MathDB

for every positive divisor d of n, exists (d-k) divisor of n

Source: Dutch IMO TST 2018 day 2 p2

8/30/2019

Find all positive integers

n

, for which there exists a positive integer

k

such that for every positive divisor

d

of

n

, the number

d - k

is also a (not necessarily positive) divisor of

n

.

number theorydivisor

tangent of circumcircle perpenciular to line, right triangle starting with

Source: Dutch IMO TST1 2018 P2

8/5/2019

Suppose a triangle

\vartriangle ABC

with

\angle C = 90^o

is given. Let

D

be the midpoint of

AC

, and let

E

be the foot of the altitude through

C

on

BD

. Show that the tangent in

C

of the circumcircle of

\vartriangle AEC

is perpendicular to

AB

.

geometrycircumcircletangentperpendicularright triangle

f(x^2) - f(y^2) <= (f(x)+y) (x-f(y))

Source: Dutch IMO TST day 3 p2

8/30/2019

Find all functions

f : R \to R

such that

f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))

for all

x, y \in R

.

Functional inequalityfunctioninequalitiesalgebra

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