Difference between revisions of "2021 GCIME Problems"
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Sugar rush (talk | contribs) (first 5 problems; will add final 10 later) |
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==Problem 1== | ==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 <math>c</math> what is the maximum possible number of solutions <math>(a, b, c)</math> but not exceeding <math>99</math>? | 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>? | ||
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+ | [[2021 GCIME Problems/Problem 1|Solution]] | ||
==Problem 2== | ==Problem 2== | ||
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>? | 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>? | ||
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+ | [[2021 GCIME Problems/Problem 2|Solution]] | ||
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==Problem 3== | ==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>? | 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>? | ||
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+ | [[2021 GCIME Problems/Problem 3|Solution]] | ||
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==Problem 4== | ==Problem 4== | ||
Define <math>H(m)</math> as the harmonic mean of all the divisors of <math>m</math>. Find the positive integer <math>n<1000</math> for which <math>\frac{H(n)}{n}</math> is the minimum amongst all <math>1<n\leq 1000</math>. | Define <math>H(m)</math> as the harmonic mean of all the divisors of <math>m</math>. Find the positive integer <math>n<1000</math> for which <math>\frac{H(n)}{n}</math> is the minimum amongst all <math>1<n\leq 1000</math>. | ||
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+ | [[2021 GCIME Problems/Problem 4|Solution]] | ||
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==Problem 5== | ==Problem 5== | ||
Let <math>x</math> be a real number such that <cmath>\frac{\sin^{4}x}{20}+\frac{\cos^{4}x}{21}=\frac{1}{41}</cmath> If the value of <cmath>\frac{\sin^{6}x}{20^{3}}+\frac{\cos^{6}x}{21^{3}}</cmath> can be expressed as <math>\tfrac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, then what is the remainder when <math>m+n</math> is divided by <math>1000</math>? | Let <math>x</math> be a real number such that <cmath>\frac{\sin^{4}x}{20}+\frac{\cos^{4}x}{21}=\frac{1}{41}</cmath> If the value of <cmath>\frac{\sin^{6}x}{20^{3}}+\frac{\cos^{6}x}{21^{3}}</cmath> can be expressed as <math>\tfrac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, then what is the remainder when <math>m+n</math> is divided by <math>1000</math>? | ||
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+ | [[2021 GCIME Problems/Problem 5|Solution]] |
Revision as of 14:33, 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 ?