MathDB

6

Part of 2008 Tournament Of Towns

Problems(4)

TT2008 Junior A-Level - P6

Source:

9/4/2010
Let ABCABC be a non-isosceles triangle. Two isosceles triangles AB′CAB'C with base ACAC and CA′BCA'B with base BCBC are constructed outside of triangle ABCABC. Both triangles have the same base angle φ\varphi. Let C1C_1 be a point of intersection of the perpendicular from CC to A′B′A'B' and the perpendicular bisector of the segment ABAB. Determine the value of ∠AC1B.\angle AC_1B.
symmetrygeometryperpendicular bisectorgeometry proposed
TT2008 Senior A-Level - P6

Source:

9/4/2010
Let P(x)P(x) be a polynomial with real coefficients so that equation P(m)+P(n)=0P(m) + P(n) = 0 has infi nitely many pairs of integer solutions (m,n)(m,n). Prove that graph of y=P(x)y = P(x) has a center of symmetry.
algebrapolynomialalgebra proposed
2008 ToT Spring Junior A P6 a/b + c/d = 1, a/d + c/b = 2008

Source:

3/7/2020
Do there exist positive integers a,b,ca,b,c and dd such that {ab+cd=1ad+cb=2008\begin{cases} \dfrac{a}{b} + \dfrac{c}{d} = 1\\ \\ \dfrac{a}{d} + \dfrac{c}{b} = 2008\end{cases} ?
number theorysystem of equations
2008 ToT Spring Senior A P6 seated in a circle are 11 wizards

Source:

3/7/2020
Seated in a circle are 1111 wizards. A different positive integer not exceeding 10001000 is pasted onto the forehead of each. A wizard can see the numbers of the other 1010, but not his own. Simultaneously, each wizard puts up either his left hand or his right hand. Then each declares the number on his forehead at the same time. Is there a strategy on which the wizards can agree beforehand, which allows each of them to make the correct declaration?
combinatoricsgame strategy