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For any information, please contact me at MATHTAM@gmail.com

Notes

USAJMO Problem 1

Given a triangle $ABC$ , let $P$ and $Q$ be points on segments $\overline{AB}$ and $\overline{AC}$ , respectively, such that $AP = AQ$ . Let $S$ and $R$ be distinct points on segment $\overline{BC}$ such that $S$ lies between $B$ and $R$ , $\angle BPS = \angle PRS$ , and $\angle CQR = \angle QSR$ . Prove that $P$ , $Q$ , $R$ , $S$ are concyclic (in other words, these four points lie on a circle).

Problem 2

Find all integers $n \ge 3$ such that among any $n$ positive real numbers $a_1$ , $a_2$ , $\dots$ , $a_n$ with $\max(a_1, a_2, \dots, a_n) \le n \cdot \min(a_1, a_2, \dots, a_n),$ there exist three that are the side lengths of an acute triangle.

Problem 3

Let $a$ , $b$ , $c$ be positive real numbers. Prove that $\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2).$

Problem 4

Let $\alpha$ be an irrational number with $0 < \alpha < 1$ , and draw a circle in the plane whose circumference has length 1. Given any integer $n \ge 3$ , define a sequence of points $P_1$ , $P_2$ , $\dots$ , $P_n$ as follows. First select any point $P_1$ on the circle, and for $2 \le k \le n$ define $P_k$ as the point on the circle for which the length of arc $P_{k - 1} P_k$ is $\alpha$ , when travelling counterclockwise around the circle from $P_{k - 1}$ to $P_k$ . Supose that $P_a$ and $P_b$ are the nearest adjacent points on either side of $P_n$ . Prove that $a + b \le n$ .

Problem 5

For distinct positive integers $a$ , $b < 2012$ , define $f(a,b)$ to be the number of integers $k$ with $1 \le k < 2012$ such that the remainder when $ak$ divided by 2012 is greater than that of $bk$ divided by 2012. Let $S$ be the minimum value of $f(a,b)$ , where $a$ and $b$ range over all pairs of distinct positive integers less than 2012. Determine $S$ .

Problem 6

Let $P$ be a point in the plane of triangle $ABC$ , and $\gamma$ a line passing through $P$ . Let $A'$ , $B'$ , $C'$ be the points where the reflections of lines $PA$ , $PB$ , $PC$ with respect to $\gamma$ intersect lines $BC$ , $AC$ , $AB$ , respectively. Prove that $A'$ , $B'$ , $C'$ are collinear.

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 For any information, please contact me at MATHTAM@gmail.com
+==Notes==
+==USAJMO Problem 1==
+Given a triangle <math>ABC</math>, let <math>P</math> and <math>Q</math> be points on segments <math>\overline{AB}</math> and <math>\overline{AC}</math>, respectively, such that <math>AP = AQ</math>.  Let <math>S</math> and <math>R</math> be distinct points on segment <math>\overline{BC}</math> such that <math>S</math> lies between <math>B</math> and <math>R</math>, <math>\angle BPS = \angle PRS</math>, and <math>\angle CQR = \angle QSR</math>.  Prove that <math>P</math>, <math>Q</math>, <math>R</math>, <math>S</math> are concyclic (in other words, these four points lie on a circle).
+==Problem 2==
+Find all integers <math>n \ge 3</math> such that among any <math>n</math> positive real numbers <math>a_1</math>, <math>a_2</math>, <math>\dots</math>, <math>a_n</math> with
+<cmath>\max(a_1, a_2, \dots, a_n) \le n \cdot \min(a_1, a_2, \dots, a_n),</cmath>
+there exist three that are the side lengths of an acute triangle.
+== Problem 3==
+Let <math>a</math>, <math>b</math>, <math>c</math> be positive real numbers.  Prove that
+<cmath>\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2).</cmath>
+== Problem 4==
+Let <math>\alpha</math> be an irrational number with <math>0 < \alpha < 1</math>, and draw a circle in the plane whose circumference has length 1.  Given any integer <math>n \ge 3</math>, define a sequence of points <math>P_1</math>, <math>P_2</math>, <math>\dots</math>, <math>P_n</math> as follows.  First select any point <math>P_1</math> on the circle, and for <math>2 \le k \le n</math> define <math>P_k</math> as the point on the circle for which the length of arc <math>P_{k - 1} P_k</math> is <math>\alpha</math>, when travelling counterclockwise around the circle from <math>P_{k - 1}</math> to <math>P_k</math>.  Supose that <math>P_a</math> and <math>P_b</math> are the nearest adjacent points on either side of <math>P_n</math>.  Prove that <math>a + b \le n</math>.
+== Problem 5==
+For distinct positive integers <math>a</math>, <math>b < 2012</math>, define <math>f(a,b)</math> to be the number of integers <math>k</math> with <math>1 \le k < 2012</math> such that the remainder when <math>ak</math> divided by 2012 is greater than that of <math>bk</math> divided by 2012.  Let <math>S</math> be the minimum value of <math>f(a,b)</math>, where <math>a</math> and <math>b</math> range over all pairs of distinct positive integers less than 2012.  Determine <math>S</math>.
+== Problem 6==
+Let <math>P</math> be a point in the plane of triangle <math>ABC</math>, and <math>\gamma</math> a line passing through <math>P</math>.  Let <math>A'</math>, <math>B'</math>, <math>C'</math> be the points where the reflections of lines <math>PA</math>, <math>PB</math>, <math>PC</math> with respect to <math>\gamma</math> intersect lines <math>BC</math>, <math>AC</math>, <math>AB</math>, respectively.  Prove that <math>A'</math>, <math>B'</math>, <math>C'</math> are collinear.

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