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Undergraduate contests
ICMC
ICMC 5
ICMC 5
Part of
ICMC
Subcontests
(6)
6
1
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Covering circle with equilateral triangles
Is it possible to cover a circle of area
1
1
1
with finitely many equilateral triangles whose areas sum to
1.01
1.01
1.01
, all pointing in the same direction?Proposed by Ethan Tan
5
2
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Tanned vectors
A tanned vector is a nonzero vector in
R
3
\mathbb R^3
R
3
with integer entries. Prove that any tanned vector of length at most
2021
2021
2021
is perpendicular to a tanned vector of length at most
100
100
100
.Proposed by Ethan Tan
A robot's random walk
A robot on the number line starts at
1
1
1
. During the first minute, the robot writes down the number
1
1
1
. Each minute thereafter, it moves by one, either left or right, with equal probability. It then multiplies the last number it wrote by
n
/
t
n/t
n
/
t
, where
n
n
n
is the number it just moved to, and
t
t
t
is the number of minutes elapsed. It then writes this number down. For example, if the robot moves right during the second minute, it would write down
2
/
2
=
1
2/2=1
2/2
=
1
.Find the expected sum of all numbers it writes down, given that it is finite.Proposed by Ethan Tan
4
2
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Set closed under multiplication and reflection mod p
Let
p
p
p
be a prime number. Find all subsets
S
⊆
Z
/
p
Z
S\subseteq\mathbb Z/p\mathbb Z
S
⊆
Z
/
p
Z
such that 1. if
a
,
b
∈
S
a,b\in S
a
,
b
∈
S
, then
a
b
∈
S
ab\in S
ab
∈
S
, and 2. there exists an
r
∈
S
r\in S
r
∈
S
such that for all
a
∈
S
a\in S
a
∈
S
, we have
r
−
a
∈
S
∪
{
0
}
r-a\in S\cup\{0\}
r
−
a
∈
S
∪
{
0
}
.Proposed by Harun Khan
Clean integers
Fix a set of integers
S
S
S
. An integer is clean if it is the sum of distinct elements of
S
S
S
in exactly one way, and dirty otherwise. Prove that the set of dirty numbers is either empty or infinite.Note: We consider the empty sum to equal
0
0
0
.Proposed by Tony Wang and Ethan Tan
3
2
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Determinants equal mod 2
Let
M
\mathcal M
M
be the set of
n
×
n
n\times n
n
×
n
matrices with integer entries. Find all
A
∈
M
A\in\mathcal M
A
∈
M
such that
det
(
A
+
B
)
+
det
(
B
)
\det(A+B)+\det(B)
det
(
A
+
B
)
+
det
(
B
)
is even for all
B
∈
M
B\in\mathcal M
B
∈
M
.Proposed by Ethan Tan
Every affine projection has point symmetry
A set of points has point symmetry if a reflection in some point maps the set to itself. Let
P
\cal P
P
be a solid convex polyhedron whose orthogonal projections onto any plane have point symmetry. Prove that
P
\cal P
P
has point symmetry.Proposed by Ethan Tan
2
2
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Subtracting rows leaves all remainders
Find all integers
n
n
n
for which there exists a table with
n
n
n
rows,
2022
2022
2022
columns, and integer entries, such that subtracting any two rows entry-wise leaves every remainder modulo
2022
2022
2022
.Proposed by Tony Wang
just a sum
Evaluate
1
/
2
1
+
2
+
1
/
4
1
+
2
4
+
1
/
8
1
+
2
8
+
1
/
16
1
+
2
16
+
⋯
\frac{1/2}{1+\sqrt2}+\frac{1/4}{1+\sqrt[4]2}+\frac{1/8}{1+\sqrt[8]2}+\frac{1/16}{1+\sqrt[16]2}+\cdots
1
+
2
1/2
+
1
+
4
2
1/4
+
1
+
8
2
1/8
+
1
+
16
2
1/16
+
⋯
Proposed by Ethan Tan
1
2
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Integer triangles
Let
T
n
T_n
T
n
be the number of non-congruent triangles with positive area and integer side lengths summing to
n
n
n
. Prove that
T
2022
=
T
2019
T_{2022}=T_{2019}
T
2022
=
T
2019
.Proposed by Constantinos Papachristoforou
integer areas
Let
S
S
S
be a set of
2022
2022
2022
lines in the plane, no two parallel, no three concurrent.
S
S
S
divides the plane into finite regions and infinite regions. Is it possible for all the finite regions to have integer area?Proposed by Tony Wang