MathDB

sequence positive

Source: ISL 2006, A2, VAIMO 2007, P4, Poland 2007

4/22/2007

The sequence of real numbers

a_0,a_1,a_2,\ldots

is defined recursively by a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0 \text{for} n\geq 1.Show that

a_{n} > 0

for all

n\geq 1

.
Proposed by Mariusz Skalba, Poland

integrationinequalitiesalgebraSequenceSummationcalculusIMO Shortlist

x is rational implies y is rational

Source: IMO Shortlist 2006, N2, VAIMO 2007, Problem 6

6/28/2007

For

x \in (0, 1)

let

y \in (0, 1)

be the number whose

n

-th digit after the decimal point is the

2^{n}

-th digit after the decimal point of

x

. Show that if

x

is rational then so is

y

.
Proposed by J.P. Grossman, Canada

number theoryrationaldecimal representationIMO Shortlist

four points lie on a circle

Source: IMO Shortlist 2006, Geometry 2, AIMO 2007, TST 1, P2

6/28/2007

Let

ABCD

be a trapezoid with parallel sides

AB > CD

. Points

K

and

L

lie on the line segments

AB

and

CD

, respectively, so that

AK/KB=DL/LC

. Suppose that there are points

P

and

Q

on the line segment

KL

satisfying \angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}. Prove that the points

P

,

Q

,

B

and

C

are concyclic.
Proposed by Vyacheslev Yasinskiy, Ukraine

geometrytrapezoidcircumcircleratioIMO ShortlisthomothetyHi

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