MathDB

Problem 1

Part of 1998 Ukraine National Math Olympiad

Problems(4)

triangle centers are lattice points (Ukraine 1998 Grade 9 P1)

Source:

6/4/2021
Is there a triangle in the coordinate plane whose vertices, centroid, orthocenter, incenter and circumcenter all have integral coordinates?
geometry
ages in a family, system (Ukraine 1998 Grade 8 P1)

Source:

6/3/2021
In a family of five members (father, mother and three children), the product of the (integral) ages of all members equals 19981998. Knowing that the father is 1010 years older than the mother, determine the ages of all members.
system of equationsalgebra
trig, absolute value equation

Source: Ukraine 1998 Grade 10 P1

6/5/2021
Find all pairs of real numbers (x,y)(x,y) satisfying the equation sin⁡2x+sin⁡2y=∣x∣x+∣y∣y.\sin^2x+\sin^2y=\frac{|x|}x+\frac{|y|}y.
trigonometryabsolute valuealgebraequation
solve floor equation in one variable

Source: Ukraine 1998 Grade 11 P1

6/6/2021
Solve the equation 1+{2x}=⌊x2⌋+2⌊x⌋+3\sqrt{1+\{2x\}}=\lfloor x^2\rfloor+2\lfloor x\rfloor+3 (where {a}=a−⌊a⌋\{a\}=a-\lfloor a\rfloor).
floor functionalgebraequationsequation