MathDB

4

Part of 2001 IMO Shortlist

Problems(4)

IMO ShortList 2001, geometry problem 4

Source: IMO ShortList 2001, geometry problem 4

9/30/2004
Let MM be a point in the interior of triangle ABCABC. Let A′A' lie on BCBC with MA′MA' perpendicular to BCBC. Define B′B' on CACA and C′C' on ABAB similarly. Define p(M)=MA′⋅MB′⋅MC′MA⋅MB⋅MC. p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}. Determine, with proof, the location of MM such that p(M)p(M) is maximal. Let μ(ABC)\mu(ABC) denote this maximum value. For which triangles ABCABC is the value of μ(ABC)\mu(ABC) maximal?
geometrytrigonometrymaximizationTriangleperpendicularIMO Shortlist
IMO ShortList 2001, number theory problem 4

Source: IMO ShortList 2001, number theory problem 4

9/30/2004
Let p≥5p \geq 5 be a prime number. Prove that there exists an integer aa with 1≤a≤p−21 \leq a \leq p-2 such that neither ap−1−1a^{p-1}-1 nor (a+1)p−1−1(a+1)^{p-1}-1 is divisible by p2p^2.
number theoryIMO Shortlist
IMO ShortList 2001, algebra problem 4

Source: IMO ShortList 2001, algebra problem 4

9/30/2004
Find all functions f:R→Rf: \mathbb{R} \rightarrow \mathbb{R}, satisfying f(xy)(f(x)−f(y))=(x−y)f(x)f(y) f(xy)(f(x) - f(y)) = (x-y)f(x)f(y) for all x,yx,y.
functionalgebrafunctional equationIMO Shortlist
IMO ShortList 2001, combinatorics problem 4

Source: IMO ShortList 2001, combinatorics problem 4

9/30/2004
A set of three nonnegative integers {x,y,z}\{x,y,z\} with x<y<zx < y < z is called historic if {z−y,y−x}={1776,2001}\{z-y,y-x\} = \{1776,2001\}. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.
combinatoricsCombinatorial Number TheorypartitionColoringIMO Shortlist