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Problems
Contests
International Contests
IMO Longlists
1970 IMO Longlists
1970 IMO Longlists
Part of
IMO Longlists
Subcontests
(47)
57
1
Hide problems
Squares in a board - Greatest possible sums of the numbers
Let the numbers
1
,
2
,
…
,
n
2
1, 2, \ldots , n^2
1
,
2
,
…
,
n
2
be written in the cells of an
n
×
n
n \times n
n
×
n
square board so that the entries in each column are arranged increasingly. What are the smallest and greatest possible sums of the numbers in the
k
t
h
k^{th}
k
t
h
row? (
k
k
k
a positive integer,
1
≤
k
≤
n
1 \leq k \leq n
1
≤
k
≤
n
.)
56
1
Hide problems
The shape and area of the territory accessible to the dog
A square hole of depth
h
h
h
whose base is of length
a
a
a
is given. A dog is tied to the center of the square at the bottom of the hole by a rope of length
L
>
2
a
2
+
h
2
L >\sqrt{2a^2+h^2}
L
>
2
a
2
+
h
2
, and walks on the ground around the hole. The edges of the hole are smooth, so that the rope can freely slide along it. Find the shape and area of the territory accessible to the dog (whose size is neglected).
55
1
Hide problems
UFO can catch the turtle for any inital position
A turtle runs away from an UFO with a speed of
0.2
m
/
s
0.2 \ m/s
0.2
m
/
s
. The UFO flies
5
5
5
meters above the ground, with a speed of
20
m
/
s
20 \ m/s
20
m
/
s
. The UFO's path is a broken line, where after flying in a straight path of length
ℓ
\ell
ℓ
(in meters) it may turn through for any acute angle
α
\alpha
α
such that
tan
α
<
ℓ
1000
\tan \alpha < \frac{\ell}{1000}
tan
α
<
1000
ℓ
. When the UFO's center approaches within
13
13
13
meters of the turtle, it catches the turtle. Prove that for any initial position the UFO can catch the turtle.
29
1
Hide problems
No Rational Solutions
Prove that the equation
4
x
+
6
x
=
9
x
4^x +6^x =9^x
4
x
+
6
x
=
9
x
has no rational solutions.
28
1
Hide problems
Prove that K is a Group
A set
G
G
G
with elements
u
,
v
,
w
.
.
.
u,v,w...
u
,
v
,
w
...
is a Group if the following conditions are fulfilled:
(
i
)
(\text{i})
(
i
)
There is a binary operation
∘
\circ
∘
defined on
G
G
G
such that
∀
{
u
,
v
}
∈
G
\forall \{u,v\}\in G
∀
{
u
,
v
}
∈
G
there is a
w
∈
G
w\in G
w
∈
G
with
u
∘
v
=
w
u\circ v = w
u
∘
v
=
w
.
(
ii
)
(\text{ii})
(
ii
)
This operation is associative; i.e.
(
u
∘
v
)
∘
w
=
u
∘
(
v
∘
w
)
(u\circ v)\circ w = u\circ (v\circ w)
(
u
∘
v
)
∘
w
=
u
∘
(
v
∘
w
)
∀
{
u
,
v
,
w
}
∈
G
\forall\{u,v,w\}\in G
∀
{
u
,
v
,
w
}
∈
G
.
(
iii
)
(\text{iii})
(
iii
)
∀
{
u
,
v
}
∈
G
\forall \{u,v\}\in G
∀
{
u
,
v
}
∈
G
, there exists an element
x
∈
G
x\in G
x
∈
G
such that
u
∘
x
=
v
u\circ x = v
u
∘
x
=
v
, and an element
y
∈
G
y\in G
y
∈
G
such that
y
∘
u
=
v
y\circ u = v
y
∘
u
=
v
.Let
K
K
K
be a set of all real numbers greater than
1
1
1
. On
K
K
K
is defined an operation by
a
∘
b
=
a
b
−
(
a
2
−
1
)
(
b
2
−
1
)
a\circ b = ab-\sqrt{(a^2-1)(b^2-1)}
a
∘
b
=
ab
−
(
a
2
−
1
)
(
b
2
−
1
)
. Prove that
K
K
K
is a Group.
27
1
Hide problems
Natural Divisible by Prime
Find a
n
∈
N
n\in\mathbb{N}
n
∈
N
such that for all primes
p
p
p
,
n
n
n
is divisible by
p
p
p
if and only if
n
n
n
is divisible by
p
−
1
p-1
p
−
1
.
26
1
Hide problems
Vectors in Space
Consider a finite set of vectors in space
{
a
1
,
a
2
,
.
.
.
,
a
n
}
\{a_1, a_2, ... , a_n\}
{
a
1
,
a
2
,
...
,
a
n
}
and the set
E
E
E
of all vectors of the form
x
=
∑
i
=
1
n
λ
i
a
i
x=\sum_{i=1}^{n}{\lambda _i a_i}
x
=
∑
i
=
1
n
λ
i
a
i
, where
λ
i
∈
R
+
∪
{
0
}
\lambda _i \in \mathbb{R}^{+}\cup \{0\}
λ
i
∈
R
+
∪
{
0
}
. Let
F
F
F
be the set consisting of all the vectors in
E
E
E
and vectors parallel to a given plane
P
P
P
. Prove that there exists a set of vectors
{
b
1
,
b
2
,
.
.
.
,
b
p
}
\{b_1, b_2, ... , b_p\}
{
b
1
,
b
2
,
...
,
b
p
}
such that
F
F
F
is the set of all vectors
y
y
y
of the form
y
=
∑
i
=
1
p
μ
i
b
i
y=\sum_{i=1}^{p}{\mu _i b_i}
y
=
∑
i
=
1
p
μ
i
b
i
, where
μ
i
∈
R
+
∪
{
0
}
\mu _i \in \mathbb{R}^{+}\cup \{0\}
μ
i
∈
R
+
∪
{
0
}
.
25
1
Hide problems
Derivatives, existence of value
A real function
f
f
f
is defined for
0
≤
x
≤
1
0\le x\le 1
0
≤
x
≤
1
, with its first derivative
f
′
f'
f
′
defined for
0
≤
x
≤
1
0\le x\le 1
0
≤
x
≤
1
and its second derivative
f
′
′
f''
f
′′
defined for
0
<
x
<
1
0<x<1
0
<
x
<
1
. Prove that if
f
(
0
)
=
f
′
(
0
)
=
f
′
(
1
)
=
f
(
1
)
−
1
=
0
f(0)=f'(0)=f'(1)=f(1)-1 =0
f
(
0
)
=
f
′
(
0
)
=
f
′
(
1
)
=
f
(
1
)
−
1
=
0
, then there exists a number
0
<
y
<
1
0<y<1
0
<
y
<
1
such that
∣
f
′
′
(
y
)
∣
≥
4
|f''(y)|\ge 4
∣
f
′′
(
y
)
∣
≥
4
.
24
1
Hide problems
Easy Factorial Inequality
Let
{
n
,
p
}
∈
N
∪
{
0
}
\{n,p\}\in\mathbb{N}\cup \{0\}
{
n
,
p
}
∈
N
∪
{
0
}
such that
2
p
≤
n
2p\le n
2
p
≤
n
. Prove that
(
n
−
p
)
!
p
!
≤
(
n
+
1
2
)
n
−
2
p
\frac{(n-p)!}{p!}\le \left(\frac{n+1}{2}\right)^{n-2p}
p
!
(
n
−
p
)!
≤
(
2
n
+
1
)
n
−
2
p
. Determine all conditions under which equality holds.
23
1
Hide problems
Finite Set, Mapping, Subsets
Let
E
E
E
be a finite set,
P
E
P_E
P
E
the family of its subsets, and
f
f
f
a mapping from
P
E
P_E
P
E
to the set of non-negative reals, such that for any two disjoint subsets
A
,
B
A,B
A
,
B
of
E
E
E
,
f
(
A
∪
B
)
=
f
(
A
)
+
f
(
B
)
f(A\cup B)=f(A)+f(B)
f
(
A
∪
B
)
=
f
(
A
)
+
f
(
B
)
. Prove that there exists a subset
F
F
F
of
E
E
E
such that if with each
A
⊂
E
A \subset E
A
⊂
E
, we associate a subset
A
′
A'
A
′
consisting of elements of
A
A
A
that are not in
F
F
F
, then
f
(
A
)
=
f
(
A
′
)
f(A)=f(A')
f
(
A
)
=
f
(
A
′
)
and
f
(
A
)
f(A)
f
(
A
)
is zero if and only if
A
A
A
is a subset of
F
F
F
.
19
1
Hide problems
Strange Inequality
Let
1
<
n
∈
N
1<n\in\mathbb{N}
1
<
n
∈
N
and
1
≤
a
∈
R
1\le a\in\mathbb{R}
1
≤
a
∈
R
and there are
n
n
n
number of
x
i
,
i
∈
N
,
1
≤
i
≤
n
x_i, i\in\mathbb{N}, 1\le i\le n
x
i
,
i
∈
N
,
1
≤
i
≤
n
such that
x
1
=
1
x_1=1
x
1
=
1
and
x
i
x
i
−
1
=
a
+
α
i
\frac{x_{i}}{x_{i-1}}=a+\alpha _ i
x
i
−
1
x
i
=
a
+
α
i
for
2
≤
i
≤
n
2\le i\le n
2
≤
i
≤
n
, where
α
i
≤
1
i
(
i
+
1
)
\alpha _i\le \frac{1}{i(i+1)}
α
i
≤
i
(
i
+
1
)
1
. Prove that
x
n
n
−
1
<
a
+
1
n
−
1
\sqrt[n-1]{x_n}< a+\frac{1}{n-1}
n
−
1
x
n
<
a
+
n
−
1
1
.
16
1
Hide problems
Proof of Non-existence of Roots - ILL 1970 - Problem 16.
Show that the equation
2
−
x
2
+
3
−
x
3
3
=
0
\sqrt{2-x^2}+\sqrt[3]{3-x^3}=0
2
−
x
2
+
3
3
−
x
3
=
0
has no real roots.
15
1
Hide problems
Excentral Triangle Inequality - IMO 1970 - Problem 15.
Given
△
A
B
C
\triangle ABC
△
A
BC
, let
R
R
R
be its circumradius and
q
q
q
be the perimeter of its excentral triangle. Prove that
q
≤
6
3
R
q\le 6\sqrt{3} R
q
≤
6
3
R
. Typesetter's Note: the excentral triangle has vertices which are the excenters of the original triangle.
14
1
Hide problems
Easy Trigonometric Identity - ILL 1970 - Problem 14.
Let
α
+
β
+
γ
=
π
\alpha + \beta +\gamma = \pi
α
+
β
+
γ
=
π
. Prove that
∑
c
y
c
sin
2
α
=
2
⋅
(
∑
c
y
c
sin
α
)
⋅
(
∑
c
y
c
cos
α
)
−
2
∑
c
y
c
sin
α
\sum_{cyc}{\sin 2\alpha} = 2\cdot \left(\sum_{cyc}{\sin \alpha}\right)\cdot\left(\sum_{cyc}{\cos \alpha}\right)- 2\sum_{cyc}{\sin \alpha}
∑
cyc
sin
2
α
=
2
⋅
(
∑
cyc
sin
α
)
⋅
(
∑
cyc
cos
α
)
−
2
∑
cyc
sin
α
.
13
1
Hide problems
Divided Sides
Each side of an arbitrary
△
A
B
C
\triangle ABC
△
A
BC
is divided into equal parts, and lines parallel to
A
B
,
B
C
,
C
A
AB,BC,CA
A
B
,
BC
,
C
A
are drawn through each of these points, thus cutting
△
A
B
C
\triangle ABC
△
A
BC
into small triangles. Points are assigned a number in the following manner:
(
1
)
(1)
(
1
)
A
,
B
,
C
A,B,C
A
,
B
,
C
are assigned
1
,
2
,
3
1,2,3
1
,
2
,
3
respectively
(
2
)
(2)
(
2
)
Points on
A
B
AB
A
B
are assigned
1
1
1
or
2
2
2
(
3
)
(3)
(
3
)
Points on
B
C
BC
BC
are assigned
2
2
2
or
3
3
3
(
4
)
(4)
(
4
)
Points on
C
A
CA
C
A
are assigned
3
3
3
or
1
1
1
Prove that there must exist a small triangle whose vertices are marked by
1
,
2
,
3
1,2,3
1
,
2
,
3
.
12
1
Hide problems
7 | Sum/Difference Combination of 6 Integers
Let
{
x
i
}
,
1
≤
i
≤
6
\{x_i\}, 1\le i\le 6
{
x
i
}
,
1
≤
i
≤
6
be a given set of six integers, none of which are divisible by
7
7
7
.
(
a
)
(a)
(
a
)
Prove that at least one of the expressions of the form
x
1
±
x
2
±
x
3
±
x
4
±
x
5
±
x
6
x_1\pm x_2\pm x_3\pm x_4\pm x_5\pm x_6
x
1
±
x
2
±
x
3
±
x
4
±
x
5
±
x
6
is divisible by
7
7
7
, where the
±
\pm
±
signs are independent of each other.
(
b
)
(b)
(
b
)
Generalize the result to every prime number.
11
1
Hide problems
Connect Midpoints of Sides of Squares to Get Another Square
Let
A
B
C
D
ABCD
A
BC
D
and
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
be two arbitrary squares in the plane that are oriented in the same direction. Prove that the quadrilateral formed by the midpoints of
A
A
′
,
B
B
′
,
C
C
′
,
D
D
′
AA',BB',CC',DD'
A
A
′
,
B
B
′
,
C
C
′
,
D
D
′
is a square.
10
1
Hide problems
Easy Trigonometric Inequality - ILL 1970 - Problem 10.
In
△
A
B
C
\triangle ABC
△
A
BC
, prove that
1
<
∑
c
y
c
cos
A
≤
3
2
1< \sum_{cyc}{\cos A}\le \frac{3}{2}
1
<
∑
cyc
cos
A
≤
2
3
.
9
1
Hide problems
Algebraic Sum - ILL 1970 - Problem 9.
For even
n
n
n
, prove that
∑
i
=
1
n
(
(
−
1
)
i
+
1
⋅
1
i
)
=
2
∑
i
=
1
n
/
2
1
n
+
2
i
\sum_{i=1}^{n}{\left((-1)^{i+1}\cdot\frac{1}{i}\right)}=2\sum_{i=1}^{n/2}{\frac{1}{n+2i}}
∑
i
=
1
n
(
(
−
1
)
i
+
1
⋅
i
1
)
=
2
∑
i
=
1
n
/2
n
+
2
i
1
.
7
1
Hide problems
Squares Constructed on Sides of Quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be an arbitrary quadrilateral. Squares with centers
M
1
,
M
2
,
M
3
,
M
4
M_1, M_2, M_3, M_4
M
1
,
M
2
,
M
3
,
M
4
are constructed on
A
B
,
B
C
,
C
D
,
D
A
AB,BC,CD,DA
A
B
,
BC
,
C
D
,
D
A
respectively, all outwards or all inwards. Prove that
M
1
M
3
=
M
2
M
4
M_1 M_3=M_2 M_4
M
1
M
3
=
M
2
M
4
and
M
1
M
3
⊥
M
2
M
4
M_1 M_3\perp M_2 M_4
M
1
M
3
⊥
M
2
M
4
.
6
1
Hide problems
Existence of Roots in Certain Intervals
There is an equation
∑
i
=
1
n
b
i
x
−
a
i
=
c
\sum_{i=1}^{n}{\frac{b_i}{x-a_i}}=c
∑
i
=
1
n
x
−
a
i
b
i
=
c
in
x
x
x
, where all
b
i
>
0
b_i >0
b
i
>
0
and
{
a
i
}
\{a_i\}
{
a
i
}
is a strictly increasing sequence. Prove that it has
n
−
1
n-1
n
−
1
roots such that
x
n
−
1
≤
a
n
x_{n-1}\le a_n
x
n
−
1
≤
a
n
, and
a
i
≤
x
i
a_i \le x_i
a
i
≤
x
i
for each
i
∈
N
,
1
≤
i
≤
n
−
1
i\in\mathbb{N}, 1\le i\le n-1
i
∈
N
,
1
≤
i
≤
n
−
1
.
5
1
Hide problems
Easy Natural Inequality - ILL 1970 - Problem 5.
Prove that
∑
i
=
1
n
i
n
+
1
n
≥
1
\sqrt[n]{\sum_{i=1}^{n}{\frac{i}{n+1}}}\ge 1
n
∑
i
=
1
n
n
+
1
i
≥
1
for
2
≤
n
∈
N
2 \le n \in \mathbb{N}
2
≤
n
∈
N
.
4
1
Hide problems
System of 2 Equations - ILL 1970 - Problem 4.
Solve the system of equations for variables
x
,
y
x,y
x
,
y
, where
{
a
,
b
}
∈
R
\{a,b\}\in\mathbb{R}
{
a
,
b
}
∈
R
are constants and
a
≠
0
a\neq 0
a
=
0
.
x
2
+
x
y
=
a
2
+
a
b
x^2 + xy = a^2 + ab
x
2
+
x
y
=
a
2
+
ab
y
2
+
x
y
=
a
2
−
a
b
y^2 + xy = a^2 - ab
y
2
+
x
y
=
a
2
−
ab
3
1
Hide problems
Very Easy Factorial Divisibility - ILL 1970 - Problem 3.
Prove that
(
a
!
⋅
b
!
)
∣
(
a
+
b
)
!
(a!\cdot b!) | (a+b)!
(
a
!
⋅
b
!)
∣
(
a
+
b
)!
∀
a
,
b
∈
N
\forall a,b\in\mathbb{N}
∀
a
,
b
∈
N
.
2
1
Hide problems
Matching Digits
Prove that the two last digits of
9
9
9
9^{9^{9}}
9
9
9
and
9
9
9
9
9^{9^{9^{9}}}
9
9
9
9
are the same in decimal representation.
1
1
Hide problems
Very Easy Inequality - ILL 1970 Problem 1.
Prove that
a
b
a
+
b
+
b
c
b
+
c
+
c
a
c
+
a
≤
a
+
b
+
c
2
\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\le \frac{a+b+c}{2}
a
+
b
ab
+
b
+
c
b
c
+
c
+
a
c
a
≤
2
a
+
b
+
c
, where
a
,
b
,
c
∈
R
+
a,b,c\in\mathbb{R}^{+}
a
,
b
,
c
∈
R
+
.
53
1
Hide problems
Dividing a square into (n-1)^2 congruent squares
A square
A
B
C
D
ABCD
A
BC
D
is divided into
(
n
−
1
)
2
(n - 1)^2
(
n
−
1
)
2
congruent squares, with sides parallel to the sides of the given square. Consider the grid of all
n
2
n^2
n
2
corners obtained in this manner. Determine all integers
n
n
n
for which it is possible to construct a non-degenerate parabola with its axis parallel to one side of the square and that passes through exactly
n
n
n
points of the grid.
54
1
Hide problems
Construct T using three polynomials P, Q, R with given roots
Let
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
be polynomials and let
S
(
x
)
=
P
(
x
3
)
+
x
Q
(
x
3
)
+
x
2
R
(
x
3
)
S(x) = P(x^3) + xQ(x^3) + x^2R(x^3)
S
(
x
)
=
P
(
x
3
)
+
x
Q
(
x
3
)
+
x
2
R
(
x
3
)
be a polynomial of degree
n
n
n
whose roots
x
1
,
…
,
x
n
x_1,\ldots, x_n
x
1
,
…
,
x
n
are distinct. Construct with the aid of the polynomials
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
a polynomial
T
T
T
of degree
n
n
n
that has the roots
x
1
3
,
x
2
3
,
…
,
x
n
3
.
x_1^3 , x_2^3 , \ldots, x_n^3.
x
1
3
,
x
2
3
,
…
,
x
n
3
.
51
1
Hide problems
(a+p)/(b+p) - a/b = 1/p^2. Find all fractions a/b (ILL 1970)
Let
p
p
p
be a prime number. A rational number
x
x
x
, with
0
<
x
<
1
0 < x < 1
0
<
x
<
1
, is written in lowest terms. The rational number obtained from
x
x
x
by adding
p
p
p
to both the numerator and the denominator differs from
x
x
x
by
1
/
p
2
1/p^2
1/
p
2
. Determine all rational numbers
x
x
x
with this property.
50
1
Hide problems
Show that 36S ≤ L^2 √3, where S is area and L is perimeter
The area of a triangle is
S
S
S
and the sum of the lengths of its sides is
L
L
L
. Prove that
36
S
≤
L
2
3
36S \leq L^2\sqrt 3
36
S
≤
L
2
3
and give a necessary and sufficient condition for equality.
49
1
Hide problems
Does there exists a real p such that f(n)/n ≥ p for all n?
For
n
∈
N
n \in \mathbb N
n
∈
N
, let
f
(
n
)
f(n)
f
(
n
)
be the number of positive integers
k
≤
n
k \leq n
k
≤
n
that do not contain the digit
9
9
9
. Does there exist a positive real number
p
p
p
such that
f
(
n
)
n
≥
p
\frac{f(n)}{n} \geq p
n
f
(
n
)
≥
p
for all positive integers
n
n
n
?
47
1
Hide problems
The ugly polynomial divisibility
Given a polynomial
P
(
x
)
=
a
b
(
a
−
c
)
x
3
+
(
a
3
−
a
2
c
+
2
a
b
2
−
b
2
c
+
a
b
c
)
x
2
+
(
2
a
2
b
+
b
2
c
+
a
2
c
+
b
3
−
a
b
c
)
x
+
a
b
(
b
+
c
)
,
P(x) = ab(a - c)x^3 + (a^3 - a^2c + 2ab^2 - b^2c + abc)x^2 +(2a^2b + b^2c + a^2c + b^3 - abc)x + ab(b + c),
P
(
x
)
=
ab
(
a
−
c
)
x
3
+
(
a
3
−
a
2
c
+
2
a
b
2
−
b
2
c
+
ab
c
)
x
2
+
(
2
a
2
b
+
b
2
c
+
a
2
c
+
b
3
−
ab
c
)
x
+
ab
(
b
+
c
)
,
where
a
,
b
,
c
≠
0
a, b, c \neq 0
a
,
b
,
c
=
0
, prove that
P
(
x
)
P(x)
P
(
x
)
is divisible by
Q
(
x
)
=
a
b
x
2
+
(
a
2
+
b
2
)
x
+
a
b
Q(x) = abx^2 + (a^2 + b^2)x + ab
Q
(
x
)
=
ab
x
2
+
(
a
2
+
b
2
)
x
+
ab
and conclude that
P
(
x
0
)
P(x_0)
P
(
x
0
)
is divisible by
(
a
+
b
)
3
(a + b)^3
(
a
+
b
)
3
for
x
0
=
(
a
+
b
+
1
)
n
,
n
∈
N
x_0 = (a + b + 1)^n, n \in \mathbb N
x
0
=
(
a
+
b
+
1
)
n
,
n
∈
N
.
46
1
Hide problems
Find a point M such that the triangles are congruent
Given a triangle
A
B
C
ABC
A
BC
and a plane
π
\pi
π
having no common points with the triangle, find a point
M
M
M
such that the triangle determined by the points of intersection of the lines
M
A
,
M
B
,
M
C
MA,MB,MC
M
A
,
MB
,
MC
with
π
\pi
π
is congruent to the triangle
A
B
C
ABC
A
BC
.
45
1
Hide problems
Volumes in Tetrahedrons
Let
M
M
M
be an interior point of tetrahedron
V
A
B
C
V ABC
V
A
BC
. Denote by
A
1
,
B
1
,
C
1
A_1,B_1, C_1
A
1
,
B
1
,
C
1
the points of intersection of lines
M
A
,
M
B
,
M
C
MA,MB,MC
M
A
,
MB
,
MC
with the planes
V
B
C
,
V
C
A
,
V
A
B
VBC,V CA,V AB
V
BC
,
V
C
A
,
V
A
B
, and by
A
2
,
B
2
,
C
2
A_2,B_2, C_2
A
2
,
B
2
,
C
2
the points of intersection of lines
V
A
1
,
V
B
1
,
V
C
1
V A_1, VB_1, V C_1
V
A
1
,
V
B
1
,
V
C
1
with the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
.(a) Prove that the volume of the tetrahedron
V
A
2
B
2
C
2
V A_2B_2C_2
V
A
2
B
2
C
2
does not exceed one-fourth of the volume of
V
A
B
C
V ABC
V
A
BC
.(b) Calculate the volume of the tetrahedron
V
1
A
1
B
1
C
1
V_1A_1B_1C_1
V
1
A
1
B
1
C
1
as a function of the volume of
V
A
B
C
V ABC
V
A
BC
, where
V
1
V_1
V
1
is the point of intersection of the line
V
M
VM
V
M
with the plane
A
B
C
ABC
A
BC
, and
M
M
M
is the barycenter of
V
A
B
C
V ABC
V
A
BC
.
44
1
Hide problems
(a+b)(b+c)(c+a) ≥ 8(a+b−c)(b+c−a)(c+a−b) (ILL 1970, P44)
If
a
,
b
,
c
a, b, c
a
,
b
,
c
are side lengths of a triangle, prove that
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
≥
8
(
a
+
b
−
c
)
(
b
+
c
−
a
)
(
c
+
a
−
b
)
.
(a + b)(b + c)(c + a) \geq 8(a + b - c)(b + c - a)(c + a - b).
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
≥
8
(
a
+
b
−
c
)
(
b
+
c
−
a
)
(
c
+
a
−
b
)
.
43
1
Hide problems
π/36 is a root of the equation - (Nice?)
Prove that the equation
x
3
−
3
tan
π
12
x
2
−
3
x
+
tan
π
12
=
0
x^3 - 3 \tan\frac{\pi}{12} x^2 - 3x + \tan\frac{\pi}{12}= 0
x
3
−
3
tan
12
π
x
2
−
3
x
+
tan
12
π
=
0
has one root
x
1
=
tan
π
36
x_1 = \tan \frac{\pi}{36}
x
1
=
tan
36
π
, and find the other roots.
41
1
Hide problems
There exists a point A on a cube - Nice one
Let a cube of side
1
1
1
be given. Prove that there exists a point
A
A
A
on the surface
S
S
S
of the cube such that every point of
S
S
S
can be joined to
A
A
A
by a path on
S
S
S
of length not exceeding
2
2
2
. Also prove that there is a point of
S
S
S
that cannot be joined with
A
A
A
by a path on
S
S
S
of length less than
2
2
2
.
40
1
Hide problems
Set of all directions in which the ball will move is finite
Let ABC be a triangle with angles
α
,
β
,
γ
\alpha, \beta, \gamma
α
,
β
,
γ
commensurable with
π
\pi
π
. Starting from a point
P
P
P
interior to the triangle, a ball reflects on the sides of
A
B
C
ABC
A
BC
, respecting the law of reflection that the angle of incidence is equal to the angle of reflection. Prove that, supposing that the ball never reaches any of the vertices
A
,
B
,
C
A,B,C
A
,
B
,
C
, the set of all directions in which the ball will move through time is finite. In other words, its path from the moment
0
0
0
to infinity consists of segments parallel to a finite set of lines.
38
1
Hide problems
The greatest A for which there exist 10 consecutive numbers
Find the greatest integer
A
A
A
for which in any permutation of the numbers
1
,
2
,
…
,
100
1, 2, \ldots , 100
1
,
2
,
…
,
100
there exist ten consecutive numbers whose sum is at least
A
A
A
.
37
1
Hide problems
The set of simultaneous equations with 5 variables
Solve the set of simultaneous equations \begin{align*} v^2+ w^2+ x^2+ y^2 &= 6 - 2u, \\ u^2+ w^2+ x^2+ y^2 &= 6 - 2v, \\ u^2+ v^2+ x^2+ y^2 &= 6- 2w, \\ u^2+ v^2+ w^2+ y^2 &= 6 - 2x, \\ u^2+ v^2+ w^2+ x^2 &= 6- 2y. \end{align*}
36
1
Hide problems
The greatest value that x^2 -yz, y^2 - xz, z^2 - xy can have
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be non-negative real numbers satisfying x^2 + y^2 + z^2 = 5 \text{ and } yz + zx + xy = 2. Which values can the greatest of the numbers
x
2
−
y
z
,
y
2
−
x
z
x^2 -yz, y^2 - xz
x
2
−
yz
,
y
2
−
x
z
and
z
2
−
x
y
z^2 - xy
z
2
−
x
y
have?
35
1
Hide problems
Any convex n-gon can be divided into p isosceles triangles
Find for every value of
n
n
n
a set of numbers
p
p
p
for which the following statement is true: Any convex
n
n
n
-gon can be divided into
p
p
p
isosceles triangles.
34
1
Hide problems
None of the circles contains the pentagon - ILL 1970, P34
In connection with a convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
we consider the set of ten circles, each of which contains three of the vertices of the pentagon on its circumference. Is it possible that none of these circles contains the pentagon? Prove your answer.
33
1
Hide problems
Is this possible if we choose another P0 point?
The vertices of a given square are clockwise lettered
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
. On the side
A
B
AB
A
B
is situated a point
E
E
E
such that
A
E
=
A
B
/
3
AE = AB/3
A
E
=
A
B
/3
. Starting from an arbitrarily chosen point
P
0
P_0
P
0
on segment
A
E
AE
A
E
and going clockwise around the perimeter of the square, a series of points
P
0
,
P
1
,
P
2
,
…
P_0, P_1, P_2, \ldots
P
0
,
P
1
,
P
2
,
…
is marked on the perimeter such that
P
i
P
i
+
1
=
A
B
/
3
P_iP_{i+1} = AB/3
P
i
P
i
+
1
=
A
B
/3
for each
i
i
i
. It will be clear that when
P
0
P_0
P
0
is chosen in
A
A
A
or in
E
E
E
, then some
P
i
P_i
P
i
will coincide with
P
0
P_0
P
0
. Does this possibly also happen if
P
0
P_0
P
0
is chosen otherwise?
32
1
Hide problems
Determine the behavior of t_i for i→∞.
Let there be given an acute angle
∠
A
O
B
=
3
α
\angle AOB = 3\alpha
∠
A
OB
=
3
α
, where
O
A
‾
=
O
B
‾
\overline{OA}= \overline{OB}
O
A
=
OB
. The point
A
A
A
is the center of a circle with radius
O
A
‾
\overline{OA}
O
A
. A line
s
s
s
parallel to
O
A
OA
O
A
passes through
B
B
B
. Inside the given angle a variable line
t
t
t
is drawn through
O
O
O
. It meets the circle in
O
O
O
and
C
C
C
and the given line
s
s
s
in
D
D
D
, where
∠
A
O
C
=
x
\angle AOC = x
∠
A
OC
=
x
. Starting from an arbitrarily chosen position
t
0
t_0
t
0
of
t
t
t
, the series
t
0
,
t
1
,
t
2
,
…
t_0, t_1, t_2, \ldots
t
0
,
t
1
,
t
2
,
…
is determined by defining
B
D
i
+
1
‾
=
O
C
i
‾
\overline{BD_{i+1}}=\overline{OC_i}
B
D
i
+
1
=
O
C
i
for each
i
i
i
(in which
C
i
C_i
C
i
and
D
i
D_i
D
i
denote the positions of
C
C
C
and
D
D
D
, corresponding to
t
i
t_i
t
i
). Making use of the graphical representations of
B
D
BD
B
D
and
O
C
OC
OC
as functions of
x
x
x
, determine the behavior of
t
i
t_i
t
i
for
i
→
∞
i\to \infty
i
→
∞
.
31
1
Hide problems
An old geometric inequality - appeared on ILL 1970, P31
Prove that for any triangle with sides
a
,
b
,
c
a, b, c
a
,
b
,
c
and area
P
P
P
the following inequality holds:
P
≤
3
4
(
a
b
c
)
2
/
3
.
P \leq \frac{\sqrt 3}{4} (abc)^{2/3}.
P
≤
4
3
(
ab
c
)
2/3
.
Find all triangles for which equality holds.
48
1
Hide problems
p(x) does not take the value 8 for any integer x [easy one]
Let a polynomial
p
(
x
)
p(x)
p
(
x
)
with integer coefficients take the value
5
5
5
for five different integer values of
x
.
x.
x
.
Prove that
p
(
x
)
p(x)
p
(
x
)
does not take the value
8
8
8
for any integer
x
.
x.
x
.