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Problems
Contests
International Contests
Final Mathematical Cup
2020 Final Mathematical Cup
2020 Final Mathematical Cup
Part of
Final Mathematical Cup
Subcontests
(3)
3
2
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Shading a Square Grid
Let
k
k
k
,
n
n
n
be positive integers,
k
,
n
>
1
k,n>1
k
,
n
>
1
,
k
<
n
k<n
k
<
n
and a
n
×
n
n \times n
n
×
n
grid of unit squares is given. Ana and Maya take turns in coloring the grid in the following way: in each turn, a unit square is colored black in such a way that no two black cells have a common side or vertex. Find the smallest positive integer
n
n
n
, such that they can obtain a configuration in which each row and column contains exactly
k
k
k
black cells. Draw one example.
Constant Sum of Numbers in a Book
Given a paper on which the numbers
1
,
2
,
3
…
,
14
,
15
1,2,3\dots ,14,15
1
,
2
,
3
…
,
14
,
15
are written. Andy and Bobby are bored and perform the following operations, Andy chooses any two numbers (say
x
x
x
and
y
y
y
) on the paper, erases them, and writes the sum of the numbers on the initial paper. Meanwhile, Bobby writes the value of
x
y
(
x
+
y
)
xy(x+y)
x
y
(
x
+
y
)
in his book. They were so bored that they both performed the operation until only
1
1
1
number remained. Then Bobby adds up all the numbers he wrote in his book, let’s call
k
k
k
as the sum.
a
a
a
. Prove that
k
k
k
is constant which means it does not matter how they perform the operation,
b
b
b
. Find the value of
k
k
k
.
4
2
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Tangent Circles
Let
A
B
C
ABC
A
BC
be a triangle such that
∡
B
A
C
=
6
0
∘
\measuredangle BAC = 60^{\circ}
∡
B
A
C
=
6
0
∘
. Let
D
D
D
and
E
E
E
be the feet of the perpendicular from
A
A
A
to the bisectors of the external angles of
B
B
B
and
C
C
C
in triangle
A
B
C
ABC
A
BC
, respectively. Let
O
O
O
be the circumcenter of the triangle
A
B
C
ABC
A
BC
. Prove that circumcircle of the triangle
B
O
C
BOC
BOC
has exactly one point in common with the circumcircle of
A
D
E
ADE
A
D
E
.
Relatively Prime Implies Prime
Find all positive integers
n
n
n
such that for all positive integers
m
m
m
,
1
<
m
<
n
1<m<n
1
<
m
<
n
, relatively prime to
n
n
n
,
m
m
m
must be a prime number.
1
2
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Divisors and Perfect Squares
Let
n
n
n
be a given positive integer. Prove that there is no positive divisor
d
d
d
of
2
n
2
2n^2
2
n
2
such that
d
2
n
2
+
d
3
d^2n^2+d^3
d
2
n
2
+
d
3
is a square of an integer.
Slightly Nested FE
Find all such functions
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
that for any real
x
,
y
x,y
x
,
y
the following equation is true.
f
(
f
(
x
)
+
y
)
+
1
=
f
(
x
2
+
y
)
+
2
f
(
x
)
+
2
y
f(f(x)+y)+1=f(x^2+y)+2f(x)+2y
f
(
f
(
x
)
+
y
)
+
1
=
f
(
x
2
+
y
)
+
2
f
(
x
)
+
2
y