MathDB

1

Part of 1999 Turkey Team Selection Test

Problems(2)

p-expansions of m and n

Source: Turkey TST 1999 - P1

7/2/2012
Let m≤nm \leq n be positive integers and pp be a prime. Let p−p-expansions of mm and nn be m=a0+a1p+⋯+arprm = a_0 + a_1p + \dots + a_rp^rn=b0+b1p+⋯+bspsn = b_0 + b_1p + \dots + b_sp^s respectively, where ar,bs≠0a_r, b_s \neq 0, for all i∈{0,1,…,r}i \in \{0,1,\dots,r\} and for all j∈{0,1,…,s}j \in \{0,1,\dots,s\}, we have 0≤ai,bj≤p−10 \leq a_i, b_j \leq p-1 . If ai≤bia_i \leq b_i for all i∈{0,1,…,r}i \in \{0,1,\dots,r\}, we write m≺pn m \prec_p n. Prove that p∤(nm)⇔m≺pnp \nmid {{n}\choose{m}} \Leftrightarrow m \prec_p n.
number theory proposednumber theory
Area/(Perimeter)^2 of cyclic quadrilateral

Source: Turkey TST 1999 - P4

7/2/2012
Let the area and the perimeter of a cyclic quadrilateral CC be ACA_C and PCP_C, respectively. If the area and the perimeter of the quadrilateral which is tangent to the circumcircle of CC at the vertices of CC are ATA_T and PTP_T , respectively, prove that ACAT≥(PCPT)2\frac{A_C}{A_T} \geq \left (\frac{P_C}{P_T}\right )^2.
geometryperimetercircumcirclecyclic quadrilateralgeometry proposed