MathDB

1

Part of 1996 Turkey Team Selection Test

Problems(2)

Turkey TST 1996 Problem 1, find a_1996

Source: Turkey TST 1996 Problem 1

9/28/2011
Let ∏n=11996(1+nx3n)=1+a1xk1+a2xk2+...+amxkm \prod_{n=1}^{1996}{(1+nx^{3^n})}= 1+ a_{1}x^{k_{1}}+ a_{2}x^{k_{2}}+...+ a_{m}x^{k_{m}}
where a1,a1,...,ama_{1}, a_{1}, . . . , a_{m} are nonzero and k1<k2<...<kmk_{1} < k_{2} <...< k_{m}. Find a1996a_{1996}.
algebra proposedalgebra
Turkey TST 1996 Problem 4, compute the ratio S_KLMN/S_ABC

Source: Turkey TST 1996 Problem 4

9/28/2011
The diagonals ACAC and BDBD of a convex quadrilateral ABCDABCD with SABC=SADCS_{ABC} = S_{ADC} intersect at EE. The lines through EE parallel to ADAD, DCDC, CBCB, BABA meet ABAB, BCBC, CDCD, DADA at KK, LL, MM, NN, respectively. Compute the ratio SKLMNSABC\frac{S_{KLMN}}{S_{ABC}}
ratiogeometryparallelogramgeometry proposed