MathDB

triangle geometry [AP^2 + PD^2 = ...]

Source: australian tst 1, IMO ShortList 2003, geometry problem 3; Indian IMOTC 2004 Day 2 Problem 1

5/12/2004

Let

ABC

be a triangle and let

P

be a point in its interior. Denote by

D

,

E

,

F

the feet of the perpendiculars from

P

to the lines

BC

,

CA

,

AB

, respectively. Suppose that

AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.

Denote by

I_A

,

I_B

,

I_C

the excenters of the triangle

ABC

. Prove that

P

is the circumcenter of the triangle

I_AI_BI_C

.
Proposed by C.R. Pranesachar, India

geometrycircumcircleCircumcenterTriangleIMO Shortlistexcentersgeometry solved

Calculus rather than inequalities

Source: German TST, IMO ShortList 2003, algebra problem 3

7/15/2004

Consider pairs of the sequences of positive real numbers

a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots

and the sums A_n = a_1 + \cdots + a_n, B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots. For any pair define

c_n = \min\{a_i,b_i\}

and

C_n = c_1 + \cdots + c_n

,

n=1,2,\ldots

.
(1) Does there exist a pair

(a_i)_{i\geq 1}

,

(b_i)_{i\geq 1}

such that the sequences

(A_n)_{n\geq 1}

and

(B_n)_{n\geq 1}

are unbounded while the sequence

(C_n)_{n\geq 1}

is bounded?
(2) Does the answer to question (1) change by assuming additionally that

b_i = 1/i

,

i=1,2,\ldots

?
Justify your answer.

calculusinequalitiesSequencesboundedalgebraIMO Shortlist

A bit of geometry, combinatorics, and number theory

Source: German TST, IMO ShortList 2003, combinatorics problem 3

5/18/2004

Let

n \geq 5

be a given integer. Determine the greatest integer

k

for which there exists a polygon with

n

vertices (convex or not, with non-selfintersecting boundary) having

k

internal right angles.
Proposed by Juozas Juvencijus Macys, Lithuania

geometrycombinatorial geometrypolygonExtremal combinatoricsright angleanglesIMO Shortlist

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