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| − | == Problem 1 ==
| + | #redirect [[2013 Mock AIME I Problems]] |
| − | Two circles <math>C_1</math> and <math>C_2</math>, each of unit radius, have centers <math>A_1</math> and <math>A_2</math> such that <math>A_1A_2=6</math>. Let <math>P</math> be the midpoint of <math>A_1A_2</math> and let <math>C_3</math> be a circle externally tangent to both <math>C_1</math> and <math>C_2</math>. <math>C_1</math> and <math>C_3</math> have a common tangent that passes through <math>P</math>. If this tangent is also a common tangent to <math>C_2</math> and <math>C_1</math>, find the radius of circle <math>C_3</math>.
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| − | [[2013 Mock AIME I Problems/Problem 1|Solution]] | |
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| − | == Problem 2 ==
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| − | Find the number of ordered positive integer pairs <math>(a,b,c)</math> such that <math>a</math> evenly divides <math>b</math>, <math>b+1</math> evenly divides <math>c</math>, and <math>c-a=10</math>.
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| − | [[2013 Mock AIME I Problems/Problem 2|Solution]]
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| − | == Problem 3 ==
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| − | Let <math>\lfloor x\rfloor</math> be the greatest integer less than or equal to <math>x</math>, and let <math>\{x\}=x-\lfloor x\rfloor</math>. If <math>x=(7+4\sqrt{3})^{2^{2013}}</math>, compute <math>x\left(1-\{x\}\right)</math>.
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| − | [[2013 Mock AIME I Problems/Problem 3|Solution]]
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| − | == Problem 4 ==
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| − | Compute the number of ways to fill in the following magic square such that:
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| − | 1. the product of all rows, columns, and diagonals are equal (the sum condition is waived),
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| − | 2. all entries are ''nonnegative'' integers less than or equal to ten, and
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| − | 3. entries CAN repeat in a column, row, or diagonal.
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| − | <asy>
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| − | size(100);
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| − | defaultpen(linewidth(0.7));
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| − | int i;
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| − | for(i=0; i<4; i=i+1) {
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| − | draw((0,2*i)--(6,2*i)^^(2*i,0)--(2*i,6));
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| − | }
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| − | label("$1$", (1,5));
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| − | label("$9$", (3,5));
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| − | label("$3$", (1,1));
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| − | </asy>
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| − | [[2013 Mock AIME I Problems/Problem 4|Solution]]
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| − | == Problem 5 ==
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| − | In quadrilateral <math>ABCD</math>, <math>AC\cap BD=M</math>. Also, <math>MA=6, MB=8, MC=4, MD=3</math>, and <math>BC=2CD</math>. The perimeter of <math>ABCD</math> can be expressed in the form <math>\frac{p\sqrt{q}}{r}</math> where <math>p</math> and <math>r</math> are relatively prime, and <math>q</math> is not divisible by the square of any prime number. Find <math>p+q+r</math>.
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| − | [[2013 Mock AIME I Problems/Problem 5|Solution]]
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| − | ==Problem 6==
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| − | Find the number of integer values <math>k</math> can have such that the equation <cmath>7\cos x+5\sin x=2k+1</cmath> has a solution.
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| − | [[2013 Mock AIME I Problems/Problem 6|Solution]]
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| − | ==Problem 7==
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| − | Let <math>S</math> be the set of all <math>7</math>th primitive roots of unity with imaginary part greater than <math>0</math>. Let <math>T</math> be the set of all <math>9</math>th primitive roots of unity with imaginary part greater than <math>0</math>. (A primitive <math>n</math>th root of unity is a <math>n</math>th root of unity that is not a <math>k</math>th root of unity for any <math>1 \le k < n</math>.)Let <math>C=\sum_{s\in S}\sum_{t\in T}(s+t)</math>. The absolute value of the real part of <math>C</math> can be expressed in the form <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime numbers. Find <math>m+n</math>.
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| − | [[2013 Mock AIME I Problems/Problem 7|Solution]]
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| − | == Problem 8 ==
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| − | Let <math>\textbf{u}=4\textbf{i}+3\textbf{j}</math> and <math>\textbf{v}</math> be two perpendicular vectors in the <math>x-y</math> plane. If there are <math>n</math> vectors <math>\textbf{r}_i</math> for <math>i=1, 2, \ldots, n</math> in the same plane having projections of <math>1</math> and <math>2</math> along <math>\textbf{u}</math> and <math>\textbf{v}</math> respectively, then find <cmath>\sum_{i=1}^{n}\|\textbf{r}_i\|^2.</cmath> (Note: <math>\textbf{i}</math> and <math>\textbf{j}</math> are unit vectors such that <math>\textbf{i}=(1,0)</math> and <math>\textbf{j}=(0,1)</math>, and the projection of a vector <math>\textbf{a}</math> onto <math>\textbf{b}</math> is the length of the vector that is formed by the origin and the foot of the perpendicular of <math>\textbf{a}</math> onto <math>\textbf{b}</math>.)
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| − | [[2013 Mock AIME I Problems/Problem 8|Solution]]
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| − | ==Problem 9==
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| − | 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 <math> E </math> is the expected number of turns it takes to short circuit all of the lights, find <math> 100E </math>.
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| − | [[2013 Mock AIME I Problems/Problem 9|Solution]]
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| − | ==Problem 10==
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| − | Let <math>T_n</math> denote the <math>n</math>th triangular number, i.e. <math>T_n=1+2+3+\cdots+n</math>. Let <math>m</math> and <math>n</math> be relatively prime positive integers so that <cmath>\sum_{i=3}^\infty \sum_{k=1}^\infty \left(\dfrac{3}{T_i}\right)^k=\dfrac{m}{n}.</cmath> Find <math>m+n</math>.
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| − | [[2013 Mock AIME I Problems/Problem 10|Solution]]
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| − | == Problem 11 ==
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| − | Let <math>a,b,</math> and <math>c</math> be the roots of the equation <math>x^3+2x-1=0</math>, and let <math>X</math> and <math>Y</math> be the two possible values of <math>\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}.</math> Find <math>(X+1)(Y+1)</math>.
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| − | [[2013 Mock AIME I Problems/Problem 11|Solution]]
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| − | == Problem 12 ==
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| − | In acute triangle <math> ABC </math>, the orthocenter <math> H </math> lies on the line connecting the midpoint of segment <math> AB </math> to the midpoint of segment <math> BC </math>. If <math> AC=24 </math>, and the altitude from <math> B </math> has length <math> 14 </math>, find <math> AB\cdot BC </math>.
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| − | [[2013 Mock AIME I Problems/Problem 12|Solution]]
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| − | == Problem 13 ==
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| − | In acute <math>\triangle ABC</math>, <math>H</math> is the orthocenter, <math>G</math> is the centroid, and <math>M</math> is the midpoint of <math>BC</math>. It is obvious that <math>AM \ge GM</math>, but <math>GM \ge HM</math> does not always hold. If <math>[ABC] = 162</math>, <math>BC=18</math>, then the value of <math>GM</math> which produces the smallest value of <math>AB</math> such that <math>GM \ge HM</math> can be expressed in the form <math>a+b\sqrt{c}</math>, for <math>b</math> squarefree. Compute <math>a+b+c</math>.
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| − | [[2013 Mock AIME I Problems/Problem 13|Solution]]
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| − | == Problem 14 ==
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| − | Let <cmath>\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*}</cmath> If <math>a_1, a_2, \cdots a_{2013}</math> are its roots, then compute the remainder when <math>a_1^{997}+a_2^{997}+\cdots + a_{2013}^{997}</math> is divided by 997.
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| − | [[2013 Mock AIME I Problems/Problem 14|Solution]]
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| − | ==Problem 15==
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| − | Let <math>S</math> be the set of integers <math>n</math> such that <math>n | (a^{n+1}-a)</math> for all integers <math>a</math>. Compute the remainder when the sum of the elements in <math>S</math> is divided by <math>1000</math>.
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| − | [[2013 Mock AIME I Problems/Problem 15|Solution]]
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