MathDB

3

Part of 2023 New Zealand MO

Problems(2)

9^{x+1} + 2187 = 3^{6x-x^2}

Source: 2023 NZMO - New Zealand Maths Olympiad Round 1 p3

9/2/2023
Find the sum of the smallest and largest possible values for xx which satisfy the following equation. 9x+1+2187=36xāˆ’x2.9^{x+1} + 2187 = 3^{6x-x^2}.
algebra
collinearity wanted, 2 squares related

Source: 2023 NZMO - New Zealand Maths Olympiad Round 2 p3

9/2/2023
Let ABCDABCD be a square (vertices labelled in clockwise order). Let ZZ be any point on diagonal ACAC between AA and CC such that AZ>ZCAZ > ZC. Points XX and YY exist such that AXYZAXY Z is a square (vertices labelled in clockwise order) and point BB lies inside AXYZAXY Z. Let MM be the point of intersection of lines BXBX and DZDZ (extended if necessary). Prove that CC, MM and YY are colinear
geometrycollinear