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CHN TST90

The document contains problems from the Chinese IMO Team Selection Tests of 1990, divided into two tests. The first test includes questions on friendship relations, polygon placements, operations on sets, and properties of real numbers. The second test features problems related to triangle inequalities, functional equations, properties of powers of 2, and combinatorial geometry involving circles and points.

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378 views1 page

CHN TST90

The document contains problems from the Chinese IMO Team Selection Tests of 1990, divided into two tests. The first test includes questions on friendship relations, polygon placements, operations on sets, and properties of real numbers. The second test features problems related to triangle inequalities, functional equations, properties of powers of 2, and combinatorial geometry involving circles and points.

Uploaded by

adityabhoj101
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Chinese IMO Team Selection Tests 1990

First Test

1. In a group of people, every m people have exactly one friend in common, where
m is given. Friendship is a symmetric and non-reflexive relation. Find the num-
ber of friends of the person having the largest number of friends.
2. Finitely many polygons are placed in the coordinate plane. We say that the poly-
gons are properly placed if for any two polygons there is a line through the origin
cutting both of them. Find the smallest positive integer m for which it is always
possible to draw m lines through the origin such that each polygon is cut at least
once.
3. An operation ◦ on set S (with a ◦ b ∈ S for all a, b ∈ S) is such that:
(i) (a ◦ b) ◦ c = a ◦ (b ◦ c) for all a, b, c ∈ S;
(ii) a ◦ b 6= b ◦ a whenever a 6= b.

(a) Prove that (a ◦ b) ◦ c = a ◦ c for all a, b, c ∈ S.


(b) Give an example of operation ◦ on the set S = {1, 2, . . ., 1990}.

4. Number a has the property that for any real numbers x1 , x2 , x3 , x4 there exist
integers k1 , k2 , k3 , k4 such that

∑ ((xi − ki ) − (x j − k j ))2 ≤ a.
1≤i< j≤4

Find the smallest a.

Second Test

1. If ABC is a triangle with ∠C ≥ 60◦, prove that


 
1 1 1 1
(a + b) + + ≥ 4+ .
a b c sin C2

2. Find all functions f , g, h : R 7→ R satisfying f (x) − g(y) = (x − y)h(x + y) for all


x, y ∈ R.
3. Prove that every integer power of 2 has a multiple whose all decimal digits are
nonzero.
4. Find the maximum possible number of different circles that pass through four of
the given seven points in the plane.

The IMO Compendium Group,


D. Djukić, V. Janković, I. Matić, N. Petrović
Typed in LATEX by Ercole Suppa
www.imomath.com

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