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Mathlinks Contests.
MathLinks Contest 5th
MathLinks Contest 5th
Part of
Mathlinks Contests.
Subcontests
(21)
7.1
1
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0571 number theory 5th edition Round 7 p1
Prove that the numbers
(
2
n
−
1
i
)
,
i
=
0
,
1
,
.
.
.
,
2
n
−
1
−
1
,
{{2^n-1} \choose {i}}, i = 0, 1, . . ., 2^{n-1} - 1,
(
i
2
n
−
1
)
,
i
=
0
,
1
,
...
,
2
n
−
1
−
1
,
have pairwise different residues modulo
2
n
2^n
2
n
7.2
1
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0572 number theory 5th edition Round 7 p2
For any positive integer
n
n
n
, let
s
(
n
)
s(n)
s
(
n
)
be the sum of its digits, written in decimal base. Prove that for each integer
n
≥
1
n \ge 1
n
≥
1
there exists a positive integer
x
x
x
such that the fraction
x
+
k
s
(
x
+
k
)
\frac{x + k}{s(x + k)}
s
(
x
+
k
)
x
+
k
is not integral, for each integer
k
k
k
with
0
≤
k
≤
n
0 \le k \le n
0
≤
k
≤
n
.
7.3
1
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0573 combo geo 5th edition Round 7 p3
Given is a square of sides
3
7
×
3
7
3\sqrt7 \times 3\sqrt7
3
7
×
3
7
. Find the minimal positive integer
n
n
n
such that no matter how we put
n
n
n
unit disks inside the given square, without overlapping, there exists a line that intersects
4
4
4
disks.
6.3
1
Hide problems
0563 inequalities 5th edition Round 6 p3
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be three positive numbers such that
(
x
+
y
−
z
)
(
1
x
+
1
y
−
1
z
)
=
4
(x + y-z) \left( \frac{1}{x}+ \frac{1}{y}- \frac{1}{z} \right)=4
(
x
+
y
−
z
)
(
x
1
+
y
1
−
z
1
)
=
4
. Find the minimal value of the expression
E
(
x
,
y
,
z
)
=
(
x
4
+
y
4
+
z
4
)
(
1
x
4
+
1
y
4
+
1
z
4
)
.
E(x, y, z) = (x^4 + y^4 + z^4) \left( \frac{1}{x^4}+ \frac{1}{y^4}+ \frac{1}{z^4} \right) .
E
(
x
,
y
,
z
)
=
(
x
4
+
y
4
+
z
4
)
(
x
4
1
+
y
4
1
+
z
4
1
)
.
6.2
1
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0562 number theory 5th edition Round 6 p2
We say that a positive integer
n
n
n
is nice if
4
n
\frac{4}{n}
n
4
cannot be written as
1
x
+
1
x
y
+
1
z
\frac{1}{x}+\frac{1}{xy}+\frac{1}{z}
x
1
+
x
y
1
+
z
1
for any positive integers
x
,
y
,
z
x, y, z
x
,
y
,
z
. Let us denote by
a
n
a_n
a
n
the number of nice numbers smaller than
n
n
n
. Prove that the sequence
n
a
n
\frac{n}{a_n}
a
n
n
is not bounded.
6.1
1
Hide problems
0561 geometry 5th edition Round 6 p1
Let
A
B
C
ABC
A
BC
be a triangle and let
C
C
C
be a circle that intersects the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
in the points
A
1
,
A
2
,
B
1
,
B
2
A_1, A_2, B_1, B_2
A
1
,
A
2
,
B
1
,
B
2
and
C
1
,
C
2
C_1, C_2
C
1
,
C
2
respectively. Prove that if
A
A
1
,
B
B
1
AA_1, BB_1
A
A
1
,
B
B
1
and
C
C
1
CC_1
C
C
1
are concurrent lines then
A
A
2
,
B
B
2
AA_2, BB_2
A
A
2
,
B
B
2
and
C
C
2
CC_2
C
C
2
are also concurrent lines.
5.3
1
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0553 combinatorics 5th edition Round 5 p3
A student wants to make his birthday party special this year. He wants to organize it such that among any groups of
4
4
4
persons at the party there is one that is friends with exactly another person in the group. Find the largest number of his friends that he can possibly invite at the party.
5.2
1
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0552 algebra 5th edition Round 5 p2
Prove or disprove the existence of a function
f
:
S
→
R
f : S \to R
f
:
S
→
R
such that for all
x
≠
y
∈
S
x \ne y \in S
x
=
y
∈
S
we have
∣
f
(
x
)
−
f
(
y
)
∣
≥
1
x
2
+
y
2
|f(x) - f(y)| \ge \frac{1}{x^2 + y^2}
∣
f
(
x
)
−
f
(
y
)
∣
≥
x
2
+
y
2
1
, in each of the cases: a)
S
=
R
S = R
S
=
R
b)
S
=
Q
S = Q
S
=
Q
.
5.1
1
Hide problems
0551 number theory 5th edition Round 5 p1
Find all real numbers
a
>
1
a > 1
a
>
1
such that there exists an integer
k
≥
1
k \ge 1
k
≥
1
such that the sequence
{
x
n
}
n
≥
1
\{x_n\}_{n\ge 1}
{
x
n
}
n
≥
1
formed with the first
k
k
k
digits of the number
⌊
a
n
⌋
\lfloor a^n\rfloor
⌊
a
n
⌋
is periodical.
4.3
1
Hide problems
0543 inequalities 5th edition Round 4 p3
Let
a
1
,
.
.
.
,
a
n
a_1,..., a_n
a
1
,
...
,
a
n
be positive reals and let
x
1
,
.
.
.
,
x
n
x_1, ... , x_n
x
1
,
...
,
x
n
be real numbers such that
a
1
x
1
+
.
.
.
+
a
n
x
n
=
0
a_1x_1 +...+ a_nx_n = 0
a
1
x
1
+
...
+
a
n
x
n
=
0
. Prove that
∑
1
≤
i
<
j
≤
n
x
i
x
j
∣
a
i
−
a
j
∣
≤
0.
\sum_{1\le i<j \le n} x_ix_j |a_i - a_j | \le 0.
1
≤
i
<
j
≤
n
∑
x
i
x
j
∣
a
i
−
a
j
∣
≤
0.
When does the equality take place?
4.2
1
Hide problems
0542 combo geo 5th edition Round 4 p2
Given is a unit cube in space. Find the maximal integer
n
n
n
such that there are
n
n
n
points, satisfying the following conditions: (a) All points lie on the surface of the cube; (b) No face contains all these points; (c) The
n
n
n
points are the vertices of a polygon.
4.1
1
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0541 geometry 5th edition Round 4 p1
Let
A
B
C
ABC
A
BC
be an acute angled triangle. Let
M
M
M
be the midpoint of
B
C
BC
BC
, and let
B
E
BE
BE
and
C
F
CF
CF
be the altitudes of the triangle. Let
D
≠
M
D \ne M
D
=
M
be a point on the circumcircle of the triangle
E
F
M
EFM
EFM
such that
D
E
=
D
F
DE = DF
D
E
=
D
F
. Prove that
A
D
⊥
B
C
AD \perp BC
A
D
⊥
BC
.
3.3
1
Hide problems
0533 inequalities 5th edition Round 3 p3
Let
x
1
,
x
2
,
.
.
.
x
n
x_1, x_2,... x_n
x
1
,
x
2
,
...
x
n
be positive numbers such that
S
=
x
1
+
x
2
+
.
.
.
+
x
n
=
1
x
1
+
.
.
.
+
1
x
n
S = x_1+x_2+...+x_n =\frac{1}{x_1}+...+\frac{1}{x_n}
S
=
x
1
+
x
2
+
...
+
x
n
=
x
1
1
+
...
+
x
n
1
Prove that
∑
i
=
1
n
1
n
−
1
+
x
i
≥
∑
i
=
1
n
1
1
+
S
−
x
i
\sum_{i=1}^{n} \frac{1}{n - 1 + x_i} \ge \sum_{i=1}^{n} \frac{1}{1+S - x_i}
i
=
1
∑
n
n
−
1
+
x
i
1
≥
i
=
1
∑
n
1
+
S
−
x
i
1
3.2
1
Hide problems
0532 a_p + a_q = a_r + a_s 5th edition Round 3 p2
Let
0
<
a
1
<
a
2
<
.
.
.
<
a
16
<
122
0 < a_1 < a_2 <... < a_{16} < 122
0
<
a
1
<
a
2
<
...
<
a
16
<
122
be
16
16
16
integers. Prove that there exist integers
(
p
,
q
,
r
,
s
)
(p, q, r, s)
(
p
,
q
,
r
,
s
)
, with
1
≤
p
<
r
≤
s
<
q
≤
16
1 \le p < r \le s < q \le 16
1
≤
p
<
r
≤
s
<
q
≤
16
, such that
a
p
+
a
q
=
a
r
+
a
s
a_p + a_q = a_r + a_s
a
p
+
a
q
=
a
r
+
a
s
.An additional
2
2
2
points will be awarded for this problem, if you can find a larger bound than
122
122
122
(with proof).
3.1
1
Hide problems
0531 sequence 5th edition Round 3 p1
Let
{
x
n
}
n
\{x_n\}_n
{
x
n
}
n
be a sequence of positive rational numbers, such that
x
1
x_1
x
1
is a positive integer, and for all positive integers
n
n
n
.
x
n
=
2
(
n
−
1
)
n
x
n
−
1
x_n = \frac{2(n - 1)}{n} x_{n-1}
x
n
=
n
2
(
n
−
1
)
x
n
−
1
, if
x
n
1
≤
1
x_{n_1} \le 1
x
n
1
≤
1
x
n
=
(
n
−
1
)
x
n
−
1
−
1
n
x_n = \frac{(n - 1)x_{n-1} - 1}{n}
x
n
=
n
(
n
−
1
)
x
n
−
1
−
1
, if
x
n
1
>
1
x_{n_1} > 1
x
n
1
>
1
. Prove that there exists a constant subsequence of
{
x
n
}
n
\{x_n\}_n
{
x
n
}
n
.
2.3
1
Hide problems
0523 inequalities 5th edition Round 2 p3
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive numbers such that
a
b
c
≤
8
abc \le 8
ab
c
≤
8
. Prove that
1
a
2
−
a
+
1
+
1
b
2
−
b
+
1
+
+
1
c
2
−
c
+
1
≥
1
\frac{1}{a^2 - a + 1} +\frac{1}{b^2 - b + 1}++\frac{1}{c^2 - c + 1} \ge 1
a
2
−
a
+
1
1
+
b
2
−
b
+
1
1
+
+
c
2
−
c
+
1
1
≥
1
2.2
1
Hide problems
0522 combo geo 5th edition Round 2 p2
Suppose that
{
D
n
}
n
≥
1
\{D_n\}_{n\ge 1}
{
D
n
}
n
≥
1
is an finite sequence of disks in the plane whose total area is less than
1
1
1
. Prove that it is possible to rearrange the disks so that they are disjoint from each other and all contained inside a disk of area
4
4
4
.
2.1
1
Hide problems
0521 functional in N 5th edition Round 2 p1
For what positive integers
k
k
k
there exists a function
f
:
N
→
N
f : N \to N
f
:
N
→
N
such that for all
n
∈
N
n \in N
n
∈
N
we have
f(f(... f(n)....))
⏟
k times
=
f
(
n
)
+
2
\underbrace{\hbox{f(f(... f(n)....))}}_{\hbox{k times}} = f(n) + 2
k times
f(f(... f(n)....))
=
f
(
n
)
+
2
?
1.3
1
Hide problems
0513 geometry 5th edition Round 1 p3
Let
A
B
C
ABC
A
BC
be a triangle and let
A
′
∈
B
C
A' \in BC
A
′
∈
BC
,
B
′
∈
C
A
B' \in CA
B
′
∈
C
A
and
C
′
∈
A
B
C' \in AB
C
′
∈
A
B
be three collinear points. a) Prove that each pair of circles of diameters
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
and
C
C
′
CC'
C
C
′
has the same radical axis; b) Prove that the circumcenter of the triangle formed by the intersections of the lines
A
A
′
,
B
B
′
AA' , BB'
A
A
′
,
B
B
′
and
C
C
′
CC'
C
C
′
lies on the common radical axis found above.
1.2
1
Hide problems
0512 number theory 5th edition Round 1 p2
Find all the integers
n
≥
5
n \ge 5
n
≥
5
such that the residue of
n
n
n
when divided by each prime number smaller than
n
2
\frac{n}{2}
2
n
is odd.
1.1
1
Hide problems
0511 x^3 - y^3 = 2005(x^2 - y^2) 5th edition Round 1 p1
Find all pairs of positive integers
x
,
y
x, y
x
,
y
such that
x
3
−
y
3
=
2005
(
x
2
−
y
2
)
x^3 - y^3 = 2005(x^2 - y^2)
x
3
−
y
3
=
2005
(
x
2
−
y
2
)
.