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4

Part of 2010 NZMOC Camp Selection Problems

Problems(2)

1/a+1/b+2007/\lcm (a, b)= 1/\gcd (a, b), 1/a+1/b+2010/\lcm (a, b)= 1/\gcd (a, b)

Source: New Zealand NZMOC Camp Selection Problems 2010 Juniors 4

9/18/2021
Find all positive integer solutions (a,b)(a, b) to the equation 1a+1b+nlcm(a,b)=1gcd(a,b)\frac{1}{a}+\frac{1}{b}+ \frac{n}{lcm(a,b)}=\frac{1}{gcd(a, b)} for (i) n=2007n = 2007; (ii) n=2010n = 2010.
LCMGCDleast common multiplegreatest common divisornumber theory
1/QD = 1/QB + 1/QC, inscribed equilateral (2012 UNSW S5 Australia)

Source:

1/1/2021
A line drawn from the vertex AA of the equilateral triangle ABCABC meets the side BCBC at DD and the circumcircle of the triangle at point QQ. Prove that 1QD=1QB+1QC\frac{1}{QD} = \frac{1}{QB} + \frac{1}{QC}.
geometryEquilateral