MathDB

1

system x^3+mx−n = 0, &, nx^3−2m2x^2 −5mnx−2m^3−n^2 = 0

x^3+mx-n = 0

nx^3-2m^2x^2 -5mnx-2m^3-n^2 = 0

have one root in common (

n \ne 0

), then the first equation has two equal roots, and find the roots of the equations in terms of

n

.

6

1

equilateral triangles on different sides of a segment, with common vertex

Let

C

be a variable interior point of a fixed segment

AB

. Equilateral triangles

ACB'

and

CBA'

are constructed on the same side and

ABC'

on the other side of the line

AB

. (a) Prove that the lines

AA' ,BB'

, and

CC'

meet at some point

P

. (b) Find the locus of

P

as

C

varies. (c) Prove that the centers

A'' ,B'' ,C''

of the three triangles form an equilateral triangle. (d) Prove that

A'' ,B'',C''

, and

P

lie on a circle.

4

1

solve p(x+y) = xy in integers, where p prime

Given a prime number

p

, find all integer solutions of

p(x+y) = xy

.

2

1

cutting off parts that overlap from paper-made disks on a table

Several paper-made disks (not necessarily equal) are put on the table so that there is some overlapping, but no disk is entirely inside another. The parts that overlap are cut off and removed. Show that the remaining parts cannot be assembled so as to form different disks.

1

smallest sum of perimeters of 100 natural, sides of non-obtuse triangles

Consider all sets

A

of one hundred different natural numbers with the property that any three elements

a,b,c \in A

(not necessarily different) are the sides of a non-obtuse triangle. Denote by

S(A)

the sum of the perimeters of all such triangles. Compute the smallest possible value of

S(A)

.

3

1

Spanish Mathematical Olympiad 1995 Day 1 Problem 3

A line through the centroid G of the triangle ABC intersects the side AB at P and the side AC at Q Show that

\frac{PB}{PA} \cdot \frac{QC}{QA} \leq \frac{1}{4}

.
Sorry for Triple-Posting. If possible, please merge the solutions to one document.
I think there was an error because it may have automatically triple-posted.

1995 Spain Mathematical Olympiad

Subcontests

system x^3+mx−n = 0, &, nx^3−2m2x^2 −5mnx−2m^3−n^2 = 0

equilateral triangles on different sides of a segment, with common vertex

solve p(x+y) = xy in integers, where p prime

cutting off parts that overlap from paper-made disks on a table

smallest sum of perimeters of 100 natural, sides of non-obtuse triangles

Spanish Mathematical Olympiad 1995 Day 1 Problem 3

1995 Spain Mathematical Olympiad

Subcontests

system x^3+mx&minus;n = 0, &amp;, nx^3&minus;2m2x^2 &minus;5mnx&minus;2m^3&minus;n^2 = 0

equilateral triangles on different sides of a segment, with common vertex

solve p(x+y) = xy in integers, where p prime

cutting off parts that overlap from paper-made disks on a table

smallest sum of perimeters of 100 natural, sides of non-obtuse triangles

Spanish Mathematical Olympiad 1995 Day 1 Problem 3

system x^3+mx−n = 0, &, nx^3−2m2x^2 −5mnx−2m^3−n^2 = 0