MathDB

1

P,Q,R are polynomials, construct a polynomial T

P,Q,R

be polynomials and let

S(x) = P(x^3) + xQ(x^3) + x^2R(x^3)

be a polynomial of degree

n

whose roots

x_1,\ldots, x_n

are distinct. Construct with the aid of the polynomials

P,Q,R

a polynomial

T

of degree

n

that has the roots

x_1^3 , x_2^3 , \ldots, x_n^3.

9

1

Inequality on 2n reals - ISL 1970

Let

u_1, u_2, \ldots, u_n, v_1, v_2, \ldots, v_n

be real numbers. Prove that

1+ \sum_{i=1}^n (u_i+v_i)^2 \leq \frac 43 \Biggr( 1+ \sum_{i=1}^n u_i^2 \Biggl) \Biggr( 1+ \sum_{i=1}^n v_i^2 \Biggl) .

7

1

Find a such that each digit of n(n+1)/2 equals a

For which digits

a

do exist integers

n \geq 4

such that each digit of

\frac{n(n+1)}{2}

equals

a \ ?

6

1

Prove that the circumcircles have a common point

In the triangle

ABC

let

B'

and

C'

be the midpoints of the sides

AC

and

AB

respectively and

H

the foot of the altitude passing through the vertex

A

. Prove that the circumcircles of the triangles

AB'C'

,

BC'H

, and

B'CH

have a common point

I

and that the line

HI

passes through the midpoint of the segment

B'C'.

5

1

M is interor point of ABCD - volume problem

Let

M

be an interior point of the tetrahedron

ABCD

. Prove that

\begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}

(

\text{vol}(PQRS)

denotes the volume of the tetrahedron

PQRS

).

1

A regular 2n-gon and the n diagonals - Prove the identity

Consider a regular

2n

-gon and the

n

diagonals of it that pass through its center. Let

P

be a point of the inscribed circle and let

a_1, a_2, \ldots , a_n

be the angles in which the diagonals mentioned are visible from the point

P

. Prove that

\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.

1970 IMO Shortlist

Subcontests

P,Q,R are polynomials, construct a polynomial T

Inequality on 2n reals - ISL 1970

Find a such that each digit of n(n+1)/2 equals a

Prove that the circumcircles have a common point

M is interor point of ABCD - volume problem

A regular 2n-gon and the n diagonals - Prove the identity