2000 Pan African MO Problems

Day 1

Problem 1

Solve for $x \in R$ : $\sin^3{x}(1+\cot{x})+\cos^3{x}(1+\tan{x})=\cos{2x}$

Problem 2

Define the polynomials $P_0, P_1, P_2 \cdots$ by: $P_0(x)=x^3+213x^2-67x-2000$ $P_n(x)=P_{n-1}(x-n), n \in N$ Find the coefficient of $x$ in $P_{21}(x)$ .

Solution

Problem 3

Let $p$ and $q$ be coprime positive integers such that: $\dfrac{p}{q}=1-\frac12+\frac13-\frac14 \cdots -\dfrac{1}{1334}+\dfrac{1}{1335}$ Prove $p$ is divisible by 2003.

Solution

Day 2

Problem 4

Let $a$ , $b$ and $c$ be real numbers such that $a^2+b^2=c^2$ , solve the system: $z^2=x^2+y^2$ $(z+c)^2=(x+a)^2+(y+b)^2$ in real numbers $x, y$ and $z$ .

Solution

Problem 5

Let $\gamma$ be circle and let $P$ be a point outside $\gamma$ . Let $PA$ and $PB$ be the tangents from $P$ to $\gamma$ (where $A, B \in \gamma$ ). A line passing through $P$ intersects $\gamma$ at points $Q$ and $R$ . Let $S$ be a point on $\gamma$ such that $BS \parallel QR$ . Prove that $SA$ bisects $QR$ .

Solution

Problem 6

A company has five directors. The regulations of the company require that any majority (three or more) of the directors should be able to open its strongroom, but any minority (two or less) should not be able to do so. The strongroom is equipped with ten locks, so that it can only be opened when keys to all ten locks are available. Find all positive integers $n$ such that it is possible to give each of the directors a set of keys to $n$ different locks, according to the requirements and regulations of the company.

Solution

2000 Pan African MO (Problems)
Preceded by First Pan African MO	1 • 2 • 3 • 4 • 5 • 6	Followed by 2001 Pan African MO
All Pan African MO Problems and Solutions

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2000 Pan African MO Problems

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6

See Also