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Simon Marais Mathematical Competition
2023 Simon Marais Mathematical Competition
2023 Simon Marais Mathematical Competition
Part of
Simon Marais Mathematical Competition
Subcontests
(7)
B4
1
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Power of r plus square root n is rational
(The following problem is open in the sense that the answer to part (b) is not currently known.)[*] Let
n
n
n
be a positive integer that is not a perfect square. Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of positive integers for which there exists a positive real number
r
r
r
, such that
r
a
+
n
and
r
b
+
n
r^a+\sqrt{n} \ \ \text{and} \ \ r^b+\sqrt{n}
r
a
+
n
and
r
b
+
n
are both rational numbers. [*] Let
n
n
n
be a positive integer that is not a perfect square. Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of positive integers for which there exists a real number
r
r
r
, such that
r
a
+
n
and
r
b
+
n
r^a+\sqrt{n} \ \ \text{and} \ \ r^b+\sqrt{n}
r
a
+
n
and
r
b
+
n
are both rational numbers.
B3
1
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Three linearly dependent bases
Let
n
n
n
be a positive integer. Let
A
,
B
,
A,B,
A
,
B
,
and
C
C
C
be three
n
n
n
-dimensional vector subspaces of
R
2
n
\mathbb{R}^{2n}
R
2
n
with the property that
A
∩
B
=
B
∩
C
=
C
∩
A
=
{
0
}
A \cap B = B \cap C = C \cap A = \{0\}
A
∩
B
=
B
∩
C
=
C
∩
A
=
{
0
}
. Prove that there exist bases
{
a
1
,
a
2
,
…
,
a
n
}
\{a_1,a_2, \dots, a_n\}
{
a
1
,
a
2
,
…
,
a
n
}
of
A
A
A
,
{
b
1
,
b
2
,
…
,
b
n
}
\{b_1,b_2, \dots, b_n\}
{
b
1
,
b
2
,
…
,
b
n
}
of
B
B
B
, and
{
c
1
,
c
2
,
…
,
c
n
}
\{c_1,c_2, \dots, c_n\}
{
c
1
,
c
2
,
…
,
c
n
}
of
C
C
C
with the property that for each
i
∈
{
1
,
2
,
…
,
n
}
i \in \{1,2, \dots, n\}
i
∈
{
1
,
2
,
…
,
n
}
, the vectors
a
i
,
b
i
,
a_i,b_i,
a
i
,
b
i
,
and
c
i
c_i
c
i
are linearly dependent.
B2
1
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Expected rank of elimination tournament
There are
256
256
256
players in a tennis tournament who are ranked from
1
1
1
to
256
256
256
, with
1
1
1
corresponding to the highest rank and
256
256
256
corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability
3
5
\frac{3}{5}
5
3
.In each round of the tournament, the player with the highest rank plays against the player with the second highest rank, the player with the third highest rank plays against the player with the fourth highest rank, and so on. At the end of the round, the players who win proceed to the next round and the players who lose exit the tournament. After eight rounds, there is one player remaining and they are declared the winner.Determine the expected value of the rank of the winner.
B1
1
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Vectors lying inside disc
Find the smallest positive real number
r
r
r
with the following property: For every choice of
2023
2023
2023
unit vectors
v
1
,
v
2
,
…
,
v
2023
∈
R
2
v_1,v_2, \dots ,v_{2023} \in \mathbb{R}^2
v
1
,
v
2
,
…
,
v
2023
∈
R
2
, a point
p
p
p
can be found in the plane such that for each subset
S
S
S
of
{
1
,
2
,
…
,
2023
}
\{1,2, \dots , 2023\}
{
1
,
2
,
…
,
2023
}
, the sum
∑
i
∈
S
v
i
\sum_{i \in S} v_i
i
∈
S
∑
v
i
lies inside the disc
{
x
∈
R
2
:
∣
∣
x
−
p
∣
∣
≤
r
}
\{x \in \mathbb{R}^2 : ||x-p|| \leq r\}
{
x
∈
R
2
:
∣∣
x
−
p
∣∣
≤
r
}
.
A4
1
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Bounded sequences for choice of first term of the sequence
Let
x
0
,
x
1
,
x
2
…
x_0, x_1, x_2 \dots
x
0
,
x
1
,
x
2
…
be a sequence of positive real numbers such that for all
n
≥
0
n \geq 0
n
≥
0
,
x
n
+
1
=
(
n
2
+
1
)
x
n
2
x
n
3
+
n
2
x_{n+1} = \dfrac{(n^2+1)x_n^2}{x_n^3+n^2}
x
n
+
1
=
x
n
3
+
n
2
(
n
2
+
1
)
x
n
2
For which values of
x
0
x_0
x
0
is this sequence bounded?
A3
1
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Minimum number of values in the union of sets
For each positive integer
n
n
n
, let
f
(
n
)
f(n)
f
(
n
)
denote the smallest possible value of
∣
A
1
∪
A
2
∪
⋯
∪
A
n
∣
|A_1 \cup A_2 \cup \dots \cup A_n|
∣
A
1
∪
A
2
∪
⋯
∪
A
n
∣
where
A
1
,
A
2
,
A
3
…
A
n
A_1, A_2, A_3 \dots A_n
A
1
,
A
2
,
A
3
…
A
n
are sets such that
A
i
⊈
A
j
A_i \not\subseteq A_j
A
i
⊆
A
j
and
∣
A
i
∣
≠
∣
A
j
∣
|A_i| \neq |A_j|
∣
A
i
∣
=
∣
A
j
∣
whenever
i
≠
j
i \neq j
i
=
j
. Determine
f
(
n
)
f(n)
f
(
n
)
for each positive integer
n
n
n
.
A2
1
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Affine and convex functions
Let
n
n
n
be a positive integer and let
f
1
(
x
)
,
f
2
(
x
)
…
f
n
(
x
)
f_1(x), f_2(x) \dots f_n(x)
f
1
(
x
)
,
f
2
(
x
)
…
f
n
(
x
)
be affine functions from
R
\mathbb{R}
R
to
R
\mathbb{R}
R
such that, amongst the graph of these functions, no two are parallel and no three are concurrent. Let
S
S
S
be the set of all convex functions
g
(
x
)
g(x)
g
(
x
)
from
R
\mathbb{R}
R
to
R
\mathbb{R}
R
such that for each
x
∈
R
x \in \mathbb{R}
x
∈
R
, there exists
i
i
i
such that
g
(
x
)
=
f
i
(
x
)
g(x) = f_i(x)
g
(
x
)
=
f
i
(
x
)
. Determine the largest and smallest possible values of
∣
S
∣
|S|
∣
S
∣
in terms of
n
n
n
.(A function
f
(
x
)
f(x)
f
(
x
)
is affine if it is of form
f
(
x
)
=
a
x
+
b
f(x) = ax + b
f
(
x
)
=
a
x
+
b
for some
a
,
b
∈
R
a, b \in \mathbb{R}
a
,
b
∈
R
. A function
g
(
x
)
g(x)
g
(
x
)
is convex if
g
(
λ
x
+
(
1
−
λ
)
y
)
≤
λ
g
(
x
)
+
(
1
−
λ
)
g
(
y
)
g(\lambda x + (1 - \lambda) y) \leq \lambda g(x) + (1-\lambda)g(y)
g
(
λ
x
+
(
1
−
λ
)
y
)
≤
λ
g
(
x
)
+
(
1
−
λ
)
g
(
y
)
for all
x
,
y
∈
R
x, y \in \mathbb{R}
x
,
y
∈
R
and
0
≤
λ
≤
1
0 \leq \lambda \leq 1
0
≤
λ
≤
1
)