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International Contests
IMO Shortlist
1986 IMO Shortlist
1986 IMO Shortlist
Part of
IMO Shortlist
Subcontests
(15)
21
1
Hide problems
Geometric inequality - ISL 1986
Let
A
B
C
D
ABCD
A
BC
D
be a tetrahedron having each sum of opposite sides equal to
1
1
1
. Prove that
r
A
+
r
B
+
r
C
+
r
D
≤
3
3
r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}
r
A
+
r
B
+
r
C
+
r
D
≤
3
3
where
r
A
,
r
B
,
r
C
,
r
D
r_A, r_B, r_C, r_D
r
A
,
r
B
,
r
C
,
r
D
are the inradii of the faces, equality holding only if
A
B
C
D
ABCD
A
BC
D
is regular.
20
1
Hide problems
Sum of face angles of a tetrahedron - ISL 1986
Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.
19
1
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Compute the least value of f(p) - ISL 1986
A tetrahedron
A
B
C
D
ABCD
A
BC
D
is given such that
A
D
=
B
C
=
a
;
A
C
=
B
D
=
b
;
A
B
⋅
C
D
=
c
2
AD = BC = a; AC = BD = b; AB\cdot CD = c^2
A
D
=
BC
=
a
;
A
C
=
B
D
=
b
;
A
B
⋅
C
D
=
c
2
. Let
f
(
P
)
=
A
P
+
B
P
+
C
P
+
D
P
f(P) = AP + BP + CP + DP
f
(
P
)
=
A
P
+
BP
+
CP
+
D
P
, where
P
P
P
is an arbitrary point in space. Compute the least value of
f
(
P
)
.
f(P).
f
(
P
)
.
18
1
Hide problems
circumscribable quadrangles - ISL 1986
Let
A
X
,
B
Y
,
C
Z
AX,BY,CZ
A
X
,
B
Y
,
CZ
be three cevians concurrent at an interior point
D
D
D
of a triangle
A
B
C
ABC
A
BC
. Prove that if two of the quadrangles
D
Y
A
Z
,
D
Z
B
X
,
D
X
C
Y
DY AZ,DZBX,DXCY
D
Y
A
Z
,
D
ZBX
,
D
XC
Y
are circumscribable, so is the third.
15
1
Hide problems
Very nice geo problem of ISL 1986
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral whose vertices do not lie on a circle. Let
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
be a quadrangle such that
A
′
,
B
′
,
C
′
,
D
′
A',B', C',D'
A
′
,
B
′
,
C
′
,
D
′
are the centers of the circumcircles of triangles
B
C
D
,
A
C
D
,
A
B
D
BCD,ACD,ABD
BC
D
,
A
C
D
,
A
B
D
, and
A
B
C
ABC
A
BC
. We write
T
(
A
B
C
D
)
=
A
′
B
′
C
′
D
′
T (ABCD) = A'B'C'D'
T
(
A
BC
D
)
=
A
′
B
′
C
′
D
′
. Let us define
A
′
′
B
′
′
C
′
′
D
′
′
=
T
(
A
′
B
′
C
′
D
′
)
=
T
(
T
(
A
B
C
D
)
)
.
A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).
A
′′
B
′′
C
′′
D
′′
=
T
(
A
′
B
′
C
′
D
′
)
=
T
(
T
(
A
BC
D
))
.
(a) Prove that
A
B
C
D
ABCD
A
BC
D
and
A
′
′
B
′
′
C
′
′
D
′
′
A''B''C''D''
A
′′
B
′′
C
′′
D
′′
are similar.(b) The ratio of similitude depends on the size of the angles of
A
B
C
D
ABCD
A
BC
D
. Determine this ratio.
14
1
Hide problems
incenters of XYZ and ABC are collinear - ISL 1986
The circle inscribed in a triangle
A
B
C
ABC
A
BC
touches the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
in
D
,
E
,
F
D,E, F
D
,
E
,
F
, respectively, and
X
,
Y
,
Z
X, Y,Z
X
,
Y
,
Z
are the midpoints of
E
F
,
F
D
,
D
E
EF, FD,DE
EF
,
F
D
,
D
E
, respectively. Prove that the centers of the inscribed circle and of the circles around
X
Y
Z
XYZ
X
Y
Z
and
A
B
C
ABC
A
BC
are collinear.
13
1
Hide problems
Find the probablity - ISL 1986
A particle moves from
(
0
,
0
)
(0, 0)
(
0
,
0
)
to
(
n
,
n
)
(n, n)
(
n
,
n
)
directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At
(
n
,
y
)
,
y
<
n
(n, y), y < n
(
n
,
y
)
,
y
<
n
, it stays there if a head comes up and at
(
x
,
n
)
,
x
<
n
(x, n), x < n
(
x
,
n
)
,
x
<
n
, it stays there if a tail comes up. Let
k
k
k
be a fixed positive integer. Find the probability that the particle needs exactly
2
n
+
k
2n+k
2
n
+
k
tosses to reach
(
n
,
n
)
.
(n, n).
(
n
,
n
)
.
11
1
Hide problems
Function defined on points - ISL 1986
Let
f
(
n
)
f(n)
f
(
n
)
be the least number of distinct points in the plane such that for each
k
=
1
,
2
,
⋯
,
n
k = 1, 2, \cdots, n
k
=
1
,
2
,
⋯
,
n
there exists a straight line containing exactly
k
k
k
of these points. Find an explicit expression for
f
(
n
)
.
f(n).
f
(
n
)
.
Simplified version.Show that
f
(
n
)
=
[
n
+
1
2
]
[
n
+
2
2
]
.
f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].
f
(
n
)
=
[
2
n
+
1
]
[
2
n
+
2
]
.
Where
[
x
]
[x]
[
x
]
denoting the greatest integer not exceeding
x
.
x.
x
.
10
1
Hide problems
Fair game - ISL 1986
Three persons
A
,
B
,
C
A,B,C
A
,
B
,
C
, are playing the following game:A
k
k
k
-element subset of the set
{
1
,
.
.
.
,
1986
}
\{1, . . . , 1986\}
{
1
,
...
,
1986
}
is randomly chosen, with an equal probability of each choice, where
k
k
k
is a fixed positive integer less than or equal to
1986
1986
1986
. The winner is
A
,
B
A,B
A
,
B
or
C
C
C
, respectively, if the sum of the chosen numbers leaves a remainder of
0
,
1
0, 1
0
,
1
, or
2
2
2
when divided by
3
3
3
.For what values of
k
k
k
is this game a fair one? (A game is fair if the three outcomes are equally probable.)
8
1
Hide problems
m distinct teams may be listed - ISL 1986
From a collection of
n
n
n
persons
q
q
q
distinct two-member teams are selected and ranked
1
,
⋯
,
q
1, \cdots, q
1
,
⋯
,
q
(no ties). Let
m
m
m
be the least integer larger than or equal to
2
q
/
n
2q/n
2
q
/
n
. Show that there are
m
m
m
distinct teams that may be listed so that : (i) each pair of consecutive teams on the list have one member in common and (ii) the chain of teams on the list are in rank order.Alternative formulation. Given a graph with
n
n
n
vertices and
q
q
q
edges numbered
1
,
⋯
,
q
1, \cdots , q
1
,
⋯
,
q
, show that there exists a chain of
m
m
m
edges,
m
≥
2
q
n
m \geq \frac{2q}{n}
m
≥
n
2
q
, each two consecutive edges having a common vertex, arranged monotonically with respect to the numbering.
7
1
Hide problems
System of equations with sequence - ISL 1986
Let real numbers
x
1
,
x
2
,
⋯
,
x
n
x_1, x_2, \cdots , x_n
x
1
,
x
2
,
⋯
,
x
n
satisfy
0
<
x
1
<
x
2
<
⋯
<
x
n
<
1
0 < x_1 < x_2 < \cdots< x_n < 1
0
<
x
1
<
x
2
<
⋯
<
x
n
<
1
and set
x
0
=
0
,
x
n
+
1
=
1
x_0 = 0, x_{n+1} = 1
x
0
=
0
,
x
n
+
1
=
1
. Suppose that these numbers satisfy the following system of equations: \sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \text{where } i = 1, 2, . . ., n. Prove that
x
n
+
1
−
i
=
1
−
x
i
x_{n+1-i} = 1- x_i
x
n
+
1
−
i
=
1
−
x
i
for
i
=
1
,
2
,
.
.
.
,
n
.
i = 1, 2, . . . , n.
i
=
1
,
2
,
...
,
n
.
6
1
Hide problems
Find four positive integers - ISL 1986 P6
Find four positive integers each not exceeding
70000
70000
70000
and each having more than
100
100
100
divisors.
3
1
Hide problems
Determine y - ISL 1986 p3
Let
A
,
B
A, B
A
,
B
, and
C
C
C
be three points on the edge of a circular chord such that
B
B
B
is due west of
C
C
C
and
A
B
C
ABC
A
BC
is an equilateral triangle whose side is
86
86
86
meters long. A boy swam from
A
A
A
directly toward
B
B
B
. After covering a distance of
x
x
x
meters, he turned and swam westward, reaching the shore after covering a distance of
y
y
y
meters. If
x
x
x
and
y
y
y
are both positive integers, determine
y
.
y.
y
.
2
1
Hide problems
Decmial Expansion - ISL 1986
Let
f
(
x
)
=
x
n
f(x) = x^n
f
(
x
)
=
x
n
where
n
n
n
is a fixed positive integer and
x
=
1
,
2
,
⋯
.
x =1, 2, \cdots .
x
=
1
,
2
,
⋯
.
Is the decimal expansion
a
=
0.
f
(
1
)
f
(
2
)
f
(
3
)
.
.
.
a = 0.f (1)f(2)f(3) . . .
a
=
0.
f
(
1
)
f
(
2
)
f
(
3
)
...
rational for any value of
n
n
n
?The decimal expansion of a is defined as follows: If
f
(
x
)
=
d
1
(
x
)
d
2
(
x
)
⋯
d
r
(
x
)
(
x
)
f(x) = d_1(x)d_2(x) \cdots d_{r(x)}(x)
f
(
x
)
=
d
1
(
x
)
d
2
(
x
)
⋯
d
r
(
x
)
(
x
)
is the decimal expansion of
f
(
x
)
f(x)
f
(
x
)
, then
a
=
0.1
d
1
(
2
)
d
2
(
2
)
⋯
d
r
(
2
)
(
2
)
d
1
(
3
)
.
.
.
d
r
(
3
)
(
3
)
d
1
(
4
)
⋯
.
a = 0.1d_1(2)d_2(2) \cdots d_{r(2)}(2)d_1(3) . . . d_{r(3)}(3)d_1(4) \cdots .
a
=
0.1
d
1
(
2
)
d
2
(
2
)
⋯
d
r
(
2
)
(
2
)
d
1
(
3
)
...
d
r
(
3
)
(
3
)
d
1
(
4
)
⋯
.
4
1
Hide problems
at least 3n + 3 different solutions
Provided the equation
x
y
z
=
p
n
(
x
+
y
+
z
)
xyz = p^n(x + y + z)
x
yz
=
p
n
(
x
+
y
+
z
)
where
p
≥
3
p \geq 3
p
≥
3
is a prime and
n
∈
N
n \in \mathbb{N}
n
∈
N
. Prove that the equation has at least
3
n
+
3
3n + 3
3
n
+
3
different solutions
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
with natural numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
and
x
<
y
<
z
x < y < z
x
<
y
<
z
. Prove the same for
p
>
3
p > 3
p
>
3
being an odd integer.