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Problems
Contests
International Contests
IMO Longlists
1979 IMO Longlists
1979 IMO Longlists
Part of
IMO Longlists
Subcontests
(55)
81
1
Hide problems
Rectangular parallelepipeds with at least one integer side
Let
Π
\Pi
Π
be the set of rectangular parallelepipeds that have at least one edge of integer length. If a rectangular parallelepiped
P
0
P_0
P
0
can be decomposed into parallelepipeds
P
1
,
P
2
,
.
.
.
,
P
N
∈
Π
P_1,P_2, . . . ,P_N\in \Pi
P
1
,
P
2
,
...
,
P
N
∈
Π
, prove that
P
0
∈
Π
P_0\in \Pi
P
0
∈
Π
.
79
1
Hide problems
Unit circle - subset of closed arcs
Let
S
S
S
be a unit circle and
K
K
K
a subset of
S
S
S
consisting of several closed arcs. Let
K
K
K
satisfy the following properties:
(
i
)
(\text{i})
(
i
)
K
K
K
contains three points
A
,
B
,
C
A,B,C
A
,
B
,
C
, that are the vertices of an acute-angled triangle
(
ii
)
(\text{ii})
(
ii
)
For every point
A
A
A
that belongs to
K
K
K
its diametrically opposite point
A
′
A'
A
′
and all points
B
B
B
on an arc of length
1
9
\frac{1}{9}
9
1
with center
A
′
A'
A
′
do not belong to
K
K
K
. Prove that there are three points
E
,
F
,
G
E,F,G
E
,
F
,
G
on
S
S
S
that are vertices of an equilateral triangle and that do not belong to
K
K
K
.
78
1
Hide problems
omega(n) is the number of prime divisors of n
Denote the number of different prime divisors of the number
n
n
n
by
ω
(
n
)
\omega (n)
ω
(
n
)
, where
n
n
n
is an integer greater than
1
1
1
. Prove that there exist infinitely many numbers
n
n
n
for which
ω
(
n
)
<
ω
(
n
+
1
)
<
ω
(
n
+
2
)
\omega (n)< \omega (n+1)<\omega (n+2)
ω
(
n
)
<
ω
(
n
+
1
)
<
ω
(
n
+
2
)
holds.
77
1
Hide problems
h(n) is the greatest prime divisor of n
By
h
(
n
)
h(n)
h
(
n
)
, where
n
n
n
is an integer greater than
1
1
1
, let us denote the greatest prime divisor of the number
n
n
n
. Are there infinitely many numbers
n
n
n
for which
h
(
n
)
<
h
(
n
+
1
)
<
h
(
n
+
2
)
h(n) < h(n+1)< h(n+2)
h
(
n
)
<
h
(
n
+
1
)
<
h
(
n
+
2
)
holds?
76
1
Hide problems
Diophantine-type geometry
Suppose that a triangle whose sides are of integer lengths is inscribed in a circle of diameter
6.25
6.25
6.25
. Find the sides of the triangle.
75
1
Hide problems
Arbitrary point in space - construct triangles
Given an equilateral triangle
A
B
C
ABC
A
BC
, let
M
M
M
be an arbitrary point in space.
(
a
)
(\text{a})
(
a
)
Prove that one can construct a triangle from the segments
M
A
,
M
B
,
M
C
MA, MB, MC
M
A
,
MB
,
MC
.
(
b
)
(\text{b})
(
b
)
Suppose that
P
P
P
and
Q
Q
Q
are two points symmetric with respect to the center
O
O
O
of
A
B
C
ABC
A
BC
. Prove that the two triangles constructed from the segments
P
A
,
P
B
,
P
C
PA,PB,PC
P
A
,
PB
,
PC
and
Q
A
,
Q
B
,
Q
C
QA,QB,QC
Q
A
,
QB
,
QC
are of equal area.
74
1
Hide problems
Identity - distance from point on circumcircle to vertices
Given an equilateral triangle
A
B
C
ABC
A
BC
of side
a
a
a
in a plane, let
M
M
M
be a point on the circumcircle of the triangle. Prove that the sum
s
=
M
A
4
+
M
B
4
+
M
C
4
s = MA^4 +MB^4 +MC^4
s
=
M
A
4
+
M
B
4
+
M
C
4
is independent of the position of the point
M
M
M
on the circle, and determine that constant value as a function of
a
a
a
.
73
1
Hide problems
Coloring finite number of equal non-intersecting circles
In a plane a finite number of equal circles are given. These circles are mutually nonintersecting (they may be externally tangent). Prove that one can use at most four colors for coloring these circles so that two circles tangent to each other are of different colors. What is the smallest number of circles that requires four colors?
72
1
Hide problems
Polynomial - f(x)=1979 four times - prove f(x) not= 2*1979
Let
f
(
x
)
f (x)
f
(
x
)
be a polynomial with integer coefficients. Prove that if
f
(
x
)
=
1979
f (x)= 1979
f
(
x
)
=
1979
for four different integer values of
x
x
x
, then
f
(
x
)
f (x)
f
(
x
)
cannot be equal to
2
×
1979
2\times 1979
2
×
1979
for any integral value of
x
x
x
.
70
1
Hide problems
1979 equilateral triangles
There are
1979
1979
1979
equilateral triangles:
T
1
,
T
2
,
.
.
.
,
T
1979
T_1,T_2, . . . ,T_{1979}
T
1
,
T
2
,
...
,
T
1979
. A side of triangle
T
k
T_k
T
k
is equal to
1
k
\frac{1}{k}
k
1
,
k
=
1
,
2
,
.
.
.
,
1979
k = 1,2, . . . ,1979
k
=
1
,
2
,
...
,
1979
. At what values of a number
a
a
a
can one place all these triangles into the equilateral triangle with side length
a
a
a
so that they don’t intersect (points of contact are allowed)?
65
1
Hide problems
Functional Equation - prove equality from inequality
Given a function
f
f
f
such that
f
(
x
)
≤
x
∀
x
∈
R
f(x)\le x\forall x\in\mathbb{R}
f
(
x
)
≤
x
∀
x
∈
R
and
f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
)
∀
{
x
,
y
}
∈
R
f(x+y)\le f(x)+f(y)\forall \{x,y\}\in\mathbb{R}
f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
)
∀
{
x
,
y
}
∈
R
, prove that
f
(
x
)
=
x
∀
x
∈
R
f(x)=x\forall x\in\mathbb{R}
f
(
x
)
=
x
∀
x
∈
R
.
64
1
Hide problems
Identity - distance from point on circumcircle to sides
From point
P
P
P
on arc
B
C
BC
BC
of the circumcircle about triangle
A
B
C
ABC
A
BC
,
P
X
PX
PX
is constructed perpendicular to
B
C
BC
BC
,
P
Y
PY
P
Y
is perpendicular to
A
C
AC
A
C
, and
P
Z
PZ
PZ
perpendicular to
A
B
AB
A
B
(all extended if necessary). Prove that
B
C
P
X
=
A
C
P
Y
+
A
B
P
Z
\frac{BC}{PX}=\frac{AC}{PY}+\frac{AB}{PZ}
PX
BC
=
P
Y
A
C
+
PZ
A
B
.
63
1
Hide problems
Classic polygon inequality
Let the sequence
{
a
i
}
\{a_i\}
{
a
i
}
of
n
n
n
positive reals denote the lengths of the sides of an arbitrary
n
n
n
-gon. Let
s
=
∑
i
=
1
n
a
i
s=\sum_{i=1}^{n}{a_i}
s
=
∑
i
=
1
n
a
i
. Prove that
2
≥
∑
i
=
1
n
a
i
s
−
a
i
≥
n
n
−
1
2\ge \sum_{i=1}^{n}{\frac{a_i}{s-a_i}}\ge \frac{n}{n-1}
2
≥
∑
i
=
1
n
s
−
a
i
a
i
≥
n
−
1
n
.
62
1
Hide problems
Triangle subdivided into triangles
T
T
T
is a given triangle with vertices
P
1
,
P
2
,
P
3
P_1,P_2,P_3
P
1
,
P
2
,
P
3
. Consider an arbitrary subdivision of
T
T
T
into finitely many subtriangles such that no vertex of a subtriangle lies strictly between two vertices of another subtriangle. To each vertex
V
V
V
of the subtriangles there is assigned a number
n
(
V
)
n(V)
n
(
V
)
according to the following rules:
(
i
)
(\text{i})
(
i
)
If
V
V
V
=
P
i
P_i
P
i
, then
n
(
V
)
=
i
n(V) = i
n
(
V
)
=
i
.
(
ii
)
(\text{ii})
(
ii
)
If
V
V
V
lies on the side
P
i
P
j
P_i P_j
P
i
P
j
of
T
T
T
, then
n
(
V
)
=
i
n(V) = i
n
(
V
)
=
i
or
j
j
j
.
(
iii
)
(\text{iii})
(
iii
)
If
V
V
V
lies inside the triangle
T
T
T
, then
n
(
V
)
n(V)
n
(
V
)
is any of the numbers
1
,
2
,
3
1,2,3
1
,
2
,
3
. Prove that there exists at least one subtriangle whose vertices are numbered
1
,
2
,
3
1, 2, 3
1
,
2
,
3
.
61
1
Hide problems
Convex function, two non-decreasing sequences
There are two non-decreasing sequences
{
a
i
}
\{a_i\}
{
a
i
}
and
{
b
i
}
\{b_i\}
{
b
i
}
of
n
n
n
real numbers each, such that
a
i
≤
a
i
+
1
a_i\le a_{i+1}
a
i
≤
a
i
+
1
for each
1
≤
i
≤
n
−
1
1\le i\le n-1
1
≤
i
≤
n
−
1
, and
b
i
≤
b
i
+
1
b_i\le b_{i+1}
b
i
≤
b
i
+
1
for each
1
≤
i
≤
n
−
1
1\le i\le n-1
1
≤
i
≤
n
−
1
, and
∑
k
=
1
m
a
k
≥
∑
k
=
1
m
b
k
\sum_{k=1}^{m}{a_k}\ge \sum_{k=1}^{m}{b_k}
∑
k
=
1
m
a
k
≥
∑
k
=
1
m
b
k
where
m
≤
n
m\le n
m
≤
n
with equality for
m
=
n
m=n
m
=
n
. For a convex function
f
f
f
defined on the real numbers, prove that
∑
k
=
1
n
f
(
a
k
)
≤
∑
k
=
1
n
f
(
b
k
)
\sum_{k=1}^{n}{f(a_k)}\le \sum_{k=1}^{n}{f(b_k)}
∑
k
=
1
n
f
(
a
k
)
≤
∑
k
=
1
n
f
(
b
k
)
.
59
1
Hide problems
Simple old inequality - find minimum
Determine the maximum value of
x
2
y
2
z
2
w
x^2 y^2 z^2 w
x
2
y
2
z
2
w
for
{
x
,
y
,
z
,
w
}
∈
R
+
∪
{
0
}
\{x,y,z,w\}\in\mathbb{R}^{+} \cup\{0\}
{
x
,
y
,
z
,
w
}
∈
R
+
∪
{
0
}
and
2
x
+
x
y
+
z
+
y
z
w
=
1
2x+xy+z+yzw=1
2
x
+
x
y
+
z
+
yz
w
=
1
.
58
1
Hide problems
Finite number of lines divide the plane in k regions
Prove that there exists a
k
0
∈
N
k_0\in\mathbb{N}
k
0
∈
N
such that for every
k
∈
N
,
k
>
k
0
k\in\mathbb{N},k>k_0
k
∈
N
,
k
>
k
0
, there exists a finite number of lines in the plane not all parallel to one of them, that divide the plane exactly in
k
k
k
regions. Find
k
0
k_0
k
0
.
57
1
Hide problems
Necessary and sufficient condition for the existence of a se
Let
M
M
M
be a set and
A
,
B
,
C
A,B,C
A
,
B
,
C
given subsets of
M
M
M
. Find a necessary and sufficient condition for the existence of a set
X
⊂
M
X\subset M
X
⊂
M
for which
(
X
∪
A
)
\
(
X
∩
B
)
=
C
(X\cup A)\backslash(X\cap B)=C
(
X
∪
A
)
\
(
X
∩
B
)
=
C
. Describe all such sets.
56
1
Hide problems
Existence of natural - involves floor function, sqrt{2}
Show that for every
n
∈
N
n\in\mathbb{N}
n
∈
N
,
n
2
−
⌊
n
2
⌋
>
1
2
n
2
n\sqrt{2}-\lfloor n\sqrt{2}\rfloor>\frac{1}{2n \sqrt{2}}
n
2
−
⌊
n
2
⌋
>
2
n
2
1
and that for every
ϵ
>
0
\epsilon >0
ϵ
>
0
, there exists an
n
∈
N
n\in\mathbb{N}
n
∈
N
such that
n
2
−
⌊
n
2
⌋
<
1
2
n
2
+
ϵ
n\sqrt{2}-\lfloor n\sqrt{2}\rfloor < \frac{1}{2n \sqrt{2}}+\epsilon
n
2
−
⌊
n
2
⌋
<
2
n
2
1
+
ϵ
.
55
1
Hide problems
Diophantine - Coprime - Infinite solutions proof
Let
a
,
b
a,b
a
,
b
be coprime integers. Show that the equation
a
x
2
+
b
y
2
=
z
3
ax^2 + by^2 =z^3
a
x
2
+
b
y
2
=
z
3
has an infinite set of solutions
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
with
{
x
,
y
,
z
}
∈
Z
\{x,y,z\}\in\mathbb{Z}
{
x
,
y
,
z
}
∈
Z
and each pair of
x
,
y
x,y
x
,
y
mutually coprime.
53
1
Hide problems
Proving that there exists finitely many sequences...
An infinite increasing sequence of positive integers
n
j
(
j
=
1
,
2
,
…
)
n_j (j = 1, 2, \ldots )
n
j
(
j
=
1
,
2
,
…
)
has the property that for a certain
c
c
c
,
1
N
∑
n
j
≤
N
n
j
≤
c
,
\frac{1}{N}\sum_{n_j\le N} n_j \le c,
N
1
n
j
≤
N
∑
n
j
≤
c
,
for every
N
>
0
N >0
N
>
0
. Prove that there exist finitely many sequences
m
j
(
i
)
(
i
=
1
,
2
,
…
,
k
)
m^{(i)}_j (i = 1, 2,\ldots, k)
m
j
(
i
)
(
i
=
1
,
2
,
…
,
k
)
such that
{
n
1
,
n
2
,
…
}
=
⋃
i
=
1
k
{
m
1
(
i
)
,
m
2
(
i
)
,
…
}
\{n_1, n_2, \ldots \} =\bigcup_{i=1}^k\{m^{(i)}_1 ,m^{(i)}_2 ,\ldots\}
{
n
1
,
n
2
,
…
}
=
i
=
1
⋃
k
{
m
1
(
i
)
,
m
2
(
i
)
,
…
}
and
m
j
+
1
(
i
)
>
2
m
j
(
i
)
(
1
≤
i
≤
k
,
j
=
1
,
2
,
…
)
.
m^{(i)}_{j+1} > 2m^{(i)}_j (1 \le i \le k, j = 1, 2,\ldots).
m
j
+
1
(
i
)
>
2
m
j
(
i
)
(
1
≤
i
≤
k
,
j
=
1
,
2
,
…
)
.
52
1
Hide problems
Writing a positive integer as sum of two terms of sequence.
Let a real number
λ
>
1
\lambda > 1
λ
>
1
be given and a sequence
(
n
k
)
(n_k)
(
n
k
)
of positive integers such that
n
k
+
1
n
k
>
λ
\frac{n_{k+1}}{n_k}> \lambda
n
k
n
k
+
1
>
λ
for
k
=
1
,
2
,
…
k = 1, 2,\ldots
k
=
1
,
2
,
…
Prove that there exists a positive integer
c
c
c
such that no positive integer
n
n
n
can be represented in more than
c
c
c
ways in the form
n
=
n
k
+
n
j
n = n_k + n_j
n
=
n
k
+
n
j
or
n
=
n
r
−
n
s
n = n_r - n_s
n
=
n
r
−
n
s
.
51
1
Hide problems
Seven circles tangent to another circle and two sides.
Let
A
B
C
ABC
A
BC
be an arbitrary triangle and let
S
1
,
S
2
,
⋯
,
S
7
S_1, S_2,\cdots, S_7
S
1
,
S
2
,
⋯
,
S
7
be circles satisfying the following conditions:
S
1
S_1
S
1
is tangent to
C
A
CA
C
A
and
A
B
AB
A
B
,
S
2
S_2
S
2
is tangent to
S
1
,
A
B
S_1, AB
S
1
,
A
B
, and
B
C
,
BC,
BC
,
S
3
S_3
S
3
is tangent to
S
2
,
B
C
S_2, BC
S
2
,
BC
, and
C
A
,
CA,
C
A
,
..............................
S
7
S_7
S
7
is tangent to
S
6
,
C
A
S_6, CA
S
6
,
C
A
and
A
B
.
AB.
A
B
.
Prove that the circles
S
1
S_1
S
1
and
S
7
S_7
S
7
coincide.
49
1
Hide problems
Two sequences of integers and condition for prime numbers.
Let there be given two sequences of integers
f
i
(
1
)
,
f
i
(
2
)
,
⋯
(
i
=
1
,
2
)
f_i(1), f_i(2), \cdots (i = 1, 2)
f
i
(
1
)
,
f
i
(
2
)
,
⋯
(
i
=
1
,
2
)
satisfying:
(
i
)
f
i
(
n
m
)
=
f
i
(
n
)
f
i
(
m
)
(i) f_i(nm) = f_i(n)f_i(m)
(
i
)
f
i
(
nm
)
=
f
i
(
n
)
f
i
(
m
)
if
gcd
(
n
,
m
)
=
1
\gcd(n,m) = 1
g
cd
(
n
,
m
)
=
1
;
(
i
i
)
(ii)
(
ii
)
for every prime
P
P
P
and all
k
=
2
,
3
,
4
,
⋯
k = 2, 3, 4, \cdots
k
=
2
,
3
,
4
,
⋯
,
f
i
(
P
k
)
=
f
i
(
P
)
f
i
(
P
k
−
1
)
−
P
2
f
(
P
k
−
2
)
.
f_i(P^k) = f_i(P)f_i(P^{k-1}) - P^2f(P^{k-2}).
f
i
(
P
k
)
=
f
i
(
P
)
f
i
(
P
k
−
1
)
−
P
2
f
(
P
k
−
2
)
.
Moreover, for every prime
P
P
P
:
(
i
i
i
)
f
1
(
P
)
=
2
P
,
(iii) f_1(P) = 2P,
(
iii
)
f
1
(
P
)
=
2
P
,
(
i
v
)
f
2
(
P
)
<
2
P
.
(iv) f_2(P) < 2P.
(
i
v
)
f
2
(
P
)
<
2
P
.
Prove that
∣
f
2
(
n
)
∣
<
f
1
(
n
)
|f_2(n)| < f_1(n)
∣
f
2
(
n
)
∣
<
f
1
(
n
)
for all
n
n
n
.
48
1
Hide problems
Distance between points of intersection of line and circle.
In the plane a circle
C
C
C
of unit radius is given. For any line
l
l
l
, a number
s
(
l
)
s(l)
s
(
l
)
is defined in the following way: If
l
l
l
and
C
C
C
intersect in two points,
s
(
l
)
s(l)
s
(
l
)
is their distance; otherwise,
s
(
l
)
=
0
s(l) = 0
s
(
l
)
=
0
. Let
P
P
P
be a point at distance
r
r
r
from the center of
C
C
C
. One defines
M
(
r
)
M(r)
M
(
r
)
to be the maximum value of the sum
s
(
m
)
+
s
(
n
)
s(m) + s(n)
s
(
m
)
+
s
(
n
)
, where
m
m
m
and
n
n
n
are variable mutually orthogonal lines through
P
P
P
. Determine the values of
r
r
r
for which
M
(
r
)
>
2
M(r) > 2
M
(
r
)
>
2
.
34
1
Hide problems
Find all fractions which satisfy the property
Notice that in the fraction
16
64
\frac{16}{64}
64
16
we can perform a simplification as
16
64
=
1
4
\cancel{\frac{16}{64}}=\frac 14
64
16
=
4
1
obtaining a correct equality. Find all fractions whose numerators and denominators are two-digit positive integers for which such a simplification is correct.
45
1
Hide problems
Number of ways of writing as sum of three integers.
For any positive integer
n
n
n
, we denote by
F
(
n
)
F(n)
F
(
n
)
the number of ways in which
n
n
n
can be expressed as the sum of three different positive integers, without regard to order. Thus, since
10
=
7
+
2
+
1
=
6
+
3
+
1
=
5
+
4
+
1
=
5
+
3
+
2
10 = 7+2+1 = 6+3+1 = 5+4+1 = 5+3+2
10
=
7
+
2
+
1
=
6
+
3
+
1
=
5
+
4
+
1
=
5
+
3
+
2
, we have
F
(
10
)
=
4
F(10) = 4
F
(
10
)
=
4
. Show that
F
(
n
)
F(n)
F
(
n
)
is even if
n
≡
2
n \equiv 2
n
≡
2
or
4
(
m
o
d
6
)
4 \pmod 6
4
(
mod
6
)
, but odd if
n
n
n
is divisible by
6
6
6
.
43
1
Hide problems
$aPA^2+bPB^2+cPC^2$ is constant for P on incircle.
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
denote the lengths of the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
, respectively, of a triangle
A
B
C
ABC
A
BC
. If
P
P
P
is any point on the circumference of the circle inscribed in the triangle, show that
a
P
A
2
+
b
P
B
2
+
c
P
C
2
aPA^2+bPB^2+cPC^2
a
P
A
2
+
b
P
B
2
+
c
P
C
2
is constant.
42
1
Hide problems
Existence of f(x) such that f(g(x))=g(f(x)).
Let a quadratic polynomial
g
(
x
)
=
a
x
2
+
b
x
+
c
g(x) = ax^2 + bx + c
g
(
x
)
=
a
x
2
+
b
x
+
c
be given and an integer
n
≥
1
n \ge 1
n
≥
1
. Prove that there exists at most one polynomial
f
(
x
)
f(x)
f
(
x
)
of
n
n
n
th degree such that
f
(
g
(
x
)
)
=
g
(
f
(
x
)
)
.
f(g(x)) = g(f(x)).
f
(
g
(
x
))
=
g
(
f
(
x
))
.
41
1
Hide problems
There doesn't exist a pyramid with conditions.
Prove the following statement: There does not exist a pyramid with square base and congruent lateral faces for which the measures of all edges, total area, and volume are integers.
40
1
Hide problems
Inequality regarding a polynomial
A polynomial
P
(
x
)
P(x)
P
(
x
)
has degree at most
2
k
2k
2
k
, where
k
=
0
,
1
,
2
,
⋯
k = 0, 1,2,\cdots
k
=
0
,
1
,
2
,
⋯
. Given that for an integer
i
i
i
, the inequality
−
k
≤
i
≤
k
-k \le i \le k
−
k
≤
i
≤
k
implies
∣
P
(
i
)
∣
≤
1
|P(i)| \le 1
∣
P
(
i
)
∣
≤
1
, prove that for all real numbers
x
x
x
, with
−
k
≤
x
≤
k
-k \le x \le k
−
k
≤
x
≤
k
, the following inequality holds:
∣
P
(
x
)
∣
<
(
2
k
+
1
)
(
2
k
k
)
|P(x)| < (2k + 1)\dbinom{2k}{k}
∣
P
(
x
)
∣
<
(
2
k
+
1
)
(
k
2
k
)
39
1
Hide problems
Amount of water required for an expedition.
A desert expedition camps at the border of the desert, and has to provide one liter of drinking water for another member of the expedition, residing on the distance of
n
n
n
days of walking from the camp, under the following conditions:
(
i
)
(i)
(
i
)
Each member of the expedition can pick up at most
3
3
3
liters of water.
(
i
i
)
(ii)
(
ii
)
Each member must drink one liter of water every day spent in the desert.
(
i
i
i
)
(iii)
(
iii
)
All the members must return to the camp. How much water do they need (at least) in order to do that?
38
1
Hide problems
There exists integer n and n polynomials satisfying equality
Prove the following statement: If a polynomial
f
(
x
)
f(x)
f
(
x
)
with real coefficients takes only nonnegative values, then there exists a positive integer
n
n
n
and polynomials
g
1
(
x
)
,
g
2
(
x
)
,
⋯
,
g
n
(
x
)
g_1(x), g_2(x),\cdots, g_n(x)
g
1
(
x
)
,
g
2
(
x
)
,
⋯
,
g
n
(
x
)
such that
f
(
x
)
=
g
1
(
x
)
2
+
g
2
(
x
)
2
+
⋯
+
g
n
(
x
)
2
f(x) = g_1(x)^2 + g_2(x)^2 +\cdots+ g_n(x)^2
f
(
x
)
=
g
1
(
x
)
2
+
g
2
(
x
)
2
+
⋯
+
g
n
(
x
)
2
36
1
Hide problems
Regular tetrahedron inscribed in another.
A regular tetrahedron
A
1
B
1
C
1
D
1
A_1B_1C_1D_1
A
1
B
1
C
1
D
1
is inscribed in a regular tetrahedron
A
B
C
D
ABCD
A
BC
D
, where
A
1
A_1
A
1
lies in the plane
B
C
D
BCD
BC
D
,
B
1
B_1
B
1
in the plane
A
C
D
ACD
A
C
D
, etc. Prove that
A
1
B
1
≥
A
B
3
A_1B_1 \ge\frac{ AB}{3}
A
1
B
1
≥
3
A
B
.
32
1
Hide problems
Number of solutions for |x_1| + |x_2| +...+ |x_k| = n
Let
n
,
k
≥
1
n, k \ge 1
n
,
k
≥
1
be natural numbers. Find the number
A
(
n
,
k
)
A(n, k)
A
(
n
,
k
)
of solutions in integers of the equation
∣
x
1
∣
+
∣
x
2
∣
+
⋯
+
∣
x
k
∣
=
n
|x_1| + |x_2| +\cdots + |x_k| = n
∣
x
1
∣
+
∣
x
2
∣
+
⋯
+
∣
x
k
∣
=
n
35
1
Hide problems
Triangle with sides as consecutive terms of a sequence.
Given a sequence
(
a
n
)
(a_n)
(
a
n
)
, with
a
1
=
4
a_1 = 4
a
1
=
4
and
a
n
+
1
=
a
n
2
−
2
(
∀
n
∈
N
)
a_{n+1} = a_n^2-2 (\forall n \in\mathbb{N})
a
n
+
1
=
a
n
2
−
2
(
∀
n
∈
N
)
, prove that there is a triangle with side lengths
a
n
−
1
,
a
n
,
a
n
+
1
,
a_{n-1}, a_n, a_{n+1},
a
n
−
1
,
a
n
,
a
n
+
1
,
and that its area is equal to an integer.
30
1
Hide problems
Axes of symmetry intersect at angle.
Let
M
M
M
be a set of points in a plane with at least two elements. Prove that if
M
M
M
has two axes of symmetry
g
1
g_1
g
1
and
g
2
g_2
g
2
intersecting at an angle
α
=
q
π
\alpha = q\pi
α
=
q
π
, where
q
q
q
is irrational, then
M
M
M
must be infinite.
26
1
Hide problems
4^n+2^n+1 is a prime => n is a power of 3 (ILL 1979 P26)
Let
n
n
n
be a positive integer. If
4
n
+
2
n
+
1
4^n + 2^n + 1
4
n
+
2
n
+
1
is a prime, prove that
n
n
n
is a power of three.
24
1
Hide problems
All integers n ≥ (a − 1)(b − 1) can be written as n=ua+vb
Let
a
a
a
and
b
b
b
be coprime integers, greater than or equal to
1
1
1
. Prove that all integers
n
n
n
greater than or equal to
(
a
−
1
)
(
b
−
1
)
(a - 1)(b - 1)
(
a
−
1
)
(
b
−
1
)
can be written in the form:
n
=
u
a
+
v
b
,
with
(
u
,
v
)
∈
N
×
N
.
n = ua + vb, \qquad \text{with} (u, v) \in \mathbb N \times \mathbb N.
n
=
u
a
+
v
b
,
with
(
u
,
v
)
∈
N
×
N
.
23
1
Hide problems
All elements of the set E of pairs of positive integers
Consider the set
E
E
E
consisting of pairs of integers
(
a
,
b
)
(a, b)
(
a
,
b
)
, with
a
≥
1
a \geq 1
a
≥
1
and
b
≥
1
b \geq 1
b
≥
1
, that satisfy in the decimal system the following properties:(i)
b
b
b
is written with three digits, as
α
2
α
1
α
0
‾
\overline{\alpha_2\alpha_1\alpha_0}
α
2
α
1
α
0
,
α
2
≠
0
\alpha_2 \neq 0
α
2
=
0
;(ii)
a
a
a
is written as
β
p
…
β
1
β
0
‾
\overline{\beta_p \ldots \beta_1\beta_0}
β
p
…
β
1
β
0
for some
p
p
p
;(iii)
(
a
+
b
)
2
(a + b)^2
(
a
+
b
)
2
is written as
β
p
…
β
1
β
0
α
2
α
1
α
0
‾
.
\overline{\beta_p\ldots \beta_1 \beta_0 \alpha_2 \alpha_1 \alpha_0}.
β
p
…
β
1
β
0
α
2
α
1
α
0
.
Find the elements of
E
E
E
.
22
1
Hide problems
Quadrilaterals with equal sides
Consider two quadrilaterals
A
B
C
D
ABCD
A
BC
D
and
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
in an affine Euclidian plane such that
A
B
=
A
′
B
′
,
B
C
=
B
′
C
′
,
C
D
=
C
′
D
′
AB = A'B', BC = B'C', CD = C'D'
A
B
=
A
′
B
′
,
BC
=
B
′
C
′
,
C
D
=
C
′
D
′
, and
D
A
=
D
′
A
′
DA = D'A'
D
A
=
D
′
A
′
. Prove that the following two statements are true:(a) If the diagonals
B
D
BD
B
D
and
A
C
AC
A
C
are mutually perpendicular, then the diagonals
B
′
D
′
B'D'
B
′
D
′
and
A
′
C
′
A'C'
A
′
C
′
are also mutually perpendicular.(b) If the perpendicular bisector of
B
D
BD
B
D
intersects
A
C
AC
A
C
at
M
M
M
, and that of
B
′
D
′
B'D'
B
′
D
′
intersects
A
′
C
′
A'C'
A
′
C
′
at
M
′
M'
M
′
, then
M
A
‾
M
C
‾
=
M
′
A
′
‾
M
′
C
′
‾
\frac{\overline{MA}}{\overline{MC}}=\frac{\overline{M'A'}}{\overline{M'C'}}
MC
M
A
=
M
′
C
′
M
′
A
′
(if
M
C
=
0
MC = 0
MC
=
0
then
M
′
C
′
=
0
M'C' = 0
M
′
C
′
=
0
).
21
1
Hide problems
All monotonic mappings such that f(t)+f^{-1}(t)=2t
Let
E
E
E
be the set of all bijective mappings from
R
\mathbb R
R
to
R
\mathbb R
R
satisfying
f
(
t
)
+
f
−
1
(
t
)
=
2
t
,
∀
t
∈
R
,
f(t) + f^{-1}(t) = 2t, \qquad \forall t \in \mathbb R,
f
(
t
)
+
f
−
1
(
t
)
=
2
t
,
∀
t
∈
R
,
where
f
−
1
f^{-1}
f
−
1
is the mapping inverse to
f
f
f
. Find all elements of
E
E
E
that are monotonic mappings.
19
1
Hide problems
Number of k-tuples is equal for odd and even k's
For
k
=
1
,
2
,
…
k = 1, 2, \ldots
k
=
1
,
2
,
…
consider the
k
k
k
-tuples
(
a
1
,
a
2
,
…
,
a
k
)
(a_1, a_2, \ldots, a_k)
(
a
1
,
a
2
,
…
,
a
k
)
of positive integers such that
a
1
+
2
a
2
+
⋯
+
k
a
k
=
1979.
a_1 + 2a_2 + \cdots + ka_k = 1979.
a
1
+
2
a
2
+
⋯
+
k
a
k
=
1979.
Show that there are as many such
k
k
k
-tuples with odd
k
k
k
as there are with even
k
k
k
.
18
1
Hide problems
The sum 1/(1+a) + 1/(1+2a) + ... + 1/(1+na) isn't an integer
Show that for no integers
a
≥
1
,
n
≥
1
a \geq 1, n \geq 1
a
≥
1
,
n
≥
1
is the sum
1
+
1
1
+
a
+
1
1
+
2
a
+
⋯
+
1
1
+
n
a
1+\frac{1}{1+a}+\frac{1}{1+2a}+\cdots+\frac{1}{1+na}
1
+
1
+
a
1
+
1
+
2
a
1
+
⋯
+
1
+
na
1
an integer.
16
1
Hide problems
Find the smallest integer n
Let
Q
Q
Q
be a square with side length
6
6
6
. Find the smallest integer
n
n
n
such that in
Q
Q
Q
there exists a set
S
S
S
of
n
n
n
points with the property that any square with side
1
1
1
completely contained in
Q
Q
Q
contains in its interior at least one point from
S
S
S
.
14
1
Hide problems
About n^2+1 intervals - Show that a subset exists
Let
S
S
S
be a set of
n
2
+
1
n^2 + 1
n
2
+
1
closed intervals (
n
n
n
a positive integer). Prove that at least one of the following assertions holds:(i) There exists a subset
S
′
S'
S
′
of
n
+
1
n+1
n
+
1
intervals from
S
S
S
such that the intersection of the intervals in
S
′
S'
S
′
is nonempty.(ii) There exists a subset
S
′
′
S''
S
′′
of
n
+
1
n + 1
n
+
1
intervals from
S
S
S
such that any two of the intervals in
S
′
′
S''
S
′′
are disjoint.
13
1
Hide problems
Interesting problem - Choose no fewer than n/4 squares
The plane is divided into equal squares by parallel lines; i.e., a square net is given. Let
M
M
M
be an arbitrary set of
n
n
n
squares of this net. Prove that it is possible to choose no fewer than
n
/
4
n/4
n
/4
squares of
M
M
M
in such a way that no two of them have a common point.
11
1
Hide problems
Prove that the pyramid is regular
Prove that a pyramid
A
1
A
2
…
A
2
k
+
1
S
A_1A_2 \ldots A_{2k+1}S
A
1
A
2
…
A
2
k
+
1
S
with equal lateral edges and equal space angles between adjacent lateral walls is regular.
9
1
Hide problems
Show that AM ≥ GM ≥ HM! Appeared on ILL 1979 (P9)
The real numbers
α
1
,
α
2
,
α
3
,
…
,
α
n
\alpha_1 , \alpha_2, \alpha_3, \ldots, \alpha_n
α
1
,
α
2
,
α
3
,
…
,
α
n
are positive. Let us denote by
h
=
n
1
/
α
1
+
1
/
α
2
+
⋯
+
1
/
α
n
h = \frac{n}{1/\alpha_1 + 1/\alpha_2 + \cdots + 1/\alpha_n}
h
=
1/
α
1
+
1/
α
2
+
⋯
+
1/
α
n
n
the harmonic mean,
g
=
α
1
α
2
⋯
α
n
n
g=\sqrt[n]{\alpha_1\alpha_2\cdots \alpha_n}
g
=
n
α
1
α
2
⋯
α
n
the geometric mean, and
a
=
α
1
+
α
2
+
⋯
+
α
n
n
a=\frac{\alpha_1+\alpha_2+\cdots + \alpha_n}{n}
a
=
n
α
1
+
α
2
+
⋯
+
α
n
the arithmetic mean. Prove that
h
≤
g
≤
a
h \leq g \leq a
h
≤
g
≤
a
, and that each of the equalities implies the other one.
8
1
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Show that a_n = [a_(n-1)^2/a_(n-2)] + 1 in the sequence
The sequence
(
a
n
)
(a_n)
(
a
n
)
of real numbers is defined as follows: a_1=1, \qquad a_2=2, \text{and} a_n=3a_{n-1}-a_{n-2} , \ \ n \geq 3. Prove that for
n
≥
3
n \geq 3
n
≥
3
,
a
n
=
[
a
n
−
1
2
a
n
−
2
]
+
1
a_n=\left[ \frac{a_{n-1}^2}{a_{n-2}} \right] +1
a
n
=
[
a
n
−
2
a
n
−
1
2
]
+
1
, where
[
x
]
[x]
[
x
]
denotes the integer
p
p
p
such that
p
≤
x
<
p
+
1
p \leq x < p + 1
p
≤
x
<
p
+
1
.
7
1
Hide problems
Find the matrix M with the properties
M
=
(
a
i
,
j
)
,
i
,
j
=
1
,
2
,
3
,
4
M = (a_{i,j} ), \ i, j = 1, 2, 3, 4
M
=
(
a
i
,
j
)
,
i
,
j
=
1
,
2
,
3
,
4
, is a square matrix of order four. Given that: [*] (i) for each
i
=
1
,
2
,
3
,
4
i = 1, 2, 3,4
i
=
1
,
2
,
3
,
4
and for each
k
=
5
,
6
,
7
k = 5, 6, 7
k
=
5
,
6
,
7
,
a
i
,
k
=
a
i
,
k
−
4
;
a_{i,k} = a_{i,k-4};
a
i
,
k
=
a
i
,
k
−
4
;
P
i
=
a
1
,
i
+
a
2
,
i
+
1
+
a
3
,
i
+
2
+
a
4
,
i
+
3
;
P_i = a_{1,}i + a_{2,i+1} + a_{3,i+2} + a_{4,i+3};
P
i
=
a
1
,
i
+
a
2
,
i
+
1
+
a
3
,
i
+
2
+
a
4
,
i
+
3
;
S
i
=
a
4
,
i
+
a
3
,
i
+
1
+
a
2
,
i
+
2
+
a
1
,
i
+
3
;
S_i = a_{4,i }+ a_{3,i+1} + a_{2,i+2} + a_{1,i+3};
S
i
=
a
4
,
i
+
a
3
,
i
+
1
+
a
2
,
i
+
2
+
a
1
,
i
+
3
;
L
i
=
a
i
,
1
+
a
i
,
2
+
a
i
,
3
+
a
i
,
4
;
L_i = a_{i,1} + a_{i,2} + a_{i,3} + a_{i,4};
L
i
=
a
i
,
1
+
a
i
,
2
+
a
i
,
3
+
a
i
,
4
;
C
i
=
a
1
,
i
+
a
2
,
i
+
a
3
,
i
+
a
4
,
i
,
C_i = a_{1,i} + a_{2,i} + a_{3,i} + a_{4,i},
C
i
=
a
1
,
i
+
a
2
,
i
+
a
3
,
i
+
a
4
,
i
,
*
for each
i
,
j
=
1
,
2
,
3
,
4
i, j = 1, 2, 3, 4
i
,
j
=
1
,
2
,
3
,
4
,
P
i
=
P
j
,
S
i
=
S
j
,
L
i
=
L
j
,
C
i
=
C
j
P_i = P_j , S_i = S_j , L_i = L_j , C_i = C_j
P
i
=
P
j
,
S
i
=
S
j
,
L
i
=
L
j
,
C
i
=
C
j
, and
*
a
1
,
1
=
0
,
a
1
,
2
=
7
,
a
2
,
1
=
11
,
a
2
,
3
=
2
a_{1,1} = 0, a_{1,2} = 7, a_{2,1} = 11, a_{2,3} = 2
a
1
,
1
=
0
,
a
1
,
2
=
7
,
a
2
,
1
=
11
,
a
2
,
3
=
2
, and
a
3
,
3
=
15
a_{3,3} = 15
a
3
,
3
=
15
.find the matrix M.
6
1
Hide problems
Trigonometric Equality - IMO LongList 1979 - P6
Prove that
1
2
⋅
4
sin
2
3
6
∘
−
1
=
cos
7
2
∘
\frac 12 \cdot \sqrt{4\sin^2 36^{\circ} - 1}=\cos 72^\circ
2
1
⋅
4
sin
2
3
6
∘
−
1
=
cos
7
2
∘
.
5
1
Hide problems
All naturals not in the form n + √n + 1/2
Describe which positive integers do not belong to the set
E
=
{
⌊
n
+
n
+
1
2
⌋
∣
n
∈
N
}
.
E = \left\{ \lfloor n+ \sqrt n +\frac 12 \rfloor | n \in \mathbb N\right\}.
E
=
{
⌊
n
+
n
+
2
1
⌋
∣
n
∈
N
}
.
3
1
Hide problems
Partition 3D space into 1979 mutually isometric subsets
Is it possible to partition
3
3
3
-dimensional Euclidean space into
1979
1979
1979
mutually isometric subsets?
2
1
Hide problems
Calculate number of 3-element subsets with the property
For a finite set
E
E
E
of cardinality
n
≥
3
n \geq 3
n
≥
3
, let
f
(
n
)
f(n)
f
(
n
)
denote the maximum number of
3
3
3
-element subsets of
E
E
E
, any two of them having exactly one common element. Calculate
f
(
n
)
f(n)
f
(
n
)
.