MathDB

Colombia TST [regular n-gon and angles summing up to 180°]

Source: IMO Shortlist 2004 geometry problem G5

6/7/2005

Let

A_1A_2A_3\ldots A_n

be a regular

n

-gon. Let

B_1

and

B_{n-1}

be the midpoints of its sides

A_1A_2

and

A_{n-1}A_n

. Also, for every

i\in\left\{2,3,4,\ldots ,n-2\right\}

. Let

S

be the point of intersection of the lines

A_1A_{i+1}

and

A_nA_i

, and let

B_i

be the point of intersection of the angle bisector bisector of the angle

\measuredangle A_iSA_{i+1}

with the segment

A_iA_{i+1}

.
Prove that

\sum_{i=1}^{n-1} \measuredangle A_1B_iA_n=180^{\circ}

.
Proposed by Dusan Dukic, Serbia and Montenegro

geometryangle bisectorIMO Shortlist

ab+bc+ca = 1

Source: IMO ShortList 2004, algebra problem 5

3/22/2005

If

a

,

b

,

c

are three positive real numbers such that

ab+bc+ca = 1

, prove that

\sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}.

inequalitiesIMO ShortlistHi

Game

Source: Colombia TST, IMO ShortList 2004, combinatorics problem 5

6/7/2005

A

and

B

play a game, given an integer

N

,

A

writes down

1

first, then every player sees the last number written and if it is

n

then in his turn he writes

n+1

or

2n

, but his number cannot be bigger than

N

. The player who writes

N

wins. For which values of

N

does

B

win?
Proposed by A. Slinko & S. Marshall, New Zealand

combinatoricsIMO Shortlistgameilostthegamegames

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