MathDB

IMO LongList 1967, Bulgaria 6

Source: IMO LongList 1967, Bulgaria 6

12/16/2004

Solve the system of equations:

\begin{matrix} |x+y| + |1-x| = 6 \\ |x+y+1| + |1-y| = 4. \end{matrix}

linear algebramatrixalgebrasystem of equationsIMO ShortlistIMO Longlist

IMO LongList 1967, Mongolia 6

Source: IMO LongList 1967, Mongolia 6

12/16/2004

Prove the identity

\sum\limits_{k=0}^n\binom{n}{k}\left(\tan\frac{x}{2}\right)^{2k}\left(1+\frac{2^k}{\left(1-\tan^2\frac{x}{2}\right)^k}\right)=\sec^{2n}\frac{x}{2}+\sec^n x

for any natural number

n

and any angle

x.

trigonometryalgebraseries summationTrigonometric IdentitiesIMO ShortlistIMO Longlist

IMO LongList 1967, Hungary 6

Source: IMO LongList 1967, Hungary 6

12/16/2004

Three disks of diameter

d

are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius

R

of the sphere in order that axis of the whole figure has an angle of

60^\circ

with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of

120^\circ

around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).

geometry3D geometrysphereanglesIMO ShortlistIMO Longlist

IMO LongList 1967, Poland 6

Source: IMO LongList 1967, Poland 6

12/16/2004

A line

l

is drawn through the intersection point

H

of altitudes of acute-angle triangles. Prove that symmetric images

l_a, l_b, l_c

of

l

with respect to the sides

BC,CA,AB

have one point in common, which lies on the circumcircle of

ABC.

geometrycircumcirclereflectionTriangleconcurrencyIMO ShortlistIMO Longlist

IMO LongList 1967, Romania 6

Source: IMO LongList 1967, Romania 6

12/16/2004

Prove the following inequality:

\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1} x^{n+k-1}_i,

where

x_i > 0,

k \in \mathbb{N}, n \in \mathbb{N}.

Inequalitypolynomialalgebran-variable inequality

IMO LongList 1967, Socialists Republic Of Czechoslovakia 6

Source: IMO LongList 1967, Socialists Republic Of Czechoslovakia 6

12/16/2004

Given a segment

AB

of the length 1, define the set

M

of points in the following way: it contains two points

A,B,

and also all points obtained from

A,B

by iterating the following rule: With every pair of points

X,Y

the set

M

contains also the point

Z

of the segment

XY

for which

YZ = 3XZ.

geometrypoint setSequenceIMO ShortlistIMO Longlist

IMO LongList 1967, Sweden 6

Source: IMO LongList 1967, Sweden 6

12/16/2004

In making Euclidean constructions in geometry it is permitted to use a ruler and a pair of compasses. In the constructions considered in this question no compasses are permitted, but the ruler is assumed to have two parallel edges, which can be used for constructing two parallel lines through two given points whose distance is at least equal to the breadth of the rule. Then the distance between the parallel lines is equal to the breadth of the ruler. Carry through the following constructions with such a ruler. Construct: a) The bisector of a given angle. b) The midpoint of a given rectilinear line segment. c) The center of a circle through three given non-collinear points. d) A line through a given point parallel to a given line.

geometryconstructionbisectormidpointCenterIMO ShortlistIMO Longlist

IMO LongList 1967, Soviet Union 6

Source: IMO LongList 1967, Soviet Union 6

12/16/2004

On the circle with center 0 and radius 1 the point

A_0

is fixed and points

A_1, A_2, \ldots, A_{999}, A_{1000}

are distributed in such a way that the angle

\angle A_00A_k = k

(in radians). Cut the circle at points

A_0, A_1, \ldots, A_{1000}.

How many arcs with different lengths are obtained. ?

combinatoricsgeometrycountingcircleIMO ShortlistIMO Longlist

6

Problems(8)