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Contests
Undergraduate contests
IMC
2018 IMC
2018 IMC
Part of
IMC
Subcontests
(10)
10
1
Hide problems
IMC 2018 P10
For
R
>
1
R>1
R
>
1
let
D
R
=
{
(
a
,
b
)
∈
Z
2
:
0
<
a
2
+
b
2
<
R
}
\mathcal{D}_R =\{ (a,b)\in \mathbb{Z}^2: 0<a^2+b^2<R\}
D
R
=
{(
a
,
b
)
∈
Z
2
:
0
<
a
2
+
b
2
<
R
}
. Compute
lim
R
→
∞
∑
(
a
,
b
)
∈
D
R
(
−
1
)
a
+
b
a
2
+
b
2
.
\lim_{R\rightarrow \infty}{\sum_{(a,b)\in \mathcal{D}_R}{\frac{(-1)^{a+b}}{a^2+b^2}}}.
R
→
∞
lim
(
a
,
b
)
∈
D
R
∑
a
2
+
b
2
(
−
1
)
a
+
b
.
Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro
9
1
Hide problems
IMC 2018 P9
Determine all pairs
P
(
x
)
,
Q
(
x
)
P(x),Q(x)
P
(
x
)
,
Q
(
x
)
of complex polynomials with leading coefficient
1
1
1
such that
P
(
x
)
P(x)
P
(
x
)
divides
Q
(
x
)
2
+
1
Q(x)^2+1
Q
(
x
)
2
+
1
and
Q
(
x
)
Q(x)
Q
(
x
)
divides
P
(
x
)
2
+
1
P(x)^2+1
P
(
x
)
2
+
1
.Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro
8
1
Hide problems
IMC 2018 P8
Let
Ω
=
{
(
x
,
y
,
z
)
∈
Z
3
:
y
+
1
⩾
x
⩾
y
⩾
z
⩾
0
}
\Omega =\{ (x,y,z)\in \mathbb{Z}^3:y+1\geqslant x\geqslant y\geqslant z\geqslant 0\}
Ω
=
{(
x
,
y
,
z
)
∈
Z
3
:
y
+
1
⩾
x
⩾
y
⩾
z
⩾
0
}
. A frog moves along the points of
Ω
\Omega
Ω
by jumps of length
1
1
1
. For every positive integer
n
n
n
, determine the number of paths the frog can take to reach
(
n
,
n
,
n
)
(n,n,n)
(
n
,
n
,
n
)
starting from
(
0
,
0
,
0
)
(0,0,0)
(
0
,
0
,
0
)
in exactly
3
n
3n
3
n
jumps.Proposed by Fedor Petrov and Anatoly Vershik, St. Petersburg State University
7
1
Hide problems
IMC 2018 P7
Let
(
a
n
)
n
=
0
∞
(a_n)_{n=0}^{\infty}
(
a
n
)
n
=
0
∞
be a sequence of real numbers such that
a
0
=
0
a_0=0
a
0
=
0
and a_{n+1}^3=a_n^2-8 \text{for} n=0,1,2,… Prove that the following series is convergent:
∑
n
=
0
∞
∣
a
n
+
1
−
a
n
∣
.
\sum_{n=0}^{\infty}{|a_{n+1}-a_n|}.
n
=
0
∑
∞
∣
a
n
+
1
−
a
n
∣
.
Proposed by Orif Ibrogimov, National University of Uzbekistan
6
1
Hide problems
IMC 2018 P6
Let
k
k
k
be a positive integer. Find the smallest positive integer
n
n
n
for which there exists
k
k
k
nonzero vectors
v
1
,
v
2
,
…
,
v
k
v_1,v_2,…,v_k
v
1
,
v
2
,
…
,
v
k
in
R
n
\mathbb{R}^n
R
n
such that for every pair
i
,
j
i,j
i
,
j
of indices with
∣
i
−
j
∣
>
1
|i-j|>1
∣
i
−
j
∣
>
1
the vectors
v
i
v_i
v
i
and
v
j
v_j
v
j
are orthogonal.Proposed by Alexey Balitskiy, Moscow Institute of Physics and Technology and M.I.T.
5
1
Hide problems
IMC 2018 P5
Let
p
p
p
and
q
q
q
be prime numbers with
p
<
q
p<q
p
<
q
. Suppose that in a convex polygon
P
1
,
P
2
,
…
,
P
p
q
P_1,P_2,…,P_{pq}
P
1
,
P
2
,
…
,
P
pq
all angles are equal and the side lengths are distinct positive integers. Prove that
P
1
P
2
+
P
2
P
3
+
⋯
+
P
k
P
k
+
1
⩾
k
3
+
k
2
P_1P_2+P_2P_3+\cdots +P_kP_{k+1}\geqslant \frac{k^3+k}{2}
P
1
P
2
+
P
2
P
3
+
⋯
+
P
k
P
k
+
1
⩾
2
k
3
+
k
holds for every integer
k
k
k
with
1
⩽
k
⩽
p
1\leqslant k\leqslant p
1
⩽
k
⩽
p
.Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Berlin
4
1
Hide problems
IMC 2018 P4
Find all differentiable functions
f
:
(
0
,
∞
)
→
R
f:(0,\infty) \to \mathbb{R}
f
:
(
0
,
∞
)
→
R
such that
f
(
b
)
−
f
(
a
)
=
(
b
−
a
)
f
’
(
a
b
)
for all
a
,
b
>
0.
f(b)-f(a)=(b-a)f’(\sqrt{ab}) \qquad \text{for all}\qquad a,b>0.
f
(
b
)
−
f
(
a
)
=
(
b
−
a
)
f
’
(
ab
)
for all
a
,
b
>
0.
Proposed by Orif Ibrogimov, National University of Uzbekistan
3
1
Hide problems
IMC 2018 P3
Determine all rational numbers
a
a
a
for which the matrix
(
a
−
a
−
1
0
a
−
a
0
−
1
1
0
a
−
a
0
1
a
−
a
)
\begin{pmatrix} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a\\ 0 & 1 & a & -a \end{pmatrix}
a
a
1
0
−
a
−
a
0
1
−
1
0
a
a
0
−
1
−
a
−
a
is the square of a matrix with all rational entries.Proposed by Daniël Kroes, University of California, San Diego
2
1
Hide problems
IMC 2018 P2
Does there exist a field such that its multiplicative group is isomorphism to its additive group?Proposed by Alexandre Chapovalov, New York University, Abu Dhabi
1
1
Hide problems
IMC 2018 P1
Let
(
a
n
)
n
=
1
∞
(a_n)_{n=1}^{\infty}
(
a
n
)
n
=
1
∞
and
(
b
n
)
n
=
1
∞
(b_n)_{n=1}^{\infty}
(
b
n
)
n
=
1
∞
be two sequences of positive numbers. Show that the following statements are equivalent:[*]There is a sequence
(
c
n
)
n
=
1
∞
(c_n)_{n=1}^{\infty}
(
c
n
)
n
=
1
∞
of positive numbers such that
∑
n
=
1
∞
a
n
c
n
\sum_{n=1}^{\infty}{\frac{a_n}{c_n}}
∑
n
=
1
∞
c
n
a
n
and
∑
n
=
1
∞
c
n
b
n
\sum_{n=1}^{\infty}{\frac{c_n}{b_n}}
∑
n
=
1
∞
b
n
c
n
both converge;[/*] [*]
∑
n
=
1
∞
a
n
b
n
\sum_{n=1}^{\infty}{\sqrt{\frac{a_n}{b_n}}}
∑
n
=
1
∞
b
n
a
n
converges.[/*] Proposed by Tomáš Bárta, Charles University, Prague