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Problems
Contests
International Contests
IMO Longlists
1982 IMO Longlists
1982 IMO Longlists
Part of
IMO Longlists
Subcontests
(35)
52
1
Hide problems
For each k, there are exactly k numbers between the two k
We are given
2
n
2n
2
n
natural numbers
1
,
1
,
2
,
2
,
3
,
3
,
…
,
n
−
1
,
n
−
1
,
n
,
n
.
1, 1, 2, 2, 3, 3, \ldots, n - 1, n - 1, n, n.
1
,
1
,
2
,
2
,
3
,
3
,
…
,
n
−
1
,
n
−
1
,
n
,
n
.
Find all
n
n
n
for which these numbers can be arranged in a row such that for each
k
≤
n
k \leq n
k
≤
n
, there are exactly
k
k
k
numbers between the two numbers
k
k
k
.
51
1
Hide problems
The inequality for n numbers - true for all 0 ≤ α ≤ 1.
Let n numbers
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots, x_n
x
1
,
x
2
,
…
,
x
n
be chosen in such a way that
1
≥
x
1
≥
x
2
≥
⋯
≥
x
n
≥
0
1 \geq x_1 \geq x_2 \geq \cdots \geq x_n \geq 0
1
≥
x
1
≥
x
2
≥
⋯
≥
x
n
≥
0
. Prove that
(
1
+
x
1
+
x
2
+
⋯
+
x
n
)
α
≤
1
+
x
1
α
+
2
α
−
1
x
2
α
+
⋯
+
n
α
−
1
x
n
α
(1 + x_1 + x_2 + \cdots + x_n)^\alpha \leq 1 + x_1^\alpha+ 2^{\alpha-1}x_2^\alpha+ \cdots+ n^{\alpha-1}x_n^\alpha
(
1
+
x
1
+
x
2
+
⋯
+
x
n
)
α
≤
1
+
x
1
α
+
2
α
−
1
x
2
α
+
⋯
+
n
α
−
1
x
n
α
if
0
≤
α
≤
1
0 \leq \alpha \leq 1
0
≤
α
≤
1
.
50
1
Hide problems
the sum of dihedral angles equals 2 pi (ILL 1982 - P50)
Let
O
O
O
be the midpoint of the axis of a right circular cylinder. Let
A
A
A
and
B
B
B
be diametrically opposite points of one base, and
C
C
C
a point of the other base circle that does not belong to the plane
O
A
B
OAB
O
A
B
. Prove that the sum of dihedral angles of the trihedral
O
A
B
C
OABC
O
A
BC
is equal to
2
π
2\pi
2
π
.
49
1
Hide problems
Simplify the sum (2n)! / (k! (n-k)!)^2
Simplify
∑
k
=
0
n
(
2
n
)
!
(
k
!
)
2
(
(
n
−
k
)
!
)
2
.
\sum_{k=0}^n \frac{(2n)!}{(k!)^2((n-k)!)^2}.
k
=
0
∑
n
(
k
!
)
2
((
n
−
k
)!
)
2
(
2
n
)!
.
48
1
Hide problems
There exists a finite sequence a_i for all sequences c_i
Given a finite sequence of complex numbers
c
1
,
c
2
,
…
,
c
n
c_1, c_2, \ldots , c_n
c
1
,
c
2
,
…
,
c
n
, show that there exists an integer
k
k
k
(
1
≤
k
≤
n
1 \leq k \leq n
1
≤
k
≤
n
) such that for every finite sequence
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
of real numbers with
1
≥
a
1
≥
a
2
≥
⋯
≥
a
n
≥
0
1 \geq a_1 \geq a_2 \geq \cdots \geq a_n \geq 0
1
≥
a
1
≥
a
2
≥
⋯
≥
a
n
≥
0
, the following inequality holds:
∣
∑
m
=
1
n
a
m
c
m
∣
≤
∣
∑
m
=
1
k
c
m
∣
.
\left| \sum_{m=1}^n a_mc_m \right| \leq \left| \sum_{m=1}^k c_m \right|.
m
=
1
∑
n
a
m
c
m
≤
m
=
1
∑
k
c
m
.
47
1
Hide problems
The sum of second derivatives of fractions k pi/4 (k odd)
Evaluate
sec
′
′
π
4
+
sec
′
′
3
π
4
+
sec
′
′
5
π
4
+
sec
′
′
7
π
4
\sec'' \frac{\pi}4 +\sec'' \frac{3\pi}4+\sec'' \frac{5\pi}4+\sec'' \frac{7\pi}4
sec
′′
4
π
+
sec
′′
4
3
π
+
sec
′′
4
5
π
+
sec
′′
4
7
π
. (Here
sec
′
′
\sec''
sec
′′
means the second derivative of
sec
\sec
sec
).
46
1
Hide problems
No matter which diagonal is drawn!
Prove that if a diagonal is drawn in a quadrilateral inscribed in a circle, the sum of the radii of the circles inscribed in the two triangles thus formed is the same, no matter which diagonal is drawn.
44
1
Hide problems
The angle between two ships
Let
A
A
A
and
B
B
B
be positions of two ships
M
M
M
and
N
N
N
, respectively, at the moment when
N
N
N
saw
M
M
M
moving with constant speed
v
v
v
following the line
A
x
Ax
A
x
. In search of help,
N
N
N
moves with speed
k
v
kv
k
v
(
k
<
1
k < 1
k
<
1
) along the line
B
y
By
B
y
in order to meet
M
M
M
as soon as possible. Denote by
C
C
C
the point of meeting of the two ships, and set
A
B
=
d
,
∠
B
A
C
=
α
,
0
≤
α
<
π
2
.
AB = d, \angle BAC = \alpha, 0 \leq \alpha < \frac{\pi}{2}.
A
B
=
d
,
∠
B
A
C
=
α
,
0
≤
α
<
2
π
.
Determine the angle
∠
A
B
C
=
β
\angle ABC = \beta
∠
A
BC
=
β
and time
t
t
t
that
N
N
N
needs in order to meet
M
M
M
.
43
1
Hide problems
partitioning a convex n-gon into triangles (ILL 1982 - P43)
(a) What is the maximal number of acute angles in a convex polygon?(b) Consider
m
m
m
points in the interior of a convex
n
n
n
-gon. The
n
n
n
-gon is partitioned into triangles whose vertices are among the
n
+
m
n + m
n
+
m
given points (the vertices of the
n
n
n
-gon and the given points). Each of the
m
m
m
points in the interior is a vertex of at least one triangle. Find the number of triangles obtained.
42
1
Hide problems
There exists f such that A and f(A) are disjoint
Let
F
\mathfrak F
F
be the family of all
k
k
k
-element subsets of the set
{
1
,
2
,
…
,
2
k
+
1
}
\{1, 2, \ldots, 2k + 1\}
{
1
,
2
,
…
,
2
k
+
1
}
. Prove that there exists a bijective function
f
:
F
→
F
f :\mathfrak F \to \mathfrak F
f
:
F
→
F
such that for every
A
∈
F
A \in \mathfrak F
A
∈
F
, the sets
A
A
A
and
f
(
A
)
f(A)
f
(
A
)
are disjoint.
40
1
Hide problems
It's impossible to reach a position with only one pawn
We consider a game on an infinite chessboard similar to that of solitaire: If two adjacent fields are occupied by pawns and the next field is empty (the three fields lie on a vertical or horizontal line), then we may remove these two pawns and put one of them on the third field. Prove that if in the initial position pawns fill a
3
k
×
n
3k \times n
3
k
×
n
rectangle, then it is impossible to reach a position with only one pawn on the board.
39
1
Hide problems
Unit circle and n points such that sum of vectors is zero
Let
S
S
S
be the unit circle with center
O
O
O
and let
P
1
,
P
2
,
…
,
P
n
P_1, P_2,\ldots, P_n
P
1
,
P
2
,
…
,
P
n
be points of
S
S
S
such that the sum of vectors
v
i
=
O
P
i
⟶
v_i=\stackrel{\longrightarrow}{OP_i}
v
i
=
O
P
i
⟶
is the zero vector. Prove that the inequality
∑
i
=
1
n
X
P
i
≥
n
\sum_{i=1}^n XP_i \geq n
∑
i
=
1
n
X
P
i
≥
n
holds for every point
X
X
X
.
38
1
Hide problems
Show that n | u_{n, k} for all n and 1 ≤ k ≤ n
Numbers
u
n
,
k
(
1
≤
k
≤
n
)
u_{n,k} \ (1\leq k \leq n)
u
n
,
k
(
1
≤
k
≤
n
)
are defined as follows u_{1,1}=1, u_{n,k}=\binom{n}{k} - \sum_{d \mid n, d \mid k, d>1} u_{n/d, k/d}. (the empty sum is defined to be equal to zero). Prove that
n
∣
u
n
,
k
n \mid u_{n,k}
n
∣
u
n
,
k
for every natural number
n
n
n
and for every
k
(
1
≤
k
≤
n
)
.
k \ (1 \leq k \leq n).
k
(
1
≤
k
≤
n
)
.
35
1
Hide problems
The radius is half of circumradius-->triangle is equilateral
If the inradius of a triangle is half of its circumradius, prove that the triangle is equilateral.
34
1
Hide problems
Fond all functions in M with a) f(1)=5/2, b) f(1)=√3
Let
M
M
M
be the set of all functions
f
f
f
with the following properties:(i)
f
f
f
is defined for all real numbers and takes only real values.(ii) For all
x
,
y
∈
R
x, y \in \mathbb R
x
,
y
∈
R
the following equality holds:
f
(
x
)
f
(
y
)
=
f
(
x
+
y
)
+
f
(
x
−
y
)
.
f(x)f(y) = f(x + y) + f(x - y).
f
(
x
)
f
(
y
)
=
f
(
x
+
y
)
+
f
(
x
−
y
)
.
(iii)
f
(
0
)
≠
0.
f(0) \neq 0.
f
(
0
)
=
0.
Determine all functions
f
∈
M
f \in M
f
∈
M
such that(a)
f
(
1
)
=
5
2
f(1)=\frac 52
f
(
1
)
=
2
5
,(b)
f
(
1
)
=
3
f(1)= \sqrt 3
f
(
1
)
=
3
.
29
1
Hide problems
Restriction of f to the set of irrationals is injective
Let
f
:
R
→
R
f : \mathbb R \to \mathbb R
f
:
R
→
R
be a continuous function. Suppose that the restriction of
f
f
f
to the set of irrational numbers is injective. What can we say about
f
f
f
? Answer the analogous question if
f
f
f
is restricted to rationals.
28
1
Hide problems
The inequality for (u1,u2,...,un) n-tuple
Let
(
u
1
,
…
,
u
n
)
(u_1, \ldots, u_n)
(
u
1
,
…
,
u
n
)
be an ordered
n
n
n
tuple. For each
k
,
1
≤
k
≤
n
k, 1 \leq k \leq n
k
,
1
≤
k
≤
n
, define
v
k
=
u
1
u
2
⋯
u
k
k
v_k=\sqrt[k]{u_1u_2 \cdots u_k}
v
k
=
k
u
1
u
2
⋯
u
k
. Prove that
∑
k
=
1
n
v
k
≤
e
⋅
∑
k
=
1
n
u
k
.
\sum_{k=1}^n v_k \leq e \cdot \sum_{k=1}^n u_k.
k
=
1
∑
n
v
k
≤
e
⋅
k
=
1
∑
n
u
k
.
(
e
e
e
is the base of the natural logarithm).
26
1
Hide problems
Determine whether there exists a pair (p, q) of naturals
Let
(
a
n
)
n
≥
0
(a_n)_{n\geq0}
(
a
n
)
n
≥
0
and
(
b
n
)
n
≥
0
(b_n)_{n \geq 0}
(
b
n
)
n
≥
0
be two sequences of natural numbers. Determine whether there exists a pair
(
p
,
q
)
(p, q)
(
p
,
q
)
of natural numbers that satisfy p < q \text{ and } a_p \leq a_q, b_p \leq b_q.
24
1
Hide problems
Infinitely Many Descendants
Prove that if a person a has infinitely many descendants (children, their children, etc.), then a has an infinite sequence
a
0
,
a
1
,
…
a_0, a_1, \ldots
a
0
,
a
1
,
…
of descendants (i.e.,
a
=
a
0
a = a_0
a
=
a
0
and for all
n
≥
1
,
a
n
+
1
n \geq 1, a_{n+1}
n
≥
1
,
a
n
+
1
is always a child of
a
n
a_n
a
n
). It is assumed that no-one can have infinitely many children. Variant 1. Prove that if
a
a
a
has infinitely many ancestors, then
a
a
a
has an infinite descending sequence of ancestors (i.e.,
a
0
,
a
1
,
…
a_0, a_1, \ldots
a
0
,
a
1
,
…
where
a
=
a
0
a = a_0
a
=
a
0
and
a
n
a_n
a
n
is always a child of
a
n
+
1
a_{n+1}
a
n
+
1
). Variant 2. Prove that if someone has infinitely many ancestors, then all people cannot descend from
A
(
d
a
m
)
A(dam)
A
(
d
am
)
and
E
(
v
e
)
E(ve)
E
(
v
e
)
.
23
1
Hide problems
Determine the um of all positive integers with the property
Determine the sum of all positive integers whose digits (in base ten) form either a strictly increasing or a strictly decreasing sequence.
21
1
Hide problems
The hexagon and the inequality
All edges and all diagonals of regular hexagon
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
are colored blue or red such that each triangle
A
j
A
k
A
m
,
1
≤
j
<
k
<
m
≤
6
A_jA_kA_m, 1 \leq j < k < m\leq 6
A
j
A
k
A
m
,
1
≤
j
<
k
<
m
≤
6
has at least one red edge. Let
R
k
R_k
R
k
be the number of red segments
A
k
A
j
,
(
j
≠
k
)
A_kA_j, (j \neq k)
A
k
A
j
,
(
j
=
k
)
. Prove the inequality
∑
k
=
1
6
(
2
R
k
−
7
)
2
≤
54.
\sum_{k=1}^6 (2R_k-7)^2 \leq 54.
k
=
1
∑
6
(
2
R
k
−
7
)
2
≤
54.
20
1
Hide problems
Interior of one of the regions meets at least three faces
Consider a cube
C
C
C
and two planes
σ
,
τ
\sigma, \tau
σ
,
τ
, which divide Euclidean space into several regions. Prove that the interior of at least one of these regions meets at least three faces of the cube.
18
1
Hide problems
(a + ab^(-1)a)^(-1) + (a + b)^(-1) = a^(-1) in every ring
You are given an algebraic system admitting addition and multiplication for which all the laws of ordinary arithmetic are valid except commutativity of multiplication. Show that
(
a
+
a
b
−
1
a
)
−
1
+
(
a
+
b
)
−
1
=
a
−
1
,
(a + ab^{-1} a)^{-1}+ (a + b)^{-1} = a^{-1},
(
a
+
a
b
−
1
a
)
−
1
+
(
a
+
b
)
−
1
=
a
−
1
,
where
x
−
1
x^{-1}
x
−
1
is the element for which
x
−
1
x
=
x
x
−
1
=
e
x^{-1}x = xx^{-1} = e
x
−
1
x
=
x
x
−
1
=
e
, where
e
e
e
is the element of the system such that for all
a
a
a
the equality
e
a
=
a
e
=
a
ea = ae = a
e
a
=
a
e
=
a
holds.
15
1
Hide problems
S is not the union of finitely many arithmetic progressions
Show that the set
S
S
S
of natural numbers
n
n
n
for which
3
n
\frac{3}{n}
n
3
cannot be written as the sum of two reciprocals of natural numbers (
S
=
{
n
∣
3
n
≠
1
p
+
1
q
for any
p
,
q
∈
N
}
S =\left\{n |\frac{3}{n} \neq \frac{1}{p} + \frac{1}{q} \text{ for any } p, q \in \mathbb N \right\}
S
=
{
n
∣
n
3
=
p
1
+
q
1
for any
p
,
q
∈
N
}
) is not the union of finitely many arithmetic progressions.
13
1
Hide problems
The relation between area of a polygon and the pyramid
A regular
n
n
n
-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by
S
1
,
S
2
S_1, S_2
S
1
,
S
2
, and
S
S
S
, respectively. Let
σ
\sigma
σ
be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that
σ
S
=
4
S
1
S
2
cos
2
π
n
.
\sigma S = 4S_1S_2 \cos^2 \frac{\pi}{n}.
σ
S
=
4
S
1
S
2
cos
2
n
π
.
12
1
Hide problems
Prove that there are exactly 1982 numbers
Let there be
3399
3399
3399
numbers arbitrarily chosen among the first
6798
6798
6798
integers
1
,
2
,
…
,
6798
1, 2, \ldots , 6798
1
,
2
,
…
,
6798
in such a way that none of them divides another. Prove that there are exactly
1982
1982
1982
numbers in
{
1
,
2
,
…
,
6798
}
\{1, 2, \ldots, 6798\}
{
1
,
2
,
…
,
6798
}
that must end up being chosen.
11
1
Hide problems
A billiard ball in a rectangular pool table
A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers
a
a
a
and
b
b
b
. A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between
a
a
a
and
b
b
b
be for this to happen?
10
1
Hide problems
The Spheres, radiis and areas
Let
r
1
,
…
,
r
n
r_1, \ldots , r_n
r
1
,
…
,
r
n
be the radii of
n
n
n
spheres. Call
S
1
,
S
2
,
…
,
S
n
S_1, S_2, \ldots , S_n
S
1
,
S
2
,
…
,
S
n
the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that
S
1
r
1
2
+
S
2
r
2
2
+
⋯
+
S
n
r
n
2
=
4
π
.
\frac{S_1}{r_1^2} + \frac{S_2}{r_2^2}+\cdots+\frac{S_n}{r_n^2} = 4 \pi.
r
1
2
S
1
+
r
2
2
S
2
+
⋯
+
r
n
2
S
n
=
4
π
.
9
1
Hide problems
Show that there exists natural m - Euler's Function
Given any two real numbers
α
\alpha
α
and
β
,
0
≤
α
<
β
≤
1
\beta , 0 \leq \alpha < \beta \leq 1
β
,
0
≤
α
<
β
≤
1
, prove that there exists a natural number
m
m
m
such that
α
<
ϕ
(
m
)
m
<
β
.
\alpha < \frac{\phi(m)}{m} < \beta.
α
<
m
ϕ
(
m
)
<
β
.
6
1
Hide problems
Construct Collinear Points X, Y, Z
On the three distinct lines
a
,
b
a, b
a
,
b
, and
c
c
c
three points
A
,
B
A, B
A
,
B
, and
C
C
C
are given, respectively. Construct three collinear points
X
,
Y
,
Z
X, Y,Z
X
,
Y
,
Z
on lines
a
,
b
,
c
a, b, c
a
,
b
,
c
, respectively, such that
B
Y
A
X
=
2
\frac{BY}{AX} = 2
A
X
B
Y
=
2
and
C
Z
A
X
=
3
\frac{CZ}{AX} = 3
A
X
CZ
=
3
.
3
1
Hide problems
Show that there exists a point 0 ≤ y ≤ 1
Given
n
n
n
points
X
1
,
X
2
,
…
,
X
n
X_1,X_2,\ldots, X_n
X
1
,
X
2
,
…
,
X
n
in the interval
0
≤
X
i
≤
1
,
i
=
1
,
2
,
…
,
n
0 \leq X_i \leq 1, i = 1, 2,\ldots, n
0
≤
X
i
≤
1
,
i
=
1
,
2
,
…
,
n
, show that there is a point
y
,
0
≤
y
≤
1
y, 0 \leq y \leq 1
y
,
0
≤
y
≤
1
, such that
1
n
∑
i
=
1
n
∣
y
−
X
i
∣
=
1
2
.
\frac{1}{n} \sum_{i=1}^{n} | y - X_i | = \frac 12.
n
1
i
=
1
∑
n
∣
y
−
X
i
∣
=
2
1
.
1
1
Hide problems
Integer for all n ≥ k
(a) Prove that
1
n
+
1
⋅
(
2
n
n
)
\frac{1}{n+1} \cdot \binom{2n}{n}
n
+
1
1
⋅
(
n
2
n
)
is an integer for
n
≥
0.
n \geq 0.
n
≥
0.
(b) Given a positive integer
k
k
k
, determine the smallest integer
C
k
C_k
C
k
with the property that
C
k
n
+
k
+
1
⋅
(
2
n
n
)
\frac{C_k}{n+k+1} \cdot \binom{2n}{n}
n
+
k
+
1
C
k
⋅
(
n
2
n
)
is an integer for all
n
≥
k
.
n \geq k.
n
≥
k
.
5
1
Hide problems
Given Perimeter - Maximal Radius of Incircle
Among all triangles with a given perimeter, find the one with the maximal radius of its incircle.
7
1
Hide problems
Solve x^3 - y^3 = 2xy + 8 in Z
Find all solutions
(
x
,
y
)
∈
Z
2
(x, y) \in \mathbb Z^2
(
x
,
y
)
∈
Z
2
of the equation
x
3
−
y
3
=
2
x
y
+
8.
x^3 - y^3 = 2xy + 8.
x
3
−
y
3
=
2
x
y
+
8.
56
1
Hide problems
Polynomial mixed with Inequality
Let
f
(
x
)
=
a
x
2
+
b
x
+
c
f(x) = ax^2 + bx+ c
f
(
x
)
=
a
x
2
+
b
x
+
c
and
g
(
x
)
=
c
x
2
+
b
x
+
a
g(x) = cx^2 + bx + a
g
(
x
)
=
c
x
2
+
b
x
+
a
. If
∣
f
(
0
)
∣
≤
1
,
∣
f
(
1
)
∣
≤
1
,
∣
f
(
−
1
)
∣
≤
1
|f(0)| \leq 1, |f(1)| \leq 1, |f(-1)| \leq 1
∣
f
(
0
)
∣
≤
1
,
∣
f
(
1
)
∣
≤
1
,
∣
f
(
−
1
)
∣
≤
1
, prove that for
∣
x
∣
≤
1
|x| \leq 1
∣
x
∣
≤
1
,(a)
∣
f
(
x
)
∣
≤
5
/
4
|f(x)| \leq 5/4
∣
f
(
x
)
∣
≤
5/4
,(b)
∣
g
(
x
)
∣
≤
2
|g(x)| \leq 2
∣
g
(
x
)
∣
≤
2
.