Difference between revisions of "2021 GCIME Problems"
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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>\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>? |
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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]] | ||
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+ | ==Problem 6== | ||
+ | Two scales used to measure temperature are degrees Fahrenheit (<math>F</math>) and degrees Celsius (<math>C</math>) and the two are related by the formula <math>F=\tfrac{9}{5}C+32</math>. When a two-digit integer degree temperature <math>n</math> in Celcius is converted to Fahrenheit and rounded to the nearest integer degree, it turns out the ones and tens digits of the original Celsius temperature | ||
+ | <math>n</math> sometimes switch places to give the rounded Fahrenheit equivalent. Find the sum of all two-digit integer values of <math>n</math> for which this happens. | ||
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+ | [[2021 GCIME Problems/Problem 6|Solution]] | ||
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+ | ==Problem 7== | ||
+ | Let <math>a_{n}</math> denote the units digit of <math>{{(4n)^{(3n)}}^{(2n)}}^{n}</math>. Then find the sum of all positive integers <math>n<1000</math> such that <cmath>\sum_{i=1}^{n}a_{i}<4n.</cmath> | ||
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+ | [[2021 GCIME Problems/Problem 7|Solution]] | ||
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+ | ==Problem 8== | ||
+ | A basketball club decided to label every basketball in the club. After labeling all <math>n</math> of the balls, the labeler noticed that exactly half of the balls had the digit <math>1</math>. Find the sum of all possible three-digit integer values of <math>n</math>. | ||
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+ | [[2021 GCIME Problems/Problem 8|Solution]] | ||
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+ | ==Problem 9== | ||
+ | <math>\triangle ABC</math> has perimeter <math>60</math>, and points <math>D, E,</math> and <math>F</math> are chosen on sides <math>BC, AC,</math> and <math>AB</math> respectively. If the circumcircles of triangles <math>\triangle AFE, \triangle BFD,</math> and <math>\triangle CED</math> all pass through the orthocenter of <math>\triangle DEF,</math> then the maximum possible area of <math>\triangle DEF</math> can be written as <math>a\sqrt{b}</math> for squarefree <math>b</math>. What is <math>a+b</math>? | ||
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+ | [[2021 GCIME Problems/Problem 9|Solution]] | ||
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+ | ==Problem 10== | ||
+ | Let <math>x, y,</math> and <math>z</math> be randomly chosen real numbers from the interval <math>[-10, 10]</math>. Let the probability that these randomly chosen <math>x, y,</math> and <math>z</math> satisfy the following inequality <cmath>10(|x|+|y|+|z|)\geq 100\geq x^{2}+y^{2}+z^{2}</cmath> be <math>\tfrac{m\pi-n}{p}</math> where <math>m, n,</math> and <math>p</math> are relatively prime positive integers and <math>m</math> and <math>p</math> are relatively prime. Find <math>m+n+p</math>. | ||
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+ | [[2021 GCIME Problems/Problem 10|Solution]] |
Latest revision as of 20:18, 6 March 2021
Contents
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 ?
Problem 6
Two scales used to measure temperature are degrees Fahrenheit () and degrees Celsius () and the two are related by the formula . When a two-digit integer degree temperature in Celcius is converted to Fahrenheit and rounded to the nearest integer degree, it turns out the ones and tens digits of the original Celsius temperature sometimes switch places to give the rounded Fahrenheit equivalent. Find the sum of all two-digit integer values of for which this happens.
Problem 7
Let denote the units digit of . Then find the sum of all positive integers such that
Problem 8
A basketball club decided to label every basketball in the club. After labeling all of the balls, the labeler noticed that exactly half of the balls had the digit . Find the sum of all possible three-digit integer values of .
Problem 9
has perimeter , and points and are chosen on sides and respectively. If the circumcircles of triangles and all pass through the orthocenter of then the maximum possible area of can be written as for squarefree . What is ?
Problem 10
Let and be randomly chosen real numbers from the interval . Let the probability that these randomly chosen and satisfy the following inequality be where and are relatively prime positive integers and and are relatively prime. Find .