MathDB

1

another 3d locus, cone and sphere

R

and a plane

\alpha

having no common points with this sphere. A point

S

moves in the plane

\alpha

, which is the vertex of a cone tangent to the sphere along a circle with center

C

. Find the locus of point

C

.
[hide=another is Polish MO 1967 p6] [url=https://artofproblemsolving.com/community/c6h3388032p31769739]here

5

1

every polygon with perimeter 2a can be covered by a disk with diameter a

Prove that every polygon with perimeter

2a

can be covered by a disk with diameter

a

.

4

1

NT a^2 + b^2 = c^2,

Prove that if the natural numbers

a

,

b

,

c

satisfy the equation

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

then:
1) at least one of the numbers

a

and

b

is divisible by

3

,
2) at least one of the numbers

a

and

b

is divisible by

4

,
3) at least one of the numbers

a

,

b

,

c

is divisible by

5

.

3

1

right triangle construction

On a straight line, three distinct points

M

,

D

,

H

are given. Construct a right-angled triangle for which

M

is the midpoint of the hypotenuse,

D

is the point of intersection of the bisector of the right angle with the hypotenuse, and

H

is the foot of the altitude to the hypotenuse.

2

1

1/ a^n + 1/ b^n+ 1/c ^n =1/ (a^n + b^n + c^n)

Prove that if

\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a + b + c}

and

n

is any odd natural number, then

\frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n} =\frac{1}{a^n + b^n + c^n}

1

x^2y^2 + x^2z^2 = axyz, y^2z^2 + y^2x^2 = bxyz, z^2x^2 + z^2y^2 = cxyz

Solve the system of equations

\begin{array}{l}<br /> x^2y^2 + x^2z^2 = axyz\\<br /> y^2z^2 + y^2x^2 = bxyz\\<br /> z^2x^2 + z^2y^2 = cxyz.<br /> \end{array}

1956 Polish MO Finals

Subcontests

another 3d locus, cone and sphere

every polygon with perimeter 2a can be covered by a disk with diameter a

NT a^2 + b^2 = c^2,

right triangle construction

1/ a^n + 1/ b^n+ 1/c ^n =1/ (a^n + b^n + c^n)

x^2y^2 + x^2z^2 = axyz, y^2z^2 + y^2x^2 = bxyz, z^2x^2 + z^2y^2 = cxyz