MathDB

3

Part of 2000 Iran MO (3rd Round)

Problems(6)

Mn visible with a constanr angle

Source: 17-th Iranian Mathematical Olympiad 1999/2000

12/14/2005
A circleΓ\Gamma with radius RR and center ω\omega, and a line dd are drawn on a plane, such that the distance of ω\omega from dd is greater than RR. Two points MM and NN vary on dd so that the circle with diameter MNMN is tangent to Γ\Gamma. Prove that there is a point PP in the plane from which all the segments MNMN are visible at a constant angle.
geometry proposedgeometry
Abc and a'|b'c'

Source: 17-th Iranian Mathematical Olympiad 1999/2000

12/14/2005
Two triangles ABC ABCand A′B′C′ A'B'C' are positioned in the space such that the length of every side of △ABC \triangle ABC is not less than a a, and the length of every side of △A′B′C′ \triangle A'B'C' is not less than a′ a'. Prove that one can select a vertex of △ABC \triangle ABC and a vertex of △A′B′C′ \triangle A'B'C' so that the distance between the two selected vertices is not less than \sqrt {\frac {a^2 \plus{} a'^2}{3}}.
vectorgeometry proposedgeometry
F:n----->n

Source: 17-th Iranian Mathematical Olympiad 1999/2000

12/14/2005
Suppose f:N⟶Nf : \mathbb{N} \longrightarrow \mathbb{N} is a function that satisfies f(1)=1f(1) = 1 and f(n + 1) =\{\begin{array}{cc} f(n)+2&\mbox{if}\ n=f(f(n)-n+1),\\f(n)+1& \mbox{Otherwise}\end {array} (a)(a) Prove that f(f(n)−n+1)f(f(n)-n+1) is either nn or n+1n+1. (b)(b) Determineff.
functionfloor functionalgebra proposedalgebra
Deck

Source: 17-th Iranian Mathematical Olympiad 1999/2000

12/14/2005
In a deck of n>1n > 1 cards, some digits from 11 to88are written on each card. A digit may occur more than once, but at most once on a certain card. On each card at least one digit is written, and no two cards are denoted by the same set of digits. Suppose that for every k=1,2,…,7k=1,2,\dots,7 digits, the number of cards that contain at least one of them is even. Find nn.
combinatorics proposedcombinatorics
Polynomial

Source: 17-th Iranian Mathematical Olympiad 1999/2000

1/3/2009
Prove that for every natural number n n there exists a polynomial p(x) p(x) with integer coefficients such thatp(1),p(2),...,p(n) p(1),p(2),...,p(n) are distinct powers of 2 2 .
algebrapolynomialalgebra proposed
N point on a circle

Source: 17-th Iranian Mathematical Olympiad 1999/2000

12/14/2005
Let nn points be given on a circle, and let nk+1nk + 1 chords between these points be drawn, where 2k+1<n2k+1 < n. Show that it is possible to select k+1k+1 of the chords so that no two of them intersect.
combinatorics proposedcombinatorics