MathDB

1

Functional equation for bounded f

f:\mathbb{R}\to\mathbb{R}

, where

\mathbb{R}

is the set of all real numbers, satisfying the following two conditions: 1) There exists a real number

M

such that for every real number

x,f(x)<M

is satisfied. 2) For every pair of real numbers

x

and

y

,

f(xf(y))+yf(x)=xf(y)+f(xy)

is satisfied.

4

1

Maximum for m when 3 conditions are satisfied

Let

n

be a fixed positive odd integer. Take

m+2

distinct points

P_0,P_1,\ldots ,P_{m+1}

(where

m

is a non-negative integer) on the coordinate plane in such a way that the following three conditions are satisfied: 1)

P_0=(0,1),P_{m+1}=(n+1,n)

, and for each integer

i,1\le i\le m

, both

x

- and

y

- coordinates of

P_i

are integers lying in between

1

and

n

(

1

and

n

inclusive). 2) For each integer

i,0\le i\le m

,

P_iP_{i+1}

is parallel to the

x

-axis if

i

is even, and is parallel to the

y

-axis if

i

is odd. 3) For each pair

i,j

with

0\le i<j\le m

, line segments

P_iP_{i+1}

and

P_jP_{j+1}

share at most

1

point. Determine the maximum possible value that

m

can take.

3

1

BPC is a right angle for angle bisectors of ABC and ACB

Let

ABC

be an acute triangle with

\angle BAC=30^{\circ}

. The internal and external angle bisectors of

\angle ABC

meet the line

AC

at

B_1

and

B_2

, respectively, and the internal and external angle bisectors of

\angle ACB

meet the line

AB

at

C_1

and

C_2

, respectively. Suppose that the circles with diameters

B_1B_2

and

C_1C_2

meet inside the triangle

ABC

at point

P

. Prove that

\angle BPC=90^{\circ}

.

2

1

The max min of angles A_iA_jA_k for 5 points in a plane

Five points

A_1,A_2,A_3,A_4,A_5

lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles

\angle A_iA_jA_k

can take where

i, j, k

are distinct integers between

1

and

5

.

1

Three numbers cannot be squares simultaneously

Let

a,b,c

be positive integers. Prove that it is impossible to have all of the three numbers

a^2+b+c,b^2+c+a,c^2+a+b

to be perfect squares.

2011 APMO

Subcontests

Functional equation for bounded f

Maximum for m when 3 conditions are satisfied

BPC is a right angle for angle bisectors of ABC and ACB

The max min of angles A_iA_jA_k for 5 points in a plane

Three numbers cannot be squares simultaneously