MathDB
Problems
Contests
International Contests
IMO Shortlist
1969 IMO Shortlist
1969 IMO Shortlist
Part of
IMO Shortlist
Subcontests
(67)
68
1
Hide problems
Prove we can form convex quad given 5 points
(
U
S
S
5
)
(USS 5)
(
U
SS
5
)
Given
5
5
5
points in the plane, no three of which are collinear, prove that we can choose
4
4
4
points among them that form a convex quadrilateral.
71
1
Hide problems
Position of points to get sum of minimum to be 180 degrees
(
Y
U
G
3
)
(YUG 3)
(
Y
U
G
3
)
Let four points
A
i
(
i
=
1
,
2
,
3
,
4
)
A_i (i = 1, 2, 3, 4)
A
i
(
i
=
1
,
2
,
3
,
4
)
in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by
S
.
S.
S
.
What is the exact placement of the points
A
i
A_i
A
i
if
S
=
18
0
∘
S = 180^{\circ}
S
=
18
0
∘
?
59
1
Hide problems
Construct a continuous function satisfying conditions
(
S
W
E
2
)
(SWE 2)
(
S
W
E
2
)
For each
λ
(
0
<
λ
<
1
\lambda (0 < \lambda < 1
λ
(
0
<
λ
<
1
and
λ
=
1
n
\lambda = \frac{1}{n}
λ
=
n
1
for all
n
=
1
,
2
,
3
,
⋯
)
n = 1, 2, 3, \cdots)
n
=
1
,
2
,
3
,
⋯
)
, construct a continuous function
f
f
f
such that there do not exist
x
,
y
x, y
x
,
y
with
0
<
λ
<
y
=
x
+
λ
≤
1
0 < \lambda < y = x + \lambda \le 1
0
<
λ
<
y
=
x
+
λ
≤
1
for which
f
(
x
)
=
f
(
y
)
.
f(x) = f(y).
f
(
x
)
=
f
(
y
)
.
56
1
Hide problems
Inequaity on natural numbers and nth root.
Let
a
a
a
and
b
b
b
be two natural numbers that have an equal number
n
n
n
of digits in their decimal expansions. The first
m
m
m
digits (from left to right) of the numbers
a
a
a
and
b
b
b
are equal. Prove that if
m
>
n
2
,
m >\frac{n}{2},
m
>
2
n
,
then
a
1
n
−
b
1
n
<
1
n
a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}
a
n
1
−
b
n
1
<
n
1
54
1
Hide problems
f(x) is divisible by 3 for k, k+1, k+2. Prove for all x
(
P
O
L
3
)
(POL 3)
(
PO
L
3
)
Given a polynomial
f
(
x
)
f(x)
f
(
x
)
with integer coefficients whose value is divisible by
3
3
3
for three integers
k
,
k
+
1
,
k, k + 1,
k
,
k
+
1
,
and
k
+
2
k + 2
k
+
2
. Prove that
f
(
m
)
f(m)
f
(
m
)
is divisible by
3
3
3
for all integers
m
.
m.
m
.
58
1
Hide problems
Prove there exists a monochromatic triangle in space
(
S
W
E
1
)
(SWE 1)
(
S
W
E
1
)
Six points
P
1
,
.
.
.
,
P
6
P_1, . . . , P_6
P
1
,
...
,
P
6
are given in
3
−
3-
3
−
dimensional space such that no four of them lie in the same plane. Each of the line segments
P
j
P
k
P_jP_k
P
j
P
k
is colored black or white. Prove that there exists one triangle
P
j
P
k
P
l
P_jP_kP_l
P
j
P
k
P
l
whose edges are of the same color.
66
1
Hide problems
Polynomial inequality
(
U
S
S
3
)
(USS 3)
(
U
SS
3
)
(
a
)
(a)
(
a
)
Prove that if
0
≤
a
0
≤
a
1
≤
a
2
,
0 \le a_0 \le a_1 \le a_2,
0
≤
a
0
≤
a
1
≤
a
2
,
then
(
a
0
+
a
1
x
−
a
2
x
2
)
2
≤
(
a
0
+
a
1
+
a
2
)
2
(
1
+
1
2
x
+
1
3
x
2
+
1
2
x
3
+
x
4
)
(a_0 + a_1x - a_2x^2)^2 \le (a_0 + a_1 + a_2)^2\left(1 +\frac{1}{2}x+\frac{1}{3}x^2+\frac{1}{2}x^3+x^4\right)
(
a
0
+
a
1
x
−
a
2
x
2
)
2
≤
(
a
0
+
a
1
+
a
2
)
2
(
1
+
2
1
x
+
3
1
x
2
+
2
1
x
3
+
x
4
)
(
b
)
(b)
(
b
)
Formulate and prove the analogous result for polynomials of third degree.
53
1
Hide problems
Find locus of M where MA^2+MB^2=MC^2+MD^2
(
P
O
L
2
)
(POL 2)
(
PO
L
2
)
Given two segments
A
B
AB
A
B
and
C
D
CD
C
D
not in the same plane, find the locus of points
M
M
M
such that
M
A
2
+
M
B
2
=
M
C
2
+
M
D
2
.
MA^2 +MB^2 = MC^2 +MD^2.
M
A
2
+
M
B
2
=
M
C
2
+
M
D
2
.
52
1
Hide problems
Odd edged regular polygon can't be partitioned into 4 pieces
Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.
51
1
Hide problems
Find number of squares cut by a curve.
(
N
E
T
6
)
(NET 6)
(
NET
6
)
A curve determined by
y
=
x
2
−
10
x
+
52
,
0
≤
x
≤
100
,
y =\sqrt{x^2 - 10x+ 52}, 0\le x \le 100,
y
=
x
2
−
10
x
+
52
,
0
≤
x
≤
100
,
is constructed in a rectangular grid. Determine the number of squares cut by the curve.
50
1
Hide problems
Construct pentagon given pentagon by external angle bisector
(
N
E
T
5
)
(NET 5)
(
NET
5
)
The bisectors of the exterior angles of a pentagon
B
1
B
2
B
3
B
4
B
5
B_1B_2B_3B_4B_5
B
1
B
2
B
3
B
4
B
5
form another pentagon
A
1
A
2
A
3
A
4
A
5
.
A_1A_2A_3A_4A_5.
A
1
A
2
A
3
A
4
A
5
.
Construct
B
1
B
2
B
3
B
4
B
5
B_1B_2B_3B_4B_5
B
1
B
2
B
3
B
4
B
5
from the given pentagon
A
1
A
2
A
3
A
4
A
5
.
A_1A_2A_3A_4A_5.
A
1
A
2
A
3
A
4
A
5
.
49
1
Hide problems
Quarter circle arcs forming a closed loop.
(
N
E
T
4
)
(NET 4)
(
NET
4
)
A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number of times through the pieces in the clockwise direction and an even number of times in the counterclockwise direction. Also, prove that the number of pieces is divisible by
4.
4.
4.
69
1
Hide problems
Inequality for 3 positive reals
(
Y
U
G
1
)
(YUG 1)
(
Y
U
G
1
)
Suppose that positive real numbers
x
1
,
x
2
,
x
3
x_1, x_2, x_3
x
1
,
x
2
,
x
3
satisfy
x
1
x
2
x
3
>
1
,
x
1
+
x
2
+
x
3
<
1
x
1
+
1
x
2
+
1
x
3
x_1x_2x_3 > 1, x_1 + x_2 + x_3 <\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}
x
1
x
2
x
3
>
1
,
x
1
+
x
2
+
x
3
<
x
1
1
+
x
2
1
+
x
3
1
Prove that:
(
a
)
(a)
(
a
)
None of
x
1
,
x
2
,
x
3
x_1, x_2, x_3
x
1
,
x
2
,
x
3
equals
1
1
1
.
(
b
)
(b)
(
b
)
Exactly one of these numbers is less than
1.
1.
1.
48
1
Hide problems
Maxima with 6 inequalities and how many ways to obtain it.
(
N
E
T
3
)
(NET 3)
(
NET
3
)
Let
x
1
,
x
2
,
x
3
,
x
4
,
x_1, x_2, x_3, x_4,
x
1
,
x
2
,
x
3
,
x
4
,
and
x
5
x_5
x
5
be positive integers satisfying
x
1
+
x
2
+
x
3
+
x
4
+
x
5
=
1000
,
x_1 +x_2 +x_3 +x_4 +x_5 = 1000,
x
1
+
x
2
+
x
3
+
x
4
+
x
5
=
1000
,
x
1
−
x
2
+
x
3
−
x
4
+
x
5
>
0
,
x_1 -x_2 +x_3 -x_4 +x_5 > 0,
x
1
−
x
2
+
x
3
−
x
4
+
x
5
>
0
,
x
1
+
x
2
−
x
3
+
x
4
−
x
5
>
0
,
x_1 +x_2 -x_3 +x_4 -x_5 > 0,
x
1
+
x
2
−
x
3
+
x
4
−
x
5
>
0
,
−
x
1
+
x
2
+
x
3
−
x
4
+
x
5
>
0
,
-x_1 +x_2 +x_3 -x_4 +x_5 > 0,
−
x
1
+
x
2
+
x
3
−
x
4
+
x
5
>
0
,
x
1
−
x
2
+
x
3
+
x
4
−
x
5
>
0
,
x_1 -x_2 +x_3 +x_4 -x_5 > 0,
x
1
−
x
2
+
x
3
+
x
4
−
x
5
>
0
,
−
x
1
+
x
2
−
x
3
+
x
4
+
x
5
>
0
-x_1 +x_2 -x_3 +x_4 +x_5 > 0
−
x
1
+
x
2
−
x
3
+
x
4
+
x
5
>
0
(
a
)
(a)
(
a
)
Find the maximum of
(
x
1
+
x
3
)
x
2
+
x
4
(x_1 + x_3)^{x_2+x_4}
(
x
1
+
x
3
)
x
2
+
x
4
(
b
)
(b)
(
b
)
In how many different ways can we choose
x
1
,
.
.
.
,
x
5
x_1, . . . , x_5
x
1
,
...
,
x
5
to obtain the desired maximum?
60
1
Hide problems
Least n such that n disks cover a set of points
(
S
W
E
3
)
(SWE 3)
(
S
W
E
3
)
Find the natural number
n
n
n
with the following properties:
(
1
)
(1)
(
1
)
Let
S
=
{
P
1
,
P
2
,
⋯
}
S = \{P_1, P_2, \cdots\}
S
=
{
P
1
,
P
2
,
⋯
}
be an arbitrary finite set of points in the plane, and
r
j
r_j
r
j
the distance from
P
j
P_j
P
j
to the origin
O
.
O.
O
.
We assign to each
P
j
P_j
P
j
the closed disk
D
j
D_j
D
j
with center
P
j
P_j
P
j
and radius
r
j
r_j
r
j
. Then some
n
n
n
of these disks contain all points of
S
.
S.
S
.
(
2
)
(2)
(
2
)
n
n
n
is the smallest integer with the above property.
70
1
Hide problems
At most 200m implies at least 100 m from each side
(
Y
U
G
2
)
(YUG 2)
(
Y
U
G
2
)
A park has the shape of a convex pentagon of area
50000
3
m
2
50000\sqrt{3} m^2
50000
3
m
2
. A man standing at an interior point
O
O
O
of the park notices that he stands at a distance of at most
200
m
200 m
200
m
from each vertex of the pentagon. Prove that he stands at a distance of at least
100
m
100 m
100
m
from each side of the pentagon.
65
1
Hide problems
Identity with lots of square roots.
(
U
S
S
2
)
(USS 2)
(
U
SS
2
)
Prove that for
a
>
b
2
,
a > b^2,
a
>
b
2
,
the identity
a
−
b
a
+
b
a
−
b
a
+
⋯
=
a
−
3
4
b
2
−
1
2
b
{\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+\cdots}}}}=\sqrt{a-\frac{3}{4}b^2}-\frac{1}{2}b}
a
−
b
a
+
b
a
−
b
a
+
⋯
=
a
−
4
3
b
2
−
2
1
b
62
1
Hide problems
Find numbers being difference of two squares
Which natural numbers can be expressed as the difference of squares of two integers?
63
1
Hide problems
Infinte numbers not expressible as sum of three squares
(
S
W
E
6
)
(SWE 6)
(
S
W
E
6
)
Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.
61
1
Hide problems
a_{n+1}=2a_n+2^n, prove a_n is power of 2 if n is.
(
S
W
E
4
)
(SWE 4)
(
S
W
E
4
)
Let
a
0
,
a
1
,
a
2
,
⋯
a_0, a_1, a_2, \cdots
a
0
,
a
1
,
a
2
,
⋯
be determined with
a
0
=
0
,
a
n
+
1
=
2
a
n
+
2
n
a_0 = 0, a_{n+1} = 2a_n + 2^n
a
0
=
0
,
a
n
+
1
=
2
a
n
+
2
n
. Prove that if
n
n
n
is power of
2
2
2
, then so is
a
n
a_n
a
n
64
1
Hide problems
Prove that (n!)!>n[(n-1)!]^{n!}
(
U
S
S
1
)
(USS 1)
(
U
SS
1
)
Prove that for a natural number
n
>
2
,
(
n
!
)
!
>
n
[
(
n
−
1
)
!
]
n
!
.
n > 2, (n!)! > n[(n - 1)!]^{n!}.
n
>
2
,
(
n
!)!
>
n
[(
n
−
1
)!
]
n
!
.
46
1
Hide problems
Positioning of points to get max and min area.
(
N
E
T
1
)
(NET 1)
(
NET
1
)
The vertices of an
(
n
+
1
)
−
(n + 1)-
(
n
+
1
)
−
gon are placed on the edges of a regular
n
−
n-
n
−
gon so that the perimeter of the
n
−
n-
n
−
gon is divided into equal parts. How does one choose these
n
+
1
n + 1
n
+
1
points in order to obtain the
(
n
+
1
)
−
(n + 1)-
(
n
+
1
)
−
gon with
(
a
)
(a)
(
a
)
maximal area;
(
b
)
(b)
(
b
)
minimal area?
44
1
Hide problems
Circumradius given sides of triangle.
(
M
O
N
5
)
(MON 5)
(
MON
5
)
Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation
x
2
−
a
x
+
b
=
0
x^2 - ax + b = 0
x
2
−
a
x
+
b
=
0
.
43
1
Hide problems
3 divides p^{2m+1}+q^{2m+1}
(
M
O
N
4
)
(MON 4)
(
MON
4
)
Let
p
p
p
and
q
q
q
be two prime numbers greater than
3.
3.
3.
Prove that if their difference is
2
n
2^n
2
n
, then for any two integers
m
m
m
and
n
,
n,
n
,
the number
S
=
p
2
m
+
1
+
q
2
m
+
1
S = p^{2m+1} + q^{2m+1}
S
=
p
2
m
+
1
+
q
2
m
+
1
is divisible by
3.
3.
3.
42
1
Hide problems
All possible subsets with condition.
(
M
O
N
3
)
(MON 3)
(
MON
3
)
Let
A
k
(
1
≤
k
≤
h
)
A_k (1 \le k \le h)
A
k
(
1
≤
k
≤
h
)
be
n
−
n-
n
−
element sets such that each two of them have a nonempty intersection. Let
A
A
A
be the union of all the sets
A
k
,
A_k,
A
k
,
and let
B
B
B
be a subset of
A
A
A
such that for each
k
(
1
≤
k
≤
h
)
k (1\le k \le h)
k
(
1
≤
k
≤
h
)
the intersection of
A
k
A_k
A
k
and
B
B
B
consists of exactly two different elements
a
k
a_k
a
k
and
b
k
b_k
b
k
. Find all subsets
X
X
X
of the set
A
A
A
with
r
r
r
elements satisfying the condition that for at least one index
k
,
k,
k
,
both elements
a
k
a_k
a
k
and
b
k
b_k
b
k
belong to
X
X
X
.
41
1
Hide problems
Expression for solution of equation.
(
M
O
N
2
)
(MON 2)
(
MON
2
)
Given reals
x
0
,
x
1
,
α
,
β
x_0, x_1, \alpha, \beta
x
0
,
x
1
,
α
,
β
, find an expression for the solution of the system
x
n
+
2
−
α
x
n
+
1
−
β
x
n
=
0
,
n
=
0
,
1
,
2
,
…
x_{n+2} -\alpha x_{n+1} -\beta x_n = 0, \qquad n= 0, 1, 2, \ldots
x
n
+
2
−
α
x
n
+
1
−
β
x
n
=
0
,
n
=
0
,
1
,
2
,
…
40
1
Hide problems
Number of five digit numbers with specific property.
(
M
O
N
1
)
(MON 1)
(
MON
1
)
Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.
39
1
Hide problems
Points on boundary of cube.
(
H
U
N
6
)
(HUN 6)
(
H
U
N
6
)
Find the positions of three points
A
,
B
,
C
A,B,C
A
,
B
,
C
on the boundary of a unit cube such that
m
i
n
{
A
B
,
A
C
,
B
C
}
min\{AB,AC,BC\}
min
{
A
B
,
A
C
,
BC
}
is the greatest possible.
38
1
Hide problems
Evaluate a trigonometric expression.
(
H
U
N
5
)
(HUN 5)
(
H
U
N
5
)
Let
r
r
r
and
m
(
r
≤
m
)
m (r \le m)
m
(
r
≤
m
)
be natural numbers and
A
k
=
2
k
−
1
2
m
π
Ak =\frac{2k-1}{2m}\pi
A
k
=
2
m
2
k
−
1
π
. Evaluate
1
m
2
∑
k
=
1
m
∑
l
=
1
m
sin
(
r
A
k
)
sin
(
r
A
l
)
cos
(
r
A
k
−
r
A
l
)
\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)
m
2
1
k
=
1
∑
m
l
=
1
∑
m
sin
(
r
A
k
)
sin
(
r
A
l
)
cos
(
r
A
k
−
r
A
l
)
37
1
Hide problems
Trigonometric equation and y=0
(
H
U
N
4
)
(HUN 4)
(
H
U
N
4
)
IMO2 If
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, . . . , a_n
a
1
,
a
2
,
...
,
a
n
are real constants, and if
y
=
cos
(
a
1
+
x
)
+
2
cos
(
a
2
+
x
)
+
⋯
+
n
cos
(
a
n
+
x
)
y = \cos(a_1 + x) +2\cos(a_2+x)+ \cdots+ n \cos(a_n + x)
y
=
cos
(
a
1
+
x
)
+
2
cos
(
a
2
+
x
)
+
⋯
+
n
cos
(
a
n
+
x
)
has two zeros
x
1
x_1
x
1
and
x
2
x_2
x
2
whose difference is not a multiple of
π
\pi
π
, prove that
y
=
0.
y = 0.
y
=
0.
36
1
Hide problems
4000 points with a special property
(
H
U
N
3
)
(HUN 3)
(
H
U
N
3
)
In the plane
4000
4000
4000
points are given such that each line passes through at most
2
2
2
of these points. Prove that there exist
1000
1000
1000
disjoint quadrilaterals in the plane with vertices at these points.
35
1
Hide problems
sigma 1/n^3 <5/4
(
H
U
N
2
)
(HUN 2)
(
H
U
N
2
)
Prove that
1
+
1
2
3
+
1
3
3
+
⋯
+
1
n
3
<
5
4
1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{5}{4}
1
+
2
3
1
+
3
3
1
+
⋯
+
n
3
1
<
4
5
34
1
Hide problems
Divisibility problem by a^2+ab+b^2
(
H
U
N
1
)
(HUN 1)
(
H
U
N
1
)
Let
a
a
a
and
b
b
b
be arbitrary integers. Prove that if
k
k
k
is an integer not divisible by
3
3
3
, then
(
a
+
b
)
2
k
+
a
2
k
+
b
2
k
(a + b)^{2k}+ a^{2k} +b^{2k}
(
a
+
b
)
2
k
+
a
2
k
+
b
2
k
is divisible by
a
2
+
a
b
+
b
2
a^2 +ab+ b^2
a
2
+
ab
+
b
2
33
1
Hide problems
Prove that we can cover a region with 8 disks
(
G
D
R
5
)
(GDR 5)
(
G
D
R
5
)
Given a ring
G
G
G
in the plane bounded by two concentric circles with radii
R
R
R
and
R
2
\frac{R}{2}
2
R
, prove that we can cover this region with
8
8
8
disks of radius
2
R
5
\frac{2R}{5}
5
2
R
. (A region is covered if each of its points is inside or on the border of some disk.)
32
1
Hide problems
Max no. of regions n circles can partition a sphere.
(
G
D
R
4
)
(GDR 4)
(
G
D
R
4
)
Find the maximal number of regions into which a sphere can be partitioned by
n
n
n
circles.
31
1
Hide problems
Recurrence and permutations.
(
G
D
R
3
)
(GDR 3)
(
G
D
R
3
)
Find the number of permutations
a
1
,
⋯
,
a
n
a_1, \cdots, a_n
a
1
,
⋯
,
a
n
of the set
{
1
,
2
,
.
.
.
,
n
}
\{1, 2, . . ., n\}
{
1
,
2
,
...
,
n
}
such that
∣
a
i
−
a
i
+
1
∣
≠
1
|a_i - a_{i+1}| \neq 1
∣
a
i
−
a
i
+
1
∣
=
1
for all
i
=
1
,
2
,
.
.
.
,
n
−
1.
i = 1, 2, . . ., n - 1.
i
=
1
,
2
,
...
,
n
−
1.
Find a recurrence formula and evaluate the number of such permutations for
n
≤
6.
n \le 6.
n
≤
6.
30
1
Hide problems
Existence of infinitely many a for which n^4+a is composite
(
G
D
R
2
)
I
M
O
1
(GDR 2)^{IMO1}
(
G
D
R
2
)
I
MO
1
Prove that there exist infinitely many natural numbers
a
a
a
with the following property: The number
z
=
n
4
+
a
z = n^4 + a
z
=
n
4
+
a
is not prime for any natural number
n
.
n.
n
.
29
1
Hide problems
Finding real numbers lambda
(
G
D
R
1
)
(GDR 1)
(
G
D
R
1
)
Find all real numbers
λ
\lambda
λ
such that the equation
sin
4
x
−
cos
4
x
=
λ
(
tan
4
x
−
cot
4
x
)
\sin^4 x - \cos^4 x = \lambda(\tan^4 x - \cot^4 x)
sin
4
x
−
cos
4
x
=
λ
(
tan
4
x
−
cot
4
x
)
(
a
)
(a)
(
a
)
has no solution,
(
b
)
(b)
(
b
)
has exactly one solution,
(
c
)
(c)
(
c
)
has exactly two solutions,
(
d
)
(d)
(
d
)
has more than two solutions (in the interval
(
0
,
π
4
)
.
(0, \frac{\pi}{4}).
(
0
,
4
π
)
.
28
1
Hide problems
Prove that b divides a(u_b-1)
(
G
B
R
5
)
(GBR 5)
(
GBR
5
)
Let us define
u
0
=
0
,
u
1
=
1
u_0 = 0, u_1 = 1
u
0
=
0
,
u
1
=
1
and for
n
≥
0
,
u
n
+
2
=
a
u
n
+
1
+
b
u
n
,
a
n\ge 0, u_{n+2} = au_{n+1}+bu_n, a
n
≥
0
,
u
n
+
2
=
a
u
n
+
1
+
b
u
n
,
a
and
b
b
b
being positive integers. Express
u
n
u_n
u
n
as a polynomial in
a
a
a
and
b
.
b.
b
.
Prove the result. Given that
b
b
b
is prime, prove that
b
b
b
divides
a
(
u
b
−
1
)
.
a(u_b -1).
a
(
u
b
−
1
)
.
27
1
Hide problems
Restricted existence of cone with axis passing through X
(
G
B
R
4
)
(GBR 4)
(
GBR
4
)
The segment
A
B
AB
A
B
perpendicularly bisects
C
D
CD
C
D
at
X
X
X
. Show that, subject to restrictions, there is a right circular cone whose axis passes through
X
X
X
and on whose surface lie the points
A
,
B
,
C
,
D
.
A,B,C,D.
A
,
B
,
C
,
D
.
What are the restrictions?
26
1
Hide problems
Cylinder and string and relation between the angles.
(
G
B
R
3
)
(GBR 3)
(
GBR
3
)
A smooth solid consists of a right circular cylinder of height
h
h
h
and base-radius
r
r
r
, surmounted by a hemisphere of radius
r
r
r
and center
O
.
O.
O
.
The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid and held down at the point
P
P
P
on the hemisphere such that
O
P
OP
OP
makes an angle
α
\alpha
α
with the horizontal. Show that if
α
\alpha
α
is small enough, the string will slacken if slightly displaced and no longer remain in a vertical plane. If then pulled tight through
P
P
P
, show that it will cross the common circular section of the hemisphere and cylinder at a point
Q
Q
Q
such that
∠
S
O
Q
=
ϕ
\angle SOQ = \phi
∠
SOQ
=
ϕ
,
S
S
S
being where it initially crossed this section, and
sin
ϕ
=
r
tan
α
h
\sin \phi = \frac{r \tan \alpha}{h}
sin
ϕ
=
h
r
t
a
n
α
.
25
1
Hide problems
Find largest number not representable by ax+by
(
G
B
R
2
)
(GBR 2)
(
GBR
2
)
Let
a
,
b
,
x
,
y
a, b, x, y
a
,
b
,
x
,
y
be positive integers such that
a
a
a
and
b
b
b
have no common divisor greater than
1
1
1
. Prove that the largest number not expressible in the form
a
x
+
b
y
ax + by
a
x
+
b
y
is
a
b
−
a
−
b
ab - a - b
ab
−
a
−
b
. If
N
(
k
)
N(k)
N
(
k
)
is the largest number not expressible in the form
a
x
+
b
y
ax + by
a
x
+
b
y
in only
k
k
k
ways, find
N
(
k
)
.
N(k).
N
(
k
)
.
24
1
Hide problems
Polynomial and divisibility
(
G
B
R
1
)
(GBR 1)
(
GBR
1
)
The polynomial
P
(
x
)
=
a
0
x
k
+
a
1
x
k
−
1
+
⋯
+
a
k
P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k
P
(
x
)
=
a
0
x
k
+
a
1
x
k
−
1
+
⋯
+
a
k
, where
a
0
,
⋯
,
a
k
a_0,\cdots, a_k
a
0
,
⋯
,
a
k
are integers, is said to be divisible by an integer
m
m
m
if
P
(
x
)
P(x)
P
(
x
)
is a multiple of
m
m
m
for every integral value of
x
x
x
. Show that if
P
(
x
)
P(x)
P
(
x
)
is divisible by
m
m
m
, then
a
0
⋅
k
!
a_0 \cdot k!
a
0
⋅
k
!
is a multiple of
m
m
m
. Also prove that if
a
,
k
,
m
a, k,m
a
,
k
,
m
are positive integers such that
a
k
!
ak!
ak
!
is a multiple of
m
m
m
, then a polynomial
P
(
x
)
P(x)
P
(
x
)
with leading term
a
x
k
ax^k
a
x
k
can be found that is divisible by
m
.
m.
m
.
23
1
Hide problems
Find about the rational number phi(d)/b
(
F
R
A
6
)
(FRA 6)
(
FR
A
6
)
Consider the integer
d
=
a
b
−
1
c
d = \frac{a^b-1}{c}
d
=
c
a
b
−
1
, where
a
,
b
a, b
a
,
b
, and
c
c
c
are positive integers and
c
≤
a
.
c \le a.
c
≤
a
.
Prove that the set
G
G
G
of integers that are between
1
1
1
and
d
d
d
and relatively prime to
d
d
d
(the number of such integers is denoted by
ϕ
(
d
)
\phi(d)
ϕ
(
d
)
) can be partitioned into
n
n
n
subsets, each of which consists of
b
b
b
elements. What can be said about the rational number
ϕ
(
d
)
b
?
\frac{\phi(d)}{b}?
b
ϕ
(
d
)
?
22
1
Hide problems
Find relation for alpha(n), beta(n).
(
F
R
A
5
)
(FRA 5)
(
FR
A
5
)
Let
α
(
n
)
\alpha(n)
α
(
n
)
be the number of pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of integers such that
x
+
y
=
n
,
0
≤
y
≤
x
x+y = n, 0 \le y \le x
x
+
y
=
n
,
0
≤
y
≤
x
, and let
β
(
n
)
\beta(n)
β
(
n
)
be the number of triples
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
such that
x
+
y
+
z
=
n
x + y + z = n
x
+
y
+
z
=
n
and
0
≤
z
≤
y
≤
x
.
0 \le z \le y \le x.
0
≤
z
≤
y
≤
x
.
Find a simple relation between
α
(
n
)
\alpha(n)
α
(
n
)
and the integer part of the number
n
+
2
2
\frac{n+2}{2}
2
n
+
2
and the relation among
β
(
n
)
,
β
(
n
−
3
)
\beta(n), \beta(n -3)
β
(
n
)
,
β
(
n
−
3
)
and
α
(
n
)
.
\alpha(n).
α
(
n
)
.
Then evaluate
β
(
n
)
\beta(n)
β
(
n
)
as a function of the residue of
n
n
n
modulo
6
6
6
. What can be said about
β
(
n
)
\beta(n)
β
(
n
)
and
1
+
n
(
n
+
6
)
12
1+\frac{n(n+6)}{12}
1
+
12
n
(
n
+
6
)
? And what about
(
n
+
3
)
2
6
\frac{(n+3)^2}{6}
6
(
n
+
3
)
2
? Find the number of triples
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
with the property
x
+
y
+
z
≤
n
,
0
≤
z
≤
y
≤
x
x+ y+ z \le n, 0 \le z \le y \le x
x
+
y
+
z
≤
n
,
0
≤
z
≤
y
≤
x
as a function of the residue of
n
n
n
modulo
6.
6.
6.
What can be said about the relation between this number and the number
(
n
+
6
)
(
2
n
2
+
9
n
+
12
)
72
\frac{(n+6)(2n^2+9n+12)}{72}
72
(
n
+
6
)
(
2
n
2
+
9
n
+
12
)
?
21
1
Hide problems
Prove that MB=MT
(
F
R
A
4
)
(FRA 4)
(
FR
A
4
)
A right-angled triangle
O
A
B
OAB
O
A
B
has its right angle at the point
B
.
B.
B
.
An arbitrary circle with center on the line
O
B
OB
OB
is tangent to the line
O
A
.
OA.
O
A
.
Let
A
T
AT
A
T
be the tangent to the circle different from
O
A
OA
O
A
(
T
T
T
is the point of tangency). Prove that the median from
B
B
B
of the triangle
O
A
B
OAB
O
A
B
intersects
A
T
AT
A
T
at a point
M
M
M
such that
M
B
=
M
T
.
MB = MT.
MB
=
MT
.
20
1
Hide problems
Relation between boundary, interior points and area
(
F
R
A
3
)
(FRA 3)
(
FR
A
3
)
A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is
S
.
S.
S
.
If
I
I
I
is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number
T
=
2
S
−
B
−
2
I
+
2.
T = 2S- B -2I + 2.
T
=
2
S
−
B
−
2
I
+
2.
19
1
Hide problems
Periodic sequence and find the minimal period.
(
F
R
A
2
)
(FRA 2)
(
FR
A
2
)
Let
n
n
n
be an integer that is not divisible by any square greater than
1.
1.
1.
Denote by
x
m
x_m
x
m
the last digit of the number
x
m
x^m
x
m
in the number system with base
n
.
n.
n
.
For which integers
x
x
x
is it possible for
x
m
x_m
x
m
to be
0
0
0
? Prove that the sequence
x
m
x_m
x
m
is periodic with period
t
t
t
independent of
x
.
x.
x
.
For which
x
x
x
do we have
x
t
=
1
x_t = 1
x
t
=
1
. Prove that if
m
m
m
and
x
x
x
are relatively prime, then
0
m
,
1
m
,
.
.
.
,
(
n
−
1
)
m
0_m, 1_m, . . . , (n-1)_m
0
m
,
1
m
,
...
,
(
n
−
1
)
m
are different numbers. Find the minimal period
t
t
t
in terms of
n
n
n
. If n does not meet the given condition, prove that it is possible to have
x
m
=
0
≠
x
1
x_m = 0 \neq x_1
x
m
=
0
=
x
1
and that the sequence is periodic starting only from some number
k
>
1.
k > 1.
k
>
1.
18
1
Hide problems
Set H(a, b) contains all numbers greater than omega
(
F
R
A
1
)
(FRA 1)
(
FR
A
1
)
Let
a
a
a
and
b
b
b
be two nonnegative integers. Denote by
H
(
a
,
b
)
H(a, b)
H
(
a
,
b
)
the set of numbers
n
n
n
of the form
n
=
p
a
+
q
b
,
n = pa + qb,
n
=
p
a
+
q
b
,
where
p
p
p
and
q
q
q
are positive integers. Determine
H
(
a
)
=
H
(
a
,
a
)
H(a) = H(a, a)
H
(
a
)
=
H
(
a
,
a
)
. Prove that if
a
≠
b
,
a \neq b,
a
=
b
,
it is enough to know all the sets
H
(
a
,
b
)
H(a, b)
H
(
a
,
b
)
for coprime numbers
a
,
b
a, b
a
,
b
in order to know all the sets
H
(
a
,
b
)
H(a, b)
H
(
a
,
b
)
. Prove that in the case of coprime numbers
a
a
a
and
b
,
H
(
a
,
b
)
b, H(a, b)
b
,
H
(
a
,
b
)
contains all numbers greater than or equal to
ω
=
(
a
−
1
)
(
b
−
1
)
\omega = (a - 1)(b -1)
ω
=
(
a
−
1
)
(
b
−
1
)
and also
ω
2
\frac{\omega}{2}
2
ω
numbers smaller than
ω
\omega
ω
17
1
Hide problems
Arithmetic progression and common difference
(
C
Z
S
6
)
(CZS 6)
(
CZS
6
)
Let
d
d
d
and
p
p
p
be two real numbers. Find the first term of an arithmetic progression
a
1
,
a
2
,
a
3
,
⋯
a_1, a_2, a_3, \cdots
a
1
,
a
2
,
a
3
,
⋯
with difference
d
d
d
such that
a
1
a
2
a
3
a
4
=
p
.
a_1a_2a_3a_4 = p.
a
1
a
2
a
3
a
4
=
p
.
Find the number of solutions in terms of
d
d
d
and
p
.
p.
p
.
16
1
Hide problems
Area of quadrilateral
(
C
Z
S
5
)
(CZS 5)
(
CZS
5
)
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with sides
A
B
=
a
,
B
C
=
b
,
C
D
=
c
,
D
A
=
d
AB = a, BC = b, CD = c, DA = d
A
B
=
a
,
BC
=
b
,
C
D
=
c
,
D
A
=
d
and angles
α
=
∠
D
A
B
,
β
=
∠
A
B
C
,
γ
=
∠
B
C
D
,
\alpha = \angle DAB, \beta = \angle ABC, \gamma = \angle BCD,
α
=
∠
D
A
B
,
β
=
∠
A
BC
,
γ
=
∠
BC
D
,
and
δ
=
∠
C
D
A
\delta = \angle CDA
δ
=
∠
C
D
A
is given. Let
s
=
a
+
b
+
c
+
d
2
s = \frac{a + b + c +d}{2}
s
=
2
a
+
b
+
c
+
d
and
P
P
P
be the area of the quadrilateral. Prove that
P
2
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
a
b
c
d
cos
2
α
+
γ
2
P^2 = (s - a)(s - b)(s - c)(s - d) - abcd \cos^2\frac{\alpha +\gamma}{2}
P
2
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
ab
c
d
cos
2
2
α
+
γ
15
1
Hide problems
Inequality in factorials
(
C
Z
S
4
)
(CZS 4)
(
CZS
4
)
Let
K
1
,
⋯
,
K
n
K_1,\cdots , K_n
K
1
,
⋯
,
K
n
be nonnegative integers. Prove that
K
1
!
K
2
!
⋯
K
n
!
≥
[
K
n
]
!
n
K_1!K_2!\cdots K_n! \ge \left[\frac{K}{n}\right]!^n
K
1
!
K
2
!
⋯
K
n
!
≥
[
n
K
]
!
n
, where
K
=
K
1
+
⋯
+
K
n
K = K_1 + \cdots + K_n
K
=
K
1
+
⋯
+
K
n
14
1
Hide problems
Quadratic equation and proof of existence
(
C
Z
S
3
)
(CZS 3)
(
CZS
3
)
Let
a
a
a
and
b
b
b
be two positive real numbers. If
x
x
x
is a real solution of the equation
x
2
+
p
x
+
q
=
0
x^2 + px + q = 0
x
2
+
p
x
+
q
=
0
with real coefficients
p
p
p
and
q
q
q
such that
∣
p
∣
≤
a
,
∣
q
∣
≤
b
,
|p| \le a, |q| \le b,
∣
p
∣
≤
a
,
∣
q
∣
≤
b
,
prove that
∣
x
∣
≤
1
2
(
a
+
a
2
+
4
b
)
|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})
∣
x
∣
≤
2
1
(
a
+
a
2
+
4
b
)
Conversely, if
x
x
x
satisfies the above inequality, prove that there exist real numbers
p
p
p
and
q
q
q
with
∣
p
∣
≤
a
,
∣
q
∣
≤
b
|p|\le a, |q|\le b
∣
p
∣
≤
a
,
∣
q
∣
≤
b
such that
x
x
x
is one of the roots of the equation
x
2
+
p
x
+
q
=
0.
x^2+px+ q = 0.
x
2
+
p
x
+
q
=
0.
13
1
Hide problems
Finding p-1 numbers whose sum of squares is divisible by sum
(
C
Z
S
2
)
(CZS 2)
(
CZS
2
)
Let
p
p
p
be a prime odd number. Is it possible to find
p
−
1
p-1
p
−
1
natural numbers
n
+
1
,
n
+
2
,
.
.
.
,
n
+
p
−
1
n + 1, n + 2, . . . , n + p -1
n
+
1
,
n
+
2
,
...
,
n
+
p
−
1
such that the sum of the squares of these numbers is divisible by the sum of these numbers?
12
1
Hide problems
Locus of centroid of tetrahedron on sides of unit cube
(
C
Z
S
1
)
(CZS 1)
(
CZS
1
)
Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.
11
1
Hide problems
Existence of unjoinable pair of points at given distance.
(
B
U
L
5
)
(BUL 5)
(
B
UL
5
)
Let
Z
Z
Z
be a set of points in the plane. Suppose that there exists a pair of points that cannot be joined by a polygonal line not passing through any point of
Z
.
Z.
Z
.
Let us call such a pair of points unjoinable. Prove that for each real
r
>
0
r > 0
r
>
0
there exists an unjoinable pair of points separated by distance
r
.
r.
r
.
10
1
Hide problems
Identity between Brocard angle and median angle
(
B
U
L
4
)
(BUL 4)
(
B
UL
4
)
Let
M
M
M
be the point inside the right-angled triangle
A
B
C
(
∠
C
=
9
0
∘
)
ABC (\angle C = 90^{\circ})
A
BC
(
∠
C
=
9
0
∘
)
such that
∠
M
A
B
=
∠
M
B
C
=
∠
M
C
A
=
ϕ
.
\angle MAB = \angle MBC = \angle MCA =\phi.
∠
M
A
B
=
∠
MBC
=
∠
MC
A
=
ϕ
.
Let
Ψ
\Psi
Ψ
be the acute angle between the medians of
A
C
AC
A
C
and
B
C
.
BC.
BC
.
Prove that
sin
(
ϕ
+
Ψ
)
sin
(
ϕ
−
Ψ
)
=
5.
\frac{\sin(\phi+\Psi)}{\sin(\phi-\Psi)}= 5.
s
i
n
(
ϕ
−
Ψ
)
s
i
n
(
ϕ
+
Ψ
)
=
5.
9
1
Hide problems
Existence of disk in square not intersecting polygons.
(
B
U
L
3
)
(BUL 3)
(
B
UL
3
)
One hundred convex polygons are placed on a square with edge of length
38
c
m
.
38 cm.
38
c
m
.
The area of each of the polygons is smaller than
π
c
m
2
,
\pi cm^2,
π
c
m
2
,
and the perimeter of each of the polygons is smaller than
2
π
c
m
.
2\pi cm.
2
π
c
m
.
Prove that there exists a disk with radius
1
1
1
in the square that does not intersect any of the polygons.
8
1
Hide problems
xf(y)+yf(x)=(x+y)f(x)f(y) - ILL BUL2
Find all functions
f
f
f
defined for all
x
x
x
that satisfy the condition
x
f
(
y
)
+
y
f
(
x
)
=
(
x
+
y
)
f
(
x
)
f
(
y
)
,
xf(y) + yf(x) = (x + y)f(x)f(y),
x
f
(
y
)
+
y
f
(
x
)
=
(
x
+
y
)
f
(
x
)
f
(
y
)
,
for all
x
x
x
and
y
.
y.
y
.
Prove that exactly two of them are continuous.
7
1
Hide problems
sqrt{x^3 + y^3 + z^3}=1969 has no solutions.
(
B
U
L
1
)
(BUL 1)
(
B
UL
1
)
Prove that the equation
x
3
+
y
3
+
z
3
=
1969
\sqrt{x^3 + y^3 + z^3}=1969
x
3
+
y
3
+
z
3
=
1969
has no integral solutions.
6
1
Hide problems
Application of Fubini's principle.
(
B
E
L
6
)
(BEL 6)
(
BE
L
6
)
Evaluate
(
cos
π
4
+
i
sin
π
4
)
10
\left(\cos\frac{\pi}{4} + i \sin\frac{\pi}{4}\right)^{10}
(
cos
4
π
+
i
sin
4
π
)
10
in two different ways and prove that
(
10
1
)
−
(
10
3
)
+
1
2
(
10
5
)
=
2
4
\dbinom{10}{1}-\dbinom{10}{3}+\frac{1}{2}\dbinom{10}{5}=2^4
(
1
10
)
−
(
3
10
)
+
2
1
(
5
10
)
=
2
4
5
1
Hide problems
Prove all conics are hyperbolas and find locus of centers.
(
B
E
L
5
)
(BEL 5)
(
BE
L
5
)
Let
G
G
G
be the centroid of the triangle
O
A
B
.
OAB.
O
A
B
.
(
a
)
(a)
(
a
)
Prove that all conics passing through the points
O
,
A
,
B
,
G
O,A,B,G
O
,
A
,
B
,
G
are hyperbolas.
(
b
)
(b)
(
b
)
Find the locus of the centers of these hyperbolas.
4
1
Hide problems
Line passes through fixed point on normal
(
B
E
L
4
)
(BEL 4)
(
BE
L
4
)
Let
O
O
O
be a point on a nondegenerate conic. A right angle with vertex
O
O
O
intersects the conic at points
A
A
A
and
B
B
B
. Prove that the line
A
B
AB
A
B
passes through a fixed point located on the normal to the conic through the point
O
.
O.
O
.
3
1
Hide problems
Construction of circle tangent to three given ones.
(
B
E
L
3
)
(BEL 3)
(
BE
L
3
)
Construct the circle that is tangent to three given circles.
2
1
Hide problems
Equations, hyperbolas, Locus and nine point circle.
(
B
E
L
2
)
(
a
)
(BEL 2) (a)
(
BE
L
2
)
(
a
)
Find the equations of regular hyperbolas passing through the points
A
(
α
,
0
)
,
B
(
β
,
0
)
,
A(\alpha, 0), B(\beta, 0),
A
(
α
,
0
)
,
B
(
β
,
0
)
,
and
C
(
0
,
γ
)
.
C(0, \gamma).
C
(
0
,
γ
)
.
(
b
)
(b)
(
b
)
Prove that all such hyperbolas pass through the orthocenter
H
H
H
of the triangle
A
B
C
.
ABC.
A
BC
.
(
c
)
(c)
(
c
)
Find the locus of the centers of these hyperbolas.
(
d
)
(d)
(
d
)
Check whether this locus coincides with the nine-point circle of the triangle
A
B
C
.
ABC.
A
BC
.
1
1
Hide problems
Locus of pole of a line with respect to parabola.
(
B
E
L
1
)
(BEL 1)
(
BE
L
1
)
A parabola
P
1
P_1
P
1
with equation
x
2
−
2
p
y
=
0
x^2 - 2py = 0
x
2
−
2
p
y
=
0
and parabola
P
2
P_2
P
2
with equation
x
2
+
2
p
y
=
0
,
p
>
0
x^2 + 2py = 0, p > 0
x
2
+
2
p
y
=
0
,
p
>
0
, are given. A line
t
t
t
is tangent to
P
2
.
P_2.
P
2
.
Find the locus of pole
M
M
M
of the line
t
t
t
with respect to
P
1
.
P_1.
P
1
.
57
1
Hide problems
Singapore Mathematical Olympiad 2009 Problem 1
Given triangle
A
B
C
ABC
A
BC
with points
M
M
M
and
N
N
N
are in the sides
A
B
AB
A
B
and
A
C
AC
A
C
respectively. If
B
M
M
A
+
C
N
N
A
=
1
\dfrac{BM}{MA} +\dfrac{CN}{NA} = 1
M
A
BM
+
N
A
CN
=
1
, then prove that the centroid of
A
B
C
ABC
A
BC
lies on
M
N
MN
MN
.