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International Contests
IMO Longlists
1972 IMO Longlists
1972 IMO Longlists
Part of
IMO Longlists
Subcontests
(35)
45
1
Hide problems
Line segment passing through intersection of diagonals.
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral whose diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at point
O
O
O
. Let a line through
O
O
O
intersect segment
A
B
AB
A
B
at
M
M
M
and segment
C
D
CD
C
D
at
N
N
N
. Prove that the segment
M
N
MN
MN
is not longer than at least one of the segments
A
C
AC
A
C
and
B
D
BD
B
D
.
43
1
Hide problems
Locus of points symmetric to a fixed point about a chord.
A fixed point
A
A
A
inside a circle is given. Consider all chords
X
Y
XY
X
Y
of the circle such that
∠
X
A
Y
\angle XAY
∠
X
A
Y
is a right angle, and for all such chords construct the point
M
M
M
symmetric to
A
A
A
with respect to
X
Y
XY
X
Y
. Find the locus of points
M
M
M
.
39
1
Hide problems
Number of tangents to y=x^3-3x and y=x^3+px
How many tangents to the curve
y
=
x
3
−
3
x
(
y
=
x
3
+
p
x
)
y = x^3-3x\:\: (y = x^3 + px)
y
=
x
3
−
3
x
(
y
=
x
3
+
p
x
)
can be drawn from different points in the plane?
38
1
Hide problems
Construct rectangles from congruent m x n rectangles
Congruent rectangles with sides
m
(
c
m
)
m(cm)
m
(
c
m
)
and
n
(
c
m
)
n(cm)
n
(
c
m
)
are given (
m
,
n
m, n
m
,
n
positive integers). Characterize the rectangles that can be constructed from these rectangles (in the fashion of a jigsaw puzzle). (The number of rectangles is unbounded.)
36
1
Hide problems
A line intersects any three parallel line segments.
A finite number of parallel segments in the plane are given with the property that for any three of the segments there is a line intersecting each of them. Prove that there exists a line that intersects all the given segments.
29
1
Hide problems
AC=BC in A_1B_1C_1 with angle bisectors A1A, B1B, C_1C
Let
A
,
B
,
C
A,B,C
A
,
B
,
C
be points on the sides
B
1
C
1
,
C
1
A
1
,
A
1
B
1
B_1C_1, C_1A_1,A_1B_1
B
1
C
1
,
C
1
A
1
,
A
1
B
1
of a triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
such that
A
1
A
,
B
1
B
,
C
1
C
A_1A,B_1B,C_1C
A
1
A
,
B
1
B
,
C
1
C
are the bisectors of angles of the triangle. We have that
A
C
=
B
C
AC = BC
A
C
=
BC
and
A
1
C
1
≠
B
1
C
1
.
A_1C_1 \neq B_1C_1.
A
1
C
1
=
B
1
C
1
.
(
a
)
(a)
(
a
)
Prove that
C
1
C_1
C
1
lies on the circumcircle of the triangle
A
B
C
ABC
A
BC
.
(
b
)
(b)
(
b
)
Suppose that
∠
B
A
C
1
=
π
6
;
\angle BAC_1 =\frac{\pi}{6};
∠
B
A
C
1
=
6
π
;
find the form of triangle
A
B
C
ABC
A
BC
.
28
1
Hide problems
Point within rectangle with integer distances to vertices.
The lengths of the sides of a rectangle are given to be odd integers. Prove that there does not exist a point within that rectangle that has integer distances to each of its four vertices.
40
1
Hide problems
Prove that sin u/ sin v lies between u/v and pi/2 times u/v
Prove the inequalities
u
v
≤
sin
u
sin
v
≤
π
2
×
u
v
,
for
0
≤
u
<
v
≤
π
2
\frac{u}{v}\le \frac{\sin u}{\sin v}\le \frac{\pi}{2}\times\frac{u}{v},\text{ for }0 \le u < v \le \frac{\pi}{2}
v
u
≤
sin
v
sin
u
≤
2
π
×
v
u
,
for
0
≤
u
<
v
≤
2
π
35
1
Hide problems
Inequality for a, b, c, d being reals and am+b=-cm+d=m
(
a
)
(a)
(
a
)
Prove that for
a
,
b
,
c
,
d
∈
R
,
m
∈
[
1
,
+
∞
)
a, b, c, d \in\mathbb{R}, m \in [1,+\infty)
a
,
b
,
c
,
d
∈
R
,
m
∈
[
1
,
+
∞
)
with
a
m
+
b
=
−
c
m
+
d
=
m
am + b =-cm + d = m
am
+
b
=
−
c
m
+
d
=
m
,
(
i
)
a
2
+
b
2
+
c
2
+
d
2
+
(
a
−
c
)
2
+
(
b
−
d
)
2
≥
4
m
2
1
+
m
2
,
and
(i)\sqrt{a^2 + b^2}+\sqrt{c^2 + d^2}+\sqrt{(a-c)^2 + (b-d)^2}\ge \frac{4m^2}{1+m^2},\text{ and}
(
i
)
a
2
+
b
2
+
c
2
+
d
2
+
(
a
−
c
)
2
+
(
b
−
d
)
2
≥
1
+
m
2
4
m
2
,
and
(
i
i
)
2
≤
4
m
2
1
+
m
2
<
4.
(ii) 2 \le \frac{4m^2}{1+m^2} < 4.
(
ii
)
2
≤
1
+
m
2
4
m
2
<
4.
(
b
)
(b)
(
b
)
Express
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
as functions of
m
m
m
so that there is equality in
(
i
)
.
(i).
(
i
)
.
25
1
Hide problems
Inequality holding for n reals always.
We consider
n
n
n
real variables
x
i
(
1
≤
i
≤
n
)
x_i(1 \le i \le n)
x
i
(
1
≤
i
≤
n
)
, where
n
n
n
is an integer and
n
≥
2
n \ge 2
n
≥
2
. The product of these variables will be denoted by
p
p
p
, their sum by
s
s
s
, and the sum of their squares by
S
S
S
. Furthermore, let
α
\alpha
α
be a positive constant. We now study the inequality
p
s
≤
S
α
ps \le S\alpha
p
s
≤
S
α
. Prove that it holds for every
n
n
n
-tuple
(
x
i
)
(x_i)
(
x
i
)
if and only if
α
=
n
+
1
2
\alpha=\frac{n+1}{2}
α
=
2
n
+
1
37
1
Hide problems
Covering an imperfect chessboard with 1x2 rectangles
On a chessboard (
8
×
8
8\times 8
8
×
8
squares with sides of length
1
1
1
) two diagonally opposite corner squares are taken away. Can the board now be covered with nonoverlapping rectangles with sides of lengths
1
1
1
and
2
2
2
?
33
1
Hide problems
Number of ways of cutting a rectangle into 1x2 rectangles
A rectangle
A
B
C
D
ABCD
A
BC
D
is given whose sides have lengths
3
3
3
and
2
n
2n
2
n
, where
n
n
n
is a natural number. Denote by
U
(
n
)
U(n)
U
(
n
)
the number of ways in which one can cut the rectangle into rectangles of side lengths
1
1
1
and
2
2
2
.
(
a
)
(a)
(
a
)
Prove that
U
(
n
+
1
)
+
U
(
n
−
1
)
=
4
U
(
n
)
;
U(n + 1)+U(n -1) = 4U(n);
U
(
n
+
1
)
+
U
(
n
−
1
)
=
4
U
(
n
)
;
(
b
)
(b)
(
b
)
Prove that
U
(
n
)
=
1
2
3
[
(
3
+
1
)
(
2
+
3
)
n
+
(
3
−
1
)
(
2
−
3
)
n
]
.
U(n) =\frac{1}{2\sqrt{3}}[(\sqrt{3} + 1)(2 +\sqrt{3})^n + (\sqrt{3} - 1)(2 -\sqrt{3})^n].
U
(
n
)
=
2
3
1
[(
3
+
1
)
(
2
+
3
)
n
+
(
3
−
1
)
(
2
−
3
)
n
]
.
24
1
Hide problems
Colouring a convex 18 gon and numbering the diagonals
The diagonals of a convex
18
18
18
-gon are colored in
5
5
5
different colors, each color appearing on an equal number of diagonals. The diagonals of one color are numbered
1
,
2
,
⋯
1, 2,\cdots
1
,
2
,
⋯
. One randomly chooses one-fifth of all the diagonals. Find the number of possibilities for which among the chosen diagonals there exist exactly
n
n
n
pairs of diagonals of the same color and with fixed indices
i
,
j
i, j
i
,
j
.
46
1
Hide problems
Symmetric squares wrt centre of 4x4 square add to 17
Numbers
1
,
2
,
⋯
,
16
1, 2,\cdots, 16
1
,
2
,
⋯
,
16
are written in a
4
×
4
4\times 4
4
×
4
square matrix so that the sum of the numbers in every row, every column, and every diagonal is the same and furthermore that the numbers
1
1
1
and
16
16
16
lie in opposite corners. Prove that the sum of any two numbers symmetric with respect to the center of the square equals
17
17
17
.
42
1
Hide problems
Last two digits in ternary representation.
The decimal number
1
3
101
13^{101}
1
3
101
is given. It is instead written as a ternary number. What are the two last digits of this ternary number?
41
1
Hide problems
Transform ternary decimal to binary decimal and vice versa
The ternary expansion
x
=
0.10101010
⋯
x = 0.10101010\cdots
x
=
0.10101010
⋯
is given. Give the binary expansion of
x
x
x
. Alternatively, transform the binary expansion
y
=
0.110110110
⋯
y = 0.110110110 \cdots
y
=
0.110110110
⋯
into a ternary expansion.
34
1
Hide problems
Prove ax^2+by^2=pz+c has integer solution for p prime
If
p
p
p
is a prime number greater than
2
2
2
and
a
,
b
,
c
a, b, c
a
,
b
,
c
integers not divisible by
p
p
p
, prove that the equation
a
x
2
+
b
y
2
=
p
z
+
c
ax^2 + by^2 = pz + c
a
x
2
+
b
y
2
=
p
z
+
c
has an integer solution.
32
1
Hide problems
Maximum product of natural numbers if their sum is constant
If
n
1
,
n
2
,
⋯
,
n
k
n_1, n_2, \cdots, n_k
n
1
,
n
2
,
⋯
,
n
k
are natural numbers and
n
1
+
n
2
+
⋯
+
n
k
=
n
n_1+n_2+\cdots+n_k = n
n
1
+
n
2
+
⋯
+
n
k
=
n
, show that
m
a
x
(
n
1
n
2
⋯
n
k
)
=
(
t
+
1
)
r
t
k
−
r
,
max(n_1n_2\cdots n_k)=(t + 1)^rt^{k-r},
ma
x
(
n
1
n
2
⋯
n
k
)
=
(
t
+
1
)
r
t
k
−
r
,
where
t
=
[
n
k
]
t =\left[\frac{n}{k}\right]
t
=
[
k
n
]
and
r
r
r
is the remainder of
n
n
n
upon division by
k
k
k
; i.e.,
n
=
t
k
+
r
,
0
≤
r
≤
k
−
1
n = tk + r, 0 \le r \le k- 1
n
=
t
k
+
r
,
0
≤
r
≤
k
−
1
.
31
1
Hide problems
Find natural n for which (3^n-2)/(2^n−3) is reducible
Find values of
n
∈
N
n\in \mathbb{N}
n
∈
N
for which the fraction
3
n
−
2
2
n
−
3
\frac{3^n-2}{2^n-3}
2
n
−
3
3
n
−
2
is reducible.
23
1
Hide problems
2n digit number such that a_{2n} ... a_1 = (a_n ... a_1)^2
Does there exist a
2
n
2n
2
n
-digit number
a
2
n
a
2
n
−
1
⋯
a
1
‾
\overline{a_{2n}a_{2n-1}\cdots a_1}
a
2
n
a
2
n
−
1
⋯
a
1
(for an arbitrary
n
n
n
) for which the following equality holds:
a
2
n
⋯
a
1
‾
=
(
a
n
⋯
a
1
‾
)
2
?
\overline{a_{2n}\cdots a_1}= (\overline{a_n \cdots a_1})^2?
a
2
n
⋯
a
1
=
(
a
n
⋯
a
1
)
2
?
14
1
Hide problems
Plane passing through vertex of regular tetrahedron
(
a
)
(a)
(
a
)
A plane
π
\pi
π
passes through the vertex
O
O
O
of the regular tetrahedron
O
P
Q
R
OPQR
OPQR
. We define
p
,
q
,
r
p, q, r
p
,
q
,
r
to be the signed distances of
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
from
π
\pi
π
measured along a directed normal to
π
\pi
π
. Prove that
p
2
+
q
2
+
r
2
+
(
q
−
r
)
2
+
(
r
−
p
)
2
+
(
p
−
q
)
2
=
2
a
2
,
p^2 + q^2 + r^2 + (q - r)^2 + (r - p)^2 + (p - q)^2 = 2a^2,
p
2
+
q
2
+
r
2
+
(
q
−
r
)
2
+
(
r
−
p
)
2
+
(
p
−
q
)
2
=
2
a
2
,
where
a
a
a
is the length of an edge of a tetrahedron.
(
b
)
(b)
(
b
)
Given four parallel planes not all of which are coincident, show that a regular tetrahedron exists with a vertex on each plane.Note: Part
(
b
)
(b)
(
b
)
is [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=49&t=60825&start=0]IMO 1972 Problem 6
6
1
Hide problems
(n+1) cos(pi/n+1)-n cos(pi/n)>1 for all natural n>1
Prove the inequality
(
n
+
1
)
cos
π
n
+
1
−
n
cos
π
n
>
1
(n + 1)\cos\frac{\pi}{n + 1}- n\cos\frac{\pi}{n}> 1
(
n
+
1
)
cos
n
+
1
π
−
n
cos
n
π
>
1
for all natural numbers
n
≥
2.
n \ge 2.
n
≥
2.
1
1
Hide problems
Integer solutions for 1 + x + x^2 + x^3 + x^4 = y^4
Find all integer solutions of the equation
1
+
x
+
x
2
+
x
3
+
x
4
=
y
4
.
1 + x + x^2 + x^3 + x^4 = y^4.
1
+
x
+
x
2
+
x
3
+
x
4
=
y
4
.
9
1
Hide problems
Possible values of kth largest angle in a convex n-gon
Given natural numbers
k
k
k
and
n
,
k
≤
n
,
n
≥
3
,
n, k \le n, n \ge 3,
n
,
k
≤
n
,
n
≥
3
,
find the set of all values in the interval
(
0
,
π
)
(0, \pi)
(
0
,
π
)
that the
k
t
h
−
k^{th}-
k
t
h
−
largest among the interior angles of a convex
n
n
n
-gon can take.
16
1
Hide problems
Smallest k for which k-subset has a number and its factor
Consider the set
S
S
S
of all the different odd positive integers that are not multiples of
5
5
5
and that are less than
30
m
,
m
30m, m
30
m
,
m
being a positive integer. What is the smallest integer
k
k
k
such that in any subset of
k
k
k
integers from
S
S
S
there must be two integers one of which divides the other? Prove your result.
18
1
Hide problems
Proving an identity for set of scores in a tournament
We have
p
p
p
players participating in a tournament, each player playing against every other player exactly once. A point is scored for each victory, and there are no draws. A sequence of nonnegative integers
s
1
≤
s
2
≤
s
3
≤
⋯
≤
s
p
s_1 \le s_2 \le s_3 \le\cdots\le s_p
s
1
≤
s
2
≤
s
3
≤
⋯
≤
s
p
is given. Show that it is possible for this sequence to be a set of final scores of the players in the tournament if and only if
(
i
)
∑
i
=
1
p
s
i
=
1
2
p
(
p
−
1
)
(i)\displaystyle\sum_{i=1}^{p} s_i =\frac{1}{2}p(p-1)
(
i
)
i
=
1
∑
p
s
i
=
2
1
p
(
p
−
1
)
and
\text{and}
and
(
i
i
)
for all
k
<
p
,
∑
i
=
1
k
s
i
≥
1
2
k
(
k
−
1
)
.
(ii)\text{ for all }k < p,\displaystyle\sum_{i=1}^{k} s_i \ge \frac{1}{2} k(k - 1).
(
ii
)
for all
k
<
p
,
i
=
1
∑
k
s
i
≥
2
1
k
(
k
−
1
)
.
17
1
Hide problems
Stretching a string along a cylinder
A solid right circular cylinder with height
h
h
h
and base-radius
r
r
r
has a solid hemisphere of radius
r
r
r
resting upon it. The center of the hemisphere
O
O
O
is on the axis of the cylinder. Let
P
P
P
be any point on the surface of the hemisphere and
Q
Q
Q
the point on the base circle of the cylinder that is furthest from
P
P
P
(measuring along the surface of the combined solid). A string is stretched over the surface from
P
P
P
to
Q
Q
Q
so as to be as short as possible. Show that if the string is not in a plane, the straight line
P
O
PO
PO
when produced cuts the curved surface of the cylinder.
13
1
Hide problems
Set of points A with a given sphere
Given a sphere
K
K
K
, determine the set of all points
A
A
A
that are vertices of some parallelograms
A
B
C
D
ABCD
A
BC
D
that satisfy
A
C
≤
B
D
AC \le BD
A
C
≤
B
D
and whose entire diagonal
B
D
BD
B
D
is contained in
K
K
K
.
12
1
Hide problems
Comparing areas of two triangles with same circumcircle
A circle
k
=
(
S
,
r
)
k = (S, r)
k
=
(
S
,
r
)
is given and a hexagon
A
A
′
B
B
′
C
C
′
AA'BB'CC'
A
A
′
B
B
′
C
C
′
inscribed in it. The lengths of sides of the hexagon satisfy
A
A
′
=
A
′
B
,
B
B
′
=
B
′
C
,
C
C
′
=
C
′
A
AA' = A'B, BB' = B'C, CC' = C'A
A
A
′
=
A
′
B
,
B
B
′
=
B
′
C
,
C
C
′
=
C
′
A
. Prove that the area
P
P
P
of triangle
A
B
C
ABC
A
BC
is not greater than the area
P
′
P'
P
′
of triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
. When does
P
=
P
′
P = P'
P
=
P
′
hold?
10
1
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Finding two obtuse triangles with five points
Given five points in the plane, no three of which are collinear, prove that there can be found at least two obtuse-angled triangles with vertices at the given points. Construct an example in which there are exactly two such triangles.
5
1
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Prove that projections of vertex of pyramid are concyclic
Given a pyramid whose base is an
n
n
n
-gon inscribable in a circle, let
H
H
H
be the projection of the top vertex of the pyramid to its base. Prove that the projections of
H
H
H
to the lateral edges of the pyramid lie on a circle.
3
1
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Translating n points so that none remain on the segments
On a line a set of segments is given of total length less than
n
n
n
. Prove that every set of
n
n
n
points of the line can be translated in some direction along the line for a distance smaller than
n
2
\frac{n}{2}
2
n
so that none of the points remain on the segments.
19
1
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Subset of real numbers with properties
Let
S
S
S
be a subset of the real numbers with the following properties:
(
i
)
(i)
(
i
)
If
x
∈
S
x \in S
x
∈
S
and
y
∈
S
y \in S
y
∈
S
, then
x
−
y
∈
S
x - y \in S
x
−
y
∈
S
;
(
i
i
)
(ii)
(
ii
)
If
x
∈
S
x \in S
x
∈
S
and
y
∈
S
y \in S
y
∈
S
, then
x
y
∈
S
xy \in S
x
y
∈
S
;
(
i
i
i
)
(iii)
(
iii
)
S
S
S
contains an exceptional number
x
′
x'
x
′
such that there is no number
y
y
y
in
S
S
S
satisfying
x
′
y
+
x
′
+
y
=
0
x'y + x' + y = 0
x
′
y
+
x
′
+
y
=
0
;
(
i
v
)
(iv)
(
i
v
)
If
x
∈
S
x \in S
x
∈
S
and
x
≠
x
′
x \neq x'
x
=
x
′
, there is a number
y
y
y
in
S
S
S
such that
x
y
+
x
+
y
=
0
xy+x+y = 0
x
y
+
x
+
y
=
0
. Show that
(
a
)
(a)
(
a
)
S
S
S
has more than one number in it;
(
b
)
(b)
(
b
)
x
′
≠
−
1
x' \neq -1
x
′
=
−
1
leads to a contradiction;
(
c
)
(c)
(
c
)
x
∈
S
x \in S
x
∈
S
and
x
≠
0
x \neq 0
x
=
0
implies
1
/
x
∈
S
1/x \in S
1/
x
∈
S
.
2
1
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System of equations with at most one real solution
Find all real values of the parameter
a
a
a
for which the system of equations
x
4
=
y
z
−
x
2
+
a
,
x^4 = yz - x^2 + a,
x
4
=
yz
−
x
2
+
a
,
y
4
=
z
x
−
y
2
+
a
,
y^4 = zx - y^2 + a,
y
4
=
z
x
−
y
2
+
a
,
z
4
=
x
y
−
z
2
+
a
,
z^4 = xy - z^2 + a,
z
4
=
x
y
−
z
2
+
a
,
has at most one real solution.
4
1
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Prove that this triangle is equilateral...
You have a triangle,
A
B
C
ABC
A
BC
. Draw in the internal angle trisectors. Let the two trisectors closest to
A
B
AB
A
B
intersect at
D
D
D
, the two trisectors closest to
B
C
BC
BC
intersect at
E
E
E
, and the two closest to
A
C
AC
A
C
at
F
F
F
. Prove that
D
E
F
DEF
D
EF
is equilateral.