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Difference between revisions of "User:Rowechen"
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[[2017 AIME II Problems/Problem 13 | Solution]]
 
[[2017 AIME II Problems/Problem 13 | Solution]]
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In triangle <math>ABC</math>, we have <math>BC = 13</math>, <math>CA = 37</math>, and <math>AB = 40</math>. Points <math>D</math>, <math>E</math>, and <math>F</math> are selected
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on <math>BC</math>, <math>CA</math>, and <math>AB</math> respectively such that <math>AD</math>, <math>BE</math>, and <math>CF</math> concur at the circumcenter of <math>ABC</math>. The value of <math>\frac{1}{AD}+\frac{1}{BE}+\frac{1}{CF}</math> can be expressed as <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine <math>m+n</math>.

Revision as of 18:35, 1 June 2020

Here's the AIME compilation I will be doing:

Problem 7

An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region $\mathcal{C}$ to the area of shaded region $\mathcal{B}$ is 11/5. Find the ratio of shaded region $\mathcal{D}$ to the area of shaded region $\mathcal{A}.$

2006AimeA7.PNG

Solution

Problem 2

A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 2 feet per second. Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 4 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the distance in feet between the start of the walkway and the middle person.

Solution

Problem 3

Let $P$ be the product of the first 100 positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k$.

Solution

Problem 9

The value of the sum \[\sum_{n=1}^{\infty} \frac{(7n+32)\cdot 3^n}{n\cdot(n+1)\cdot 4^n}\] can be expressed in the form $\frac{p}{q}$, for some relatively prime positive integers $p$ and $q$. Compute the value of $p + q$.

Problem 8

Determine the remainder obtained when the expression \[2004^{2003^{2002^{2001}}}\] is divided by $1000$.

Problem 9

Let \[(a+x^3)(a+2x^{3^2}) ... (1+kx^{3^k}) ... (1+1997x^{3^{1997}}) = 1+a_1x^{k_1}+a_2x^{k_2}+...+a_mx^{k_m}\] where $a_i \neq 0$ and $k_1 < k_2 < ... < k_m$. Determine the remainder obtained when $a_1997$ is divided by $1000$.

Problem 11

A sequence is defined as follows $a_1=a_2=a_3=1,$ and, for all positive integers $n, a_{n+3}=a_{n+2}+a_{n+1}+a_n.$ Given that $a_{28}=6090307, a_{29}=11201821,$ and $a_{30}=20603361,$ find the remainder when $\sum^{28}_{k=1} a_k$ is divided by 1000.

Solution

Problem 10

$p, q$, and $r$ are positive real numbers such that \[p^2+pq+q^2 = 211\] \[q^2+qr+r^2 = 259\] \[r^2+rp+p^2 = 307\] Compute the value of $pq + qr + rp$.

Problem 11

$x_1$, $x_2$, and $x_3$ are complex numbers such that \[x_1 + x_2 + x_3 = 0\] \[x_1^2+x_2^2+x_3^2 = 16\] \[x_1^3+x_2^3+x_3^3 = -24\]

Let $\gamma = min(|x1| , |x2| , |x3|)$, where $|a + bi| = \sqrt{a^2+b^2}$. Determine the value of $\gamma^6-15\gamma^4+\gamma^3+56\gamma^2$.

Problem 12

$ABC$ is a scalene triangle. The circle with diameter $AB$ intersects $BC$ at $D$, and $E$ is the foot of the altitude from $C$. $P$ is the intersection of $AD$ and $CE$. Given that $AP = 136$, $BP = 80$, and $CP = 26$, determine the circumradius of $ABC$.

Problem 13

Point $D$ lies on side $\overline{BC}$ of $\triangle ABC$ so that $\overline{AD}$ bisects $\angle BAC.$ The perpendicular bisector of $\overline{AD}$ intersects the bisectors of $\angle ABC$ and $\angle ACB$ in points $E$ and $F,$ respectively. Given that $AB=4,BC=5,$ and $CA=6,$ the area of $\triangle AEF$ can be written as $\tfrac{m\sqrt{n}}p,$ where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p.$

Solution

Problem 15

Let $\triangle ABC$ be an acute triangle with circumcircle $\omega,$ and let $H$ be the intersection of the altitudes of $\triangle ABC.$ Suppose the tangent to the circumcircle of $\triangle HBC$ at $H$ intersects $\omega$ at points $X$ and $Y$ with $HA=3,HX=2,$ and $HY=6.$ The area of $\triangle ABC$ can be written as $m\sqrt{n},$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n.$

Solution

Problem 14

Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1.$ Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b.$ Find the sum of all possible values of $(a+b)^2.$

Solution

Problem 13

For each integer $n\geq3$, let $f(n)$ be the number of $3$-element subsets of the vertices of a regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$.

Solution

In triangle $ABC$, we have $BC = 13$, $CA = 37$, and $AB = 40$. Points $D$, $E$, and $F$ are selected on $BC$, $CA$, and $AB$ respectively such that $AD$, $BE$, and $CF$ concur at the circumcenter of $ABC$. The value of $\frac{1}{AD}+\frac{1}{BE}+\frac{1}{CF}$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Determine $m+n$.

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