2019 IMC

Part of IMC

Subcontests

(10)

1

Random points in the unit circle

2019

points are chosen at random, independently, and distributed uniformly in the unit disc

\{(x,y)\in\mathbb R^2: x^2+y^2\le 1\}

C

be the convex hull of the chosen points. Which probability is larger: that

C

is a polygon with three vertices, or a polygon with four vertices?
Proposed by Fedor Petrov, St. Petersburg State University

1

AB-BA=B^2A

Determine all positive integers

n

for which there exist

n\times n

real invertible matrices

A

B

AB-BA=B^2A

.
Proposed by Karen Keryan, Yerevan State University & American University of Armenia, Yerevan

1

s(I)=2019

x_1,\ldots,x_n

be real numbers. For any set

I\subset\{1,2,…,n\}

s(I)=\sum_{i\in I}x_i

. Assume that the function

I\to s(I)

takes on at least

1.8^n

I

2^n

\{1,2,…,n\}

. Prove that the number of sets

I\subset \{1,2,…,n\}

s(I)=2019

does not exceed

1.7^n

.
Proposed by Fedor Part and Fedor Petrov, St. Petersburg State University

1

A series over composite numbers

C=\{4,6,8,9,10,\ldots\}

be the set of composite positive integers. For each

n\in C

a_n

be the smallest positive integer

k

k!

is divisible by

n

. Determine whether the following series converges:

\sum_{n\in C}\left(\frac{a_n}{n}\right)^n.

Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan

1

Two continuous functions

f,g:\mathbb R\to\mathbb R

be continuous functions such that

g

is differentiable. Assume that

(f(0)-g'(0))(g'(1)-f(1))>0

. Show that there exists a point

c\in (0,1)

f(c)=g'(c)

.
Proposed by Fereshteh Malek, K. N. Toosi University of Technology

1

(n^3+3n)^2/(n^6-64)

Evaluate the product

\prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64}.

Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karen Keryan, Yerevan State University and American University of Armenia, Yerevan

1

Very good YEAR

A four-digit number

YEAR

is called very good if the system \begin{align*} Yx+Ey+Az+Rw& =Y\\ Rx+Yy+Ez+Aw & = E\\\ Ax+Ry+Yz+Ew & = A\\ Ex+Ay+Rz+Yw &= R \end{align*} of linear equations in the variables

x,y,z

w

has at least two solutions. Find all very good

YEAR

s in the 21st century.
(The

21

st century starts in

2001

2100

.)
Proposed by Tomáš Bárta, Charles University, Prague

1

Twice differentiable function

f:(-1,1)\to \mathbb{R}

be a twice differentiable function such that 2f’(x)+xf''(x)\geqslant 1 \text{ for } x\in (-1,1). Prove that

\int_{-1}^{1}xf(x)dx\geqslant \frac{1}{3}.

Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karim Rakhimov, Scuola Normale Superiore and National University of Uzbekistan

1

Matrices satisfy three conditions

Determine whether there exist an odd positive integer

n

n\times n

A

B

with integer entries, that satisfy the following conditions:
[*]

\det (B)=1

AB=BA

A^4+4A^2B^2+16B^4=2019I

I

n\times n

identity matrix.)
Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan

1

IMC 2019 day 1 problem 4

(n+3)a_{n+2}=(6n+9)a_{n+1}-na_n

a_0=1

a_1=2

prove that all the terms of the sequence are integers