MathDB

3

Part of 2003 Tournament Of Towns

Problems(7)

Find the value of angle KNL

Source: Tournament of Towns Spring 2003 - Junior O-Level - Problem 3

6/14/2011
Points KK and LL are chosen on the sides ABAB and BCBC of the isosceles △ABC\triangle ABC (AB=BCAB = BC) so that AK+LC=KLAK +LC = KL. A line parallel to BCBC is drawn through midpoint MM of the segment KLKL, intersecting side ACAC at point NN. Find the value of ∠KNL\angle KNL.
geometrytrapezoidgeometry proposed
One “odd” Game in a tournament

Source: Tournament of Towns Spring 2003 - Junior A-Level - Problem 3

6/14/2011
In a tournament, each of 1515 teams played with each other exactly once. Let us call the game “odd” if the total number of games previously played by both competing teams was odd.
(a) Prove that there was at least one “odd” game.
(b) Could it happen that there was exactly one “odd” game?
combinatorics unsolvedcombinatorics
Prove that all four radii are equal

Source: Tournament of Towns Spring 2003 - Senior O-Level - Problem 3

6/14/2011
Point MM is chosen in triangle ABCABC so that the radii of the circumcircles of triangles AMC,BMCAMC, BMC, and BMABMA are no smaller than the radius of the circumcircle of ABCABC. Prove that all four radii are equal.
geometrycircumcirclegeometry proposed
Can one cover a cube by three paper triangles?

Source: Tournament of Towns Spring 2003 - Senior A-Level - Problem 3

6/15/2011
Can one cover a cube by three paper triangles (without overlapping)?
geometry3D geometrycombinatorics unsolvedcombinatorics
Sum of greatest odd divisors equal n^2

Source: ToT 2003-JO-3, SO-1

6/18/2011
For any integer n+1,…,2nn+1,\ldots, 2n (nn is a natural number) consider its greatest odd divisor. Prove that the sum of all these divisors equals n2.n^2.
number theory proposednumber theory
Find all k such that m(m+k)=n(n+1) has natural solutions.

Source: ToT 2003-JA-3

6/19/2011
Find all positive integers kk such that there exist two positive integers mm and nn satisfying m(m+k)=n(n+1).m(m + k) = n(n + 1).
number theory unsolvednumber theory
Buying a cat and getting the correct change...

Source: ToT 2003 SO-3

6/26/2011
A salesman and a customer altogether have 19991999 rubles in coins and bills of 1,5,10,50,100,500,10001, 5, 10, 50, 100, 500 , 1000 rubles. The customer has enough money to buy a Cat in the Bag which costs the integer number of rubles. Prove that the customer can buy the Cat and get the correct change.
combinatorics unsolvedcombinatorics