MathDB

2

Part of 1988 Irish Math Olympiad

Problems(2)

Square and Circumcircle Problem

Source:

5/1/2012
A; B; C; D are the vertices of a square, and P is a point on the arc CD of its circumcircle. Prove that ∣PA∣2−∣PB∣2=∣PB∣.∣PD∣−∣PA∣.∣PC∣ |PA|^2 - |PB|^2 = |PB|.|PD| -|PA|.|PC| Can anyone here find the solution? I'm not great with geometry, so i tried turning it into co-ordinate geometry equations, but sadly to no avail. Thanks in advance.
geometrycircumcircletrigonometrygeometry unsolved
this question be real

Source: IrMO 1988 paper 2 Q2

11/14/2015
2. Let x1,...,xnx_1, . . . , x_n be nn integers, and let pp be a positive integer, with p<np < n. Put S1=x1+x2+...+xpS_1 = x_1 + x_2 + . . . + x_p T1=xp+1+xp+2+...+xnT_1 = x_{p+1} + x_{p+2} + . . . + x_n S2=x2+x3+...+xp+1S_2 = x_2 + x_3 + . . . + x_{p+1} T2=xp+2+xp+3+...+xn+x1T_2 = x_{p+2} + x_{p+3} + . . . + x_n + x_1 ...... Sn=xn+x1+...+xp−1S_n=x_n+x_1+...+x_{p-1} Tn=xp+xp+1+...+xn−1T_n=x_p+x_{p+1}+...+x_{n-1} For a=0,1,2,3a = 0, 1, 2, 3, and b=0,1,2,3b = 0, 1, 2, 3, let m(a,b)m(a, b) be the number of numbers ii, 1≤i≤n1 \leq i \leq n, such that SiS_i leaves remainder aa on division by 44 and TiT_i leaves remainder bb on division by 44. Show that m(1,3)m(1, 3) and m(3,1)m(3, 1) leave the same remainder when divided by 44 if, and only if, m(2,2)m(2, 2) is even.
algebra