MathDB

functional equation

Source: 2016 Korea Winter Program Test1 Day1 #3

1/25/2016

Determine all the functions

f : \mathbb{R}\rightarrow\mathbb{R}

that satisfies the following.

f(xf(y)+yf(z)+zf(x))=yf(x)+zf(y)+xf(z)

functional equationalgebraalgebra proposed

Writing Number on Balls

Source: 2016 Korea Winter Program Test2 Day1 #3

1/25/2016

p, q, r

are natural numbers greater than 1.
There are

pq

balls placed on a circle, and one number among

0, 1, 2, \cdots , pr-1

is written on each ball, satisfying following conditions.
(1) If

i

and

j

is written on two adjacent balls,

|i-j|=1

or

|i-j|=pr-1

. (2)

i

is written on a ball

A

. If we skip

q-1

balls clockwise from

A

and see

q^{th}

ball,

i+r

or

i-(p-1)r

is written on it. (This condition is satisfied for every ball.)
If

p

is even, prove that the number of pairs of two adjacent balls with

1

and

2

written on it is odd.

combinatoricscombinatorics proposed

Lots and lots of circles

Source: 2016 Korea Winter Camp 1st Test #7

1/25/2016

There are three circles

w_1, w_2, w_3

. Let

w_{i+1} \cap w_{i+2} = A_i, B_i

, where

A_i

lies insides of

w_i

. Let

\gamma

be the circle that is inside

w_1,w_2,w_3

and tangent to the three said circles at

T_1, T_2, T_3

. Let

T_iA_{i+1}T_{i+2}

's circumcircle and

T_iA_{i+2}T_{i+1}

's circumcircle meet at

S_i

. Prove that the circumcircles of

A_iB_iS_i

meet at two points. (

1 \le i \le 3

, indices taken modulo

3

)
If one of

A_i,B_i,S_i

are collinear - the intersections of the other two circles lie on this line. Prove this as well.

geometry

Weighted Fermat

Source: 2016 Korean Winter Camp 2nd Test #7

1/25/2016

Let there be a triangle

\triangle ABC

with

BC=a

,

CA=b

,

AB=c

. Let

T

be a point not inside

\triangle ABC

and on the same side of

A

with respect to

BC

, such that

BT-CT=c-b

. Let

n=BT

and

m=CT

. Find the point

P

that minimizes

f(P)=-a \cdot AP + m \cdot BP + n \cdot CP

.

geometryalgebraconics

3

Problems(4)