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Contests
International Contests
IMO Shortlist
1985 IMO Shortlist
1985 IMO Shortlist
Part of
IMO Shortlist
Subcontests
(21)
3
1
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Polynomial inequality - IMO Long List 1986
For any polynomial
P
(
x
)
=
a
0
+
a
1
x
+
…
+
a
k
x
k
P(x)=a_0+a_1x+\ldots+a_kx^k
P
(
x
)
=
a
0
+
a
1
x
+
…
+
a
k
x
k
with integer coefficients, the number of odd coefficients is denoted by
o
(
P
)
o(P)
o
(
P
)
. For
i
−
0
,
1
,
2
,
…
i-0,1,2,\ldots
i
−
0
,
1
,
2
,
…
let
Q
i
(
x
)
=
(
1
+
x
)
i
Q_i(x)=(1+x)^i
Q
i
(
x
)
=
(
1
+
x
)
i
. Prove that if
i
1
,
i
2
,
…
,
i
n
i_1,i_2,\ldots,i_n
i
1
,
i
2
,
…
,
i
n
are integers satisfying
0
≤
i
1
<
i
2
<
…
<
i
n
0\le i_1<i_2<\ldots<i_n
0
≤
i
1
<
i
2
<
…
<
i
n
, then:
o
(
Q
i
1
+
Q
i
2
+
…
+
Q
i
n
)
≥
o
(
Q
i
1
)
.
o(Q_{i_{1}}+Q_{i_{2}}+\ldots+Q_{i_{n}})\ge o(Q_{i_{1}}).
o
(
Q
i
1
+
Q
i
2
+
…
+
Q
i
n
)
≥
o
(
Q
i
1
)
.
8
1
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There exists a polygonal line
Let
A
A
A
be a set of
n
n
n
points in the space. From the family of all segments with endpoints in
A
A
A
,
q
q
q
segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of
m
m
m
yellow segments, where
m
≥
2
q
n
m \geq \frac{2q}{n}
m
≥
n
2
q
, arranged in order of increasing length.
21
1
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Prove that BAM=CAX and AM/AX=cos(BAC)
The tangents at
B
B
B
and
C
C
C
to the circumcircle of the acute-angled triangle
A
B
C
ABC
A
BC
meet at
X
X
X
. Let
M
M
M
be the midpoint of
B
C
BC
BC
. Prove that(a)
∠
B
A
M
=
∠
C
A
X
\angle BAM = \angle CAX
∠
B
A
M
=
∠
C
A
X
, and(b)
A
M
A
X
=
cos
∠
B
A
C
.
\frac{AM}{AX} = \cos\angle BAC.
A
X
A
M
=
cos
∠
B
A
C
.
18
1
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n variables ineq. IMO Long List 1986
Let
x
1
,
x
2
,
⋯
,
x
n
x_1, x_2, \cdots , x_n
x
1
,
x
2
,
⋯
,
x
n
be positive numbers. Prove that
x
1
2
x
1
2
+
x
2
x
3
+
x
2
2
x
2
2
+
x
3
x
4
+
⋯
+
x
n
−
1
2
x
n
−
1
2
+
x
n
x
1
+
x
n
2
x
n
2
+
x
1
x
2
≤
n
−
1
\frac{x_1^2}{x_1^2+x_2x_3} + \frac{x_2^2}{x_2^2+x_3x_4} + \cdots +\frac{x_{n-1}^2}{x_{n-1}^2+x_nx_1} +\frac{x_n^2}{x_n^2+x_1x_2} \leq n-1
x
1
2
+
x
2
x
3
x
1
2
+
x
2
2
+
x
3
x
4
x
2
2
+
⋯
+
x
n
−
1
2
+
x
n
x
1
x
n
−
1
2
+
x
n
2
+
x
1
x
2
x
n
2
≤
n
−
1
2
1
Hide problems
Find the ratio x/y
A polyhedron has
12
12
12
faces and is such that:(i) all faces are isosceles triangles, (ii) all edges have length either
x
x
x
or
y
y
y
, (iii) at each vertex either
3
3
3
or
6
6
6
edges meet, and (iv) all dihedral angles are equal.Find the ratio
x
/
y
.
x/y.
x
/
y
.
15
1
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Is it possible to decompose K ?
Let
K
K
K
and
K
′
K'
K
′
be two squares in the same plane, their sides of equal length. Is it possible to decompose
K
K
K
into a finite number of triangles
T
1
,
T
2
,
…
,
T
p
T_1, T_2, \ldots, T_p
T
1
,
T
2
,
…
,
T
p
with mutually disjoint interiors and find translations
t
1
,
t
2
,
…
,
t
p
t_1, t_2, \ldots, t_p
t
1
,
t
2
,
…
,
t
p
such that
K
′
=
⋃
i
=
1
p
t
i
(
T
i
)
?
K'=\bigcup_{i=1}^{p} t_i(T_i) \ ?
K
′
=
i
=
1
⋃
p
t
i
(
T
i
)
?
6
1
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IMO Long List 1986 inequality for x_i
Let
x
n
=
2
+
3
+
⋯
+
n
n
3
2
.
x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.
x
n
=
2
2
+
3
3
+
⋯
+
n
n
.
Prove that x_{n+1}-x_n <\frac{1}{n!} n=2,3,\cdots
16
1
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Construct an equilateral triangle if it's possible
If possible, construct an equilateral triangle whose three vertices are on three given circles.
14
1
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There must be at least one good point
A set of
1985
1985
1985
points is distributed around the circumference of a circle and each of the points is marked with
1
1
1
or
−
1
-1
−
1
. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. Show that if the number of points marked with
−
1
-1
−
1
is less than
662
662
662
, there must be at least one good point.
10
1
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Point M on surface of a regular tetrahedron
Prove that for every point
M
M
M
on the surface of a regular tetrahedron there exists a point
M
′
M'
M
′
such that there are at least three different curves on the surface joining
M
M
M
to
M
′
M'
M
′
with the smallest possible length among all curves on the surface joining
M
M
M
to
M
′
M'
M
′
.
12
1
Hide problems
Sequence of polynomials
A sequence of polynomials
P
m
(
x
,
y
,
z
)
,
m
=
0
,
1
,
2
,
⋯
P_m(x, y, z), m = 0, 1, 2, \cdots
P
m
(
x
,
y
,
z
)
,
m
=
0
,
1
,
2
,
⋯
, in
x
,
y
x, y
x
,
y
, and
z
z
z
is defined by
P
0
(
x
,
y
,
z
)
=
1
P_0(x, y, z) = 1
P
0
(
x
,
y
,
z
)
=
1
and by
P
m
(
x
,
y
,
z
)
=
(
x
+
z
)
(
y
+
z
)
P
m
−
1
(
x
,
y
,
z
+
1
)
−
z
2
P
m
−
1
(
x
,
y
,
z
)
P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)
P
m
(
x
,
y
,
z
)
=
(
x
+
z
)
(
y
+
z
)
P
m
−
1
(
x
,
y
,
z
+
1
)
−
z
2
P
m
−
1
(
x
,
y
,
z
)
for
m
>
0
m > 0
m
>
0
. Prove that each
P
m
(
x
,
y
,
z
)
P_m(x, y, z)
P
m
(
x
,
y
,
z
)
is symmetric, in other words, is unaltered by any permutation of
x
,
y
,
z
.
x, y, z.
x
,
y
,
z
.
9
1
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The radius of the sphere S
Determine the radius of a sphere
S
S
S
that passes through the centroids of each face of a given tetrahedron
T
T
T
inscribed in a unit sphere with center
O
O
O
. Also, determine the distance from
O
O
O
to the center of
S
S
S
as a function of the edges of
T
.
T.
T
.
1
1
Hide problems
Popular problem - A set with 1985 members
Given a set
M
M
M
of
1985
1985
1985
positive integers, none of which has a prime divisor larger than
26
26
26
, prove that the set has four distinct elements whose geometric mean is an integer.
13
1
Hide problems
m boxes with some balls in each box
Let
m
m
m
boxes be given, with some balls in each box. Let
n
<
m
n < m
n
<
m
be a given integer. The following operation is performed: choose
n
n
n
of the boxes and put
1
1
1
ball in each of them. Prove:(a) If
m
m
m
and
n
n
n
are relatively prime, then it is possible, by performing the operation a finite number of times, to arrive at the situation that all the boxes contain an equal number of balls.(b) If
m
m
m
and
n
n
n
are not relatively prime, there exist initial distributions of balls in the boxes such that an equal distribution is not possible to achieve.
5
1
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The function is strictly convex
Let
D
D
D
be the interior of the circle
C
C
C
and let
A
∈
C
A \in C
A
∈
C
. Show that the function
f
:
D
→
R
,
f
(
M
)
=
∣
M
A
∣
∣
M
M
′
∣
f : D \to \mathbb R, f(M)=\frac{|MA|}{|MM'|}
f
:
D
→
R
,
f
(
M
)
=
∣
M
M
′
∣
∣
M
A
∣
where
M
′
=
A
M
∩
C
M' = AM \cap C
M
′
=
A
M
∩
C
, is strictly convex; i.e.,
f
(
P
)
<
f
(
M
1
)
+
f
(
M
2
)
2
,
∀
M
1
,
M
2
∈
D
,
M
1
≠
M
2
f(P) <\frac{f(M_1)+f(M_2)}{2}, \forall M_1,M_2 \in D, M_1 \neq M_2
f
(
P
)
<
2
f
(
M
1
)
+
f
(
M
2
)
,
∀
M
1
,
M
2
∈
D
,
M
1
=
M
2
where
P
P
P
is the midpoint of the segment
M
1
M
2
.
M_1M_2.
M
1
M
2
.
7
1
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P/p^k is not less than n!
The positive integers
x
1
,
⋯
,
x
n
x_1, \cdots , x_n
x
1
,
⋯
,
x
n
,
n
≥
3
n \geq 3
n
≥
3
, satisfy
x
1
<
x
2
<
⋯
<
x
n
<
2
x
1
x_1 < x_2 <\cdots< x_n < 2x_1
x
1
<
x
2
<
⋯
<
x
n
<
2
x
1
. Set
P
=
x
1
x
2
⋯
x
n
.
P = x_1x_2 \cdots x_n.
P
=
x
1
x
2
⋯
x
n
.
Prove that if
p
p
p
is a prime number,
k
k
k
a positive integer, and
P
P
P
is divisible by
p
k
pk
p
k
, then
P
p
k
≥
n
!
.
\frac{P}{p^k} \geq n!.
p
k
P
≥
n
!
.
4
1
Hide problems
All numbers in N must receive the same color
Each of the numbers in the set
N
=
{
1
,
2
,
3
,
⋯
,
n
−
1
}
N = \{1, 2, 3, \cdots, n - 1\}
N
=
{
1
,
2
,
3
,
⋯
,
n
−
1
}
, where
n
≥
3
n \geq 3
n
≥
3
, is colored with one of two colors, say red or black, so that:(i)
i
i
i
and
n
−
i
n - i
n
−
i
always receive the same color, and(ii) for some
j
∈
N
j \in N
j
∈
N
, relatively prime to
n
n
n
,
i
i
i
and
∣
j
−
i
∣
|j - i|
∣
j
−
i
∣
receive the same color for all
i
∈
N
,
i
≠
j
.
i \in N, i \neq j.
i
∈
N
,
i
=
j
.
Prove that all numbers in
N
N
N
must receive the same color.
17
1
Hide problems
Prove that there exist one number a
The sequence
f
1
,
f
2
,
⋯
,
f
n
,
⋯
f_1, f_2, \cdots, f_n, \cdots
f
1
,
f
2
,
⋯
,
f
n
,
⋯
of functions is defined for
x
>
0
x > 0
x
>
0
recursively by f_1(x)=x , f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right) Prove that there exists one and only one positive number
a
a
a
such that
0
<
f
n
(
a
)
<
f
n
+
1
(
a
)
<
1
0 < f_n(a) < f_{n+1}(a) < 1
0
<
f
n
(
a
)
<
f
n
+
1
(
a
)
<
1
for all integers
n
≥
1.
n \geq 1.
n
≥
1.
20
1
Hide problems
Prove that EB+CD = ED
A circle whose center is on the side
E
D
ED
E
D
of the cyclic quadrilateral
B
C
D
E
BCDE
BC
D
E
touches the other three sides. Prove that
E
B
+
C
D
=
E
D
.
EB+CD = ED.
EB
+
C
D
=
E
D
.
19
1
Hide problems
Find integers n so that n-gon exist
For which integers
n
≥
3
n \geq 3
n
≥
3
does there exist a regular
n
n
n
-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system?
11
1
Hide problems
Find a method-IMO Long List P7
Find a method by which one can compute the coefficients of
P
(
x
)
=
x
6
+
a
1
x
5
+
⋯
+
a
6
P(x) = x^6 + a_1x^5 + \cdots+ a_6
P
(
x
)
=
x
6
+
a
1
x
5
+
⋯
+
a
6
from the roots of
P
(
x
)
=
0
P(x) = 0
P
(
x
)
=
0
by performing not more than
15
15
15
additions and
15
15
15
multiplications.