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Difference between revisions of "2021 GCIME Problems"
(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 $\pi(n)$ denote the number of primes less than or equal to $n$. Suppose $\pi(a)^{\pi(b)}=\pi(b)^{\pi(a)}=c$. For some fixed $c$ what is the maximum possible number of solutions $(a, b, c)$ but not exceeding $99$?

Solution

Problem 2

Let $N$ denote the number of solutions to the given equation: \[\lfloor\sqrt{n}\rfloor+\lfloor\sqrt[3]{n}\rfloor+\lfloor\sqrt[4]{n}\lfloor+\lfloor\sqrt[5]{n}\rfloor=100\] What is the value of $N$?

Solution


Problem 3

Let $ABCD$ be a cyclic kite. Let $r\in\mathbb{N}$ be the inradius of $ABCD$. Suppose $AB\cdot BC\cdot r$ is a perfect square. What is the smallest value of $AB\cdot BC\cdot r$?

Solution


Problem 4

Define $H(m)$ as the harmonic mean of all the divisors of $m$. Find the positive integer $n<1000$ for which $\frac{H(n)}{n}$ is the minimum amongst all $1<n\leq 1000$.

Solution


Problem 5

Let $x$ be a real number such that \[\frac{\sin^{4}x}{20}+\frac{\cos^{4}x}{21}=\frac{1}{41}\] If the value of \[\frac{\sin^{6}x}{20^{3}}+\frac{\cos^{6}x}{21^{3}}\] can be expressed as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, then what is the remainder when $m+n$ is divided by $1000$?

Solution

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