MathDB

2

Part of 2003 Tournament Of Towns

Problems(8)

Show that R / r > a / h in any triangle ABC

Source: Tournament of Towns Spring 2003 - Junior A-Level - Problem 2

6/14/2011
Triangle ABCABC is given. Prove that Rr>ah\frac{R}{r} > \frac{a}{h}, where RR is the radius of the circumscribed circle, rr is the radius of the inscribed circle, aa is the length of the longest side, hh is the length of the shortest altitude.
trigonometryinequalitiesgeometrycircumcircletriangle inequalitygeometry unsolved
Values of n that the second player has a winning strategy

Source: Tournament of Towns Spring 2003 - Junior O-Level - Problem 2

6/14/2011
Two players in turns color the sides of an nn-gon. The first player colors any side that has 00 or 22 common vertices with already colored sides. The second player colors any side that has exactly 11 common vertex with already colored sides. The player who cannot move, loses. For which nn the second player has a winning strategy?
inductioncombinatorics unsolvedcombinatorics
Construct a polygon from any lesser number of these sticks

Source:

6/14/2011
100100-gon made of 100100 sticks. Could it happen that it is not possible to construct a polygon from any lesser number of these sticks?
Determine the possible values of degree of P(x)

Source: Tournament of Towns Spring 2003 - Senior A-Level - Problem 2

6/15/2011
P(x)P(x) is a polynomial with real coefficients such that P(a1)=0,P(ai+1)=aiP(a_1) = 0, P(a_{i+1}) = a_i (i=1,2,…i = 1, 2,\ldots) where {ai}i=1,2,…\{a_i\}_{i=1,2,\ldots} is an infinite sequence of distinct natural numbers. Determine the possible values of degree of P(x)P(x).
algebrapolynomialalgebra proposed
Two sets of diagonals in 7 gon are equal

Source: ToT 2003-JO-2

6/19/2011
In 77-gon A1A2A3A4A5A6A7A_1A_2A_3A_4A_5A_6A_7 diagonals A1A3,A2A4,A3A5,A4A6,A5A7,A6A1A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_7, A_6A_1 and A7A2A_7A_2 are congruent to each other and diagonals A1A4,A2A5,A3A6,A4A7,A5A1,A6A2A_1A_4, A_2A_5, A_3A_6, A_4A_7, A_5A_1, A_6A_2 and A7A3A_7A_3 are also congruent to each other. Is the polygon necessarily regular?
geometry unsolvedgeometry
Doing two arrangements at the same time.

Source: ToT 2003-JA-2, SA-1

6/19/2011
Smallville is populated by unmarried men and women, some of them are acquainted. Two city’s matchmakers are aware of all acquaintances. Once, one of matchmakers claimed: “I could arrange that every brunette man would marry a woman he was acquainted with”. The other matchmaker claimed “I could arrange that every blonde woman would marry a man she was acquainted with”. An amateur mathematician overheard their conversation and said “Then both arrangements could be done at the same time! ” Is he right?
combinatorics unsolvedcombinatorics
Least number of unit squares to draw to get a picture.

Source: ToT 2003 SO-2

6/26/2011
What least possible number of unit squares (1×1)(1\times1) must be drawn in order to get a picture of 25×2525 \times 25-square divided into 625625 of unit squares?
inductioncombinatorics unsolvedcombinatorics
Writing every integer in form of sum of product of powers

Source: ToT 2003-SA-2

6/19/2011
Prove that every positive integer can be represented in the form 3u1…2v1+3u2…2v2+…+3uk…2vk3^{u_1} \ldots 2^{v_1} + 3^{u_2} \ldots 2^{v_2} + \ldots + 3^{u_k} \ldots 2^{v_k} with integers u1,u2,…,uk,v1,…,vku_1, u_2, \ldots , u_k, v_1, \ldots, v_k such that u1>u2>…>uk≥0u_1 > u_2 >\ldots > u_k\ge 0 and 0≤v1<v2<…<vk0 \le v_1 < v_2 <\ldots < v_k.
inductionnumber theory unsolvednumber theory