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1. Given a straight line and a circle in the plane, on every chord of this circle parallel

to the given line we construct a circle having this chord as a diameter. Which set is formed by all the circles constructed?
2. A real square matrix

(aij )

of order

n2

is called narrow if

1 i, j n - 1.

Suppose that four distinct narrow matrices 4 same straight line in the space of all matrices and that A = 7 Prove that D = 0.
3. We are given two triangles

aij = 0 whenever A, B , C, D belong to the B 5 = C 6 = 0. a1 b 1 c
1,

1 , 2

with the corresponding side lengths

a2 b 2 c
(ii)

2 . Prove that
side lengths where

(i) there is a triangle

with S () S (1 ) + S (2 ),

S ()

2 b2 + b2 , c 2 2, 1 2 1 is the area of the triangle

a2 + a 1

+ c2 ; 2 , and n3

that the persons

equality is possible only for similar triangles.
4. At the meeting of the round table Math and Magic, where

are sitting at a round table, a magician and his assistant enter and have a short conversation, after which the magician leaves the room. His assistant announces that a trick will be shown. The participants of the meeting take seats around the table in an arbitrary order and then one of them leaves the table. The assistant asks one of the two neighbors of that participant to leave his place too. The magician returns to the room, looks at the two standing persons and which these two have been n 1 5. Let f : R R be a n a R , where U is a ball centered at the origin. 1 (i) Prove that f (c · U ) [-1, 1] if c = . 6 (ii) For which c > 0 the assertion is valid independently of

n - 2 sitting and announces the sitting. For which n such a trick is possible? homogeneous polynomial and f (U + a) [-1, 1]

order in for some

n, f , a

, and

U

?