A
i
A
j
A_iA_j
A
i
A
j
. Suppose also that
U
U
U
is tangent to all edges
A
i
A
j
A_iA_j
A
i
A
j
and
V
V
V
is tangent to the spheres
S
1
,
S
2
,
S
3
,
S
4
S_1, S_2, S_3, S_4
S
1
,
S
2
,
S
3
,
S
4
. Prove that
A
1
A
2
A
3
A
4
A_1A_2A_3A_4
A
1
A
2
A
3
A
4
is a regular tetrahedron.
10
1
Hide problems
Probablity with dice
A set of
n
n
n
standard dice are shaken and randomly placed in a straight line. If
n
<
2
r
n < 2r
n
<
2
r
and
r
<
s
r < s
r
<
s
, then the probability that there will be a string of at least
r
r
r
, but not more than
s
s
s
, consecutive
1
1
1
's can be written as
P
6
s
+
2
\frac{P}{6^{s+2}}
6
s
+
2
P
. Find an explicit expression for
P
P
P
.
9
1
Hide problems
Prove that AD + DB = BC
In a triangle
A
B
C
ABC
A
BC
,
∠
B
A
C
=
10
0
∘
,
A
B
=
A
C
\angle BAC = 100^{\circ}, AB = AC
∠
B
A
C
=
10
0
∘
,
A
B
=
A
C
. A point
D
D
D
is chosen on the side
A
C
AC
A
C
such that
∠
A
B
D
=
∠
C
B
D
\angle ABD = \angle CBD
∠
A
B
D
=
∠
CB
D
. Prove that
A
D
+
D
B
=
B
C
.
AD + DB = BC.
A
D
+
D
B
=
BC
.
6
1
Hide problems
Find the probability-IMO Long List P6
In an urn there are one ball marked
1
1
1
, two balls marked
2
2
2
, and so on, up to
n
n
n
balls marked
n
n
n
. Two balls are randomly drawn without replacement. Find the probability that the two balls are assigned the same number.
5
1
Hide problems
There exist P such that X attains a minimum-IMO Long List P5
Let
A
B
C
ABC
A
BC
and
D
E
F
DEF
D
EF
be acute-angled triangles. Write
d
=
E
F
,
e
=
F
D
,
f
=
D
E
.
d = EF, e = FD, f = DE.
d
=
EF
,
e
=
F
D
,
f
=
D
E
.
Show that there exists a point
P
P
P
in the interior of
A
B
C
ABC
A
BC
for which the value of the expression
X
=
d
⋅
A
P
+
e
⋅
B
P
+
f
⋅
C
P
X=d \cdot AP +e \cdot BP +f \cdot CP
X
=
d
⋅
A
P
+
e
⋅
BP
+
f
⋅
CP
attains a minimum.
4
1
Hide problems
Binary development of 27^{1986} -IMO Long List P4
Find the last eight digits of the binary development of
2
7
1986
.
27^{1986}.
2
7
1986
.
3
1
Hide problems
Circles intersect on altitude of ABC -IMO Long List P3
A line parallel to the side
B
C
BC
BC
of a triangle
A
B
C
ABC
A
BC
meets
A
B
AB
A
B
in
F
F
F
and
A
C
AC
A
C
in
E
E
E
. Prove that the circles on
B
E
BE
BE
and
C
F
CF
CF
as diameters intersect in a point lying on the altitude of the triangle
A
B
C
ABC
A
BC
dropped from
A
A
A
to
B
C
.
BC.
BC
.
2
1
Hide problems
Parallel bisectors -IMO Long List P2
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral.
D
A
DA
D
A
and
C
B
CB
CB
meet at
F
F
F
and
A
B
AB
A
B
and
D
C
DC
D
C
meet at
E
E
E
. The bisectors of the angles
D
F
C
DFC
D
FC
and
A
E
D
AED
A
E
D
are perpendicular. Prove that these angle bisectors are parallel to the bisectors of the angles between the lines
A
C
AC
A
C
and
B
D
.
BD.
B
D
.
1
1
Hide problems
Inequality holds for all real b ≥ 0 -IMO Long List P1
Let
k
k
k
be one of the integers
2
,
3
,
4
2, 3,4
2
,
3
,
4
and let
n
=
2
k
−
1
n = 2^k -1
n
=
2
k
−
1
. Prove the inequality
1
+
b
k
+
b
2
k
+
⋯
+
b
n
k
≥
(
1
+
b
n
)
k
1+ b^k + b^{2k} + \cdots+ b^{nk} \geq (1 + b^n)^k
1
+
b
k
+
b
2
k
+
⋯
+
b
nk
≥
(
1
+
b
n
)
k
for all real
b
≥
0.
b \geq 0.
b
≥
0.
77
1
Hide problems
x^3+y^3+z^3 = x+y+z = 8 in integers
Find all integers
x
,
y
,
z
x,y,z
x
,
y
,
z
such that
x
3
+
y
3
+
z
3
=
x
+
y
+
z
=
8
x^3+y^3+z^3=x+y+z=8
x
3
+
y
3
+
z
3
=
x
+
y
+
z
=
8