MathDB

1

Part of 2001 Tournament Of Towns

Problems(8)

Can 2001 be Obtained?

Source: ToT - 2001 Spring Junior O-Level #1

8/17/2011
The natural number nn can be replaced by abab if a+b=na + b = n, where aa and bb are natural numbers. Can the number 20012001 be obtained from 2222 after a sequence of such replacements?
inductionnumber theory unsolvednumber theory
A Region's Salary

Source: ToT - 2001 Spring Junior A-Level #1

8/17/2011
In a certain country 10%10\% of the employees get 90%90\% of the total salary paid in this country. Supposing that the country is divided in several regions, is it possible that in every region the total salary of any 10% of the employees is no greater than 11%11\% of the total salary paid in this region?
algebra unsolvedalgebra
Bus Problem

Source: ToT - 2001 Spring Senior O-Level #1

8/17/2011
A bus that moves along a 100 km route is equipped with a computer, which predicts how much more time is needed to arrive at its final destination. This prediction is made on the assumption that the average speed of the bus in the remaining part of the route is the same as that in the part already covered. Forty minutes after the departure of the bus, the computer predicts that the remaining travelling time will be 1 hour. And this predicted time remains the same for the next 5 hours. Could this possibly occur? If so, how many kilometers did the bus cover when these 5 hours passed? (Average speed is the number of kilometers covered divided by the time it took to cover them.)
functionalgebra unsolvedalgebra
Find the Polynomial

Source: ToT - 2001 Spring Senior A-Level #1

8/17/2011
Find at least one polynomial P(x)P(x) of degree 2001 such that P(x)+P(1−x)=1P(x)+P(1- x)=1 holds for all real numbers xx.
algebrapolynomialalgebra unsolved
Proof of Intersection

Source: ToT - Fall Junior O-Level #1

8/17/2011
In the quadrilateral ABCDABCD, ADAD is parallel to BCBC. KK is a point on ABAB. Draw the line through AA parallel to KCKC and the line through BB parallel to KDKD.
Prove that these two lines intersect at some point on CDCD.
geometry unsolvedgeometry
Does there Exist?

Source: ToT - 2001 Fall Junior A-Level #1

8/17/2011
Do there exist postive integers a1<a2<⋯<a100a_1<a_2<\cdots<a_{100} such that for 2≤k≤1002\le k\le100 the greatest common divisor of ak−1a_{k-1} and aka_k is greater than the greatest common divisor of aka_k and ak+1a_{k+1}?
greatest common divisornumber theory unsolvednumber theory
Regular Pentagon

Source: ToT - 2001 Fall Senior O-Level #1

8/17/2011
An altitude of a pentagon is the perpendicular drop from a vertex to the opposite side. A median of a pentagon is the line joining a vertex to the midpoint of the opposite side. If the five altitudes and the five medians all have the same length, prove that the pentagon is regular.
geometry unsolvedgeometry
Circles and Triangles

Source: ToT - 2001 Fall Senior A-Level #1

8/17/2011
On the plane is a triangle with red vertices and a triangle with blue vertices. OO is a point inside both triangles such that the distance from OO to any red vertex is less than the distance from OO to any blue vertex. Can the three red vertices and the three blue vertices all lie on the same circle?
geometry unsolvedgeometry