MathDB

1

Sum with letters - Paenza Olympiad 2010

a) Replace each letter in the following sum by a digit from

0

to

9

, in such a way that the sum is correct.
\tab \tab

ABC

\tab \tab

DEF

+GHI

\tab \tab \tab

J J J

Different letters must be replaced by different digits, and equal letters must be replaced by equal digits. Numbers

ABC

,

DEF

,

GHI

and

JJJ

cannot begin by

0

.
b) Determine how many triples of numbers

(ABC,DEF,GHI)

can be formed under the conditions given in a).

6

1

Two tetrahedra and octahedra - Paenza 2010

In space are given two tetrahedra with the same barycenter such that one of them contains the other. For each tetrahedron, we consider the octahedron whose vertices are the midpoints of the tetrahedron's edges. Prove that one of this octahedra contains the other.

5

1

Bricks in 4-dimensional space - Paenza 2010

In

4

-dimensional space, a set of

1 \times 2 \times 3 \times 4

bricks is given. Decide whether it is possible to build boxes of the following sizes using these bricks: i)

2 \times 5 \times 7 \times 12

ii)

5 \times 5 \times 10 \times 12

iii)

6 \times 6 \times 6 \times 6

.

3

1

Compute limit - Paenza 2010

Let

(x_n)_{n \in \mathbb{N}}

be the sequence defined as

x_n = \sin(2 \pi n! e)

for all

n \in \mathbb{N}

. Compute

\lim_{n \to \infty} x_n

.

2

1

Polynomial with integer coefficients - Paenza 2010

A polynomial

f

with integer coefficients is written on the blackboard. The teacher is a mathematician who has

3

kids: Andrew, Beth and Charles. Andrew, who is

7

, is the youngest, and Charles is the oldest. When evaluating the polynomial on his kids' ages he obtains:

f(7) = 77

f(b) = 85

, where

b

is Beth's age,

f(c) = 0

, where

c

is Charles' age. How old is each child?

2010 Paenza

Subcontests

Sum with letters - Paenza Olympiad 2010

Two tetrahedra and octahedra - Paenza 2010

Bricks in 4-dimensional space - Paenza 2010

Compute limit - Paenza 2010

Polynomial with integer coefficients - Paenza 2010