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International Contests
IMO Longlists
1974 IMO Longlists
1974 IMO Longlists
Part of
IMO Longlists
Subcontests
(40)
52
1
Hide problems
Fox catching a rabbit [ILL 1974]
A fox stands in the centre of the field which has the form of an equilateral triangle, and a rabbit stands at one of its vertices. The fox can move through the whole field, while the rabbit can move only along the border of the field. The maximal speeds of the fox and rabbit are equal to
u
u
u
and
v
v
v
, respectively. Prove that: (a) If
2
u
>
v
2u>v
2
u
>
v
, the fox can catch the rabbit, no matter how the rabbit moves. (b) If
2
u
≤
v
2u\le v
2
u
≤
v
, the rabbit can always run away from the fox.
50
1
Hide problems
Inequality for integers m,n [ILL 1974]
Let
m
m
m
and
n
n
n
be natural numbers with
m
>
n
m>n
m
>
n
. Prove that
2
(
m
−
n
)
2
(
m
2
−
n
2
+
1
)
≥
2
m
2
−
2
m
n
+
1
2(m-n)^2(m^2-n^2+1)\ge 2m^2-2mn+1
2
(
m
−
n
)
2
(
m
2
−
n
2
+
1
)
≥
2
m
2
−
2
mn
+
1
34
1
Hide problems
How many inequivalent diagrams exist? [ILL 1974]
Consider infinite diagrams [asy] import graph; size(90); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; label("
n
00
n
01
n
02
…
n_{00} \ n_{01} \ n_{02} \ldots
n
00
n
01
n
02
…
", (1.14,1.38), SE*lsf); label("
n
10
n
11
n
12
…
n_{10} \ n_{11} \ n_{12} \ldots
n
10
n
11
n
12
…
", (1.2,1.8), SE*lsf); label("
n
20
n
21
n
22
…
n_{20} \ n_{21} \ n_{22} \ldots
n
20
n
21
n
22
…
", (1.2,2.2), SE*lsf); label("\vdots \vdots \qquad \vdots ", (1.32,2.72), SE*lsf); draw((1,1)--(3,1)); draw((1,1)--(1.02,2.62)); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle); [/asy] where all but a finite number of the integers
n
i
j
,
i
=
0
,
1
,
2
,
…
,
j
=
0
,
1
,
2
,
…
,
n_{ij} , i = 0, 1, 2, \ldots, j = 0, 1, 2, \ldots ,
n
ij
,
i
=
0
,
1
,
2
,
…
,
j
=
0
,
1
,
2
,
…
,
are equal to
0
0
0
. Three elements of a diagram are called adjacent if there are integers
i
i
i
and
j
j
j
with
i
≥
0
i \geq 0
i
≥
0
and
j
≥
0
j \geq 0
j
≥
0
such that the three elements are(i)
n
i
j
,
n
i
,
j
+
1
,
n
i
,
j
+
2
,
n_{ij}, n_{i,j+1}, n_{i,j+2},
n
ij
,
n
i
,
j
+
1
,
n
i
,
j
+
2
,
or(ii)
n
i
j
,
n
i
+
1
,
j
,
n
i
+
2
,
j
,
n_{ij}, n_{i+1,j}, n_{i+2,j} ,
n
ij
,
n
i
+
1
,
j
,
n
i
+
2
,
j
,
or(iii)
n
i
+
2
,
j
,
n
i
+
1
,
j
+
1
,
n
i
,
j
+
2
.
n_{i+2,j}, n_{i+1,j+1}, n_{i,j+2}.
n
i
+
2
,
j
,
n
i
+
1
,
j
+
1
,
n
i
,
j
+
2
.
An elementary operation on a diagram is an operation by which three adjacent elements
n
i
j
n_{ij}
n
ij
are changed into
n
i
j
′
n_{ij}'
n
ij
′
in such a way that
∣
n
i
j
−
n
i
j
′
∣
=
1.
|n_{ij}-n_{ij}'|=1.
∣
n
ij
−
n
ij
′
∣
=
1.
Two diagrams are called equivalent if one of them can be changed into the other by a finite sequence of elementary operations. How many inequivalent diagrams exist?
46
1
Hide problems
Half-equilateral triangles are constructed [ILL 1974]
Outside an arbitrary triangle
A
B
C
ABC
A
BC
, triangles
A
D
B
ADB
A
D
B
and
B
C
E
BCE
BCE
are constructed such that
∠
A
D
B
=
∠
B
E
C
=
9
0
∘
\angle ADB=\angle BEC=90^{\circ}
∠
A
D
B
=
∠
BEC
=
9
0
∘
and
∠
D
A
B
=
∠
E
B
C
=
3
0
∘
\angle DAB=\angle EBC=30^{\circ}
∠
D
A
B
=
∠
EBC
=
3
0
∘
. On the segment
A
C
AC
A
C
the point
F
F
F
with
A
F
=
3
F
C
AF=3FC
A
F
=
3
FC
is chosen. Prove that
∠
D
F
E
=
9
0
∘
\angle DFE=90^{\circ}
∠
D
FE
=
9
0
∘
and
∠
F
D
E
=
3
0
∘
\angle FDE=30^{\circ}
∠
F
D
E
=
3
0
∘
.
44
1
Hide problems
Eventually centre of masses coincide [ILL 1974]
We are given
n
n
n
mass points of equal mass in space. We define a sequence of points
O
1
,
O
2
,
O
3
,
…
O_1,O_2,O_3,\ldots
O
1
,
O
2
,
O
3
,
…
as follows:
O
1
O_1
O
1
is an arbitrary point (within the unit distance of at least one of the
n
n
n
points);
O
2
O_2
O
2
is the centre of gravity of all the
n
n
n
given points that are inside the unit sphere centred at
O
1
O_1
O
1
;
O
3
O_3
O
3
is the centre of gravity of all of the
n
n
n
given points that are inside the unit sphere centred at
O
2
O_2
O
2
; etc. Prove that starting from some
m
m
m
, all points
O
m
,
O
m
+
1
,
O
m
+
2
,
…
O_m,O_{m+1},O_{m+2},\ldots
O
m
,
O
m
+
1
,
O
m
+
2
,
…
coincide.
36
1
Hide problems
All binomials are even [ILL 1974]
Consider the binomial coefficients
(
n
k
)
=
n
!
k
!
(
n
−
k
)
!
(
k
=
1
,
2
,
…
n
−
1
)
\binom{n}{k}=\frac{n!}{k!(n-k)!}\ (k=1,2,\ldots n-1)
(
k
n
)
=
k
!
(
n
−
k
)!
n
!
(
k
=
1
,
2
,
…
n
−
1
)
. Determine all positive integers
n
n
n
for which
(
n
1
)
,
(
n
2
)
,
…
,
(
n
n
−
1
)
\binom{n}{1},\binom{n}{2},\ldots ,\binom{n}{n-1}
(
1
n
)
,
(
2
n
)
,
…
,
(
n
−
1
n
)
are all even numbers.
51
1
Hide problems
Prove that there exist two vectors [ILL 1974]
There are
n
n
n
points on a flat piece of paper, any two of them at a distance of at least
2
2
2
from each other. An inattentive pupil spills ink on a part of the paper such that the total area of the damaged part equals
3
2
\frac 32
2
3
. Prove that there exist two vectors of equal length less than
1
1
1
and with their sum having a given direction, such that after a translation by either of these two vectors no points of the given set remain in the damaged area.
49
1
Hide problems
Third degree polynomial with sin roots [ILL 1974]
Determine an equation of third degree with integral coefficients having roots
sin
π
14
,
sin
5
π
14
\sin \frac{\pi}{14}, \sin \frac{5 \pi}{14}
sin
14
π
,
sin
14
5
π
and
sin
−
3
π
14
.
\sin \frac{-3 \pi}{14}.
sin
14
−
3
π
.
47
1
Hide problems
Find the positions of P [ILL 1974]
Given two points
A
,
B
A,B
A
,
B
outside of a given plane
P
,
P,
P
,
find the positions of points
M
M
M
in the plane
P
P
P
for which the ratio
M
A
M
B
\frac{MA}{MB}
MB
M
A
takes a minimum or maximum.
32
1
Hide problems
Familiar inequality.. [ILL 1971]
Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots ,a_n
a
1
,
a
2
,
…
,
a
n
be
n
n
n
real numbers such that
0
<
a
≤
a
k
≤
b
0<a\le a_k\le b
0
<
a
≤
a
k
≤
b
for
k
=
1
,
2
,
…
,
n
k=1,2,\ldots ,n
k
=
1
,
2
,
…
,
n
. If
m
1
=
1
n
(
a
1
+
a
2
+
⋯
+
a
n
)
m_1=\frac{1}{n}(a_1+a_2+\cdots+a_n)
m
1
=
n
1
(
a
1
+
a
2
+
⋯
+
a
n
)
and
m
2
=
1
n
(
a
1
2
+
a
2
2
+
⋯
+
a
n
2
)
m_2=\frac{1}{n}(a_1^2+a_2^2+\cdots + a_n^2)
m
2
=
n
1
(
a
1
2
+
a
2
2
+
⋯
+
a
n
2
)
, prove that
m
2
≤
(
a
+
b
)
2
4
a
b
m
1
2
m_2\le\frac{(a+b)^2}{4ab}m_1^2
m
2
≤
4
ab
(
a
+
b
)
2
m
1
2
and find a necessary and sufficient condition for equality.
43
1
Hide problems
Number does not exceed (n + 1)(n^2 + n + 1) [ILL 1974]
An
(
n
2
+
n
+
1
)
×
(
n
2
+
n
+
1
)
(n^2+n+1) \times (n^2+n+1)
(
n
2
+
n
+
1
)
×
(
n
2
+
n
+
1
)
matrix of zeros and ones is given. If no four ones are vertices of a rectangle, prove that the number of ones does not exceed
(
n
+
1
)
(
n
2
+
n
+
1
)
.
(n + 1)(n^2 + n + 1).
(
n
+
1
)
(
n
2
+
n
+
1
)
.
41
1
Hide problems
Equality with circumradius [ILL 1974]
Through the circumcenter
O
O
O
of an arbitrary acute-angled triangle, chords
A
1
A
2
,
B
1
B
2
,
C
1
C
2
A_1A_2,B_1B_2, C_1C_2
A
1
A
2
,
B
1
B
2
,
C
1
C
2
are drawn parallel to the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
of the triangle respectively. If
R
R
R
is the radius of the circumcircle, prove that
A
1
O
⋅
O
A
2
+
B
1
O
⋅
O
B
2
+
C
1
O
⋅
O
C
2
=
R
2
.
A_1O \cdot OA_2 + B_1O \cdot OB_2 + C_1O \cdot OC_2 = R^2.
A
1
O
⋅
O
A
2
+
B
1
O
⋅
O
B
2
+
C
1
O
⋅
O
C
2
=
R
2
.
28
1
Hide problems
Operation on the set P of partitions of M [ILL 1974]
Let
M
M
M
be a finite set and
P
=
{
M
1
,
M
2
,
…
,
M
l
}
P=\{ M_1,M_2,\ldots ,M_l\}
P
=
{
M
1
,
M
2
,
…
,
M
l
}
a partition of
M
M
M
(i.e.,
⋃
i
=
1
k
M
i
,
M
i
≠
∅
,
M
i
∩
M
j
=
∅
\bigcup_{i=1}^k M_i, M_i\not=\emptyset, M_i\cap M_j =\emptyset
⋃
i
=
1
k
M
i
,
M
i
=
∅
,
M
i
∩
M
j
=
∅
for all
i
,
j
∈
{
1
,
2
,
…
,
k
}
,
i
≠
j
)
i,j\in\{1,2, \ldots ,k\} ,i\not= j)
i
,
j
∈
{
1
,
2
,
…
,
k
}
,
i
=
j
)
. We define the following elementary operation on
P
P
P
: Choose
i
,
j
∈
{
1
,
2
,
…
,
k
}
i,j\in\{1,2,\ldots ,k\}
i
,
j
∈
{
1
,
2
,
…
,
k
}
, such that
i
=
j
i=j
i
=
j
and
M
i
M_i
M
i
has a elements and
M
j
M_j
M
j
has
b
b
b
elements such that
a
≥
b
a\ge b
a
≥
b
. Then take
b
b
b
elements from
M
i
M_i
M
i
and place them into
M
j
M_j
M
j
, i.e.,
M
j
M_j
M
j
becomes the union of itself and a
b
b
b
-element subset of
M
i
M_i
M
i
, while the same subset is subtracted from
M
i
M_i
M
i
(if
a
=
b
a=b
a
=
b
,
M
i
M_i
M
i
is thus removed from the partition). Let a finite set
M
M
M
be given. Prove that the property “for every partition
P
P
P
of
M
M
M
there exists a sequence
P
=
P
1
,
P
2
,
…
,
P
r
P=P_1,P_2,\ldots ,P_r
P
=
P
1
,
P
2
,
…
,
P
r
such that
P
i
+
1
P_{i+1}
P
i
+
1
is obtained from
P
i
P_i
P
i
by an elementary operation and
P
r
=
{
M
}
P_r=\{M\}
P
r
=
{
M
}
” is equivalent to “the number of elements of
M
M
M
is a power of
2
2
2
.”
39
1
Hide problems
Determine all complex numbers x [ILL 1974]
Let
n
n
n
be a positive integer,
n
≥
2
n \geq 2
n
≥
2
, and consider the polynomial equation
x
n
−
x
n
−
2
−
x
+
2
=
0.
x^n - x^{n-2} - x + 2 = 0.
x
n
−
x
n
−
2
−
x
+
2
=
0.
For each
n
,
n,
n
,
determine all complex numbers
x
x
x
that satisfy the equation and have modulus
∣
x
∣
=
1.
|x| = 1.
∣
x
∣
=
1.
37
1
Hide problems
For what values of theta will the ball strike? [ILL 1974]
Let
a
,
b
a, b
a
,
b
, and
c
c
c
denote the three sides of a billiard table in the shape of an equilateral triangle. A ball is placed at the midpoint of side
a
a
a
and then propelled toward side
b
b
b
with direction defined by the angle
θ
\theta
θ
. For what values of
θ
\theta
θ
will the ball strike the sides
b
,
c
,
a
b, c, a
b
,
c
,
a
in that order?
35
1
Hide problems
Determine the maximum value of b - a [ILL 1974]
If
p
p
p
and
q
q
q
are distinct prime numbers, then there are integers
x
0
x_0
x
0
and
y
0
y_0
y
0
such that
1
=
p
x
0
+
q
y
0
.
1 = px_0 + qy_0.
1
=
p
x
0
+
q
y
0
.
Determine the maximum value of
b
−
a
b - a
b
−
a
, where
a
a
a
and
b
b
b
are positive integers with the following property: If
a
≤
t
≤
b
a \leq t \leq b
a
≤
t
≤
b
, and
t
t
t
is an integer, then there are integers
x
x
x
and
y
y
y
with
0
≤
x
≤
q
−
1
0 \leq x \leq q - 1
0
≤
x
≤
q
−
1
and
0
≤
y
≤
p
−
1
0 \leq y \leq p - 1
0
≤
y
≤
p
−
1
such that
t
=
p
x
+
q
y
.
t = px + qy.
t
=
p
x
+
q
y
.
33
1
Hide problems
Prove that there exists a real number A [ILL 1974]
Let a be a real number such that
0
<
a
<
1
0 < a < 1
0
<
a
<
1
, and let
n
n
n
be a positive integer. Define the sequence
a
0
,
a
1
,
a
2
,
…
,
a
n
a_0, a_1, a_2, \ldots, a_n
a
0
,
a
1
,
a
2
,
…
,
a
n
an recursively by a_0 = a, a_{k+1} = a_k +\frac 1n a_k^2 \text{ for } k = 0, 1, \ldots, n - 1. Prove that there exists a real number
A
A
A
, depending on
a
a
a
but independent of
n
n
n
, such that
0
<
n
(
A
−
a
n
)
<
A
3
.
0 < n(A - a_n) < A^3.
0
<
n
(
A
−
a
n
)
<
A
3
.
31
1
Hide problems
n variables inequality with α, λ [ILL 1974]
Let
y
α
=
∑
i
=
1
n
x
i
α
y^{\alpha}=\sum_{i=1}^n x_i^{\alpha}
y
α
=
∑
i
=
1
n
x
i
α
where
α
≠
0
,
y
>
0
,
x
i
>
0
\alpha \neq 0, y > 0, x_i > 0
α
=
0
,
y
>
0
,
x
i
>
0
are real numbers, and let
λ
≠
α
\lambda \neq \alpha
λ
=
α
be a real number. Prove that
y
λ
>
∑
i
=
1
n
x
i
λ
y^{\lambda} > \sum_{i=1}^n x_i^{\lambda}
y
λ
>
∑
i
=
1
n
x
i
λ
if
α
(
λ
−
α
)
>
0
,
\alpha (\lambda - \alpha) > 0,
α
(
λ
−
α
)
>
0
,
and
y
λ
<
∑
i
=
1
n
x
i
λ
y^{\lambda} < \sum_{i=1}^n x_i^{\lambda}
y
λ
<
∑
i
=
1
n
x
i
λ
if
α
(
λ
−
α
)
<
0.
\alpha (\lambda - \alpha) < 0.
α
(
λ
−
α
)
<
0.
29
1
Hide problems
Prove that A, B, C, D lie on the same line [ILL 1974]
Let
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
be points in space. If for every point
M
M
M
on the segment
A
B
AB
A
B
the sum
S
A
M
C
+
S
C
M
D
+
S
D
M
B
S_{AMC}+S_{CMD}+S_{DMB}
S
A
MC
+
S
CM
D
+
S
D
MB
Is constant show that the points
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
lie in the same plane.[hide="Note."] Note.
S
X
S_X
S
X
denotes the area of triangle
X
.
X.
X
.
26
1
Hide problems
Partitions of M inequality [ILL 1974]
Let
g
(
k
)
g(k)
g
(
k
)
be the number of partitions of a
k
k
k
-element set
M
M
M
, i.e., the number of families
{
A
1
,
A
2
,
…
,
A
s
}
\{ A_1,A_2,\ldots ,A_s\}
{
A
1
,
A
2
,
…
,
A
s
}
of nonempty subsets of
M
M
M
such that
A
i
∩
A
j
=
∅
A_i\cap A_j=\emptyset
A
i
∩
A
j
=
∅
for
i
≠
j
i\not= j
i
=
j
and
⋃
i
=
1
n
A
i
=
M
\bigcup_{i=1}^n A_i=M
⋃
i
=
1
n
A
i
=
M
. Prove that, for every
n
n
n
,
n
n
≤
g
(
2
n
)
≤
(
2
n
)
2
n
n^n\le g(2n)\le (2n)^{2n}
n
n
≤
g
(
2
n
)
≤
(
2
n
)
2
n
27
1
Hide problems
Find the locus of the centroid of the triangle [ILL 1974]
Let
C
1
C_1
C
1
and
C
2
C_2
C
2
be circles in the same plane,
P
1
P_1
P
1
and
P
2
P_2
P
2
arbitrary points on
C
1
C_1
C
1
and
C
2
C_2
C
2
respectively, and
Q
Q
Q
the midpoint of segment
P
1
P
2
.
P_1P_2.
P
1
P
2
.
Find the locus of points
Q
Q
Q
as
P
1
P_1
P
1
and
P
2
P_2
P
2
go through all possible positions. Alternative version. Let
C
1
,
C
2
,
C
3
C_1, C_2, C_3
C
1
,
C
2
,
C
3
be three circles in the same plane. Find the locus of the centroid of triangle
P
1
P
2
P
3
P_1P_2P_3
P
1
P
2
P
3
as
P
1
,
P
2
,
P_1, P_2,
P
1
,
P
2
,
and
P
3
P_3
P
3
go through all possible positions on
C
1
,
C
2
C_1, C_2
C
1
,
C
2
, and
C
3
C_3
C
3
respectively.
20
1
Hide problems
Finding n such that sum of a_i^-2 is 1 [ILL 1971]
For which natural numbers
n
n
n
do there exist
n
n
n
natural numbers
a
i
(
1
≤
i
≤
n
)
a_i\ (1\le i\le n)
a
i
(
1
≤
i
≤
n
)
such that
∑
i
=
1
n
a
i
−
2
=
1
\sum_{i=1}^n a_i^{-2}=1
∑
i
=
1
n
a
i
−
2
=
1
?
25
1
Hide problems
Prove that there exists an integer k [ILL 1974]
Let
f
:
R
→
R
f : \mathbb R \to \mathbb R
f
:
R
→
R
be of the form
f
(
x
)
=
x
+
ϵ
sin
x
,
f(x) = x + \epsilon \sin x,
f
(
x
)
=
x
+
ϵ
sin
x
,
where
0
<
∣
ϵ
∣
≤
1.
0 < |\epsilon| \leq 1.
0
<
∣
ϵ
∣
≤
1.
Define for any
x
∈
R
,
x \in \mathbb R,
x
∈
R
,
x
n
=
f
o
…
o
f
⏟
n
times
(
x
)
.
x_n=\underbrace{f \ o \ \ldots \ o \ f}_{n \text{ times}} (x).
x
n
=
n
times
f
o
…
o
f
(
x
)
.
Show that for every
x
∈
R
x \in \mathbb R
x
∈
R
there exists an integer
k
k
k
such that
lim
n
→
∞
x
n
=
k
π
.
\lim_{n\to \infty } x_n = k\pi.
lim
n
→
∞
x
n
=
kπ
.
21
1
Hide problems
Prove that M =ℤ+ [ILL 1974]
Let
M
M
M
be a nonempty subset of
Z
+
\mathbb Z^+
Z
+
such that for every element
x
x
x
in
M
,
M,
M
,
the numbers
4
x
4x
4
x
and
⌊
x
⌋
\lfloor \sqrt x \rfloor
⌊
x
⌋
also belong to
M
.
M.
M
.
Prove that
M
=
Z
+
.
M = \mathbb Z^+.
M
=
Z
+
.
16
1
Hide problems
Dealers and receivers - 2n cards [ILL 1974]
A pack of
2
n
2n
2
n
cards contains
n
n
n
different pairs of cards. Each pair consists of two identical cards, either of which is called the twin of the other. A game is played between two players
A
A
A
and
B
B
B
. A third person called the dealer shuffles the pack and deals the cards one by one face upward onto the table. One of the players, called the receiver, takes the card dealt, provided he does not have already its twin. If he does already have the twin, his opponent takes the dealt card and becomes the receiver.
A
A
A
is initially the receiver and takes the first card dealt. The player who first obtains a complete set of
n
n
n
different cards wins the game. What fraction of all possible arrangements of the pack lead to
A
A
A
winning? Prove the correctness of your answer.
19
1
Hide problems
A set of 3n equal circles [ILL 1974]
Prove that there exists, for
n
≥
4
n \geq 4
n
≥
4
, a set
S
S
S
of
3
n
3n
3
n
equal circles in space that can be partitioned into three subsets
s
5
,
s
4
s_5, s_4
s
5
,
s
4
, and
s
3
s_3
s
3
, each containing
n
n
n
circles, such that each circle in
s
r
s_r
s
r
touches exactly
r
r
r
circles in
S
.
S.
S
.
14
1
Hide problems
Sum of a_i is 1 inequality [ILL 1974]
Let
n
n
n
and
k
k
k
be natural numbers and
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots ,a_n
a
1
,
a
2
,
…
,
a
n
be positive real numbers satisfying
a
1
+
a
2
+
⋯
+
a
n
=
1
a_1+a_2+\cdots +a_n=1
a
1
+
a
2
+
⋯
+
a
n
=
1
. Prove that
1
a
1
k
+
1
a
2
k
+
⋯
+
1
a
n
k
≥
n
k
+
1
.
\dfrac {1} {a_1^{k}}+\dfrac {1} {a_2^{k}}+\cdots +\dfrac {1} {a_n^{k}} \ge n^{k+1}.
a
1
k
1
+
a
2
k
1
+
⋯
+
a
n
k
1
≥
n
k
+
1
.
17
1
Hide problems
Show that there exists a set of circles [ILL 1974]
Show that there exists a set
S
S
S
of
15
15
15
distinct circles on the surface of a sphere, all having the same radius and such that
5
5
5
touch exactly
5
5
5
others,
5
5
5
touch exactly
4
4
4
others, and
5
5
5
touch exactly
3
3
3
others.[General Problem: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=384764]
12
1
Hide problems
Construct the parallelogram [ILL 1974]
A circle
K
K
K
with radius
r
r
r
, a point
D
D
D
on
K
K
K
, and a convex angle with vertex
S
S
S
and rays
a
a
a
and
b
b
b
are given in the plane. Construct a parallelogram
A
B
C
D
ABCD
A
BC
D
such that
A
A
A
and
B
B
B
lie on
a
a
a
and
b
b
b
respectively,
S
A
+
S
B
=
r
SA+SB=r
S
A
+
SB
=
r
, and
C
C
C
lies on
K
K
K
.
13
1
Hide problems
Prove that 2^{147}-1 is divisible 343 [ILL 1974]
Prove that
2
147
−
1
2^{147} - 1
2
147
−
1
is divisible by
343.
343.
343.
11
1
Hide problems
Construct an equilateral triangle [ILL 1974]
Given a line
p
p
p
and a triangle
Δ
\Delta
Δ
in the plane, construct an equilateral triangle one of whose vertices lies on the line
p
p
p
, while the other two halve the perimeter of
Δ
.
\Delta.
Δ.
9
1
Hide problems
Solve the system of equations [ILL 1974]
Solve the following system of linear equations with unknown
x
1
,
x
2
…
,
x
n
(
n
≥
2
)
x_1,x_2 \ldots, x_n \ (n \geq 2)
x
1
,
x
2
…
,
x
n
(
n
≥
2
)
and parameters
c
1
,
c
2
,
…
,
c
n
:
c_1,c_2, \ldots , c_n:
c
1
,
c
2
,
…
,
c
n
:
2
x
1
−
x
2
=
c
1
;
2x_1 -x_2 = c_1;
2
x
1
−
x
2
=
c
1
;
−
x
1
+
2
x
2
−
x
3
=
c
2
;
-x_1 +2x_2 -x_3 = c_2;
−
x
1
+
2
x
2
−
x
3
=
c
2
;
−
x
2
+
2
x
3
−
x
4
=
c
3
;
-x_2 +2x_3 -x_4 = c_3;
−
x
2
+
2
x
3
−
x
4
=
c
3
;
⋯
⋯
⋯
\cdots \qquad \cdots \qquad \cdots \qquad
⋯
⋯
⋯
−
x
n
−
2
+
2
x
n
−
1
−
x
n
=
c
n
−
1
;
-x_{n-2} +2x_{n-1} -x_n = c_{n-1};
−
x
n
−
2
+
2
x
n
−
1
−
x
n
=
c
n
−
1
;
−
x
n
−
1
+
2
x
n
=
c
n
.
-x_{n-1} +2x_n = c_n.
−
x
n
−
1
+
2
x
n
=
c
n
.
10
1
Hide problems
Triangles, Octagons, 11-gons and 16-gons [ILL 1974]
A regular octagon
P
P
P
is given whose incircle
k
k
k
has diameter
1
1
1
. About
k
k
k
is circumscribed a regular
16
16
16
-gon, which is also inscribed in
P
P
P
, cutting from
P
P
P
eight isosceles triangles. To the octagon
P
P
P
, three of these triangles are added so that exactly two of them are adjacent and no two of them are opposite to each other. Every
11
11
11
-gon so obtained is said to be
P
′
P'
P
′
. Prove the following statement: Given a finite set
M
M
M
of points lying in
P
P
P
such that every two points of this set have a distance not exceeding
1
1
1
, one of the
11
11
11
-gons
P
′
P'
P
′
contains all of
M
M
M
.
7
1
Hide problems
The product is integer [ILL 1974]
Let
p
p
p
be a prime number and
n
n
n
a positive integer. Prove that the product
N
=
1
p
n
2
∏
i
=
1
;
2
∤
i
2
n
−
1
[
(
(
p
−
1
)
!
)
(
p
2
i
p
i
)
]
{N=\frac{1}{p^{n^2}}} \prod_{i=1;2 \nmid i}^{2n-1} \biggl[ \left( (p-1)! \right) \binom{p^2 i}{pi}\biggr]
N
=
p
n
2
1
i
=
1
;
2
∤
i
∏
2
n
−
1
[
(
(
p
−
1
)!
)
(
p
i
p
2
i
)
]
Is a positive integer that is not divisible by
p
.
p.
p
.
5
1
Hide problems
Geometric Inequality [ILL 1974]
A straight cone is given inside a rectangular parallelepiped
B
B
B
, with the apex at one of the vertices, say
T
T
T
, of the parallelepiped, and the base touching the three faces opposite to
T
.
T .
T
.
Its axis lies at the long diagonal through
T
.
T.
T
.
If
V
1
V_1
V
1
and
V
2
V_2
V
2
are the volumes of the cone and the parallelepiped respectively, prove that
V
1
≤
3
π
V
2
27
.
V_1 \leq \frac{\sqrt 3 \pi V_2}{27}.
V
1
≤
27
3
π
V
2
.
3
1
Hide problems
Parallelograms have two vertices in common [ILL 1974]
Let
A
B
C
D
ABCD
A
BC
D
be an arbitrary quadrilateral. Let squares
A
B
B
1
A
2
,
B
C
C
1
B
2
,
C
D
D
1
C
2
,
D
A
A
1
D
2
ABB_1A_2, BCC_1B_2, CDD_1C_2, DAA_1D_2
A
B
B
1
A
2
,
BC
C
1
B
2
,
C
D
D
1
C
2
,
D
A
A
1
D
2
be constructed in the exterior of the quadrilateral. Furthermore, let
A
A
1
P
A
2
AA_1PA_2
A
A
1
P
A
2
and
C
C
1
Q
C
2
CC_1QC_2
C
C
1
Q
C
2
be parallelograms. For any arbitrary point
P
P
P
in the interior of
A
B
C
D
ABCD
A
BC
D
, parallelograms
R
A
S
C
RASC
R
A
SC
and
R
P
T
Q
RPTQ
RPTQ
are constructed. Prove that these two parallelograms have two vertices in common.
6
1
Hide problems
xy(x+y) cannot be a cube [ILL 1974]
Prove that the product of two natural numbers with their sum cannot be the third power of a natural number.
4
1
Hide problems
Lines from excentres are concurrent [ILL 1974]
Let
K
a
,
K
b
,
K
c
K_a,K_b,K_c
K
a
,
K
b
,
K
c
with centres
O
a
,
O
b
,
O
c
O_a,O_b,O_c
O
a
,
O
b
,
O
c
be the excircles of a triangle
A
B
C
ABC
A
BC
, touching the interiors of the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at points
T
a
,
T
b
,
T
c
T_a,T_b,T_c
T
a
,
T
b
,
T
c
respectively. Prove that the lines
O
a
T
a
,
O
b
T
b
,
O
c
T
c
O_aT_a,O_bT_b,O_cT_c
O
a
T
a
,
O
b
T
b
,
O
c
T
c
are concurrent in a point
P
P
P
for which
P
O
a
=
P
O
b
=
P
O
c
=
2
R
PO_a=PO_b=PO_c=2R
P
O
a
=
P
O
b
=
P
O
c
=
2
R
holds, where
R
R
R
denotes the circumradius of
A
B
C
ABC
A
BC
. Also prove that the circumcentre
O
O
O
of
A
B
C
ABC
A
BC
is the midpoint of the segment
P
I
PI
P
I
, where
I
I
I
is the incentre of
A
B
C
ABC
A
BC
.
2
1
Hide problems
Primes p dividing F_p-1 [ILL 1974]
Let
u
n
{u_n}
u
n
be the Fibonacci sequence, i.e.,
u
0
=
0
,
u
1
=
1
,
u
n
=
u
n
−
1
+
u
n
−
2
u_0=0,u_1=1,u_n=u_{n-1}+u_{n-2}
u
0
=
0
,
u
1
=
1
,
u
n
=
u
n
−
1
+
u
n
−
2
for
n
>
1
n>1
n
>
1
. Prove that there exist infinitely many prime numbers
p
p
p
that divide
u
p
−
1
u_{p-1}
u
p
−
1
.
38
1
Hide problems
Find number of squares and rectangles - [IMO LongList 1971]
The points
S
(
i
,
j
)
S(i, j)
S
(
i
,
j
)
with integer Cartesian coordinates
0
<
i
≤
n
,
0
<
j
≤
m
,
m
≤
n
0 < i \leq n, 0 < j \leq m, m \leq n
0
<
i
≤
n
,
0
<
j
≤
m
,
m
≤
n
, form a lattice. Find the number of:(a) rectangles with vertices on the lattice and sides parallel to the coordinate axes;(b) squares with vertices on the lattice and sides parallel to the coordinate axes;(c) squares in total, with vertices on the lattice.