MathDB

1

\sqrt{ xxx...x - yy...y}= zzz...z, for digits

Determine all triples of nonzero decimal digits

(x,y,z)

for which the equality

\sqrt{ \underbrace{xxx...x}_{2n}- \underbrace{yy...y}_{n}}= \underbrace{zzz...z}_{n}

holds for at least two different natural numbers

n

.

6

1

cos \alpha + \cos\beta + \cos\beta <= \frac{3}{2}

Let

\alpha,\beta, \gamma

be the angles of a triangle. Prove that

\cos\alpha, + \cos\beta + \cos\gamma \le \frac{3}{2}

and find the cases of equality.

4

1

locus sum of distances from sides of regular polygon is 5 times the inradius

Find the locus of points in the plane, the sum of whose distances from the sides of a regular polygon is five times the inradius of the pentagon.

3

1

parallelogram in space, same ratio

Two parallelograms

ABCD

and

A'B'C'D'

are given in space. Points

A'',B'',C'',D''

divide the segments

AA',BB',CC',DD'

in the same ratio. What can be said about the quadrilateral

A''B''C''D''

?

2

1

which of 2,3,5,7,11,13,109,151,491 divides z =1963^{1965} -1963

Determine which of the prime numbers

2,3,5,7,11,13,109,151,491

divide

z =1963^{1965} -1963

.

1

\sqrt{p+x}+\sqrt{p-x}= x

For a given positive real parameter

p

, solve the equation

\sqrt{p+x}+\sqrt{p-x }= x

.

1965 German National Olympiad

Subcontests

\sqrt{ xxx...x - yy...y}= zzz...z, for digits

cos \alpha + \cos\beta + \cos\beta <= \frac{3}{2}

locus sum of distances from sides of regular polygon is 5 times the inradius

parallelogram in space, same ratio

which of 2,3,5,7,11,13,109,151,491 divides z =1963^{1965} -1963

\sqrt{p+x}+\sqrt{p-x}= x

1965 German National Olympiad

Subcontests

\sqrt{ xxx...x - yy...y}= zzz...z, for digits

cos \alpha + \cos\beta + \cos\beta &lt;= \frac{3}{2}

locus sum of distances from sides of regular polygon is 5 times the inradius

parallelogram in space, same ratio

which of 2,3,5,7,11,13,109,151,491 divides z =1963^{1965} -1963

\sqrt{p+x}+\sqrt{p-x}= x

cos \alpha + \cos\beta + \cos\beta <= \frac{3}{2}