MathDB

2

Part of 2006 Taiwan National Olympiad

Problems(5)

Number of problems

Source: Taiwan NMO 2006

3/21/2006
Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. At least how many problems are needed?
combinatorics proposedcombinatorics
Mod problem: xy = a mod z and cyclic

Source: Taiwan NMO 2006

3/21/2006
x,y,z,a,b,cx,y,z,a,b,c are positive integers that satisfy xy≡a(modz)xy \equiv a \pmod z, yz≡b(modx)yz \equiv b \pmod x, zx≡c(mody)zx \equiv c \pmod y. Prove that min⁡{x,y,z}≤ab+bc+ca\min{\{x,y,z\}} \le ab+bc+ca.
modular arithmeticnumber theory unsolvednumber theory
Triangle inequality

Source: Taiwan NMO 2006 oral test

3/21/2006
In triangle ABCABC, DD is the midpoint of side ABAB. EE and FF are points arbitrarily chosen on segments ACAC and BCBC, respectively. Show that [DEF]<[ADE]+[BDF][DEF] < [ADE] + [BDF].
inequalitiesgeometry proposedgeometry
Equilateral triangles (quite easy)

Source: Taiwan NMO 2006

3/21/2006
Given a line segment AB=7AB=7, CC is constructed on ABAB so that AC=5AC=5. Two equilateral triangles are constructed on the same side of ABAB with ACAC and BCBC as a side. Find the length of the segment connecting their two circumcenters.
trigonometrygeometry proposedgeometry
Floor function

Source: Taiwan NMO 2006

3/21/2006
Find all reals xx satisfying 0≤x≤50 \le x \le 5 and ⌊x2−2x⌋=⌊x⌋2−2⌊x⌋\lfloor x^2-2x \rfloor = \lfloor x \rfloor ^2 - 2 \lfloor x \rfloor.
functionfloor functionalgebra proposedalgebra