https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Mathlete-of-the-century&feedformat=atom AoPS Wiki - User contributions [en] 2022-05-17T18:48:55Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_10B_Problems/Problem_22&diff=99969 2003 AMC 10B Problems/Problem 22 2019-01-02T22:05:59Z <p>Mathlete-of-the-century: /* Solution */</p> <hr /> <div>==Problem==<br /> <br /> A clock chimes once at &lt;math&gt;30&lt;/math&gt; minutes past each hour and chimes on the hour according to the hour. For example, at &lt;math&gt;1 \text{PM}&lt;/math&gt; there is one chime and at noon and midnight there are twelve chimes. Starting at &lt;math&gt;11:15 \text{AM}&lt;/math&gt; on &lt;math&gt;\text{February 26, 2003},&lt;/math&gt; on what date will the &lt;math&gt;2003^{\text{rd}}&lt;/math&gt; chime occur?<br /> <br /> &lt;math&gt;\textbf{(A) } \text{March 8} \qquad\textbf{(B) } \text{March 9} \qquad\textbf{(C) } \text{March 10} \qquad\textbf{(D) } \text{March 20} \qquad\textbf{(E) } \text{March 21}&lt;/math&gt;<br /> <br /> ==Solution==<br /> First, find how many chimes will have already happened before midnight (the beginning of the day) of &lt;math&gt;\text{February 27, 2003}.&lt;/math&gt; &lt;math&gt;13&lt;/math&gt; half-hours have passed, and the number of chimes according to the hour is &lt;math&gt;1+2+3+\cdots+12.&lt;/math&gt; The total number of chimes is &lt;math&gt;13+78=91&lt;/math&gt;<br /> <br /> Every day, there will be &lt;math&gt;24&lt;/math&gt; half-hours and &lt;math&gt;2(1+2+3+\cdots+12)&lt;/math&gt; chimes according to the arrow, resulting in &lt;math&gt;24+156=180&lt;/math&gt; total chimes.<br /> <br /> On &lt;math&gt;\text{February 27},&lt;/math&gt; the number of chimes that still need to occur is &lt;math&gt;2003-91=1912.&lt;/math&gt; &lt;math&gt;1912 \div 180=10 \text{R}112.&lt;/math&gt; Rounding up, it is &lt;math&gt;11&lt;/math&gt; days past &lt;math&gt;\text{February 27},&lt;/math&gt; which is &lt;math&gt;\boxed{\textbf{(C) \ } \text{March 10}}&lt;/math&gt;<br /> <br /> ==See Also==<br /> {{AMC10 box|year=2003|ab=B|num-b=21|num-a=23}}<br /> {{MAA Notice}}</div> Mathlete-of-the-century