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Difference between revisions of "1985 IMO Problems"

 
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=== Problem 1 ===
 
=== Problem 1 ===
  
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A circle has center on the side <math>\displaystyle AB</math> of the cyclic quadrilateral <math>\displaystyle ABCD</math>.  The other three sides are tangent to the circle.  Prove that <math>\displaystyle AB + BC = AB</math>.
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A circle has center on the side <math>\displaystyle AB</math> of the cyclic quadrilateral <math>\displaystyle ABCD</math>.  The other three sides are tangent to the circle.  Prove that <math>\displaystyle AD + BC = AB</math>.
  
 
[[1985 IMO Problems/Problem 1 | Solution]]
 
[[1985 IMO Problems/Problem 1 | Solution]]

Revision as of 12:33, 5 November 2006

Problems of the 26th IMO Finland.

Day I

Problem 1

A circle has center on the side $\displaystyle AB$ of the cyclic quadrilateral $\displaystyle ABCD$. The other three sides are tangent to the circle. Prove that $\displaystyle AD + BC = AB$.

Solution

Problem 2

Let $\displaystyle n$ and $\displaystyle k$ be given relatively prime natural numbers, $\displaystyle n < k$. Each number in the set $\displaystyle M = \{ 1,2, \ldots , n-1 \}$ is colored either blue or white. It is given that

(i) for each $i \in M$, both $\displaystyle i$ and $\displaystyle n-i$ have the same color;

(ii) for each $i \in M, i \neq k$, both $\displaystyle i$ and $\displaystyle |i-j|$ have the same color.

Prove that all number in $\displaystyle M$ have the same color.

Solution

Problem 3

For any polynomial $P(x) = a_0 + a_1 x + \cdots + a_k x^k$ with integer coefficients, the number of coefficients which are odd is denoted by $\displaystyle w(P)$. For $i = 0, 1, \ldots$, let $\displaystyle Q_i (x) = (1+x)^i$. Prove that if $i_1, i_2, \ldots , i_n$ are integers such that $0 \leq i_1 < i_2 < \cdots < i_n$, then

$\displaystyle w(Q_{i_1} + Q_{i_2} + \cdots + Q_{i_n}) \ge w(Q_{i_1})$.

Solution

Day II

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Resources

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