MathDB

Find all functions f:N--> N

Source: Saudi Arabia IMO TST Day I Problem 4

7/22/2014

Find all functions

f:\mathbb{N}\rightarrow\mathbb{N}

such that

f(n+1)>\frac{f(n)+f(f(n))}{2}

for all

n\in\mathbb{N}

, where

\mathbb{N}

is the set of strictly positive integers.

functioninductionalgebra unsolvedalgebra

Aws is playing solitaire!

Source: Saudi Arabia IMO TST Day II Problem 4

7/22/2014

Aws plays a solitaire game on a fifty-two card deck: whenever two cards of the same color are adjacent, he can remove them. Aws wins the game if he removes all the cards. If Aws starts with the cards in a random order, what is the probability for him to win?

probabilitycombinatorics unsolvedcombinatorics

Prove that MS=MT iff X,Y,Z,W are concylic

Source: Saudi Arabia IMO TST Day III Problem 4

7/22/2014

Let

\omega_1

and

\omega_2

with center

O_1

and

O_2

respectively, meet at points

A

and

B

. Let

X

and

Y

be points on

\omega_1

. Lines

XA

and

Y A

meet

\omega_2

at

Z

and

W

, respectively, such that

A

lies between

X

and

Z

and between

Y

and

W

. Let

M

be the midpoint of

O_1O_2

,

S

be the midpoint of

XA

and

T

be the midpoint of

W A

. Prove that

MS = MT

if and only if

X,~ Y ,~ Z

and

W

are concyclic.

geometry unsolvedgeometry

Circumcenter of incenter triangle

Source: Saudi Arabia IMO TST Day IV Problem 4

7/22/2014

Points

A_1,~ B_1,~ C_1

lie on the sides

BC,~ AC

and

AB

of a triangle

ABC

, respectively, such that

AB_1 -AC_1 = CA_1 -CB_1 = BC_1 -BA_1

. Let

I_A,~ I_B,~ I_C

be the incenters of triangles

AB_1C_1,~ A_1BC_1

and

A_1B_1C

respectively. Prove that the circumcenter of triangle

I_AI_BI_C

, is the incenter of triangle

ABC

.

geometrycircumcircleincentergeometric transformationreflectionperpendicular bisectorangle bisector

4

Problems(4)