Difference between revisions of "2021 GCIME Problems"
Sugar rush (talk | contribs) (Created page with "==Problem 1== Let <math>\pi(n)</math> denote the number of primes less than or equal to <math>n</math>. Suppose <math>\pi(a)^{\pi(b)}=\pi(b)^{\pi(a)}=c</math>. For some fixed...") |
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==Problem 2== | ==Problem 2== | ||
− | Let <math>N</math> denote the number of solutions to the given equation: <cmath>\sqrt{n}+\sqrt[3]{n}+\sqrt[4]{n}+\sqrt[5]{n}=100</cmath> What is the value of <math>N</math>? | + | Let <math>N</math> denote the number of solutions to the given equation: <cmath>\lfloor\sqrt{n}\rfloor+\lfloor\sqrt[3]{n}\rfloor+\lfloor\sqrt[4]{n}\lfloor+\lfloor\sqrt[5]{n}\rfloor=100</cmath> What is the value of <math>N</math>? |
==Problem 3== | ==Problem 3== |
Revision as of 14:32, 6 March 2021
Problem 1
Let denote the number of primes less than or equal to
. Suppose
. For some fixed
what is the maximum possible number of solutions
but not exceeding
?
Problem 2
Let denote the number of solutions to the given equation:
What is the value of
?
Problem 3
Let be a cyclic kite. Let
be the inradius of
. Suppose
is a perfect square. What is the smallest value of
?
Problem 4
Define as the harmonic mean of all the divisors of
. Find the positive integer
for which
is the minimum amongst all
.
Problem 5
Let be a real number such that
If the value of
can be expressed as
where
and
are relatively prime positive integers, then what is the remainder when
is divided by
?