2015 Turkey EGMO TST

Part of Turkey EGMO TST

Subcontests

(6)

1

Number of Handshakes

In a party attended by

2015

guests among any

5

6

handshakes had been exchanged. Determine the maximal possible total number of handshakes.

1

Permutations of (1,2,...,2015)

2015

(a_1,a_2,\ldots,a_{2015})

in each step we choose two indices

1\le k,l\le 2015

a_k

even and transform the

2015

(a_1,\ldots,\dfrac{a_k}{2},\ldots,a_l+\dfrac{a_k}{2},\ldots,a_{2015})

. Prove that starting from

(1,2,\ldots,2015)

in finite number of steps one can reach any permutation of

(1,2,\ldots,2015)

1

Point Inside Triangle

D

be the midpoint of the side

BC

ABC

P

be a point inside the

ABD

\angle PAD=90^\circ - \angle PBD=\angle CAD

\angle PQB=\angle BAC

Q

is the intersection point of the lines

PC

AD

1

Solve Equation (Turkey EGMO TST 2015)

a

is a real number. Find the all

(x,y)

real number pairs satisfy;

y^2=x^3+(a-1)x^2+a^2x

x^2=y^3+(a-1)y^2+a^2y

1

Coordinate Plane Algebra (Turkey EGMO TST 2015)

a \ge b \ge 0

be real numbers. Find the area of the region defined as;

K=\{(x,y): x\ge y\ge0

\forall n

positive integers satisfy

a^n+b^n\ge x^n+y^n\}

in the cordinate plane.

1

Solve Equation NT (Turkey EGMO TST 2015)

(m,n)

integer pairs satisfying

m^4+2n^3+1=mn^3+n