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Problems
Contests
International Contests
IMO Shortlist
1984 IMO Shortlist
1984 IMO Shortlist
Part of
IMO Shortlist
Subcontests
(14)
20
1
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Solve the logarithmic inequality (ISL84)
Determine all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of positive real numbers with
a
≠
1
a \neq 1
a
=
1
such that
log
a
b
<
log
a
+
1
(
b
+
1
)
.
\log_a b < \log_{a+1} (b + 1).
lo
g
a
b
<
lo
g
a
+
1
(
b
+
1
)
.
19
1
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Find the harmonic mean of the 1985th row
The harmonic table is a triangular array:
1
1
1
1
2
1
2
\frac 12 \qquad \frac 12
2
1
2
1
1
3
1
6
1
3
\frac 13 \qquad \frac 16 \qquad \frac 13
3
1
6
1
3
1
1
4
1
12
1
12
1
4
\frac 14 \qquad \frac 1{12} \qquad \frac 1{12} \qquad \frac 14
4
1
12
1
12
1
4
1
Where
a
n
,
1
=
1
n
a_{n,1} = \frac 1n
a
n
,
1
=
n
1
and
a
n
,
k
+
1
=
a
n
−
1
,
k
−
a
n
,
k
a_{n,k+1} = a_{n-1,k} - a_{n,k}
a
n
,
k
+
1
=
a
n
−
1
,
k
−
a
n
,
k
for
1
≤
k
≤
n
−
1.
1 \leq k \leq n-1.
1
≤
k
≤
n
−
1.
Find the harmonic mean of the
198
5
t
h
1985^{th}
198
5
t
h
row.
18
1
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Determine the radius of k
Inside triangle
A
B
C
ABC
A
BC
there are three circles
k
1
,
k
2
,
k
3
k_1, k_2, k_3
k
1
,
k
2
,
k
3
each of which is tangent to two sides of the triangle and to its incircle
k
k
k
. The radii of
k
1
,
k
2
,
k
3
k_1, k_2, k_3
k
1
,
k
2
,
k
3
are
1
,
4
1, 4
1
,
4
, and
9
9
9
. Determine the radius of
k
.
k.
k
.
17
1
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Discordant pairs of the permutation
In a permutation
(
x
1
,
x
2
,
…
,
x
n
)
(x_1, x_2, \dots , x_n)
(
x
1
,
x
2
,
…
,
x
n
)
of the set
1
,
2
,
…
,
n
1, 2, \dots , n
1
,
2
,
…
,
n
we call a pair
(
x
i
,
x
j
)
(x_i, x_j )
(
x
i
,
x
j
)
discordant if
i
<
j
i < j
i
<
j
and
x
i
>
x
j
x_i > x_j
x
i
>
x
j
. Let
d
(
n
,
k
)
d(n, k)
d
(
n
,
k
)
be the number of such permutations with exactly
k
k
k
discordant pairs. Find
d
(
n
,
2
)
d(n, 2)
d
(
n
,
2
)
and
d
(
n
,
3
)
.
d(n, 3).
d
(
n
,
3
)
.
15
1
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Equilateral triangles in the exterior of ABC
Angles of a given triangle
A
B
C
ABC
A
BC
are all smaller than
12
0
∘
120^\circ
12
0
∘
. Equilateral triangles
A
F
B
,
B
D
C
AFB, BDC
A
FB
,
B
D
C
and
C
E
A
CEA
CE
A
are constructed in the exterior of
A
B
C
ABC
A
BC
.(a) Prove that the lines
A
D
,
B
E
AD, BE
A
D
,
BE
, and
C
F
CF
CF
pass through one point
S
.
S.
S
.
(b) Prove that
S
D
+
S
E
+
S
F
=
2
(
S
A
+
S
B
+
S
C
)
.
SD + SE + SF = 2(SA + SB + SC).
S
D
+
SE
+
SF
=
2
(
S
A
+
SB
+
SC
)
.
13
1
Hide problems
Volume of a tetrahedron
Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume
1
1
1
does not exceed
2
3
π
.
\frac{2}{3 \pi}.
3
π
2
.
11
1
Hide problems
Solve the equation
Let
n
n
n
be a positive integer and
a
1
,
a
2
,
…
,
a
2
n
a_1, a_2, \dots , a_{2n}
a
1
,
a
2
,
…
,
a
2
n
mutually distinct integers. Find all integers
x
x
x
satisfying
(
x
−
a
1
)
⋅
(
x
−
a
2
)
⋯
(
x
−
a
2
n
)
=
(
−
1
)
n
(
n
!
)
2
.
(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.
(
x
−
a
1
)
⋅
(
x
−
a
2
)
⋯
(
x
−
a
2
n
)
=
(
−
1
)
n
(
n
!
)
2
.
10
1
Hide problems
Product of five consecutive positive integers (ISL84)
Prove that the product of five consecutive positive integers cannot be the square of an integer.
9
1
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System has exactly one real solution
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive numbers with
a
+
b
+
c
=
3
2
\sqrt a +\sqrt b +\sqrt c = \frac{\sqrt 3}{2}
a
+
b
+
c
=
2
3
. Prove that the system of equations
y
−
a
+
z
−
a
=
1
,
\sqrt{y-a}+\sqrt{z-a}=1,
y
−
a
+
z
−
a
=
1
,
z
−
b
+
x
−
b
=
1
,
\sqrt{z-b}+\sqrt{x-b}=1,
z
−
b
+
x
−
b
=
1
,
x
−
c
+
y
−
c
=
1
\sqrt{x-c}+\sqrt{y-c}=1
x
−
c
+
y
−
c
=
1
has exactly one solution
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
in real numbers.
7
1
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Numbering fields of chessboard
(a) Decide whether the fields of the
8
×
8
8 \times 8
8
×
8
chessboard can be numbered by the numbers
1
,
2
,
…
,
64
1, 2, \dots , 64
1
,
2
,
…
,
64
in such a way that the sum of the four numbers in each of its parts of one of the formshttp://www.artofproblemsolving.com/Forum/download/file.php?id=28446is divisible by four.(b) Solve the analogous problem forhttp://www.artofproblemsolving.com/Forum/download/file.php?id=28447
6
1
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Sequence of positive integers
Let
c
c
c
be a positive integer. The sequence
{
f
n
}
\{f_n\}
{
f
n
}
is defined as follows: f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 (n \geq 2). Show that for each
k
∈
N
k \in \mathbb N
k
∈
N
there exists
r
∈
N
r \in \mathbb N
r
∈
N
such that
f
k
f
k
+
1
=
f
r
.
f_kf_{k+1}= f_r.
f
k
f
k
+
1
=
f
r
.
3
1
Hide problems
On divisors of n
Find all positive integers
n
n
n
such that
n
=
d
6
2
+
d
7
2
−
1
,
n=d_6^2+d_7^2-1,
n
=
d
6
2
+
d
7
2
−
1
,
where
1
=
d
1
<
d
2
<
⋯
<
d
k
=
n
1 = d_1 < d_2 < \cdots < d_k = n
1
=
d
1
<
d
2
<
⋯
<
d
k
=
n
are all positive divisors of the number
n
.
n.
n
.
2
1
Hide problems
Diophantine equations
Prove:(a) There are infinitely many triples of positive integers
m
,
n
,
p
m, n, p
m
,
n
,
p
such that
4
m
n
−
m
−
n
=
p
2
−
1.
4mn - m- n = p^2 - 1.
4
mn
−
m
−
n
=
p
2
−
1.
(b) There are no positive integers
m
,
n
,
p
m, n, p
m
,
n
,
p
such that
4
m
n
−
m
−
n
=
p
2
.
4mn - m- n = p^2.
4
mn
−
m
−
n
=
p
2
.
1
1
Hide problems
System of equations with n variables
Find all solutions of the following system of
n
n
n
equations in
n
n
n
variables:
x
1
∣
x
1
∣
−
(
x
1
−
a
)
∣
x
1
−
a
∣
=
x
2
∣
x
2
∣
,
x
2
∣
x
2
∣
−
(
x
2
−
a
)
∣
x
2
−
a
∣
=
x
3
∣
x
3
∣
,
⋮
x
n
∣
x
n
∣
−
(
x
n
−
a
)
∣
x
n
−
a
∣
=
x
1
∣
x
1
∣
\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}
x
1
∣
x
1
∣
−
(
x
1
−
a
)
∣
x
1
−
a
∣
=
x
2
∣
x
2
∣
,
x
2
∣
x
2
∣
−
(
x
2
−
a
)
∣
x
2
−
a
∣
=
x
3
∣
x
3
∣
,
⋮
x
n
∣
x
n
∣
−
(
x
n
−
a
)
∣
x
n
−
a
∣
=
x
1
∣
x
1
∣
where
a
a
a
is a given number.