2010-2011 Mock USAJMO Problems/Solutions: Difference between revisions

Revision as of 21:04, 27 September 2010

Problem 1

Given two fixed, distinct points $B$ and $C$ on plane $\mathcal{P}$ , find the locus of all points $A$ belonging to $\mathcal{P}$ such that the quadrilateral formed by point $A$ , the midpoint of $AB$ , the centroid of $\triangle ABC$ , and the midpoint of $AC$ (in that order) can be inscribed in a circle.\\

Problem 2

Let $x, y, z$ be positive real numbers such that $x+y+z = 1$ . Prove that $\frac{3x+1}{y+z}+\frac{3y+1}{z+x}+\frac{3z+1}{x+y}\ge\frac{4}{2x+y+z}+\frac{4}{x+2y+z}+\frac{4}{x+y+2z},$ with equality if and only if $x=y=z$ .

Problem 3

At a certain (lame) conference with $n$ people, each group of three people wishes to meet exactly once. On each of the $d$ days of the conference, every person participates in at most one meeting with two other people, with no limit on the number of groups that can meet. Determine $\min d$ for a) $n=6$ b) $n=8$

Problem 4

The sequence $\{a_i\}_{i\ge0}$ satisfies $a_0=1$ and $a_n=\sum_{i=0}^{n-1}(n-i)a_i$ for $n\ge1$ . Prove that for all nonnegative integers $m$ , we have $\sum_{k=0}^{m}\frac{a_k}{4^k} < \frac{9}{5}.$

Problem 5

Find all solutions to $5\cdot4^{3m+1}-4^{2m+1}-1=15n^{2m},$ where $m$ and $n$ are positive integers.

Problem 6

In convex, tangential quadrilateral $ABCD$ , let the incircle touch the four sides $AB, BC, CD, DA$ at points $E, F, G, H$ , respectively. Prove that $\frac{[EFGH]}{[ABCD]} \le \frac{1}{2}.$ \textbf{Note:} A tangential quadrilateral has concurrent angle bisectors, and $[X]$ denotes the area of figure $X$ .

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 ==Problem 1==
+Given two fixed, distinct points <math>B</math> and <math>C</math> on plane <math>\mathcal{P}</math>, find the locus of all points <math>A</math> belonging to <math>\mathcal{P}</math> such that the quadrilateral formed by point <math>A</math>, the midpoint of <math>AB</math>, the centroid of <math>\triangle ABC</math>, and the midpoint of <math>AC</math> (in that order) can be inscribed in a circle.\\
 ==Problem 2==
+Let <math>x, y, z</math> be positive real numbers such that <math>x+y+z = 1</math>. Prove that
+<cmath>\frac{3x+1}{y+z}+\frac{3y+1}{z+x}+\frac{3z+1}{x+y}\ge\frac{4}{2x+y+z}+\frac{4}{x+2y+z}+\frac{4}{x+y+2z},</cmath>
+with equality if and only if <math>x=y=z</math>.
 ==Problem 3==
+At a certain (lame) conference with <math>n</math> people, each group of three people wishes to meet exactly once. On each of the <math>d</math> days of the conference, every person participates in at most one meeting with two other people, with no limit on the number of groups that can meet. Determine <math>\min d</math> for
+a) <math>n=6</math>
+b) <math>n=8</math>
 ==Problem 4==
+The sequence <math>\{a_i\}_{i\ge0}</math> satisfies <math>a_0=1</math> and <math>a_n=\sum_{i=0}^{n-1}(n-i)a_i</math> for <math>n\ge1</math>. Prove that for all nonnegative integers <math>m</math>, we have
+<cmath>\sum_{k=0}^{m}\frac{a_k}{4^k} < \frac{9}{5}.</cmath>
 ==Problem 5==
+Find all solutions to
+<cmath>5\cdot4^{3m+1}-4^{2m+1}-1=15n^{2m},</cmath>
+where <math>m</math> and <math>n</math> are positive integers.
 ==Problem 6==
+In convex, tangential quadrilateral <math>ABCD</math>, let the incircle touch the four sides <math>AB, BC, CD, DA</math> at points <math>E, F, G, H</math>, respectively. Prove that
+<cmath>\frac{[EFGH]}{[ABCD]} \le \frac{1}{2}.</cmath>
+\textbf{Note:} A tangential quadrilateral has concurrent angle bisectors, and <math>[X]</math> denotes the area of figure <math>X</math>.

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