MathDB

2

Part of 2005 International Zhautykov Olympiad

Problems(4)

Perfect Square

Source: IZO 1 Junior Problem 2

12/16/2008
Let m,n m,n be integers such that 0≤m≤2n 0\le m\le 2n. Then prove that the number 2^{2n \plus{} 2} \plus{} 2^{m \plus{} 2} \plus{} 1 is perfect square iff m \equal{} n.
number theorygreatest common divisornumber theory proposed
Nice problem for you!

Source: IZO 1 Junior, Problem 5

1/22/2008
Let the circle (I;r) (I; r) be inscribed in the triangle ABC ABC. Let D D be the point of contact of this circle with BC BC. Let E E and F F be the midpoints of BC BC and AD AD, respectively. Prove that the three points I I, E E, F F are collinear.
geometryperimeterconicsanalytic geometrylinear algebraparallelogram
Zhautykov Olympiad Sequence

Source: First Zhautykov Olympiad 2005, Problem 2

12/22/2008
Let r r be a real number such that the sequence (an)n≥1 (a_{n})_{n\geq 1} of positive real numbers satisfies the equation a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{m \plus{} 1} \leq r a_{m} for each positive integer m m. Prove that r≥4 r \geq 4.
functionalgebra unsolvedalgebra
A k-point if it is observable from the exactly k sides

Source: First Zhautykov Olympiad 2005, Problem 5

12/22/2008
The inner point X X of a quadrilateral is observable from the side YZ YZ if the perpendicular to the line YZ YZ meet it in the colosed interval [YZ]. [YZ]. The inner point of a quadrilateral is a k\minus{}point if it is observable from the exactly k k sides of the quadrilateral. Prove that if a convex quadrilateral has a 1-point then it has a k\minus{}point for each k\equal{}2,3,4.
geometry unsolvedgeometry