https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Dark+adonis&feedformat=atom AoPS Wiki - User contributions [en] 2021-04-18T09:54:07Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2020_CIME_II_Problems/Problem_8&diff=142797 2020 CIME II Problems/Problem 8 2021-01-19T18:59:03Z <p>Dark adonis: </p> <hr /> <div>==Problem 8==<br /> A committee has an oligarchy, consisting of &lt;math&gt;A\%&lt;/math&gt; of the members of the committee. Suppose that &lt;math&gt;B\%&lt;/math&gt; of the work is done by the oligarchy. If the average amount of work done by a member of the oligarchy is &lt;math&gt;16&lt;/math&gt; times the amount of work done by a nonmember of the oligarchy, find the maximum possible value of &lt;math&gt;B-A&lt;/math&gt;.<br /> <br /> ==Solution==<br /> Average work done sets up an equation:<br /> &lt;cmath&gt; \frac{B}{A} = 16\frac{100-B}{100-A}&lt;/cmath&gt;<br /> &lt;cmath&gt; (100-A)B = 16(100-B)A&lt;/cmath&gt;<br /> &lt;cmath&gt; 100B - AB = 1600 A - 16AB&lt;/cmath&gt;<br /> Let &lt;math&gt;B-A = C&lt;/math&gt; and &lt;math&gt;A+B = D&lt;/math&gt;:<br /> &lt;cmath&gt; 50C + 50D - \frac{D^2-C^2}{4} = 800D - 800C - 4(D^2-C^2)&lt;/cmath&gt;<br /> &lt;cmath&gt; 15D^2 -3000D = 15C^2-3400C&lt;/cmath&gt;<br /> Complete the squares:<br /> &lt;cmath&gt; 15(D-100)^2 - 1500^2 = 15(C-1700/15)^2 - 1700^2/15&lt;/cmath&gt;<br /> &lt;cmath&gt; (D-100)^2 + \frac{(17^2-15^2) \times 100^2}{15^2} = (C-1700/15)^2&lt;/cmath&gt;<br /> &lt;cmath&gt; C = \frac{1700}{15} \pm \sqrt{(D-100)^2 + \frac{8^2 \times 100^2}{15^2}}&lt;/cmath&gt;<br /> <br /> Note that &lt;math&gt;\frac{1700+800}{15}&gt;100&lt;/math&gt; so must use minus. This means that C is maximized if &lt;math&gt;D=100&lt;/math&gt;<br /> &lt;cmath&gt; C = \frac{1700}{15} - \frac{8 \times 100}{15} = 900/15=60&lt;/cmath&gt;<br /> &lt;math&gt;B-A&lt;/math&gt; is at a maximum &lt;math&gt;60&lt;/math&gt;</div> Dark adonis https://artofproblemsolving.com/wiki/index.php?title=2020_CIME_II_Problems/Problem_8&diff=142796 2020 CIME II Problems/Problem 8 2021-01-19T18:58:50Z <p>Dark adonis: </p> <hr /> <div>==Problem 8==<br /> A committee has an oligarchy, consisting of &lt;math&gt;A\%&lt;/math&gt; of the members of the committee. Suppose that &lt;math&gt;B\%&lt;/math&gt; of the work is done by the oligarchy. If the average amount of work done by a member of the oligarchy is &lt;math&gt;16&lt;/math&gt; times the amount of work done by a nonmember of the oligarchy, find the maximum possible value of &lt;math&gt;B-A&lt;/math&gt;.<br /> <br /> ==Solution==<br /> Average work done sets up an equation:<br /> &lt;cmath&gt; \frac{B}{A} = 16\frac{100-B}{100-A}&lt;/cmath&gt;<br /> &lt;cmath&gt; (100-A)B = 16(100-B)A&lt;/cmath&gt;<br /> &lt;cmath&gt; 100B - AB = 1600 A - 16AB&lt;/cmath&gt;<br /> Let &lt;math&gt;B-A = C&lt;/math&gt; and &lt;math&gt;A+B = D&lt;/math&gt;:<br /> &lt;cmath&gt; 50C + 50D - \frac{D^2-C^2}{4} = 800D - 800C - 4(D^2-C^2)&lt;/cmath&gt;<br /> &lt;cmath&gt; 15D^2 -3000D = 15C^2-3400C&lt;/cmath&gt;<br /> Complete the squares:<br /> &lt;cmath&gt; 15(D-100)^2 - 1500^2 = 15(C-1700/15)^2 - 1700^2/15&lt;/cmath&gt;<br /> &lt;cmath&gt; (D-100)^2 + \frac{(17^2-15^2) \times 100^2}{15^2} = (C-1700/15)^2&lt;/cmath&gt;<br /> &lt;cmath&gt; C = \frac{1700}{15} \pm \sqrt{(D-100)^2 + \frac{8^2 \times 100^2}{15^2}}&lt;/cmath&gt;<br /> <br /> Note that &lt;math&gt;\frac{1700+800}{15}&gt;1&lt;/math&gt; so must use minus. This means that C is maximized if &lt;math&gt;D=100&lt;/math&gt;<br /> &lt;cmath&gt; C = \frac{1700}{15} - \frac{8 \times 100}{15} = 900/15=60&lt;/cmath&gt;<br /> &lt;math&gt;B-A&lt;/math&gt; is at a maximum &lt;math&gt;60&lt;/math&gt;</div> Dark adonis https://artofproblemsolving.com/wiki/index.php?title=2020_CIME_II_Problems/Problem_8&diff=142795 2020 CIME II Problems/Problem 8 2021-01-19T18:58:23Z <p>Dark adonis: Created page with &quot;==Problem 8== A committee has an oligarchy, consisting of &lt;math&gt;A\%&lt;/math&gt; of the members of the committee. Suppose that &lt;math&gt;B\%&lt;/math&gt; of the work is done by the oligarchy....&quot;</p> <hr /> <div>==Problem 8==<br /> A committee has an oligarchy, consisting of &lt;math&gt;A\%&lt;/math&gt; of the members of the committee. Suppose that &lt;math&gt;B\%&lt;/math&gt; of the work is done by the oligarchy. If the average amount of work done by a member of the oligarchy is &lt;math&gt;16&lt;/math&gt; times the amount of work done by a nonmember of the oligarchy, find the maximum possible value of &lt;math&gt;B-A&lt;/math&gt;.<br /> <br /> ==Solution==<br /> Average work done sets up an equation:<br /> &lt;cmath&gt; \frac{B}{A} = 16\frac{100-B}{100-A}&lt;/cmath&gt;<br /> &lt;cmath&gt; (100-A)B = 16(100-B)A&lt;/cmath&gt;<br /> &lt;cmath&gt; 100B - AB = 1600 A - 16AB&lt;/cmath&gt;<br /> Let &lt;math&gt;B-A = C&lt;/math&gt; and &lt;math&gt;A+B = D&lt;/math&gt;:<br /> &lt;cmath&gt; 50C + 50D - \frac{D^2-C^2}{4} = 800D - 800C - 4(D^2-C^2)&lt;/cmath&gt;<br /> &lt;cmath&gt; 15D^2 -3000D = 15C^2-3400C&lt;/cmath&gt;<br /> Complete the squares:<br /> &lt;cmath&gt; 15(D-100)^2 - 1500^2 = 15(C-1700/15)^2 - 1700^2/15&lt;/cmath&gt;<br /> &lt;cmath&gt; (D-100)^2 + \frac{(17^2-15^2) \times 100^2}{15^2} = (C-1700/15)^2&lt;/cmath&gt;<br /> &lt;cmath&gt; C = \frac{1700}{15} \pm \sqrt{(D-100)^2 + \frac{8^2 \times 100^2}{15^2}}&lt;/cmath&gt;<br /> <br /> Note that &lt;math&gt;\frac{1700+800}{15}&gt;1&lt;/math&gt; so must use minus. This means that C is maximized if &lt;math&gt;D=100&lt;/math&gt;<br /> &lt;cmath&gt; C = \frac{1700}{15} - \frac{8 \times 100}{15}} = 900/15=60&lt;/cmath&gt;<br /> &lt;math&gt;B-A&lt;/math&gt; is at a maximum &lt;math&gt;60&lt;/math&gt;</div> Dark adonis https://artofproblemsolving.com/wiki/index.php?title=2021_CIME_I_Problems/Problem_2&diff=142736 2021 CIME I Problems/Problem 2 2021-01-19T13:30:25Z <p>Dark adonis: Created page with &quot;==Problem 2== For digits &lt;math&gt;a, b, c,&lt;/math&gt; with &lt;math&gt;a\neq 0,&lt;/math&gt; the positive integer &lt;math&gt;N&lt;/math&gt; can be written as &lt;math&gt;\underline{a}\underline{a}\underline{b}\u...&quot;</p> <hr /> <div>==Problem 2==<br /> For digits &lt;math&gt;a, b, c,&lt;/math&gt; with &lt;math&gt;a\neq 0,&lt;/math&gt; the positive integer &lt;math&gt;N&lt;/math&gt; can be written as &lt;math&gt;\underline{a}\underline{a}\underline{b}\underline{b}&lt;/math&gt; in base &lt;math&gt;9,&lt;/math&gt; and &lt;math&gt;\underline{a}\underline{a}\underline{b}\underline{b}\underline{c}&lt;/math&gt; in base &lt;math&gt;5&lt;/math&gt;. Find the base-&lt;math&gt;10&lt;/math&gt; representation of &lt;math&gt;N&lt;/math&gt;.<br /> <br /> ==Solution==<br /> Consider the different representations of the number and equate them:<br /> &lt;cmath&gt; (9^3 + 9^2) a +(9+1)b = (5^4+5^3) a + (5^2+5)b+c&lt;/cmath&gt;<br /> &lt;cmath&gt; (810)a+10b = (750)a+30b+c&lt;/cmath&gt;<br /> &lt;cmath&gt; 60 a - 20b-c=0&lt;/cmath&gt;<br /> <br /> Note that c can't contribute since it is less than 5 so &lt;math&gt;c=0&lt;/math&gt;<br /> Next note that &lt;math&gt;b = 3a&lt;/math&gt; since &lt;math&gt;b&lt;5&lt;/math&gt; and &lt;math&gt;a&gt;0&lt;/math&gt; the only solution is &lt;math&gt;b=3&lt;/math&gt;,&lt;math&gt;a=1&lt;/math&gt;<br /> Thus in base 10 the number is &lt;math&gt;810+30=840&lt;/math&gt;</div> Dark adonis https://artofproblemsolving.com/wiki/index.php?title=2020_CIME_II_Problems/Problem_12&diff=142735 2020 CIME II Problems/Problem 12 2021-01-19T13:20:05Z <p>Dark adonis: Putting in an actual solution</p> <hr /> <div>Note that the &lt;math&gt;lcm(gcd(a,b),c)&lt;/math&gt; and its iterations are all divisible by 180. This implies that 2 of &lt;math&gt;a,b,c&lt;/math&gt; are divisible by 4, 2 are divisible by 9 and 2 are divisible by 5. <br /> &lt;cmath&gt;a,b,c = d 20, e 36, f 45&lt;/cmath&gt;<br /> <br /> Next we note that iterations are 180,&lt;math&gt;2\times 180&lt;/math&gt;,&lt;math&gt;3 \times 180&lt;/math&gt;. This implies that &lt;math&gt;d&lt;/math&gt; or &lt;math&gt;e&lt;/math&gt; must have an additional factor of 2 and &lt;math&gt;e&lt;/math&gt; or &lt;math&gt;f&lt;/math&gt; must have an additional factor of 3. The sum is minimized if d=2 and e=3 and f=1.<br /> <br /> &lt;cmath&gt;a,b,c = 40,45,108&lt;/cmath&gt;<br /> &lt;cmath&gt;a+b+c = 193&lt;/cmath&gt;</div> Dark adonis