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International Contests
IMO Shortlist
1973 IMO Shortlist
1973 IMO Shortlist
Part of
IMO Shortlist
Subcontests
(11)
13
1
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Find the sphere of maximal radius - ISL 1973
Find the sphere of maximal radius that can be placed inside every tetrahedron that has all altitudes of length greater than or equal to
1.
1.
1.
16
1
Hide problems
factorize P(x) as a product of m real quadratic polynomials
Given
a
,
θ
∈
R
,
m
∈
N
a, \theta \in \mathbb R, m \in \mathbb N
a
,
θ
∈
R
,
m
∈
N
, and
P
(
x
)
=
x
2
m
−
2
∣
a
∣
m
x
m
cos
θ
+
a
2
m
P(x) = x^{2m}- 2|a|^mx^m \cos \theta +a^{2m}
P
(
x
)
=
x
2
m
−
2∣
a
∣
m
x
m
cos
θ
+
a
2
m
, factorize
P
(
x
)
P(x)
P
(
x
)
as a product of
m
m
m
real quadratic polynomials.
15
1
Hide problems
Prove the identity on sines
Prove that for all
n
∈
N
n \in \mathbb N
n
∈
N
the following is true:
2
n
∏
k
=
1
n
sin
k
π
2
n
+
1
=
2
n
+
1
2^n \prod_{k=1}^n \sin \frac{k \pi}{2n+1} = \sqrt{2n+1}
2
n
k
=
1
∏
n
sin
2
n
+
1
kπ
=
2
n
+
1
12
1
Hide problems
Prove that B cannot be obtain ed from A with some operations
Consider the two square matrices A=\begin{bmatrix} +1 & +1 &+1& +1 &+1 \\+1 &+1 &+1&-1 &-1 \\ +1 &-1&-1 &+1& +1 \\ +1 & -1 & -1 & -1 & +1 \\ +1 &+1&-1 &+1&-1 \end{bmatrix} \text{ and } B=\begin{bmatrix} +1 & +1 &+1& +1 &+1 \\+1 &+1 &+1&-1 &-1 \\ +1 &+1&-1& +1&-1 \\ +1 &-1& -1& +1& +1 \\ +1 & -1& +1&-1 &+1 \end{bmatrix}with entries
+
1
+1
+
1
and
−
1
-1
−
1
. The following operations will be called elementary:(1) Changing signs of all numbers in one row;(2) Changing signs of all numbers in one column;(3) Interchanging two rows (two rows exchange their positions);(4) Interchanging two columns.Prove that the matrix
B
B
B
cannot be obtained from the matrix
A
A
A
using these operations.
9
1
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Tetrahedron OABC with minimal perimeter
Let
O
x
,
O
y
,
O
z
Ox, Oy, Oz
O
x
,
O
y
,
O
z
be three rays, and
G
G
G
a point inside the trihedron
O
x
y
z
Oxyz
O
x
yz
. Consider all planes passing through
G
G
G
and cutting
O
x
,
O
y
,
O
z
Ox, Oy, Oz
O
x
,
O
y
,
O
z
at points
A
,
B
,
C
A,B,C
A
,
B
,
C
, respectively. How is the plane to be placed in order to yield a tetrahedron
O
A
B
C
OABC
O
A
BC
with minimal perimeter ?
8
1
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Number of terms of non-negative sequence
Prove that there are exactly
(
k
[
k
/
2
]
)
\binom{k}{[k/2]}
(
[
k
/2
]
k
)
arrays
a
1
,
a
2
,
…
,
a
k
+
1
a_1, a_2, \ldots , a_{k+1}
a
1
,
a
2
,
…
,
a
k
+
1
of nonnegative integers such that
a
1
=
0
a_1 = 0
a
1
=
0
and
∣
a
i
−
a
i
+
1
∣
=
1
|a_i-a_{i+1}| = 1
∣
a
i
−
a
i
+
1
∣
=
1
for
i
=
1
,
2
,
…
,
k
.
i = 1, 2, \ldots , k.
i
=
1
,
2
,
…
,
k
.
7
1
Hide problems
Prove that there exist a triangle with sides x,y,z
Given a tetrahedron
A
B
C
D
ABCD
A
BC
D
, let
x
=
A
B
⋅
C
D
x = AB \cdot CD
x
=
A
B
⋅
C
D
,
y
=
A
C
⋅
B
D
y = AC \cdot BD
y
=
A
C
⋅
B
D
, and
z
=
A
D
⋅
B
C
z = AD \cdot BC
z
=
A
D
⋅
BC
. Prove that there exists a triangle with edges
x
,
y
,
z
.
x, y, z.
x
,
y
,
z
.
5
1
Hide problems
Find the locus of centers of such circles - ISL 1973
A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles.
4
1
Hide problems
How many such partitions of C are there ?
Let
P
P
P
be a set of
7
7
7
different prime numbers and
C
C
C
a set of
28
28
28
different composite numbers each of which is a product of two (not necessarily different) numbers from
P
P
P
. The set
C
C
C
is divided into
7
7
7
disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of
C
C
C
are there ?
2
1
Hide problems
Find the locus of vertices A
Given a circle
K
K
K
, find the locus of vertices
A
A
A
of parallelograms
A
B
C
D
ABCD
A
BC
D
with diagonals
A
C
≤
B
D
AC \leq BD
A
C
≤
B
D
, such that
B
D
BD
B
D
is inside
K
K
K
.
1
1
Hide problems
Find the locus of the point P
Let a tetrahedron
A
B
C
D
ABCD
A
BC
D
be inscribed in a sphere
S
S
S
. Find the locus of points
P
P
P
inside the sphere
S
S
S
for which the equality
A
P
P
A
1
+
B
P
P
B
1
+
C
P
P
C
1
+
D
P
P
D
1
=
4
\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4
P
A
1
A
P
+
P
B
1
BP
+
P
C
1
CP
+
P
D
1
D
P
=
4
holds, where
A
1
,
B
1
,
C
1
A_1,B_1, C_1
A
1
,
B
1
,
C
1
, and
D
1
D_1
D
1
are the intersection points of
S
S
S
with the lines
A
P
,
B
P
,
C
P
AP,BP,CP
A
P
,
BP
,
CP
, and
D
P
DP
D
P
, respectively.