MathDB

1997*5^1998 mod 10000 (Ukraine 1998 Grade 8 P4)

Source:

6/4/2021

Determine the last four decimal digits of the number

1997\cdot5^{1998}

.

number theory

floor(x/y)=floor(nx)/floor(ny) for all n in N (Ukraine 1998 Grade 9 P4)

Source:

6/4/2021

Real numbers

x

and

y

not less than

1

have the property that

\left\lfloor\frac xy\right\rfloor=\frac{\lfloor nx\rfloor}{\lfloor ny\rfloor}\enspace\text{for any }n\in\mathbb N.

Prove that either

x=y

or

x

and

y

are integers, one dividing the other.

number theoryalgebrafloor function

functional inequality, bounding f

Source: Ukraine 1998 Grade 10 P4

6/5/2021

A function

f

defined on the interval

[1,+\infty)

satisfies

\frac{f(2x)}{\sqrt2}\le f(x)\le x\enspace\text{for }x\ge1.

Prove that

f(x)<\sqrt{2x}

for all

x\ge1

.

inequalitiesfefunctional equationfunctional inequalitiesfunctionFunctional Equations

f(f(x)+y)=f(x)+f(y), prove f(f(x))=f(x)

Source: Ukraine 1998 Grade 11 P4

6/6/2021

Consider a function

f:[0,1]\to[0,1]

. Suppose that there is a real number

0<\lambda<1

such that

f(\lambda)\notin\{0,\lambda\}

and the equality

f(f(x)+y)=f(x)+f(y)

holds whenever the function is defined on the arguments.
(a) Give an example of such a function. (b) Prove that for some

x\in[0,1]

,

\underbrace{f(f(\cdots f(}_{19}x)\cdots))=\underbrace{f(f(\cdots f(}_{98}x)\cdots)).

fefunctional equationFunctional Equationsalgebra

Problem 4