Problem 2

Part of 2000 Slovenia National Olympiad

Problems(4)

|equation| real parameter (Slovenia National MO 2000 1st Grade P2)

Source:

Find all real numbers

a

for which the following equation has a unique real solution:

|x-1|+|x-2|+\ldots+|x-99|=a.

parameterizationabsolute value

rate problem (Slovenia National MO 2000 2nd Grade P2)

Source:

Three students start walking with constant speeds at the same time, each along a straight line in the plane. Prove that if the students are not on the same line at the beginning, then they will be on the same line at most twice during their journey.

polynomial inequalities (Slovenia National MO 2000 3rd Grade P2)

Source:

Consider the polynomial

p(x)=a_nx^n+\ldots+a_1x+a_0

with real coefficients such that

0\le a_i\le a_0

i=1,2,\ldots,n

a

is the coefficient of

x^{n+1}

in the polynomial

q(x)=p(x)^2

2a\le p(1)^2

algebrapolynomialinequalities

real valued FE (Slovenia National MO 2000 4th Grade P2)

Source:

Find all functions

f:\mathbb R\to\mathbb R

such that for all

x,y\in\mathbb R

f(x-f(y))=1-x-y.

functional equationfe