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Difference between revisions of "2011 USAJMO Problems"

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=Day 1=
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==Day 1==
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==Problem 1==
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===Problem 1===
 
Find, with proof, all positive integers <math>n</math> for which <math>2^n + 12^n + 2011^n</math> is a perfect square.
 
Find, with proof, all positive integers <math>n</math> for which <math>2^n + 12^n + 2011^n</math> is a perfect square.
  
 
[[2011 USAJMO Problems/Problem 1|Solution]]
 
[[2011 USAJMO Problems/Problem 1|Solution]]
  
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==Problem 2==
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===Problem 2===
 
Let <math>a</math>, <math>b</math>, <math>c</math> be positive real numbers such that <math>a^2 + b^2 + c^2 + (a + b + c)^2 \le 4</math>.  Prove that
 
Let <math>a</math>, <math>b</math>, <math>c</math> be positive real numbers such that <math>a^2 + b^2 + c^2 + (a + b + c)^2 \le 4</math>.  Prove that
 
<cmath>\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.</cmath>
 
<cmath>\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.</cmath>
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[[2011 USAMO Problems/Problem 1|Solution]]
 
[[2011 USAMO Problems/Problem 1|Solution]]
  
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==Problem 3==
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===Problem 3===
 
For a point <math>P = (a,a^2)</math> in the coordinate plane, let <math>\ell(P)</math> denote the line passing through <math>P</math> with slope <math>2a</math>.  Consider the set of triangles with vertices of the form <math>P_1 = (a_1, a_1^2)</math>, <math>P_2 = (a_2, a_2^2)</math>, <math>P_3 = (a_3, a_3^2)</math>, such that the intersections of the lines <math>\ell(P_1)</math>, <math>\ell(P_2)</math>, <math>\ell(P_3)</math> form an equilateral triangle <math>\Delta</math>.  Find the locus of the center of <math>\Delta</math> as <math>P_1 P_2 P_3</math> ranges over all such triangles.
 
For a point <math>P = (a,a^2)</math> in the coordinate plane, let <math>\ell(P)</math> denote the line passing through <math>P</math> with slope <math>2a</math>.  Consider the set of triangles with vertices of the form <math>P_1 = (a_1, a_1^2)</math>, <math>P_2 = (a_2, a_2^2)</math>, <math>P_3 = (a_3, a_3^2)</math>, such that the intersections of the lines <math>\ell(P_1)</math>, <math>\ell(P_2)</math>, <math>\ell(P_3)</math> form an equilateral triangle <math>\Delta</math>.  Find the locus of the center of <math>\Delta</math> as <math>P_1 P_2 P_3</math> ranges over all such triangles.
  
 
[[2011 USAJMO Problems/Problem 3|Solution]]
 
[[2011 USAJMO Problems/Problem 3|Solution]]
  
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=Day 2=
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==Day 2==
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==Problem 4==
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===Problem 4===
 
A ''word'' is defined as any finite string of letters.  A word is a ''palindrome'' if it reads the same backwards as forwards.  Let a sequence of words <math>W_0</math>, <math>W_1</math>, <math>W_2</math>, <math>\dots</math> be defined as follows: <math>W_0 = a</math>, <math>W_1 = b</math>, and for <math>n \ge 2</math>, <math>W_n</math> is the word formed by writing <math>W_{n - 2}</math> followed by <math>W_{n - 1}</math>.  Prove that for any <math>n \ge 1</math>, the word formed by writing <math>W_1</math>, <math>W_2</math>, <math>\dots</math>, <math>W_n</math> in succession is a palindrome.
 
A ''word'' is defined as any finite string of letters.  A word is a ''palindrome'' if it reads the same backwards as forwards.  Let a sequence of words <math>W_0</math>, <math>W_1</math>, <math>W_2</math>, <math>\dots</math> be defined as follows: <math>W_0 = a</math>, <math>W_1 = b</math>, and for <math>n \ge 2</math>, <math>W_n</math> is the word formed by writing <math>W_{n - 2}</math> followed by <math>W_{n - 1}</math>.  Prove that for any <math>n \ge 1</math>, the word formed by writing <math>W_1</math>, <math>W_2</math>, <math>\dots</math>, <math>W_n</math> in succession is a palindrome.
  
 
[[2011 USAJMO Problems/Problem 4|Solution]]
 
[[2011 USAJMO Problems/Problem 4|Solution]]
  
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==Problem 5==
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===Problem 5===
 
Points <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, <math>E</math> lie on a circle <math>\omega</math> and point <math>P</math> lies outside the circle.  The given points are such that (i) lines <math>PB</math> and <math>PD</math> are tangent to <math>\omega</math>, (ii) <math>P</math>, <math>A</math>, <math>C</math> are collinear, and (iii) <math>\overline{DE} \parallel \overline{AC}</math>.  Prove that <math>\overline{BE}</math> bisects <math>\overline{AC}</math>.
 
Points <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, <math>E</math> lie on a circle <math>\omega</math> and point <math>P</math> lies outside the circle.  The given points are such that (i) lines <math>PB</math> and <math>PD</math> are tangent to <math>\omega</math>, (ii) <math>P</math>, <math>A</math>, <math>C</math> are collinear, and (iii) <math>\overline{DE} \parallel \overline{AC}</math>.  Prove that <math>\overline{BE}</math> bisects <math>\overline{AC}</math>.
  
 
[[2011 USAJMO Problems/Problem 5|Solution]]
 
[[2011 USAJMO Problems/Problem 5|Solution]]
  
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==Problem 6==
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===Problem 6===
 
Consider the assertion that for each positive integer <math>n \ge 2</math>, the remainder upon dividing <math>2^{2^n}</math> by <math>2^n - 1</math> is a power of 4.  Either prove the assertion or find (with proof) a counterexample.
 
Consider the assertion that for each positive integer <math>n \ge 2</math>, the remainder upon dividing <math>2^{2^n}</math> by <math>2^n - 1</math> is a power of 4.  Either prove the assertion or find (with proof) a counterexample.
  
 
[[2011 USAMO Problems/Problem 4|Solution]]
 
[[2011 USAMO Problems/Problem 4|Solution]]
  
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= See also =
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== See Also ==
 
*[[USAJMO Problems and Solutions]]
 
*[[USAJMO Problems and Solutions]]
 +
 +
{{USAJMO box|year=2011|before=[[2010 USAJMO Problems]]|after=[[2012 USAJMO Problems]]}}
 +
{{MAA Notice}}

Latest revision as of 15:40, 5 August 2023

Day 1

Problem 1

Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.

Solution

Problem 2

Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \le 4$. Prove that \[\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.\]

Solution

Problem 3

For a point $P = (a,a^2)$ in the coordinate plane, let $\ell(P)$ denote the line passing through $P$ with slope $2a$. Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2)$, $P_2 = (a_2, a_2^2)$, $P_3 = (a_3, a_3^2)$, such that the intersections of the lines $\ell(P_1)$, $\ell(P_2)$, $\ell(P_3)$ form an equilateral triangle $\Delta$. Find the locus of the center of $\Delta$ as $P_1 P_2 P_3$ ranges over all such triangles.

Solution

Day 2

Problem 4

A word is defined as any finite string of letters. A word is a palindrome if it reads the same backwards as forwards. Let a sequence of words $W_0$, $W_1$, $W_2$, $\dots$ be defined as follows: $W_0 = a$, $W_1 = b$, and for $n \ge 2$, $W_n$ is the word formed by writing $W_{n - 2}$ followed by $W_{n - 1}$. Prove that for any $n \ge 1$, the word formed by writing $W_1$, $W_2$, $\dots$, $W_n$ in succession is a palindrome.

Solution

Problem 5

Points $A$, $B$, $C$, $D$, $E$ lie on a circle $\omega$ and point $P$ lies outside the circle. The given points are such that (i) lines $PB$ and $PD$ are tangent to $\omega$, (ii) $P$, $A$, $C$ are collinear, and (iii) $\overline{DE} \parallel \overline{AC}$. Prove that $\overline{BE}$ bisects $\overline{AC}$.

Solution

Problem 6

Consider the assertion that for each positive integer $n \ge 2$, the remainder upon dividing $2^{2^n}$ by $2^n - 1$ is a power of 4. Either prove the assertion or find (with proof) a counterexample.

Solution

See Also

2011 USAJMO (Problems • Resources)
Preceded by
2010 USAJMO Problems 		Followed by
2012 USAJMO Problems
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions

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