MathDB

1

Part of 2004 Estonia National Olympiad

Problems(4)

(x + 6)/y+ 13/xy=(4-y)/x

Source: 2004 Estonia National Olympiad Final Round grade 9 p1

3/25/2020
Find all pairs of real numbers (x,y)(x, y) that satisfy the equation x+6y+13xy=4−yx\frac{x + 6}{y}+\frac{13}{xy}=\frac{4-y}{x}
equationalgebra
gcd(x, y) = 6, gcd(y, z) = 10, gcd(z, x) = 8, lcm(x, y,z) = 2400

Source: 2004 Estonia National Olympiad Final Round grade 10 p1

3/25/2020
Find all triples of positive integers (x,y,z)(x, y, z) satisfying x<y<zx < y < z, gcd(x,y)=6,gcd(y,z)=10,gcd(z,x)=8gcd(x, y) = 6, gcd(y, z) = 10, gcd(z, x) = 8 and lcm(x,y,z)=2400lcm(x, y,z) = 2400.
GCDLCMnumber theory
a^2 + ab + b^2 = 1 diophantine

Source: 2004 Estonia National Olympiad Final Round grade 11 p1

3/25/2020
Find all pairs of integers (a,b)(a, b) such that a2+ab+b2=1a^2 + ab + b^2 = 1
Diophantine equationdiophantinenumber theory
4 pieces with equal area in a circle, &lt;KOL &lt; 75^o

Source: 2004 Estonia National Olympiad Final Round grade 12 p 1

3/25/2020
Inside a circle, point KK is taken such that the ray drawn from KK through the centre OO of the circle and the chord perpendicular to this ray passing through KK divide the circle into three pieces with equal area. Let LL be one of the endpoints of the chord mentioned. Does the inequality ∠KOL<75o\angle KOL < 75^o hold?
geometrycircleequal areaschord