MathDB

3

Part of 2015 Dutch IMO TST

Problems(2)

c_1 + c_2 + ... + c_k = n^2 - 1, when c_1=a_i if a_i = a_{i-1} etc

Source: Dutch IMO TST 2015 day 1 p3

8/30/2019
Let nn be a positive integer. Consider sequences a0,a1,...,aka_0, a_1, ..., a_k and b0,b1,,..,bkb_0, b_1,,..,b_k such that a0=b0=1a_0 = b_0 = 1 and ak=bk=na_k = b_k = n and such that for all ii such that 1≤i≤k1 \le i \le k , we have that (ai,bi)(a_i, b_i) is either equal to (1+ai−1,bi−1)(1 + a_{i-1}, b_{i-1}) or (ai−1;1+bi−1)(a_{i-1}; 1 + b_{i-1}). Consider for 1≤i≤k1 \le i \le k the number ci={ai   if   ai=ai−1bi   if   bi=bi−1c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\ b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases} Show that c1+c2+...+ck=n2−1c_1 + c_2 + ... + c_k = n^2 - 1.
algebraSequenceSum
equilateral given and wanted, IMO TST problem

Source: Dutch IMO TST2 2015 p3

8/5/2019
An equilateral triangle ABCABC is given. On the line through BB parallel to ACAC there is a point DD, such that DD and CC are on the same side of the line ABAB. The perpendicular bisector of CDCD intersects the line ABAB in EE. Prove that triangle CDECDE is equilateral.
geometryEquilateralEquilateral Triangle