MathDB

Integers satisfying squared and product sum

Source: IMO Shortlist 1995, A2

8/10/2008

Let

a

and

b

be non-negative integers such that

ab \geq c^2,

where

c

is an integer. Prove that there is a number

n

and integers

x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n

such that \sum^n_{i\equal{}1} x^2_i \equal{} a, \sum^n_{i\equal{}1} y^2_i \equal{} b, \text{ and } \sum^n_{i\equal{}1} x_iy_i \equal{} c.

algebrasystem of equationsequationsIMO Shortlist

Unique point X

Source: IMO Shortlist 1995, G

8/3/2008

Let

A, B

and

C

be non-collinear points. Prove that there is a unique point

X

in the plane of

ABC

such that XA^2 \plus{} XB^2 \plus{} AB^2 \equal{} XB^2 \plus{} XC^2 \plus{} BC^2 \equal{} XC^2 \plus{} XA^2 \plus{} CA^2.

geometrycircumcirclereflectioncomplex numbersperpendicular bisectorIMO Shortlist

{x^2 + Ax + B} and {2x^2 + 2x + C} do not intersect

Source: IMO Shortlist 1995, N2

8/10/2008

Let

\mathbb{Z}

denote the set of all integers. Prove that for any integers

A

and

B,

one can find an integer

C

for which M_1 \equal{} \{x^2 \plus{} Ax \plus{} B : x \in \mathbb{Z}\} and M_2 \equal{} {2x^2 \plus{} 2x \plus{} C : x \in \mathbb{Z}} do not intersect.

quadraticsmodular arithmeticnumber theorypartitionIMO Shortlist

2

Problems(3)