MathDB

4

Part of 2013 Saudi Arabia GMO TST

Problems(3)

perpenducilar wanted, circle with diamater an altitude related

Source: 2013 Saudi Arabia GMO TST day I p4

7/31/2020
In acute triangle ABCABC, points DD and EE are the feet of the perpendiculars from AA to BCBC and BB to CACA, respectively. Segment ADAD is a diameter of circle ω\omega. Circle ω\omega intersects sides ACAC and ABAB at FF and GG (other than AA), respectively. Segment BEBE intersects segments GDGD and GFGF at XX and YY respectively. Ray DYDY intersects side ABAB at ZZ. Prove that lines XZXZ and BCBC are perpendicular
geometryperpendicularcirclealtitudes
F_n + 2 = F_{n+1} + 1 = F_{n+2} mod m, Fibonacci

Source: 2013 Saudi Arabia GMO TST II p4

7/26/2020
Let F0=0,F1=1F_0 = 0, F_1 = 1 and Fn+1=Fn+Fn−1F_{n+1} = F_n + F_{n-1}, for all positive integer nn, be the Fibonacci sequence. Prove that for any positive integer mm there exist infinitely many positive integers nn such that Fn+2≡Fn+1+1≡Fn+2F_n + 2 \equiv F_{n+1} + 1 \equiv F_{n+2} mod mm .
fibonacci numberremaindernumber theory
a^2 + b^2 divides both a^3 + 1 and b^3 + 1

Source: 2013 Saudi Arabia GMO TST III p4

7/26/2020
Find all pairs of positive integers (a,b)(a,b) such that a2+b2a^2 + b^2 divides both a3+1a^3 + 1 and b3+1b^3 + 1.
number theorydividesdivisible