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Problems
Contests
National and Regional Contests
Poland Contests
Poland - First Round
1993 Poland - First Round
1993 Poland - First Round
Part of
Poland - First Round
Subcontests
(12)
12
1
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Sum of opposite dihedral angles in a tetrahedron
Prove that the sums of the opposite dihedral angles of a tetrahedron are equal if and only if the sums of the opposite edges of this tetrahedron are equal.
11
1
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Triangle with perimeter 2p inscribed and circumscribed on circles
A triangle with perimeter
2
p
2p
2
p
is inscribed in a circle of radius
R
R
R
and also circumscribed on a circle of radius
r
r
r
. Prove that
p
<
2
(
R
+
r
)
p < 2(R+r)
p
<
2
(
R
+
r
)
.
10
1
Hide problems
On values of (1-p^m)^n+(1-q^n)^m
Given positive real numbers
p
,
q
p,q
p
,
q
with
p
+
q
=
1
p+q=1
p
+
q
=
1
. Prove that for all positive integers
m
,
n
m,n
m
,
n
the following inequality holds
(
1
−
p
m
)
n
+
(
1
−
q
n
)
m
≥
1
(1-p^m)^n+(1-q^n)^m \geq 1
(
1
−
p
m
)
n
+
(
1
−
q
n
)
m
≥
1
.
9
1
Hide problems
Fun with acquaintaces
In a conference
2
n
2n
2
n
personalities take apart. Each person has at least
n
n
n
acquaintaces among the others. Prove that it is possible to quarter the participants into two-person rooms, so that each participant would share the room with his/her acquaintace.
8
1
Hide problems
When is (a+b+c)^13 divisible by abc?
Given positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
a
3
a^3
a
3
is divisible by
b
b
b
,
b
3
b^3
b
3
is divisible by
c
c
c
,
c
3
c^3
c
3
is divisible by
a
a
a
. Prove that
(
a
+
b
+
c
)
13
(a+b+c)^{13}
(
a
+
b
+
c
)
13
is divisible by
a
b
c
abc
ab
c
.
7
1
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Relationships between triangles and parallelograms
Given convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. We construct the similar triangles
A
P
B
,
B
Q
C
,
C
R
D
,
D
S
A
APB, BQC, CRD, DSA
A
PB
,
BQC
,
CR
D
,
D
S
A
outside
A
B
C
D
ABCD
A
BC
D
so that
∠
P
A
B
=
∠
Q
B
C
=
∠
R
C
D
=
∠
S
D
A
,
∠
P
B
A
=
∠
Q
C
B
=
∠
R
D
C
=
∠
S
A
D
\angle PAB = \angle QBC = \angle RCD = \angle SDA, \angle PBA = \angle QCB = \angle RDC = \angle SAD
∠
P
A
B
=
∠
QBC
=
∠
RC
D
=
∠
S
D
A
,
∠
PB
A
=
∠
QCB
=
∠
R
D
C
=
∠
S
A
D
. Prove that if
P
Q
R
S
PQRS
PQRS
is a parallelogram, so is
A
B
C
D
ABCD
A
BC
D
.
6
1
Hide problems
f^n(x) = 1 implies f(1) = 1
The function
f
:
R
⟶
R
f: R \longrightarrow R
f
:
R
⟶
R
is continuous. Prove that if for every real number
x
x
x
, there exists a positive integer
n
n
n
, such that
(
f
∘
f
∘
.
.
.
∘
f
)
⏟
n
(
x
)
=
1
\underbrace{(f \circ f \circ ... \circ f)}_{n}(x) = 1
n
(
f
∘
f
∘
...
∘
f
)
(
x
)
=
1
, then
f
(
1
)
=
1
f(1) = 1
f
(
1
)
=
1
.
5
1
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Polynomials with 3 real roots
Prove that if the polynomial
x
3
+
a
x
2
+
b
x
+
c
x^3 + ax^2 + bx + c
x
3
+
a
x
2
+
b
x
+
c
has three distinct real roots, so does the polynomial
x
3
+
a
x
2
+
1
4
(
a
2
+
b
)
x
+
1
8
(
a
b
−
c
)
x^3 + ax^2 + \frac{1}{4}(a^2 + b)x + \frac{1}{8}(ab-c)
x
3
+
a
x
2
+
4
1
(
a
2
+
b
)
x
+
8
1
(
ab
−
c
)
.
4
1
Hide problems
Circles, cords, and symmetry
Given is a circle with center
O
O
O
, point
A
A
A
inside the circle and a chord
P
Q
PQ
PQ
which is not a diameter and passing through
A
A
A
. The lines
p
p
p
and
q
q
q
are tangent to the given circle at
P
P
P
and
Q
Q
Q
respectively. The line
l
l
l
passing through
A
A
A
and perpendicular to
O
A
OA
O
A
intersects the lines
p
p
p
and
q
q
q
at
K
K
K
and
L
L
L
respectively. Prove that
∣
A
K
∣
=
∣
A
L
∣
|AK| = |AL|
∣
A
K
∣
=
∣
A
L
∣
.
3
1
Hide problems
Inequality between the lengths of the sides of a triangle
Prove that if
a
,
b
,
c
a,b,c
a
,
b
,
c
are the lengths of the sides of a triangle, then
1
a
+
1
b
+
1
c
≤
1
a
+
b
−
c
+
1
c
+
a
−
b
+
1
b
+
c
−
a
\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \leq \frac{1}{a+b-c}+\frac{1}{c+a-b}+\frac{1}{b+c-a}
a
1
+
b
1
+
c
1
≤
a
+
b
−
c
1
+
c
+
a
−
b
1
+
b
+
c
−
a
1
.
2
1
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Recursive functions with absolute value
The sequence of functions
f
0
,
f
1
,
f
2
,
.
.
.
f_0,f_1,f_2,...
f
0
,
f
1
,
f
2
,
...
is given by the conditions:
f
0
(
x
)
=
∣
x
∣
f_0(x) = |x|
f
0
(
x
)
=
∣
x
∣
for all
x
∈
R
x \in R
x
∈
R
f
n
+
1
(
x
)
=
∣
f
n
(
x
)
−
2
∣
f_{n+1}(x) = |f_n(x)-2|
f
n
+
1
(
x
)
=
∣
f
n
(
x
)
−
2∣
for
n
=
0
,
1
,
2
,
.
.
.
n=0,1,2,...
n
=
0
,
1
,
2
,
...
and all
x
∈
R
x \in R
x
∈
R
. For each positive integer
n
n
n
, solve the equation
f
n
(
x
)
=
1
f_n(x)=1
f
n
(
x
)
=
1
.
1
1
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System of equations has no integer solutions
Prove that the system of equations
{
a
2
−
b
=
c
2
b
2
−
a
=
d
2
\begin{cases} \ a^2 - b = c^2 \\ \ b^2 - a = d^2 \\ \end{cases}
{
a
2
−
b
=
c
2
b
2
−
a
=
d
2
has no integer solutions
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
.