MathDB
Problems
Contests
International Contests
Final Mathematical Cup
2019 Final Mathematical Cup
2019 Final Mathematical Cup
Part of
Final Mathematical Cup
Subcontests
(4)
1
1
Hide problems
AP bisects BC, BD // CE, 3 circumcircles related
Let
A
B
C
ABC
A
BC
be a triangle and let
D
,
E
D, E
D
,
E
are points on its circumscribed circle, such that
D
D
D
lies on arc
A
B
,
E
AB, E
A
B
,
E
lies on arc
A
C
AC
A
C
(smaller arcs) and
B
D
∥
C
E
BD \parallel CE
B
D
∥
CE
. Let the point F be the intersection of the lines
D
A
DA
D
A
and
C
E
CE
CE
, and the intersection of the lines
E
A
EA
E
A
and
B
D
BD
B
D
is
G
G
G
. Let
P
P
P
be the second intersection of the circumscribed circles of
△
A
B
G
\vartriangle ABG
△
A
BG
and
△
A
C
F
\vartriangle ACF
△
A
CF
. Prove that the line
A
P
AP
A
P
passes through the mid point of the side
B
C
BC
BC
.
2
1
Hide problems
a_0+a_1+a_2+...+a_n is divisible by 2 , P(m) =2018, a_i >0, m=(-1+\sqrt{17})/2
Let
m
=
−
1
+
17
2
m=\frac{-1+\sqrt{17}}{2}
m
=
2
−
1
+
17
. Let the polynomial
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
1
x
+
a
0
P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
...
+
a
1
x
+
a
0
is given, where
n
n
n
is a positive integer, the coefficients
a
0
,
a
1
,
a
2
,
.
.
.
,
a
n
a_0,a_1,a_2,...,a_n
a
0
,
a
1
,
a
2
,
...
,
a
n
are positive integers and
P
(
m
)
=
2018
P(m) =2018
P
(
m
)
=
2018
. Prove that the sum
a
0
+
a
1
+
a
2
+
.
.
.
+
a
n
a_0+a_1+a_2+...+a_n
a
0
+
a
1
+
a
2
+
...
+
a
n
is divisible by
2
2
2
.
3
1
Hide problems
pq | (5^p - 2^p )(7^q - 2^q ) if p <= q
Determine every prime numbers
p
p
p
and
q
,
p
≤
q
q , p \le q
q
,
p
≤
q
for which
p
q
∣
(
5
p
−
2
p
)
(
7
q
−
2
q
)
pq | (5^p - 2^ p )(7^q -2 ^q )
pq
∣
(
5
p
−
2
p
)
(
7
q
−
2
q
)
4
2
Hide problems
a grasshopper moves along a nxn square net
Let
n
≥
2
n \ge 2
n
≥
2
be a positive integer. A grasshopper is moving along the sides of an
n
×
n
n \times n
n
×
n
square net, which is divided on
n
2
n^2
n
2
unit squares. It moves so that а) in every
1
×
1
1 \times 1
1
×
1
unit square of the net, it passes only through one side b) when it passes one side of
1
×
1
1 \times1
1
×
1
unit square of the net, it jumps on a vertex on another arbitrary
1
×
1
1 \times 1
1
×
1
unit square of the net, which does not have a side on which the grasshopper moved along. The grasshopper moves until the conditions can be fulfilled. What is the shortest and the longest path that the grasshopper can go through if it moves according to the condition of the problem? Calculate its length and draw it on the net.
2 sheets of paper, n numbers on the one, m numbers on the other, sums
On two sheets of paper are written more than one positive integers. On the first paper
n
n
n
numbers are written and on the second paper
m
m
m
numbers are written. If one number is written on any of the papers then on the first paper is written also the sum of that number and
13
13
13
, and on the second paper the difference of that number and
23
23
23
. Calculate the value of
m
n
\frac{m}{n}
n
m
. .