MathDB

OMK 2015 Sulong, Section B Problem 2

Source: OMK 2015 Sulong, Section B Problem 3

6/21/2021

A subset of

\{1, 2, 3, ... ... , 2015\}

is called good if the following condition is fulfilled: for any element

x

of the subset, the sum of all the other elements in the subset has the same last digit as

x

. For example,

\{10, 20, 30\}

is a good subset since

10

has the same last digit as

20 + 30 = 50

,

20

has the same last digit as

10 + 30 = 40

, and

30

has the same last digit as

10 + 20 = 30

. (a) Find an example of a good subset with 400 elements. (b) Prove that there is no good subset with 405 elements.

SetsProofnumber theory

OMK 2018 Muda, Section B Problem 2

Source: OMK 2018 Muda, Section B Problem 2

6/21/2021

Let

a

and

b

be positive integers such that (i) both

a

and

b

have at least two digits; (ii)

a + b

is divisible by

10

; (iii)

a

can be changed into

b

by changing its last digit. Prove that the hundreds digit of the product

ab

is even.

number theoryDigits

OMK 2018 Bongsu, Section B Problem 2

Source: OMK 2018 Bongsu, Section B Problem 2

6/20/2021

Prove that the number

9^{(a_1 + a_2)(a_2 + a_3)(a_3 + a_4)...(a_{98} + a_{99})(a_{99} + a_1)}

−

1

is divisible by

10

, for any choice of positive integers

a_1, a_2, a_3, . . . , a_{99}

.

Proofnumber theorycontests

B2