MathDB

1

Bound on real root of polynomial with non-increasing coefficients

n \geq 1

be an integer, and

a_0, a_1, \dots, a_{n-1}

be real numbers such that

1 \geq a_{n-1} \geq a_{n-2} \geq \dots \geq a_1 \geq a_0 \geq 0.

We assume that

\lambda

is a real root of the polynomial

x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0.

Prove that

|\lambda| \leq 1

.

G3

1

Lengths involving regular pentagon and a point on its circumcircle

Let

ABCDE

be a regular pentagon, and

F

some point on the arc

AB

of the circumcircle of

ABCDE

. Show that

\frac{FD}{FE + FC} = \frac{FB + FA}{FD} = \frac{-1 + \sqrt{5}}{2},

and that

FD + FB + FA = FE + FC

.

N2

1

2017 PAMO Shortlsit: Power of a prime is a sum of cubes

For which prime numbers

p

can we find three positive integers

n

,

x

and

y

such that

p^n = x^3 + y^3

?

C1

1

Number of sequences of coin tosses with odd number of heads followed by tails

Abimbola plays a game with a coin. He tosses the coin a number of times, and records whether each toss was a "heads" or "tails". He stops tossing the coin as soon as he tosses an odd number of heads in a row, followed by a tails. (Note that he stops if the number of heads since the previous time that he tosses tails is odd, and he then tosses another tails. If he has not tossed tails previously, then he stops if the total number of heads is odd, and he then tosses tails.) How many different sequences of coin tosses are there such that he stops after the

n^\text{th}

coin toss?

I4

1

PAMO 2017 Shortlst: Sum of maxima of adjacent pairs in permutation

Find the maximum and minimum of the expression

\max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1),

where

(a_1, a_2, \dots, a_n)

runs over the set of permutations of

(1, 2, \dots, n)

.

C2

1

PAMO 2017 Shortlist: Extrema of sum of numbers on a chessboard

On a

50 \times 50

chessboard, we put, in the lower left corner, a die whose faces are numbered from

1

to

6

. By convention, the sum of digits on two opposite side of the die equals

7

. Adama wants to move the die to the diagonally opposite corner using the following rule: at each step, Adama can roll the die only on to its right side, or to its top side. We suppose that whenever the die lands on a square, the number on its bottom face is printed on the square. By the end of these operations, Adama wants to find the sum of the

99

numbers appearing on the chessboard. What are the maximum and minimum possible values of this sum?

G1

1

PAMO 2017 Shortlist: Angle bisector in a square

We consider a square

ABCD

and a point

E

on the segment

CD

. The bisector of

\angle EAB

cuts the segment

BC

in

F

. Prove that

BF + DE = AE

.

A4

1

f ( f (x)+y) = f (x^2 -y)+4 f (x)y

Find all functions

f : R\rightarrow R

such that

f ( f (x)+y) = f (x^2 -y)+4 f (x)y

for all

x,y \in R

.

A2

1

a+b+c=abc

Find all integers

a,b,c

such that

a+b+c=abc

N1

1

A 28

Prove that the expression

\frac{\gcd(m, n)}{n}{n \choose m}

is an integer for all pairs of positive integers

(m, n)

with

n \ge m \ge 1

.

2017 Pan-African Shortlist

Subcontests

Bound on real root of polynomial with non-increasing coefficients

Lengths involving regular pentagon and a point on its circumcircle

2017 PAMO Shortlsit: Power of a prime is a sum of cubes

Number of sequences of coin tosses with odd number of heads followed by tails

PAMO 2017 Shortlst: Sum of maxima of adjacent pairs in permutation

PAMO 2017 Shortlist: Extrema of sum of numbers on a chessboard

PAMO 2017 Shortlist: Angle bisector in a square

f ( f (x)+y) = f (x^2 -y)+4 f (x)y

a+b+c=abc

A 28