Difference between revisions of "2021 GCIME"

Latest revision as of 14:25, 6 March 2021

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-==Problem 1==
+#redirect[[2021 GCIME Problems]]
-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 <math>c</math> what is the maximum possible number of solutions <math>(a, b, c)</math> but not exceeding <math>99</math>?
-==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>?
-==Problem 3==
-Let <math>ABCD</math> be a cyclic kite. Let <math>r\in\mathbb{N}</math> be the inradius of <math>ABCD</math>. Suppose <math>AB\cdot BC\cdot r</math> is a perfect square. What is the smallest value of <math>AB\cdot BC\cdot r</math>?