MathDB
Problems
Contests
Undergraduate contests
IMC
2015 IMC
2015 IMC
Part of
IMC
Subcontests
(10)
10
1
Hide problems
IMC2015, problem 10
Let
n
n
n
be a positive integer, and let
p
(
x
)
p(x)
p
(
x
)
be a polynomial of degree
n
n
n
with integer coefficients. Prove that \max_{0\le x\le1} \big|p(x)\big| > \frac1{e^n}. Proposed by Géza Kós, Eötvös University, Budapest
9
1
Hide problems
IMC2015, problem 9
An
n
×
n
n \times n
n
×
n
complex matrix
A
A
A
is called \emph{t-normal} if
A
A
t
=
A
t
A
AA^t = A^t A
A
A
t
=
A
t
A
where
A
t
A^t
A
t
is the transpose of
A
A
A
. For each
n
n
n
, determine the maximum dimension of a linear space of complex
n
×
n
n \times n
n
×
n
matrices consisting of t-normal matrices.Proposed by Shachar Carmeli, Weizmann Institute of Science
8
1
Hide problems
IMC2015, problem 8
Consider all
2
6
26
26^{26}
2
6
26
words of length 26 in the Latin alphabet. Define the \emph{weight} of a word as
1
/
(
k
+
1
)
1/(k+1)
1/
(
k
+
1
)
, where
k
k
k
is the number of letters not used in this word. Prove that the sum of the weights of all words is
3
75
3^{75}
3
75
.Proposed by Fedor Petrov, St. Petersburg State University
7
1
Hide problems
IMC2015, problem 7
Compute
lim
A
→
+
∞
1
A
∫
1
A
A
1
x
d
x
.
\lim_{A\to+\infty}\frac1A\int_1^A A^{\frac1x}\, dx .
A
→
+
∞
lim
A
1
∫
1
A
A
x
1
d
x
.
Proposed by Jan Šustek, University of Ostrava
6
1
Hide problems
IMC2015, problem 6
Prove that \sum\limits_{n = 1}^{\infty}\frac{1}{\sqrt{n}\left(n+1\right)} < 2.Proposed by Ivan Krijan, University of Zagreb
5
1
Hide problems
IMC2015, problem 5
Let
n
≥
2
n\ge2
n
≥
2
, let
A
1
,
A
2
,
…
,
A
n
+
1
A_1,A_2,\ldots,A_{n+1}
A
1
,
A
2
,
…
,
A
n
+
1
be
n
+
1
n+1
n
+
1
points in the
n
n
n
-dimensional Euclidean space, not lying on the same hyperplane, and let
B
B
B
be a point strictly inside the convex hull of
A
1
,
A
2
,
…
,
A
n
+
1
A_1,A_2,\ldots,A_{n+1}
A
1
,
A
2
,
…
,
A
n
+
1
. Prove that \angle A_iBA_j>90^\circ holds for at least
n
n
n
pairs
(
i
,
j
)
(i,j)
(
i
,
j
)
with \displaystyle{1\le i<j\le n+1}.Proposed by Géza Kós, Eötvös University, Budapest
4
1
Hide problems
IMC2015, problem 4
Determine whether or not there exist 15 integers
m
1
,
…
,
m
15
m_1,\ldots,m_{15}
m
1
,
…
,
m
15
such that~ \displaystyle \sum_{k=1}^{15}\,m_k\cdot\arctan(k) = \arctan(16). \eqno(1)(Proposed by Gerhard Woeginger, Eindhoven University of Technology)
3
1
Hide problems
IMC2015, problem 3
Let
F
(
0
)
=
0
F(0)=0
F
(
0
)
=
0
,
F
(
1
)
=
3
2
F(1)=\frac32
F
(
1
)
=
2
3
, and
F
(
n
)
=
5
2
F
(
n
−
1
)
−
F
(
n
−
2
)
F(n)=\frac{5}{2}F(n-1)-F(n-2)
F
(
n
)
=
2
5
F
(
n
−
1
)
−
F
(
n
−
2
)
for
n
≥
2
n\ge2
n
≥
2
.Determine whether or not
∑
n
=
0
∞
1
F
(
2
n
)
\displaystyle{\sum_{n=0}^{\infty}\, \frac{1}{F(2^n)}}
n
=
0
∑
∞
F
(
2
n
)
1
is a rational number.(Proposed by Gerhard Woeginger, Eindhoven University of Technology)
2
1
Hide problems
IMC2015, problem 2
For a positive integer
n
n
n
, let
f
(
n
)
f(n)
f
(
n
)
be the number obtained by writing
n
n
n
in binary and replacing every 0 with 1 and vice versa. For example,
n
=
23
n=23
n
=
23
is 10111 in binary, so
f
(
n
)
f(n)
f
(
n
)
is 1000 in binary, therefore
f
(
23
)
=
8
f(23) =8
f
(
23
)
=
8
. Prove that
∑
k
=
1
n
f
(
k
)
≤
n
2
4
.
\sum_{k=1}^n f(k) \leq \frac{n^2}{4}.
k
=
1
∑
n
f
(
k
)
≤
4
n
2
.
When does equality hold?(Proposed by Stephan Wagner, Stellenbosch University)
1
1
Hide problems
IMC2015, problem 1
For any integer
n
≥
2
n\ge 2
n
≥
2
and two
n
×
n
n\times n
n
×
n
matrices with real entries
A
,
B
A,\; B
A
,
B
that satisfy the equation
A
−
1
+
B
−
1
=
(
A
+
B
)
−
1
A^{-1}+B^{-1}=(A+B)^{-1}\;
A
−
1
+
B
−
1
=
(
A
+
B
)
−
1
prove that
det
(
A
)
=
det
(
B
)
\det (A)=\det(B)
det
(
A
)
=
det
(
B
)
. Does the same conclusion follow for matrices with complex entries?(Proposed by Zbigniew Skoczylas, Wroclaw University of Technology)