MathDB

Find the number of functions

Source: Turkish TST 2012 Problem 1

3/26/2012

Let

A=\{1,2,\ldots,2012\}, \: B=\{1,2,\ldots,19\}

and

S

be the set of all subsets of

A.

Find the number of functions

f : S\to B

satisfying

f(A_1\cap A_2)=\min\{f(A_1),f(A_2)\}

for all

A_1, A_2 \in S.

functioninductioncombinatorics proposedcombinatorics

A circumradius equation

Source: Turkish TST 2012 Problem 4

3/26/2012

In a triangle

ABC,

incircle touches the sides

BC, CA, AB

at

D, E, F,

respectively. A circle

\omega

passing through

A

and tangent to line

BC

at

D

intersects the line segments

BF

and

CE

at

K

and

L,

respectively. The line passing through

E

and parallel to

DL

intersects the line passing through

F

and parallel to

DK

at

P.

If

R_1, R_2, R_3, R_4

denotes the circumradius of the triangles

AFD, AED, FPD, EPD,

respectively, prove that

R_1R_4=R_2R_3.

geometrycircumcircletrigonometrygeometric transformationangle bisectortrig identitiesLaw of Sines

Sum of powers

Source: Turkish TST 2012 Problem 7

3/26/2012

Let

S_r(n)=1^r+2^r+\cdots+n^r

where

n

is a positive integer and

r

is a rational number. If

S_a(n)=(S_b(n))^c

for all positive integers

n

where

a, b

are positive rationals and

c

is positive integer then we call

(a,b,c)

as nice triple. Find all nice triples.

limitcalculusintegrationfunctionalgebra proposedalgebra

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