2016 South African National Olympiad

Part of South Africa National Olympiad

Subcontests

(6)

1

Divisibility of Binomial Differences

k

m

be integers with

1 < k < m

. For a positive integer

i

L_i

be the least common multiple of

1,2,\ldots,i

k

is a divisor of

L_i \cdot [\binom{m}{i} - \binom{m-k}{i}]

i \geq 1

\binom{n}{i} = \frac{n!}{i!(n-i)!}

denotes a binomial coefficient. Note that

\binom{n}{i} = 0

n < i

1

Maximize the quotient

For every positive integer

n

, determine the greatest possible value of the quotient

\frac{1-x^{n}-(1-x)^{n}}{x(1-x)^n+(1-x)x^n}

0 < x < 1

1

Lines with triple intersection points

For which integers

n \geq 2

is it possible to draw

n

straight lines in the plane in such a way that there are at least

n - 2

points where exactly three of the lines meet?

1

Points on a straight line

The inscribed circle of triangle

ABC

I

, touches sides

BC

CA

AB

D

E

F

, respectively. Let

P

be a point, on the same side of

FE

A

\angle PFE = \angle BCA

\angle PEF = \angle ABC

P

I

D

lie on a straight line.

1

Simultaneous quadratic equations

Determine all pairs of real numbers

a

b

b > 0

, such that the solutions to the two equations

x^2 + ax + a = b \qquad \text{and} \qquad x^2 + ax + a = -b

are four consecutive integers.

1

Football Jerseys

At the start of the Mighty Mathematicians Football Team's first game of the season, their coach noticed that the jersey numbers of the 22 players on the field were all the numbers from 1 to 22. At halftime, the coach substituted her goal-keeper, with jersey number 1, for a reserve player. No other substitutions were made by either team at or before halftime. The coach noticed that after the substitution, no two players on the field had the same jersey number and that the sums of the jersey numbers of each of the teams were exactly equal. Determine * the greatest possible jersey number of the reserve player, * the smallest possible (positive) jersey number of the reserve player.