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International Contests
IMO Longlists
1971 IMO Longlists
1971 IMO Longlists
Part of
IMO Longlists
Subcontests
(36)
29
1
Hide problems
Radii of circles tangent to incircle of rhombus [ILL 1971]
A rhombus with its incircle is given. At each vertex of the rhombus a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii
r
1
,
r
2
r_1,r_2
r
1
,
r
2
, while the incircle has radius
r
r
r
. Given that
r
1
r_1
r
1
and
r
2
r_2
r
2
are natural numbers and that
r
1
r
2
=
r
r_1r_2=r
r
1
r
2
=
r
, find
r
1
,
r
2
,
r_1,r_2,
r
1
,
r
2
,
and
r
r
r
.
30
1
Hide problems
There cannot be 5 distinct solutions [ILL 1971]
Prove that the system of equations
2
y
z
+
x
−
y
−
z
=
a
,
2
x
z
−
x
+
y
−
z
=
a
,
2
x
y
−
x
−
y
+
z
=
a
,
2yz+x-y-z=a,\\ 2xz-x+y-z=a,\\ 2xy-x-y+z=a,
2
yz
+
x
−
y
−
z
=
a
,
2
x
z
−
x
+
y
−
z
=
a
,
2
x
y
−
x
−
y
+
z
=
a
,
a
a
a
being a parameter, cannot have five distinct solutions. For what values of
a
a
a
does this system have four distinct integer solutions?
32
1
Hide problems
Geometry... and limits [ILL 1971]
Two half-lines
a
a
a
and
b
b
b
, with the common endpoint
O
O
O
, make an acute angle
α
\alpha
α
. Let
A
A
A
on
a
a
a
and
B
B
B
on
b
b
b
be points such that
O
A
=
O
B
OA=OB
O
A
=
OB
, and let
b
b
b
be the line through
A
A
A
parallel to
b
b
b
. Let
β
\beta
β
be the circle with centre
B
B
B
and radius
B
O
BO
BO
. We construct a sequence of half-lines
c
1
,
c
2
,
c
3
,
…
c_1,c_2,c_3,\ldots
c
1
,
c
2
,
c
3
,
…
, all lying inside the angle
α
\alpha
α
, in the following manner: (i)
c
i
c_i
c
i
is given arbitrarily; (ii) for every natural number
k
k
k
, the circle
β
\beta
β
intercepts on
c
k
c_k
c
k
a segment that is of the same length as the segment cut on
b
′
b'
b
′
by
a
a
a
and
c
k
+
1
c_{k+1}
c
k
+
1
. Prove that the angle determined by the lines
c
k
c_k
c
k
and
b
b
b
has a limit as
k
k
k
tends to infinity and find that limit.
33
1
Hide problems
Paths in a 2n x 2n grid [ILL 1971]
A square
2
n
×
2
n
2n\times 2n
2
n
×
2
n
grid is given. Let us consider all possible paths along grid lines, going from the centre of the grid to the border, such that (1) no point of the grid is reached more than once, and (2) each of the squares homothetic to the grid having its centre at the grid centre is passed through only once. (a) Prove that the number of all such paths is equal to
4
∏
i
=
2
n
(
16
i
−
9
)
4\prod_{i=2}^n(16i-9)
4
∏
i
=
2
n
(
16
i
−
9
)
. (b) Find the number of pairs of such paths that divide the grid into two congruent figures. (c) How many quadruples of such paths are there that divide the grid into four congruent parts?
53
1
Hide problems
The limit of the multiplicity of p [ILL 1971]
Denote by
x
n
(
p
)
x_n(p)
x
n
(
p
)
the multiplicity of the prime
p
p
p
in the canonical representation of the number
n
!
n!
n
!
as a product of primes. Prove that
x
n
(
p
)
n
<
1
p
−
1
\frac{x_n(p)}{n}<\frac{1}{p-1}
n
x
n
(
p
)
<
p
−
1
1
and
lim
n
→
∞
x
n
(
p
)
n
=
1
p
−
1
\lim_{n \to \infty}\frac{x_n(p)}{n}=\frac{1}{p-1}
lim
n
→
∞
n
x
n
(
p
)
=
p
−
1
1
.
37
1
Hide problems
α and β are two coverings of S [ILL 1971]
Let
S
S
S
be a circle, and
α
=
{
A
1
,
…
,
A
n
}
\alpha =\{A_1,\ldots ,A_n\}
α
=
{
A
1
,
…
,
A
n
}
a family of open arcs in
S
S
S
. Let
N
(
α
)
=
n
N(\alpha )=n
N
(
α
)
=
n
denote the number of elements in
α
\alpha
α
. We say that
α
\alpha
α
is a covering of
S
S
S
if
⋃
k
=
1
n
A
k
⊃
S
\bigcup_{k=1}^n A_k\supset S
⋃
k
=
1
n
A
k
⊃
S
. Let
α
=
{
A
1
,
…
,
A
n
}
\alpha=\{A_1,\ldots ,A_n\}
α
=
{
A
1
,
…
,
A
n
}
and
β
=
{
B
1
,
…
,
B
m
}
\beta =\{B_1,\ldots ,B_m\}
β
=
{
B
1
,
…
,
B
m
}
be two coverings of
S
S
S
. Show that we can choose from the family of all sets
A
i
∩
B
j
,
i
=
1
,
2
,
…
,
n
,
j
=
1
,
2
,
…
,
m
,
A_i\cap B_j,\ i=1,2,\ldots ,n,\ j=1, 2,\ldots ,m,
A
i
∩
B
j
,
i
=
1
,
2
,
…
,
n
,
j
=
1
,
2
,
…
,
m
,
a covering
γ
\gamma
γ
of
S
S
S
such that
N
(
γ
)
≤
N
(
α
)
+
N
(
β
)
N(\gamma )\le N(\alpha)+N(\beta)
N
(
γ
)
≤
N
(
α
)
+
N
(
β
)
.
26
1
Hide problems
Set of rectangles - [IMO LongList 1971]
An infinite set of rectangles in the Cartesian coordinate plane is given. The vertices of each of these rectangles have coordinates
(
0
,
0
)
,
(
p
,
0
)
,
(
p
,
q
)
,
(
0
,
q
)
(0, 0), (p, 0), (p, q), (0, q)
(
0
,
0
)
,
(
p
,
0
)
,
(
p
,
q
)
,
(
0
,
q
)
for some positive integers
p
,
q
p, q
p
,
q
. Show that there must exist two among them one of which is entirely contained in the other.
25
1
Hide problems
PQR is equilateral - [IMO LongList 1971]
Let
A
B
C
,
A
A
1
A
2
,
B
B
1
B
2
,
C
C
1
C
2
ABC,AA_1A_2,BB_1B_2, CC_1C_2
A
BC
,
A
A
1
A
2
,
B
B
1
B
2
,
C
C
1
C
2
be four equilateral triangles in the plane satisfying only that they are all positively oriented (i.e., in the counterclockwise direction). Denote the midpoints of the segments
A
2
B
1
,
B
2
C
1
,
C
2
A
1
A_2B_1,B_2C_1, C_2A_1
A
2
B
1
,
B
2
C
1
,
C
2
A
1
by
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
in this order. Prove that the triangle
P
Q
R
PQR
PQR
is equilateral.
38
1
Hide problems
The ratio AB·BC·CA/R inequality [ILL 1971]
Let
A
,
B
,
C
A,B,C
A
,
B
,
C
be three points with integer coordinates in the plane and
K
K
K
a circle with radius
R
R
R
passing through
A
,
B
,
C
A,B,C
A
,
B
,
C
. Show that
A
B
⋅
B
C
⋅
C
A
≥
2
R
AB\cdot BC\cdot CA\ge 2R
A
B
⋅
BC
⋅
C
A
≥
2
R
, and if the centre of
K
K
K
is in the origin of the coordinates, show that
A
B
⋅
B
C
⋅
C
A
≥
4
R
AB\cdot BC\cdot CA\ge 4R
A
B
⋅
BC
⋅
C
A
≥
4
R
.
40
1
Hide problems
The maximum of S for the subset σ [ILL 1971]
Consider the set of grid points
(
m
,
n
)
(m,n)
(
m
,
n
)
in the plane,
m
,
n
m,n
m
,
n
integers. Let
σ
\sigma
σ
be a finite subset and define
S
(
σ
)
=
∑
(
m
,
n
)
∈
σ
(
100
−
∣
m
∣
−
∣
n
∣
)
S(\sigma)=\sum_{(m,n)\in\sigma}(100-|m|-|n|)
S
(
σ
)
=
(
m
,
n
)
∈
σ
∑
(
100
−
∣
m
∣
−
∣
n
∣
)
Find the maximum of
S
S
S
, taken over the set of all such subsets
σ
\sigma
σ
.
24
1
Hide problems
Prove that the triangle is right angled - [ILL 1971]
Let
A
,
B
,
A, B,
A
,
B
,
and
C
C
C
denote the angles of a triangle. If
sin
2
A
+
sin
2
B
+
sin
2
C
=
2
\sin^2 A + \sin^2 B + \sin^2 C = 2
sin
2
A
+
sin
2
B
+
sin
2
C
=
2
, prove that the triangle is right-angled.
41
1
Hide problems
Area of M(L) <= Area of M(L*) [ILL 1971]
Let
L
i
,
i
=
1
,
2
,
3
L_i,\ i=1,2,3
L
i
,
i
=
1
,
2
,
3
, be line segments on the sides of an equilateral triangle, one segment on each side, with lengths
l
i
,
i
=
1
,
2
,
3
l_i,\ i=1,2,3
l
i
,
i
=
1
,
2
,
3
. By
L
i
∗
L_i^{\ast}
L
i
∗
we denote the segment of length
l
i
l_i
l
i
with its midpoint on the midpoint of the corresponding side of the triangle. Let
M
(
L
)
M(L)
M
(
L
)
be the set of points in the plane whose orthogonal projections on the sides of the triangle are in
L
1
,
L
2
L_1,L_2
L
1
,
L
2
, and
L
3
L_3
L
3
, respectively;
M
(
L
∗
)
M(L^{\ast})
M
(
L
∗
)
is defined correspondingly. Prove that if
l
1
≥
l
2
+
l
3
l_1\ge l_2+l_3
l
1
≥
l
2
+
l
3
, we have that the area of
M
(
L
)
M(L)
M
(
L
)
is less than or equal to the area of
M
(
L
∗
)
M(L^{\ast})
M
(
L
∗
)
.
19
1
Hide problems
Prove that L1=L2=L3 - [IMO LongList 1971]
In a triangle
P
1
P
2
P
3
P_1P_2P_3
P
1
P
2
P
3
let
P
i
Q
i
P_iQ_i
P
i
Q
i
be the altitude from
P
i
P_i
P
i
for
i
=
1
,
2
,
3
i = 1, 2,3
i
=
1
,
2
,
3
(
Q
i
Q_i
Q
i
being the foot of the altitude). The circle with diameter
P
i
Q
i
P_iQ_i
P
i
Q
i
meets the two corresponding sides at two points different from
P
i
.
P_i.
P
i
.
Denote the length of the segment whose endpoints are these two points by
l
i
.
l_i.
l
i
.
Prove that
l
1
=
l
2
=
l
3
.
l_1 = l_2 = l_3.
l
1
=
l
2
=
l
3
.
42
1
Hide problems
I know it's only Power Mean but bear with me.. [ILL 1971]
Show that for nonnegative real numbers
a
,
b
a,b
a
,
b
and integers
n
≥
2
n\ge 2
n
≥
2
,
a
n
+
b
n
2
≥
(
a
+
b
2
)
n
\frac{a^n+b^n}{2}\ge\left(\frac{a+b}{2}\right)^n
2
a
n
+
b
n
≥
(
2
a
+
b
)
n
When does equality hold?
44
1
Hide problems
Let's use nu (n) instead of pi (n)! [ILL 1971]
Let
m
m
m
and
n
n
n
denote integers greater than
1
1
1
, and let
ν
(
n
)
\nu (n)
ν
(
n
)
be the number of primes less than or equal to
n
n
n
. Show that if the equation
n
ν
(
n
)
=
m
\frac{n}{\nu(n)}=m
ν
(
n
)
n
=
m
has a solution, then so does the equation
n
ν
(
n
)
=
m
−
1
\frac{n}{\nu(n)}=m-1
ν
(
n
)
n
=
m
−
1
.
23
1
Hide problems
Solve the equation x^2+y^2=(x-y)^3 - [IMO LongList 1971]
Find all integer solutions of the equation
x
2
+
y
2
=
(
x
−
y
)
3
.
x^2+y^2=(x-y)^3.
x
2
+
y
2
=
(
x
−
y
)
3
.
47
1
Hide problems
Is x_50000<1? [ILL 1971]
A sequence of real numbers
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots ,x_n
x
1
,
x
2
,
…
,
x
n
is given such that
x
i
+
1
=
x
i
+
1
30000
1
−
x
i
2
,
i
=
1
,
2
,
…
,
x_{i+1}=x_i+\frac{1}{30000}\sqrt{1-x_i^2},\ i=1,2,\ldots ,
x
i
+
1
=
x
i
+
30000
1
1
−
x
i
2
,
i
=
1
,
2
,
…
,
and
x
1
=
0
x_1=0
x
1
=
0
. Can
n
n
n
be equal to
50000
50000
50000
if
x
n
<
1
x_n<1
x
n
<
1
?
22
1
Hide problems
The old +1 and -1 problem on the board - [IMO LongList 1971]
We are given an
n
×
n
n \times n
n
×
n
board, where
n
n
n
is an odd number. In each cell of the board either
+
1
+1
+
1
or
−
1
-1
−
1
is written. Let
a
k
a_k
a
k
and
b
k
b_k
b
k
denote them products of numbers in the
k
t
h
k^{th}
k
t
h
row and in the
k
t
h
k^{th}
k
t
h
column respectively. Prove that the sum
a
1
+
a
2
+
⋯
+
a
n
+
b
1
+
b
2
+
⋯
+
b
n
a_1 +a_2 +\cdots+a_n +b_1 +b_2 +\cdots+b_n
a
1
+
a
2
+
⋯
+
a
n
+
b
1
+
b
2
+
⋯
+
b
n
cannot be equal to zero.
20
1
Hide problems
The product is equal two 2 - [IMO LongList 1971]
Let
M
M
M
be the circumcenter of a triangle
A
B
C
.
ABC.
A
BC
.
The line through
M
M
M
perpendicular to
C
M
CM
CM
meets the lines
C
A
CA
C
A
and
C
B
CB
CB
at
Q
Q
Q
and
P
,
P,
P
,
respectively. Prove that
C
P
‾
C
M
‾
⋅
C
Q
‾
C
M
‾
⋅
A
B
‾
P
Q
‾
=
2.
\frac{\overline{CP}}{\overline{CM}} \cdot \frac{\overline{CQ}}{\overline{CM}}\cdot \frac{\overline{AB}}{\overline{PQ}}= 2.
CM
CP
⋅
CM
CQ
⋅
PQ
A
B
=
2.
48
1
Hide problems
O in ABCD making angles of 30 and 45 degrees [ILL 1971]
The diagonals of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at a point
O
O
O
. Find all angles of this quadrilateral if
∡
O
B
A
=
3
0
∘
,
∡
O
C
B
=
4
5
∘
,
∡
O
D
C
=
4
5
∘
\measuredangle OBA=30^{\circ},\measuredangle OCB=45^{\circ},\measuredangle ODC=45^{\circ}
∡
OB
A
=
3
0
∘
,
∡
OCB
=
4
5
∘
,
∡
O
D
C
=
4
5
∘
, and
∡
O
A
D
=
3
0
∘
\measuredangle OAD=30^{\circ}
∡
O
A
D
=
3
0
∘
.
51
1
Hide problems
Areas of triangles OKM and OLN are different [ILL 1971]
Suppose that the sides
A
B
AB
A
B
and
D
C
DC
D
C
of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
are not parallel. On the sides
B
C
BC
BC
and
A
D
AD
A
D
, pairs of points
(
M
,
N
)
(M,N)
(
M
,
N
)
and
(
K
,
L
)
(K,L)
(
K
,
L
)
are chosen such that
B
M
=
M
N
=
N
C
BM=MN=NC
BM
=
MN
=
NC
and
A
K
=
K
L
=
L
D
AK=KL=LD
A
K
=
K
L
=
L
D
. Prove that the areas of triangles
O
K
M
OKM
O
K
M
and
O
L
N
OLN
O
L
N
are different, where
O
O
O
is the intersection point of
A
B
AB
A
B
and
C
D
CD
C
D
.
18
1
Hide problems
Arithmetic and Geometric mean - [IMO LongList 1971]
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
be positive numbers,
m
g
=
(
a
1
a
2
⋯
a
n
)
n
m_g = \sqrt[n]{(a_1a_2 \cdots a_n)}
m
g
=
n
(
a
1
a
2
⋯
a
n
)
their geometric mean, and
m
a
=
(
a
1
+
a
2
+
⋯
+
a
n
)
n
m_a = \frac{(a_1 + a_2 + \cdots + a_n)}{n}
m
a
=
n
(
a
1
+
a
2
+
⋯
+
a
n
)
their arithmetic mean. Prove that
(
1
+
m
g
)
n
≤
(
1
+
a
1
)
⋯
(
1
+
a
n
)
≤
(
1
+
m
a
)
n
.
(1 + m_g)^n \leq (1 + a_1) \cdots(1 + a_n) \leq (1 + m_a)^n.
(
1
+
m
g
)
n
≤
(
1
+
a
1
)
⋯
(
1
+
a
n
)
≤
(
1
+
m
a
)
n
.
15
1
Hide problems
Convex Quadrilateral - [IMO LongList 1971]
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral whose diagonals intersect at
O
O
O
at an angle
θ
\theta
θ
. Let us set
O
A
=
a
,
O
B
=
b
,
O
C
=
c
OA = a, OB = b, OC = c
O
A
=
a
,
OB
=
b
,
OC
=
c
, and
O
D
=
d
,
c
>
a
>
0
OD = d, c > a > 0
O
D
=
d
,
c
>
a
>
0
, and
d
>
b
>
0.
d > b > 0.
d
>
b
>
0.
Show that if there exists a right circular cone with vertex
V
V
V
, with the properties:(1) its axis passes through
O
O
O
, and(2) its curved surface passes through
A
,
B
,
C
A,B,C
A
,
B
,
C
and
D
,
D,
D
,
then
O
V
2
=
d
2
b
2
(
c
+
a
)
2
−
c
2
a
2
(
d
+
b
)
2
c
a
(
d
−
b
)
2
−
d
b
(
c
−
a
)
2
.
OV^2=\frac{d^2b^2(c + a)^2 - c^2a^2(d + b)^2}{ca(d - b)^2 - db(c - a)^2}.
O
V
2
=
c
a
(
d
−
b
)
2
−
d
b
(
c
−
a
)
2
d
2
b
2
(
c
+
a
)
2
−
c
2
a
2
(
d
+
b
)
2
.
Show also that if
c
+
a
d
+
b
\frac{c+a}{d+b}
d
+
b
c
+
a
lies between
c
a
d
b
\frac{ca}{db}
d
b
c
a
and
c
a
d
b
,
\sqrt{\frac{ca}{db}},
d
b
c
a
,
and
c
−
a
d
−
b
=
c
a
d
b
,
\frac{c-a}{d-b}=\frac{ca}{db},
d
−
b
c
−
a
=
d
b
c
a
,
then for a suitable choice of
θ
\theta
θ
, a right circular cone exists with properties (1) and (2).
14
1
Hide problems
The square of the sum - [IMO LongList 1971]
Note that
8
3
−
7
3
=
169
=
1
3
2
8^3 - 7^3 = 169 = 13^2
8
3
−
7
3
=
169
=
1
3
2
and
13
=
2
2
+
3
2
.
13 = 2^2 + 3^2.
13
=
2
2
+
3
2
.
Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.
13
1
Hide problems
Martian, Venusian, and the Human! - [IMO LongList 1971]
One Martian, one Venusian, and one Human reside on Pluton. One day they make the following conversation:Martian : I have spent
1
/
12
1/12
1/12
of my life on Pluton. Human : I also have. Venusian : Me too. Martian : But Venusian and I have spend much more time here than you, Human. Human : That is true. However, Venusian and I are of the same age. Venusian : Yes, I have lived
300
300
300
Earth years. Martian : Venusian and I have been on Pluton for the past
13
13
13
years.It is known that Human and Martian together have lived
104
104
104
Earth years. Find the ages of Martian, Venusian, and Human.*[hide="*"]*: Note that the numbers in the problem are not necessarily in base
10.
10.
10.
54
1
Hide problems
Choosing a subset P of n+1 men [ILL 1971]
A set
M
M
M
is formed of
(
2
n
n
)
\binom{2n}{n}
(
n
2
n
)
men,
n
=
1
,
2
,
…
n=1,2,\ldots
n
=
1
,
2
,
…
. Prove that we can choose a subset
P
P
P
of the set
M
M
M
consisting of
n
+
1
n+1
n
+
1
men such that one of the following conditions is satisfied:
(
1
)
(1)
(
1
)
every member of the set
P
P
P
knows every other member of the set
P
P
P
;
(
2
)
(2)
(
2
)
no member of the set
P
P
P
knows any other member of the set
P
P
P
.
12
1
Hide problems
Product of x_k is equal to 1 - [IMO LongList 1971]
A system of n numbers
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots, x_n
x
1
,
x
2
,
…
,
x
n
is given such that
x
1
=
log
x
n
−
1
x
n
,
x
2
=
log
x
n
x
1
,
…
,
x
n
=
log
x
n
−
2
x
n
−
1
.
x_1 = \log_{x_{n-1}} x_n, x_2 = \log_{x_{n}} x_1, \ldots, x_n = \log_{x_{n-2}} x_{n-1}.
x
1
=
lo
g
x
n
−
1
x
n
,
x
2
=
lo
g
x
n
x
1
,
…
,
x
n
=
lo
g
x
n
−
2
x
n
−
1
.
Prove that
∏
k
=
1
n
x
k
=
1.
\prod_{k=1}^n x_k =1.
∏
k
=
1
n
x
k
=
1.
55
1
Hide problems
No real roots if coefficients sum to less than root 2
Prove that the polynomial
x
4
+
λ
x
3
+
μ
x
2
+
ν
x
+
1
x^4+\lambda x^3+\mu x^2+\nu x+1
x
4
+
λ
x
3
+
μ
x
2
+
νx
+
1
has no real roots if
λ
,
μ
,
ν
\lambda, \mu , \nu
λ
,
μ
,
ν
are real numbers satisfying
∣
λ
∣
+
∣
μ
∣
+
∣
ν
∣
≤
2
|\lambda |+|\mu |+|\nu |\le \sqrt{2}
∣
λ
∣
+
∣
μ
∣
+
∣
ν
∣
≤
2
10
1
Hide problems
Three different knights - [IMO LongList 1971]
In how many different ways can three knights be placed on a chessboard so that the number of squares attacked would be maximal?
9
1
Hide problems
Determine r1r2+r2r3+r1r3 - [IMO LongList 1971]
The base of an inclined prism is a triangle
A
B
C
ABC
A
BC
. The perpendicular projection of
B
1
B_1
B
1
, one of the top vertices, is the midpoint of
B
C
BC
BC
. The dihedral angle between the lateral faces through
B
C
BC
BC
and
A
B
AB
A
B
is
α
\alpha
α
, and the lateral edges of the prism make an angle
β
\beta
β
with the base. If
r
1
,
r
2
,
r
3
r_1, r_2, r_3
r
1
,
r
2
,
r
3
are exradii of a perpendicular section of the prism, assuming that in
A
B
C
,
cos
2
A
+
cos
2
B
+
cos
2
C
=
1
,
∠
A
<
∠
B
<
∠
C
,
ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1, \angle A < \angle B < \angle C,
A
BC
,
cos
2
A
+
cos
2
B
+
cos
2
C
=
1
,
∠
A
<
∠
B
<
∠
C
,
and
B
C
=
a
BC = a
BC
=
a
, calculate
r
1
r
2
+
r
1
r
3
+
r
2
r
3
.
r_1r_2 + r_1r_3 + r_2r_3.
r
1
r
2
+
r
1
r
3
+
r
2
r
3
.
7
1
Hide problems
Cosines Relation with H, O and R - [IMO LongList 1971]
In a triangle
A
B
C
ABC
A
BC
, let
H
H
H
be its orthocenter,
O
O
O
its circumcenter, and
R
R
R
its circumradius. Prove that:(a)
∣
O
H
∣
=
R
1
−
8
cos
α
⋅
cos
β
⋅
cos
γ
|OH| = R \sqrt{1-8 \cos \alpha \cdot \cos \beta \cdot \cos \gamma}
∣
O
H
∣
=
R
1
−
8
cos
α
⋅
cos
β
⋅
cos
γ
where
α
,
β
,
γ
\alpha, \beta, \gamma
α
,
β
,
γ
are angles of the triangle
A
B
C
;
ABC;
A
BC
;
(b)
O
≡
H
O \equiv H
O
≡
H
if and only if
A
B
C
ABC
A
BC
is equilateral.
6
1
Hide problems
Prove that S = 8S1 − 4S2 - [IMO LongList 1971]
Let squares be constructed on 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
, all to the outside of the triangle, and let
A
1
,
B
1
,
C
1
A_1,B_1, C_1
A
1
,
B
1
,
C
1
be their centers. Starting from the triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
one analogously obtains a triangle
A
2
B
2
C
2
A_2B_2C_2
A
2
B
2
C
2
. If
S
,
S
1
,
S
2
S, S_1, S_2
S
,
S
1
,
S
2
denote the areas of triangles
A
B
C
,
A
1
B
1
C
1
,
A
2
B
2
C
2
ABC,A_1B_1C_1,A_2B_2C_2
A
BC
,
A
1
B
1
C
1
,
A
2
B
2
C
2
, respectively, prove that
S
=
8
S
1
−
4
S
2
.
S = 8S_1 - 4S_2.
S
=
8
S
1
−
4
S
2
.
4
1
Hide problems
Find the last digit of the number m - [IMO LongList 1971]
Let
x
n
=
2
2
n
+
1
x_n=2^{2^{n}}+1
x
n
=
2
2
n
+
1
and let
m
m
m
be the least common multiple of
x
2
,
x
3
,
…
,
x
1971
.
x_2, x_3, \ldots, x_{1971}.
x
2
,
x
3
,
…
,
x
1971
.
Find the last digit of
m
.
m.
m
.
3
1
Hide problems
For any positive reals x, y, z it holds-[IMO LongList 1971]
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers,
0
<
a
≤
b
≤
c
0 < a \leq b \leq c
0
<
a
≤
b
≤
c
. Prove that for any positive real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
the following inequality holds:
(
a
x
+
b
y
+
c
z
)
(
x
a
+
y
b
+
z
c
)
≤
(
x
+
y
+
z
)
2
⋅
(
a
+
c
)
2
4
a
c
.
(ax+by+cz) \left( \frac xa + \frac yb+\frac zc \right) \leq (x+y+z)^2 \cdot \frac{(a+c)^2}{4ac}.
(
a
x
+
b
y
+
cz
)
(
a
x
+
b
y
+
c
z
)
≤
(
x
+
y
+
z
)
2
⋅
4
a
c
(
a
+
c
)
2
.
2
1
Hide problems
Inequality on sum of divisors of n - [IMO LongList 1971]
Let us denote by
s
(
n
)
=
∑
d
∣
n
d
s(n)= \sum_{d|n} d
s
(
n
)
=
∑
d
∣
n
d
the sum of divisors of a positive integer
n
n
n
(
1
1
1
and
n
n
n
included). If
n
n
n
has at most
5
5
5
distinct prime divisors, prove that
s
(
n
)
<
77
16
n
.
s(n) < \frac{77}{16} n.
s
(
n
)
<
16
77
n
.
Also prove that there exists a natural number
n
n
n
for which
s
(
n
)
<
76
16
n
s(n) < \frac{76}{16} n
s
(
n
)
<
16
76
n
holds.
11
1
Hide problems
1!+2!+...+n! is a perfect power - [Iran Second Round 1986]
Find all positive integers
n
n
n
for which the number
1
!
+
2
!
+
3
!
+
⋯
+
n
!
1!+2!+3!+\cdots+n!
1
!
+
2
!
+
3
!
+
⋯
+
n
!
is a perfect power of an integer.