2017 IMC

Part of IMC

Subcontests

(10)

1

IMC 2017 Problem 10

K

be an equilateral triangle in the plane. Prove that for every

p>0

there exists an

\varepsilon>0

with the following property: If

n

is a positive integer, and

T_1,\ldots,T_n

are non-overlapping triangles inside

K

such that each of them is homothetic to

K

with a negative ratio, and

\sum_{\ell=1}^n \textrm{area}(T_\ell) > \textrm{area}(K)-\varepsilon,

\sum_{\ell=1}^n \textrm{perimeter}(T_\ell) > p.

1

IMC 2017 Problem 9

Define the sequence

f_1,f_2,\ldots :[0,1)\to \mathbb{R}

of continuously differentiable functions by the following recurrence: f_1=1; \qquad f_{n+1}'=f_nf_{n+1} \text{on $(0,1)$}, \text{and} f_{n+1}(0)=1.
Show that

\lim\limits_{n\to \infty}f_n(x)

exists for every

x\in [0,1)

and determine the limit function.

1

IMC 2017 Problem 8

Define the sequence

A_1,A_2,\ldots

of matrices by the following recurrence: A_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}, A_{n+1} = \begin{pmatrix} A_n & I_{2^n} \\ I_{2^n} & A_n \\ \end{pmatrix} (n=1,2,\ldots) where

I_m

m\times m

identity matrix.
Prove that

A_n

n+1

distinct integer eigenvalues

\lambda_0< \lambda_1<\ldots <\lambda_n

with multiplicities

\binom{n}{0},\binom{n}{1},\ldots,\binom{n}{n}

, respectively.

1

IMC 2017 Problem 7

p(x)

be a nonconstant polynomial with real coefficients. For every positive integer~

n

q_n(x) = (x+1)^np(x)+x^n p(x+1) .

Prove that there are only finitely many numbers

n

such that all roots of

q_n(x)

1

IMC 2017 Problem 6

f:[0;+\infty)\to \mathbb R

be a continuous function such that

\lim\limits_{x\to +\infty} f(x)=L

exists (it may be finite or infinite). Prove that

\lim\limits_{n\to\infty}\int\limits_0^{1}f(nx)\,\mathrm{d}x=L.

1

IMC 2017 Problem 5

k

n

be positive integers with

n\geq k^2-3k+4

f(z)=z^{n-1}+c_{n-2}z^{n-2}+\dots+c_0

be a polynomial with complex coefficients such that

c_0c_{n-2}=c_1c_{n-3}=\dots=c_{n-2}c_0=0

f(z)

z^n-1

n-k

1

IMC 2017 Problem 4

n

people in a city, and each of them has exactly

1000

friends (friendship is always symmetric). Prove that it is possible to select a group

S

of people such that at least

\frac{n}{2017}

S

have exactly two friends in

S

1

IMC 2017 Problem 3

For any positive integer

m

P(m)

the product of positive divisors of

m

P(6)=36

). For every positive integer

n

define the sequence a_1(n)=n,\qquad a_{k+1}(n)=P(a_k(n)) (k=1,2,\dots,2016) Determine whether for every set

S\subset\{1,2,\dots,2017\}

, there exists a positive integer

n

such that the following condition is satisfied:
For every

k

1\leq k\leq 2017

a_k(n)

is a perfect square if and only if

k\in S

1

IMC 2017 Problem 2

f:\mathbb R\to(0,\infty)

be a differentiabe function, and suppose that there exists a constant

L>0

|f'(x)-f'(y)|\leq L|x-y|

x,y

(f'(x))^2<2Lf(x)

x

1

IMC 2017 Problem 1

Determine all complex numbers

\lambda

for which there exists a positive integer

n

n\times n

A

A^2=A^T

\lambda

is an eigenvalue of

A