https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Ihatepie&feedformat=atom AoPS Wiki - User contributions [en] 2021-04-20T06:52:20Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_10A_Problems/Problem_25&diff=103979 2019 AMC 10A Problems/Problem 25 2019-03-02T06:05:39Z <p>Ihatepie: /* Solution */</p> <hr /> <div>{{duplicate|[[2019 AMC 10A Problems|2019 AMC 10A #25]] and [[2019 AMC 12A Problems|2019 AMC 12A #24]]}}<br /> <br /> ==Problem==<br /> <br /> For how many integers &lt;math&gt;n&lt;/math&gt; between &lt;math&gt;1&lt;/math&gt; and &lt;math&gt;50&lt;/math&gt;, inclusive, is &lt;cmath&gt;\frac{(n^2-1)!}{(n!)^n}&lt;/cmath&gt; an integer? (Recall that &lt;math&gt;0! = 1&lt;/math&gt;.)<br /> <br /> &lt;math&gt;\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35&lt;/math&gt;<br /> <br /> ==Solution==<br /> The main insight is that <br /> <br /> &lt;cmath&gt;\frac{(n^2)!}{(n!)^{n+1}}&lt;/cmath&gt; <br /> <br /> is always an integer. This is true because it is precisely the number of ways to split up &lt;math&gt;n^2&lt;/math&gt; objects into &lt;math&gt;n&lt;/math&gt; unordered groups of size &lt;math&gt;n&lt;/math&gt;. Thus,<br /> <br /> &lt;cmath&gt;\frac{(n^2-1)!}{(n!)^n}=\frac{(n^2)!}{(n!)^{n+1}}\cdot\frac{n!}{n^2}&lt;/cmath&gt;<br /> <br /> is an integer if &lt;math&gt;n^2 \mid n!&lt;/math&gt;, or in other words, if &lt;math&gt;n \mid (n-1)!&lt;/math&gt;. This condition is false precisely when &lt;math&gt;n=4&lt;/math&gt; or &lt;math&gt;n&lt;/math&gt; is prime, by Wilson's Theorem. There are &lt;math&gt;15&lt;/math&gt; primes between &lt;math&gt;1&lt;/math&gt; and &lt;math&gt;50&lt;/math&gt;, inclusive, so there are 15 + 1 = 16 terms for which<br /> <br /> &lt;cmath&gt;\frac{(n^2-1)!}{(n!)^{n}}&lt;/cmath&gt;<br /> <br /> is potentially not an integer. It can be easily verified that the above expression is not an integer for &lt;math&gt;n=4&lt;/math&gt; as there are more factors of 2 in the denominator than the numerator. Similarly, it can be verified that the above expression is not an integer for any prime &lt;math&gt;n=p&lt;/math&gt;, as there are more factors of p in the denominator than the numerator. Thus all 16 values of n make the expression not an integer and the answer is &lt;math&gt;50-16=\boxed{\mathbf{(D)}\ 34}&lt;/math&gt;.<br /> <br /> ==See Also==<br /> <br /> {{AMC10 box|year=2019|ab=A|num-b=24|after=Last Problem}}<br /> {{AMC12 box|year=2019|ab=A|num-b=23|num-a=25}}<br /> {{MAA Notice}}</div> Ihatepie https://artofproblemsolving.com/wiki/index.php?title=1998_PMWC_Problems/Problem_I1&diff=32612 1998 PMWC Problems/Problem I1 2009-08-08T20:37:13Z <p>Ihatepie: /* Solution */</p> <hr /> <div>== Problem I1 ==<br /> Calculate: &lt;math&gt;\frac{1*2*3+2*4*6+3*6*9+4*8*12+5*10*15}{1*3*5+2*6*10+3*9*15+4*12*20+5*15*25}&lt;/math&gt;<br /> <br /> == Solution ==<br /> <br /> If you factor the top, you get &lt;math&gt;(1*2*3)(1^{3}+2^{3}+3^{3}+4^{3}+5^{3})&lt;/math&gt;<br /> <br /> <br /> If you factor the bottom, you get &lt;math&gt;(1*3*5)(1^{3}+2^{3}+3^{3}+4^{3}+5^{3})&lt;/math&gt;<br /> <br /> Dividing out the common factor, you get &lt;math&gt;\frac {1*2*3}{1*3*5}=\frac {6}{15}= \frac {2}{5}&lt;/math&gt;</div> Ihatepie https://artofproblemsolving.com/wiki/index.php?title=1998_PMWC_Problems/Problem_I1&diff=32611 1998 PMWC Problems/Problem I1 2009-08-08T20:36:53Z <p>Ihatepie: /* Solution */</p> <hr /> <div>== Problem I1 ==<br /> Calculate: &lt;math&gt;\frac{1*2*3+2*4*6+3*6*9+4*8*12+5*10*15}{1*3*5+2*6*10+3*9*15+4*12*20+5*15*25}&lt;/math&gt;<br /> <br /> == Solution ==<br /> <br /> If you factor the top, you get &lt;math&gt;(1*2*3)(1^{3}+2^{3}+3^{3}+4^{3}+5^{3})&lt;/math&gt;<br /> <br /> <br /> If you factor the bottom, you get &lt;math&gt;(1*3*5)(1^{3}+2^{3}+3^{3}+4^{3}+5^{3})&lt;/math&gt;<br /> <br /> Dividing out the common factor, you get &lt;math&gt;\frac {1*2*3}{1*3*5}=\frac {6}{15}= \frac {2}{5}&lt;/math&gt;\$</div> Ihatepie https://artofproblemsolving.com/wiki/index.php?title=1998_PMWC_Problems/Problem_I1&diff=32610 1998 PMWC Problems/Problem I1 2009-08-08T20:36:18Z <p>Ihatepie: Created page with '== Problem I1 == Calculate: &lt;math&gt;\frac{1*2*3+2*4*6+3*6*9+4*8*12+5*10*15}{1*3*5+2*6*10+3*9*15+4*12*20+5*15*25}&lt;/math&gt; == Solution == If you factor the top, you get &lt;math&gt;(1*2*3…'</p> <hr /> <div>== Problem I1 ==<br /> Calculate: &lt;math&gt;\frac{1*2*3+2*4*6+3*6*9+4*8*12+5*10*15}{1*3*5+2*6*10+3*9*15+4*12*20+5*15*25}&lt;/math&gt;<br /> <br /> == Solution ==<br /> <br /> If you factor the top, you get &lt;math&gt;(1*2*3)(1^{3}+2^{3}+3^{3}+4^{3}+5^{3})&lt;/math&gt;<br /> <br /> <br /> If you factor the bottom, you get &lt;math&gt;(1*3*5)(1^{3}+2^{3}+3^{3}+4^{3}+5^{3})&lt;/math&gt;<br /> <br /> Dividing out the common factor, you get &lt;cmath&gt;\frac {1*2*3}{1*3*5}&lt;/cmath&gt; &lt;cmath&gt;\frac {6}{15}&lt;/cmath&gt; &lt;cmath&gt;\frac {2}{5}&lt;/cmath&gt;</div> Ihatepie https://artofproblemsolving.com/wiki/index.php?title=1997_PMWC_Problems/Problem_I12&diff=32591 1997 PMWC Problems/Problem I12 2009-08-08T04:52:31Z <p>Ihatepie: /* Solution 2 */</p> <hr /> <div>== Problem ==<br /> In a die, 1 and 6, 2 and 5, 3 and 4 appear on opposite faces. When 2 dice are thrown, product of numbers appearing on the top and bottom faces of the 2 dice are formed as follows: <br /> *number on top face of 1st die x number on top face of 2nd die<br /> *number on top face of 1st die x number on bottom face of 2nd die<br /> *number on bottom face of 1st die x number on top face of 2nd die<br /> *number on bottom face of 1st die x number on bottom face of 2nd die<br /> What is the sum of these 4 products ? <br /> <br /> <br /> == Solution ==<br /> Let &lt;math&gt;x,y&lt;/math&gt; be the two numbers at the top of the two dice. Then<br /> <br /> &lt;cmath&gt;xy + x(7-y) + (7-x)y + (7-x)(7-y) = (x + (7-x))(y + (7 - y)) = 49&lt;/cmath&gt;<br /> <br /> <br /> == Solution 2 ==<br /> <br /> Let &lt;math&gt;a&lt;/math&gt;, &lt;math&gt;b&lt;/math&gt;, &lt;math&gt;c&lt;/math&gt;, and &lt;math&gt;d&lt;/math&gt; be the numbers on the top of dies one and two and the numbers on the bottom of dies one and two respectively.<br /> <br /> Therefore, you are trying to find the sum of &lt;math&gt;ab+ad+cb+cd&lt;/math&gt;. This factored is &lt;math&gt;(a+c)(b+d)&lt;/math&gt;. Since the sum of opposite faces is &lt;math&gt;7&lt;/math&gt;, the answer is &lt;math&gt;7*7&lt;/math&gt; or &lt;math&gt;49&lt;/math&gt;.<br /> <br /> == See also ==<br /> {{PMWC box|year=1997|num-b=I11|num-a=I13}}<br /> <br /> [[Category:Introductory Combinatorics Problems]]</div> Ihatepie https://artofproblemsolving.com/wiki/index.php?title=1997_PMWC_Problems/Problem_I12&diff=32590 1997 PMWC Problems/Problem I12 2009-08-08T04:52:16Z <p>Ihatepie: /* Solution 2 */</p> <hr /> <div>== Problem ==<br /> In a die, 1 and 6, 2 and 5, 3 and 4 appear on opposite faces. When 2 dice are thrown, product of numbers appearing on the top and bottom faces of the 2 dice are formed as follows: <br /> *number on top face of 1st die x number on top face of 2nd die<br /> *number on top face of 1st die x number on bottom face of 2nd die<br /> *number on bottom face of 1st die x number on top face of 2nd die<br /> *number on bottom face of 1st die x number on bottom face of 2nd die<br /> What is the sum of these 4 products ? <br /> <br /> <br /> == Solution ==<br /> Let &lt;math&gt;x,y&lt;/math&gt; be the two numbers at the top of the two dice. Then<br /> <br /> &lt;cmath&gt;xy + x(7-y) + (7-x)y + (7-x)(7-y) = (x + (7-x))(y + (7 - y)) = 49&lt;/cmath&gt;<br /> <br /> <br /> == Solution 2 ==<br /> Let &lt;math&gt;a&lt;/math&gt;, &lt;math&gt;b&lt;/math&gt;, &lt;math&gt;c&lt;/math&gt;, and &lt;math&gt;d&lt;/math&gt; be the numbers on the top of dies one and two and the numbers on the bottom of dies one and two respectively.<br /> <br /> Therefore, you are trying to find the sum of &lt;math&gt;ab+ad+cb+cd&lt;/math&gt;. This factored is &lt;math&gt;(a+c)(b+d)&lt;/math&gt;. Since the sum of opposite faces is &lt;math&gt;7&lt;/math&gt;, the answer is &lt;math&gt;7*7&lt;/math&gt; or &lt;math&gt;49&lt;/math&gt;.<br /> <br /> == See also ==<br /> {{PMWC box|year=1997|num-b=I11|num-a=I13}}<br /> <br /> [[Category:Introductory Combinatorics Problems]]</div> Ihatepie https://artofproblemsolving.com/wiki/index.php?title=1997_PMWC_Problems/Problem_I12&diff=32589 1997 PMWC Problems/Problem I12 2009-08-08T04:51:49Z <p>Ihatepie: /* Solution 2 */</p> <hr /> <div>== Problem ==<br /> In a die, 1 and 6, 2 and 5, 3 and 4 appear on opposite faces. When 2 dice are thrown, product of numbers appearing on the top and bottom faces of the 2 dice are formed as follows: <br /> *number on top face of 1st die x number on top face of 2nd die<br /> *number on top face of 1st die x number on bottom face of 2nd die<br /> *number on bottom face of 1st die x number on top face of 2nd die<br /> *number on bottom face of 1st die x number on bottom face of 2nd die<br /> What is the sum of these 4 products ? <br /> <br /> <br /> == Solution ==<br /> Let &lt;math&gt;x,y&lt;/math&gt; be the two numbers at the top of the two dice. Then<br /> <br /> &lt;cmath&gt;xy + x(7-y) + (7-x)y + (7-x)(7-y) = (x + (7-x))(y + (7 - y)) = 49&lt;/cmath&gt;<br /> <br /> <br /> == Solution 2 ==<br /> Let &lt;math&gt;a&lt;/math&gt;, &lt;math&gt;b&lt;/math&gt;, &lt;math&gt;c&lt;/math&gt;, and &lt;math&gt;d&lt;/math&gt; be the numbers on the top of dies one and two and the numbers on the bottom of dies one and two respectively.<br /> <br /> Therefore, you are trying to find the sum of &lt;cmath&gt;ab+ad+cb+cd&lt;/cmath&gt;. This factored is &lt;cmath&gt;(a+c)(b+d)&lt;/cmath&gt;. Since the sum of opposite faces is &lt;cmath&gt;7&lt;/cmath&gt;, the answer is &lt;cmath&gt;7*7&lt;/cmath&gt; or &lt;cmath&gt;49&lt;/cmath&gt;.<br /> <br /> == See also ==<br /> {{PMWC box|year=1997|num-b=I11|num-a=I13}}<br /> <br /> [[Category:Introductory Combinatorics Problems]]</div> Ihatepie https://artofproblemsolving.com/wiki/index.php?title=1997_PMWC_Problems/Problem_I12&diff=32588 1997 PMWC Problems/Problem I12 2009-08-08T04:50:40Z <p>Ihatepie: /* Solution */</p> <hr /> <div>== Problem ==<br /> In a die, 1 and 6, 2 and 5, 3 and 4 appear on opposite faces. When 2 dice are thrown, product of numbers appearing on the top and bottom faces of the 2 dice are formed as follows: <br /> *number on top face of 1st die x number on top face of 2nd die<br /> *number on top face of 1st die x number on bottom face of 2nd die<br /> *number on bottom face of 1st die x number on top face of 2nd die<br /> *number on bottom face of 1st die x number on bottom face of 2nd die<br /> What is the sum of these 4 products ? <br /> <br /> <br /> == Solution ==<br /> Let &lt;math&gt;x,y&lt;/math&gt; be the two numbers at the top of the two dice. Then<br /> <br /> &lt;cmath&gt;xy + x(7-y) + (7-x)y + (7-x)(7-y) = (x + (7-x))(y + (7 - y)) = 49&lt;/cmath&gt;<br /> <br /> <br /> == Solution 2 ==<br /> Let &lt;math&gt;a&lt;/math&gt;, &lt;math&gt;b&lt;/math&gt;, &lt;math&gt;c&lt;/math&gt;, and &lt;math&gt;d&lt;/math&gt; be the numbers on the top of dies one and two and the numbers on the bottom of dies one and two respectively.<br /> <br /> Therefore, you are trying to find the sum of &lt;math&gt;ab+ad+cb+cd&lt;/math&gt;. This factored is &lt;math&gt;(a+c)(b+d)&lt;/math&gt;. Since the sum of opposite faces is &lt;math&gt;7&lt;/math&gt;, the answer is &lt;math&gt;7*7&lt;/math&gt; or &lt;math&gt;49&lt;/math&gt;<br /> <br /> == See also ==<br /> {{PMWC box|year=1997|num-b=I11|num-a=I13}}<br /> <br /> [[Category:Introductory Combinatorics Problems]]</div> Ihatepie https://artofproblemsolving.com/wiki/index.php?title=1997_PMWC_Problems&diff=32587 1997 PMWC Problems 2009-08-08T04:35:10Z <p>Ihatepie: /* Problem I8 */</p> <hr /> <div>== Problem I1 ==<br /> Evaluate &lt;math&gt;29 \dfrac{27}{28} \times 27 \frac{14}{15}&lt;/math&gt;<br /> <br /> [[1997 PMWC Problems/Problem I1|Solution]]<br /> <br /> == Problem I2 ==<br /> In the multiplication in the image, each letter and each box represent a single digit. Different letters represent different digits but a box can represent any digit. What does the five-digit number HAPPY stand for?<br /> <br /> [[Image:1997 PMWC individual problem 2.png]]<br /> <br /> [[1997 PMWC Problems/Problem I2|Solution]]<br /> <br /> == Problem I3 ==<br /> Peter is ill. He has to take medicine A every 8 hours,<br /> medicine B every 5 hours and medicine C every 10 hours.<br /> If he took all three medicines at 7 a.m. on Tuesday, when will he take them altogether again?<br /> <br /> [[1997 PMWC Problems/Problem I3|Solution]]<br /> <br /> == Problem I4 ==<br /> Each of the three diagrams in the image show a balance of weights using different objects. How many squares will balance a circle?<br /> <br /> [[Image:1997 PMWC individual 4.png]]<br /> <br /> [[1997 PMWC Problems/Problem I4|Solution]]<br /> <br /> == Problem I5 ==<br /> Two squares of different sizes overlap as shown in the given figure. What is the difference between the non-overlapping areas?<br /> <br /> [[Image:1997 PMWC individual 5.png]]<br /> <br /> [[1997 PMWC Problems/Problem I5|Solution]]<br /> <br /> == Problem I6 ==<br /> John and Mary went to a book shop and bought some exercise books. They had &lt;dollar/&gt;100 each. John could buy 7 large and 4 small ones. Mary could buy 5 large and 6 small ones and had &lt;dollar/&gt;5 left. How much was a small exercise book?<br /> <br /> [[1997 PMWC Problems/Problem I6|Solution]]<br /> <br /> == Problem I7 ==<br /> 40% of girls and 50% of boys in a class got an 'A'. If there<br /> are only 12 students in the class who got 'A's and the ratio of<br /> boys and girls in the class is 4:5, how many students are<br /> there in the class?<br /> <br /> [[1997 PMWC Problems/Problem I7|Solution]]<br /> <br /> == Problem I8 ==<br /> &lt;math&gt;997-996-995+994+993-992+991-990-989+988+987-986+\cdots+7-6-5+4+3-2+1=?&lt;/math&gt;<br /> <br /> [[1997 PMWC Problems/Problem I8|Solution]]<br /> <br /> == Problem I9 ==<br /> A chemist mixed an acid of 48% concentration with the<br /> same acid of 80% concentration, and then added 2 litres of<br /> distilled water to the mixed acid. As a result, he got 10<br /> litres of the acid of 40% concentration. How many<br /> millilitre of the acid of 48% concentration that the chemist<br /> had used? (1 litre = 1000 millilitres)<br /> <br /> [[1997 PMWC Problems/Problem I9|Solution]]<br /> <br /> == Problem I10 ==<br /> Mary took 24 chickens to the market. In the morning she<br /> sold the chickens at &lt;math&gt;\&lt;/math&gt;7 each and she only sold out less than<br /> half of them. In the afternoon she discounted the price of<br /> each chicken but the price was still an integral number in<br /> dollar. In the afternoon she could sell all the chickens, and<br /> she got totally &lt;math&gt;\&lt;/math&gt;132 for the whole day. How many<br /> chickens were sold in the morning?<br /> <br /> [[1997 PMWC Problems/Problem I10|Solution]]<br /> <br /> == Problem I11 ==<br /> A rectangle &lt;math&gt;ABCD&lt;/math&gt; is made up of five small congruent rectangles as shown in the given figure. Find the perimeter, in cm, of &lt;math&gt;ABCD&lt;/math&gt; if its area is &lt;math&gt;6750 \text{cm}^2&lt;/math&gt;. <br /> [[Image:ABCD.gif]]<br /> <br /> [[1997 PMWC Problems/Problem I11|Solution]]<br /> <br /> == Problem I12 ==<br /> In a die, 1 and 6,2 and 5,3 and 4 appear on opposite faces.<br /> When 2 dice are thrown, product of numbers appearing on<br /> the top and bottom faces of the 2 dice are formed as follows:<br /> number on top face of 1st die x number on top face of 2nd die<br /> number on top face of 1st die x number on bottom face of 2nd die<br /> number on bottom face of 1st die x number on top face of 2nd die<br /> number on bottom face of 1st die x number on bottom face of 2nd die<br /> What is the sum of these 4 products ?<br /> <br /> [[1997 PMWC Problems/Problem I12|Solution]]<br /> <br /> == Problem I13 ==<br /> A truck moved from A to B at a speed of &lt;math&gt;50&lt;/math&gt; km/h and returns from B to A at &lt;math&gt;70&lt;/math&gt; km/h. It traveled &lt;math&gt;3&lt;/math&gt; rounds within 18 hours. What is the distance between A and B?<br /> <br /> [[1997 PMWC Problems/Problem I13|Solution]]<br /> <br /> == Problem I14 ==<br /> If we make five two-digit numbers using the digits &lt;math&gt;0, 1, 2,...9&lt;/math&gt; exactly once, and the product of the five numbers is maximized, find the greatest number among them.<br /> <br /> [[1997 PMWC Problems/Problem I14|Solution]]<br /> <br /> == Problem I15 ==<br /> How many paths from A to B consist of exactly six line<br /> segments (vertical, horizontal or inclined)?<br /> [[Image:1997_PMWC-I15.png]]<br /> <br /> [[1997 PMWC Problems/Problem I15|Solution]]<br /> <br /> == Problem T1 ==<br /> Let &lt;math&gt;PQR&lt;/math&gt; be an equilateral triangle with sides of length three units. &lt;math&gt;U&lt;/math&gt;, &lt;math&gt;V&lt;/math&gt;, &lt;math&gt;W&lt;/math&gt;, &lt;math&gt;X&lt;/math&gt;, &lt;math&gt;Y&lt;/math&gt;, and &lt;math&gt;Z&lt;/math&gt; divide the sides into lengths of one unit. Find the ratio of the area of the shaded quadrilateral &lt;math&gt;UWXY&lt;/math&gt; to the area of the triangle &lt;math&gt;PQR&lt;/math&gt;.<br /> <br /> [[Image:1997 PMWC team 1.png]]<br /> <br /> [[1997 PMWC Problems/Problem T1|Solution]]<br /> <br /> == Problem T2 == <br /> Evaluate<br /> <br /> &lt;cmath&gt;\begin{eqnarray*}<br /> &amp;&amp; 1 \left(\dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right) \\<br /> &amp;+&amp; 3\left(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\<br /> &amp;+&amp;5\left(\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\<br /> &amp;+&amp;7\left(\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\<br /> &amp;+&amp;9\left(\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)+11\left(\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\<br /> &amp;+&amp;13\left(\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)+15\left(\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\<br /> &amp;+&amp;17\left(\dfrac{1}{9}+\dfrac{1}{10}\right)+19\left(\dfrac{1}{10}\right)&lt;/cmath&gt;<br /> <br /> [[1997 PMWC Problems/Problem T2|Solution]]<br /> <br /> == Problem T3 ==<br /> To type all the integers from &lt;tt&gt;1&lt;/tt&gt; to &lt;tt&gt;1997&lt;/tt&gt; using a typewriter on a piece of paper, how many does the key '&lt;tt&gt;9&lt;/tt&gt;' needed to be pressed?<br /> <br /> [[1997 PMWC Problems/Problem T3|Solution]]<br /> <br /> == Problem T4 == <br /> In one morning, a ferry traveled from Hong Kong to Kowloon and another ferry traveled from Kowloon to Hong Kong at a different speed. They started at the same time and met first time at 8:20. The two ferries then sailed to their destinations, stopped for 15 minutes and returned. The two ferries met again at 9:11. Suppose the two ferries traveled at a uniform speed throughout the whole journey, what time did the two ferries start their journey?<br /> <br /> [[1997 PMWC Problems/Problem T4|Solution]]<br /> <br /> == Problem T5 ==<br /> During recess, one of five pupils wrote something nasty on the chalkboard. When questioned by the class teacher, the following ensued:<br /> <br /> 'A': It was 'B' or 'C'<br /> <br /> 'B': Neither 'E' nor I did it.<br /> <br /> 'C': You are both lying.<br /> <br /> 'D': No, either A or B is telling the truth.<br /> <br /> 'E': No, 'D', that's not true.<br /> <br /> The class teacher knows that three of them never lie while the other two cannot be trusted. Who was the culprit?<br /> <br /> [[1997 PMWC Problems/Problem T5|Solution]]<br /> <br /> == Problem T6 ==<br /> During a rebuilding project by contractors 'A', 'B' and 'C', there was a shortage of tractors. The contractors lent each other tractors as needed. At first, 'A' lent 'B' and 'C' as many tractors as they each already had. A few months later, 'B' lent 'A' and 'C' as many as they each already had. Still later, 'C' lent 'A' and 'B' as many as they each already had. By then each contractor had 24 tractors. How many tractors did each contractor originally have?<br /> <br /> [[1997 PMWC Problems/Problem T6|Solution]]<br /> <br /> == Problem T7 == <br /> Color the surfaces of a cube of dimension 5*5*5 red, and then cut the cube into smaller cubes of dimension 1*1*1. Take out all the smaller cubes which have at least one red surface and fix a cuboid, keeping the surfaces of the cuboid red. Now what is the maximum possible volume of the cuboid?<br /> <br /> [[1997 PMWC Problems/Problem T7|Solution]]<br /> <br /> == Problem T8 ==<br /> Among the integers 1, 2, ..., 1997, what is the maximum number of integers that can be selected such that the sum of any two selected numbers is not a multiple of 7?<br /> <br /> [[1997 PMWC Problems/Problem T8|Solution]]<br /> <br /> == Problem T9 ==<br /> Find the two 10-digit numbers which become nine times as large if the order of the digits is reversed.<br /> <br /> [[1997 PMWC Problems/Problem T9|Solution]]<br /> <br /> == Problem T10 ==<br /> The twelve integers 1, 2, 3,..., 12 are arranged in a circle such that the difference of any two adjacent numbers is either 2, 3 or 4. What is the maximum number of the difference '4' can occur in any such arrangement?<br /> <br /> [[1997 PMWC Problems/Problem T10|Solution]]</div> Ihatepie https://artofproblemsolving.com/wiki/index.php?title=1997_PMWC_Problems&diff=32586 1997 PMWC Problems 2009-08-08T04:32:08Z <p>Ihatepie: /* Problem I7 */</p> <hr /> <div>== Problem I1 ==<br /> Evaluate &lt;math&gt;29 \dfrac{27}{28} \times 27 \frac{14}{15}&lt;/math&gt;<br /> <br /> [[1997 PMWC Problems/Problem I1|Solution]]<br /> <br /> == Problem I2 ==<br /> In the multiplication in the image, each letter and each box represent a single digit. Different letters represent different digits but a box can represent any digit. What does the five-digit number HAPPY stand for?<br /> <br /> [[Image:1997 PMWC individual problem 2.png]]<br /> <br /> [[1997 PMWC Problems/Problem I2|Solution]]<br /> <br /> == Problem I3 ==<br /> Peter is ill. He has to take medicine A every 8 hours,<br /> medicine B every 5 hours and medicine C every 10 hours.<br /> If he took all three medicines at 7 a.m. on Tuesday, when will he take them altogether again?<br /> <br /> [[1997 PMWC Problems/Problem I3|Solution]]<br /> <br /> == Problem I4 ==<br /> Each of the three diagrams in the image show a balance of weights using different objects. How many squares will balance a circle?<br /> <br /> [[Image:1997 PMWC individual 4.png]]<br /> <br /> [[1997 PMWC Problems/Problem I4|Solution]]<br /> <br /> == Problem I5 ==<br /> Two squares of different sizes overlap as shown in the given figure. What is the difference between the non-overlapping areas?<br /> <br /> [[Image:1997 PMWC individual 5.png]]<br /> <br /> [[1997 PMWC Problems/Problem I5|Solution]]<br /> <br /> == Problem I6 ==<br /> John and Mary went to a book shop and bought some exercise books. They had &lt;dollar/&gt;100 each. John could buy 7 large and 4 small ones. Mary could buy 5 large and 6 small ones and had &lt;dollar/&gt;5 left. How much was a small exercise book?<br /> <br /> [[1997 PMWC Problems/Problem I6|Solution]]<br /> <br /> == Problem I7 ==<br /> 40% of girls and 50% of boys in a class got an 'A'. If there<br /> are only 12 students in the class who got 'A's and the ratio of<br /> boys and girls in the class is 4:5, how many students are<br /> there in the class?<br /> <br /> [[1997 PMWC Problems/Problem I7|Solution]]<br /> <br /> == Problem I8 ==<br /> &lt;math&gt;997-996-995+994+993-992+991-990-989+988+989-986+\cdots+7-6-5+4+3-2+1=?&lt;/math&gt;<br /> <br /> [[1997 PMWC Problems/Problem I8|Solution]]<br /> <br /> == Problem I9 ==<br /> A chemist mixed an acid of 48% concentration with the<br /> same acid of 80% concentration, and then added 2 litres of<br /> distilled water to the mixed acid. As a result, he got 10<br /> litres of the acid of 40% concentration. How many<br /> millilitre of the acid of 48% concentration that the chemist<br /> had used? (1 litre = 1000 millilitres)<br /> <br /> [[1997 PMWC Problems/Problem I9|Solution]]<br /> <br /> == Problem I10 ==<br /> Mary took 24 chickens to the market. In the morning she<br /> sold the chickens at &lt;math&gt;\&lt;/math&gt;7 each and she only sold out less than<br /> half of them. In the afternoon she discounted the price of<br /> each chicken but the price was still an integral number in<br /> dollar. In the afternoon she could sell all the chickens, and<br /> she got totally &lt;math&gt;\&lt;/math&gt;132 for the whole day. How many<br /> chickens were sold in the morning?<br /> <br /> [[1997 PMWC Problems/Problem I10|Solution]]<br /> <br /> == Problem I11 ==<br /> A rectangle &lt;math&gt;ABCD&lt;/math&gt; is made up of five small congruent rectangles as shown in the given figure. Find the perimeter, in cm, of &lt;math&gt;ABCD&lt;/math&gt; if its area is &lt;math&gt;6750 \text{cm}^2&lt;/math&gt;. <br /> [[Image:ABCD.gif]]<br /> <br /> [[1997 PMWC Problems/Problem I11|Solution]]<br /> <br /> == Problem I12 ==<br /> In a die, 1 and 6,2 and 5,3 and 4 appear on opposite faces.<br /> When 2 dice are thrown, product of numbers appearing on<br /> the top and bottom faces of the 2 dice are formed as follows:<br /> number on top face of 1st die x number on top face of 2nd die<br /> number on top face of 1st die x number on bottom face of 2nd die<br /> number on bottom face of 1st die x number on top face of 2nd die<br /> number on bottom face of 1st die x number on bottom face of 2nd die<br /> What is the sum of these 4 products ?<br /> <br /> [[1997 PMWC Problems/Problem I12|Solution]]<br /> <br /> == Problem I13 ==<br /> A truck moved from A to B at a speed of &lt;math&gt;50&lt;/math&gt; km/h and returns from B to A at &lt;math&gt;70&lt;/math&gt; km/h. It traveled &lt;math&gt;3&lt;/math&gt; rounds within 18 hours. What is the distance between A and B?<br /> <br /> [[1997 PMWC Problems/Problem I13|Solution]]<br /> <br /> == Problem I14 ==<br /> If we make five two-digit numbers using the digits &lt;math&gt;0, 1, 2,...9&lt;/math&gt; exactly once, and the product of the five numbers is maximized, find the greatest number among them.<br /> <br /> [[1997 PMWC Problems/Problem I14|Solution]]<br /> <br /> == Problem I15 ==<br /> How many paths from A to B consist of exactly six line<br /> segments (vertical, horizontal or inclined)?<br /> [[Image:1997_PMWC-I15.png]]<br /> <br /> [[1997 PMWC Problems/Problem I15|Solution]]<br /> <br /> == Problem T1 ==<br /> Let &lt;math&gt;PQR&lt;/math&gt; be an equilateral triangle with sides of length three units. &lt;math&gt;U&lt;/math&gt;, &lt;math&gt;V&lt;/math&gt;, &lt;math&gt;W&lt;/math&gt;, &lt;math&gt;X&lt;/math&gt;, &lt;math&gt;Y&lt;/math&gt;, and &lt;math&gt;Z&lt;/math&gt; divide the sides into lengths of one unit. Find the ratio of the area of the shaded quadrilateral &lt;math&gt;UWXY&lt;/math&gt; to the area of the triangle &lt;math&gt;PQR&lt;/math&gt;.<br /> <br /> [[Image:1997 PMWC team 1.png]]<br /> <br /> [[1997 PMWC Problems/Problem T1|Solution]]<br /> <br /> == Problem T2 == <br /> Evaluate<br /> <br /> &lt;cmath&gt;\begin{eqnarray*}<br /> &amp;&amp; 1 \left(\dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right) \\<br /> &amp;+&amp; 3\left(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\<br /> &amp;+&amp;5\left(\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\<br /> &amp;+&amp;7\left(\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\<br /> &amp;+&amp;9\left(\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)+11\left(\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\<br /> &amp;+&amp;13\left(\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)+15\left(\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\<br /> &amp;+&amp;17\left(\dfrac{1}{9}+\dfrac{1}{10}\right)+19\left(\dfrac{1}{10}\right)&lt;/cmath&gt;<br /> <br /> [[1997 PMWC Problems/Problem T2|Solution]]<br /> <br /> == Problem T3 ==<br /> To type all the integers from &lt;tt&gt;1&lt;/tt&gt; to &lt;tt&gt;1997&lt;/tt&gt; using a typewriter on a piece of paper, how many does the key '&lt;tt&gt;9&lt;/tt&gt;' needed to be pressed?<br /> <br /> [[1997 PMWC Problems/Problem T3|Solution]]<br /> <br /> == Problem T4 == <br /> In one morning, a ferry traveled from Hong Kong to Kowloon and another ferry traveled from Kowloon to Hong Kong at a different speed. They started at the same time and met first time at 8:20. The two ferries then sailed to their destinations, stopped for 15 minutes and returned. The two ferries met again at 9:11. Suppose the two ferries traveled at a uniform speed throughout the whole journey, what time did the two ferries start their journey?<br /> <br /> [[1997 PMWC Problems/Problem T4|Solution]]<br /> <br /> == Problem T5 ==<br /> During recess, one of five pupils wrote something nasty on the chalkboard. When questioned by the class teacher, the following ensued:<br /> <br /> 'A': It was 'B' or 'C'<br /> <br /> 'B': Neither 'E' nor I did it.<br /> <br /> 'C': You are both lying.<br /> <br /> 'D': No, either A or B is telling the truth.<br /> <br /> 'E': No, 'D', that's not true.<br /> <br /> The class teacher knows that three of them never lie while the other two cannot be trusted. Who was the culprit?<br /> <br /> [[1997 PMWC Problems/Problem T5|Solution]]<br /> <br /> == Problem T6 ==<br /> During a rebuilding project by contractors 'A', 'B' and 'C', there was a shortage of tractors. The contractors lent each other tractors as needed. At first, 'A' lent 'B' and 'C' as many tractors as they each already had. A few months later, 'B' lent 'A' and 'C' as many as they each already had. Still later, 'C' lent 'A' and 'B' as many as they each already had. By then each contractor had 24 tractors. How many tractors did each contractor originally have?<br /> <br /> [[1997 PMWC Problems/Problem T6|Solution]]<br /> <br /> == Problem T7 == <br /> Color the surfaces of a cube of dimension 5*5*5 red, and then cut the cube into smaller cubes of dimension 1*1*1. Take out all the smaller cubes which have at least one red surface and fix a cuboid, keeping the surfaces of the cuboid red. Now what is the maximum possible volume of the cuboid?<br /> <br /> [[1997 PMWC Problems/Problem T7|Solution]]<br /> <br /> == Problem T8 ==<br /> Among the integers 1, 2, ..., 1997, what is the maximum number of integers that can be selected such that the sum of any two selected numbers is not a multiple of 7?<br /> <br /> [[1997 PMWC Problems/Problem T8|Solution]]<br /> <br /> == Problem T9 ==<br /> Find the two 10-digit numbers which become nine times as large if the order of the digits is reversed.<br /> <br /> [[1997 PMWC Problems/Problem T9|Solution]]<br /> <br /> == Problem T10 ==<br /> The twelve integers 1, 2, 3,..., 12 are arranged in a circle such that the difference of any two adjacent numbers is either 2, 3 or 4. What is the maximum number of the difference '4' can occur in any such arrangement?<br /> <br /> [[1997 PMWC Problems/Problem T10|Solution]]</div> Ihatepie https://artofproblemsolving.com/wiki/index.php?title=1990_AIME_Problems&diff=32566 1990 AIME Problems 2009-08-06T04:10:18Z <p>Ihatepie: /* Problem 3 */</p> <hr /> <div>{{AIME Problems|year=1990}}<br /> <br /> == Problem 1 ==<br /> The [[increasing sequence]] &lt;math&gt;2,3,5,6,7,10,11,\ldots&lt;/math&gt; consists of all [[positive integer]]s that are neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence.<br /> <br /> [[1990 AIME Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> Find the value of &lt;math&gt;(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}&lt;/math&gt;.<br /> <br /> [[1990 AIME Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> Let &lt;math&gt;P_1^{}&lt;/math&gt; be a regular &lt;math&gt;r~\mbox{gon}&lt;/math&gt; and &lt;math&gt;P_2^{}&lt;/math&gt; be a regular &lt;math&gt;s~\mbox{gon}&lt;/math&gt; &lt;math&gt;(r\geq s\geq 3)&lt;/math&gt; such that each interior angle of &lt;math&gt;P_1^{}&lt;/math&gt; is &lt;math&gt;\frac{59}{58}&lt;/math&gt; as large as each interior angle of &lt;math&gt;P_2^{}&lt;/math&gt;. What's the largest possible value of &lt;math&gt;s_{}^{}&lt;/math&gt;?<br /> <br /> [[1990 AIME Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> Find the positive solution to<br /> &lt;center&gt;&lt;math&gt;\frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0&lt;/math&gt;&lt;/center&gt;<br /> <br /> [[1990 AIME Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> Let &lt;math&gt;n^{}_{}&lt;/math&gt; be the smallest positive integer that is a multiple of &lt;math&gt;75_{}^{}&lt;/math&gt; and has exactly &lt;math&gt;75_{}^{}&lt;/math&gt; positive integral divisors, including &lt;math&gt;1_{}^{}&lt;/math&gt; and itself. Find &lt;math&gt;n/75^{}_{}&lt;/math&gt;.<br /> <br /> [[1990 AIME Problems/Problem 5|Solution]]<br /> <br /> == Problem 6 ==<br /> A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that 25% of these fish are no longer in the lake on September 1 (because of death and emigrations), that 40% of the fish were not in the lake May 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the September 1 sample are representative of the total population. What does the biologist calculate for the number of fish in the lake on May 1? <br /> <br /> [[1990 AIME Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> A triangle has vertices &lt;math&gt;P_{}^{}=(-8,5)&lt;/math&gt;, &lt;math&gt;Q_{}^{}=(-15,-19)&lt;/math&gt;, and &lt;math&gt;R_{}^{}=(1,-7)&lt;/math&gt;. The equation of the bisector of &lt;math&gt;\angle P&lt;/math&gt; can be written in the form &lt;math&gt;ax+2y+c=0_{}^{}&lt;/math&gt;. Find &lt;math&gt;a+c_{}^{}&lt;/math&gt;.<br /> <br /> [[1990 AIME Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> In a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules: <br /> <br /> 1) The marksman first chooses a column from which a target is to be broken. <br /> <br /> 2) The marksman must then break the lowest remaining target in the chosen column. <br /> <br /> If the rules are followed, in how many different orders can the eight targets be broken? <br /> <br /> [[1990 AIME Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> A fair coin is to be tossed &lt;math&gt;10_{}^{}&lt;/math&gt; times. Let &lt;math&gt;i/j^{}_{}&lt;/math&gt;, in lowest terms, be the probability that heads never occur on consecutive tosses. Find &lt;math&gt;i+j_{}^{}&lt;/math&gt;. <br /> <br /> [[1990 AIME Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> The sets &lt;math&gt;A = \{z : z^{18} = 1\}&lt;/math&gt; and &lt;math&gt;B = \{w : w^{48} = 1\}&lt;/math&gt; are both sets of complex roots of unity. The set &lt;math&gt;C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}&lt;/math&gt; is also a set of complex roots of unity. How many distinct elements are in &lt;math&gt;C^{}_{}&lt;/math&gt;?<br /> <br /> [[1990 AIME Problems/Problem 10|Solution]]<br /> <br /> == Problem 11 ==<br /> Someone observed that &lt;math&gt;6! = 8 \cdot 9 \cdot 10&lt;/math&gt;. Find the largest positive integer &lt;math&gt;n^{}_{}&lt;/math&gt; for which &lt;math&gt;n^{}_{}!&lt;/math&gt; can be expressed as the product of &lt;math&gt;n - 3_{}^{}&lt;/math&gt; consecutive positive integers.<br /> <br /> [[1990 AIME Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form<br /> &lt;center&gt;&lt;math&gt;a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},&lt;/math&gt;&lt;/center&gt;<br /> where &lt;math&gt;a^{}_{}&lt;/math&gt;, &lt;math&gt;b^{}_{}&lt;/math&gt;, &lt;math&gt;c^{}_{}&lt;/math&gt;, and &lt;math&gt;d^{}_{}&lt;/math&gt; are positive integers. Find &lt;math&gt;a + b + c + d^{}_{}&lt;/math&gt;.<br /> <br /> [[1990 AIME Problems/Problem 12|Solution]]<br /> <br /> == Problem 13 ==<br /> Let &lt;math&gt;T = \{9^k : k ~ \mbox{is an integer}, 0 \le k \le 4000\}&lt;/math&gt;. Given that &lt;math&gt;9^{4000}_{}&lt;/math&gt; has 3817 digits and that its first (leftmost) digit is 9, how many elements of &lt;math&gt;T_{}^{}&lt;/math&gt; have 9 as their leftmost digit?<br /> <br /> [[1990 AIME Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> The rectangle &lt;math&gt;ABCD^{}_{}&lt;/math&gt; below has dimensions &lt;math&gt;AB^{}_{} = 12 \sqrt{3}&lt;/math&gt; and &lt;math&gt;BC^{}_{} = 13 \sqrt{3}&lt;/math&gt;. Diagonals &lt;math&gt;\overline{AC}&lt;/math&gt; and &lt;math&gt;\overline{BD}&lt;/math&gt; intersect at &lt;math&gt;P^{}_{}&lt;/math&gt;. If triangle &lt;math&gt;ABP^{}_{}&lt;/math&gt; is cut out and removed, edges &lt;math&gt;\overline{AP}&lt;/math&gt; and &lt;math&gt;\overline{BP}&lt;/math&gt; are joined, and the figure is then creased along segments &lt;math&gt;\overline{CP}&lt;/math&gt; and &lt;math&gt;\overline{DP}&lt;/math&gt;, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid.<br /> <br /> [[Image:AIME_1990_Problem_14.png]]<br /> <br /> [[1990 AIME Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> Find &lt;math&gt;a_{}^{}x^5 + b_{}y^5&lt;/math&gt; if the real numbers &lt;math&gt;a_{}^{}&lt;/math&gt;, &lt;math&gt;b_{}^{}&lt;/math&gt;, &lt;math&gt;x_{}^{}&lt;/math&gt;, and &lt;math&gt;y_{}^{}&lt;/math&gt; satisfy the equations<br /> &lt;cmath&gt;ax + by = 3^{}_{},&lt;/cmath&gt;<br /> &lt;cmath&gt;ax^2 + by^2 = 7^{}_{},&lt;/cmath&gt;<br /> &lt;cmath&gt;ax^3 + by^3 = 16^{}_{},&lt;/cmath&gt;<br /> &lt;cmath&gt;ax^4 + by^4 = 42^{}_{}.&lt;/cmath&gt;<br /> <br /> <br /> [[1990 AIME Problems/Problem 15|Solution]]<br /> <br /> == See also ==<br /> * [[American Invitational Mathematics Examination]]<br /> * [[AIME Problems and Solutions]]<br /> * [[Mathematics competition resources]]<br /> <br /> [[Category:AIME Problems]]</div> Ihatepie https://artofproblemsolving.com/wiki/index.php?title=1990_AIME_Problems&diff=32565 1990 AIME Problems 2009-08-06T04:04:50Z <p>Ihatepie: /* Problem 3 */</p> <hr /> <div>{{AIME Problems|year=1990}}<br /> <br /> == Problem 1 ==<br /> The [[increasing sequence]] &lt;math&gt;2,3,5,6,7,10,11,\ldots&lt;/math&gt; consists of all [[positive integer]]s that are neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence.<br /> <br /> [[1990 AIME Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> Find the value of &lt;math&gt;(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}&lt;/math&gt;.<br /> <br /> [[1990 AIME Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> Let &lt;math&gt;P_1^{}&lt;/math&gt; be a regular &lt;math&gt;n~\mbox{gon}&lt;/math&gt; and &lt;math&gt;P_2^{}&lt;/math&gt; be a regular &lt;math&gt;s~\mbox{gon}&lt;/math&gt; &lt;math&gt;(n\geq s\geq 3)&lt;/math&gt; such that each interior angle of &lt;math&gt;P_1^{}&lt;/math&gt; is &lt;math&gt;\frac{59}{58}&lt;/math&gt; as large as each interior angle of &lt;math&gt;P_2^{}&lt;/math&gt;. What's the largest possible value of &lt;math&gt;s_{}^{}&lt;/math&gt;?<br /> <br /> [[1990 AIME Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> Find the positive solution to<br /> &lt;center&gt;&lt;math&gt;\frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0&lt;/math&gt;&lt;/center&gt;<br /> <br /> [[1990 AIME Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> Let &lt;math&gt;n^{}_{}&lt;/math&gt; be the smallest positive integer that is a multiple of &lt;math&gt;75_{}^{}&lt;/math&gt; and has exactly &lt;math&gt;75_{}^{}&lt;/math&gt; positive integral divisors, including &lt;math&gt;1_{}^{}&lt;/math&gt; and itself. Find &lt;math&gt;n/75^{}_{}&lt;/math&gt;.<br /> <br /> [[1990 AIME Problems/Problem 5|Solution]]<br /> <br /> == Problem 6 ==<br /> A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that 25% of these fish are no longer in the lake on September 1 (because of death and emigrations), that 40% of the fish were not in the lake May 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the September 1 sample are representative of the total population. What does the biologist calculate for the number of fish in the lake on May 1? <br /> <br /> [[1990 AIME Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> A triangle has vertices &lt;math&gt;P_{}^{}=(-8,5)&lt;/math&gt;, &lt;math&gt;Q_{}^{}=(-15,-19)&lt;/math&gt;, and &lt;math&gt;R_{}^{}=(1,-7)&lt;/math&gt;. The equation of the bisector of &lt;math&gt;\angle P&lt;/math&gt; can be written in the form &lt;math&gt;ax+2y+c=0_{}^{}&lt;/math&gt;. Find &lt;math&gt;a+c_{}^{}&lt;/math&gt;.<br /> <br /> [[1990 AIME Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> In a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules: <br /> <br /> 1) The marksman first chooses a column from which a target is to be broken. <br /> <br /> 2) The marksman must then break the lowest remaining target in the chosen column. <br /> <br /> If the rules are followed, in how many different orders can the eight targets be broken? <br /> <br /> [[1990 AIME Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> A fair coin is to be tossed &lt;math&gt;10_{}^{}&lt;/math&gt; times. Let &lt;math&gt;i/j^{}_{}&lt;/math&gt;, in lowest terms, be the probability that heads never occur on consecutive tosses. Find &lt;math&gt;i+j_{}^{}&lt;/math&gt;. <br /> <br /> [[1990 AIME Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> The sets &lt;math&gt;A = \{z : z^{18} = 1\}&lt;/math&gt; and &lt;math&gt;B = \{w : w^{48} = 1\}&lt;/math&gt; are both sets of complex roots of unity. The set &lt;math&gt;C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}&lt;/math&gt; is also a set of complex roots of unity. How many distinct elements are in &lt;math&gt;C^{}_{}&lt;/math&gt;?<br /> <br /> [[1990 AIME Problems/Problem 10|Solution]]<br /> <br /> == Problem 11 ==<br /> Someone observed that &lt;math&gt;6! = 8 \cdot 9 \cdot 10&lt;/math&gt;. Find the largest positive integer &lt;math&gt;n^{}_{}&lt;/math&gt; for which &lt;math&gt;n^{}_{}!&lt;/math&gt; can be expressed as the product of &lt;math&gt;n - 3_{}^{}&lt;/math&gt; consecutive positive integers.<br /> <br /> [[1990 AIME Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form<br /> &lt;center&gt;&lt;math&gt;a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},&lt;/math&gt;&lt;/center&gt;<br /> where &lt;math&gt;a^{}_{}&lt;/math&gt;, &lt;math&gt;b^{}_{}&lt;/math&gt;, &lt;math&gt;c^{}_{}&lt;/math&gt;, and &lt;math&gt;d^{}_{}&lt;/math&gt; are positive integers. Find &lt;math&gt;a + b + c + d^{}_{}&lt;/math&gt;.<br /> <br /> [[1990 AIME Problems/Problem 12|Solution]]<br /> <br /> == Problem 13 ==<br /> Let &lt;math&gt;T = \{9^k : k ~ \mbox{is an integer}, 0 \le k \le 4000\}&lt;/math&gt;. Given that &lt;math&gt;9^{4000}_{}&lt;/math&gt; has 3817 digits and that its first (leftmost) digit is 9, how many elements of &lt;math&gt;T_{}^{}&lt;/math&gt; have 9 as their leftmost digit?<br /> <br /> [[1990 AIME Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> The rectangle &lt;math&gt;ABCD^{}_{}&lt;/math&gt; below has dimensions &lt;math&gt;AB^{}_{} = 12 \sqrt{3}&lt;/math&gt; and &lt;math&gt;BC^{}_{} = 13 \sqrt{3}&lt;/math&gt;. Diagonals &lt;math&gt;\overline{AC}&lt;/math&gt; and &lt;math&gt;\overline{BD}&lt;/math&gt; intersect at &lt;math&gt;P^{}_{}&lt;/math&gt;. If triangle &lt;math&gt;ABP^{}_{}&lt;/math&gt; is cut out and removed, edges &lt;math&gt;\overline{AP}&lt;/math&gt; and &lt;math&gt;\overline{BP}&lt;/math&gt; are joined, and the figure is then creased along segments &lt;math&gt;\overline{CP}&lt;/math&gt; and &lt;math&gt;\overline{DP}&lt;/math&gt;, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid.<br /> <br /> [[Image:AIME_1990_Problem_14.png]]<br /> <br /> [[1990 AIME Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> Find &lt;math&gt;a_{}^{}x^5 + b_{}y^5&lt;/math&gt; if the real numbers &lt;math&gt;a_{}^{}&lt;/math&gt;, &lt;math&gt;b_{}^{}&lt;/math&gt;, &lt;math&gt;x_{}^{}&lt;/math&gt;, and &lt;math&gt;y_{}^{}&lt;/math&gt; satisfy the equations<br /> &lt;cmath&gt;ax + by = 3^{}_{},&lt;/cmath&gt;<br /> &lt;cmath&gt;ax^2 + by^2 = 7^{}_{},&lt;/cmath&gt;<br /> &lt;cmath&gt;ax^3 + by^3 = 16^{}_{},&lt;/cmath&gt;<br /> &lt;cmath&gt;ax^4 + by^4 = 42^{}_{}.&lt;/cmath&gt;<br /> <br /> <br /> [[1990 AIME Problems/Problem 15|Solution]]<br /> <br /> == See also ==<br /> * [[American Invitational Mathematics Examination]]<br /> * [[AIME Problems and Solutions]]<br /> * [[Mathematics competition resources]]<br /> <br /> [[Category:AIME Problems]]</div> Ihatepie https://artofproblemsolving.com/wiki/index.php?title=Euclidean_algorithm&diff=32268 Euclidean algorithm 2009-07-08T21:29:56Z <p>Ihatepie: /* Main idea and Informal Description */</p> <hr /> <div>The '''Euclidean algorithm''' (also known as the '''Euclidean division algorithm''' or '''Euclid's algorithm''') is an algorithm that finds the [[greatest common divisor]] (GCD) of two elements of a [[Euclidean domain]], the most common of which is the [[nonnegative]] [[integer]]s &lt;math&gt;\mathbb{Z}{\geq 0}&lt;/math&gt;, without [[factoring]] them.<br /> <br /> ==Main idea and Informal Description==<br /> The basic idea is to repeatedly use the fact that &lt;math&gt;\gcd({a,b}) \equiv \gcd({b,a - b})&lt;/math&gt;<br /> <br /> If we have two non-negative integers &lt;math&gt;a,b&lt;/math&gt; with &lt;math&gt;a\ge b&lt;/math&gt; and &lt;math&gt;b=0&lt;/math&gt;, then the greatest common divisor is &lt;math&gt;{a}&lt;/math&gt;. If &lt;math&gt;a\ge b&gt;0&lt;/math&gt;, then the set of common divisors of &lt;math&gt;{a}&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; is the same as the set of common divisors of &lt;math&gt;b&lt;/math&gt; and &lt;math&gt;r&lt;/math&gt; where &lt;math&gt;r&lt;/math&gt; is the [[remainder]] of division of &lt;math&gt;{a}&lt;/math&gt; by &lt;math&gt;b&lt;/math&gt;. Indeed, we have &lt;math&gt;a=mb+r&lt;/math&gt; with some integer&lt;math&gt;m&lt;/math&gt;, so, if &lt;math&gt;{d}&lt;/math&gt; divides both &lt;math&gt;{a}&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt;, it must divide both &lt;math&gt;{a}&lt;/math&gt; and &lt;math&gt;mb&lt;/math&gt; and, thereby, their difference &lt;math&gt;r&lt;/math&gt;. Similarly, if &lt;math&gt;{d}&lt;/math&gt; divides both &lt;math&gt;b&lt;/math&gt; and &lt;math&gt;r&lt;/math&gt;, it should divide &lt;math&gt;{a}&lt;/math&gt; as well. Thus, the greatest common divisors of &lt;math&gt;{a}&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; and of &lt;math&gt;b&lt;/math&gt; and &lt;math&gt;r&lt;/math&gt; coincide: &lt;math&gt;GCD(a,b)=GCD(b,r)&lt;/math&gt;. But the pair &lt;math&gt;(b,r)&lt;/math&gt; consists of smaller numbers than the pair &lt;math&gt;(a,b)&lt;/math&gt;! So, we reduced our task to a simpler one. And we can do this reduction again and again until the smaller number becomes &lt;math&gt;0&lt;/math&gt;<br /> <br /> == General Form ==<br /> Start with any two elements &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; of a [[Euclidean Domain]]<br /> <br /> * If &lt;math&gt;b=0&lt;/math&gt;, then &lt;math&gt;\gcd(a,b)=a&lt;/math&gt;.<br /> * Otherwise take the remainder when &lt;math&gt;{a}&lt;/math&gt; is divided by &lt;math&gt;a \pmod{b}&lt;/math&gt;, and find &lt;math&gt;\gcd(a,a \bmod {b})&lt;/math&gt;.<br /> * Repeat this until the remainder is 0.&lt;br&gt;<br /> <br /> &lt;math&gt;a \pmod{b} \equiv r_1&lt;/math&gt;&lt;br&gt;<br /> &lt;math&gt;b (\bmod r_1) \equiv r_2&lt;/math&gt;&lt;br&gt;<br /> &lt;math&gt; \vdots&lt;/math&gt; &lt;br&gt;<br /> &lt;math&gt;r_{n-1} (\bmod r_n) \equiv 0&lt;/math&gt;&lt;br&gt;<br /> Then &lt;math&gt;\gcd({a,b}) = r_n&lt;/math&gt;&lt;br&gt;<br /> <br /> Usually the Euclidean algorithm is written down just as a chain of divisions with remainder:<br /> <br /> for &lt;math&gt;r_{k+1} &lt; r_k &lt; r_{k-1}&lt;/math&gt;&lt;br&gt;<br /> &lt;math&gt;a = b \cdot q_1+r_1&lt;/math&gt;&lt;br&gt;<br /> &lt;math&gt;b = r_1 \cdot q_2 + r_2&lt;/math&gt;&lt;br&gt;<br /> &lt;math&gt;r_1 = r_2 \cdot q_3 + r_3&lt;/math&gt;&lt;br&gt;<br /> &lt;math&gt;\vdots&lt;/math&gt;&lt;br&gt;<br /> &lt;math&gt;r_{n-1} = r_n \cdot q_{n+2} +0&lt;/math&gt;&lt;br&gt;<br /> and so &lt;math&gt;\gcd({a,b}) = r_n&lt;/math&gt;&lt;br&gt;<br /> <br /> == Simple Example ==<br /> To see how it works, just take an example. Say &lt;math&gt;a=112,b=42&lt;/math&gt;. We have &lt;math&gt;112\equiv 28\pmod {42}&lt;/math&gt;, so &lt;math&gt;{\gcd(112,42)}=\gcd(42,28)&lt;/math&gt;. Similarly, &lt;math&gt;42\equiv 14\pmod {28}&lt;/math&gt;, so &lt;math&gt;\gcd(42,28)=\gcd(28,14)&lt;/math&gt;. Then &lt;math&gt;28\equiv {0}\pmod {14}&lt;/math&gt;, so &lt;math&gt;{\gcd(28,14)}={\gcd(14,0)} = 14&lt;/math&gt;. Thus &lt;math&gt;\gcd(112,42)=14&lt;/math&gt;.<br /> <br /> * &lt;math&gt;{112 = 2 \cdot 42 + 28 \qquad (1)}&lt;/math&gt;<br /> * &lt;math&gt;42 = 1\cdot 28+14\qquad (2)&lt;/math&gt;<br /> * &lt;math&gt;28 = 2\cdot 14+0\qquad (3)&lt;/math&gt;<br /> <br /> == Linear Representation ==<br /> An added bonus of the Euclidean algorithm is the &quot;linear representation&quot; of the greatest common divisor. This allows us to write &lt;math&gt;\gcd(a,b)=ax+by&lt;/math&gt;, where &lt;math&gt;x,y&lt;/math&gt; are some elements from the same [[Euclidean Domain]] as &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; that can be determined using the algorithm. We can work backwards from whichever step is the most convenient.<br /> <br /> In the previous example, we can work backwards from equation &lt;math&gt;(2)&lt;/math&gt;:<br /> <br /> &lt;math&gt;14 = 42-1\cdot 28&lt;/math&gt;&lt;br&gt;<br /> &lt;math&gt;14 = 42-1\cdot (112-2\cdot 42)&lt;/math&gt;&lt;br&gt;<br /> &lt;math&gt;14 = 3\cdot 42-1\cdot 112.&lt;/math&gt;&lt;br&gt;<br /> <br /> == Problems ==<br /> ===Introductory===<br /> ===Intermediate===<br /> * [[1985 AIME Problems/Problem 13]]<br /> ===Olympiad===<br /> * [[1959 IMO Problems/Problem 1]]<br /> <br /> ==See Also==<br /> *[[Divisibility]]<br /> <br /> [[Category:Algorithms]]<br /> [[Category:Number theory]]</div> Ihatepie https://artofproblemsolving.com/wiki/index.php?title=User:USAMO13&diff=31531 User:USAMO13 2009-05-04T04:14:10Z <p>Ihatepie: Removing all content from page</p> <hr /> <div></div> Ihatepie https://artofproblemsolving.com/wiki/index.php?title=1966_AHSME_Problems/Problem_4&diff=30898 1966 AHSME Problems/Problem 4 2009-03-21T04:26:52Z <p>Ihatepie: New page: Make half of the square's side x. Now the radius of the smaller circle is x, so it's area is pi*x^2 Now find the diameter of the bigger circle. Since half of the square's side is x, the f...</p> <hr /> <div>Make half of the square's side x. Now the radius of the smaller circle is x, so it's area is pi*x^2<br /> <br /> Now find the diameter of the bigger circle. Since half of the square's side is x, the full side is 2x. Using the Pythagorean theorem, you get the diagonal to be 2sqrt2*x. Half of that is the radius, or xsqrt2. Using the same equation as before, you get the area of the larger circle to be 2x^2*pi. Putting one over the other and dividing, you get two as the answer: or<br /> <br /> (B)</div> Ihatepie