MathDB

a_k is smallest integer > a_{k-1} for which a_k +a_{k-1} is perfect square

Source: Dutch IMO TST 2018 day 1 p3

8/30/2019

Let

n \ge 0

be an integer. A sequence

a_0,a_1,a_2,...

of integers is defined as follows: we have

a_0 = n

and for

k \ge 1, a_k

is the smallest integer greater than

a_{k-1}

for which

a_k +a_{k-1}

is the square of an integer. Prove that there are exactly

\lfloor \sqrt{2n}\rfloor

positive integers that cannot be written in the form

a_k - a_{\ell}

with

k > \ell\ge 0

.

number theorySumfloor functionPerfect Square

AE=BE, AF =CF, <BTE= <CTF=90^o, prove TA^2 =TB \cdot TC

Source: Dutch IMO TST2 2018 P3

8/5/2019

Let

ABC

be an acute triangle, and let

D

be the foot of the altitude through

A

. On

AD

, there are distinct points

E

and

F

such that

|AE| = |BE|

and

|AF| =|CF|

. A point

T \ne D

satises

\angle BTE = \angle CTF = 90^o

. Show that

|TA|^2 =|TB| \cdot |TC|

.

geometryright angleequal segments

(a+b)^3-2a^3-2b^3 is a power of two

Source: Dutch IMO TST 2018 day 3 p3

8/30/2019

Determine all pairs

(a,b)

of positive integers such that

(a+b)^3-2a^3-2b^3

is a power of two.

number theorypower of 2

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