MathDB

1

r_1+r_2+r_3+...+r_{p-1}=\frac{p^2(p-1)}{2} where r_k = remainder k^p : p^2

p> 2

is a prime number. For each integer

k = 1, 2,..., p-1

, denote

r_k

as the remainder of the division

k^p

by

p^2

. Prove that

r_1+r_2+r_3+...+r_{p-1}=\frac{p^2(p-1)}{2}

3

1

5 boxes in a circles, moving balls until each box has 17^{5^{2016}} balls ?

There are

5

boxes arranged in a circle. At first, there is one a box containing one ball, while the other boxes are empty. At each step, we can do one of the following two operations: i. select one box that is not empty, remove one ball from the box and add one ball into both boxes next to the box, ii. select an empty box next to a non-empty box, from the box the non-empty one moves one ball to the empty box. Is it possible, that after a few steps, obtained conditions where each box contains exactly

17^{5^{2016}}

balls?

1

angle chasing with an orthodiagonal ABCD

Let

ABCD

be a cyclic quadrilateral wih both diagonals perpendicular to each other and intersecting at point

O

. Let

E,F,G,H

be the orthogonal projections of

O

on sides

AB,BC,CD,DA

respectively. a. Prove that

\angle EFG + \angle GHE = 180^o

b. Prove that

OE

bisects angle

\angle FEH

.

2

1

2^c + 2^{2016} = a^b with b>1, and a,b,c naturals

Determine all triples of natural numbers

(a,b, c)

with

b> 1

such that

2^c + 2^{2016} = a^b

.

4

1

Equation with cosinus

Given triangle

ABC

such that angles

A

,

B

,

C

satisfy

\frac{\cos A}{20}+\frac{\cos B}{21}+\frac{\cos C}{29}=\frac{29}{420}

Prove that

ABC

is right angled triangle

8

1

Indonesian MO 2016 Last Problem

Determine with proof, the number of permutations

a_1,a_2,a_3,...,a_{2016}

of

1,2,3,...,2016

such that the value of

|a_i-i|

is fixed for all

i=1,2,3,...,2016

, and its value is an integer multiple of

3

.

6

1

Square geometry

For a quadrilateral

ABCD

, we call a square

amazing

if all of its sides(extended if necessary) pass through distinct vertices of

ABCD

(no side passing through 2 vertices). Prove that for an arbitrary

ABCD

such that its diagonals are not perpendicular, there exist at least 6

amazing

squares

2016 Indonesia MO

Subcontests

r_1+r_2+r_3+...+r_{p-1}=\frac{p^2(p-1)}{2} where r_k = remainder k^p : p^2

5 boxes in a circles, moving balls until each box has 17^{5^{2016}} balls ?

angle chasing with an orthodiagonal ABCD

2^c + 2^{2016} = a^b with b>1, and a,b,c naturals

Equation with cosinus

Indonesian MO 2016 Last Problem

Square geometry

2016 Indonesia MO

Subcontests

r_1+r_2+r_3+...+r_{p-1}=\frac{p^2(p-1)}{2} where r_k = remainder k^p : p^2

5 boxes in a circles, moving balls until each box has 17^{5^{2016}} balls ?

angle chasing with an orthodiagonal ABCD

2^c + 2^{2016} = a^b with b&gt;1, and a,b,c naturals

Equation with cosinus

Indonesian MO 2016 Last Problem

Square geometry

2^c + 2^{2016} = a^b with b>1, and a,b,c naturals