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Part of 1998 Estonia National Olympiad

Problems(4)

last two digits of 11^{1998}

Source: 1998 Estonia National Olympiad Final Round grade 9 p1

3/13/2020
Find the last two digits of 11199811^{1998}
Digitsnumber theory
a^2(b - c) + b^2(c - a) + c^2(a - b)> 0

Source: 1998 Estonia National Olympiad Final Round grade 10 p1

3/14/2020
Prove that for any reals a>b>ca> b> c, the inequality a2(b−c)+b2(c−a)+c2(a−b)>0a^2(b - c) + b^2(c - a) + c^2(a - b)> 0.
inequalitiesalgebra
gcd (d_1, n/d_2)=gcd (d_2, n/d_1) => d_1=d_2 , where d_1,d_2 pos. divisors of n

Source: 1998 Estonia National Olympiad Final Round grade 11 p1

3/11/2020
Let d1d_1 and d2d_2 be divisors of a positive integer nn. Suppose that the greatest common divisor of d1d_1 and n/d2n/d_2 and the greatest common divisor of d2d_2 and n/d1n/d_1 are equal. Show that d1=d2d_1 = d_2.
number theorygreatest common divisorDivisors
x^2+1 = log_2(x+2)- 2x

Source: 1998 Estonia National Olympiad Final Round grade 12 p1

3/11/2020
Solve the equation x2+1=log2(x+2)−2xx^2+1 = log_2(x+2)- 2x.
algebralogarithm