Maths Olympiad Daily Problems
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For any positive integer n, let τ (n) denote the number of positive divisors of n. If n is a positive integer such that τ(n^2)/τ(n)= 3, compute τ(n^7)/τ(n).
Ari repeatedly rolls a standard, fair, six-sided die. Let R(n) be the n
th number rolled, and let
Q(n) = R(1)· R(2)· . . . · R(n). Let the probability that there exists an n such that Q(n) = 100 and for
all m < n, Q(m) is not a perfect square be p/q
where p, q are relatively prime positive integers. Find
the largest prime factor of p + q.
Let D be a point on the side AC of triangle ABC. Let E and F be points on the segments BD and BC respectively, such that
A circle having radius r1 centered at point N is tangent to a circle of radius r2 centered at
M. Let l and j be the two common external tangent lines to the two circles. A circle centered at
P with radius r2 is externally tangent to circle N at the point at which l coincides with circle N,
and line k is externally tangent to P and N such that points M, N, and P all lie on the same side
of k. For what ratio r1/r2 are j and k parallel?
