MathDB

n+ \lfloor \sqrt{n} \rfloor+\lfloor \sqrt[3]{n} \rfloor=2014

Source: JBMO Shortlist 2014 A1

4/24/2019

Solve in positive real numbers:

n+ \lfloor \sqrt{n} \rfloor+\lfloor \sqrt[3]{n} \rfloor=2014

algebrafloor functionFloorequationpositive real

AZE JBMO TST

Source: AZE JBMO TST

5/2/2015

There are some real numbers on the board (at least two). In every step we choose two of them, for example

a

and

b

, and then we replace them with

\frac{ab}{a+b}

. We continue until there is one number. Prove that the last number does not depend on which order we choose the numbers to erase.

combinatorics

2014 JBMO Shortlist G1

Source: 2014 JBMO Shortlist G1

10/8/2017

Let

{ABC}

be a triangle with

m\left( \angle B \right)=m\left( \angle C \right)={{40}^{{}^\circ }}

Line bisector of

{\angle{B}}

intersects

{AC}

at point

{D}

. Prove that

BD+DA=BC

.

JBMOgeometry

AZE JBMO TST

Source: AZE JBMO TST

5/2/2015

All letters in the word

VUQAR

are different and chosen from the set

\{1,2,3,4,5\}

. Find all solutions to the equation

\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.

number theoryAZE JBMO TST

1