https://artofproblemsolving.com/wiki/index.php?title=2013_Mock_AIME_I_Problems&feed=atom&action=history 2013 Mock AIME I Problems - Revision history 2021-10-24T16:53:34Z Revision history for this page on the wiki MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2013_Mock_AIME_I_Problems&diff=139723&oldid=prev Sugar rush: Created page with "== Problem 1 == Two circles $C_1$ and $C_2$, each of unit radius, have centers $A_1$ and $A_2$ such that $A_1A_2=6$. Le..." 2020-12-15T21:16:48Z <p>Created page with &quot;== Problem 1 == Two circles &lt;math&gt;C_1&lt;/math&gt; and &lt;math&gt;C_2&lt;/math&gt;, each of unit radius, have centers &lt;math&gt;A_1&lt;/math&gt; and &lt;math&gt;A_2&lt;/math&gt; such that &lt;math&gt;A_1A_2=6&lt;/math&gt;. Le...&quot;</p> <p><b>New page</b></p><div>== Problem 1 ==<br /> Two circles &lt;math&gt;C_1&lt;/math&gt; and &lt;math&gt;C_2&lt;/math&gt;, each of unit radius, have centers &lt;math&gt;A_1&lt;/math&gt; and &lt;math&gt;A_2&lt;/math&gt; such that &lt;math&gt;A_1A_2=6&lt;/math&gt;. Let &lt;math&gt;P&lt;/math&gt; be the midpoint of &lt;math&gt;A_1A_2&lt;/math&gt; and let &lt;math&gt;C_3&lt;/math&gt; be a circle externally tangent to both &lt;math&gt;C_1&lt;/math&gt; and &lt;math&gt;C_2&lt;/math&gt;. &lt;math&gt;C_1&lt;/math&gt; and &lt;math&gt;C_3&lt;/math&gt; have a common tangent that passes through &lt;math&gt;P&lt;/math&gt;. If this tangent is also a common tangent to &lt;math&gt;C_2&lt;/math&gt; and &lt;math&gt;C_1&lt;/math&gt;, find the radius of circle &lt;math&gt;C_3&lt;/math&gt;.<br /> <br /> [[2013 Mock AIME I Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> Find the number of ordered positive integer triplets &lt;math&gt;(a,b,c)&lt;/math&gt; such that &lt;math&gt;a&lt;/math&gt; evenly divides &lt;math&gt;b&lt;/math&gt;, &lt;math&gt;b+1&lt;/math&gt; evenly divides &lt;math&gt;c&lt;/math&gt;, and &lt;math&gt;c-a=10&lt;/math&gt;.<br /> <br /> [[2013 Mock AIME I Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> Let &lt;math&gt;\lfloor x\rfloor&lt;/math&gt; be the greatest integer less than or equal to &lt;math&gt;x&lt;/math&gt;, and let &lt;math&gt;\{x\}=x-\lfloor x\rfloor&lt;/math&gt;. If &lt;math&gt;x=(7+4\sqrt{3})^{2^{2013}}&lt;/math&gt;, compute &lt;math&gt;x\left(1-\{x\}\right)&lt;/math&gt;.<br /> <br /> [[2013 Mock AIME I Problems/Problem 3|Solution]]<br /> <br /> <br /> == Problem 4 ==<br /> Compute the number of ways to fill in the following magic square such that:<br /> <br /> 1. the product of all rows, columns, and diagonals are equal (the sum condition is waived),<br /> <br /> 2. all entries are ''nonnegative'' integers less than or equal to ten, and<br /> <br /> 3. entries CAN repeat in a column, row, or diagonal.<br /> <br /> &lt;asy&gt;<br /> size(100);<br /> defaultpen(linewidth(0.7));<br /> int i;<br /> for(i=0; i&lt;4; i=i+1) {<br /> draw((0,2*i)--(6,2*i)^^(2*i,0)--(2*i,6));<br /> }<br /> label(&quot;$1$&quot;, (1,5));<br /> label(&quot;$9$&quot;, (3,5));<br /> label(&quot;$3$&quot;, (1,1));<br /> &lt;/asy&gt;<br /> <br /> [[2013 Mock AIME I Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> In quadrilateral &lt;math&gt;ABCD&lt;/math&gt;, &lt;math&gt;AC\cap BD=M&lt;/math&gt;. Also, &lt;math&gt;MA=6, MB=8, MC=4, MD=3&lt;/math&gt;, and &lt;math&gt;BC=2CD&lt;/math&gt;. The perimeter of &lt;math&gt;ABCD&lt;/math&gt; can be expressed in the form &lt;math&gt;\frac{p\sqrt{q}}{r}&lt;/math&gt; where &lt;math&gt;p&lt;/math&gt; and &lt;math&gt;r&lt;/math&gt; are relatively prime, and &lt;math&gt;q&lt;/math&gt; is not divisible by the square of any prime number. Find &lt;math&gt;p+q+r&lt;/math&gt;.<br /> <br /> [[2013 Mock AIME I Problems/Problem 5|Solution]]<br /> <br /> ==Problem 6==<br /> Find the number of integer values &lt;math&gt;k&lt;/math&gt; can have such that the equation &lt;cmath&gt;7\cos x+5\sin x=2k+1&lt;/cmath&gt; has a solution.<br /> <br /> [[2013 Mock AIME I Problems/Problem 6|Solution]]<br /> <br /> ==Problem 7==<br /> Let &lt;math&gt;S&lt;/math&gt; be the set of all &lt;math&gt;7&lt;/math&gt;th primitive roots of unity with imaginary part greater than &lt;math&gt;0&lt;/math&gt;. Let &lt;math&gt;T&lt;/math&gt; be the set of all &lt;math&gt;9&lt;/math&gt;th primitive roots of unity with imaginary part greater than &lt;math&gt;0&lt;/math&gt;. (A primitive &lt;math&gt;n&lt;/math&gt;th root of unity is a &lt;math&gt;n&lt;/math&gt;th root of unity that is not a &lt;math&gt;k&lt;/math&gt;th root of unity for any &lt;math&gt;1 \le k &lt; n&lt;/math&gt;.)Let &lt;math&gt;C=\sum_{s\in S}\sum_{t\in T}(s+t)&lt;/math&gt;. The absolute value of the real part of &lt;math&gt;C&lt;/math&gt; can be expressed in the form &lt;math&gt;\frac{m}{n}&lt;/math&gt; where &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; are relatively prime numbers. Find &lt;math&gt;m+n&lt;/math&gt;.<br /> <br /> [[2013 Mock AIME I Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> Let &lt;math&gt;\textbf{u}=4\textbf{i}+3\textbf{j}&lt;/math&gt; and &lt;math&gt;\textbf{v}&lt;/math&gt; be two perpendicular vectors in the &lt;math&gt;x-y&lt;/math&gt; plane. If there are &lt;math&gt;n&lt;/math&gt; vectors &lt;math&gt;\textbf{r}_i&lt;/math&gt; for &lt;math&gt;i=1, 2, \ldots, n&lt;/math&gt; in the same plane having projections of &lt;math&gt;1&lt;/math&gt; and &lt;math&gt;2&lt;/math&gt; along &lt;math&gt;\textbf{u}&lt;/math&gt; and &lt;math&gt;\textbf{v}&lt;/math&gt; respectively, then find &lt;cmath&gt;\sum_{i=1}^{n}\|\textbf{r}_i\|^2.&lt;/cmath&gt; (Note: &lt;math&gt;\textbf{i}&lt;/math&gt; and &lt;math&gt;\textbf{j}&lt;/math&gt; are unit vectors such that &lt;math&gt;\textbf{i}=(1,0)&lt;/math&gt; and &lt;math&gt;\textbf{j}=(0,1)&lt;/math&gt;, and the projection of a vector &lt;math&gt;\textbf{a}&lt;/math&gt; onto &lt;math&gt;\textbf{b}&lt;/math&gt; is the length of the vector that is formed by the origin and the foot of the perpendicular of &lt;math&gt;\textbf{a}&lt;/math&gt; onto &lt;math&gt;\textbf{b}&lt;/math&gt;.)<br /> <br /> [[2013 Mock AIME I Problems/Problem 8|Solution]]<br /> <br /> ==Problem 9==<br /> In a magic circuit, there are six lights in a series, and if one of the lights short circuit, then all lights after it will short circuit as well, without affecting the lights before it. Once a turn, a random light that isn’t already short circuited is short circuited. If &lt;math&gt; E &lt;/math&gt; is the expected number of turns it takes to short circuit all of the lights, find &lt;math&gt; 100E &lt;/math&gt;.<br /> <br /> [[2013 Mock AIME I Problems/Problem 9|Solution]]<br /> <br /> ==Problem 10==<br /> Let &lt;math&gt;T_n&lt;/math&gt; denote the &lt;math&gt;n&lt;/math&gt;th triangular number, i.e. &lt;math&gt;T_n=1+2+3+\cdots+n&lt;/math&gt;. Let &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; be relatively prime positive integers so that &lt;cmath&gt;\sum_{i=3}^\infty \sum_{k=1}^\infty \left(\dfrac{3}{T_i}\right)^k=\dfrac{m}{n}.&lt;/cmath&gt; Find &lt;math&gt;m+n&lt;/math&gt;.<br /> <br /> [[2013 Mock AIME I Problems/Problem 10|Solution]]<br /> <br /> == Problem 11 ==<br /> Let &lt;math&gt;a,b,&lt;/math&gt; and &lt;math&gt;c&lt;/math&gt; be the roots of the equation &lt;math&gt;x^3+2x-1=0&lt;/math&gt;, and let &lt;math&gt;X&lt;/math&gt; and &lt;math&gt;Y&lt;/math&gt; be the two possible values of &lt;math&gt;\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}.&lt;/math&gt; Find &lt;math&gt;(X+1)(Y+1)&lt;/math&gt;.<br /> <br /> [[2013 Mock AIME I Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> In acute triangle &lt;math&gt; ABC &lt;/math&gt;, the orthocenter &lt;math&gt; H &lt;/math&gt; lies on the line connecting the midpoint of segment &lt;math&gt; AB &lt;/math&gt; to the midpoint of segment &lt;math&gt; BC &lt;/math&gt;. If &lt;math&gt; AC=24 &lt;/math&gt;, and the altitude from &lt;math&gt; B &lt;/math&gt; has length &lt;math&gt; 14 &lt;/math&gt;, find &lt;math&gt; AB\cdot BC &lt;/math&gt;.<br /> <br /> [[2013 Mock AIME I Problems/Problem 12|Solution]]<br /> <br /> == Problem 13 ==<br /> In acute &lt;math&gt;\triangle ABC&lt;/math&gt;, &lt;math&gt;H&lt;/math&gt; is the orthocenter, &lt;math&gt;G&lt;/math&gt; is the centroid, and &lt;math&gt;M&lt;/math&gt; is the midpoint of &lt;math&gt;BC&lt;/math&gt;. It is obvious that &lt;math&gt;AM \ge GM&lt;/math&gt;, but &lt;math&gt;GM \ge HM&lt;/math&gt; does not always hold. If &lt;math&gt;[ABC] = 162&lt;/math&gt;, &lt;math&gt;BC=18&lt;/math&gt;, then the value of &lt;math&gt;GM&lt;/math&gt; which produces the smallest value of &lt;math&gt;AB&lt;/math&gt; such that &lt;math&gt;GM \ge HM&lt;/math&gt; can be expressed in the form &lt;math&gt;a+b\sqrt{c}&lt;/math&gt;, for &lt;math&gt;b&lt;/math&gt; squarefree. Compute &lt;math&gt;a+b+c&lt;/math&gt;.<br /> <br /> [[2013 Mock AIME I Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> Let &lt;cmath&gt;\begin{align*}P(x) = x^{2013}+4x^{2012}+9x^{2011}+16x^{2010}+\cdots + 4052169x + 4056196 = \sum_{j=1}^{2014}j^2x^{2014-j}.\end{align*}&lt;/cmath&gt; If &lt;math&gt;a_1, a_2, \cdots a_{2013}&lt;/math&gt; are its roots, then compute the remainder when &lt;math&gt;a_1^{997}+a_2^{997}+\cdots + a_{2013}^{997}&lt;/math&gt; is divided by 997.<br /> <br /> [[2013 Mock AIME I Problems/Problem 14|Solution]]<br /> <br /> ==Problem 15==<br /> Let &lt;math&gt;S&lt;/math&gt; be the set of integers &lt;math&gt;n&lt;/math&gt; such that &lt;math&gt;n | (a^{n+1}-a)&lt;/math&gt; for all integers &lt;math&gt;a&lt;/math&gt;. Compute the remainder when the sum of the elements in &lt;math&gt;S&lt;/math&gt; is divided by &lt;math&gt;1000&lt;/math&gt;.<br /> <br /> [[2013 Mock AIME I Problems/Problem 15|Solution]]</div> Sugar rush