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find real a such that 3 < a < 4 and a(a-3{a}) is an integer.

Source: CRMO 2015 region 1 p6

9/30/2018

Find all real numbers

a

such that

3 < a < 4

and

a(a-3\{a\})

is an integer.
(Here

\{a\}

denotes the fractional part of

a

.)

number theoryfractional partInteger

Question 6

Source:

12/6/2015

Find all real numbers

a

such that

4 < a < 5

and

a(a-3\{a\})

is an integer. ({x} represents the fractional part of x)

RMO 2015

Greatest Integer Function

Source: RMO 2015 P6

9/2/2018

For how many integer values of

m

, (i)

1\le m \le 5000

(ii)

[\sqrt{m}] =[\sqrt{m+125}]

Note:

[x]

is the greatest integer function

algebrafloor functionfunction

RMO Delhi Q.6

Source:

12/20/2015

Show that there are infinitely many positive real numbers a which are not integers such that a(a-3{a}) is an integer.

Summing up cardinality of intersection of sets with union=S

Source: RMO (Mumbai Region) 2015 P6

12/6/2015

Let

S=\{1,2,\cdots, n\}

and let

T

be the set of all ordered triples of subsets of

S

, say

(A_1, A_2, A_3)

, such that

A_1\cup A_2\cup A_3=S

. Determine, in terms of

n

,

\sum_{(A_1,A_2,A_3)\in T}|A_1\cap A_2\cap A_3|

combinatoricscombinatorics proposedEnumerative Combinatoricscounting

6