2014 Paenza

Part of Paenza

Subcontests

(6)

1

CIMA 2014 Problem 6

(a) Show that if

f:[-1,1]\to \mathbb{R}

is a convex and

C^2

function such that

f(1),f(-1)\geq 0

\min_{x\in[-1,1]} \{f(x)\} \geq - \int_{-1}^1 f''

B\subset \mathbb{R}^2

the closed ball with center

0

1

f: B \to \mathbb{R}

is a convex and

C^2

f\geq 0

\partial B

f(0)\geq -\frac{1}{\sqrt{\pi}} \left( \int_{B} (f_{xx}f_{yy}-f_{xy}^2) \right)^{1/2}

1

CIMA 2014 Problem 5

\mathbb{A}

be the least subset of finite sequences of nonnegative integers that satisfies the following two properties:
-

(0,0) \in \mathbb{A}

(a_1,\ldots,a_n)\in \mathbb{A}

(a_1,\ldots,a_{i-2},a_{i-1}+1,1,a_{i}+1,a_{i+1},\ldots,a_n)\in \mathbb{A}

i\in \{2,\ldots,n\}

n\geq 2

\mathbb{B}(n)

be the set of sequences in

\mathbb{A}

n

terms. Find the number of elements of

\mathbb{B}

1

CIMA 2014 Problem 4

\mathcal{C}

be the family of circumferences in

\mathbb{R}^2

that satisfy the following properties:
(i) if

C_n

is the circumference with center

(n,1/2)

1/2

C_n\in \mathcal{C}

n\in \mathbb{Z}

C

C'

\mathcal{C}

, are externally tangent, then the circunference externally tangent to

C

C'

x

-axis also belongs to

\mathcal{C}

\mathcal{C}

is the least family which these properties.
Determine the set of the real numbers which are obtained as the first coordinate of the points of intersection between the elements of

\mathcal{C}

x

1

CIMA 2014 Problem 3

(m,n)

\mathbb{N}^2

m\mid n^2+1

n\mid m^2+1

1

CIMA 2014 Problem 2

n

cards numbered and stacked in increasing order from up to down (i.e. the card in the top is the number 1, the second is the 2, and so on...). With this deck, the next steps are followed:
-the first card (from the top) is put in the bottom of the deck. -the second card (from the top) is taken away of the deck. -the third card (from the top) is put in the bottom of the deck. -the fourth card (from the top) is taken away of the deck. - ...
The proccess goes on always the same way: the card in the top is put at the end of the deck and the next is taken away of the deck, until just one card is left. Determine which is that card.

1

CIMA 2014 Problem 1

\{a_n\}_{n\geq 1}

be a sequence of real numbers which satisfies the following relation:

a_{n+1}=10^n a_n^2

(a) Prove that if

a_1

is small enough, then

\displaystyle\lim_{n\to\infty} a_n =0

. (b) Find all possible values of

a_1\in \mathbb{R}

a_1\geq 0

\displaystyle\lim_{n\to\infty} a_n =0